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# Reduction and integrability: a geometric perspective
José F. Cariñena111E-mail<EMAIL_ADDRESS>
Departamento de Física Teórica
Universidad de Zaragoza, 50009 Zaragoza, Spain
Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza
###### Abstract
A geometric approach to integrability and reduction of dynamical system is
developed from a modern perspective. The main ingredients in such analysis are
the infinitesimal symmetries and the tensor fields that are invariant under
the given dynamics. Particular emphasis is given to the existence of invariant
volume forms and the associated Jacobi multiplier theory, and then the Hojman
symmetry theory is developed as a complement to Noether theorem and non-
Noether constants of motion. The geometric approach to Hamilton-Jacobi
equation is shown to be a particular example of the search for related field
in a lower dimensional manifold.
Mathematics Subject Classifications (2010): 34A34, 37N05, 53C15
PACS numbers: 02.30.Hq, 02.40.Yy, 02.40.Hw, 02.40.Ky, 45.10.Na
### Keywords:
Reduction; Integrability; the Quadratures; Symmetry; First-integrals; Hojmam
symmetry.
## 1 Introduction
Systems of (may be partial) differential equations play a relevant rôle in the
development of science and technology, and they quite often appear in many
different branches of science ranging from pure mathematics to classical and
quantum physics, control theory, economy, biology, etc.. For instance, the
dynamical evolution of deterministic systems is described by systems of (may
be time-dependent) first-order systems. However, the solution of such systems
is not an easy task. The study of their integrability is a subject of
significant interest that has been an active field of research in recent years
and appears very often in physics and mathematics and with very different
meanings. Here by integrability of an autonomous system we mean that we are
able to determine the general solution of the system. Such solutions can be
searched in a specific family, for instance polynomial, rational functions or
more generally, the so-called integrability by quadratures, that means that it
is possible to determine the solutions by means of a finite number of
algebraic operations, use of the Implicit Function Theorem and quadratures of
some appropriate fundamental functions, including rational, trigonometric and
exponential functions, as well as their inverses.
To solve specific problems reduction techniques are used. Students in physics
relate the reduction process with infinitesimal symmetries and existence of
constants of motion, mainly in the framework of Lagrangian or Hamiltonian
mechanics. However reduction processes can be developed in a more general
framework, allowing therefore other interesting cases also relevant in
physics, and need not to be related with symmetry properties but only aim to
find simpler solvable systems providing, at least partially, information on
the solution of the original problem. In other words, given a difficult to
solve problem we can look for related simpler problems whose solutions provide
at least partial information on the original one. Sometimes we find geometric
structures difficult to handle, for instance because they have some kind of
degeneracy. We look then for a related structure of the same kind but easier
to manage, for instance a non-degenerate one.
The existence of additional compatible geometric structures, like symplectic
or Poisson structures may be useful in the search for solutions, and therefore
when the original problem admits some geometric structures, the reduction
procedure tries to preserve such structures. There is no systematic way of
selecting the related reduced systems and sometimes there are different
possibilities. We generally assume that a system with less unknown variables
or of lower-order differential equations is simpler than other with more
unknown variables or of higher-order differential equations, but reduction
processes from linear systems produce nonlinear interacting ones.
The simplest case is that of an autonomous system of first-order differential
equations (the higher-order case can be related to some first-order one)
$\dot{x}^{i}=X^{i}(x^{1},\ldots,x^{n})\ ,\qquad i=1,\ldots,n,$ (1.1)
while non-autonomous systems correspond to the case in which the functions
$X^{i}$ also depend on the independent variable $t$. In order to incorporate
holonomic constraints, such a system (1.1) is geometrically interpreted in
terms of a vector field $X$ in a $n$-dimensional manifold $M$ with a
coordinate expression in a local chart
$X=\sum_{i=1}^{n}X^{i}(x^{1},\ldots,x^{n})\frac{\partial}{\partial x^{i}}\ ,$
(1.2)
in such a way that its integral curves are the solutions of the given system.
In this sense, integrability by quadratures of a vector field means that you
can determine its integral curves (i.e. the flow of $X$) by quadratures, i.e.
by means of a finite number of algebraic operations and quadratures of some
functions. In a general case the flows of the vector fields cannot be found in
an explicit way using fundamental functions, and we are happy if, at least, we
can express the solutions in terms of quadratures. The purpose of
integrability is therefore the characterisation of systems admitting such type
of solutions and it has being receiving a lot of attention. The answer is
always based on the use of appropriated Lie algebras of vector fields
containing the given vector field.
From the geometric viewpoint by reduction of the dynamics given by the vector
field $X\in\mathfrak{X}(M)$ we mean to find a manifold $N$, a differentiable
map $F:M\to N$ and a simpler $F$-related vector field $Y\in\mathfrak{X}(N)$,
i.e. such that $TF\circ X=Y\circ F$, and hence the integral curves of $Y$ are
images under $F$ of the integral curves of $X$. Sometimes it remains the
problem of the reconstruction of the integral curves of $X\in\mathfrak{X}(M)$
from the integral curves of $Y\in\mathfrak{X}(N)$.
The first of the two more important cases is when $i:N\to M$ is an immersed
submanifold of $M$ (later on we will study when $F:M\to N$ is a surjective
submersion). Note that $X\in\mathfrak{X}(M)$ must be such that $X_{|i(N)}$ is
tangent to $i(N)$ and therefore there exists a vector field $\bar{X}$ in the
lower-dimensional manifold $N$ such that is $i$-related to $X$ and whose
integral curves provide us some integral curves of $X$. We can suppose that
the $(n-r)$-submanifold $N$ is (at least locally) defined by $r$ functions
$F_{1},\ldots,F_{r}$, which should be constants of motion for $X$ because of
the assumed tangency condition. Then a relevant technique in the reduction
process consists on determining $r$ functionally independent constants of
motion, because this reduces the original problem to another problem of a
$r$-parameter family involving only $(n-r)$ variables. This provides us
foliations such that the vector field $X$ is tangent to the leaves, and the
problem is reduced to those of a family of lower dimensional ones, one on each
leaf.
More specifically, if $r$ functionally independent constants of motion
$F_{1},\ldots,F_{r}$ (i.e. such that $dF_{1}\wedge\cdots\wedge dF_{r}\neq 0$)
are known, it allows us to reduce the problem to that of a family of vector
fields $\widetilde{X}_{c}$, with $c\in\mathbb{R}^{r}$, defined in the $n-r$
dimensional submanifolds $M_{c}$ given by the level sets of the vector valued
function of rank $r$, $(F_{1},\ldots,F_{r}):M\to\mathbb{R}^{r}$. Of course the
best situation is when $r=n-1$: the leaves are 1-dimensional, giving us the
solutions to the problem, up to a reparametrisation.
The second important case is when there exists a manifold $N$ and a surjective
submersion $F:M\to N$ such that $X\in\mathfrak{X}(M)$ is a projectable vector
field. The map $F$ defines an equivalence relation in $M$, and then $N$ is the
space of such equivalence classes. Sometimes the starting point is the
equivalence relation and $F$ is the projection map. For instance, usually one
considers a transformation Lie group $G$ of $M$ that is a symmetry group for
$X$ and the equivalence relation associated to the action: $N$ is the space of
orbits, assumed to be a differentiable manifold, the function $F$ being the
natural projection of each element of $M$ on its orbit. The symmetry condition
implies that $X$ is a projectable vector field. A particular example of
symmetry is given by the local flow of a vector field $Y$ that is an
infinitesimal symmetry of $X$, i.e. a vector field $Y\in\mathfrak{X}(M)$ such
that $[Y,X]=0$. Then in a neighbourhood of a point where $Y$ is different from
zero we can choose adapted coordinates, $(y^{1},\ldots,y^{n})$, for which $Y$
is written (Straightening out Theorem) $Y=\partial/\partial{y^{n}}$. Then
$[Y,X]=0$ implies that $X$ has the form
$X=\bar{X}^{1}(y^{1},\ldots,y^{n-1})\,\frac{\partial}{\partial
y^{1}}+\ldots+\bar{X}^{n-1}(y^{1},\ldots,y^{n-1})\,\frac{\partial}{\partial
y^{n-1}}+\bar{X}^{n}(y^{1},\ldots,y^{n-1})\frac{\partial}{\partial y^{n}}\ ,$
and its integral curves are obtained by solving the system
$\left\\{\begin{array}[]{ccl}{\displaystyle\frac{dy^{i}}{dt}}&=&\bar{X}^{i}(y^{1},\ldots,y^{n-1}),\qquad
i=1,\ldots,n-1,\cr&&\cr{\displaystyle\frac{dy^{n}}{dt}}&=&\bar{X}^{n}(y^{1},\ldots,y^{n-1})\end{array}\
.\right.$
We have reduced the problem to a subsystem involving only the first $n-1$
equations, and once this has been solved, the last equation is used to obtain
the function $y^{n}(t)$ by means of one quadrature. Note however that the new
coordinates $y^{1},\ldots,y^{n-1}$, are local constants of the motion for $X$
and therefore we cannot find in an easy way such coordinates in a general
case. Note also that the information provided by two different symmetry vector
fields, $Y_{1}$ and $Y_{2}$, cannot be used simultaneously in the general
case, because it is not possible to find local coordinates
$(y^{1},\ldots,y^{n})$ such that $Y_{1}=\partial/\partial{y^{n-1}}$ and
$Y_{2}=\partial/\partial{y^{n}}$, unless that $[Y_{1},Y_{2}]=0$.
In other situations the equivalence relation is defined by a foliation
(involutive distribution) and the equivalence classes are the leaves. The set
of leaves is assumed to have a manifold structure. If the original system has
an invariance group we can consider invariant foliations in such a way that
leaves are preserved.
Other reduction procedures for systems can be developed when we know some
particular solutions of them or related systems. For instance, one particular
example is Riccati equation, of a fundamental importance both in physics (for
instance factorisation of second order differential operators, Darboux
transformations and in general Supersymmetry in Quantum Mechanics) and in
mathematics. It is well known that if one particular solution of Riccati
equation is known, we can find the general solution by means of two
quadratures, while if two particular solutions are known, we can find the
general solution by means of just one quadrature, and finally if three
particular solutions are known we can explicitly write the solution without
any quadrature, by means of a superposition function. Actually there is a kind
of systems admitting such a superposition function, the so-called Lie systems,
which appear very often in many problems in science and engineering (see [1,
2] for a review). In the solution of such non-autonomous systems of first-
order differential equations we can use techniques imported from group theory,
for instance Wei-Norman method [3, 4, 5], and reduction techniques coming from
the theory of connections.
Throughout the paper we shall take advantage of results in previous
publications, where more details can be found. Section 2 is devoted to Lie
fundamental result of integrability by quadratures of a system without
recourse to any related invariant tensor field but only to properties of a Lie
algebra of vector fields containing the given vector field. In section 3 the
interest of invariant tensorial fields is pointed out and the particular case
of invariant (1,1)-tensor fields giving rise to recursion operators, also
called symmetry generators, and Lax equations is analysed. In Section 4, for
the sake of completeness we review some basic aspects of symplectic geometry,
and the classical first Noether Theorems in both Hamiltonian and Lagrangian
approach are summarised. The non-Noether constants of motion are derived in
Section 5 from the existence of alternative invariant geometric structures for
the dynamical vector field. As we want to add some comments on a different
method of finding constants of motion, as a prelude to deal in Section 7 with
the theory of Hojman symmetry we develop in Section 6 some remarkable points
on invariant volume forms and the associated theory of Jacobi multipliers, a
relevant ingredient in integrability of classical mechanical systems. Section
8 is a short summary of how to use the reduction theory to introduce Hamilton-
Jacobi equation that appears here as a method of finding lower-dimensional
systems related to a Hamiltonian one.
## 2 Lie integrability by quadratures
We first summarise the pioneer work of Lie on integrability of some particular
types of vector fields, without any recourse to the existence of additional
compatible structures, but only using modern tools of algebra and geometry, in
particular Lie algebras of symmetry vector fields, and more specifically,
solvable Lie algebras [6].
###### Lemma 2.1
If $n$ vector fields $X_{1}$,…,$X_{n}$, which are linearly independent in each
point of an open set $U\subset\mathbb{R}^{n}$, generate a solvable Lie algebra
and are such that $[X_{1},X_{i}]=\lambda_{i}\,X_{1}$ with
$\lambda_{i}\in\mathbb{R}$, then the differential equation $\dot{x}=X_{1}(x)$
is solvable by quadratures in $U$.
Proof.- We only prove here the simplest case $n=2$. Then, the derived algebra
is 1-dimensional, and therefore the Lie algebra is solvable. The differential
equation can be integrated if we are able to find a first-integral $F$, i.e.
$X_{1}F=0$, such that $dF\neq 0$ in an open set $U$. In this case, for each
real number $c\in\mathbb{R}$, we can implicitly define one variable, for
instance $x_{2}$, in terms of the other one by $F(x_{1},\phi(x_{1}))=k$, and
the differential equation determining the integral curves of $X_{1}$ is in
separate variables, i.e. integrable by quadratures. Note that the condition
$[X_{1},X_{2}]=\lambda_{2}\,X_{1}$, with $\lambda_{2}\in\mathbb{R}$, shows
that if $F$ is a first-integral for $X_{1}$ then $X_{2}F$ is a first-integral
for $X_{1}$ too, because $X_{1}(X_{2}F)=X_{2}(X_{1}F)+\lambda_{2}\,X_{1}F=0$.
Note that as $n=2$ there exists a 1-form $\alpha_{0}$, which is defined up to
multiplication by a function, such that $i(X_{1})\alpha_{0}=0$. Obviously as
$X_{2}$ is linearly independent of $X_{1}$ at each point,
$i(X_{2})\alpha_{0}\neq 0$. We can see that the 1-form
$\alpha=(i(X_{2})\alpha_{0})^{-1}\alpha_{0}$ satisfies the condition
$i(X_{2})\alpha=1$, by definition of $\alpha$, and that it is closed, because
$X_{1}$ and $X_{2}$ generate $\mathfrak{X}(\mathbb{R}^{2})$ and
$d\alpha(X_{1},X_{2})=X_{1}\alpha(X_{2})-X_{2}\alpha(X_{1})+\alpha([X_{1},X_{2}])=\alpha([X_{1},X_{2}])=\lambda_{2}\,\alpha(X_{1})=0.$
Therefore, there exists, at least locally, a function $F$ such that
$\alpha=dF$, and the condition $i(X_{1})\alpha=0$ means that the function $F$
is a first-integral for $X_{1}$. $\Box$
A generalisation of these results was given in [7] where the vector fields
$X_{1},\ldots,X_{n}$ are assumed to close on a real Lie algebra, i.e.
$[X_{i},X_{j}]={\displaystyle\sum_{k=1}^{n}}c_{ij}\,^{k}X_{k}$, with
$c_{ij}\,^{k}\in\mathbb{R}$, and it was proved that if such Lie algebra is
solvable and $A$ is an Abelian ideal, then the vector field $X$ is integrable
for each vector field $X$ in the ideal $A$, and in particular, if the Lie
algebra is nilpotent, any vector field of the Lie algebra is integrable. This
result was extended in [8] where it was proved that solvability of a Lie
algebra of vector fields implies their integrability by quadratures.
## 3 Invariant tensor fields and integrability of a vector field
We have indicated how the knowledge of a first-integral, i.e. an invariant
function, can be used to reduce the integrability of the vector field to a
family of lower dimensional problems. This is a particular example of a more
general case, the rôle of tensor fields invariant under a vector field $X$ on
a $n$-dimensional manifold $M$ in its integrability (see e.g. [9]). For
instance, invariant vector fields $Y$ give rise to infinitesimal symmetries of
$X$, whose flows transform solutions into solutions. Moreover, if $F$ is a
first-integral for $X$, then $Y(F)$ is also a first-integral, because
$\mathcal{L}_{X}(Y(F))=Y\left(\mathcal{L}_{X}(F)\right)=0$. The same property
holds when the vector field $Y$ is not a symmetry of the vector field $X$ but
only of the 1-dimensional distribution spanned by $X$, i.e. there exists a
function $h\in C^{\infty}(M)$ such that $[Y,X]=h\,X$, because then
$\mathcal{L}_{X}(Y(F))=Y\left(\mathcal{L}_{X}(F)\right)-h\,X(F)=0.$
Later on, we will see how to find new constants of motion when additional
tensor fields enter in the game. Of course some invariant multivector fields
are also interesting but particularly the case of Poisson bivector fields is
very relevant. Differential forms that are invariant under $X$ give rise to
absolutely integral invariants, a theory developed by Poincaré in [10]. In
particular, invariant 2-forms as for instance the relevant case in mechanics
of symplectic forms, are very relevant in the study of integrability of $X$
and Arnold-Liouville integrability is based on the existence of an
appropriated invariant symplectic form. Even if we have a $(1,1)$-tensor field
$\mathcal{R}$ invariant under $X$, it may be used as a generator of symmetry,
in the sense that if $Y$ is a symmetry of $X$, then
$\mathcal{L}_{X}\mathcal{R}=0$ implies that
$\mathcal{L}_{X}(\mathcal{R}(Y))=0$. Furthermore, if we choose in this case a
local basis of vector fields $\\{X_{1},\ldots,X_{n}\\}$, then if the
(1,1)-tensor field $\mathcal{R}$ is invariant under $X\in\mathfrak{X}(M)$ and
the matrices $A$ and $B$ are the matrix representatives of $\mathcal{R}$ and
$\mathcal{L}_{X}$ in such a basis, respectively, then the invariance condition
$\mathcal{L}_{X}\mathcal{R}=0$ is written as
$\dot{A}=[B,A],$ (3.1)
where $\dot{A}_{i}\,^{j}$ denotes $X(A_{i}\,^{j})$, because if
$\mathcal{R}(X_{i})={\displaystyle\sum_{j=1}^{n}}A_{i}\,^{j}X_{j}$ and
$\mathcal{L}_{X}X_{i}={\displaystyle\sum_{j=1}^{n}}B_{i}\,^{j}X_{j}$, as we
have
$\mathcal{L}_{X}[\mathcal{R}(X_{i})]={\displaystyle\sum_{j=1}^{n}}\mathcal{L}_{X}[A_{i}\,^{j}X_{j}]={\displaystyle\sum_{j=1}^{n}}\mathcal{L}_{X}(A_{i}\,^{j})X_{j}+{\displaystyle\sum_{j,k=1}^{n}}A_{i}\,^{j}B_{j}\,^{k}X_{k},$
and
$\mathcal{R}(\mathcal{L}_{X}X_{i})={\displaystyle\sum_{j,k=1}^{n}}B_{i}\,^{j}A_{j}\,^{k}X_{k},$
then, taking into account that $X$-invariance of $\mathcal{R}$ is equivalent
to $\mathcal{L}_{X}[\mathcal{R}(X_{i})]=\mathcal{R}[\mathcal{L}_{X}(X_{i})]$,
we find that $\mathcal{L}_{X}\mathcal{R}=0$ is equivalent to
$\mathcal{L}_{X}(A_{i}\,^{j})+{\displaystyle\sum_{k=1}^{n}}\
A_{i}\,^{k}B_{k}\,^{j}={\displaystyle\sum_{k=1}^{n}}B_{i}\,^{k}A_{k}\,^{j}$,
and therefore to the matrix equation (3.1), which is called Lax equation [11,
12, 13]. Note also that as the matrix representative of $\mathcal{R}^{2}$ is
$A^{2}$ and $\mathcal{L}_{X}\mathcal{R}=0$ implies that
$\mathcal{L}_{X}\mathcal{R}^{k}=0$, with $k\in\mathbb{N}$, we also have
$\dot{A^{k}}=[B,A^{k}],\quad k=1,\ldots n.$ (3.2)
The importance of these equations is that, as for any pair of matrices the
trace of the commutator is zero, we have that
${\displaystyle\frac{d}{dt}\textrm{Tr}A^{k}=0}$, and consequently a
$(1,1)$-tensor field $\mathcal{R}$ invariant under $X$ provides us $n$
constants of motion, in principle not all of them functionally independent, as
indicated by the Hamilton-Cayley theorem. Another remark is that, as the
coefficients of the characteristic equation $\det(A-\lambda I)=0$ can be
reconstructed from the traces of powers of $A$ (Le Verrier method of
determining the characteristic equation of a matrix, see e.g. [14]), the roots
of such characteristic equation, the eigenvalues of $A$, are constants of
motion.
The search of these invariant $(1,1)$-tensor fields $\mathcal{R}$ is not an
easy task and use to come from the existence of alternative structures [15,
16, 17], the associated constants of motion being called non-Noether constants
of motion (see.e.g. [15, 18, 19]), but they may have a different origin [12,
20]. We will be back on this point in Section 5.
Another interesting case is when there exists a volume form invariant under
$X$. This case and an alternative method to find first-integrals associated to
symmetries of 1-dimensional distribution spanned by the vector field $X$ will
be analysed in Sections 6 and 7.
## 4 Invariant symplectic structures
We recall that $(M,\omega)$ is a symplectic manifold if $M$ is a finite-
dimensional differentiable manifold and $\omega$ is a nondegenerate 2–form
which satisfies $d\omega=0$ (that is, $\omega\in Z^{2}(M)$). The dimension of
$M$ is necessarily even, $\dim M=2n$. For general results, reference textbooks
are, for instance, [21] and [22]. By nondegeneracy of $\omega$ we mean that
for every point $u\in M$ the map $\widehat{\omega}_{u}:T_{u}M\to T_{u}^{*}M$,
given by:
$\langle\widehat{\omega}_{u}(v),v^{\prime}\rangle=\omega_{u}(v,v^{\prime})$
with $v,v^{\prime}\in T_{u}M$, has a maximal rank, i.e. $(\omega)^{\wedge
n}\neq 0$. Such a map $\widehat{\omega}$ is a base-preserving fibered map, and
hence it induces a mapping between the linear spaces of sections which, with a
slight abuse of notation, we will also write
$\widehat{\omega}:\mathfrak{X}(M)\to\bigwedge^{1}(M)$.
The following well-known result completely characterises these symplectic
manifolds, from the local point of view:
###### Theorem 4.1
(Darboux) Around each point $u\in M$ with $\dim M=2n$ there is a local chart
$(U,\phi)$ such that if $\phi=(q^{1},\dots,q^{n};p_{1},\dots,p_{n})$, then
$\omega|_{U}={\displaystyle\sum_{i=1}^{n}}dq^{i}\wedge dp_{i}$.
$\Box$
Such coordinates are called Darboux coordinates.
Since closed, and in particular exact, 1-forms are distinguished elements in
$\bigwedge^{1}(M)$, the corresponding vector fields are called locally
Hamiltonian vector fields and Hamiltonian vector fields, respectively. If
$H\in C^{\infty}(M)$, the Hamiltonian vector field $X_{H}$ associated with the
Hamiltonian $H$ is the unique vector field satisfying
$\widehat{\omega}(X_{H})=i(X_{H})\omega=dH$. The set of Hamiltonian vector
fields will be denoted $\mathfrak{X}_{{\rm H}}(M,\omega)$ and that of locally
Hamiltonian vector fields, $\mathfrak{X}_{{\rm LH}}(M,\omega)$, i.e.
$\mathfrak{X}_{{\rm LH}}(M,\omega)=\widehat{\omega}^{-1}(Z^{1}(M))$ and
$\mathfrak{X}_{{\rm H}}(M,\omega)=\widehat{\omega}^{-1}(B^{1}(M))$. Observe
that $\widehat{\omega}^{-1}$ is an isomorphism of real vector spaces.
The Cartan homotopy identity, i.e.
$\mathcal{L}_{X}\alpha=i(X)d\alpha+d(i(X)\alpha)$, for any
$\alpha\in\bigwedge^{p}(M)$, shows that, given $X\in\mathfrak{X}(M)$,
$\mathcal{L}_{X}\omega=0$ if and only if $i(X)\omega$ is a closed 1–form, i.e.
$X\in\mathfrak{X}_{{\rm LH}}(M,\omega)$. In particular,
$\mathcal{L}_{X_{H}}\omega=0$. In Darboux coordinates the Hamiltonian vector
field $X_{H}$ corresponding to the function $H$ is given by
$X_{H}=\sum_{i=1}^{N}\left(\frac{\partial H}{\partial
p_{i}}\frac{\partial}{\partial q^{i}}-\frac{\partial H}{\partial
q^{i}}\frac{\partial}{\partial p_{i}}\right)\,,$
and therefore, the equations determining its integral curves are similar to
Hamilton equations.
Remark that if $X,Y\in\mathfrak{X}_{{\rm LH}}(M,\omega)$ the commutator
$[X,Y]$ is a Hamiltonian vector field, with Hamiltonian $\omega(Y,X)$, because
from the relation
$i(X)\mathcal{L}_{Y}\alpha-\mathcal{L}_{Y}i(X)\alpha=i([X,Y])\alpha$, valid
for any form $\alpha$, we obtain:
$i([X,Y])\omega=i(X)\mathcal{L}_{Y}\omega-\mathcal{L}_{Y}i(X)\omega=-\mathcal{L}_{Y}i(X)\omega=-i(Y)d(i(X)\omega)-d(i(Y)i(X)\omega)\,,$
and then,
$i([X,Y])\omega=-d(\omega(X,Y))\,.$ (4.1)
Consequently the set $\mathfrak{X}_{{\rm LH}}(M,\omega)$ is a Lie algebra and
$\mathfrak{X}_{{\rm H}}(M,\omega)$ is an ideal in $\mathfrak{X}_{{\rm
LH}}(M,\omega)$.
As an important property, if $(M,\omega)$ is a symplectic manifold of
dimension $2n$ we define the Poisson bracket of two functions $F,G\in
C^{\infty}(M)$ as being the function $\\{F,G\\}$ given by:
$\\{F,G\\}=\omega(X_{F},X_{G})=dF(X_{G})=-dG(X_{F})\ .$
In Darboux coordinates for $\omega$ the expression for $\\{F,G\\}$ is the
usual one:
$\\{F,G\\}=\sum_{i=1}^{N}\left(\frac{\partial F}{\partial q^{i}}\frac{\partial
G}{\partial p_{i}}-\frac{\partial F}{\partial p_{i}}\frac{\partial G}{\partial
q^{i}}\right).$
The aforementioned property (4.1) means that if $F,G\in C^{\infty}(M)$, then
we have that $d\\{F,G\\}=-i([X_{F},X_{G}])\omega$, i.e.
$[X_{F},X_{G}]=X_{\\{G,F\\}}$.
This Poisson bracket $\\{\cdot,\cdot\\}$ is a skewsymmetric
$\mathbb{R}$–bilinear map on $C^{\infty}(M)$ such that it satisfies Jacobi’s
identity, as a consequence of $\omega$ being closed. In fact, if $F,G,H\in
C^{\infty}(M)$,
$\displaystyle(d\omega)(X_{F},X_{G},X_{H})$
$\displaystyle=X_{F}\omega(X_{G},X_{H})-X_{G}\omega(X_{F},X_{H})+X_{H}\omega(X_{F},X_{G})$
$\displaystyle-\omega([X_{F},X_{G}],X_{H})+\omega([X_{F},X_{H}],X_{G})-\omega([X_{G},X_{H}],X_{F})\
,$
and taking into account that
$X_{F}\omega(X_{G},X_{H})=X_{F}\\{G,H\\}=\\{\\{G,H\\},F\\}$ and that
$\omega([X_{F},X_{G}],X_{H})$ can also be rewritten as
$\omega([X_{F},X_{G}],X_{H})=\omega(X_{\\{G,F\\}},X_{H})=\\{\\{G,F\\},H\\}$,
as well as the corresponding expressions for each cyclic reordering, we find
that
$(d\omega)(X_{F},X_{G},X_{H})=2[\\{\\{G,H\\},F\\}+\\{\\{H,F\\},G\\}+\\{\\{F,G\\},H\\}]\
.$
Finally, we recall that in each point of a neighborhood of every point of $M$
the values of the set of Hamiltonian vector fields generate the tangent space,
and therefore $d\omega=0$ if and only if Jacobi identity holds. Hence, the
Poisson bracket endowes $C^{\infty}(M)$ with a real Lie algebra structure, the
Jacobi identity being a consequence of the closedness of $\omega$, and the
above mentioned property shows that $-\widehat{\omega}^{-1}\circ
d:C^{\infty}(M\,\\{\cdot,\cdot\\}\to\mathfrak{X}_{{\rm H}}(M,\omega)$ is a Lie
algebra homomorphism.
Given a Hamiltonian system $(M,\omega,H)$ one usually look for vector fields
whose flows are symplectomorphisms and such that are also symmetries of $H$
and, therefore, symmetries of $X_{H}$. Then for each $F\in C^{\infty}(M)$, the
relation $X_{H}F=\\{F,H\\}=-X_{F}H$ shows that $X_{F}$ is a symmetry of $H$ if
and only if $F$ is a constant of motion (a result usually known as Noether’s
theorem). It is to be remarked that in the case of a Hamiltonian dynamical
system the constants of motion $F$ are playing a double rôle in the reduction
process, either as constants of motion, or as generating infinitesimal
symmetries $X_{F}$, because if $F$ is a constant of the motion, $\\{F,H\\}=0$,
then the Hamiltonian vector field $X_{F}$ is a symmetry of $H$.
Moreover, both $X_{H}$ and $X_{F}$ are tangent to the level sets of $F$,
because $X_{H}F=\\{F,H\\}=0$, and $X_{F}F=\\{F,F\\}=0$. The restriction of
$X_{F}$ on each leaf of the foliation $\mathcal{F}_{H}$ whose leaves are the
level sets of $F$ can be used to determine adapted coordinates in such a way
that the problem of determining the integral curves of $X_{H}$ is reduced not
just in one but in two degrees of freedom.
In order to be able to use simultaneously the information given by different
constants of motion, $F_{1},\ldots,F_{r}$, it is sufficient that
$\\{F_{i},F_{j}\\}=0,\,\forall i,j=1,\ldots,r$, because
$[X_{F_{i}},X_{F_{j}}]=X_{\\{F_{j},F_{i}\\}}$. If the condition is satisfied,
then $[X_{F_{i}},X_{F_{j}}]=0$, for any pair of indices $i$ and $j$, and we
can find adapted coordinates such that $X_{F_{i}}=\partial/\partial y_{i}$,
for $i=1,\ldots,r$. The condition is not necessary because if, for instance,
$\\{F_{i},F_{j}\\}$ is constant for each pair of indices, then it is also true
that $[X_{F_{i}},X_{F_{j}}]=0$.
The simplest case will be that of Hamiltonian systems in a space of dimension
$N=2\,n$ for which such $n$ constants of motion, $F_{1},\ldots,F_{n}$, in
involution are known: they are called completely integrable systems.
We are now ready to recall the notion of Liouville-Arnold integrability:
###### Definition 4.1
The Hamiltonian dynamical system $(M,\omega,H)$, with $\,\dim M=2n$, is said
to be completely integrable if there exists a set of $n$ functions
$\\{F_{j}\mid j=1,\ldots,n\\}$, where $F_{j}\in C^{\infty}(M)$, with
$F_{1}=H$, which are constants of the motion, i.e.
$\frac{dF_{k}}{dt}=\mathcal{L}_{X_{H}}F_{k}=\\{F_{k},H\\}=0,\qquad\forall
k=2,\ldots,n,$
that are functionally independent, i.e. $dF_{1}\land\cdots\land dF_{n}\neq 0$,
and are pairwise in involution, i.e.
$\\{F_{k},F_{j}\\}=0,\qquad\forall j,k=1,\ldots n.$
A completely integrable system that admits more than $n$ functionally
independent constants of motion is said to be superintegrable and when the
number is the maximum number, $2n-1$, the system is called maximally
superintegrable [23]. Many of such systems can be found in the physics
literature (see e.g. the recent papers [24, 25] and references therein).
The two main examples of Hamiltonian dynamical systems are those of
Hamiltonian systems on the cotangent bundle $T^{*}Q$ of a manifold $Q$ and
those defined by regular Lagrangians on the tangent bundle $TQ$. In fact the
cotangent bundle $\pi:T^{*}Q\to Q$ is endowed with a canonical 1-form
$\theta\in\bigwedge^{1}(T^{*}Q)$ defined by
$\theta_{\alpha}=\pi^{*}_{\alpha}\alpha$, for all covectors $\alpha$ in $Q$.
Then $\omega=-d\theta$ is a symplectic form on $T^{*}Q$. It is to be remarked
that if $\alpha$ is a 1-form in $Q$, then as $\alpha^{*}\theta=\alpha$ we have
that $\alpha^{*}\omega=\alpha^{*}(-d\theta)=-d(\alpha^{*}\theta)=-d\alpha$.
The geometric framework for the study of Lagrangian mechanics is that of
tangent bundles [26, 27, 28]. The tangent bundle $\tau:TQ\to Q$ is
characterised by two geometric tensors, the vertical endomorphism $S$, a
(1,1)-tensor field on $TQ$, also called tangent structure, which satisfies
$\text{Im}\,S=\ker S$ and an integrability condition, the vanishing of the
Nijenhuis tensor, $N_{S}=0$, and the Liouville vector field $\Delta$
generating dilations along fibres in $TQ$ [29].
If $(U,\varphi)$ is a local chart on $Q$ and ${\rm
pr}^{i}:\mathbb{R}^{n}\to\mathbb{R}$ are the natural projections on the
$i$-th-factor and $q^{i}={\rm pr}^{i}\circ\varphi$ we define the coordinate
system $(U,q^{1},\ldots,q^{n})$ on $Q$, and the corresponding chart in
$\mathcal{U}=\tau^{-1}(U)$ given by $(\mathcal{U},\varphi,\varphi_{*})$,
defines a coordinate system $(q^{1},\ldots,q^{n},v^{1},\ldots,v^{n})$ on the
open set $\mathcal{U}=\tau^{-1}(U)$ of $TQ$. Correspondingly, we consider the
coordinate basis of $\mathfrak{X}(U)$ usually denoted $\\{\partial/\partial
q^{j}\mid j=1,\ldots,n\\}$ and its dual basis for $\bigwedge^{1}(U)$,
$\\{dq^{j}\mid j=1,\ldots,n\\}$. Then a vector $v$ in a point $q\in U$ is
$v={\displaystyle\sum_{i=1}^{n}}v^{j}\,(\partial/\partial q^{j})_{|q}$, i.e.
$v^{i}=dq^{i}(v)$, while a covector $\alpha$ in a point $q\in U$ is
$\alpha={\displaystyle\sum_{j=1}^{n}}p_{j}\,dq^{j}\,_{|q}$, with
$p_{i}=\alpha((\partial/\partial q^{i})_{|q})$.
With this notation the coordinate expressions of the vertical endomorphism $S$
and the Liouville vector field $\Delta$, are [26, 27]:
$S(q,v)=\sum_{i=1}^{n}\frac{\partial}{\partial v^{i}}\otimes
dq^{i},\qquad\Delta(q,v)=\sum_{i=1}^{n}v^{i}\frac{\partial}{\partial v^{i}}.$
(4.2)
Similarly we can introduce a coordinate system
$(q^{1},\ldots,q^{n},p_{1},\ldots,p_{n})$ in the open set
$\bar{\mathcal{U}}=\pi^{-1}(U)$ of the cotangent bundle $\pi:T^{*}Q\to Q$ and
the coordinate expression of the 1-form $\theta$ is
$\theta(q,p)=\sum_{i=1}^{n}p_{i}\,dq^{i},$ (4.3)
with shows that $\omega=-d\theta$ is a canonical symplectic structure on
$T^{*}Q$.
The vector field $\Delta$ and the tensor field $S$ can be used to select a
special kind of vector fields whose integral curves are given by lifting
solutions of a second order differential equations. These vector fields called
second order differential equation vector fields $D\in\mathfrak{X}(TQ)$
(hereafter shortened as SODE vector fields) are characterised by
$S(D)=\Delta$.
Recall also that given a Lagrangian $L\in C^{\infty}(TQ)$ we can define a
1-form $\theta_{L}=dL\circ S$ and the exact 2-form $\omega_{L}=-d\theta_{L}$.
When $\omega_{L}$ is regular the Lagrangian $L$ is said to be regular and then
the dynamics is given by the uniquely defined SODE vector field $\Gamma_{L}$
such that
$i(\Gamma_{L})\omega_{L}=dE_{L}\Longleftrightarrow\mathcal{L}_{\Gamma_{L}}\theta_{L}-dL=0,$
(4.4)
where the energy function $E_{L}\in C^{\infty}(TQ)$ is defined by
$E_{L}=\Delta(L)-L$. Moreover, this implies that
$\mathcal{L}_{\Gamma_{L}}\omega_{L}=d(i(\Gamma_{L})\omega_{L})=0$
In usual local tangent bundle coordinates the expressions of $\theta_{L}$ and
$E_{L}$ are
$\theta_{L}=\sum_{i=1}^{n}\frac{\partial L}{\partial v^{i}}\,dq^{i},\qquad
E_{L}=\sum_{i=1}^{n}v^{i}\frac{\partial L}{\partial v^{i}}-L,$ (4.5)
while that of $\omega_{L}$ is:
$\omega_{L}=\sum_{i,j=1}^{n}\frac{\partial^{2}L}{\partial{q}^{j}\partial
v^{i}}d{q}^{i}\wedge dq^{j}+\sum_{j,k=1}^{n}\frac{\partial^{2}L}{\partial
v^{k}\partial v^{j}}dq^{j}\wedge dv^{k}.$ (4.6)
It may be of be interest under what conditions two regular Lagrange functions
$L,L^{\prime}\in C^{\infty}(TQ)$ in $TQ$ produce the same symplectic structure
and the same energy function, i.e. $L_{0}=L-L^{\prime}$ is such that
$\omega_{L_{0}}\equiv 0$ and $E_{L_{0}}=0$.
Now, if $\alpha$ is a 1-form on $Q$, $\alpha\in\bigwedge^{1}(Q)$, then
$\widehat{\alpha}$ denotes the function $\widehat{\alpha}\in C^{\infty}(TQ)$
defined by $\widehat{\alpha}(u)=\alpha_{\pi(u)}(u)$. If the local coordinate
expression of a 1-form $\alpha$ is
$\alpha={\displaystyle\sum_{i=1}^{n}}\alpha_{i}(q)\,dq^{i}$, then
$\widehat{\alpha}(q,v)={\displaystyle\sum_{i=1}^{n}}\alpha_{i}(q)\,v^{i}$.
Consequently $\Delta\widehat{\alpha}=\widehat{\alpha}$, because, by definition
$\widehat{\alpha}$ is a homogeneous of degree one function. On the other hand,
as
$(d\widehat{\alpha}\circ S)\left(\frac{\partial}{\partial
q^{i}}\right)=\alpha_{i}(q),\qquad(d\widehat{\alpha}\circ
S)\left(\frac{\partial}{\partial v^{i}}\right)=0,$
we see that $d\widehat{\alpha}\circ S=\tau^{*}\alpha$.
Then, a function $L_{0}\in C^{\infty}(TQ)$ is such that $\omega_{L_{0}}\equiv
0$ if and only if there exist a closed 1-form $\alpha\in Z^{1}(Q)$ and a
function $h\in C^{\infty}(Q)$ such that $L_{0}=\widehat{\alpha}+h\circ\tau$
(see e.g. [30]). But as for such a function $L_{0}$ we have that
$E_{L_{0}}=-h$, we see that $L,L^{\prime}\in C^{\infty}(TQ)$ in $TQ$ produce
the same Hamiltonian dynamical system, i.e. $\omega_{L}=\omega_{L^{\prime}}$
and $E_{L}=E_{L^{\prime}}$, if and only if there exists a closed 1-form
$\alpha\in Z^{1}(Q)$ such that $L^{\prime}=L+\widehat{\alpha}$, and both
Lagrangians are then said to be gauge equivalent.
It is also possible to identify a complete lift vector field
$X^{c}\in\mathfrak{X}(TQ)$ that is a symmetry of the Hamiltonian dynamical
system $(TQ,\omega_{L},E_{L})$ in terms of symmetries of $L$ itself. Recall
that if $X\in\mathfrak{X}(Q)$, its complete lift, denoted $X^{c}$, is the
vector field on $TQ$ whose flow is given by $\phi_{t*}$ where $\phi_{t}$ is
the local flow of the vector field $X$. If the local coordinate expression of
the vector field $X$ in a chart of $Q$ is
$X(q)={\displaystyle\sum_{i=1}^{n}}X^{i}(q)\partial/\partial q^{i}$, then the
corresponding expression for $X^{c}$ in the associated chart of $TQ$ is
$X^{c}(q,v)={\displaystyle\sum_{i=1}^{n}}X^{i}(q)\partial/\partial
q^{i}+{\displaystyle\sum_{i,j=1}^{n}}(\partial X^{i}/\partial
q^{j})v^{j}\partial/\partial v^{i}$.
It is also to be remarked that if $\Phi\in{\rm Diff}(TQ)$ is a lift of a
diffeomorphism $\varphi$ of the manifold $Q$, then
$\Phi^{*}\theta_{L}=\theta_{\Phi^{*}L}$ as well as
$\Phi^{*}E_{L}=E_{\Phi^{*}L}$ as a consequence of $\Phi_{*}\Delta=\Delta$ and
$[\Phi_{*},S]=0$, because
$\Phi^{*}\theta_{L}=\Phi^{*}(dL\circ S)=dL\circ
S\circ\Phi_{*}=dL\circ\Phi_{*}\circ S=d(L\circ\Phi)\circ
S=\theta_{\Phi^{*}L},$
while, as $\Delta$ is $\Phi$-related with itself, $(\Delta
L)\circ\Phi=\Delta(L\circ\Phi)$, we have
$\Phi^{*}E_{L}=\Phi^{*}(\Delta
L)-\Phi^{*}L=\Delta(\Phi^{*}L)-\Phi^{*}L=E_{\Phi^{*}L}.$
At the infinitesimal level this means that if $X^{c}\in\mathfrak{X}(TQ)$ is
the complete lift of $X\in\mathfrak{X}(Q)$ we have that
$\mathcal{L}_{X^{c}}\theta_{L}=\theta_{X^{c}L},\qquad X^{c}E_{L}=E_{X^{c}L}.$
All these properties can be used to derive the Noether’s theorem in the
Lagrangian formalism:
###### Theorem 4.2
(Noether) If the vector field $X\in\mathfrak{X}(Q)$ is such that there exists
a function $h\in C^{\infty}(Q)$ such that $X^{c}L=\widehat{dh}$, then the
function $f=i(X^{c})\theta_{L}-\tau^{*}h$ is a constant of motion.
Proof.- First, the vector field $X^{c}$ is Hamiltonian because
$\mathcal{L}_{X^{c}}\theta_{L}=\theta_{X^{c}L}=\theta_{\widehat{dh}}=\tau^{*}(dh)=d(\tau^{*}h),$
and then
$i(X^{c})d\theta_{L}+d(i(X^{c})\theta_{L})=d(\tau^{*}h)\Longleftrightarrow
i(X^{c})\omega_{L}=d((i(X^{c})\theta_{L}-h)=df.$
Moreover, $X^{c}E_{L}=E_{X^{c}L}=E_{\widehat{dh}}=0$, and then if $\Gamma_{L}$
is the dynamical vector field determined by $L$, i.e.
$i(\Gamma_{L})\omega_{L}=dE_{L}$, we have
$\Gamma_{L}f=\\{f,E_{L}\\}=-\\{E_{L},f\\}=-X^{c}E_{L}=0.$
$\Box$
Particularly interesting cases are the geodesic Lagrangians defined through a
Riemann structure $g$ by $L=\frac{1}{2}g(v,v)$ (see e.g. [31]) and the so
called natural Lagrangians defined by means of a Riemann structure $g$ and a
function $V$ in $Q$ as follows: $L=\frac{1}{2}g(v,v)-V(q)$.
## 5 Non-Noether constants of motion
Given a locally Hamiltonian vector field $X$ in a $2n$-dimensional symplectic
manifold $(M,\omega)$, if $\omega^{\prime}$ is another closed 2-form invariant
under $X$, then we can consider the pencil of invariant under $X$ closed
2-forms $\\{\omega^{\prime}-\lambda\,\omega\mid\lambda\in\mathbb{R}\\}$, which
defines a function $f\in\mathbb{R}\times M\to\mathbb{R}$ such that
$(\omega^{\prime}-\lambda\,\omega)^{\wedge n}=f\,(\omega)^{\wedge n}$, called
characteristic function of the pencil, and that, by construction, is a
constant of the motion for each value of $\lambda$, because from
$\mathcal{L}_{X}(\omega^{\prime}-\lambda\,\omega)^{\wedge n}=0$ we find
$\mathcal{L}_{X}f=0$. Moreover, the function $f$ is a polynomial function in
$\lambda$ whose coefficients are constants of the motion. Note also that
$\widehat{\omega}^{-1}\circ\widehat{\omega}^{\prime}:\mathfrak{X}(M)\to\mathfrak{X}(M)$
is $C^{\infty}(M)$-linear and define then a $(1,1)$-tensor field that is
$X$-invariant and such that the characteristic polynomial of such composition
coincides with the characteristic function of the pencil, and therefore the
mentioned constants of the motion for $X$ coincide with those associated to
the invariant $(1,1)$-tensor field
$\mathcal{R}=\widehat{\omega}^{-1}\circ\widehat{\omega}^{\prime}$, and hence
they are related to the traces of different powers of $\mathcal{R}$.
The point is how to find such a $X$-invariant 2-form $\omega^{\prime}$. We
mention now two particular cases. First, if a vector field $Y$ is such that
$[Y,X]=0$ but it is not a locally Hamiltonian vector field, then
$\omega^{\prime}=\mathcal{L}_{Y}(\omega)$ is a $X$-invariant closed 2-form,
because
$\mathcal{L}_{X}(\mathcal{L}_{Y}(\omega))=\mathcal{L}_{Y}(\mathcal{L}_{X}(\omega))=0$.
On the other side, if $\phi\in{\rm Diff}(M)$ is a noncanonical symmetry of the
Hamiltonian vector field defined by the Hamiltonian dynamical system
$(M,\omega,H)$, from the relation
$i(\phi_{*}(X_{H}))((\phi^{-1})^{*}(\omega))=d((\phi^{-1})^{*}H)$, we see that
if $\phi_{*}(X_{H})=X_{H}$ then
$i(X_{H})((\phi^{-1})^{*}(\omega))=d((\phi^{-1})^{*}H)$, and therefore
$\mathcal{L}_{X_{H}}((\phi^{-1})^{*}(\omega))=0$, and equivalently
$\mathcal{L}_{X_{H}}(\phi^{*}(\omega))=0$. As it was assumed that
$\omega^{\prime}=\phi^{*}(\omega)\neq\omega$, we can choose such a
$X$-invariant closed 2-form $\omega^{\prime}$ to define, together with
$\omega$, the pencil. We also recall that $\phi\in{\rm Diff}(M)$ is said to be
a canonoid transformation of the Hamiltonian dynamical system $(M,\omega,H)$
when $\phi_{*}(X_{H})$ is also a Hamiltonian vector field, or equivalently
when $X_{H}$ is Hamiltonian with respect to the transformed 2-form
$\phi^{*}(\omega)$, and in this case $\mathcal{L}_{X_{H}}(\phi^{*}(\omega))=0$
[16, 17], i.e. we can also choose $\omega^{\prime}=\phi^{*}(\omega)$ as
invariant $X_{H}$-form.
## 6 Invariant volume forms and Jacobi multipliers
A particularly interesting case with many applications not only in the theory
of differential equations but in classical mechanics is that of volume forms
invariant under a given vector field $X$ on an oriented manifold $(M,\Omega)$.
Each volume form is of the form $R\,\Omega$ with $R\in C^{\infty}(M)$ and the
invariance condition is $\mathcal{L}_{X}(R\,\Omega)=0$, and if we take into
account that
$\mathcal{L}_{X}(R\,\Omega)=d(i(X)(R\,\Omega))=d(i(R\,X)(\Omega))=\mathcal{L}_{R\,X}(\Omega)$
we see that the invariance condition of $R\,\Omega$ under $X$ is equivalent to
the invariance condition of $\Omega$ under $R\,X$. Remark also that for each
vector field $X\in\mathfrak{X}(M)$ the volume form $\mathcal{L}_{X}\Omega$ is
proportional to $\Omega$, and the proportionality function is called
divergence of $X$, i.e.
$\mathcal{L}_{X}(\Omega)={\rm div}(X)\,\Omega\,,$ (6.1)
and that if local coordinates are chosen such that
$\Omega=dx^{1}\wedge\cdots\wedge dx^{n}$ and the local expression of $X$ is
$X={\displaystyle\sum_{i=1}^{n}X^{i}\frac{\partial}{\partial x^{i}}}$, then
the local expression of ${\rm div}(X)$ is:
${\rm div}(X)=\sum_{i=1}^{n}\frac{\partial X^{i}}{\partial x^{i}}.$
Vector fields $X$ such that ${\rm div}(X)=0$ are called divergence-free vector
fields and enjoy interesting properties.
Recalling the properties of Lie derivatives, we find that, as for each $f\in
C^{\infty}(M)$,
$\mathcal{L}_{f\,X}(\Omega)=\mathcal{L}_{X}(f\,\Omega)=f\,\mathcal{L}_{X}(\Omega)+(\mathcal{L}_{X}f)\,\Omega$,
using the definition (6.1) of ${\rm div}(X)$, we have that
${\rm div}(fX)=f\,{\rm div}(X)+X(f)\,.$ (6.2)
The nonvanishing functions $R$ such that
$\mathcal{L}_{X}(R\,\Omega)=\mathcal{L}_{R\,X}(\Omega)=0$, i.e. ${\rm
div}(R\,X)=0$, are called Jacobi multipliers, and (6.2) shows that the
nonvanishing function $R$ is a Jacobi multiplier if and only if (see [32, 33]
and references therein)
$R\,{\rm div}(X)+X(R)=0\Longleftrightarrow{\rm div}(X)+X(\log R)=0\,.$ (6.3)
Locally defined Jacobi multipliers, obtained as particular solutions of (6.3),
always exist, but in some particular cases global solutions, giving rise to
invariant volume forms, do not exist [34, 35].
It is also to be remarked that it follows from the relation
$\mathcal{L}_{X}(R\,\Omega)=d(i(X)(R\,\Omega))=d(i(R\,X)\Omega)$ that the
function $R$ is a Jacobi multiplier for $X$ in the oriented manifold
$(M,\Omega)$ iff and only if $f\,R$ is a Jacobi multiplier for $X$ in the
oriented manifold $(M,f^{-1}\,\Omega)$, for each positive function $f$.
As an interesting case we can consider a SODE vector field
$\Gamma\in\mathfrak{X}(TQ)$ which admits a Lagrangian formulation $L$, i.e.
there exists a regular Lagrange function $L\in C^{\infty}(TQ)$, such that
$i(\Gamma)\omega_{L}=dE_{L}$, or equivalently
$\mathcal{L}_{\Gamma}\theta_{L}=dL$. Remark first that $TQ$ is orientable and
a local chart of $TQ$ induced from one in $Q$ induces an orientation. As
$\mathcal{L}_{\Gamma}\omega_{L}=0$, we see that the volume form
$(\omega_{L})^{\wedge n}$ is $\Gamma$-invariant and therefore if a volume form
was previously fixed, there will be a Jacobi multiplier $R$ such that
$(\omega_{L})^{\wedge n}=R\,\Omega$. If, for instance, $\Omega$ is the volume
form determined by a tangent bundle local chart,
$\Omega=dq^{1}\wedge\cdots\wedge dq^{n}\wedge dv^{1}\wedge\cdots\wedge
dv^{n}$, we obtain, according to (4.6), that the determinant of the Hessian
matrix $W$ with elements $W_{ij}=\partial^{2}L/\partial v^{i}\partial v^{j}$
is a Jacobi multiplier, because $(\omega_{L})^{\wedge n}$ is a real multiple
of $\det W\,dq^{1}\wedge\cdots\wedge dq^{n}\wedge dv^{1}\wedge\cdots\wedge
dv^{n}$.
## 7 Hojman symmetry and constants of motion
In the preceding sections 4 and 5 we have first summarised the usual way of
searching first-integrals by means of infinitesimal symmetries via the first
Noether theorem and presented then a second procedure for finding non-Noether
constants of motion, which is not based on symmetries but on the existence of
alternative geometric structures for the description of the vector field, what
leads to the existence of a recursion operator. We mention next a third
approach started by Hojman [36] and González-Gascón in [37] and that it is
becoming more and more important during the last years for its applications in
$f(R)$-gravity and FRW cosmology [38, 39, 40, 41, 42, 43, 44].
The main result is very general and a direct consequence of a simple geometric
relation that when considering particular cases contains many results
scattered in the physics literature. So, different relations among vector
fields and their divergences can be used to establish first-integrals and
integral invariants for vector fields in a manifold $M$.
The following geometric relation plays a fundamental rôle: If $X,Y$ is an
arbitrary pair of vector fields in an oriented manifold $(M,\Omega)$, then
$\mathcal{L}_{X}({\rm div\,}(Y))-\mathcal{L}_{Y}({\rm div\,}(X))={\rm
div\,}([X,Y]),$ (7.1)
because as the Lie derivatives of a $p$-form $\alpha$ satisfy the relation
$\mathcal{L}_{X}(\mathcal{L}_{Y}\alpha)-\mathcal{L}_{Y}(\mathcal{L}_{X}\alpha)=\mathcal{L}_{[X,Y]}\alpha,$
(7.2)
and in the particular case $\alpha=\Omega$, we have
$\mathcal{L}_{X}(\mathcal{L}_{Y}\Omega)-\mathcal{L}_{Y}(\mathcal{L}_{X}\Omega)=\mathcal{L}_{[X,Y]}\Omega,$
(7.3)
and from difference of
$\mathcal{L}_{X}(\mathcal{L}_{Y}\Omega)=\mathcal{L}_{X}({\rm
div}(Y)\,\Omega)=\mathcal{L}_{X}({\rm div}(Y))\,\Omega+{\rm div}(Y)\,{\rm
div}(X)\,\Omega$ and the corresponding relation
$\mathcal{L}_{Y}(\mathcal{L}_{X}\Omega)=\mathcal{L}_{Y}({\rm
div}(X)\,\Omega)=\mathcal{L}_{Y}({\rm div}(X))\,\Omega+{\rm div}(X)\,{\rm
div}(Y)\,\Omega$ we obtain (7.1).
As a consequence of this relation (7.1) we see that if the vector field
$X\in\mathfrak{X}(M)$ in an oriented manifold $(M,\Omega)$ is divergence-free,
then if the vector field $Y$ is an infinitesimal symmetry of $X$, i.e.
$[X,Y]=0$, we have that $\textrm{div\,}(Y)$ is a constant of the motion for
$X$. Something similar happens when the vector field $Y$ is an infinitesimal
symmetry of the 1-dimensional distribution generated by $X$, i.e. there exists
a function $h$ such that $[Y,X]=h\,X$, because then in this case the function
$\textrm{div\,}(Y)+h$ is a constant of the motion for $X$. Actually, if ${\rm
div\,}(X)=0$, then $\mathcal{L}_{X}\Omega=0$, and hence, as $[X,Y]=-h\,X$,
$\mathcal{L}_{X}(\mathcal{L}_{Y}\Omega)=\mathcal{L}_{Y}(\mathcal{L}_{X}\Omega)+\mathcal{L}_{-h\,X}\Omega=-\mathcal{L}_{X}(h\,\Omega)=-X(h)\,\Omega,$
and then, from
$\mathcal{L}_{X}(\mathcal{L}_{Y}\Omega)=\mathcal{L}_{X}({\rm
div\,}(Y)\,\Omega)=\mathcal{L}_{X}({\rm div\,}(Y))\Omega,$
we obtain that $\mathcal{L}_{X}({\rm div\,}(Y)+h)=0$, and therefore the
following function $I$ is a constant of the motion for $X$.
$I={\rm div\,}(Y)+h.$ (7.4)
If the vector field is not divergence-free, remark that for any nonvanishing
function $R$ the constants of motion of $X$ coincide with those of $R\,X$, and
in particular if $R$ is a Jacobi multiplier for $X$ the vector field
$\bar{X}=R\,X$ is such that when $[Y,X]=h\,X$, we have that
$[Y,\bar{X}]=\bar{h}\,\bar{X}$ with $\bar{h}=(Y(R)/R)+h$, and hence we have
the constant of motion for $\bar{X}$, and therefore for $X$, given by
$I={\rm div\,}(Y)+\bar{h}={\rm div\,}(Y)+Y(\log R)+h.$ (7.5)
These general results can be applied to specific examples, and we recover as
particular examples many previously found constants of motion. For instance
one can apply the general theory to both Hamltonian and Lagrangian
formulations of autonomous systems, or generic second order differential
equations. In the particular case of a system on an Euclidean configuration
space admitting a Lagrangian formulation we mentioned at the end of the
preceding section that a Jacobi multiplier is given by the determinant of the
Hessian matrix $W$, and this explains the result obtained by Lutzky [45] as a
particular case of the constant of motion given by (7.5). In the more general
case of $Q$ being a $n$-dimensional manifold, a local chart of coordinates
$(x^{1},\ldots,x^{n})$ for $Q$ induces a local chart of coordinates in its
tangent bundle, denoted $(x^{1},\ldots,x^{n},v^{1},\ldots,v^{n})$, as
indicated in Section 4, and an associated volume element in such a chart given
by $\Omega=dx^{1}\wedge\cdots\wedge dx^{n}\wedge dv^{1}\wedge\cdots\wedge
dv^{n}$. If the dynamical vector field $\Gamma$ is given in such coordinates
by
$\Gamma=\sum_{i=1}^{n}\left(v^{i}\frac{\partial}{\partial
x^{i}}+F^{i}(x,v)\frac{\partial}{\partial v^{i}}\right)$ (7.6)
is determined by the Lagrangian $L$, i.e. $i(\Gamma)\omega_{L}=dE_{L}$, which
implies $\mathcal{L}_{\Gamma}\omega_{L}=0$ (see e.g [28]), it has been shown
(see e.g. [32] and references therein) that a particular Jacobi multiplier for
$\Gamma$ with respect to the volume form $\Omega$ is given by the determinant
of the Hessian matrix $W$ in the velocities, with elements
$W_{ij}=\partial^{2}L/{\partial v^{i}\partial v^{j}}$. Actually, from
$\mathcal{L}_{\Gamma}\omega_{L}=0$, we see that $(\omega_{L})^{\wedge n}$ is
an invariant volume under $\Gamma$, and the proportionality factor of
$(\omega_{L})^{\wedge n}$ and $\Omega$ is a real multiple of $\det W$.
Therefore, the constant of motion obtained in [45] is just a particular case
of the expression (7.5) for $R$ equal to the determinant of the Hessian matrix
$W$. As far as nonautonomous systems are concerned, the dynamical vector
fields must be replaced by 1-dimensional distributions and therefore the more
general condition $[Y,X]=h\,X$, which means that the vector field $Y$
preserves the 1-dimensional distribution generated by $X$, is relevant in the
context of symmetry for such systems. As in the autonomous cases, we can study
particular examples of non-autonomous systems of first-order differential
equations [36, 37] but also of Hamiltonian systems as in [46], or even non-
autonomous systems of second-order differential equations [36, 37, 47, 48],
and in particular systems admitting a Lagrangian formulation [45, 48]. For a
recent geometric presentation of all these results see e.g. [49].
## 8 Hamilton-Jacobi equation as a reduction procedure
As a last example of the reduction theory we analyse the well-know Hamilton-
Jacobi equation from a geometric perspective. There is a very recent nice
review paper on the subject [50] based on [51] to which I refer for more
general results and only the reduction technique for the particular case of a
non-autonomous Hamiltonian systems in the phase space $T^{*}Q$ is presented
here. Recall that 1-forms on $Q$ are sections of the cotangent bundle
$\pi_{Q}:T^{*}Q\to Q$ while vector fields in $Q$ are sections for the
projection map $\tau_{Q}:TQ\to Q$ of the tangent bundle. Then, given a
Hamiltonian dynamical system $(T^{*}Q,\omega,H)$ which defines a Hamiltonian
vector field $X_{H}\in\mathfrak{X}(T^{*}Q)$ by the relation
$i(X_{H})\omega=dH$, where $\omega$ is the canonical symplectic structure on
$T^{*}Q$, the aim is to find a vector field $Z\in\mathfrak{X}(Q)$ and a 1-form
$\alpha\in\bigwedge^{1}(T^{*}Q)$ such that $X_{H}\circ\alpha=T\alpha\circ Z$.
Such a pair $(Z,\alpha)$ is said to be a generalised solution of the Hamilton-
Jacobi problem, and when $\alpha$ is closed we say that $(Z,\alpha)$ is a
standard solution the Hamilton-Jacobi problem. Note that in these conditions
if a curve $\gamma$ in $Q$ is an integral curve of $Z$, then the curve
$\alpha\circ\gamma$ in $T^{*}Q$ is an integral curve of $X_{H}$. This shows
that the vector field $X_{H}$ is tangent to the image of $\alpha$. Moreover,
as a consequence, as $X_{H}$ is a vector field in $T^{*}Q$ and $T\pi_{Q}\circ
T\alpha={\rm id}_{TQ}$, we have that $T\pi_{Q}\circ
X_{H}\circ\alpha=T\pi_{Q}\circ T\alpha\circ Z=Z$, i.e. the vector field
restriction of $X_{H}$ on the image of $\alpha$ is projectable on $Z$. In this
case the vector field $Z$ in the pair $(Z,\alpha)$ can be derived from
$\alpha$.
Observe that for each vector field $Y\in\mathfrak{X}(Q)$, if we recall that
$\alpha^{*}\omega=d\alpha$, the 1-form $i(Z)d\alpha$ is such that
$(i(Z)d\alpha)(Y)=d\alpha(Z,Y)=-\alpha^{*}\omega(Z,Y)=-\omega(T\alpha\circ
Z,T\alpha\circ Y)\circ\alpha,$
while
$(\alpha^{*}(dH))(Y)=\alpha^{*}(i(X_{H})\omega)(Y)=(i(T\alpha\circ
Y)i(X_{H})\omega)\circ\alpha=\omega(X_{H}\circ\alpha,T\alpha\circ
Y)\circ\alpha,$
and consequently the relatedness condition $X_{H}\circ\alpha=T\alpha\circ Z$
is equivalent to
$i(Z)d\alpha+d(\alpha^{*}H)=0.$ (8.1)
Remark that for a standard solution as the 1-form $\alpha$ is closed,
$d\alpha=0$, there exists a locally defined function $S$ in $Q$ such that
$\alpha=dS$ and the relatedness condition (8.1) can be replaced by
$d(\alpha^{*}H)=0$, i.e $\alpha^{*}H$ is locally constant, or more explicitly
$(dS)^{*}H=E\Longleftrightarrow H\left(q^{1},\ldots,q^{n},\frac{\partial
S}{\partial q^{i}},\ldots,\frac{\partial S}{\partial q^{i}}\right)=E,$
which is the usually known as Hamilton-Jacobi equation. Once a solution of
this equation has been obtained we can define the vector field
$X^{S}\in\mathfrak{X}(Q)$ by $X^{S}=T\pi\circ X_{H}\circ dS$, and then $dS(Q)$
is an invariant under $X_{H}$ submanifold such that for each integral curve
$\gamma$ of $X^{S}$, the curve $dS\circ\gamma$ is an integral curve of
$X_{H}$, and in this sense the vector field $X^{S}$ in $Q$ is a reduction of
$X_{H}$ which describes, at least partially, the dynamics defined by $X_{H}$
in $T^{*}Q$. In this sense the integral curves of $X_{H}$ are completely in
one of such invariant submanifolds.
In order to fully solve the problem we need an appropriate $n$-parameter
family of functions $S_{\lambda}$, with
$\lambda=(\lambda_{1},\ldots,\lambda_{n})$, $\lambda_{i}\in\mathbb{R}$, what
is usually known as a complete solution of the Hamilton-Jacobi equation. More
details and applications can be found in [50].
## 9 Conclusions
The theory of integrability of systems of differential equations has been
analysed from a geometric perspective through the properties of their
associated vector fields. The reduction process consists on finding a more
easily integrable related system whose solutions give us an at least partial
information on the solutions of the original system. The search for such
simpler related systems is based on the use of first integrals and
infinitesimal symmetries of the vector field or, more generally, invariant
tensor fields. So, in Section 2 we have revisited the well-known classical Lie
theorem and a more recent generalised result, without recourse to additional
compatible structures, while in Section 3 the rôle of invariant tensor fields
in integrability has been discussed with a special emphasis on invariant
(1,1)-tensor fields leading to symmetry generators and Lax equations with
their correspondig first integrals. In Section 4, after a brief summary of the
geometric Hamiltonian and Lagrangian mechanics as well as the explicit
formulation of Noether theorem in both frameworks, the Arnold-Liouville
integrability has been described. The meaning of non-Noether symmetries and
how to find such symmetries have been discussed in Section 5. The usefulness
of invariant volume forms and its relation to Jacobi multipliers on oriented
manifolds has been displayed in Section 6 as a prelude to the geometric
approach to Hojman symmetry briefly presented in Section 7. Finally, the
geometric approach to Hamilton-Jacobi equation has also been analysed from the
perspective of the search of related systems. according to the approach to
reduction and integrability developed in this paper.
## Acknowledgements
Financial support from the projects of Spanish Ministerio de Ciencia,
Innovación y Universidades PGC2018-098265-B-C31 and Gobierno de Aragón
$48\\_20$R Análisis y Física Matemática, is acknowledged.
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|
# Chaotic dynamics of the sugarcane borer-two parasitoid agroecosystem with
seasonality
Marat Rafikov 1, Alexandre Molter2, João Inácio Moreira Bezerra2, Elvira
Rafikova1, Maria Cristina Varriale3 1<EMAIL_ADDRESS>Federal
University of ABC, Brazil.
1<EMAIL_ADDRESS>Federal University of ABC, Brazil.
2<EMAIL_ADDRESS>Federal University of Pelotas, Brazil.
2<EMAIL_ADDRESS>Federal University of Pelotas, Brazil.
3<EMAIL_ADDRESS>Federal University of Rio Grande do Sul, Brazil.
###### Abstract
Sugarcane production is a significant and profitable agribusiness sector in
many countries. Nevertheless, this industry suffers significant losses from
the sugarcane pests, among which the most important one is the sugarcane borer
(Diatraea saccharalis). This pest population is hard to be controlled due to
its different life stages, thus biological control ( with more than one
predator species) can be applied. Therefore, in this work, we present and
analyze a mathematical model that describes the dynamics of the sugarcane
borer and its two different life stages parasitoids: eggs’ (Trichogramma
galloi) and larval (Cotesia flavipes). First, a host-parasitoid model is used
to obtain the population dynamics, which also considers the influence of
seasonal variations. Then, system simulations and bifurcation diagrams show
that the introduction of seasonality perturbations causes complex dynamics and
results in limit cycles and strange attractors.
###### keywords:
Sugarcane borer , Parasitoids , Dynamics , Seasonality , Chaos
††journal: Applied Mathematics and Computation
[table]capposition=bottom
## 1 Introduction
Sugarcane is a global commodity produced throughout the tropics and used for
sweeteners, biofuels, and a growing range of bioproducts (including
bioplastics) [WinNT]. The production of sugarcane suffers from insect pests
such as sugarcane borer Diatraea sacharalis, which is reportedly the most
significant sugarcane pest in Brazil and other countries [parraetal2002,
postali2019applied]. This peculiar insect lays eggs on the surface of the
sugarcane leaves. The larvae, however, live inside the sugarcane stalk,
carving internal galleries and causing significant damages that can lead to
the death of the plant. Because the larvae live hidden from the surface, the
pesticide control becomes inefficient and biological control is the
alternative for this case.
Biological control is the pest populations reduction by their natural enemies
(predators, parasitoids, and pathogens). The sugarcane borer larvae population
has been controlled since the 1970s by the parasitoid, Cotesia flavipes.
Recently, the Trichogramma galloi parasitoid is a new alternative for
biological control of it’s egg population [parraetal2002, postali2019applied].
The applications of prey-predator and host-parasitoid models for biological
control were reviewed in [gamez2009observation, gamez2010open,
venturino2007diseases, venturino2008biological, chattopadhyay2015chaos].
In biological systems with more than two species, prey-predator interactions
become complex and consequently harder to model prey invasions. Mathematical
modeling has an important role in the decision-making of the prey’s population
control, providing information about the natural systems’ stability
[goh2012management], along with computational simulations, revealing the
behavior of these complex systems and understanding how the prey interacts
with other species in the environment.
Mathematical models for biological control, with only one predator population,
are considered in [rafikov2012mathematical] and [rafikov2014dynamical] for egg
and larval parasitoid, respectively. The mathematical model of interactions
between the sugarcane borer and its eggs and larval parasitoids was proposed
by Rafikov and Silveira [rafikov2013dynamics]. In Molnár et al.
[molnar2016two], this model was used for the formulation of some scenarios of
biological pest control. In both publications, the populations are described
by four population compartments: sugarcane borer’s eggs population, sugarcane
borer’s larvae population, Trichogramma galloi(eggs parasitoid) population,
Cotesia Flavipes(larval parasitoid) population.
As pointed out by [rinaldi1993multiple], periodic external forces are of great
importance in ecological systems since environments of the population
communities vary periodically. There are many systems that have a very simple
dynamic behavior in the constant parameter case but become very complex
(multiplicity of attractors, catastrophes, and chaos) when they are
periodically perturbed.
Seasonality in biological systems has already been addressed in several works,
such as [rinaldi1993multiple, gakkhar2003chaos, gakkhar2003seasonally,
altizer2006seasonality, zhang2012complexity, white2020seasonality,
bezerra2021biological, stollenwerk2017hopf]. Meanwhile, the inclusion of two
parasitoids (eggs and larval) populations, as well as seasonal variations in
the sugarcane borer agroecosystem dynamics are novelties. Bezerra et al. in
[bezerra2021biological], models sugarcane borer and its larval parasitoid
(Cotesia flavipes) interaction, considering the influence of the seasonal
variations on the dynamics of the system. Their results show that this
variation generates chaotic dynamics in the system.
The system model in the present paper is an extension of the mathematical
model in [rafikov2013dynamics], with the addition of the parasitized egg and
larvae population of the sugarcane borer. This addition improves the
estimation and observation of the system parameters, as it is much easier to
monitor sugarcane borer’s parasitized eggs and parasitized larvae than adult
parasitoid population in real conditions. Moreover, seasonality variations are
introduced into the system dynamics resulting in chaotic behavior.
The paper is organized as follows. In Section 2, our six-dimensional,
continuous-time, dynamical system is proposed. Section 3 is dedicated to the
study of the system’s local and global dynamics. Seasonal dynamics of the
sugarcane borer parasitoid agroecosystem is considered in Section 4. Section 5
discusses the results of previous Sections and concludes this paper.
## 2 Mathematical Model
The proposed continuous-time mathematical model describes the interactions
between the sugarcane borer, its egg and larval parasitoid, considering six
population densities: the un-parasitized egg population density of the
sugarcane borer, $x_{1}$; the parasitized egg population density of the
sugarcane borer,, $x_{2}$; the density of the adult egg parasitoid
Trichogramma galloi, $x_{3}$; the unparasitized larvae density of the
sugarcane borer, $x_{4}$; the parasitized larvae density of the sugarcane
borer, $x_{5}$; the density of the adult larval parasitoid Cotesia flavipes,
$x_{6}$. So, the mathematical model has the following form:
$\displaystyle\frac{dx_{1}}{dt}$
$\displaystyle=rx_{1}\left(1-\frac{x_{1}}{K}\right)-m_{1}x_{1}-n_{1}x_{1}-\alpha
x_{1}x_{3},$ $\displaystyle\frac{dx_{2}}{dt}$ $\displaystyle=\alpha
x_{1}x_{3}-m_{2}x_{2}-n_{2}x_{2},$ $\displaystyle\frac{dx_{3}}{dt}$
$\displaystyle=\gamma_{1}n_{2}x_{2}-m_{3}x_{3},$
$\displaystyle\frac{dx_{4}}{dt}$
$\displaystyle=n_{1}x_{1}-m_{4}x_{4}-n_{3}x_{4}-\beta x_{4}x_{6},$ (1)
$\displaystyle\frac{dx_{5}}{dt}$ $\displaystyle=\beta
x_{4}x_{6}-m_{5}x_{5}-n_{4}x_{5},$ $\displaystyle\frac{dx_{6}}{dt}$
$\displaystyle=\gamma_{2}n_{4}x_{5}-m_{6}x_{6}.$
In the system of differential equations (2), the 16 parameters are defined as
follows. $r$ is the intrinsic oviposition rate of female sugarcane borer; $K$
is the potential maximum of oviposition rate of female sugarcane borer;
$m_{1},m_{2},m_{3},m_{4},m_{5}$ and $m_{6}$ are the mortality rates of the un-
parasitized egg, parasitized egg, egg parasitoid, un- parasitized larvae,
parasitized larvae and larvae parasitoid populations, respectively; $n_{1}$ is
the fraction of the sugarcane borer larvae population which emerges from the
eggs per unit of time; $n_{2}$ is the fraction of the parasitized egg
population from which larval parasitoids emerge in a time unit; $n_{3}$ is the
fraction of the un-parasitized sugarcane borer larvae from which pupae emerge
in a time unit; $n_{4}$ is the fraction of the parasitized sugarcane borer
larvae from which larvae parasitoids emerge in a time unit; $\alpha$ and
$\beta$ are are the intrinsic parasitism rate of the egg and larvae
parasitoids, respectively; $\gamma_{1}$ and $\gamma_{2}$ are numbers of adult
parasitoids which emerge from a unit of parasitized eggs and larvae,
respectively.
## 3 System Equilibrium states and their stability
### 3.1 Equilibrium states
The equilibrium points can be obtained by equalling to zero the right-hand
sides of the system (2). We obtain five following equilibrium by sequences of
${x_{i},i=1,2,\cdots,6}$.
* 1.
Extinction of all populations: $E_{1}=(0,0,0,0,0,0)$.
* 2.
Extinction of all parasitoid and parasitized populations:
$E_{2}=\left(\frac{K}{r}(r-m_{1}-n_{1}),0,0,\frac{Kn_{1}(r-m_{1}-n_{1})}{r(m_{4}+n_{3})},0,0\right)$.
* 3.
Extinction of the egg parasitoid and parasitized egg populations:
$E_{3}=\left(\frac{K}{r}(r-m_{1}-n_{1}),0,0,p_{4}^{\ast},p_{5}^{\ast},p_{6}^{\ast}\right)$.
* 4.
Extinction of the larvae parasitoid and parasitized larvae populations:
$E_{4}=(x_{1}^{\ast},x_{2}^{\ast},x_{3}^{\ast},q^{\ast},0,0)$.
* 5.
Coexistence of all populations:
$E_{5}=(x_{1}^{\ast},x_{2}^{\ast},x_{3}^{\ast},x_{4}^{\ast},x_{5}^{\ast},x_{6}^{\ast}).$
The values cited above are given as follows.
$\displaystyle x_{1}^{\ast}$
$\displaystyle=\frac{m_{3}(m_{2}+n_{2})}{\alpha\gamma_{1}n_{2}},$
$\displaystyle x_{3}^{\ast}$
$\displaystyle=\frac{1}{\alpha}\left[r\left(1-\frac{m_{3}(m_{2}+n_{2})}{\alpha\gamma_{1}n_{2}K}\right)-m_{1}-n_{1}\right],$
$\displaystyle x_{2}^{\ast}$
$\displaystyle=\frac{m_{3}}{\gamma_{1}n_{2}}x_{3}^{\ast},$ $\displaystyle
x_{4}^{\ast}$ $\displaystyle=\frac{m_{6}(m_{5}+n_{4})}{\beta\gamma_{2}n_{4}},$
$\displaystyle x_{5}^{\ast}$
$\displaystyle=\frac{m_{3}n_{1}(m_{2}+n_{2})}{\alpha\gamma_{1}n_{2}(m_{5}+n_{4})}-\frac{m_{6}(m_{4}+n_{3})}{\beta\gamma_{2}n_{4}},$
$\displaystyle x_{6}^{\ast}$
$\displaystyle=\frac{\gamma_{2}n_{4}}{m_{6}}x_{5}^{\ast},$ $\displaystyle
p_{4}^{\ast}$ $\displaystyle=\frac{m_{6}(m_{5}+n_{4})}{\beta\gamma_{2}n_{4}},$
$\displaystyle p_{5}^{\ast}$
$\displaystyle=\frac{n_{1}}{m_{5}+n_{4}}\left[\frac{K}{r}(r-m_{1}-n_{1})\right]-\frac{m_{6}(m_{4}+n_{3})}{\beta\gamma_{2}n_{4}},$
$\displaystyle p_{6}^{\ast}$
$\displaystyle=\frac{\gamma_{2}n_{4}}{m_{6}}p_{5}^{\ast},$ $\displaystyle
q^{\ast}$ $\displaystyle=\frac{n_{1}}{m_{4}+n_{3}}x_{1}^{\ast}.$
Since we are modelling a biological system, where the dependent variables are
populations, the space phase includes only positive or zero values for all
coordinates, thus defining their biological viability. When analyzing, in next
section, each one of these equilibrium points, conditions will be set for
their biological viability.
### 3.2 Local stability analysis of the equilibrium points
In order to study the local the stability of these equilibrium states, system
(2) is linearized in a small neighborhood of each equilibrium state, and the
Jacobian matrix being computed as:
$J=\begin{bmatrix}a_{11}&0&a_{13}&0&0&0\\\ a_{21}&a_{22}&a_{23}&0&0&0\\\
0&a_{32}&a_{33}&0&0&0\\\ a_{41}&0&0&a_{44}&0&a_{46}\\\
0&0&0&a_{54}&a_{55}&a_{56}\\\ 0&0&0&0&a_{65}&a_{66}\\\ \end{bmatrix},$ (2)
in which:
$\displaystyle a_{11}$ $\displaystyle=r-\frac{2rx_{1}}{K}-m_{1}-n_{1}-\alpha
x_{3},a_{13}=-\alpha x_{1},a_{21}=\alpha x_{3},a_{22}=-m_{2}-n_{2},$
$\displaystyle a_{23}$ $\displaystyle=\alpha
x_{1},a_{32}=\gamma_{1}n_{2},a_{33}=-m_{3},a_{41}=n_{1},a_{44}=-m_{4}-n_{3}-\beta
x_{6},$ $\displaystyle a_{46}$ $\displaystyle=-\beta x_{4},a_{54}=\beta
x_{6},a_{55}=-m_{5}-n_{4},a_{56}=\beta
x_{4},a_{65}=\gamma_{2}n_{4},a_{66}=-m_{6}.$
The matrix (2) can be written as a block matrix:
$\begin{bmatrix}A&0\\\ C&B\\\ \end{bmatrix},$ (3)
in which:
$\displaystyle A$ $\displaystyle=\begin{bmatrix}a_{11}&0&a_{13}\\\
a_{21}&a_{22}&a_{23}\\\ 0&a_{32}&a_{33}\\\
\end{bmatrix},C=\begin{bmatrix}a_{41}&0&0\\\ 0&0&0\\\ 0&0&0\\\
\end{bmatrix},B=\begin{bmatrix}a_{44}&0&a_{46}\\\ a_{54}&a_{55}&a_{56}\\\
0&a_{65}&a_{66}\\\ \end{bmatrix},$
and 0 is a matrix with all elements equal to zero.
Matrices of the form (3) are called block-lower-triangular. The full
explanation on the block-triangular matrix determinant computation method can
be found in [gantmachertheory], and briefly described below:
If D is a block-triangular matrix, then the determinant of the matrix is equal
to the product of determinant of diagonal cells:
$\det{(D)}=\det{(D_{11})}\det{(D_{22})}\cdots\det{(D_{nn})}.$ (4)
The matrix $D=J-\lambda I$, where $I$ is the identity matrix of dimensions
$n\times n$, is block-lower-triangular too. Using the above mentioned rule,
the characteristic equation is given as follows:
$\det{(D)}=\det{(A-\lambda I)}\det{(B-\lambda I)}=0.$ (5)
Hence, the characteristic equation (5) is given by:
$\begin{vmatrix}a_{11}-\lambda&0&a_{13}\\\ a_{21}&a_{22}-\lambda&a_{23}\\\
0&a_{32}&a_{33}-\lambda\\\
\end{vmatrix}\begin{vmatrix}a_{44}-\lambda&0&a_{46}\\\
a_{54}&a_{55}-\lambda&a_{56}\\\ 0&a_{65}&a_{66}-\lambda\\\ \end{vmatrix}=0.$
(6)
From (6), we get that:
$\begin{vmatrix}a_{11}-\lambda&0&a_{13}\\\ a_{21}&a_{22}-\lambda&a_{23}\\\
0&a_{32}&a_{33}-\lambda\\\ \end{vmatrix}=0,$ (7)
$\begin{vmatrix}a_{44}-\lambda&0&a_{46}\\\ a_{54}&a_{55}-\lambda&a_{56}\\\
0&a_{65}&a_{66}-\lambda\\\ \end{vmatrix}=0.$ (8)
Now, applying this rule to the equilibrium points found above, we obtain for
the equilibrium point $E_{1}=(0,0,0,0,0,0)$, the matrix (2) has a triangular
form, and the eigenvalues are given by:
$\lambda_{1}=r-m_{1}-n_{1},\,\lambda_{2}=-m_{2}-n_{2},\,\lambda_{3}=-m_{3},\,\lambda_{4}=-m_{4}-n_{3},\,\lambda_{5}=-m_{5}-n_{4},\,\text{and}\,\lambda_{6}=-m_{6}.$
Therefore, it follows that equilibrium $E_{1}$ is asymptotically stable if
$r<m_{1}+n_{1}$.
For the equilibrium
$E_{2}=\left(\frac{K}{r}(r-m_{1}-n_{1}),0,0,\frac{Kn_{1}(r-m_{1}-n_{1})}{r(m_{4}+n_{3})},0,0\right)$,
which is biologically viable if $r_{1}>m_{1}+n_{1}$, the parameters of the
characteristic equation (5) are given as:
$\displaystyle a_{11}$
$\displaystyle=-(r-m_{1}-n_{1}),a_{13}=-\frac{aK}{r}(r-m_{1}-n_{1}),a_{21}=0,a_{22}=-m_{2}-n_{2},$
$\displaystyle a_{23}$
$\displaystyle=\frac{aK}{r}(r-m_{1}-n_{1}),a_{32}=\gamma_{1}n_{2},a_{33}=-m_{3},a_{44}=-m_{4}-n_{3},$
$\displaystyle a_{46}$ $\displaystyle=-\frac{\beta
Kn_{1}(r-m_{1}-n_{1})}{r(m_{4}+n_{3})},a_{54}=0,a_{55}=-m_{5}-n_{4},a_{56}=\frac{\beta
Kn_{1}(r-m_{1}-n_{1})}{r(m_{4}+n_{3})},$ $\displaystyle a_{65}$
$\displaystyle=\gamma_{2}n_{4},a_{66}=-m_{6}.$
From (7) and (8), we obtain:
$\displaystyle(a_{11}-\lambda)[\lambda^{2}-(a_{22}+a_{33})\lambda+a_{22}a_{33}-a_{23}a_{32}]$
$\displaystyle=0$ (9)
$\displaystyle(a_{44}-\lambda)[\lambda^{2}-(a_{55}+a_{66})\lambda+a_{55}a_{66}-a_{56}a_{65}]$
$\displaystyle=0$ (10)
According to the Routh-Hurwitz criterion the eigenvalues of the second-degree
polynomial have negative real parts if, and only if, both coefficients are
positive. Analyzing (7) and (8), we can conclude that when
(a) $a_{11}<0$, (b) $a_{22}+a_{33}-a_{23}a_{32}>0$, (c)
$a_{55}+a_{66}-a_{56}a_{65}>0$,
then all the eigenvalues of the equation (6) have negative real parts.
From conditions (a), (b) and (c), we obtain:
$\displaystyle\alpha$
$\displaystyle<\frac{rm_{3}(m_{2}+n_{2})}{\gamma_{1}n_{2}K(r-m_{1}-n_{1})}$
(11) $\displaystyle\beta$
$\displaystyle<\frac{rm_{6}(m_{4}+n_{3})(m_{5}+n_{4})}{\gamma_{2}n_{1}n_{4}K(r-m_{1}-n_{1})}$
(12) $\displaystyle r$ $\displaystyle>m_{1}+n_{1}$ (13)
Similarly, the equilibrium point $E_{2}$ is asymptotically stable if, and only
if, the inequalities (11), (12) and (13) are satisfied.
The equilibrium point
$E_{3}=\left(\frac{K}{r}(r-m_{1}-n_{1}),0,0,p_{4}^{\ast},p_{5}^{\ast},p_{6}^{\ast}\right)$
is biologically viable if:
$\displaystyle\beta$
$\displaystyle>\frac{rm_{6}(m_{4}+n_{3})(m_{5}+n_{4})}{\gamma_{2}n_{1}n_{4}K(r-m_{1}-n_{1})},$
(14) $\displaystyle r$ $\displaystyle>m_{1}+n_{1}.$
The characteristic equation at $E_{3}$ can be written in the form (6), where:
$\displaystyle a_{11}$
$\displaystyle=-(r-m_{1}-n_{1}),a_{13}=-\frac{aK}{r}(r-m_{1}-n_{1}),a_{21}=0,a_{22}=-m_{2}-n_{2},$
$\displaystyle a_{23}$
$\displaystyle=\frac{aK}{r}(r-m_{1}-n_{1}),a_{32}=\gamma_{1}n_{2},a_{33}=-m_{3},a_{44}=-m_{4}-n_{3}-\beta
p_{6}^{\ast},$ $\displaystyle a_{46}$ $\displaystyle=-\beta
p_{4}^{\ast},a_{54}=\beta p_{6}^{\ast},a_{55}=-m_{5}-n_{4},a_{56}=\beta
p_{4}^{\ast},a_{65}=\gamma_{2}n_{4},a_{66}=-m_{6}.$
The parameter values of determinant (7) are the same as $E_{2}$, and the
conditions (11) and (13) are satisfied.
Considering the determinant (8)
$\begin{vmatrix}a_{44}-\lambda&0&a_{46}\\\ a_{54}&a_{55}-\lambda&a_{56}\\\
0&a_{65}&a_{66}-\lambda\\\ \end{vmatrix}=0,$ (15)
we have
$\lambda^{3}+b_{1}\lambda^{2}+b_{2}\lambda+b_{3}=0,$ (16)
where
$\displaystyle b_{1}$ $\displaystyle=m_{4}+n_{3}+\beta
p_{6}^{\ast}>0,\;\;\;\;\;b_{2}=(m_{4}+n_{3}+\beta
p_{6}^{\ast})(m_{5}+n_{4})>0,$ $\displaystyle b_{3}$ $\displaystyle=\beta
m_{6}(m_{5}+n_{4})p_{6}^{\ast}>0,\;\;\;\;\;b_{1}b_{2}-b_{3}>0.$ (17)
Therefore, we obtain that the equilibrium point $E_{3}$ is asymptotically
stable if, and only if, the inequalities (11), (14) and (13) are satisfied.
The equilibrium point
$E_{4}=(x_{1}^{\ast},x_{2}^{\ast},x_{3}^{\ast},q^{\ast},0,0)$ is biologically
viable if
$\displaystyle\alpha$
$\displaystyle>\frac{rm_{3}(m_{2}+n_{2})}{\gamma_{1}n_{2}K(r-m_{1}-n_{1})},$
(18) $\displaystyle r$ $\displaystyle>m_{1}+n_{1}.$
The characteristic equation at $E_{4}$ can be written in the form (6) where:
$\displaystyle a_{11}$
$\displaystyle=-\frac{rx_{1}^{\ast}}{K},a_{13}=-ax_{1}^{\ast},a_{21}=ax_{3}^{\ast},a_{22}=-m_{2}-n_{2},$
$\displaystyle a_{23}$
$\displaystyle=ax_{1}^{\ast},a_{32}=\gamma_{1}n_{2},a_{33}=-m_{3},a_{44}=-m_{4}-n_{3},$
(19) $\displaystyle a_{46}$ $\displaystyle=-\beta
q^{\ast},a_{54}=0,a_{55}=-m_{5}-n_{4},a_{56}=\beta
q^{\ast},a_{65}=\gamma_{2}n_{4},a_{66}=-m_{6}.$
Considering the determinant (7) with parameter values (3.2) we have:
$\lambda^{3}+c_{1}\lambda^{2}+c_{2}\lambda+c_{3}=0.$ (20)
where
$c_{1}=m_{2}+m_{3}+n_{2}+\frac{rx_{1}^{\ast}}{K}>0,\;c_{2}=(m_{2}+m_{3}+n_{2})\frac{rx_{1}^{\ast}}{K}>0,\;c_{3}=\alpha
m_{3}(m_{2}+n_{2})x_{3}^{\ast}>0.$
From $c_{1}c_{2}-c_{3}>0$, we obtain:
$\alpha<\frac{rm_{3}(m_{2}+n_{2})}{\gamma_{1}n_{2}zK},$ (21)
where
$\displaystyle z$
$\displaystyle=-\frac{h_{1}}{2}+\sqrt{\frac{h_{1}^{2}}{4}+h_{2}}\,,$
$\displaystyle h_{1}$
$\displaystyle=m_{2}+m_{3}+n_{2}+\frac{m_{3}(m_{2}+n_{2})}{m_{2}+m_{3}+n_{2}},$
$\displaystyle h_{2}$
$\displaystyle=\frac{m_{3}(m_{2}+n_{2})(r-m_{1}-n_{1})}{m_{2}+m_{3}+n_{2}}.$
Considering the determinant (8) with parameter values (3.2) we have:
$(a_{44}-\lambda)[\lambda^{2}-(a_{55}+a_{66})\lambda+a_{55}a_{66}-a_{56}a_{65}]=0,$
(22)
where $a_{44}=-m_{4}-n_{3}<0,-(a_{55}+a_{66})>0.$
From $a_{55}a_{66}-a_{56}a_{65}>0$ we obtain:
$\beta<\frac{m_{6}(m_{5}+n_{4})}{\gamma_{2}n_{4}q^{\ast}}=\frac{\alpha\gamma_{1}n_{2}m_{6}(m_{4}+n_{3})(m_{5}+n_{4})}{\gamma_{2}n_{1}n_{4}m_{3}(m_{2}+n_{2})}.$
(23)
The equilibrium point $E_{4}$ is asymptotically stable if, and only if, the
following inequalities are satisfied:
$\frac{rm_{3}(m_{2}+n_{2})}{\gamma_{1}n_{2}K(r-n_{1}-m_{1})}<\alpha<\frac{rm_{3}(m_{2}+n_{2})}{\gamma_{1}n_{2}zK},$
(24)
$\beta<\frac{\alpha\gamma_{1}n_{2}m_{6}(m_{4}+n_{3})(m_{5}+n_{4})}{\gamma_{2}n_{1}n_{4}m_{3}(m_{2}+n_{2})}.$
(25)
Consider the equilibrium point
$E_{5}=(x_{1}^{\ast},x_{2}^{\ast},x_{3}^{\ast},x_{4}^{\ast},x_{5}^{\ast},x_{6}^{\ast})$
where
$\displaystyle x_{1}^{\ast}$
$\displaystyle=\frac{m_{3}(m_{2}+n_{2})}{\alpha\gamma_{1}n_{2}},x_{3}^{\ast}=\frac{1}{\alpha}\left[r\left(1-\frac{m_{3}(m_{2}+n_{2})}{\alpha\gamma_{1}n_{2}K}-m_{1}-n_{1}\right)\right],x_{2}^{\ast}=\frac{m_{3}}{\gamma_{1}x_{2}}x_{3}^{\ast},$
(26) $\displaystyle x_{4}^{\ast}$
$\displaystyle=\frac{m_{6}(m_{5}+n_{4})}{\beta\gamma_{2}n_{4}},x_{5}^{\ast}=\frac{m_{3}n_{1}(m_{2}+n_{2})}{\alpha\gamma_{1}n_{2}(m_{5}+n_{4})}-\frac{m_{6}(m_{4}+n_{3})}{\beta\gamma_{2}n_{4}},x_{6}^{\ast}=\frac{\gamma_{2}n_{4}}{m_{6}}x_{5}^{\ast}.$
The equilibrium point $E_{5}$ is biologically viable if $x_{3}^{\ast}>0$ and
$x_{5}^{\ast}>0$. So, we get:
$\displaystyle\alpha$
$\displaystyle>\frac{rm_{3}(m_{2}+n_{2})}{\gamma_{1}n_{2}K(r-n_{1}-m_{1})},$
$\displaystyle\beta$
$\displaystyle>\frac{\alpha\gamma_{1}n_{2}m_{6}(m_{4}+n_{3})(m_{5}+n_{4})}{\gamma_{2}n_{1}n_{4}m_{3}(m_{2}+n_{2})},$
(27) $\displaystyle r$ $\displaystyle>m_{1}+n_{1}.$
The characteristic equation at $E_{5}$ can be written in the form (6) where:
$\displaystyle a_{11}$
$\displaystyle=-\frac{rx_{1}^{\ast}}{K},a_{13}=-ax_{1}^{\ast},a_{21}=ax_{3}^{\ast},a_{22}=-m_{2}-n_{2},$
$\displaystyle a_{23}$
$\displaystyle=ax_{1}^{\ast},a_{32}=\gamma_{1}n_{2},a_{33}=-m_{3},a_{44}=-m_{4}-n_{3}-\beta
x_{6}^{\ast},$ (28) $\displaystyle a_{46}$ $\displaystyle=-\beta
x_{4}^{\ast},a_{54}=\beta x_{6}^{\ast},a_{55}=-m_{5}-n_{4},a_{56}=\beta
x_{4}^{\ast},a_{65}=\gamma_{2}n_{4},a_{66}=-m_{6}.$
The parameter values of determinant (7) are the same of $E_{4}$, and the
inequality (21) is satisfied.
Considering the determinant (8), we obtain:
$\lambda^{3}+g_{1}\lambda^{2}+g_{2}\lambda+g_{3}=0,$ (29)
where
$\displaystyle g_{1}$ $\displaystyle=m_{4}+m_{5}+m_{6}+n_{3}+n_{4}+\beta
x_{6}^{\ast}>0,$ $\displaystyle g_{2}$ $\displaystyle=(m_{4}+n_{3}+\beta
x_{6}^{\ast})(m_{5}+n_{4}+m_{6})>0,$ (30) $\displaystyle g_{3}$
$\displaystyle=\beta
m_{6}(m_{5}+n_{4})x_{6}^{\ast}>0,\;\;\;g_{1}g_{2}-g_{3}>0.$
Therefore, we obtain that equilibrium point $E_{5}$ is asymptotically stable
if, and only if, the following inequalities are satisfied:
$\frac{rm_{3}(m_{2}+n_{2})}{\gamma_{1}n_{2}K(r-m_{1}-n_{1})}<\alpha<\frac{rm_{3}(m_{2}+n_{2})}{\gamma_{1}n_{2}zK},$
(31)
$\beta>\frac{\alpha\gamma_{1}n_{2}m_{6}(m_{4}+n_{3})(m_{5}+n_{4})}{\gamma_{2}n_{1}n_{4}m_{3}(m_{2}+n_{2})}.$
(32)
We can now summarize the results of this local stability analysis, after
defining the following dimensionless parameters related to our resulting
conditions (regions in the parameter space) for biological viability (b. v.)
and for local stability (l.s.) of each equilibrium point:
$\displaystyle A_{1}$ $\displaystyle\equiv\frac{r}{m_{1}+n_{1}};\
A_{2}\equiv\frac{\alpha\gamma_{1}n_{2}K(r-m_{1}-n_{1})}{rm_{3}(m_{2}+n_{2})};\
A_{3}\equiv\frac{\beta\gamma_{2}n_{1}n_{4}K(r-m_{1}-n_{1})}{rm_{6}(m_{4}+n_{3})(m_{5}+n_{4})};$
(33) $\displaystyle A_{4}$
$\displaystyle\equiv\frac{\beta\gamma_{2}n_{1}n_{4}m_{3}(m_{2}+n_{2})}{\alpha\gamma_{1}n_{2}m_{6}(m_{4}+n_{3})(m_{5}+n_{4})}=\frac{A_{3}}{A_{2}};\
A_{5}\equiv\frac{\alpha\gamma_{1}n_{2}zK}{rm_{3}(m_{2}+n_{2})}=\frac{A_{2}z}{r-m_{1}-n_{1}}.$
With these dimensionless parameters, whose critical value 1 is associated with
a bifurcation in the behavior of the system, all conditions previously deduced
can be presented in a summary and complete form as shown in Table 1.
| | | | $E_{1}$ | $E_{2}$ | $E_{3}$ | $E_{4}$ | $E_{5}$
---|---|---|---|---|---|---|---|---
$A_{1}<1$ | b.v. | l.s. | Not b.v. | Not b.v. | Not b.v. | Not b.v.
$A_{1}>1$ | $A_{2}<1$ | $A_{3}<1$ | b.v. | un. | b.v. | l.s. | Not b.v. | Not b.v. | Not b.v.
$A_{3}>1$ | b.v. | un. | b.v. | un. | b.v. | l.s. | Not b.v. | Not b.v.
$A_{2}>1$ | $A_{5}<1$ | $A_{4}<1$ | b.v. | un. | b.v. | un. | b.v. | un. | b.v. | l.s. | Not b.v.
$A_{4}>1$ | b.v. | un. | b.v. | un. | b.v. | un. | b.v. | un. | b.v. | l.s.
$A_{5}>1$ | $A_{4}>1$ | b.v. | un. | b.v. | un. | b.v. | un. | b.v. | un. | b.v. | un.
Table 1: Local stability and biological viability of the system equilibria, in
which b.v. means biologically viable, l.s. means locally stable and un. means
unstable.
### 3.3 Global stability analysis of the coexistence equilibrium
From local stability analysis, we get the equilibrium point $E_{5}$ is the
only one with no null population density. Therefore this is the point of
interest for the forthcoming global stability analysis.
We define a Lyapunov function as follows:
$V(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})=\int_{x_{1}^{\ast}}^{x_{1}}\frac{y-x_{1}^{\ast}}{y}dy+\sum_{i=2}^{6}\frac{(x_{i}-x_{1}^{\ast})^{2}}{2}.$
(34)
It can be easily verified that the function $V$ is zero at the equilibrium
point $E_{5}$ and is positive for all other biologically viable equilibria.
The function is also radially unbounded, which means that $V\rightarrow\infty$
when $x\rightarrow\infty$.
The time derivative of $V$ along (2) can be written as
$\dot{V}=e^{T}Pe,$ (35)
where the matrix $P$ is
$\begin{bmatrix}-\frac{r}{K}&0&-\alpha&0&0&0\\\ \alpha
x_{3}&-m_{2}-n_{2}&\alpha x_{1}^{\ast}&0&0&0\\\
0&\gamma_{1}n_{2}&-m_{3}&0&0&0\\\ n_{1}&0&0&-m_{4}-n_{3}-\beta x_{6}&0&-\beta
x_{4}^{\ast}\\\ 0&0&0&\beta x_{6}&-m_{5}-n_{4}&\beta x_{4}^{\ast}\\\
0&0&0&0&\gamma_{2}n_{4}&-m_{6}\\\ \end{bmatrix},$
and the elements of vector $e$ are: $e_{i}=x_{i}-x_{i}^{\ast},i=1,2,...,6.$
If conditions (31) and (32) are satisfied then the matrix $P$ in (35) is
negative definite, as is the time derivative of $V$ along the trajectories of
(2), and consequently the equilibrium point $E_{5}$ is globally asymptotically
stable.
### 3.4 Hopf bifurcation analysis
From Table 1, it is evident the role of the dimensionless parameter $A_{5}$ in
determining the region in the parameter space in which each one of the
equilibrium points $E_{4}$ and $E_{5}$ is asymptotically stable. Furthermore,
from the definition of $A_{5}$ in (33), it can be seen that the condition
$A_{5}<1$ can be written equivalently in the form of $\alpha<\alpha_{c}$,
where the critical value $\alpha_{c}$ is defined by
$\alpha_{c}\equiv\frac{rm_{3}(m_{2}+n_{2})}{\gamma_{1}n_{2}zK}.$ (36)
In (36) we observe that the condition is exactly the same one as we got in
(21), from the characteristic (20), for the local stability of the equilibria
$E_{4}$ and $E_{5}$.
When $A_{5}>1$, that is, $\alpha>\alpha_{c}$, the positive coexistence
equilibrium $E_{5}$ becomes unstable and a Hopf bifurcation occurs.
Now we can analyze the bifurcation of the model (2) assuming $\alpha$ as the
bifurcation parameter and considering only three first equations of the system
(2) which are not dependent on the variables $x_{4},x_{5}$ and $x_{6}$. The
traditional Hopf bifurcation criterion is stated in terms of the properties of
the eigenvalues. Alternatively, Liu (1994) [liu1994criterion] presented a
criterion of Hopf bifurcation without using the eigenvalues of the
characteristic equation. Liu’s approach is the one that is applied in the
present Hopf bifurcation analysis, as follows:
Liu’s criterion. If the characteristic equation of the positive equilibrium
point is given by:
$\lambda^{3}+c_{1}(\alpha)\lambda^{2}+c_{2}(\alpha)\lambda+c_{3}(\alpha)=0,$
where $c_{1}(\alpha),c_{2}(\alpha)$ and $c_{3}(\alpha)$ are smooth functions
of $\alpha$ in an open interval about $\alpha_{c}\in\mathcal{R}$ such that:
1. (a)
$c_{1}(\alpha_{c})>0,\Delta(\alpha_{c})=c_{1}(\alpha_{c})c_{2}(\alpha_{c})-c_{3}(\alpha_{c})=0,c_{3}(\alpha_{c})>0$.
2. (b)
$\left(\frac{d\Delta}{d\alpha}\right)_{\alpha=\alpha_{c}}\neq 0$,
then a simple Hopf bifurcation occurs at $\alpha=\alpha_{c}$.
Applying the Liu’s criterion to the characteristic equation (20), we observe
that
$c_{1}=m_{2}+m_{3}+n_{2}+\frac{rx_{1}^{\ast}}{K}>0,\;\;c_{2}=(m_{2}+m_{3}+n_{2})\frac{rx_{1}^{\ast}}{K}>0,\;\;c_{3}=\alpha(m_{2}+n_{2})x_{3}^{\ast}>0,$
for all positive values of $\alpha$.
Solving the equation $c_{1}(\alpha_{c})c_{2}(\alpha_{c})-c_{3}(\alpha_{c})=0,$
we obtain
$\alpha_{c}=\frac{rm_{3}(m_{2}+n_{2})}{\gamma_{1}n_{2}zK},$ (37)
where
$\displaystyle z$
$\displaystyle=-\frac{h_{1}}{2}+\sqrt{\frac{h_{1}^{2}}{4}+h_{2}}\,,$
$\displaystyle h_{1}$
$\displaystyle=m_{2}+m_{3}+n_{2}+\frac{m_{3}(m_{2}+n_{2})}{m_{2}+m_{3}+n_{2}},$
$\displaystyle h_{2}$
$\displaystyle=\frac{m_{3}(m_{2}+n_{2})(r-m_{1}-n_{1})}{m_{2}+m_{3}+n_{2}}.$
Considering condition (b) of the Liu’s criterion, we have
$\left(\frac{d\Delta}{d\alpha}\right)_{\alpha=\alpha_{c}}=-\frac{B_{1}}{\alpha_{c}^{2}}-\frac{2B_{2}}{\alpha_{c}^{3}}<0,$
where
$\displaystyle B_{1}$
$\displaystyle=\frac{rm_{3}(m_{2}+n_{2}+m_{3})^{2}(m_{2}+n_{2})+rm_{3}(m_{2}+n_{2})^{2}}{\gamma_{1}n_{2}K},$
$\displaystyle B_{2}$
$\displaystyle=\frac{r^{2}(m_{2}+n_{2}+m_{3})(m_{2}+n_{2})^{2}+m_{3}^{2}}{\gamma_{1}^{2}n_{2}^{2}K^{2}}.$
Hence, according to Liu’s criterion, a simple Hopf bifurcation occurs at
$\alpha=\alpha_{c}$, that is, $A_{5}=1$.
## 4 Seasonal dynamics of the sugarcane borer-parasitoid agroecosystem
Several environmental parameters (such as air temperature, air humidity,
rainfall dispersion, among others) fluctuate periodically affecting an
ecological system dynamics. Thus, they can be represented as periodic-time
functions. In this section, the intrinsic growth rate $r$ in system (2) is
considered as a sinusoidal function representing these seasonal perturbations.
The parameter $r$ being defined by the following function
[rinaldi1993multiple, gakkhar2003chaos, gakkhar2003seasonally,
altizer2006seasonality]:
$r(t)=r_{0}\left(1+r_{1}\sin\left(\frac{2\pi t}{365}\right)\right),$ (38)
where $t$ is measured in days, so $r_{0}$ is the average value of $r$ over an
integer number of years. The parameter $r_{1}$ represents the degree of
seasonality, hence $r_{0}r_{1}$ is the magnitude of the perturbation in $r$.
Next, we are interested in the seasonal dynamics of the equilibrium point
$E_{5}$, where there is coexistence of the parasitoid and pest populations.
For this, we will keep the values of the parameters fixed as follows
[parraetal2002, rafikov2012mathematical, rafikov2014dynamical].
$\displaystyle m_{1}$ $\displaystyle=0,\ m_{2}=0.03566,\ m_{3}=\frac{1}{4},\
m_{4}=0.00257,\ m_{5}=m_{4},\ m_{6}=\frac{1}{5},\ $ $\displaystyle n_{1}$
$\displaystyle=\frac{1}{8},\ n_{2}=\frac{1}{9},\ n_{3}=\frac{1}{50},\
n_{4}=\frac{1}{16},\ r_{0}=0.19,\ $ (39) $\displaystyle\beta$
$\displaystyle=0.000009,\ \gamma_{1}=2.29,\ \gamma_{2}=40,\ K=25000.$
Setting the parameter values as specified in (4), it can be shown from (3.2)
that without seasonality, that is, $r_{1}=0$ in (38), depending on the value
of $\alpha>0.169\times 10^{-4}$, the attractor in the phase space can be an
equilibrium point or a limit cycle, namely:
* 1.
For $0.169\times 10^{-4}<\alpha<\alpha_{c}$, where $\alpha_{c}=0.9135\times
10^{-4}$, the corresponding attractor is the coexistence equilibrium point
$E_{5}$, whose components depend on the value of $\alpha$, as specified in
(26);
* 2.
For $\alpha>\alpha_{c}$, that is, beyond the Hopf bifurcation, the attractor
is a period one limit-cycle and, the amplitude of this limit cycle in the
phase space increases with increasing the value of $\alpha$.
The addition of seasonality to the model, through the population growth rate
according to (38), induces a destabilizing effect and may even trigger chaotic
behavior. This destabilizing effect will be confirmed further by computer
simulations with the parameter values fixed according to (4).
First, we investigate the effect of seasonality for a value of $\alpha$ when
the attractor is the equilibrium point $E_{5}$ in the 6D phase space, as shown
in Fig. 1 for $\alpha=0.6\times 10^{-4}$. Considering this value of $\alpha$,
the bifurcation diagram for $0\leq r_{1}\leq 0.35$ is shown in Fig. 2, where
it can be immediately verified that the value of $x_{1}(t)$ for $r_{1}=0$ is,
as expected, the same value of this component of the equilibrium point
obtained without seasonality, given by (26), and shown in Fig. 1. Increasing
the value of $r_{1}$, there is a periodic solution followed by a period
doubling sequence, which constitutes a route to chaos that occurs for
$r_{1}>0.33$. The projections of the 6D strange attractor, in the phase space,
with $r_{1}=0.35$, are plotted in Fig. 3. Thus, the seasonality destabilizes
the $E_{5}$ equilibrium, changing the attractor from equilibrium point to
limit cycle, and then to more complex dynamics as $r_{1}$ increases.
Figure 1: The equilibrium $E_{5}$, reached by the populations of system (2),
without seasonality, for the parameters fixed in (4) and $\alpha=0.6\times
10^{-4}<\alpha_{c}$. Figure 2: Keeping the parameter values fixed in (4) and
$\alpha=0.6\times 10^{-4}$, the bifurcation diagram of $x_{1}(t)$ for $0\leq
r_{1}\leq 0.35$. Figure 3: Keeping the parameter values fixed according to (4)
and $\alpha=0.6\times 10^{-4}$, the projections of the 6D strange attractor in
the phase space, corresponding to $r_{1}=0.35$, in the subspaces
$x_{1}x_{2}x_{3}$ and $x_{4}x_{5}x_{6}$.
Now, the range $\alpha>\alpha_{c}$ is considered, with the system having a
period one limit cycle in the phase space, whose amplitude increases as
$\alpha$ increases. We consider $\alpha=1\times 10^{-4}$ in Fig. 4, and the
maximum and minimum values of $x_{1}$ as the same that as we identify for
$r_{1}=0$ in the bifurcation diagram of $x_{1}(t)$ plotted with seasonality in
Fig 5 for $0\leq r_{1}\leq 0.35$. Increasing the value of $r_{1}$, a period
doubling sequence can be noted, which constitutes a route to chaos that occurs
for $r_{1}>0.2$. The projections of the 6D strange attractor in the phase
space, corresponding to $r_{1}=0.25$, are plotted in Figure 6. Increasing even
more the value of $r_{1}$, periodic attractors emerge, as the one plotted in
Fig. 7, corresponding to $r_{1}=0.28$, and similar periodic attractors occur
for $0.26\leq r_{1}\leq 0.3$. Therefore, the seasonality destabilizes the
attractor from period one limit cycle, changing the behavior of our system to
more complex dynamics as $r_{1}$ increases.
Regarding to the value of $r_{1}$ at which chaos is observed to occur, the
comparison of bifurcation diagrams in Figs. 2 and 5 show that if
$\alpha>\alpha_{c}$, chaos occurs at a lower value of $r_{1}$ than if
$\alpha<\alpha_{c}$.
Figure 4: The projection of the 6D limit cycle reached by the populations of
system (2), without seasonality, for the parameters fixed in (4) and
$\alpha=1\times 10^{-4}>\alpha_{c}$, in the subspaces $x_{1}x_{2}x_{3}$ and
$x_{4}x_{5}x_{6}$. Figure 5: Keeping the parameter values fixed in (4) and
$\alpha=1\times 10^{-4}$, the bifurcation diagram of $x_{1}(t)$ for $0\leq
r_{1}\leq 0.35$. Figure 6: Keeping the parameter values fixed in (4) and
$\alpha=1\times 10^{-4}$, the projections of the 6D strange attractor in the
phase space, corresponding to $r_{1}=0.25$, in the subspaces $x_{1}x_{2}x_{3}$
and $x_{4}x_{5}x_{6}$. Figure 7: Keeping the parameter values fixed in (4) and
$\alpha=1\times 10^{-4}$, the projections of the 6D periodic attractor in the
phase space, corresponding to $r_{1}=0.28$, in the subspaces $x_{1}x_{2}x_{3}$
and $x_{4}x_{5}x_{6}$.
Additionally, we can investigate the effect of seasonality for fixed values of
its degree $r_{1}$, while varying the parameter $\alpha$. For that,
bifurcation diagrams of $x_{1}(t)$ are plotted in Figures 8 and 9, keeping the
parameter values fixed in (4) for $0.169\times 10^{-4}\leq\alpha\leq 1.2\times
10^{-4}$, and setting values for the parameter $r_{1}$. The diagram presents
in Fig. 8 corresponds to $r_{1}=0.25$, whose strange attractor considering
$\alpha=1\times 10^{-4}$ was visualized in Fig. 6, while the diagram in Fig. 9
the degree of seasonality is $r_{1}=0.35$, whose strange attractor for
$\alpha=0.6\times 10^{-4}$ was visualized in Fig. 3. Comparing these two
bifurcation diagrams, we conclude that the higher value of $r_{1}$, the lower
the value of $\alpha$ at which chaos is established, as it occurs at
$\alpha=0.8\times 10{-4}$ for $r_{1}=0.25$ and at $\alpha=0.6\times 10{-4}$
for $r_{1}=0.35$.
Figure 8: Bifurcation diagram of $x_{1}(t)$ for $0.169\times
10^{-4}\leq\alpha\leq 1.2\times 10^{-4}$, and keeping the parameter values
fixed according to 4, for $r_{1}=0.25$. Figure 9: Bifurcation diagram of
$x_{1}(t)$ for $0.169\times 10^{-4}\leq\alpha\leq 1.2\times 10^{-4}$, and
keeping the parameter values fixed according to 4, for $r_{1}=0.35$.
## 5 Conclusions
According to [white2020seasonality], seasonality is a significant feature in
ecological systems driven by periodic climatic conditions, but it is often not
explicitly included in either empirical or theoretical studies. Therefore this
article is an effort toward the integration of the complex dynamics involving
such seasonal influences into the sugarcane borer agroecosystem and its
parasitoids. In this work, we have proposed a novel, six-dimensional
continuous-time dynamical system, modeling interactions between the sugarcane
borer and its egg and larval parasitoid. On the analytical side, five
equilibrium states and conditions for their local stability were found out.
Moreover, the Lyapunov function stability analysis ensured the global
asymptotical stability of the equilibrium state in which all considered
populations coexist. Then, the occurrence of a Hopf bifurcation was
investigated applying Liu’s theorem. Numerical simulations revealed the
chaotic behavior of the system with seasonality. These results show how
seasonality changes considerably the agroecosystem dynamics leading an
asymptotically stable system, as shown in Fig 1 to the period-doubling and
subsequently to a chaotic attractor shown in Fig 2. For a real system, this
means sudden changes can occur in populations that without seasonality could
coexist in an equilibrium, in the presence of seasonal conditions. Moreover,
when populations exhibit periodic oscillations, as shown in Fig. 4, the
introduction of seasonality can transform these oscillations into chaos, even
for smaller values of $r_{1}$ then in the case of Fig. 5. Finally, bifurcation
diagrams of the maximum and minimum population values, in the presence of
seasonal influences, show that an increase in the parasitism coefficient
$\alpha$ can lead the stabilized system to a chaotic regime, as shown by
Figures 8 and 9.
The present results help understand the dynamics of the six-dimensional agro-
ecological system with the seasonal forcing. Using these results the
biological control strategies can be investigated in future research.
|
# Eigenvalue asymptotics for the one-particle density matrix
Alexander V. Sobolev Department of Mathematics
University College London
Gower Street
London
WC1E 6BT UK<EMAIL_ADDRESS>
###### Abstract.
The one-particle density matrix $\gamma(x,y)$ for a bound state of an atom or
molecule is one of the key objects in the quantum-mechanical approximation
schemes. We prove the asymptotic formula $\lambda_{k}\sim(Ak)^{-8/3}$, $A\geq
0$, as $k\to\infty$, for the eigenvalues $\lambda_{k}$ of the self-adjoint
operator $\boldsymbol{\Gamma}\geq 0$ with kernel $\gamma(x,y)$.
###### Key words and phrases:
Multi-particle Schrödinger operator, one-particle density matrix, eigenvalues,
spectral asymptotics
###### 2010 Mathematics Subject Classification:
Primary 35J10; Secondary 47G10, 81Q10
## 1\. Introduction
Consider on $\textup{{{L}}}^{2}(\mathbb{R}^{3N})$ the Schrödinger operator
(1.1)
$\displaystyle\mathcal{H}=\sum_{k=1}^{N}\bigg{(}-\Delta_{k}-\frac{Z}{|x_{k}|}\bigg{)}+\sum_{1\leq
j<k\leq N}\frac{1}{|x_{j}-x_{k}|},$
describing an atom with $N$ particles (e.g. electrons) with coordinates
$\mathbf{x}=(x_{1},x_{2},\dots,x_{N})$, $x_{k}\in\mathbb{R}^{3}$,
$k=1,2,\dots,N$, and a nucleus with charge $Z>0$. The notation $\Delta_{k}$ is
used for the Laplacian w.r.t. the variable $x_{k}$. The operator $\mathcal{H}$
acts on the Hilbert space $\textup{{{L}}}^{2}(\mathbb{R}^{3N})$ and it is
self-adjoint on the domain
$D(\mathcal{H})=\textup{{{H}}}^{2}(\mathbb{R}^{3N})$, since the potential in
(1.1) is an infinitesimal perturbation relative to the unperturbed operator
$-\Delta=-\sum_{k}\Delta_{k}$, see e.g. [20, Theorem X.16]. Let
$\psi=\psi(\mathbf{x})$, be an eigenfunction of the operator $\mathcal{H}$
with an eigenvalue $E\in\mathbb{R}$, i.e. $\psi\in D(\mathcal{H})$ and
$\displaystyle(\mathcal{H}-E)\psi=0.$
For each $j=1,\dots,N$, we represent
$\displaystyle\mathbf{x}=(\hat{\mathbf{x}}_{j},x_{j}),\quad\textup{where}\
\hat{\mathbf{x}}_{j}=(x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{N}),$
with obvious modifications if $j=1$ or $j=N$. The one-particle density matrix
is defined as the function
(1.2)
$\displaystyle\gamma(x,y)=\sum_{j=1}^{N}\int\limits_{\mathbb{R}^{3N-3}}\overline{\psi(\hat{\mathbf{x}}_{j},x)}\psi(\hat{\mathbf{x}}_{j},y)\
d\hat{\mathbf{x}}_{j},\quad(x,y)\in\mathbb{R}^{3}\times\mathbb{R}^{3}.$
This function is one of the key objects in the multi-particle quantum
mechanics, see [9], [10], [18], [19] for details and futher references. If one
assumes that all $N$ particles are spinless fermions (resp. bosons), i.e. that
the function $\psi$ is antisymmetric (resp. symmetric) under the permutations
$x_{j}\leftrightarrow x_{k}$, then the definition (1.2) simplifies:
(1.3)
$\displaystyle\gamma(x,y)=N\int_{\mathbb{R}^{3N-3}}\overline{\psi(\hat{\mathbf{x}},x)}\psi(\hat{\mathbf{x}},y)d\hat{\mathbf{x}},\
\quad\textup{where}\ \hat{\mathbf{x}}=\hat{\mathbf{x}}_{N}.$
Our main result however does not require any symmetry assumptions. For the
sake of completeness mention that, as found in [16], the function (1.2) is
real-analytic for all $x\not=0,y\not=0,x\not=y$. In the current paper our
focus is on spectral properties of the self-adjoint non-negative operator
$\boldsymbol{\Gamma}$ with the kernel $\gamma(x,y)$, which we call the one-
particle density operator. The operator $\boldsymbol{\Gamma}$ is easily shown
to be trace class, and in [13] it was shown that $\boldsymbol{\Gamma}$ has
infinite rank. However no sharp results on the behaviour of the eigenvalues
$\lambda_{k}(\boldsymbol{\Gamma})>0$ as $k\to\infty$ had been available until
paper [22] (see however [6], [7] for relevant quantum chemistry calculations),
where it was shown that $\lambda_{k}(\boldsymbol{\Gamma})=O(k^{-8/3})$. We
always label eigenvalues in non-increasing order counting multiplicity. The
purpose of the paper is to prove the asymptotic formula (1.4), which confirms
the sharpness of the bound from [22]. Apart from being a mathematically
interesting and challenging question, spectral asymptotics for the operator
$\boldsymbol{\Gamma}$ are important for electronic structure computations as
it limits accuracy of electronic properties computed with finite basis sets,
see e.g. [6], [8], [13] and [15] for discussion.
We assume throughout that $\psi$ decays exponentially as
$|\mathbf{x}|\to\infty$:
$\displaystyle|\psi(\mathbf{x})|\lesssim
e^{-\varkappa_{\scaleto{0}{3pt}}|\mathbf{x}|},\ \mathbf{x}\in\mathbb{R}^{3N}$
Here $\varkappa_{0}>0$ is a constant, and the notation “$\lesssim$” means that
the left-hand side is bounded from above by the right-hand side times some
positive constant whose precise value is of no importance for us. This
notation is used throughout the paper. The property (1) holds for the
eigenfunctions associated with discrete eigenvalues (i.e. the ones below the
essential spectrum), and in particular, for the ground state. For references
and detailed discussion we quote [21].
The next theorem contains a concise version of the main result.
###### Theorem 1.1.
Suppose that the eigenfunction $\psi$ satisfies the bound (1). Then the
eigenvalues $\lambda_{k}(\boldsymbol{\Gamma}),k=1,2,\dots$, of the operator
$\boldsymbol{\Gamma}$ with kernel (1.2) satisfy the relation
(1.4)
$\displaystyle\lim_{k\to\infty}k^{\frac{8}{3}}\lambda_{k}(\boldsymbol{\Gamma})=A^{\frac{8}{3}},$
with an explicit constant $A\geq 0$.
The complete statement includes a formula for the coefficient $A$, and it is
given as Theorem 2.3.
###### Remark.
Theorem 1.1 extends to the case of a molecule with several nuclei whose
positions are fixed, i.e. the operator (1.1) can be replaced by
$\displaystyle\mathcal{H}=\sum_{k=1}^{N}\bigg{(}-\Delta_{k}-\sum_{l=1}^{N_{0}}\frac{Z_{l}}{|x_{k}-R_{l}|}\bigg{)}+\sum_{1\leq
j<k\leq N}\frac{1}{|x_{j}-x_{k}|},$
with constant $R_{l}\in\mathbb{R}^{3}$ and nuclear charges $Z_{l}>0$,
$l=1,2,\dots,N_{0}$. The modifications are straightforward.
Let us outline the main ideas of the proof. First we represent the operator
$\boldsymbol{\Gamma}$ as the product
$\boldsymbol{\Gamma}=\boldsymbol{\Psi}^{*}\boldsymbol{\Psi}$, where the
operator
$\boldsymbol{\Psi}:\textup{{{L}}}^{2}(\mathbb{R}^{3})\to\textup{{{L}}}^{2}(\mathbb{R}^{3N-3})$
with a vector-valued kernel is defined in Subsect. 2.2. Therefore we have
$\lambda_{k}(\boldsymbol{\Gamma})=s_{k}(\boldsymbol{\Psi})^{2},k=1,2,\dots$,
where $s_{k}(\boldsymbol{\Psi})$ are the singular values ($s$-values) of the
operator $\boldsymbol{\Psi}$. As a consequence, the asymptotic formula (1.4)
rewrites as
(1.5)
$\displaystyle\lim_{k\to\infty}k^{\frac{4}{3}}s_{k}(\boldsymbol{\Psi})=A^{\frac{4}{3}}.$
For the sake of discussion consider the fermionic (or bosonic) case, in which
the kernel $\gamma(x,y)$ is given by (1.3). Then it is straightforward that
$\boldsymbol{\Gamma}=\boldsymbol{\Psi}^{*}\boldsymbol{\Psi}$ with the operator
(1.6)
$\displaystyle(\boldsymbol{\Psi}u)(\hat{\mathbf{x}})=\sqrt{N}\int_{\mathbb{R}^{3}}\psi(\hat{\mathbf{x}},x)u(x)dx,\
u\in\textup{{{L}}}^{2}(\mathbb{R}^{3}).$
For integral operators the rate of decay of singular values increases with the
smoothness of their kernels, and the appropriate estimates via suitable
Sobolev norms can be found in [3]. Such estimates, together with the recent
regularity estimates for $\psi$ obtained in [11], were used in [22] to prove
the bound $s_{k}(\boldsymbol{\Psi})\lesssim k^{-4/3}$, $k=1,2,\dots$.
The study of spectral asymptotics of the operator (1.6) requires more precise
information on the singularities of $\psi$. By elliptic regularity, the
function $\psi$ is real analytic away from the coalescence points of the
particles, i.e. for $x_{j}\not=x_{k},1\leq j<k\leq N$ and $x_{j}\not=0$,
$j=1,2,\dots,N$, and hence only the coalescence points contribute to the
asymptotics (1.5). As shown by T. Kato in [17], the function $\psi$ is
Lipschitz. Of course, this fact alone is not sufficient to obtain an
asymptotic formula for $\boldsymbol{\Psi}$ – one needs to know the precise
shape of the function $\psi$ near the coalescence points. A suitable
representation formula for the function $\psi$ was obtained in [12]. To
explain in more detail we make a further simplifying assumption and consider
the special case $N=2$, so that
$\mathbf{x}=(t,x)\in\mathbb{R}^{3}\times\mathbb{R}^{3}$, and the operator
$\boldsymbol{\Psi}$ acts from $\textup{{{L}}}^{2}(\mathbb{R}^{3})$ into
$\textup{{{L}}}^{2}(\mathbb{R}^{3})$. According to [12], there exists a
neighbourhood (open connected set)
$\Omega_{1,2}\subset\big{(}\mathbb{R}^{3}\setminus\\{0\\}\big{)}\times\big{(}\mathbb{R}^{3}\setminus\\{0\\}\big{)}$
of the diagonal set $\\{(x,x):x\in\mathbb{R}^{3}\setminus\\{0\\}\\}$ and two
functions $\xi_{1,2},\eta_{1,2}$, real analytic in $\Omega_{1,2}$, such that
the eigenfunction $\psi=\psi(t,x)$ admits the representation
(1.7)
$\displaystyle\psi(t,x)=\xi_{1,2}(t,x)+|t-x|\,\eta_{1,2}(t,x),\quad\textup{for
all}\quad(t,x)\in\Omega_{1,2}.$
The form of the second term is in line with Kato’s observation (see [17]) that
$\psi$ is Lipschitz. The representation (1.7) is ideally suited for the study
of spectral asymptotics. Indeed, the Lipschitz factor on the right-hand side
of (1.7) is homogeneous of order one. The behaviour of eigenvalues for a wide
class of integral operators including those with homogeneous kernels, was
studied by M. Birman and M. Solomyak in [1],[2] and [4], see also [3].
However, the existing results are not directly applicable, since the functions
$\xi_{1,2}$ and $\eta_{1,2}$ may not be smooth on the closure
$\overline{\Omega_{1,2}}$. Moreover, there is no information on the
integrability of $\xi_{1,2}$ and $\eta_{1,2}$ over $\Omega_{1,2}$. To
circumvent this difficulty we approximate $\xi_{1,2},\eta_{1,2}$ by suitable
$\textup{{{C}}}^{\infty}_{0}$-functions supported inside $\Omega_{1,2}$. The
error incurred is controlled with the help of the bounds obtained in [22].
Using the Birman-Solomyak results and subsequently taking the limit of these
smooth approximations we arrive at the formula (1.5) with the coefficient
$\displaystyle
A=\frac{1}{3}\bigg{(}\frac{2}{\pi}\bigg{)}^{\frac{5}{4}}\int_{\mathbb{R}^{3}}|2^{1/2}\eta_{1,2}(x,x)|^{3/4}dx.$
The finiteness of the above integral is a by-product of the proof. Note that
the coalescence points $x=0$ and $t=0$ do not affect the asymptotics.
For $N\geq 3$ application of the existing results on spectral asymptotics for
integral operators is not immediate. It relies on the reduction to a certain
model operator whose kernel includes the functions $\eta_{j,k}$ describing the
eigenfunction $\psi$ in a neighbourhood of all pair coalescence points
$x_{j}=x_{k}$, $j,k=1,2,\dots,N$, $j\not=k$. We emphasize that neither the
points $x_{j}=0,j=1,2,\dots,N$, nor the coalescence points of higher orders
(e.g. $x_{j}=x_{k}=x_{l}$ with pair-wise distinct $j,k,l$) contribute to the
asymptotics (1.5).
The paper is organized as follows. In Section 2 we describe the representation
of the function $\psi$ near the pair coalescence points (see (1.7) for the
case $N=2$), state the main result in its complete form as Theorem 2.3, which
includes the formula (2.7) for the coefficient $A$, and give the details of
the factorization
$\boldsymbol{\Gamma}=\boldsymbol{\Psi}^{*}\boldsymbol{\Psi}$. Section 3
contains necessary facts about compact operators, and it includes asymptotic
formulas for spectra of integral operators with homogeneous kernels. Section 4
is focused on spectral asymptotics of the model integral operator that is
instrumental to the case $N\geq 3$. Using the factorization
$\boldsymbol{\Gamma}=\boldsymbol{\Psi}^{*}\boldsymbol{\Psi}$, in Sections 5
and 6 the main Theorem 2.3 is restated in terms of the operator
$\boldsymbol{\Psi}$, see Theorem 5.1. Here we also construct suitable
approximations for $\boldsymbol{\Psi}$, to which one can apply the results of
Sect. 4. Section. 7 completes the proof of Theorem 5.1 and hence that of
Theorem 2.3.
We conclude the introduction with some general notational conventions.
Coordinates. As mentioned earlier, we use the following standard notation for
the coordinates: $\mathbf{x}=(x_{1},x_{2},\dots,x_{N})$, where
$x_{j}\in\mathbb{R}^{3}$, $j=1,2,\dots,N$. In order to write formulas in a
more compact and unified way, we sometimes use the notation $x_{0}=0$.
The vector $\mathbf{x}$ is often represented in the form
$\displaystyle\mathbf{x}=(\hat{\mathbf{x}}_{j},x_{j})\quad\textup{with}\quad\hat{\mathbf{x}}_{j}=(x_{1},x_{2},\dots,x_{j-1},x_{j+1},\dots,x_{N})\in\mathbb{R}^{3N-3},$
for arbitrary $j=1,2,\dots,N$. Most frequently we use this notation with
$j=N$, and write $\hat{\mathbf{x}}=\hat{\mathbf{x}}_{N}$, so that
$\mathbf{x}=(\hat{\mathbf{x}},x_{N})$.
For $N\geq 3$ it is also useful to introduce the notation for $\mathbf{x}$
with $x_{j}$ and $x_{k}$ taken out:
(1.8)
$\displaystyle\begin{cases}\tilde{\mathbf{x}}_{j,k}=(x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{k-1},x_{k+1},\dots,x_{N}),\quad\textup{if}\
j<k,\\\\[5.69046pt] \qquad\textup{and}\
\tilde{\mathbf{x}}_{j,k}=\tilde{\mathbf{x}}_{k,j},\quad\textup{if}\
j>k.\end{cases}$
If $j<k$, then we write $\mathbf{x}=(\tilde{\mathbf{x}}_{j,k},x_{j},x_{k})$.
For any $j\leq N-1$ the vector $\hat{\mathbf{x}}$ can be represented as
$\hat{\mathbf{x}}=(\tilde{\mathbf{x}}_{j,N},x_{j})$.
The notation $B_{R}$ is used for the ball $\\{x\in\mathbb{R}^{3}:|x|<R\\}$.
Derivatives. Let $\mathbb{N}_{0}=\mathbb{N}\cup\\{0\\}$. If
$x=(x^{\prime},x^{\prime\prime},x^{\prime\prime\prime})\in\mathbb{R}^{3}$ and
$m=(m^{\prime},m^{\prime\prime},m^{\prime\prime\prime})\in\mathbb{N}_{0}^{3}$,
then the derivative $\partial_{x}^{m}$ is defined in the standard way:
$\displaystyle\partial_{x}^{m}=\partial_{x^{\prime}}^{m^{\prime}}\partial_{x^{\prime\prime}}^{m^{\prime\prime}}\partial_{x^{\prime\prime\prime}}^{m^{\prime\prime\prime}}.$
Cut-off functions. We systematically use the following smooth cut-off
functions. Let
(1.9)
$\displaystyle\theta\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}),\quad\zeta(t)=1-\theta(t),$
be functions such that $0\leq\theta\leq 1$ and
(1.10) $\displaystyle\theta(t)=0,\quad\textup{if}\quad|t|>1;\
\quad\theta(t)=1,\quad\textup{if}\quad|t|<\frac{1}{2}.\ $
Integral operators. The notation $\operatorname{{\sf Int}}(\mathcal{K})$ is
used for the integral operator with kernel $\mathcal{K}$, e.g.
$\boldsymbol{\Gamma}=\operatorname{{\sf Int}}(\gamma)$. The functional spaces,
where $\operatorname{{\sf Int}}(\mathcal{K})$ acts are obvious from the
context.
Bounds. As explained earlier, for two non-negative numbers (or functions) $X$
and $Y$ depending on some parameters, we write $X\lesssim Y$ (or $Y\gtrsim X$)
if $X\leq CY$ with some positive constant $C$ independent of those parameters.
To avoid confusion we often make explicit comments on the nature of (implicit)
constants in the bounds.
## 2\. Representation formula. Details of the main result
### 2.1. Representation formula
Our approach is built on the sharp qualitative result for $\psi$ obtained in
[12]. In order to write all the formulas in a more compact and unified way, we
use the notation $x_{0}=0$. As before,
$\mathbf{x}=(x_{1},x_{2},\dots,x_{N})\in\mathbb{R}^{3N}$. Thus, unless
otherwise stated, the indices labeling the particles, run from $0$ to $N$.
Denote
(2.1) $\displaystyle{\sf
S}_{l,s}=\\{\mathbf{x}\in\mathbb{R}^{3N}:x_{l}\not=x_{s}\\},\ l\not=s.$
The function $\psi$ is real-analytic on the set
$\displaystyle{\sf{U}}=\bigcap_{0\leq l<s\leq N}{\sf S}_{l,s}.$
For each pair $j,k:j\not=k$, we are interested in the behaviour of $\psi$ on
the set
(2.2) $\displaystyle{\sf{U}}_{j,k}=\bigcap_{\begin{subarray}{c}l\not=s\\\
(l,s)\not=(j,k)\end{subarray}}{\sf S}_{l,s}.$
In words, ${\sf{U}}_{j,k}$ includes the coalescence point $x_{j}=x_{k}$, but
excludes all the others. Our main focus will be on the function $\psi$ near
the “diagonal” set
(2.3) $\displaystyle{\sf{U}}^{(\rm
d)}_{j,k}=\\{\mathbf{x}\in{\sf{U}}_{j,k}:x_{j}=x_{k}\\}.$
The sets introduced above are obviously symmetric with respect to permutations
of indices, e.g. ${\sf{U}}_{j,k}={\sf{U}}_{k,j}$, ${\sf{U}}^{(\rm
d)}_{j,k}={\sf{U}}^{(\rm d)}_{k,j}$. Observe also that the sets
${\sf{U}}_{j,k}$, ${\sf{U}}_{j,k}^{(\operatorname{d})}$ are of full measure in
$\mathbb{R}^{3N}$ and $\mathbb{R}^{3N-3}$ respectively, and that they are
connected.
The following property follows from [12, Theorem 1.4].
###### Proposition 2.1.
For each pair of indices $j,k=0,1,\dots,N$ such that $j\not=k$, there exists
an open connected set $\Omega_{j,k}=\Omega_{k,j}\subset\mathbb{R}^{3N}$, such
that
(2.4)
$\displaystyle{\sf{U}}_{j,k}^{(d)}\subset\Omega_{j,k}\subset{\sf{U}}_{j,k},$
and two uniquely defined functions $\xi_{j,k},\eta_{j,k}$, real analytic on
$\Omega_{j,k}$, such that for all $\mathbf{x}\in\Omega_{j,k}$ the following
representation holds:
(2.5)
$\displaystyle\psi(\mathbf{x})=\xi_{j,k}(\mathbf{x})+|x_{j}-x_{k}|\eta_{j,k}(\mathbf{x}).$
Due to the uniqueness of functions $\xi_{j,k},\eta_{j,k}$, we have the
symmetry $\xi_{j,k}=\xi_{k,j}$, $\eta_{j,k}=\eta_{k,j}$ for all $j\not=k$.
The asymptotic coefficient $A$ in the formula (1.4) is defined via the
functions $\eta_{j,k}$, $j,k=1,2,\dots,N,j<k$, on the sets (2.3). Using the
notation (1.8) we write the function $\eta_{j,k}(\mathbf{x})$ on
${\sf{U}}_{j,k}^{(\operatorname{d})}$ as
$\eta_{j,k}(\tilde{\mathbf{x}}_{j,k},x,x)$. As a by-product of the proof we
obtain the following integrability properties.
###### Theorem 2.2.
If $N\geq 3$, then each function $\eta_{j,k}(\ \cdot\ ,x,x)$, $1\leq j<k\leq
N$, belongs to $\textup{{{L}}}^{2}(\mathbb{R}^{3N-6})$ for a.e.
$x\in\mathbb{R}^{3}$ and the function
(2.6) $\displaystyle H(x):=\bigg{[}2\sum\limits_{1\leq j<k\leq
N}\int_{\mathbb{R}^{3N-6}}\big{|}\eta_{j,k}(\tilde{\mathbf{x}}_{j,k},x,x)\big{|}^{2}d\tilde{\mathbf{x}}_{j,k}\bigg{]}^{\frac{1}{2}},$
belongs to $\textup{{{L}}}^{\frac{3}{4}}(\mathbb{R}^{3})$.
If $N=2$, then the function $H(x):=\sqrt{2}|\eta_{1,2}(x,x)|$ belongs to
$\textup{{{L}}}^{\frac{3}{4}}(\mathbb{R}^{3})$.
Having at our disposal this theorem, we can now state the main result of the
paper in its complete form.
###### Theorem 2.3.
Suppose that the eigenfunction $\psi$ satisfies the bound (1). Then the
eigenvalues $\lambda_{k}(\boldsymbol{\Gamma}),k=1,2,\dots,$ of the operator
$\boldsymbol{\Gamma}$ satisfy the asymptotic formula (1.4) with the constant
(2.7) $\displaystyle
A=\frac{1}{3}\bigg{(}\frac{2}{\pi}\bigg{)}^{\frac{5}{4}}\int_{\mathbb{R}^{3}}H(x)^{\frac{3}{4}}dx.$
###### Remark 2.4.
The coefficient $A$ can be equal to zero for some eigenfunctions $\psi$. For
example, if we assume that the particles are spinless fermions, i.e. the
function $\psi$ is antisymmetric, then it is immediate to see that for all
$j,k$, $j\not=k$, both components $\xi_{j,k}$ and $\eta_{j,k}$ in (2.5) vanish
on the diagonal ${\sf{U}}_{j,k}^{(\operatorname{d})}$, and as a consequence
$A=0$. This means that $\lambda_{k}(\boldsymbol{\Gamma})=o(k^{-8/3})$. This
fact can be interpreted by saying that antisymmetric eigenfunctions possess
better than Lipschitz smoothness at the coalescence points, and hence the
eigenvalues of $\boldsymbol{\Gamma}$ decay faster.
The fermionic nature of particles may manifest itself differently if we
introduce the spin variable. In this case the antisymmetry of the full
eigenfunction comes either from the spatial component $\psi(\mathbf{x})$ or
from the spin component. For illustration first consider the case of two
electrons, i.e. $N=2$. In the triplet configuration the antisymmetry is
carried by the spatial component $\psi(x_{1},x_{2})$, see [15, Subsect.
3.3.2], and then, as pointed out a few lines above, we have $A=0$. If the
electrons are in the singlet configuration, then the spin component is
antisymmetric, whereas the function $\psi=\psi(x_{1},x_{2})$ is symmetric and
the diagonal value $\eta_{1,2}(x,x)$ is not identically zero, see [15,
Subsect. 3.3.1]. Thus $A>0$.
In the case $N\geq 3$ different electron pairs may form different
configurations, in which case the triplet coalescences will not contribute to
the coefficient $A$.
###### Remark 2.5.
If we assume that the function $\psi$ is symmetric or antisymmetric, then both
the proof of the main asymptotic formula (1.4), and the formula (2.6) can be
simplified. Indeed, as we have seen, the factorization
$\boldsymbol{\Gamma}=\boldsymbol{\Psi}^{*}\boldsymbol{\Psi}$ holds with the
simple looking integral operator $\boldsymbol{\Psi}$ given by (1.6). This is
in contrast with the general case, as will be evident from the next
subsection. Furthermore, as discussed in Remark 2.4, for the antisymmetric
$\psi$ we have $A=0$. Assume that $N\geq 3$ and that $\psi$ is totally
symmetric. It follows that for all
$\tilde{\mathbf{y}}=(y_{1},y_{2},\dots,y_{N-2})\in\mathbb{R}^{3N-6}$,
$x,t\in\mathbb{R}^{3}$ and $j\not=k,l\not=s$, we have
$\displaystyle\psi(y_{1},\dots,y_{j-1},x,y_{j},$ $\displaystyle\
\dots,y_{k-2},t,y_{k-1},\dots,y_{N-2})$ $\displaystyle=$ $\displaystyle\
\psi(y_{1},\dots,y_{l-1},x,y_{l},\dots,y_{s-2},t,y_{s-1},\dots,y_{N-2}).$
Due to the uniqueness of functions $\xi_{j,k},\eta_{j,k}$ in Proposition 2.1,
the above equality leads to
$\displaystyle\eta_{j,k}(y_{1},\dots,y_{j-1},x,y_{j},$ $\displaystyle\
\dots,y_{k-2},t,y_{k-1},\dots,y_{N-2})$ $\displaystyle=$ $\displaystyle\
\eta_{l,s}(y_{1},\dots,y_{l-1},x,y_{l},\dots,y_{s-2},t,y_{s-1},\dots,y_{N-2}).$
As a consequence, the formula (2.6) rewrites as
$\displaystyle
H(x)=\bigg{[}N(N-1)\int_{\mathbb{R}^{3N-6}}\big{|}\eta_{N-1,N}(\tilde{\mathbf{y}},x,x)\big{|}^{2}\,d\tilde{\mathbf{y}}\bigg{]}^{\frac{1}{2}}.$
### 2.2. Factorization of $\boldsymbol{\Gamma}$: change of variables
$(\hat{\mathbf{x}}_{j},x)\mapsto(\hat{\mathbf{x}},x)$
In the general case (i.e. without any symmetry assumptions on $\psi$) the
operator $\boldsymbol{\Psi}$ in the identity
$\boldsymbol{\Gamma}=\boldsymbol{\Psi}^{*}\boldsymbol{\Psi}$ looks more
complicated compared to (1.6). The purpose of this subsection is to describe
this factorization and the associated change of variables. Rewrite the
definition (1.2) in the form:
$\displaystyle\gamma(x,y)=$ $\displaystyle\
\sum_{j=1}^{N}\int_{\mathbb{R}^{3N-3}}\overline{\psi_{j}(\hat{\mathbf{x}},x)}\psi_{j}(\hat{\mathbf{x}},y)d\hat{\mathbf{x}},\quad\textup{where}$
(2.8) $\displaystyle\psi_{j}(\hat{\mathbf{x}},x)=$ $\displaystyle\
\psi(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1}),\quad j=1,2,\dots,N.$
Therefore $\boldsymbol{\Gamma}$ can be represented as a product
$\boldsymbol{\Gamma}=\boldsymbol{\Psi}^{*}\boldsymbol{\Psi}$, where
$\boldsymbol{\Psi}:\textup{{{L}}}^{2}(\mathbb{R}^{3})\to\textup{{{L}}}^{2}(\mathbb{R}^{3N-3};\mathbb{C}^{N})$
is the integral operator with the vector-valued kernel
(2.9)
$\displaystyle\boldsymbol{\Psi}(\hat{\mathbf{x}},x)=\\{\psi_{j}(\hat{\mathbf{x}},x)\\}_{j=1}^{N}.$
As explained in the Introduction, given this factorization, the asymptotic
relation (1.4) translates to the formula (1.5). Later we state this fact again
as Theorem 5.1 using a more convenient notation.
The change of variables $(\hat{\mathbf{x}}_{j},x)\mapsto(\hat{\mathbf{x}},x)$
plays an important role throughout the paper. In particular, it is crucial to
recast Proposition 2.1 in terms of the new variables $(\hat{\mathbf{x}},x)$,
which is done below.
Let $j\not=k$, and let $\Omega_{j,k}$ be the sets and $\xi_{j,k}(\mathbf{x})$,
$\eta_{j,k}(\mathbf{x})$ be the functions from Proposition 2.1. For all
$j=1,2,\dots,N$ and all $k=0,1,\dots,N-1,$ denote
$\displaystyle\tilde{\Omega}_{j,k}=\begin{cases}\\{(\hat{\mathbf{x}},x)\in\mathbb{R}^{3N}:(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1})\in\Omega_{j,k}\\},\quad\textup{if}\
j\geq k+1,\\\
\\{(\hat{\mathbf{x}},x)\in\mathbb{R}^{3N}:(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1})\in\Omega_{j,k+1}\\},\quad\textup{if}\
j\leq k.\end{cases}$
According to (2.4) we have
(2.10)
$\displaystyle{\sf{U}}_{N,k}^{(\operatorname{d})}\subset\tilde{\Omega}_{j,k}\subset{\sf{U}}_{N,k},$
for all $k\leq N-1$ and $j=1,2,\dots,N$, $j\not=k$. Together with functions
$\xi_{j,k},\eta_{j,k}$ define
$\displaystyle\tilde{\xi}_{j,k}(\hat{\mathbf{x}},x)=\begin{cases}\xi_{j,k}(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1}),\quad\textup{if}\
j\geq k+1,\\\\[5.69046pt]
\xi_{j,k+1}(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1}),\quad\textup{if}\ j\leq
k,\end{cases}$
and
(2.11)
$\displaystyle\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x)=\begin{cases}\eta_{j,k}(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1}),\quad\textup{if}\
j\geq k+1,\\\\[5.69046pt]
\eta_{j,k+1}(x_{1},\dots,x_{j-1},x,x_{j},\dots,x_{N-1}),\quad\textup{if}\
j\leq k.\end{cases}$
By Proposition 2.1, for each $j=1,2,\dots,N$, and each $k=0,1,\dots,N-1$, we
have
(2.12)
$\displaystyle\psi_{j}(\hat{\mathbf{x}},x)=\tilde{\xi}_{j,k}(\hat{\mathbf{x}},x)+|x_{k}-x|\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x),\quad\textup{for
all}\ (\hat{\mathbf{x}},x)\in\tilde{\Omega}_{j,k}.$
Observe that the newly introduced sets $\tilde{\Omega}_{j,k}$ and the
functions $\tilde{\xi}_{j,k},\tilde{\eta}_{j,k}$ are not symmetric under the
permutation $j\leftrightarrow k$.
The function (2.6) can be easily rewritten via the new functions
$\tilde{\eta}_{j,k}$:
(2.13) $\displaystyle
H(x)=\begin{cases}\big{(}|\tilde{\eta}_{1,1}(x,x)|^{2}+|\tilde{\eta}_{2,1}(x,x)|^{2}\big{)}^{1/2},\quad\textup{if}\
N=2;\\\\[8.5359pt]
\bigg{[}\sum\limits_{j=1}^{N}\sum\limits_{k=1}^{N-1}\int_{\mathbb{R}^{3N-6}}\big{|}\tilde{\eta}_{j,k}(\tilde{\mathbf{x}}_{k,N},x,x)\big{|}^{2}d\tilde{\mathbf{x}}_{k,N}\bigg{]}^{\frac{1}{2}},\quad\textup{if}\
N\geq 3.\end{cases}$
For $N=2$ the above formula is a consequence of the symmetry relation
$\eta_{1,2}=\eta_{2,1}$ and equalities
$\eta_{1,2}(x,x)=\tilde{\eta}_{1,1}(x,x)$,
$\eta_{2,1}(x,x)=\tilde{\eta}_{2,1}(x,x)$, which follow from the definition
(2.11).
Now assume that $N\geq 3$. In view of the symmetry $\eta_{j,k}=\eta_{k,j}$ we
can rewrite (2.6) extending the summation to all $j,k$ such that $j\not=k$:
$\displaystyle
H(x)^{2}=\sum_{j=1}^{N-1}\sum_{k=j+1}^{N}\int_{\mathbb{R}^{3N-6}}\big{|}\eta_{j,k}(\tilde{\mathbf{x}}_{j,k},x,x)\big{|}^{2}d\tilde{\mathbf{x}}_{j,k}+\sum_{j=1}^{N}\sum_{k=1}^{j-1}\int_{\mathbb{R}^{3N-6}}\big{|}\eta_{j,k}(\tilde{\mathbf{x}}_{j,k},x,x)\big{|}^{2}d\tilde{\mathbf{x}}_{j,k}.$
By (2.11), the second sum coincides with
$\displaystyle\sum_{j=1}^{N}\sum_{k=1}^{j-1}\int_{\mathbb{R}^{3N-6}}\big{|}\tilde{\eta}_{j,k}(\tilde{\mathbf{x}}_{k,N},x,x)\big{|}^{2}d\tilde{\mathbf{x}}_{k,N},$
and the first one coincides with
$\displaystyle\sum_{j=1}^{N-1}\sum_{k=j+1}^{N}\int_{\mathbb{R}^{3N-6}}\big{|}\tilde{\eta}_{j,k-1}(\tilde{\mathbf{x}}_{k-1,N},x,x)\big{|}^{2}d\tilde{\mathbf{x}}_{k-1,N}=\sum_{j=1}^{N-1}\sum_{k=j}^{N-1}\int_{\mathbb{R}^{3N-6}}\big{|}\tilde{\eta}_{j,k}(\tilde{\mathbf{x}}_{k,N},x,x)\big{|}^{2}d\tilde{\mathbf{x}}_{k,N}.$
Adding the first and second sums together we obtain (2.13), as claimed.
## 3\. Compact operators
### 3.1. Compact operators
For information on compact operators we use mainly Chapter 11 of the book [5],
where one can also find further references. Let $\mathcal{H}$ and
$\mathcal{G}$ be separable Hilbert spaces. Let $T:\mathcal{H}\to\mathcal{G}$
be a compact operator. If $\mathcal{H}=\mathcal{G}$ and $T=T^{*}\geq 0$, then
$\lambda_{k}(T)$, $k=1,2,\dots$, denote the positive eigenvalues of $T$
numbered in descending order counting multiplicity. For arbitrary spaces
$\mathcal{H}$, $\mathcal{G}$ and compact $T$, by $s_{k}(T)>0$, $k=1,2,\dots$,
we denote the singular values of $T$ defined by
$s_{k}(T)^{2}=\lambda_{k}(T^{*}T)=\lambda_{k}(TT^{*})$.
We classify compact operators by the rate of decay of their singular values.
If $s_{k}(T)\lesssim k^{-1/p},k=1,2,\dots$, with some $p>0$, then we say that
$T\in\mathbf{S}_{p,\infty}$ and denote
$\displaystyle\|T\|_{p,\infty}=\sup_{k}s_{k}(T)k^{\frac{1}{p}}.$
These classes are discussed in detail in [5, §11.6]. The class
$\mathbf{S}_{p,\infty}$ is a complete linear space with the quasi-norm
$\|T\|_{p,\infty}$. For all $p>0$ the quasi-norm satisfies the following
“triangle” inequality for operators $T_{1},T_{2}\in\mathbf{S}_{p,\infty}$:
(3.1)
$\displaystyle\|T_{1}+T_{2}\|_{{\scaleto{p}{4pt}},{\scaleto{\infty}{3pt}}}^{{\frac{{\scaleto{p}{3pt}}}{{\scaleto{p+1}{4pt}}}}}\leq\|T_{1}\|_{{\scaleto{p}{4pt}},{\scaleto{\infty}{3pt}}}^{{\frac{{\scaleto{p}{3pt}}}{{\scaleto{p+1}{4pt}}}}}+\|T_{2}\|_{{\scaleto{p,\infty}{4pt}}}^{{\frac{{\scaleto{p}{3pt}}}{{\scaleto{p+1}{4pt}}}}}.$
For $T\in\mathbf{S}_{p,\infty}$ the following numbers are finite:
(3.2)
$\displaystyle\begin{cases}{\sf{G}}_{p}(T)=\big{(}\limsup\limits_{k\to\infty}k^{\frac{1}{p}}s_{k}(T)\big{)}^{p}=\limsup\limits_{s\to
0}s^{p}n(s,T),\\\\[8.5359pt]
{\sf{g}}_{p}(T)=\big{(}\liminf\limits_{k\to\infty}k^{\frac{1}{p}}s_{k}(T)\big{)}^{p}=\liminf\limits_{s\to
0}s^{p}n(s,T),\end{cases}$
and they clearly satisfy the inequalities
$\displaystyle{\sf{g}}_{p}(T)\leq{\sf{G}}_{p}(T)\leq\|T\|_{p,\infty}^{p}.$
Note that ${\sf{G}}_{q}(T)=0$ for all $q>p$. Observe that
(3.3)
$\displaystyle{\sf{g}}_{p}(TT^{*})={\sf{g}}_{p}(T^{*}T)={\sf{g}}_{2p}(T),\quad{\sf{G}}_{p}(TT^{*})={\sf{G}}_{p}(T^{*}T)={\sf{G}}_{2p}(T).$
If ${\sf{G}}_{p}(T)={\sf{g}}_{p}(T)$, then the singular values of $T$ satisfy
the asymptotic formula
$\displaystyle
s_{n}(T)=\big{(}{\sf{G}}_{p}(T)\big{)}^{\frac{1}{p}}n^{-\frac{1}{p}}+o(n^{-\frac{1}{p}}),\
n\to\infty.$
The functionals ${\sf{g}}_{p}(T)$, ${\sf{G}}_{p}(T)$ also satisfy the
inequalities of the type (3.1):
(3.4)
$\displaystyle\begin{cases}{\sf{G}}_{p}(T_{1}+T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}\leq{\sf{G}}_{p}(T_{1})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}+{\sf{G}}_{p}(T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}},\\\\[8.5359pt]
{\sf{g}}_{p}(T_{1}+T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}\leq{\sf{g}}_{p}(T_{1})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}+{\sf{G}}_{p}(T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}.\end{cases}$
It follows from these inequalities that the functionals ${\sf{G}}_{p}$ and
${\sf{g}}_{p}$ are continuous on $\mathbf{S}_{p,\infty}$:
$\displaystyle\big{|}{\sf{G}}_{p}(T_{1})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}-{\sf{G}}_{p}(T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}\big{|}\leq$
$\displaystyle\
{\sf{G}}_{p}(T_{1}-T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}},$
$\displaystyle\big{|}{\sf{g}}_{p}(T_{1})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}-{\sf{g}}_{p}(T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}\big{|}\leq$
$\displaystyle\
{\sf{G}}_{p}(T_{1}-T_{2})^{{\frac{{\scaleto{1}{3pt}}}{{\scaleto{p+1}{4pt}}}}}.$
We need the following two corollaries of this fact:
###### Corollary 3.1.
Suppose that ${\sf{G}}_{p}(T_{1}-T_{2})=0$. Then
$\displaystyle{\sf{G}}_{p}(T_{1})={\sf{G}}_{p}(T_{2}),\quad{\sf{g}}_{p}(T_{1})={\sf{g}}_{p}(T_{2}).$
The next corollary is more general:
###### Corollary 3.2.
Suppose that $T\in\mathbf{S}_{p,\infty}$ and that for every $\nu>0$ there
exists an operator $T_{\nu}\in\mathbf{S}_{p,\infty}$ such that
${\sf{G}}_{p}(T-T_{\nu})\to 0$, $\nu\to 0$. Then the functionals
${\sf{G}}_{p}(T_{\nu}),{\sf{g}}_{p}(T_{\nu})$ have limits as $\nu\to 0$ and
$\displaystyle\lim_{\nu\to
0}{\sf{G}}_{p}(T_{\nu})={\sf{G}}_{p}(T),\quad\lim_{\nu\to
0}{\sf{g}}_{p}(T_{\nu})={\sf{g}}_{p}(T).$
### 3.2. Estimates for singular values of integral operators
The final ingredients of the proof are the results due to M.S. Birman and M.Z.
Solomyak, investigating the membership of integral operators in various
classes of compact operators.
For estimates of the singular values we rely on [3, Corollaries 4.2, 4.4,
Theorem 4.4], which we state here in a form convenient for our purposes. Below
we use the following notation which is standard in the theory of Sobolev
spaces:
$\textup{{{H}}}^{l}(\mathbb{R}^{d})=\textup{{{W}}}^{2,l}(\mathbb{R}^{d})$.
###### Proposition 3.3.
Let $a\in\textup{{{L}}}^{\infty}(\mathbb{R}^{d})$,
$b\in\textup{{{L}}}^{2}_{\textup{\tiny{\rm loc}}}(\mathbb{R}^{n})$. Assume
that the function $a$ has compact support. Suppose that $T(t,x)$,
$t\in\mathbb{R}^{n}$, $x\in\mathbb{R}^{d}$, is a kernel such that $T(t,\
\cdot\ )\in\textup{{{H}}}^{l}(\mathbb{R}^{d})$ with some $l=0,1,\dots$, for
a.e. $t\in\mathbb{R}^{n}$, and the function $\|T(t,\ \cdot\
)\|_{\textup{{{H}}}^{l}}$ is in
$\textup{{{L}}}^{2}(\mathbb{R}^{n},|b(t)|^{2}dt)$.
Let
$T_{ba}:\textup{{{L}}}^{2}(\mathbb{R}^{d})\to\textup{{{L}}}^{2}(\mathbb{R}^{n})$,
be the integral operator
$\displaystyle(T_{ba}u)(t)=b(t)\int T(t,x)a(x)u(x)\,dx,\quad
u\in\textup{{{L}}}^{2}(\mathbb{R}^{d}).$
Then
(3.5)
$\displaystyle\sum_{k=0}^{\infty}k^{\frac{2l}{d}}s_{k}(T_{ba})^{2}<\infty,$
and hence $s_{k}(T_{ba})=o(k^{-1/q})$, where $1/q=1/2+l/d$. In other words,
${\sf{G}}_{q}(T_{ba})=0$.
The original results in [3, Corollaries 4.2, 4.4, Theorem 4.4] are
considerably more general and more precise: instead of just the finiteness
statement (3.5), they contain estimates depending explicitly on the kernel $T$
and weights $a,b$. These estimates have slightly different form for different
cases $2l>d,2l=d$ and $2l<d$, and therefore, to avoid cumbersome formulations
we chose not to quote them in detail.
The next group of results is concerned with spectral asymptotics for integral
operators.
### 3.3. Integral operators with homogeneous kernels
First we consider pseudo - differential operators with asymptotically
homogeneous matrix-valued symbols. Spectral asymptotics for such operators
were studied in [2], [4]. In fact, these papers allow for more general
operators, but we need only a relatively simple special case of those results.
Precisely, let $\mathcal{A}(x),\mathcal{B}(x),X(\xi)$, where
$x,y,\xi\in\mathbb{R}^{d}$, be rectangular matrix-valued functions of matching
dimensions, so that the product
(3.6) $\displaystyle\mathcal{B}(x)X(\xi)\mathcal{A}(y)$
is again a rectangular matrix. Assume that
(3.7)
$\displaystyle\mathcal{B}\in\textup{{{C}}}_{0}(\mathbb{R}^{d}),\quad\mathcal{A}\in\textup{{{C}}}_{0}(\mathbb{R}^{d}).$
We do not reflect the matrix nature of the functional spaces in the notation
to avoid cumbersome formulas, and this should not cause confusion. Suppose
that $X(\xi)$ is a bounded function which is asymptotically homogeneous of
negative order, i.e. there exists a matrix-valued function
$X_{\infty}\in\textup{{{C}}}^{\infty}(\mathbb{R}^{d}\setminus\\{0\\})$ such
that for some $\tau>0$,
(3.8) $\displaystyle X_{\infty}(t\xi)=t^{-\tau}X_{\infty}(\xi),\ \xi\not=0,$
for all $t>0$, and
(3.9) $\displaystyle
X(\xi)-X_{\infty}(\xi)=o(|\xi|^{-\tau}),\quad|\xi|\to\infty.$
Define the matrix-valued function
$\displaystyle\mathcal{T}_{\infty}(x,\xi)=\mathcal{B}(x)X_{\infty}(\xi)\mathcal{A}(x).$
###### Proposition 3.4.
Let the above conditions on $\mathcal{A},\mathcal{B},X$ be satisfied and let
$p=d\tau^{-1}$. Then the pseudo-differential operator
$T:\textup{{{L}}}^{2}(\mathbb{R}^{d})\to\textup{{{L}}}^{2}(\mathbb{R}^{d})$
defined by the formula
(3.10)
$\displaystyle(Tu)(x)=\frac{1}{(2\pi)^{d}}\iint\mathcal{B}(x)e^{i\xi(x-y)}X(\xi)\mathcal{A}(y)u(y)dyd\xi,$
is compact, it belongs to $\mathbf{S}_{p,\infty}$ and satisfies the asymptotic
formula
(3.11)
$\displaystyle{\sf{G}}_{p}(T)={\sf{g}}_{p}(T)=\frac{1}{d(2\pi)^{d}}\int\limits_{\mathbb{R}^{d}}\int\limits_{\mathbb{S}^{d-1}}\sum\limits_{k}\big{[}s_{k}\big{(}\mathcal{T}_{\infty}(x,\omega)\big{)}\big{]}^{p}\,d\omega
dx.$
This proposition is a consequence of Theorem 2 from [2] and Remark 3 following
this theorem.
We apply Proposition 3.4 to integral operators with homogeneous kernels. Let
$\Phi\in\textup{{{C}}}^{\infty}(\mathbb{R}^{d}\setminus\\{0\\})$ be a matrix-
valued function such that
(3.12) $\displaystyle\Phi(tx)=t^{\alpha}\Phi(x),\quad x\not=0,\alpha>-d,$
for all $t>0$. Consider the integral operator $W$ with the matrix-valued
kernel
$\displaystyle W(x,y)=\mathcal{B}(x)\Phi(x-y)\mathcal{A}(y)$
with $\mathcal{A},\mathcal{B}$ satisfying the conditions (3.7), and assuming
that the matrix dimensions are matched in the same way as for the symbol
(3.6). We study spectral asymptotics of the operator $W$ by reducing it to the
operator of the form (3.10). Let $\theta$ be as defined in (1.9), (1.10), and
let $R_{0}>0$ be a number such that
$\displaystyle W(x,y)=W(x,y)\theta\big{(}|x-y|R^{-1}\big{)},\quad\textup{for
all}\quad R\geq R_{0}.$
Consequently, the operator $W$ has the form (3.10) with the function
(3.13) $\displaystyle X(\xi)=X_{R}(\xi)=\int e^{-i\xi
x}\theta\big{(}|x|R^{-1}\big{)}\Phi(x)dx.$
Integrating by parts, we conclude that for each $\xi\not=0$ the function
$X_{R}(\xi)$ converges as $R\to\infty$ to a
$\textup{{{C}}}^{\infty}(\mathbb{R}^{d}\setminus\\{0\\})$-function
(3.14) $\displaystyle X_{\infty}(\xi)=\lim_{R\to\infty}X_{R}(\xi).$
The function $X_{\infty}$ satisfies (3.8) with $\tau=\alpha+d$. Indeed, using
(3.12) write for $t>0$:
$\displaystyle X_{R}(t\xi)=$ $\displaystyle\ t^{-\alpha-d}\int e^{-i\xi
x}\theta\big{(}|x|(Rt)^{-1}\big{)}\Phi(x)dx$ (3.15) $\displaystyle=$
$\displaystyle\ t^{-\alpha-d}X_{Rt}(\xi).$
Passing to the limit as $R\to\infty$, we get (3.8) with $\tau=\alpha+d$, as
claimed. The equality (3.3) also implies that
$\displaystyle
X_{R}(t\xi)-X_{\infty}(t\xi)=t^{-\alpha-d}\big{(}X_{Rt}(\xi)-X_{\infty}(\xi)\big{)}=o(t^{-\alpha-d}),\quad
t\to\infty,$
for each $\xi\in\mathbb{R}^{d}$ and $R>0$, which entails (3.9). Thus, applying
Proposition 3.4, we obtain the spectral asymptotics for the operator $W$.
###### Corollary 3.5.
The operator $W$ has the form (3.10) with the function
$X\in\textup{{{C}}}^{\infty}(\mathbb{R}^{d})$ defined in (3.13). The singular
values of $W$ satisfy the relation (3.11) with $p^{-1}=1+\alpha d^{-1}$.
We need this result for the special case of scalar
$\mathcal{A}=a\in\textup{{{C}}}_{0}(\mathbb{R}^{d}),\mathcal{B}=b\in\textup{{{C}}}_{0}(\mathbb{R}^{d})$,
and
(3.16) $\displaystyle\Phi(x)=\\{\phi_{j}(x)\\}_{j=1}^{m}$
with scalar $\alpha$-homogeneous functions $\phi_{j}$, $j=1,2,\dots,m$. As the
next assertion shows, in this case the right-hand side of (3.11) can be easily
evaluated.
###### Corollary 3.6.
Suppose that $\Phi$ is given as in (3.16) with some $\alpha$-homogeneous
scalar functions $\phi_{j}$, $j=1,2,\dots,k,$ with $\alpha>-d$. Then
$\displaystyle{\sf{G}}_{p}(W)={\sf{g}}_{p}(W)=\frac{1}{d(2\pi)^{d}}\int_{\mathbb{S}^{d-1}}|X_{\infty}(\omega)|^{p}\,d\omega\int_{\mathbb{R}^{d}}|a(x)b(x)|^{p}\,dx,$
where $p^{-1}=1+\alpha d^{-1}$.
###### Proof.
The matrix $\mathcal{T}_{\infty}(x,\xi)$ is rank one and
$\displaystyle
s_{1}\big{(}\mathcal{T}_{\infty}(x,\xi)\big{)}=|a(x)|\,|b(x)|\,|X_{\infty}(\xi)|.$
The required formula follows from Corollary 3.5. ∎
Consider two examples in which the above formula can be simplified further.
The first example is crucial for the proof of Theorem 2.3.
###### Example 3.7.
Let $m=1$, and let $\Phi(x)=\phi(x)=|x|^{\alpha}$, $\alpha>-d$, be a scalar
function. Then (see, e.g. [14, Ch. 2, Sect. 3.3])
$\displaystyle
X_{\infty}(\xi)=2^{d+\alpha}\pi^{\frac{d}{2}}\frac{\Gamma\big{(}\frac{d+\alpha}{2}\big{)}}{\Gamma\big{(}-\frac{\alpha}{2}\big{)}}|\xi|^{-(d+\alpha)},\quad\alpha\not=0,2,4,\dots,$
and $X_{\infty}(\xi)=0$ for $\alpha=0,2,4,\dots$. Thus, for $1/p=1+\alpha/d$
and $\alpha\not=0,2,4,\dots,$ we have
(3.17)
$\displaystyle\mu_{\alpha,d}:=\frac{1}{d(2\pi)^{d}}\int_{\mathbb{S}^{d-1}}|X_{\infty}(\omega)|^{p}d\omega=\bigg{[}\frac{\Gamma\big{(}\frac{d+\alpha}{2}\big{)}}{\pi^{\frac{\alpha}{2}}|\Gamma\big{(}-\frac{\alpha}{2}\big{)}|}\bigg{]}^{p}\frac{1}{\Gamma\big{(}\frac{d}{2}+1\big{)}}.$
Now Corollary 3.6 yields
$\displaystyle{\sf{G}}_{p}(W)={\sf{g}}_{p}(W)=\mu_{\alpha,d}\int_{\mathbb{R}^{d}}|a(x)b(x)|^{p}dx,\quad\frac{1}{p}=1+\frac{\alpha}{d}.$
Note that the case of scalar functions $\Phi$ was studied in [1], see also [3,
Theorem 10.9]. Next we consider an important example of a vector-valued
function $\Phi$. We do not need it for the current paper but prepare it for
future use.
###### Example 3.8.
Let $m=d$, and let $\Phi(x)=\nabla|x|^{\alpha+1}=(\alpha+1)|x|^{\alpha-1}x$,
$\alpha>-d$. This vector-valued function is homogeneous of order $\alpha$ and
(similarly to [14, Ch. 2, Sect. 3.3])
$\displaystyle
X_{\infty}(\xi)=-i(\alpha+1)2^{d+\alpha}\pi^{\frac{d}{2}}\frac{\Gamma\big{(}\frac{d+\alpha+1}{2}\big{)}}{\Gamma\big{(}\frac{-\alpha+1}{2}\big{)}}|\xi|^{-(\alpha+1+d)}\xi,\quad\alpha\not=1,3,5,\dots,$
and $X_{\infty}(\xi)=0$ for $\alpha=1,3,5,\dots$. Thus, for $1/p=1+\alpha/d$
and $\alpha\not=1,3,5,\dots,$ we have
(3.18)
$\displaystyle\nu_{\alpha,d}:=\frac{1}{d(2\pi)^{d}}\int_{\mathbb{S}^{d-1}}|X_{\infty}(\omega)|^{p}d\omega=\bigg{[}\frac{(\alpha+1)\Gamma\big{(}\frac{d+\alpha+1}{2}\big{)}}{\pi^{\frac{\alpha}{2}}|\Gamma\big{(}\frac{-\alpha+1}{2}\big{)}|}\bigg{]}^{p}\frac{1}{\Gamma\big{(}\frac{d}{2}+1\big{)}}.$
Now Corollary 3.6 yields
$\displaystyle{\sf{G}}_{p}(W)={\sf{g}}_{p}(W)=\nu_{\alpha,d}\int_{\mathbb{R}^{d}}|a(x)b(x)|^{p}dx,\quad\frac{1}{p}=1+\frac{\alpha}{d}.$
## 4\. Spectral asymptotics for the model problem
The objective of this section is to find the spectral asymptotics for a model
integral operator. Recall that for any function
$\mathcal{K}=\mathcal{K}(x,y)$, $x\in\mathbb{R}^{n},y\in\mathbb{R}^{d}$, we
denote by $\operatorname{{\sf Int}}(\mathcal{K})$ the integral operator acting
from $\textup{{{L}}}^{2}(\mathbb{R}^{d})$ into
$\textup{{{L}}}^{2}(\mathbb{R}^{n})$. In each case the values of $n$ and $d$
are clear from the context. If $\mathcal{K}(x,y)$ is $\mathbb{C}^{s}$-valued
then the “target” space $\textup{{{L}}}^{2}(\mathbb{R}^{n})$ is replaced by
$\textup{{{L}}}^{2}(\mathbb{R}^{n};\mathbb{C}^{s})$.
### 4.1. The model operator
Let $a,b_{j,k},\beta_{j,k}$, $j=1,2,\dots,N$, $k=1,2,\dots,N-1$, be scalar
functions such that
(4.1)
$\displaystyle\begin{cases}a\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3}),&\
\quad b_{j,k}\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3N-3}),\\\\[5.69046pt]
\beta_{j,k}\in\textup{{{C}}}^{\infty}(\mathbb{R}^{3N}),\end{cases}$
for all $j=1,2,\dots,N$, $k=1,2,\dots,N-1$. Let
$\Phi\in\textup{{{C}}}^{\infty}(\mathbb{R}^{3}\setminus\\{0\\})$ be a vector-
valued function with $m$ scalar components, homogeneous of order $\alpha>-3$,
as defined in (3.16). Consider the vector-valued kernel
$\mathcal{M}(\hat{\mathbf{x}},x)$ with $mN$ components:
(4.2)
$\displaystyle\begin{cases}\mathcal{M}(\hat{\mathbf{x}},x)=\\{\mathcal{M}_{j}(\hat{\mathbf{x}},x)\\}_{j=1}^{N},\
\quad\mathcal{M}_{j}(\hat{\mathbf{x}},x)=\sum_{k=1}^{N-1}\mathcal{M}_{j,k}(\hat{\mathbf{x}},x),\\\\[8.5359pt]
\mathcal{M}_{j,k}(\hat{\mathbf{x}},x)=b_{j,k}(\hat{\mathbf{x}})\Phi(x_{k}-x)a(x)\beta_{j,k}(\hat{\mathbf{x}},x).\end{cases}$
Our aim is to find an asymptotic formula for the singular values of the
operator $\operatorname{{\sf
Int}}(\mathcal{M}):\textup{{{L}}}^{2}(\mathbb{R}^{3})\to\textup{{{L}}}^{2}(\mathbb{R}^{3N-3};\mathbb{C}^{mN})$.
Although the function $\Phi(x)$ is homogeneous, the results on homogeneous
kernels, notably Corollary 3.6, are not applicable directly, since the number
of “target” variables (i.e. $3N-3$) is greater than the number of the input
variables (i.e. $3$), unless $N=2$. The proof of Theorem 4.1 below amounts to
reducing the operator $\operatorname{{\sf Int}}(\mathcal{M})$ to a form for
which Corollary 3.6 can be used. Recall that the weights $\mathcal{A}$ and
$\mathcal{B}$ in Corollary 3.6 are only required to be continuous (with
compact support). Thus the smoothness restrictions on the functions $a$,
$b_{j,k}$ $\beta_{j,k}$ in the definition (4.2) can be relaxed, but for our
purposes it suffices to assume conditions (4.1). Moreover, this assumption
allows us to avoid unnecessary technical complications.
We use the representations
$(\hat{\mathbf{x}},x)=(\tilde{\mathbf{x}}_{k,N},x_{k},x)$ introduced in (1.8).
Denote
(4.3)
$\displaystyle\begin{cases}h(t)=\bigg{[}\sum_{j=1}^{N}\sum_{k=1}^{N-1}\int_{\mathbb{R}^{3N-6}}|b_{j,k}(\tilde{\mathbf{x}}_{k,N},t)\beta_{j,k}(\tilde{\mathbf{x}}_{k,N},t,t)|^{2}d\tilde{\mathbf{x}}_{k}\bigg{]}^{\frac{1}{2}},\
\textup{if}\ N\geq 3;\\\\[8.5359pt]
h(t)=\big{(}|b_{1,1}(t)\beta_{1,1}(t,t)|^{2}+|b_{2,1}(t)\beta_{2,1}(t,t)|^{2}\big{)}^{\frac{1}{2}},\
\textup{if}\ N=2.\end{cases}$
Let $X_{\infty}(\xi),\xi\in\mathbb{R}^{3}$, be the function defined by (3.13)
and (3.14).
###### Theorem 4.1.
Let $\mathcal{M}$ be the operator defined above, where
$\Phi\in\textup{{{C}}}^{\infty}(\mathbb{R}^{3}\setminus\\{0\\})$ is a
homogeneous vector function of order $\alpha>-5/2$. Then the operator
$\operatorname{{\sf Int}}(\mathcal{M})$ belongs to $\mathbf{S}_{p,\infty}$,
$1/p=1+\alpha/3$, and
(4.4) $\displaystyle{\sf{G}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}={\sf{g}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}=\frac{1}{24\pi^{3}}\int_{\mathbb{S}^{2}}|X_{\infty}(\omega)|^{p}\,d\omega\int_{\mathbb{R}^{3}}\big{(}|a(x)h(x)|\big{)}^{p}\,dx.$
Throughout the proof we assume that $N\geq 3$. For $N=2$ the argument
simplifies, and we omit it.
We begin the proof with the following lemma.
###### Lemma 4.2.
For each $j=1,2,\dots,N$ and each pair $k,l=1,2,\dots,N-1$, $k\not=l$, we have
$\displaystyle{\sf{G}}_{p/2}\big{(}\operatorname{{\sf
Int}}(\mathcal{M}_{j,k})^{*}\operatorname{{\sf
Int}}(\mathcal{M}_{j,l})\big{)}=0.$
###### Proof.
Fix a $j=1,2,\dots,N$ and write the kernel of the operator $\operatorname{{\sf
Int}}(\mathcal{M}_{j,k})^{*}\operatorname{{\sf Int}}(\mathcal{M}_{j,l})$:
$\displaystyle\mathcal{P}_{k,l}(x,y)=$ $\displaystyle\
\overline{a(x)}a(y)\int\overline{\mathcal{M}_{j,k}(\hat{\mathbf{x}},x)}\mathcal{M}_{j,l}(\hat{\mathbf{x}},y)d\hat{\mathbf{x}}$
$\displaystyle=$ $\displaystyle\
\overline{a(x)}a(y)\int\overline{\Phi(x-x_{k})}\cdot\Phi(x_{l}-y)\,\overline{b_{j,k}(\hat{\mathbf{x}})\beta_{j,k}(\hat{\mathbf{x}},x)}b_{j,l}(\hat{\mathbf{x}})\beta_{j,l}(\hat{\mathbf{x}},y)\,d\hat{\mathbf{x}}.$
Write $\hat{\mathbf{x}}=(\tilde{\mathbf{x}}_{l,N},x_{l})$,
$d\hat{\mathbf{x}}=d\tilde{\mathbf{x}}_{l,N}dx_{l}$ and change $x_{l}$ to
$x_{l}+y$, so that
$\displaystyle\mathcal{P}_{k,l}(x,y)=\overline{a(x)}a(y)\int\overline{\Phi(x-x_{k})}\cdot$
$\displaystyle\
\Phi(x_{l})\overline{b_{j,k}(\tilde{\mathbf{x}}_{l,N},x_{l}+y)\beta_{j,k}(\tilde{\mathbf{x}}_{j,N},x_{j}+y,x)}$
$\displaystyle\qquad\qquad\times
b_{j,l}(\tilde{\mathbf{x}}_{l,N},x_{l}+y)\beta_{j,l}(\tilde{\mathbf{x}}_{l,N},x_{l}+y,y)d\tilde{\mathbf{x}}_{l,N}dx_{l}.$
Because of the conditions (4.1) for all $x\in\mathbb{R}^{3}$ the kernel
$\mathcal{P}_{k,l}$ is a $\textup{{{C}}}^{\infty}_{0}$-function of
$y\in\mathbb{R}^{3}$. Hence by Proposition 3.3 the singular values of the
operator $\operatorname{{\sf Int}}(\mathcal{P}_{k,l})$ decay faster than any
negative power of their number. In particular,
${\sf{G}}_{p/2}\big{(}\operatorname{{\sf Int}}(\mathcal{P}_{k,l})\big{)}=0$,
as required. ∎
### 4.2. Proof of Theorem 4.1 for $\beta_{jk}=1$
First we prove Theorem 4.1 for the simpler case $\beta_{j,k}=1$. It follows
from Lemma 4.2 and from the inequality (3.4) that
$\displaystyle{\sf{G}}_{p/2}\bigg{(}\sum_{j=1}^{N}\sum_{k\not=l}\operatorname{{\sf
Int}}(\mathcal{M}_{j,k})^{*}\operatorname{{\sf
Int}}(\mathcal{M}_{j,l})\bigg{)}=0.$
By Corollary 3.1 this implies that
$\displaystyle{\sf{G}}_{p/2}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})^{*}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}={\sf{G}}_{p/2}\bigg{(}\sum_{j=1}^{N}\sum_{k=1}^{N-1}\operatorname{{\sf
Int}}(\mathcal{M}_{j,k})^{*}\operatorname{{\sf
Int}}(\mathcal{M}_{j,k})\bigg{)},$
and the same equality holds for the functional ${\sf{g}}_{p/2}$. Let us write
the kernel $\mathcal{F}(x,y)$ of the operator on the right-hand side,
remembering that $\beta_{j,k}=1$:
$\displaystyle\mathcal{F}(x,y)=\overline{a(x)}a(y)$ $\displaystyle\
\sum_{j=1}^{N}\sum_{k=1}^{N-1}\int\overline{\Phi(x-x_{k})}\cdot\Phi(x_{k}-y)|b_{j,k}(\hat{\mathbf{x}})|^{2}d\hat{\mathbf{x}}$
$\displaystyle=$ $\displaystyle\
\overline{a(x)}a(y)\sum_{j=1}^{N}\sum_{k=1}^{N-1}\int_{\mathbb{R}^{3}}\overline{\Phi(x-t)}\cdot\Phi(t-y)\int_{\mathbb{R}^{3N-6}}|b_{j,k}(\tilde{\mathbf{x}}_{k,N},t)|^{2}d\tilde{\mathbf{x}}_{k,N}dt$
$\displaystyle=$ $\displaystyle\
\overline{a(x)}a(y)\int_{\mathbb{R}^{3}}\overline{\Phi(x-t)}\cdot\Phi(t-y)\
h(t)^{2}dt,$
where the function
$\displaystyle
h(t)=\bigg{[}\sum_{j=1}^{N}\sum_{k=1}^{N-1}\int_{\mathbb{R}^{3N-6}}|b_{j,k}(\tilde{\mathbf{x}}_{k,N},t)|^{2}d\tilde{\mathbf{x}}_{k,N}\bigg{]}^{\frac{1}{2}}$
coincides with (4.3) for $\beta_{j,k}=1$. Define the vector-valued kernel
$\mathcal{G}$ by
$\displaystyle\mathcal{G}(x,y)=h(x)\Phi(x-y)a(y),$
so that $\operatorname{{\sf Int}}(\mathcal{F})=\operatorname{{\sf
Int}}(\mathcal{G})^{*}\operatorname{{\sf Int}}(\mathcal{G})$. Thus the
functionals ${\sf{G}}_{p/2}$ for the operators $\operatorname{{\sf
Int}}(\mathcal{M})^{*}\operatorname{{\sf Int}}(\mathcal{M})$ and
$\operatorname{{\sf Int}}(\mathcal{G})^{*}\operatorname{{\sf
Int}}(\mathcal{G})$ coincide with each other, and the same applies to the
functionals ${\sf{g}}_{p/2}$. Consequently, by virtue of (3.3),
(4.5) $\displaystyle{\sf{G}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}={\sf{G}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{G})\big{)},\quad{\sf{g}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}={\sf{g}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{G})\big{)}.$
Since $b_{j,k}\in\textup{{{C}}}^{\infty}_{0}$, the function $h$ belongs to
$\textup{{{C}}}_{0}$. Thus, to find ${\sf{G}}_{p}$ and ${\sf{g}}_{p}$ for the
operator $\operatorname{{\sf Int}}(\mathcal{G})$ we can apply Corollary 3.6
with $d=3$ and with the weights $b=h\in\textup{{{C}}}_{0}$ and
$a\in\textup{{{C}}}^{\infty}_{0}$, which gives
$\displaystyle{\sf{G}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{G})\big{)}={\sf{g}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{G})\big{)}=\frac{1}{24\pi^{3}}\int_{\mathbb{R}^{3}}\int_{\mathbb{S}^{2}}\big{(}|a(x)h(x)||X_{\infty}(\omega)|\big{)}^{p}\,d\omega
dx,$
with $1/p=1+\alpha/3$. By (4.5), this equality implies (4.4), which completes
the proof of Theorem 4.1 for $\beta_{j,k}=1$.
### 4.3. Proof of Theorem 4.1 for arbitrary
$\beta_{j,k}\in\textup{{{C}}}^{\infty}$
We reduce the general case to the one considered in Subsect. 4.2. Since
$b_{j,k}$ and $a$ are compactly supported, without loss of generality we may
assume that $\beta_{j,k}\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3N})$. For
each $j=1,2,\dots,N$ represent
$\displaystyle\mathcal{M}_{j}=\mathcal{A}_{j}+\sum_{k=1}^{N-1}\mathcal{F}_{j,k},$
where
$\displaystyle\mathcal{A}_{j}(\hat{\mathbf{x}},x)=$ $\displaystyle\
\sum_{k=1}^{N-1}\Phi(x_{k}-x)b_{j,k}(\hat{\mathbf{x}})a(x)\beta_{j,k}(\hat{\mathbf{x}},x_{j}),$
$\displaystyle\mathcal{F}_{j,k}(\hat{\mathbf{x}},x)=$ $\displaystyle\
\Phi(x_{k}-x)b_{j,k}(\hat{\mathbf{x}})a(x)\big{(}\beta_{j,k}(\hat{\mathbf{x}},x)-\beta_{j,k}(\hat{\mathbf{x}},x_{k})\big{)},\quad
j=1,2,\dots,N-1.$
Representing
$\displaystyle\beta_{j,k}(\hat{\mathbf{x}},x)-\beta_{j,k}(\hat{\mathbf{x}},x_{k})=(x-x_{k})\cdot\int_{0}^{1}\nabla_{x}\beta_{j,k}(\hat{\mathbf{x}},x_{k}+s(x-x_{k}))ds=:(x_{k}-x)\cdot\sigma_{j,k}(\hat{\mathbf{x}},x),$
we can rewrite $\mathcal{F}_{j,k}$ as
$\displaystyle\mathcal{F}_{j,k}(\hat{\mathbf{x}},x)=\Xi_{j,k}(\hat{\mathbf{x}},x)b_{j,k}(\hat{\mathbf{x}})a(x),\quad\textup{where}\quad\Xi_{j,k}(\hat{\mathbf{x}},x)=\Phi(x_{k}-x)\,\big{[}(x_{k}-x)\cdot\sigma_{j,k}(\hat{\mathbf{x}},x)\big{]}.$
Remembering that $\Phi$ is homogeneous of order $\alpha$ and that
$\sigma_{j,k}\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3N})$, we conclude
that
$\displaystyle\big{|}\partial^{m}_{x}\Xi_{j,k}(\hat{\mathbf{x}},x)\big{|}\lesssim|x-x_{j}|^{\alpha+1-|m|},\quad
m\in\mathbb{N}_{0}^{3}.$
Since $\sigma_{j,k}$ is compactly supported, the kernel
$\Xi_{j,k}(\hat{\mathbf{x}},x)$, as a function of $x\in\mathbb{R}^{3}$,
belongs to $\textup{{{H}}}^{l}(\mathbb{R}^{3})$ for all $0\leq l<\alpha+5/2$.
As $\alpha>-5/2$ the set of such values $l$ is non-empty. Moreover, the
$\textup{{{H}}}^{l}$-norm of the kernel, as a function of
$\hat{\mathbf{x}}\in\mathbb{R}^{3N-3}$, is uniformly bounded, and hence it
trivially belongs to
$\textup{{{L}}}^{2}(\mathbb{R}^{3N-3},|b_{j,k}(\hat{\mathbf{x}})|^{2}d\hat{\mathbf{x}})$.
By virtue of Proposition 3.3, we obtain that ${\sf{G}}_{q}(\operatorname{{\sf
Int}}(\mathcal{F}_{j,k}))=0$, $1/q=1/2+l/3$. Note that
$\displaystyle\frac{1}{q}\geq\frac{1}{p}=1+\frac{\alpha}{3},\quad\textup{for}\quad
l\geq\alpha+\frac{3}{2}.$
Consequently, taking $l$ to be the only non-negative integer in the interval
$[\alpha+3/2,\alpha+5/2)$, we conclude that ${\sf{G}}_{p}(\operatorname{{\sf
Int}}(\mathcal{F}_{j,k}))=0$, and, by (3.4),
$\displaystyle{\sf{G}}_{p}\bigg{(}\sum_{k=1}^{N-1}\operatorname{{\sf
Int}}(\mathcal{F}_{j,k})\bigg{)}=0.$
By Corollary 3.1,
(4.6) $\displaystyle{\sf{G}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}={\sf{G}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{A})\big{)},\quad{\sf{g}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}={\sf{g}}_{p}\big{(}\operatorname{{\sf
Int}}(\mathcal{A})\big{)},$
where $\mathcal{A}$ is the vector function with components
$\mathcal{A}_{j}(\hat{\mathbf{x}},x)$, $j=1,2,\dots,N$. To find ${\sf{G}}_{p}$
and ${\sf{g}}_{p}$ for the operator $\operatorname{{\sf Int}}(\mathcal{A})$,
we observe that each kernel $\mathcal{A}_{j}(\hat{\mathbf{x}},x)$ has the form
$\displaystyle\sum_{k=1}^{N-1}\Phi(x_{k}-x)\tilde{b}_{j,k}(\hat{\mathbf{x}})a(x)\quad\textup{with}\quad\tilde{b}_{j,k}(\hat{\mathbf{x}})=b_{j,k}(\hat{\mathbf{x}})\beta_{j,k}(\hat{\mathbf{x}},x_{k}),$
Using the result of Subsect 4.2 we obtain the formula (4.4) for the operator
$\operatorname{{\sf Int}}(\mathcal{A})$ with the function $h$ defined in
(4.3). In view of (4.6) this implies (4.4) for the operator
$\operatorname{{\sf Int}}(\mathcal{M})$, as claimed. ∎
###### Corollary 4.3.
Let $\Phi(x)$ be as in Example 3.7 with $\alpha=1,d=3$, i.e. $\Phi(x)=|x|$ and
$1/p=1+\alpha/d=4/3$. According to (4.4) and (3.17),
(4.7) $\displaystyle{\sf{G}}_{3/4}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}={\sf{g}}_{3/4}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}=\mu_{1,3}\int_{\mathbb{R}^{3}}\big{(}|a(x)h(x)|\big{)}^{\frac{3}{4}}\,dx,$
where $\mu_{\alpha,d}$ is defined in (3.17).
###### Corollary 4.4.
Let $\Phi(x)$ be as in Example 3.8 with $\alpha=0,d=3$, i.e.
$\Phi(x)=\nabla|x|=|x|^{-1}x$ and $1/p=1+\alpha/d=1$. According to (4.4) and
(3.18),
$\displaystyle{\sf{G}}_{1}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}={\sf{g}}_{1}\big{(}\operatorname{{\sf
Int}}(\mathcal{M})\big{)}=\nu_{0,3}\int_{\mathbb{R}^{3}}|a(x)h(x)|\,dx,$
where $\nu_{\alpha,d}$ is defined in (3.18).
In the current paper we need only Corollary 4.3. Corollary 4.4 is needed for
future use.
## 5\. Factorization of $\boldsymbol{\Gamma}$: operator $\boldsymbol{\Psi}$
### 5.1. Reformulation of the problem
Using the functionals (3.2), one can rewrite the sought formula (1.4) as
$\displaystyle{\sf{G}}_{3/8}(\boldsymbol{\Gamma})={\sf{g}}_{3/8}(\boldsymbol{\Gamma})=A.$
Since $\boldsymbol{\Gamma}=\boldsymbol{\Psi}^{*}\boldsymbol{\Psi}$ with the
operator
$\boldsymbol{\Psi}:\textup{{{L}}}^{2}(\mathbb{R}^{3})\to\textup{{{L}}}^{2}(\mathbb{R}^{3N-3})$
defined in (2.9), by (3.3) the above equalities rewrite as
(5.1)
$\displaystyle{\sf{G}}_{3/4}(\boldsymbol{\Psi})={\sf{g}}_{3/4}(\boldsymbol{\Psi})=A.$
Thus the main Theorem 2.3 can be recast as follows:
###### Theorem 5.1.
Under the conditions of Theorem 2.3 the formula (5.1) holds with the constant
$A$ which is defined in (2.7).
The rest of the paper is focused on the proof of Theorem 5.1.
As explained in the Introduction, at the heart of the proof is the formula
(2.5) for the function $\psi$, which translates to the representation (2.12)
for the kernels $\psi_{j}$ defined in (2.2). This representation allows us to
reduce the problem to the model operator considered in Sect. 4 with the
function $\Phi(x)=|x|$. At the first stage of this reduction we construct
$\textup{{{C}}}^{\infty}_{0}$ approximations of the functions
$\tilde{\xi}_{j,k}$ and $\tilde{\eta}_{j,k}$ from (2.12).
### 5.2. Cut-off functions
Firt we construct appropriate cut-offs. Fix a $\delta>0$. Along with the sets
(2.1) introduce
(5.2) $\displaystyle{\sf S}_{l,s}(\delta)={\sf
S}_{s,l}(\delta)=\\{\mathbf{x}\in\mathbb{R}^{3N}:|x_{l}-x_{s}|>\delta\\},\
0\leq l<s\leq N,$
and for all $k=0,1,\dots,N-1$, define
(5.3) $\displaystyle{\sf{U}}_{k}(\delta)=\bigg{(}\bigcap_{0\leq l<s\leq
N-1}{\sf
S}_{l,s}(\delta)\bigg{)}\bigcap\bigg{(}\bigcap_{\begin{subarray}{c}0\leq s\leq
N-1\\\ s\not=k\end{subarray}}{\sf S}_{s,N}(\delta)\bigg{)}.$
Comparing with (2.2) we see that ${\sf{U}}_{k}(\delta)\subset{\sf{U}}_{k,N}$,
and for $\mathbf{x}\in{\sf{U}}_{k}(\delta)$ all the coordinate pairs, except
for $x_{k}$ and $x_{N}$, are separated by a distance $\delta$. Similarly to
(2.3) define the diagonal set
$\displaystyle{\sf{U}}^{(\rm
d)}_{k}(\delta)=\\{\mathbf{x}\in{\sf{U}}_{k}(\delta):x_{j}=x_{N}\\}\subset{\sf{U}}_{k,N}^{(\operatorname{d})}.$
Recall that the representation (2.12) holds on the domain
$\tilde{\Omega}_{j,k}$ which satisfies (2.10) for all $j=1,2,\dots,N$,
$k=0,1,\dots,N-1$. We construct a compact subset of $\tilde{\Omega}_{j,k}$ in
the following way. For $R>0$ let
$\displaystyle{\sf{U}}_{k}(\delta,R)=$ $\displaystyle\
{\sf{U}}_{k}(\delta)\bigcap\ (B_{R})^{N},$ $\displaystyle{\sf{U}}^{(\rm
d)}_{k}(\delta,R)=$ $\displaystyle\
\\{\mathbf{x}\in{\sf{U}}_{k}(\delta,R):x_{k}=x_{N}\\},$
where $B_{R}=\\{x\in\mathbb{R}^{3}:|x|<R\\}$. The set ${\sf{U}}^{(\rm
d)}_{k}(\delta,R)$ is bounded and its closure belongs to
$\tilde{\Omega}_{j,k}$ for all $\delta>0,R>0$. Therefore, there exists an
$\varepsilon_{0}=\varepsilon_{0}(\delta,R)>0$ such that the
$\varepsilon$-neighbourhood
(5.4)
$\displaystyle\tilde{\Omega}_{k}(\delta,R,\varepsilon):=\\{\mathbf{x}\in{\sf{U}}_{k}(\delta,R):|x_{k}-x_{N}|<\varepsilon\\},$
together with its closure, belongs to $\tilde{\Omega}_{j,k}$ for all
$\varepsilon\in(0,\varepsilon_{0})$:
(5.5)
$\displaystyle\overline{\tilde{\Omega}_{k}(\delta,R,\varepsilon)}\subset\tilde{\Omega}_{j,k},\quad\forall\varepsilon\in(0,\varepsilon_{0}).$
Now we specify $\textup{{{C}}}^{\infty}_{0}$ cutoffs supported on the domains
$\tilde{\Omega}_{k}(\delta,R,\varepsilon)$. Let
$\theta\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R})$ and $\zeta=1-\theta$ be as
defined in (1.9), (1.10). Denote
(5.6) $\displaystyle Y_{\delta}(\hat{\mathbf{x}})=\prod_{0\leq l<s\leq
N-1}\zeta\big{(}|x_{l}-x_{s}|(4\delta)^{-1}\big{)}.$
By the definition of $\zeta$,
(5.7) $\displaystyle\operatorname{{supp}}Y_{\delta}\subset\bigcap_{0\leq
l<s\leq N-1}{\sf S}_{l,s}(2\delta),$
where ${\sf S}_{l,s}(\ \cdot\ )$ is defined in (5.2). Define also cut-offs at
infinity. Denote
(5.8) $\displaystyle Q_{R}(\hat{\mathbf{x}})=\prod_{1\leq l\leq
N-1}\theta\big{(}|x_{l}|R^{-1}\big{)},\quad
K_{R}(x)=\theta\big{(}|x|R^{-1}\big{)}.$
###### Lemma 5.2.
Let $\tilde{\Omega}_{k}(\delta,R,\varepsilon)$ be the set introduced in (5.4).
Then for all $\varepsilon<\min\\{\varepsilon_{0},\delta\\}$ the support of the
function
(5.9) $\displaystyle
Q_{R}(\hat{\mathbf{x}})K_{R}(x)Y_{\delta}(\hat{\mathbf{x}})\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}$
belongs to $\tilde{\Omega}_{k}(\delta,R,\varepsilon)$ for all
$k=0,1,\dots,N-1$.
###### Proof.
Assume that $\mathbf{x}$ belongs to the support of the function (5.9). In view
of (5.7), for such $\mathbf{x}$ we have
(5.10) $\displaystyle|x_{l}-x_{s}|>2\delta,\ 0\leq l<s\leq
N-1,\quad\textup{and}\quad|x-x_{k}|<\varepsilon.$
As $\varepsilon<\delta$, for all $s=0,1,\dots,N-1$, $s\not=k$, we can write
$\displaystyle|x-x_{s}|\geq|x_{k}-x_{s}|-|x-x_{k}|>2\delta-\varepsilon>\delta,$
By definition (5.3), together with (5.10) this gives
$\mathbf{x}\in{\sf{U}}_{k}(\delta)$. Moreover, since
$\operatorname{{supp}}(Q_{R}K_{R})\subset(B_{R})^{N}$, this means that
$\mathbf{x}\in{\sf{U}}_{k}(\delta,R)$. Now the claimed inclusion follows from
the definition (5.4). ∎
### 5.3.
Using the cut-offs introduced above we construct a convenient approximation
for the kernels $\psi_{j}(\hat{\mathbf{x}},x)$. Taking if necessary, a smaller
$\varepsilon_{0}$ in (5.5), we will assume that
$\varepsilon_{0}(\delta,R)\leq\delta$, and hence for all
$\varepsilon<\varepsilon_{0}(\delta,R)$, apart from the inclusion (5.5) we
have Lemma 5.2. Thus, for these values of $\varepsilon$ the real analytic
functions $\tilde{\xi}_{j,k},\tilde{\eta}_{j,k}$ are well-defined on the
support of (5.9), and hence the kernel
(5.11)
$\displaystyle\Upsilon_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})K_{R}(x)\sum_{k=1}^{N-1}\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}|x-x_{k}|\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x),$
is well-defined for all $(\hat{\mathbf{x}},x)\in\mathbb{R}^{3N}$, and each of
the functions
$\displaystyle
Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})K_{R}(x)\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x),\quad
k=1,2,\dots,N-1,$
is $\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3N})$. Our objective is to prove
that the vector-valued kernel
$\displaystyle\boldsymbol{\Upsilon}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=\big{\\{}\Upsilon_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)\big{\\}}_{j=1}^{N}$
is an approximation for $\boldsymbol{\Psi}(\hat{\mathbf{x}},x)$(see (2.9)) in
the following sense.
###### Lemma 5.3.
The following relations hold:
$\displaystyle{\sf{G}}_{3/4}(\boldsymbol{\Psi})=\lim\limits_{\begin{subarray}{c}\delta\to
0\\\ R\to\infty\end{subarray}}\lim_{\varepsilon\to
0}{\sf{G}}_{3/4}\big{(}\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])\big{)},\quad{\sf{g}}_{3/4}(\boldsymbol{\Psi})=\lim\limits_{\begin{subarray}{c}\delta\to
0\\\ R\to\infty\end{subarray}}\lim_{\varepsilon\to
0}{\sf{g}}_{3/4}\big{(}\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])\big{)},$
where the limits on the right-hand side exist.
The proof of this lemma is given in the next section.
## 6\. Proof of Lemma 5.3
### 6.1. Spectral estimates for $\boldsymbol{\Psi}$
Our proof of Lemma 5.3 relies on the bounds obtained in [22]. Let
$\mathcal{C}_{n}=(0,1)^{3}+n$, $n\in\mathbb{Z}^{3}$. Assume that
$b\in\textup{{{L}}}^{\infty}(\mathbb{R}^{3N-3})$ and that
$a\in\textup{{{L}}}^{2}_{\textup{\tiny loc}}(\mathbb{R}^{3})$ is such that
$\displaystyle\sup_{n\in\mathbb{Z}^{3}}\|a\|_{\textup{{{L}}}^{2}(\mathcal{C}_{n})}<\infty.$
Then the functionals
$\displaystyle
S_{\varkappa}(a)=\bigg{[}\sum_{n\in\mathbb{Z}^{3}}e^{-\frac{3}{4}\varkappa|n|}\|a\|_{\textup{{{L}}}^{2}(\mathcal{C}_{n})}^{\frac{3}{4}}\bigg{]}^{\frac{4}{3}}$
and
$\displaystyle
M_{\varkappa}(b)=\biggl{[}\int_{\mathbb{R}^{3N-3}}|b(\hat{\mathbf{x}})|^{2}e^{-2\varkappa|\hat{\mathbf{x}}|}d\hat{\mathbf{x}}\biggr{]}^{\frac{1}{2}},$
are both finite for all $\varkappa>0$. Recall that the functional
${\sf{G}}_{p}$ is defined in (3.2), and $\psi_{j}$ – in (2.2). The next bound
for the operators $b\,\operatorname{{\sf Int}}(\psi_{j})a$ follows from [22,
Theorem 3.1].
###### Proposition 6.1.
Assume that $\psi$ satisfies (1), and let $j=1,2,\dots,N$. Let the functions
$a$ and $b$ be as described above. Then $b\,\operatorname{{\sf
Int}}(\psi_{j})a\in\mathbf{S}_{3/4,\infty}$ and for some
$\varkappa\leq\varkappa_{0}$ we have
(6.1) $\displaystyle{\sf{G}}_{3/4}(b\,\operatorname{{\sf
Int}}(\psi_{j})a)\lesssim\big{(}M_{\varkappa}(b)S_{\varkappa}(a)\big{)}^{\frac{3}{4}}.$
### 6.2. Proof of Lemma 5.3
The strategy of the proof is to “trim down” the kernel (2.9) in several steps,
by multiplying it by appropriate cut-offs including the functions (5.6) and
(5.8), or dropping some of the components, until it reduces to the kernel
(5.11). At every step of this process we justify the trimming using either
Corollary 3.1 or Corollary 3.2.
The first stage is described in the next lemma.
###### Lemma 6.2.
The following relations hold:
(6.2)
$\displaystyle{\sf{G}}_{3/4}(\boldsymbol{\Psi})=\lim\limits_{\begin{subarray}{c}\delta\to
0\\\
R\to\infty\end{subarray}}{\sf{G}}_{3/4}(Q_{R}Y_{\delta}\boldsymbol{\Psi}K_{R}),\quad{\sf{g}}_{3/4}(\boldsymbol{\Psi})=\lim\limits_{\begin{subarray}{c}\delta\to
0\\\
R\to\infty\end{subarray}}{\sf{g}}_{3/4}(Q_{R}Y_{\delta}\boldsymbol{\Psi}K_{R}),$
where the limits on the right-hand side exist.
###### Proof.
First we check that
(6.3) $\displaystyle\begin{cases}\lim\limits_{\delta\to
0}{\sf{G}}_{3/4}\big{(}(I-Y_{\delta})\boldsymbol{\Psi}\big{)}=0,\\\\[8.5359pt]
\lim\limits_{R\to\infty}{\sf{G}}_{3/4}\big{(}(I-Q_{R})\boldsymbol{\Psi}\big{)}=0,\
\lim\limits_{R\to\infty}{\sf{G}}_{3/4}\big{(}\boldsymbol{\Psi}(I-K_{R})\big{)}=0.\end{cases}$
It suffices to check the above relations for each operator $\operatorname{{\sf
Int}}(\psi_{j})$, $j=1,2,\dots,N$. Consider first
$(I-Y_{\delta})\operatorname{{\sf Int}}(\psi_{j})$. Since
$\displaystyle 1-Y_{\delta}(\hat{\mathbf{x}})\leq\sum_{0\leq l<s\leq
N-1}\theta\big{(}|x_{l}-x_{s}|(4\delta)^{-1}\big{)},$
it follows from (6.1) that
$\displaystyle{\sf{G}}_{3/4}\big{(}(1-Y_{\delta})\operatorname{{\sf
Int}}(\psi_{j})\big{)}\lesssim$ $\displaystyle\
\big{(}M_{\varkappa}(1-Y_{\delta})\big{)}^{3/4}$ $\displaystyle\lesssim$
$\displaystyle\ \sum_{0\leq l<s\leq
N-1}\bigg{[}\int\theta\big{(}|x_{l}-x_{s}|(4\delta)^{-1}\big{)}^{2}e^{-2\varkappa|\hat{\mathbf{x}}|}d\hat{\mathbf{x}}\bigg{]}^{3/8}\lesssim\delta^{9/8}\to
0,\ \delta\to 0,$
and hence the first relation in (6.3) holds.
In a similar way one estimates $(I-Q_{R})\operatorname{{\sf Int}}(\psi_{j})$
and $\operatorname{{\sf Int}}(\psi_{j})(I-K_{R})$. Estimate, for example, the
first of these operators. Since
$\displaystyle 1-Q_{R}(\hat{\mathbf{x}})\leq\sum_{1\leq l\leq
N-1}\zeta\big{(}|x_{l}|R^{-1}\big{)},$
it follows from (6.1) again that
$\displaystyle{\sf{G}}_{3/4}\big{(}(I-Q_{R})\operatorname{{\sf
Int}}(\psi_{j})\big{)}\lesssim$ $\displaystyle\
\big{(}M_{\varkappa}(1-Q_{R})\big{)}^{3/4}$ $\displaystyle\lesssim$
$\displaystyle\ \sum_{0\leq l\leq
N-1}\bigg{[}\int_{\mathbb{R}^{3N-3}}\zeta(|x_{l}|R^{-1})^{2}e^{-2\varkappa|\hat{\mathbf{x}}|}\,d\hat{\mathbf{x}}\bigg{]}^{3/8}\lesssim
e^{-3\varkappa R/8}\to 0,\ R\to\infty,$
whence the second equality in (6.3).
Represent $\boldsymbol{\Psi}$ in the form
$\displaystyle\boldsymbol{\Psi}=Q_{R}Y_{d}\boldsymbol{\Psi}K_{R}+(I-Q_{R})\boldsymbol{\Psi}+Q_{R}(1-Y_{\delta})\boldsymbol{\Psi}+Q_{R}Y_{\delta}\boldsymbol{\Psi}(I-K_{R}),$
According to (3.4),
$\displaystyle{\sf{G}}_{3/4}\big{(}\boldsymbol{\Psi}-Q_{R}Y_{d}\boldsymbol{\Psi}K_{R}\big{)}^{\frac{{\scaleto{3}{3pt}}}{{\scaleto{7}{3pt}}}}\leq$
$\displaystyle\
{\sf{G}}_{3/4}\big{(}(I-Q_{R})\boldsymbol{\Psi}\big{)}^{\frac{{\scaleto{3}{3pt}}}{{\scaleto{7}{3pt}}}}$
$\displaystyle\
+{\sf{G}}_{3/4}\big{(}Q_{R}(1-Y_{\delta})\boldsymbol{\Psi}\big{)}^{\frac{{\scaleto{3}{3pt}}}{{\scaleto{7}{3pt}}}}+{\sf{G}}_{3/4}\big{(}Q_{R}Y_{\delta}\boldsymbol{\Psi}(I-K_{R})\big{)}^{\frac{{\scaleto{3}{3pt}}}{{\scaleto{7}{3pt}}}}$
$\displaystyle\leq$ $\displaystyle\
{\sf{G}}_{3/4}\big{(}(I-Q_{R})\boldsymbol{\Psi}\big{)}^{\frac{{\scaleto{3}{3pt}}}{{\scaleto{7}{3pt}}}}$
$\displaystyle\
+{\sf{G}}_{3/4}\big{(}(1-Y_{\delta})\boldsymbol{\Psi}\big{)}^{\frac{{\scaleto{3}{3pt}}}{{\scaleto{7}{3pt}}}}+{\sf{G}}_{3/4}\big{(}\boldsymbol{\Psi}(I-K_{R})\big{)}^{\frac{{\scaleto{3}{3pt}}}{{\scaleto{7}{3pt}}}}.$
By virtue of (6.3) the right-hand side tends to zero as $\delta\to
0,R\to\infty$. By Corollary 3.2 this implies (6.2). ∎
At the next stage we partition the kernel
(6.4) $\displaystyle
Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\boldsymbol{\Psi}(\hat{\mathbf{x}},x)K_{R}(x)$
of the operator $Q_{R}Y_{\delta}\boldsymbol{\Psi}K_{R}$ on the right-hand side
of the formulas (6.2). We do this by introducing the cut-offs
$\theta\big{(}|x-~{}x_{k}|\varepsilon^{-1}\big{)}$, $k=0,1,\dots,N-1$,
assuming that $\varepsilon<\delta$. In view of the definition (5.6) it is
straightforward to check that under this condition, we have
$\displaystyle
Y_{\delta}(\hat{\mathbf{x}})\sum_{k=0}^{N-1}\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}+Y_{\delta}(\hat{\mathbf{x}})\prod_{k=0}^{N-1}\zeta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}=Y_{\delta}(\hat{\mathbf{x}}),$
and hence the $j$’th component of (6.4) can be represented as follows:
(6.5) $\displaystyle
Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\psi_{j}(\hat{\mathbf{x}},x)K_{R}(x)=\sum_{k=0}^{N-1}\phi_{j,k}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)+\tau_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)$
with
$\displaystyle\phi_{j,k}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=$
$\displaystyle\
Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\theta\big{(}|x-~{}x_{k}|\varepsilon^{-1}\big{)}\psi_{j}(\hat{\mathbf{x}},x)K_{R}(x),\quad
k=0,1,\dots,N-1,$
$\displaystyle\tau_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=$
$\displaystyle\
Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\prod_{k=0}^{N-1}\zeta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}\psi_{j}(\hat{\mathbf{x}},x)K_{R}(x).$
First we show that the kernels $\tau_{j}[\delta,R,\varepsilon]$ and
$\phi_{j,0}[\delta,R,\varepsilon]$ give negligible contributions to the
asymptotics.
###### Lemma 6.3.
For each $\delta>0,R>0$ and $\varepsilon<\delta$ one has
(6.6) $\displaystyle{\sf{G}}_{3/4}\big{(}\operatorname{{\sf
Int}}\big{(}\tau_{j}[\delta,R,\varepsilon]\big{)}\big{)}=0,\ j=1,2,\dots,N.$
###### Proof.
By the definitions (5.6) and (1.10), the support of the kernel
$\tau_{j}[\delta,R,\varepsilon]$ belongs to the bounded domain
$\displaystyle{\bigcap_{0\leq l<s\leq N}{\sf
S}_{l,s}(\varepsilon/2)\cap(B_{R})^{N}}.$
The function $\psi_{j}$ is real-analytic on this domain and it is uniformly
bounded together with all its derivatives, so that
$\tau_{j}[\delta,R,\varepsilon]\in\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3N})$.
By Proposition 3.3, ${\sf{G}}_{p}(\operatorname{{\sf
Int}}(\tau_{j}[\delta,\mathbb{R},\varepsilon]))=0$ for all $p>0$, and in
particular, for $p=3/4$, as claimed. ∎
###### Lemma 6.4.
For each $\delta>0,R>0$ one has
(6.7) $\displaystyle\lim_{\varepsilon\to
0}{\sf{G}}_{3/4}\big{(}\operatorname{{\sf
Int}}\big{(}\phi_{j,0}[\delta,R,\varepsilon]\big{)}\big{)}=0,\ j=1,2,\dots,N.$
###### Proof.
As $x_{0}=0$ by definition, the kernel $\phi_{j,0}[\delta,R,\varepsilon]$ has
the form
$\displaystyle\phi_{j,0}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\psi_{j}(\hat{\mathbf{x}},x)\theta\big{(}|x|\varepsilon^{-1}\big{)}K_{R}(x).$
Estimating $Q_{R}Y_{\delta}\leq 1$, $K_{R}\leq 1$, one sees that the singular
values of $\operatorname{{\sf Int}}(\phi_{j,0}[\delta,R,\varepsilon])$ do not
exceed those of the operator $\operatorname{{\sf Int}}(\psi_{j})a$ with the
weight $a(x)=\theta(|x|\varepsilon^{-1})$. By (6.1),
$\displaystyle{\sf{G}}_{3/4}(\operatorname{{\sf Int}}(\psi_{j})a)\lesssim
S_{\varkappa}(a)^{3/4}\lesssim\bigg{(}\int_{\mathbb{R}^{3}}\theta\big{(}|x|\varepsilon^{-1}\big{)}^{2}dx\bigg{)}^{3/8}\lesssim\varepsilon^{9/8}\to
0,\ \varepsilon\to 0.$
This implies (6.7). ∎
###### Corollary 6.5.
Denote by
$\boldsymbol{\alpha}[\delta,R,\varepsilon](\mathbf{x},x)=\\{\alpha_{j}[\delta,R,\varepsilon]\\}_{j=1}^{N}$
the vector-valued kernel with the components
$\displaystyle\alpha_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=\sum_{k=1}^{N-1}\phi_{j,k}[\delta,R,\varepsilon](\hat{\mathbf{x}},x).$
Then for all $\delta>0$ and $R>0$, we have
(6.8)
$\displaystyle\begin{cases}{\sf{G}}_{3/4}(Q_{R}Y_{\delta}\boldsymbol{\Psi}K_{R})=&\
\lim\limits_{\varepsilon\to 0}{\sf{G}}_{3/4}(\operatorname{{\sf
Int}}(\boldsymbol{\alpha}[\delta,R,\varepsilon])),\\\\[5.69046pt]
{\sf{g}}_{3/4}(Q_{R}Y_{\delta}\boldsymbol{\Psi}K_{R})=&\
\lim\limits_{\varepsilon\to 0}{\sf{g}}_{3/4}(\operatorname{{\sf
Int}}(\boldsymbol{\alpha}[\delta,R,\varepsilon])),\end{cases}$
where the limits on the right-hand side exist.
###### Proof.
By (6.5), the kernel $Q_{R}Y_{\delta}\psi_{j}K_{R}$ has the form
$\displaystyle\alpha_{j}[\delta,R,\varepsilon]+\phi_{j,0}[\delta,R,\varepsilon]+\tau_{j}[\delta,R,\varepsilon].$
By virtue of (3.4) and (6.6), (6.7), we have
$\displaystyle\lim_{\varepsilon\to 0}{\sf{G}}_{3/4}(\operatorname{{\sf
Int}}\big{(}\phi_{j,0}[\delta,R,\varepsilon]+\tau_{j}[\delta,R,\varepsilon]\big{)}=0.$
Now (6.8) follows from Corollary 3.2. ∎
###### Completion of the proof of Lemma 5.3.
According to Lemma 5.2, under the condition
$\varepsilon<\varepsilon_{0}(\delta,R)$, the support of each kernel
$\displaystyle\phi_{j,k}[\delta,R,\varepsilon],\quad j=1,2,\dots,N,\quad
k=1,2,\dots,N-1,$
belongs to $\tilde{\Omega}_{k}(\delta,R,\varepsilon)$, see (5.4) for the
definition. Therefore one can use the representation (2.12) for the function
$\psi_{j}$:
$\displaystyle\alpha_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=\sum_{k=1}^{N-1}$
$\displaystyle\
\phi_{j,k}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)=\sum_{k=1}^{N-1}Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}\tilde{\xi}_{j,k}(\hat{\mathbf{x}},x)K_{R}(x)$
$\displaystyle\
\quad+\sum_{k=1}^{N-1}Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}|x_{k}-x|\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x)K_{R}(x).$
Each term in the first sum on the right-hand side is
$\textup{{{C}}}^{\infty}_{0}(\mathbb{R}^{3N})$. Thus, by Proposition 3.3, the
functional ${\sf{G}}_{p}$ for the associated operator equals zero for all
$p>0$, and in particular, for $p=3/4$. The second sum coincides with the
kernel $\Upsilon_{j}[\delta,R,\varepsilon](\hat{\mathbf{x}},x)$, defined in
(5.11). Therefore, by Corollary 3.1,
(6.9) $\displaystyle\begin{cases}{\sf{G}}_{3/4}(\operatorname{{\sf
Int}}(\boldsymbol{\alpha}[\delta,R,\varepsilon]))={\sf{G}}_{3/4}(\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])),\\\\[5.69046pt]
{\sf{g}}_{3/4}(\operatorname{{\sf
Int}}(\boldsymbol{\alpha}[\delta,R,\varepsilon]))={\sf{g}}_{3/4}(\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])),\end{cases}$
for each $\delta>0,R>0$ and $\varepsilon<\varepsilon_{0}(\delta,R)$. Putting
together (6.2), (6.8) and (6.9), and using Corollary 3.2, we conclude the
proof of Lemma 5.3. ∎
## 7\. Proof of Theorems 2.2 and 5.1, 2.3
###### Lemma 7.1.
The operator $\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])$ belongs to
$\mathbf{S}_{3/4,\infty}$ for all
$\delta>0,R>0,\varepsilon<\varepsilon_{0}(\delta,R)$ and
(7.1) $\displaystyle{\sf{G}}_{3/4}\big{(}\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])\big{)}={\sf{g}}_{3/4}\big{(}\operatorname{{\sf
Int}}(\boldsymbol{\Upsilon}[\delta,R,\varepsilon])\big{)}=\mu_{1,3}\int\big{(}K_{R}(t)H_{\delta,R}(t)\big{)}^{\frac{3}{4}}dt,$
where
$\displaystyle
H_{\delta,R}(t)=Q_{R}(t)Y_{\delta}(t)\,\big{(}|\tilde{\eta}_{1,1}(t,t)|^{2}+\tilde{\eta}_{1,2}(t,t)|^{2}\big{)}^{1/2},\
\textup{if}\ N=2,$
and
(7.2) $\displaystyle
H_{\delta,R}(t)=\bigg{[}\sum_{j=1}^{N}\sum_{k=1}^{N-1}\int_{\mathbb{R}^{3N-6}}\big{|}Q_{R}(\tilde{\mathbf{x}}_{k,N},t)Y_{\delta}(\tilde{\mathbf{x}}_{k,N},t)\tilde{\eta}_{j,k}(\tilde{\mathbf{x}}_{k,N},t,t)\big{|}^{2}d\tilde{\mathbf{x}}_{k,N}\bigg{]}^{\frac{1}{2}},\
\textup{if}\ N\geq 3,$
and $\mu_{\alpha,d}$ is defined in (3.17).
###### Proof.
The kernel $\boldsymbol{\Upsilon}[\delta,R,\varepsilon]$ (see (5.11)) has the
form (4.2) with
$\displaystyle a(x)=K_{2R}(x),\ $ $\displaystyle\
b_{j,k}(\hat{\mathbf{x}})=Q_{2R}(\hat{\mathbf{x}})Y_{\delta/2}(\hat{\mathbf{x}}),$
$\displaystyle\beta_{j,k}(\hat{\mathbf{x}},x)=$ $\displaystyle\
\theta\big{(}|x-x_{k}|\varepsilon^{-1}\big{)}\tilde{\eta}_{j,k}(\hat{\mathbf{x}},x)Q_{R}(\hat{\mathbf{x}})Y_{\delta}(\hat{\mathbf{x}})K_{R}(x),$
and the homogeneous function $\Phi(x)=|x|$. Here we have used the fact that
$\displaystyle
Q_{R}(\hat{\mathbf{x}})Q_{2R}(\hat{\mathbf{x}})=Q_{R}(\hat{\mathbf{x}}),\quad
Y_{\delta}(\hat{\mathbf{x}})Y_{\delta/2}(\hat{\mathbf{x}})=Y_{\delta}(\hat{\mathbf{x}})\quad\textup{and}\quad
K_{R}(x)K_{2R}(x)=K_{R}(x).$
Therefore we can use Corollary 4.3. It is immediate to see that in this case
the function $h$ defined in (4.3), coincides with $H_{\delta,R}$, so that
(4.7) entails (7.1), as required. ∎
###### Proof of Theorems 2.2, 5.1 and 2.3.
By Lemma 5.3, each term in the relation (7.1) has a limit as $\delta\to
0,R\to\infty$. Therefore the integral on the right-hand side of (7.1) is
bounded uniformly in $\delta>0,R>0$. Assume for convenience that the function
$\theta$ defined in (1.10) is monotone decreasing for $t\geq 0$. Therefore the
pointwise convergencies
$\displaystyle Y_{\delta}(\tilde{\mathbf{x}}_{k,N},t)\to 1,\ \delta\to
0\quad\textup{and}\quad K_{R}(t)\to 1,Q_{R}(\tilde{\mathbf{x}}_{k,N},t)\to 1,\
R\to\infty,$
are monotone increasing. By the Monotone Convergence Theorem, the integrand
$K_{R}(t)H_{\delta,R}(t)$ on the right-hand side of (7.1) converges for a.e.
$t\in\mathbb{R}^{3}$ as $\delta\to 0,R\to\infty$ to an
$\textup{{{L}}}^{3/4}(\mathbb{R})$-function, which we denote by
$\tilde{H}(t)$, and the integral in (7.1) converges to
(7.3) $\displaystyle\mu_{1,3}\int\big{(}\tilde{H}(t)\big{)}^{3/4}dt.$
If $N=2$, then this concludes the proof of Theorem 2.2, since in this case
$\displaystyle
H_{\delta,R}(t)\to\big{(}|\tilde{\eta}_{1,1}(t,t)|^{2}+\tilde{\eta}_{2,1}(t,t)|^{2}\big{)}^{1/2},$
a.e. $t\in\mathbb{R}^{3}$, and by virtue of (2.13) this limit coincides with
$H(t)$.
If $N\geq 3$, then the convergence to $\tilde{H}(t)$ implies that for a.e.
$t\in\mathbb{R}^{3}$ the function $K_{R}(t)H_{\delta,R}(t)$, and hence
$H_{\delta,R}(t)$, is bounded uniformly in $\delta$ and $R$. Applying the
Monotone Convergence Theorem to the integral (7.2), we conclude that the
a.e.-limit
$\displaystyle|\tilde{\eta}_{j,k}(\tilde{\mathbf{x}}_{k,N},t,t)|=\lim_{\delta\to
0,R\to\infty}\big{|}Q_{R}(\tilde{\mathbf{x}}_{k,N},t)Y_{\delta}(\tilde{\mathbf{x}}_{k,N},t)\tilde{\eta}_{j,k}(\tilde{\mathbf{x}}_{k,N},t,t)\big{|},\
$
belongs to $\textup{{{L}}}^{2}(\mathbb{R}^{3N-6})$, a.e. $t\in\mathbb{R}^{3}$,
and
$\displaystyle\lim_{\delta\to
0,R\to\infty}H_{\delta,R}(t)=H(t),\quad\textup{a.e.}\quad t\in\mathbb{R}^{3},$
where we have used the formula (2.13) for $H$. Thus
$H=\tilde{H}\in\textup{{{L}}}^{3/4}(\mathbb{R}^{3})$. As (2.13) is equivalent
to (2.6), this completes the proof of Theorem 2.2.
An easy calculation shows that $\mu_{1,3}=3^{-1}(2/\pi)^{5/4}$, so that the
limit (7.3) coincides with the coefficient $A$ in (2.7). Together with Lemma
5.3 this completes the proof of Theorem 5.1. As explained before, Theorem 5.1
is equivalent to Theorem 2.3. This completes the proof. ∎
Acknowledgments. The author is grateful to S. Fournais, T. Hoffmann-Ostenhof,
M. Lewin and T. Ø. Sørensen for stimulating discussions and advice. The author
thanks J. Cioslowski for his comments and for bringing to the author’s
attention papers [6], [7] and [15].
The author was supported by the EPSRC grant EP/P024793/1.
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|
# Quantum phenomena inside a black hole: quantization of the scalar field
iniside horizon in Schwarzschild spacetime
Pawel Gusin1, Andrzej Radosz1, Andy T. Augousti2 Janos Polonyi3, Oleg B.
Zaslavskii4 and Romuald J. Ściborski5
1Department of Quantum Technologies,
Wroclaw University of Science and Technology, Wroclaw, Poland
2Faculty of Science, Engineering and Computing,
Kingston University London, London, UK
3CNRS-IPHC, Strasbourg University, 23 r. du Loess BP28 67037,
Strasbourg Cedex 2, France
4Department of Physics and Technology, Kharkov V.N. Karazin
National University, 4 Svoboda Square, Kharkov 61022, Ukrainey
5Jaramogi Oginga Odinga University of Science and Technology in
Bondo (Kenya)
###### Abstract
We discuss the problem of the quantization and dynamic evolution of a scalar
free field in the interior of a Schwarzschild black hole. A unitary approach
to the dynamics of the quantized field is proposed: a time-dependent
Hamiltonian governing the Heisenberg equations is derived. It is found that
the system is represented by a set of harmonic oscillators coupled via terms
corresponding to the creation and annihilation of pairs of particles and that
the symmetry properties of the spacetime, homogeneity and isotropy are obeyed
by the coupling terms in the Hamiltonian. It is shown that Heisenberg
equations for annihilation and creation operators are transformed into
ordinary differential equations for appropriate Bogolyubov coefficients. Such
a formulation leads to a general question concerning the possibility of
gravitationally driven instability, that is however excluded in this case.
## 1 Introduction
The horizon of a black hole (BH) may be regarded as a geometrical singularity
(”fake geometrical singularity”). Indeed, considering a Schwarzschild BH in
Schwarzschild coordinates one finds the metric tensor exhibiting an on-horizon
singularity that is absent in other, singularity-free coordinate systems.
There are a variety of singularity-free coordinate systems in this case, e.g.
Kruskal-Szekers, Eddington-Finkelstein, Novikov and others [1-2]. Two
interesting observations might be made here. The first is to notice that the
presence of the event horizon is manifested both in coordinates revealing the
horizon’s singularity as well as in the singularity-free systems. The second
one is to note the surprising similarities and/or analogies for phenomena
taking place outside and inside black holes. A rather well-known example of
such a property is the so-called BSW effect [3]. Two-particle collisions
occuring in the vicinity of the black hole’s horizon may lead to a high-energy
outcome according to two scenarios [4-5]. These two scenarios turn out to be
the same in the exterior as well as in the interior of BH. A variety of other
aspects of the Exterior vs Interior (a)symmetry have been discussed in Ref.
[6].
It was shown by Doran et al. [7] that the interior of a Schwarzschild BH’s,
which is a dynamically changing spacetime, may be regarded as a solution of
Einstein’s equation. This interior spacetime, also called ”T-sphere” (see [8])
which is globally hyperbolic, gains then the status of a cosmological model.
Its 3D spatial-like section is a hypercylinder $\mathbf{R}^{1}\times S^{2}$,
expanding longitudinally, along the homogeneity direction $\mathbf{R}^{1},$
(see also [6-8]) and contracting transversally, perpendicularly to this
direction in the angular coordinates of the sphere $S^{2}$. However, as shown
in Ref. [7], such a process may be preceded by a process of expansion of the
sphere and collapsing of the cylinder to its base sphere of radius $r_{S}$.
Such an expansion followed by a contraction constitutes the full cycle for the
cosmological model introduced in [7].
Various phenomena and processess have been considered both in the interior of
the Schwarzschild BH [8, 10-13] and in its extension [7] to which we will
hereafter refer to as the ”T-model”, an anisotropic cosmological model. In
particular the Yang-Mills and Higgs fields in the Kantowski-Sachs anisotropic,
cigar-like – referred to above as a hypercylinder – cosmological model were
discussed in [14] (see also [15]). Canonical quantization of the scalar field
inside a Schwarzschild BH was presented by Yajnik and Narayan [16], where a
so-called tortoise coordinate was used, in consequence leading to a
Hamiltonian of diagonal form and, as claimed by the authors, to ”QFT set up by
the freely falling observer”. Other studies of the quantum properties of
scalar field were given for instance in Refs. [17-18] and the investigations
of the interior of the Schwarzschild BH were presented in Refs. [19-20]. The
most recent results have been given by Almeida and Rodrigues in Ref. [21]
where the quantization of the BH gravity was discussed and by Giddings and
Perkins in Ref. [22], in which the quantum evolution of the Hawking state in
Schwarzschild spacetime was investigated.
In this paper we will present a particular quantum aspect of the ”T-model”.
Namely the problem of dynamics, i.e. the temporal evolution of the quantized
scalar field in the case of such a cosmology will be introduced and briefly
discussed within a unitary approach. The Hamiltonian of the system,
represented by a set of harmonic oscillators, coupled via creation and
annihilation of pairs of particles, revealing interesting symmetry properties,
will be derived. The Heisenberg equations of motion for appropriate
annihilation and creation operators will be converted into ordinary
differential equations for Bogolyubov coefficients and will be shown to reveal
the possibility of an instability that is referred to as a gravitationally
driven instability.
The paper is organized as follows. In Sec. 2 we discuss the properties of the
Schwarzschild BH and a T-model is formulated. In Sec. 3 a scalar field and its
quantization are discussed. In Sec. 4. the Hamiltonian of the scalar field is
derived and a discussion is presented in the final section, Sec.5; Appendix is
devoted for a derivation of explicit form the temporal part of (factorized)
Klein-Gordon equation.
## 2 ”T-sphere” model - an anisotropic cosmological model
The metric $g_{\mu\nu}$ for the exterior of the Schwarzchild black hole,
diagonal in the Schwarzschild coordinates $\left(t,r,\theta,\varphi\right)$,
reveals the singularity on the horizon:
$ds^{2}=g_{t}\left(r\right)dt^{2}-g_{r}\left(r\right)dr^{2}-g_{2}\left(r\right)d\Omega^{2}.$
(2.1)
where
$g_{t}=1-\frac{2M}{r}=g_{r}^{-1}$ (2.2)
$g_{2}\left(r\right)=r^{2},$ and $d\Omega^{2}$ denotes the metric on the two-
dimensional unit sphere $S^{2}$ with the coordinates
$\left(\theta,\phi\right):$
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}.$ (2.3)
The geometrical singularity at the horizon, $r_{S}=2M$ may be removed by a
transformation to a singularity-free coordinate system, such as Kruskal-
Szekeres, Eddington-Finkelstein, Novikov, Lemaitre or other systems [1-2].
The coordinate system (2.1), though ill-defined on the horizon, may be applied
inside the horizon (see e.g. [6-7]). The interior of a BH, $r<r_{S}$
possesses, apart from some well-known, some not so well-known, properties too
(see [9]). The Killing vector $\partial_{t}$ becomes a spatial one that
results in momentum conservation instead of energy conservation, as obeyed
outside BH (see below). This is accompanied by the interchange of the roles of
the coordinates: $t$ and $r$ play the role of the spatial- and temporal-like
coordinates, respectively.
The interesting feature of the interior of a Schwarzschild BH is that it may
be regarded as a unique spacetime, a cosmological anisotropic model called a
”T-sphere” model or simply T-model [8]. It is described by the line element
(2.1) for $r<r_{S}$ but now expressed in terms of $T\left(=-r\right)$
(temporal) and $z$ (spatial) coordinates instead of $r$ and $t$, coordinates,
respectively
$ds_{-}^{2}=g_{T}dT^{2}-g_{z}dz^{2}-g_{2}\left(T\right)\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),$ (2.4)
where, $T\in\left\langle-r_{S},0\right\rangle,$
$z\in\left(-\infty,+\infty\right),$
$g_{T}=\left(\frac{r_{S}}{T}-1\right)^{-1}=g_{z}^{-1}$. At each instant of
$T_{0}$ the spatial slice is a hypercylinder $\mathbf{R}^{1}\times S^{2}$,
longitudinally expanding and transversally, a two-sphere of radius
$\left|T_{0}\right|,$ contracting (see e.g. [6]). Along the cylinder axis $z$
the system is homogeneous and that represents the momentum $z$-component
conservation.
Phenomena of a classical nature have been considered in the T-model both
within a more traditional approach (see e.g.[10-13]) as well as from other
specific perspectives (see [9], [23-25]). Here we will consider a special
quantum phenomenon, namely the problem of dynamics of the quantized scalar
field in the case T-model will be introduced and briefly discussed within a
unitary approach.
## 3 Scalar free field in a T-model
A scalar free field $\Phi$ in a space-time $M$ with a metric $g_{\mu\nu}$ is
described in terms of Lagrangian density $\mathcal{L}$:
$\mathcal{L}=\frac{1}{2}\sqrt{-g}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-\left(\mu^{2}+\xi
R\right)\Phi^{2},$ (3.1)
where $-g=\det\left[g_{\alpha\beta}\right],$ the parameter $\mu$ can be
interpreted as the mass only in asymptomatically flat space-time, $R$ is the
scalar curvature of $M$ and $\xi$ is the field coupling to the spacetime
curvature.
In the case of the spacetime (2.4) the coupling with gravitational field
vanishes (as $R=0$) and the action of the scalar free field (3.1) takes the
form
$S=\frac{1}{2}\int dT\int\limits_{\mathbf{\Sigma}}dzd\Omega
T^{2}\left[\frac{1}{g_{T}}\left(\partial_{T}\Phi\right)^{2}-\frac{1}{g_{z}}\left(\partial_{z}\Phi\right)^{2}+\frac{1}{T^{2}}\Phi\Delta_{S^{2}}\Phi-\mu^{2}\Phi^{2}\right],$
(3.2)
where $\Sigma=\mathbf{R}^{1}\times S^{2},$ $d\Omega=\sin\theta d\varphi
d\theta$ and we have integrated by parts in the sector $S^{2}$ which resulted
in the Laplace operator $\Delta_{S^{2}}$ on $S^{2}$:
$\Delta_{S^{2}}\Phi=\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\Phi}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}\Phi}{\partial\varphi^{2}}.$
(3.3)
The Klein-Gordon (or Euler-Lagrange) equation
$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\right)+\mu^{2}\Phi=0,$
(3.4)
takes in this case the following form:
$\partial_{T}\left(T^{2}g_{z}\partial_{T}\Phi\right)-\frac{T^{2}}{g_{z}}\partial_{z}^{2}\Phi-\Delta_{S^{2}}\Phi+\mu^{2}T^{2}\Phi=0,$
(3.5)
Taking the field $\Phi$ in the form of a product:
$\Phi\left(T,z,\theta,\phi\right)=R\left(T\right)u\left(z\right)Y\left(\theta,\phi\right).$
(3.6)
it follows that the wave equation ( 3.5) separates into the following
equations:
$\Delta_{S^{2}}Y=-l\left(l+1\right)Y,$ (3.7)
$\frac{d^{2}u_{\varepsilon}}{dz^{2}}=-\varepsilon^{2}u_{\varepsilon},$ (3.8)
$\frac{d}{dT}\left(T^{2}g_{z}\frac{dR_{\varepsilon
l}}{dT}\right)+T^{2}\left(\frac{\varepsilon^{2}}{g_{z}}+\mu^{2}+\frac{l\left(l+1\right)}{T^{2}}\right)R_{\varepsilon
l}=0,$ (3.9)
where $\varepsilon$ is a (separation) constant. The solution of Eq.(3.7) is
given by the spherical harmonics $Y_{lm}\left(\theta,\phi\right),$
$\int\limits_{S^{2}}d\Omega
Y_{lm}\left(\theta,\varphi\right)Y_{l^{\prime}m^{\prime}}^{\ast}\left(\theta,\varphi\right)=\delta_{ll^{\prime}}\delta_{mm^{\prime}},$
(3.10)
$\int\limits_{S^{2}}d\Omega
Y_{lm}\left(\theta,\varphi\right)Y_{l^{\prime}-m^{\prime}}\left(\theta,\varphi\right)=\delta_{ll^{\prime}}\delta_{m,-m^{\prime}}$
(3.11)
where $m=-l,-\left(l-1\right),...0....,l$. The solution of equation (3.8) is
$u\left(z\right)=e^{\pm i\varepsilon z}.$ (3.12)
One can decompose the field $\Phi$ into the complete system of functions on
$\mathbf{R}^{1}$ and $S^{2}$. Thus, the real field $\Phi=\Phi^{\ast}$ is
represented as:
$\Phi\left(T,z,\theta,\varphi\right)=\sum\limits_{\varepsilon,l,m}\left[R_{\varepsilon
l}\left(T\right)e^{i\varepsilon
z}Y_{lm}\left(\theta,\varphi\right)A_{\varepsilon lm}+R_{\varepsilon
l}^{\ast}\left(T\right)e^{-i\varepsilon
z}Y_{lm}^{\ast}\left(\theta,\varphi\right)A_{\varepsilon lm}^{\ast}\right],$
(3.13)
where $R_{\varepsilon l}\left(T\right)$ are the functions of the temporal
variable $T$ satisfying second order differential equation (3.9) and
$A_{\varepsilon lm}$ are Fourier-like coefficients.
The scalar product $\left(\cdot,\cdot\right)$ (Klein-Gordon) is in general
defined as :
$\left(\Phi,\Psi\right)=i\int\limits_{\Sigma_{t}}\left(\Phi^{\ast}\partial_{\mu}\Psi-\Psi\partial_{\mu}\Phi^{\ast}\right)n^{\mu}dvol\left(\Sigma_{t}\right),$
(3.14)
where $n=n^{\mu}\partial_{\mu}$ denotes the unit time-like vector field
orthogonal to a space-like hypersurface (slice) $\Sigma_{t}$ and $\Phi,\Psi$
are the solutions of the Klein-Gordon equation. In this case $\Sigma_{t}\simeq
A\times S^{2}$ and the scalar product takes the form (see [17], [26]):
$\left(\Phi,\Psi\right)=iT^{2}g_{z}\int\limits_{S^{2}}\sin\theta d\theta
d\phi\int\limits_{A}\left(\Phi^{\ast}\partial_{T}\Psi-\Psi\partial_{T}\Phi^{\ast}\right)dz.$
(3.15)
There is the following normalization condition
$A_{\varepsilon lm}=\left(R_{\varepsilon l}\left(T\right)e^{i\varepsilon
z}Y_{lm}\left(\theta,\varphi\right),\Phi\right)$ (3.16)
where $\Phi$ is given by (3.6), which is equivalent to the claim of the
canonical commutation relations (see also below).
After some (lengthy but simple) algebra one finds that condition (3.16) is
satisfied iff
$T^{2}g_{z}\left[R_{\varepsilon l}^{\ast}\overset{\cdot}{R}_{\varepsilon
l}-\overset{\cdot}{R}_{\varepsilon l}^{\ast}R_{\varepsilon l}\right]=-i,$
(3.17)
$R_{\varepsilon l}^{\ast}\overset{\cdot}{R}_{-\varepsilon
l}^{\ast}-R_{-\varepsilon l}^{\ast}\overset{\cdot}{R}_{\varepsilon
l}^{\ast}=0.$ (3.18)
The condition (3.17) is derived from the differential equation (3.9). First,
one writes Eq.(3.9) for the complex conjugated function $R_{\varepsilon
l}^{\ast}$; then one multiplies it by $R_{\varepsilon l}$ and Eq.(3.9) by
$R_{\varepsilon l}^{\ast}$; finally one subtracts the former from the latter
obtaining
$d_{T}\left(T^{2}g_{z}\left[R_{\varepsilon
l}^{\ast}\overset{\cdot}{R}_{\varepsilon l}-\overset{\cdot}{R}_{\varepsilon
l}^{\ast}R_{\varepsilon l}\right]\right)=0.$ (3.19)
Therefore, (3.17) turns out to be a normalization condition for
$R_{\varepsilon l}$ i.e. the Wronskian in this case, as it should be. On the
other hand Eq. (3.18) is just an equivalence.
### 3.1 Quantization
Quantization of the field (3.1-2) is performed in a canonical way. Namely, one
introduces the momentum field as the field canonically conjugated to
$\Phi\left(T,z,\theta,\varphi\right),$i.e.
$\pi=\frac{\partial\mathcal{L}}{\partial\left(\partial_{T}\Phi\right)}=\frac{T^{2}}{g_{T}}\partial_{T}\Phi.$
(3.20)
Then one imposes canonical commutation relations
where $\mathbf{x,y}\in\Sigma_{t}$. In our case the slice $\Sigma_{t}$ has the
topology of the product space of the set $A\subset\mathbf{R}^{1}$ and the two-
dimensional sphere $S^{2}$. The momentum field given in its Fourier decomposed
form is:
$\widehat{\pi}\left(t,r,\theta,\phi\right)=\frac{T^{2}}{g_{T}}\sum\limits_{\varepsilon,l,m}\left[\widehat{A}_{\varepsilon
lm}\overset{\cdot}{R}_{\varepsilon l}\left(T\right)e^{i\varepsilon
z}Y_{lm}\left(\theta,\phi\right)+\widehat{A}_{\varepsilon
lm}^{{\dagger}}\overset{\cdot}{R}_{\varepsilon
l}^{\ast}\left(T\right)e^{-i\varepsilon
z}Y_{lm}^{\ast}\left(\theta,\phi\right)\right]$ (3.22)
The canonical commutation relations Eqs. (3.21) turn out to be satisfied under
the following conditions:
a) $\widehat{A}_{\varepsilon lm}$, $\widehat{A}_{\varepsilon lm}^{{\dagger}},$
are the annihilation and creation operators, respectively, i.e. the only
nonvanishing commutator is
$\left[\widehat{A}_{\varepsilon
lm},\widehat{A}_{\varepsilon^{\prime}l^{\prime}m^{\prime}}^{{\dagger}}\right]=\delta_{\varepsilon\varepsilon^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}}$
(3.22)
b) the Wronskian (3.17) must hold.
## 4 Hamiltonian of the scalar field in a T-model
The Hamiltonian of the field described by the Lagrangian density $\mathcal{L}$
is determined as an integral over the spatial part $\mathbf{\Sigma}$ of the
spacetime
$H=\int\limits_{\mathbf{\Sigma}}d^{3}x\left[\pi\partial_{T}\Phi-\mathcal{L}\right],$
(4.1)
and this expression is equivalent to the (integrated) $T_{TT}$ element of the
stress-energy tensor. Applying formula (4.1) for the case (2.4) and (3.1) one
obtains
$H=\frac{1}{2}\int\limits_{\mathbf{\Sigma}}dzd\theta d\varphi
T^{2}\sin\theta\left[\frac{1}{g_{T}}\left(\partial_{T}\Phi\right)^{2}+\frac{1}{g_{z}}\left(\partial_{z}\Phi\right)^{2}-\Phi\Delta_{S^{2}}\Phi+\mu^{2}\Phi^{2}\right]$
(4.2)
Using the Fourier decomposition of the quantized field and momentum (see Eqs.
(3.13), (3.22)) one finds the Hamiltonian of the quantized scalar field as
expressed in terms of annihilation and creation operators:
$H=\frac{1}{2}\sum\limits_{\varkappa}\left[\omega_{\varkappa}\widehat{A}_{\varkappa}\widehat{A}_{\varkappa}^{{\dagger}}+\gamma_{\varkappa\varkappa^{\prime}}\widehat{A}_{\varkappa}\widehat{A}_{\varkappa^{\prime}}+\left(c.c\right)\right]$
(4.3)
where indices $\varkappa,\varkappa^{\prime}$ correspond to the appropriate
three letter sets $\varepsilon lm.$ The parameters
$\omega_{\varkappa},\gamma_{\varkappa\varkappa^{\prime}}$ are given as
$\gamma_{\varepsilon
lm/\varepsilon^{\prime}lm^{\prime}}=\left[T^{2}g_{z}\overset{\cdot}{R}_{\varepsilon
l}\overset{\cdot}{R}_{-\varepsilon
l}+T^{2}\left\\{\frac{\varepsilon^{2}}{g_{z}}+\frac{l\left(l+1\right)}{T^{2}}+\mu^{2}\right\\}R_{\varepsilon
l}\left(T\right)R_{-\varepsilon
l}\left(T\right)\right]\delta_{\varepsilon,-\varepsilon^{\prime}}\delta_{m,-m^{\prime}}$
(4.4)
$\omega_{\varepsilon lm}=\left[T^{2}g_{z}\overset{\cdot}{R}_{\varepsilon
l}\overset{\cdot}{R}_{\varepsilon
l}^{\ast}+T^{2}\left\\{\frac{\varepsilon^{2}}{g_{z}}+\frac{l\left(l+1\right)}{T^{2}}+\mu^{2}\right\\}R_{\varepsilon
l}\left(T\right)R_{\varepsilon l}^{\ast}\left(T\right)\right].$ (4.5)
Therefore, the Hamiltonian of the scalar field in the T-model, i.e.
anisotropic cosmological model representing interior of the Schwarzschild BH,
turns out to be
$H=\frac{1}{2}\sum\limits_{\varepsilon lm}\left[\omega_{\varepsilon
lm}\left(\widehat{A}_{\varepsilon lm}\widehat{A}_{\varepsilon
lm}^{{\dagger}}+\widehat{A}_{\varepsilon
lm}^{{\dagger}}\widehat{A}_{\varepsilon lm}\right)+\gamma_{\varepsilon
lm/-\varepsilon l-m}\widehat{A}_{\varepsilon lm}\widehat{A}_{-\varepsilon
l-m}+\gamma_{\varepsilon lm/-\varepsilon l-m}^{\ast}\widehat{A}_{\varepsilon
lm}^{{\dagger}}\widehat{A}_{-\varepsilon l-m}^{{\dagger}}\right].$ (4.6)
representing the set of interacting, time-dependent harmonic oscillators.
On this basis one can study the dynamics of the quantized scalar field. The
evolution of the system is described by the Heisenberg equation of motion for
the operators $\widehat{A}_{\varepsilon lm}$
$i\frac{d}{dt}\widehat{A}_{\varepsilon lm}=\left[\widehat{A}_{\varepsilon
lm},\widehat{H}\right]=\omega_{\varepsilon
lm}\left(t\right)\widehat{A}_{\varepsilon
lm}\left(t\right)+\gamma_{\varepsilon
lm}^{\ast}\left(t\right)\widehat{A}_{-\varepsilon
l-m}^{{\dagger}}\left(t\right)$ (4.7)
where, $\gamma_{\varepsilon lm/-\varepsilon l-m}\equiv\gamma_{\varepsilon
lm}.$ One can search for the solutions of the above equations by using the
following ansatz:
$\widehat{A}_{\varepsilon lm}\left(t\right)=\alpha_{\varepsilon
lm}\left(t\right)\widehat{A}_{\varepsilon lm}+\beta_{\varepsilon
lm}\left(t\right)\widehat{A}_{-\varepsilon l-m}^{{\dagger}},$ (4.8)
where $\alpha_{\varepsilon lm}\left(t\right)$ and $\beta_{\varepsilon
lm}\left(t\right)$ are some unknown complex functions and
$\widehat{A}_{\varepsilon lm}$ and $\widehat{A}_{-\varepsilon
l-m}^{{\dagger}}$ are time independent operators. By definition the relation
(4.8) preserves the commutation relations (3.22), hence it turns out to be the
Bogolyubov transformation,
$\left|\alpha_{\varepsilon
lm}\left(t\right)\right|^{2}-\left|\beta_{\varepsilon
lm}\left(t\right)\right|^{2}=1.$ (4.9)
Then, the Heisenberg equations (4.7) are converted into differential equations
for the Bogolyubov coefficients
$i\frac{d}{dt}\alpha_{\varepsilon lm}\left(t\right)=\omega_{\varepsilon
lm}\left(t\right)\alpha_{\varepsilon lm}\left(t\right)+\gamma_{\varepsilon
lm}^{\ast}\left(t\right)\beta_{\varepsilon lm}^{\ast}\left(t\right),$ (4.10)
$i\frac{d}{dt}\beta_{\varepsilon lm}\left(t\right)=\omega_{\varepsilon
lm}\left(t\right)\beta_{\varepsilon lm}\left(t\right)+\gamma_{\varepsilon
lm}^{\ast}\left(t\right)\alpha_{\varepsilon lm}^{\ast}\left(t\right).$ (4.11)
In general, one can’t expect exact solutions of the equations (4.10-11) and
approximate schemes would therefore be proposed. Our forthcoming paper will be
devoted to the comprehenssive discussion of this problem.
## 5 Discussion
Considering the interior of a Schwarzschild BH as a unique spacetime, an
anisotropic cosmological model, we have performed the quantization of the free
(noninteracting) scalar field by imposing the canonical commutation relations.
One decomposes the field and momentum in terms of the complete set of
solutions of the Klein-Gordon (or in fact Euler-Lagrange equations) with the
coefficients of expansion being annihilation and creation operators. This
procedure leads to the Hamiltonian of the quantized scalar field taking the
form of the set of harmonic, time-dependent oscillators coupled in a special
way: there are terms in the Hamiltonian corresponding to creation,
$\gamma_{\varepsilon lm}\widehat{A}_{\varepsilon lm}\widehat{A}_{-\varepsilon
l-m}$ and annihilation $\gamma_{\varepsilon lm}^{\ast}\widehat{A}_{\varepsilon
lm}^{{\dagger}}\widehat{A}_{-\varepsilon l-m}^{{\dagger}}$ particles in pairs.
Such a picture, peculiar at first sight, appears to have a deeper sense. The
spacetime considered is a dynamic one - there is no energy conservation there,
hence the Hamiltonian contains terms representing spontaneous creation and
annihilation pairs of particles. Homogeneity of the spacetime along the
$z$-direction results in the presence of a spatial-like Killing vector
representing, $z$-momentum-component conservation. Hamiltonian (4.6) reflects
this symmetry property: pairs of the particles with opposite $z$-component
momenta may be created, $\widehat{A}_{\varepsilon
lm}^{{\dagger}}\widehat{A}_{-\varepsilon l-m}^{{\dagger}}$ and annihilated
$\widehat{A}_{\varepsilon lm}\widehat{A}_{-\varepsilon l-m}$; the Hamiltonian
of the system also obeys rotational invariance.
The conservation of the $z$-momentum component in the terms represented by
$\gamma_{\varepsilon lm}$ and $\gamma_{\varepsilon lm}^{\ast}$ in the
Hamiltonian is an analogue of energy conservation outside the BH, i.e. the
particles in a pair carry positive/negative energy; the one with negative
energy cannot survive outside the BH but only within the horizon of the BH.
There is a more or less obvious interpretation of the $\beta_{\varepsilon
lm}\left(t\right)$ coefficient of the Bogolyubov’s transformation (4.8): it is
proportional to the number of the particles created during the evolution of
the system,
$\left\langle 0\left(t\right)\right|\widehat{A}_{\varepsilon
lm}^{{\dagger}}\widehat{A}_{\varepsilon
lm}\left|0\left(t\right)\right\rangle=\left\langle
0\right|\widehat{A}_{\varepsilon
lm}^{{\dagger}}\left(t\right)\widehat{A}_{\varepsilon
lm}\left(t\right)\left|0\right\rangle=\left|\beta_{\varepsilon
lm}\left(t\right)\right|^{2},$ (5.1)
where $\left|0\right\rangle$ is the vacuum state for fixed time $t=0$ and
annihilation operators $\widehat{A}_{\varepsilon lm}$ while
$\left|0\left(t\right)\right\rangle$ is the vacuum state for later time $t$
and annihilation operators $\widehat{A}_{\varepsilon lm}\left(t\right).$ Due
to the violent dynamics of the background spacetime, one may expect the
dynamics of the creation and annihilation of the (pairs of) particles to be
violent, and conventional adiabatic-like approaches (see e.g. [17-18]) could
hardly be regarded as a working scheme. Therefore, attempts to find an
approximate solution within a treatment here proposed that might be called a
”unitary approach” as based on a unitarity of the evolution of the system,
will be discussed in our following paper.
An interesting aspect of the dynamics of the model (3.1) will be however
briefly discussed here. That is the question of the possible instability of
the system of interacting harmonic oscillators (4.6) (see [27-28]). The
oscillators interact in pairs, $\left(\varepsilon lm\right)/\left(-\varepsilon
l-m\right)$ and one can consider diagonalization (at an arbitrary instant
$T^{\prime}$) of the Hamiltonian corresponding to such a subsystem. Then the
frequency in such a diagonalized case is given as:
$\Omega_{\varepsilon lm}^{2}=\omega_{\varepsilon
lm}^{2}-\left|\gamma_{\varepsilon lm/-\varepsilon l-m}\right|^{2}.$ (5.2)
This expression should be positive, otherwise the system is unstable (see
[27]) (this problem will be discussed in detail in our following paper) - this
would be named a ”gravitationally driven instability”. One can check that in
this case, Eqs. (4.4-5) the right hand side of Eq.(5.2)
$\Omega_{\varepsilon
lm}^{2}=\frac{1}{g_{z}}\left[\frac{\varepsilon^{2}}{g_{z}}+\frac{l\left(l+1\right)}{T^{2}}+\mu^{2}\right]$
(5.3)
is positive: there is no gravitational instability in the scalar field
quantized in Doran et al. [7] spacetime. An interesting issue is that, apart
from the possible instability of type (5.2), that might be referred to as ”a
restoring force instability” there is also another possible instability,
namely ”a friction driven instability” but the problem of its origin and
character will be discussed elsewhere.
## 6 Appendix
Let us briefly analyze the form of the temporal part of Klein-Gordon equation
in this case, i.e. Eq. (3.9):
$\frac{1}{T^{2}}\frac{d}{dT}\left(T^{2}g_{z}\frac{dR}{dT}\right)+\left(\frac{\varepsilon^{2}}{g_{z}}+\mu^{2}+\frac{l\left(l+1\right)}{T^{2}}\right)R=0,$
(A.1)
where $g_{z}=\frac{r_{S}}{T}-1,$ and lower labels have been omitted here.
Making the substitution, $R=f\eta,$ one finds
$\frac{1}{T^{2}}\frac{d}{dT}\left(T^{2}g_{z}\frac{dR}{dT}\right)=\frac{1}{T^{2}}\left[\left(r_{S}-2T\right)\left(f^{\prime}\eta+f\eta^{\prime}\right)+\left(r_{S}T-T^{2}\right)\left(f^{\prime\prime}\eta+f\eta^{\prime\prime}+2f^{\prime}\eta^{\prime}\right)\right]$
(A.2)
and prime means differentiation with respect to $T$. Claiming
$\left(r_{S}-2T\right)f+2\left(r_{S}T-T^{2}\right)f^{\prime}=0,$ (A.3)
one gets $R\left(T\right)$ in the form
$R=\frac{\eta}{\sqrt{T\left(r_{S}-T\right)}},$ (A.4)
and $\eta\left(T\right)$ satisfies the following confluent Heun equation
$\left[\frac{d^{2}}{dT^{2}}+\nu^{2}\left(T\right)\right]\eta=0,$ (A.5)
where
$\nu^{2}\left(T\right)=A+\frac{B}{T}+\frac{C}{\left(r_{S}-T\right)}+\frac{D}{T^{2}}+\frac{E}{\left(r_{S}-T\right)^{2}},$
(A.6)
and the five coefficients $A,...,E$ are equal to:
$\displaystyle A=\left(\varepsilon^{2}-\mu^{2}\right),\text{ \ \
}B=\frac{1}{2r_{S}}\left(2l\left(l+1\right)+1\right),$ $\displaystyle
C=r_{S}\left(\mu^{2}+2\varepsilon^{2}\right)+B,\text{ \ \ }D=\frac{1}{4},$
$\displaystyle E=D-2\left(1+r_{S}^{2}\varepsilon^{2}\right).$
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|
Identifying diffusive length scales in one-dimensional Bose gases
Frederik Møller1$\star$, Federica Cataldini1 and Jörg Schmiedmayer1
1 Vienna Center for Quantum Science and Technology (VCQ), Atominstitut, TU
Wien, Vienna, Austria
⋆<EMAIL_ADDRESS>
## Abstract
In the hydrodynamics of integrable models, diffusion is a subleading
correction to ballistic propagation. Here we quantify the diffusive
contribution for one-dimensional Bose gases and find it most influential in
the crossover between the main thermodynamic regimes of the gas. Analysing the
experimentally measured dynamics of a single density mode, we find diffusion
to be relevant only for high wavelength excitations. Instead, the observed
relaxation is solely caused by a ballistically driven dephasing process, whose
time scale is related to the phonon lifetime of the system and is thus useful
to evaluate the applicability of the phonon bases typically used in quantum
field simulators.
###### Contents
1. 1 Introduction
2. 2 Propagation of quasi-particles in the 1D Bose gas
1. 2.1 Generalized Hydrodynamics at the diffuse scale
2. 2.2 Linearization of the diffusive GHD equation
3. 2.3 Critical length scale of diffusion
3. 3 Magnitude of diffusive corrections in thermal states
4. 4 Quasi-particle dynamics following a single-mode quench in a quasi-condensate
1. 4.1 Experimental setup and protocol
2. 4.2 Analysis of quasi-particle propagation
1. 4.2.1 Comparing experimental observations with linearized GHD
2. 4.2.2 Quasi-particle dephasing and applicability of phonon basis
3. 4.2.3 Diffusive length scales
3. 4.3 Single-mode quench in the maximal diffusive regime
5. 5 Conclusion
6. A Thermodynamic Bethe Ansatz
7. B Quasi-particle propagation versus energy
## 1 Introduction
The last few decades have seen significant advances in techniques for
realizing and manipulating quantum many-body systems, particularly in the form
of gases of ultracold atoms [1]. The behavior of interacting quantum many-body
systems is, however, notoriously difficult to describe [2, 3], spurring the
development of new theoretical models. In continuous systems with large number
of particles, microscopic descriptions are generally intractable; instead,
models describing the emergent, large-scale behavior of the system are more
appealing [4]. For such purpose, hydrodynamics provides a powerful framework,
describing the large-scale flow of densities of conserved quantities [5, 6].
However, integrable systems, such as the one-dimensional (1D) Bose gas,
feature an infinite number of conserved quantities, thus complicating the
formulation of a hydrodynamic theory; this infinite number of conservation
laws poses a severe constraint of dynamics and inhibits thermalization [7].
The breakthrough came with the advent of Generalized Hydrodynamics (GHD) [8,
9], which parameterizes the hydrodynamics in terms of quasi-particles given by
the Bethe Ansatz solution to integrable models.
At lowest order (Euler scale), the working GHD equation is a collisionless
Boltzmann equation [10], where quasi-particles propagate ballistically at an
effective velocity modified by collisions with other particles. Beyond the
Euler scale one must account for diffusive effects [11], which arise from the
number of collisions experienced by the particle to be subject to equilibrium
thermal fluctuations [12]. The result is a Gaussian broadening of the quasi-
particle trajectories [13].
Unlike for most non-integrable systems, diffusion in GHD arises as a
subleading correction to Euler-scale hydrodynamics. Thus, diffusive effects
become evident in dynamics mainly at long time scales. For instance, in the
presence of an inhomogeneous potential, diffusion has been shown to cause a
gradual thermalization of a 1D Bose gas [14]. In real systems, however, other
mechanisms of integrability breaking [15], such as noise [16], losses [17], or
violation of one-dimensionality [18, 19, 20, 21], are likely stronger and may
therefore mask signatures of diffusion. Indeed, in experimental tests of GHD
with 1D Bose gases, no clear signs of diffusion have been observed thus far
[22, 23].
In this work, we study and quantify the influence of diffusion on the dynamics
of one-dimensional Bose gases, particularly on time and length scales
accessible in experimental setups. First, we calculate the diffusive spreading
of quasi-particle trajectories relative to their ballistic propagation
velocity in thermal states. We consider interaction strengths and temperatures
ranging across several orders of magnitude, thus allowing the identification
of the thermodynamic regimes where diffusive effects are most prominent. Next,
we analyse the experimental results of Ref. [24] in order to identify the
length scales at which diffusive effects become relevant in a quasi-
condensate; the setup of Ref. [24] is especially well-suited for such
analysis, as the high degree of control over the potential enabled the
excitation of a single momentum mode, thus creating excitations at a desired
length scale. Our analysis also provides a proxy for the phonon lifetime,
enabling one to gauge the applicability of a phonon basis to describe
dynamics. Phonon bases have been used extensively to describe quantum field
simulators realized with ultracold atoms [25, 26, 27, 28, 29, 30, 31];
identifying the time scales at which a low-energy effective theory remains
valid is thus beneficial for a number of applications.
## 2 Propagation of quasi-particles in the 1D Bose gas
Upon confinement to a single spatial dimension, an ultracold gas of $N$
repulsively interacting bosons of mass $m$ is well described by the Lieb-
Lininger model [32, 33], whose Hamiltonian reads
${\mathcal{H}}=-\sum_{i}^{N}\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
z_{i}^{2}}+g\sum_{i<j}^{N}\delta\left(z_{i}-z_{j}\right)\;.$ (1)
Here, $z_{i}$ is the position of the $i$’th boson and $g>0$ is their coupling
strength, which in an experimental system depends on the s-wave scattering
length of the atoms and the potential confining the atoms to 1D [34]. Often,
the coupling strength is parameterized as $c=mg/\hbar^{2}$, in units of
inverse length.
By virtue of integrability, the 1D Bose gas can be solved exactly using the
Bethe Ansatz, which identifies the elementary excitations as fermionic quasi-
particles uniquely labelled by their quasi-momentum, or rapidity, $\theta$
[32, 33]. In the thermodynamics limit, the density of occupied rapidities,
i.e. the density of quasi-particles, $\rho_{\mathrm{p}}(\theta)$ fully
characterises the thermodynamic properties of the local equilibrium
macrostate. Similarly, a density of holes $\rho_{\mathrm{h}}(\theta)$ can be
introduced, which describes the density of unoccupied rapidities; together the
two densities form the density of states $\rho_{\mathrm{s}}(\theta)$. The
occupational fraction of allowed rapidity states, dubbed the filling function
$\vartheta(\theta)=\rho_{\mathrm{p}}(\theta)/\rho_{\mathrm{s}}(\theta)$,
equivalently characterises the macrostate.
At thermal equilibrium, the state of a homogeneous 1D Bose gas with density
$n$ is characterized by only two dimensionless parameters: the interaction
strength $\gamma=c/n$ and the reduced temperature
$\mathcal{T}=\frac{2\hbar^{2}k_{\mathrm{B}}T}{mg^{2}}$, where $T$ is the
temperature and $k_{\mathrm{B}}$ Boltzmann’s constant. The interaction
strength and temperature span an entire phase-diagram of the 1D Bose gas,
whose regimes can be identified via the $g^{(2)}$-function [35]. At asymptotic
values of $\gamma$ and $\mathcal{T}$, the description of the Bose gas
drastically simplifies. However, in order to describe the state at arbitrary
values of these parameters, the Thermodynamic Bethe Ansatz (TBA) is generally
needed [36]. For more detail on the TBA and GHD of the 1D Bose gas, see
Appendix A and Refs. [37, 38, 39]).
### 2.1 Generalized Hydrodynamics at the diffuse scale
Hydrodynamics is an expansion of dynamics in spatial derivatives. At lowest
order in derivatives, where currents only depend on the local densities of
conserved quantities, one obtains the Euler scaling limit, which in GHD
captures the ballistic propagation of quasi-particles. Semi-classically, or in
soliton-gas interpretation of GHD [40], the bare propagation velocity of
quasi-particles is modified though collision with other particles, as each
two-body elastic collision is associated with a Wigner time delay [41]. Thus,
as a particle with rapidity $\theta$ traverses a thermodynamic state with
quasi-particle density $\rho_{\mathrm{p}}$, averaging over all the Wigner time
delays experienced by the particle results in an effective propagation
velocity $v^{\mathrm{eff}}(\theta)$ (see figure 1). On the quantum mechanical
level, the effective velocity can be viewed as a modified group velocity of
the quasi-particles, following local interactions with other particles,
whereby properties, such as the momentum $p(\theta)$ and energy
$\epsilon(\theta)$, of the particle $\theta$ become modified or "dressed" [42,
43].
Figure 1: Illustration of quasi-particle propagation. A ’tagged’ quasi-
particle with rapidity $\theta_{i}$ (blue) propagates ballistically with an
average velocity $v^{\mathrm{eff}}(\theta_{i})$ through a system with quasi-
particle density $\rho_{\mathrm{p}}(\theta)$. The effective velocity is a
modification of the bare velocity
$v^{\mathrm{bare}}(\theta_{i})=\hbar\theta_{i}/m$ following collision with
other particles, as each collision is associated with an apparent delay of the
particles (or equivalently a positional shift $\Delta$). Thermal fluctuations
of $\rho_{\mathrm{p}}(\theta)$, and thus the number of collisions experienced
by the particle, result in diffusive corrections, which manifest as a
broadening of the quasi-particle trajectory on the order of $\sqrt{t}$. Figure
inspired by Ref. [44].
At the next order of spatial derivatives, diffusive contributions to the
dynamics are accounted for [11]. Quasi-particle diffusion arises from thermal
fluctuations [12, 13]. For the duration $t$, a quasi-particle will traverse a
region of length $L\propto t$, through which $\rho_{\mathrm{p}}$ exhibits
thermal fluctuations of order $1/\sqrt{L}$. Thus, the number of collisions
experienced by the particle, and in turn the distance it travels, fluctuates
by $\sqrt{t}$ [13]. The result is a diffusive broadening of the quasi-particle
front, as illustrated in figure 1. Further, the Euler scale terms of the GHD
equation are unaltered by scaling $z\to zL$ and $t\to tL$, whilst the
diffusive terms are rescaled by a factor $1/L$ [14], thus highlighting
diffusive effects in GHD as subleading to the ballistic quasi-particle
propagation.
Assuming only large scale variations in space and time, the quasi-particle
distribution of an inhomogeneous system becomes time and space dependent
$\rho_{\mathrm{p}}=\rho_{\mathrm{p}}(z,t,\theta)$; the GHD equation describes
the evolution of this distribution. Note, in the following all
$(z,t)$-dependencies have been omitted for a more compact notation. On the
diffusive scale, the advective form of the GHD equation (describing the
evolution of the filling $\vartheta$) reads
$\partial_{t}\vartheta(\theta)+v^{\mathrm{eff}}(\theta)\,\partial_{z}\vartheta(\theta)=\frac{1}{2\rho_{\mathrm{s}}(\theta)}\left(\mathbf{1}-\frac{\vartheta\mathbf{\Delta}}{2\pi}\right)\>\partial_{z}\left(\left(\mathbf{1}-\frac{\vartheta\mathbf{\Delta}}{2\pi}\right)^{-1}\rho_{s}(\theta)\>\mathbf{D}\>\partial_{z}\vartheta(\theta)\right)\;,$
(2)
where the effective velocity is given by
$v^{\mathrm{eff}}(\theta)=\frac{(\partial_{\theta}\epsilon)^{\mathrm{dr}}(\theta)}{(\partial_{\theta}p)^{\mathrm{dr}}(\theta)}\;,$
(3)
with $\epsilon(\theta)=\hbar^{2}\theta^{2}/(2m)$ and $p(\theta)=\hbar\theta$
being the single-particle energy and momentum, respectively. The superscript
’dr’ denotes that the function has been dressed, following the equation
$g^{\mathrm{dr}}(\theta)=g(\theta)-\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\theta^{\prime}\;\Delta(\theta,\theta^{\prime})\vartheta(\theta^{\prime})g^{\mathrm{dr}}(\theta^{\prime})\;,$
(4)
where
$\Delta(\theta,\theta^{\prime})=-\frac{2c}{c^{2}+(\theta-\theta^{\prime})^{2}}$
is the two-body scattering kernel of the Lieb-Liniger model. Further, in eq.
(2) we have employed bold symbols to indicate integral operators and the
following shorthand notation for their action
$\left(\mathbf{K}g\right)(\theta)\coloneqq\int_{-\infty}^{\infty}\mathrm{d}\theta^{\prime}\>K\left(\theta,\theta^{\prime}\right)g\left(\theta^{\prime}\right)\;,$
(5)
where $K\left(\theta,\theta^{\prime}\right)$ is the associated kernel of the
generic integral operator $\mathbf{K}$. The kernel of the integral operator
$\mathbf{D}$ in eq. (2) reads
$D(\theta,\theta^{\prime})=\frac{1}{\rho_{\mathrm{s}}^{2}(\theta)}\left(\delta(\theta-\theta^{\prime})w(\theta^{\prime})-W(\theta,\theta^{\prime})\right)\;,$
(6)
where the off-diagonal and diagonal elements, respectively, are given by
$\displaystyle W(\theta,\theta^{\prime})$
$\displaystyle=\rho_{\mathrm{p}}(\theta)(1-\vartheta(\theta))\left[\frac{\Delta^{\mathrm{dr}}(\theta,\theta^{\prime})}{2\pi}\right]^{2}\left|v^{\mathrm{eff}}(\theta)-v^{\mathrm{eff}}(\theta^{\prime})\right|\;,$
(7) $\displaystyle w(\theta)$
$\displaystyle=\int_{-\infty}^{\infty}\mathrm{d}\theta^{\prime}\>W(\theta^{\prime},\theta)\;.$
(8)
The diffusive effects arise mainly from fluctuations of the state $\vartheta$,
which are diagonal in rapidity and given by the expression
$\langle\delta\vartheta(\theta)\>\delta\vartheta(\theta^{\prime})\rangle=\delta(\theta-\theta^{\prime})\frac{\vartheta(\theta^{\prime})\left(1-\vartheta(\theta^{\prime})\right)}{\rho_{\mathrm{s}}(\theta^{\prime})}\;.$
(9)
Indeed, we see that the diffusion kernel $D(\theta,\theta^{\prime})$ of eq.
(6) is proportional to the filling fluctuations.
### 2.2 Linearization of the diffusive GHD equation
Simulating diffusive GHD according to eq. (2) generally requires powerful
numerics [45, 46]. However, following the approach of Ref. [47], one can
diagonalize the GHD propagation kernel, enabling a direct study of diffusive
length scales. Here we briefly summarize the approach.
Consider the time-dependent filling function consisting of a stationary,
homogeneous background $\vartheta_{0}$ and an evolving perturbation
$\delta\vartheta$, such that
$\vartheta(z,t,\theta)=\vartheta_{0}(\theta)+\delta\vartheta(z,t,\theta)\;.$
(10)
Assuming a small perturbation $\delta\vartheta\ll\vartheta_{0}$, we can
neglect interactions within the perturbation itself and only treat the
interactions between the perturbation and the stationary background. Thus,
both the effective velocity and the diffusion kernel are evaluated with
respect to the background filling $\vartheta_{0}$, whereby each Fourier mode
of the filling function perturbation evolves independently. A perturbation
consisting of only a single Fourier mode
$\delta\vartheta(z,t,\theta)=\delta\vartheta_{j}(t,\theta)e^{ik_{j}z}$, where
$k_{j}=2\pi j/L$ with $L$ being the system size, has the time-dependent
solution
$\partial_{t}\delta\vartheta_{j}(t,\theta)+\int_{-\infty}^{\infty}\mathrm{d}\theta^{\prime}\>\mathfrak{D}_{j}(\theta,\theta^{\prime})\>\delta\vartheta_{j}(t.\theta)=0\;,$
(11)
where we have introduced $\mathfrak{D}_{j}$, which is kernel of the
propagation operator of the $j$’th mode and is given by
$\mathfrak{D}_{j}(\theta,\theta^{\prime})=ik_{j}v_{0}^{\mathrm{eff}}(\theta)\delta(\theta-\theta^{\prime})+\frac{k_{j}^{2}}{2}{D}_{0}(\theta,\theta^{\prime})\;.$
(12)
The propagation operator is a linear integral operator with eigenstates
$f_{\omega,j}(\theta)$ and corresponding eigenvalues
$\lambda_{\omega,j}\in\mathbb{C}$, where the index $\omega$ enumerates the
eigenvalues. The solution to eq. (11) can be expressed in terms of the
eigenstates as
$\delta\vartheta_{j}(\theta,t)=\sum_{\omega}\eta_{\omega,j}f_{\omega,j}(\theta)e^{-\lambda_{\omega,j}t}\;,$
(13)
where the coefficients $\eta_{\omega,j}$ are determined from the initial
state, such that
$\delta\vartheta(z,t=0,\theta)=\sum_{j}e^{ik_{j}z}\sum_{\omega}\eta_{\omega,j}f_{\omega,j}(\theta)\;.$
(14)
According to eqs. (13) and (14), each $k$-mode (Fourier mode) of the filling
can be expressed as a sum of eigenstates $f_{\omega,j}(\theta)$, whose
evolution is determined by their corresponding eigenvalue
$\lambda_{\omega,j}$. The imaginary part of $\lambda$ describes ballistic
propagation, here manifesting as an oscillation of the eigenstate; the real
part of $\lambda$ represents diffusion, which dampens the oscillation.
Following the non-degeneracy of the spectrum of $\mathfrak{D}_{j}$, each
eigenstate $f_{\omega}$ evolves ballistically at a different frequency and
eventually dephase with respect to one-another, causing an apparent relaxation
of the mode $\delta\vartheta_{j}$. On shorter time scales, accounting for
diffusion merely leads to (slightly) faster apparent relaxation of the mode,
thus highlighting the difficulty in experimentally observing diffusion. On
long time scales, however, the redistributes of rapidities via diffusion
causes the $k$-mode to relax from a decoherent superposition of waves to a
stationary state.
The single-mode picture above has clear analogies to that of phonons in
Bogoliubov theory for quasi-condensates [48] (or more generally in Luttinger
liquid theory [49, 50]): For these effective low-energy Hamiltonians,
collective phase-density excitations in the form of phonons are the exact
eigenstates. In analog quantum field simulators, basis of phonons have been
used extensively to analyze experimental results (see e.g. Refs. [27, 28, 29,
30, 31]). However, when considering the underlying microscopic Hamiltonian
(here the Lieb-Liniger Hamiltonian), phonons are superpositions of the true
eigenstates and therefore do not have infinite lifetime. In Ref. [51], the
phonon lifetime was derived by expressing the phonon as a coherent
superposition of Bethe particle-hole states; over time, dephasing of the
particle-hole states leads to an apparent relaxation of the phonon, similarly
to the filling of a single $k$-mode $\delta\vartheta_{j}$.
### 2.3 Critical length scale of diffusion
The propagation operator kernel (12) explicitly shows that ballistic
propagation scales linearly with momentum $k$, whereas diffusion scales
quadratically. Thus, we can estimate the length scale at which diffusive
effects become important by diagonalizing the propagation kernel
$\mathfrak{D}_{j}$ and analysing its spectrum. First, we write the eigenvalues
of $\mathfrak{D}_{j}$ as
$\lambda_{\omega,j}\equiv i\alpha_{\omega,j}+\beta_{\omega,j}\;,$ (15)
where $\alpha_{\omega,j}$ and $\beta_{\omega,j}$ are both real. From the
expression (12), we see that $\alpha_{\omega,j}$ and $\beta_{\omega,j}$
describes the ballistic propagation and diffusive propagation of the
eigenstates, respectively. In the numerical study conducted in Ref. [47], it
was demonstrated that the spectrum of $\mathfrak{D}_{j}$ is non-degenerate and
can be ordered with respect to the value of $\alpha_{\omega}$. Further, the
eigenvalues are, to a good approximation, smooth functions of $j$ scaling as
$\displaystyle\alpha_{\omega,j}$ $\displaystyle\approx j\>\alpha_{\omega,j=1}$
(16) $\displaystyle\beta_{\omega,j}$ $\displaystyle\approx
j^{2}\>\beta_{\omega,j=1}$
Thus, in order to estimate whether a given mode $j$ propagates mainly
ballistically or diffusively, we compute the ratio
$\frac{\alpha_{\omega,j}}{\beta_{\omega,j}}\approx\frac{1}{j}\frac{\alpha_{\omega,j=1}}{\beta_{\omega,j=1}}\coloneqq\frac{j_{\omega}^{\star}}{j}\;,$
(17)
where we have defined the critical mode
$j_{\omega}^{\star}\coloneqq\frac{\alpha_{\omega,j=1}}{\beta_{\omega,j=1}}\;.$
(18)
Eigenstates of modes with $j>j_{\omega}^{\star}$ propagate mainly diffusively,
while eigenstates of modes with $j<j_{\omega}^{\star}$ propagate mainly
ballistically.
## 3 Magnitude of diffusive corrections in thermal states
Figure 2: Propagation metric $\Gamma$ scaled by the coupling strength $c$ for
thermal states at different dimensionless temperatures $\mathcal{T}$ and
interaction strengths $\gamma$. Lower values signify a relative increased
contribution of diffusion to quasi-particle propagation. The solid lines
indicate the boundary between the three main thermodynamic regimes of the
Lieb-Liniger model, namely the quasi-condensate (QC), ideal Bose gas (IBG),
and and Tonks-Girardeau gas (TG). The dashed lines mark additional sub-
regimes, namely the thermal-quantum statistics boundary for the fluctuations
of the quasi-condensate and the degeneracy transition of the ideal Bose gas.
For the purpose of quantifying the contribution of diffusion to the overall
propagation of quasi-particle in the state $\rho_{\mathrm{p}}$, we introduce
the measure
$\Gamma=\frac{1}{n}\int_{-\infty}^{\infty}\mathrm{d}\theta\>\rho_{\mathrm{p}}(\theta)\frac{|v^{\mathrm{eff}}(\theta)|}{D^{\mathrm{diag}}(\theta)}\;,$
(19)
where $D^{\mathrm{diag}}(\theta)=w(\theta)\rho_{\mathrm{s}}^{-2}(\theta)$ are
the diagonal elements of the diffusion kernel and $n$ the atomic density. The
measure $\Gamma$ relates the ballistic propagation velocity to the diagonal of
the diffusion kernel weighted by the quasi-particle density. For fixed values
of $\gamma=c/n$ we find that it scales linearly with the coupling strength
$c$. Hence, we calculate $\Gamma/c$ for a large number of thermal states for
various combinations of reduced temperatures $\mathcal{T}$ and interaction
strength $\gamma$. The results are plotted in figure 2, where the different
regimes of the Lieb-Liniger phase diagram have been indicated by solid lines.
We find that a minimum of $\Gamma/c$, indicating a maximal broadening of
quasi-particle trajectories relative to their distance travelled, is located
near the crossover between the three main regimes of the Lieb-Liniger model,
corresponding to $\mathcal{T}\approx\gamma\approx 1$. At higher temperatures,
diffusive effects appear to be most relevant for temperatures
$\mathcal{T}\sim\gamma^{-3/2}$, which coincides with the crossover between the
quasi-condensate and ideal Bose gas regimes. Interestingly, we find that
diffusion persists even at rather low temperatures; a similar observation was
made in Ref. [52], where diffusion at very low temperatures was shown to
result in an effective viscous hydrodynamics.
Upon approaching any of the asymptotic regimes, the quasi-particle propagation
becomes completely ballistic, indicated by a very large value of $\Gamma$. In
the Tonks-Girardeau and ideal Bose gas limits this behavior is expected; free
particles do not diffuse, and in these two regimes the gas behaves as free
fermions and bosons, respectively. Meanwhile, in the quasi-condensate regime,
the interaction energy dominates the average energy per atom, thus making
density fluctuations energetically costly, which in turn represses diffusion.
The 1D Bose gas is essentially maximally diffusive in the thermodynamic regime
furthest from any of the asymptotic regimes above. Note, that we find no
direct relation between $\Gamma$ and the $g^{(2)}$-function; instead,
diffusion appears maximal for kinetic and interaction energies around
$E_{\mathrm{int}}\sim E_{\mathrm{kin}}\sim Nk_{B}T/2$. See Appendix B for
more.
((a))
((b))
((c))
Figure 3: Temperature scaling of diffusive and ballistic propagation. (a)
Vertical cuts of the diagram 2 for three different interaction strengths
$\gamma$. (b) Diagonal elements of the diffusion kernel weighted by the quasi-
particle density: Diffusive contributions scale with increasing thermal
fluctuations, until the gas starts transitioning into a regime of free
particles. (c) Effective velocity weighted by the quasi-particle density. The
ballistic contribution plateaus at lower temperatures as the filling function
approaches a Fermi sea.
To unravel the individual scaling behavior of ballistic and diffusive
propagation, we plot their weighted contribution as function of temperature
for different values of $\gamma$ in figure 3. For reference, figure 3(a) shows
the corresponding values of $\Gamma/c$ (equivalent to vertical cuts of figure
2). Starting with the diffusive contribution, plotted in figure 3(b), we find
that it is maximal at a temperature $\mathcal{T}$ inversely proportional to
interaction $\gamma$; for lower temperatures, the diffusive contribution
increase linearly with $\mathcal{T}$, while for higher temperatures it
decreases again, scaling as $\mathcal{T}^{-1/2}$. The vanishing of diffusion
at very low temperatures follows from the suppression of thermal fluctuations
(see eq. (9)), as the filling $\vartheta$ approaches a Fermi sea state.
Meanwhile, for very high temperatures and fixed particle number, the
occupation of individual rapidity states becomes very low as quasi-particles
occupy increasingly higher rapidities; such behavior is indicative of the
transition towards free bosons, resulting in the eventual decrease of
diffusion.
In contrast, the ballistic contribution, plotted in figure 3(c), is almost
constant for lower temperatures until it suddenly starts increasing
proportionally to $\mathcal{T}^{1/2}$. Near the ground state (Fermi sea of
rapidity states), the effective velocity evaluated at the Fermi point is equal
to the sound velocity of the gas [53], while for rapidities between the two
Fermi points, $v^{\mathrm{eff}}(\theta)$ is approximately a linear function.
Thus, approaching lower temperatures, the fermionic statistics of the quasi-
particles ensure that rapidity states up to the Fermi point remain populated,
resulting in the plateau of the weighted ballistic contribution. As the
temperature increases and the Fermi sea melts, interactions in the gas become
less relevant, and the effective velocity approaches $\hbar\theta/m$; for
large rapidity this velocity is far greater than the elements of the diffusion
kernel, thus leading to a quasi-particle propagation completely dominated by
ballistic motion.
Finally, it should be stressed that $\Gamma$ can be computed for any state
$\vartheta$, not just thermal states. Indeed, explicitly constructing a state
which maximizes fluctuations will result in a larger diffusive broadening of
the quasi-particle trajectory. An important non-thermal state, is the initial
state of the seminal quantum Newton’s cradle experiment [54]; here, two halves
of a (typically thermal) state have been boosted to large, opposite rapidities
[55]. Notably, boosting the state substantially increases the quasi-particle
velocities while merely shifting $D^{\mathrm{diag}}(\theta)$ towards higher
rapidities, thus resulting in an increase in $\Gamma$. Nevertheless, diffusive
effects are important to the long time-scale dynamics of the quantum Newton’s
cradle, as purely ballistic propagation can produce quasi-particle
distributions featuring very fine structures in the ($z,\theta$)-phase-space
[56]. Such small wavelength features are much more susceptible to diffusive
effects, which over time lead to the features being washes out and the system
thermalizing [46].
## 4 Quasi-particle dynamics following a single-mode quench in a quasi-
condensate
Diffusive effects become increasingly important at shorter length scales. To
quantify these length scales, we can calculate the critical mode $j^{\star}$
of eq. (18). Note that $\Gamma$ and $j^{\star}$ are closely related; states
with low $\Gamma$ will likewise feature lower critical modes. One particular
system, where this sort of analysis is very insightful, is the single-mode
quench in a quasi-condensate realized in the recent experiment of Ref. [24].
Although the Bose gas was realized in a parameter regime where diffusive
effects are rather weak (based on our previous analysis), the ability to
excite a single $k$-mode (of the atomic density) makes such an experimental
platform ideal for studying diffusive effects.
### 4.1 Experimental setup and protocol
The experimental setup is already described in detail in Ref. [24]. Hence, we
will only summarize the details most relevant to this study.
In the experiment, a Bose gas of 87Rb atoms are trapped on an atom chip [57],
which creates a strong magnetic potential along two (transverse) axes, while
providing weak trapping along the third (longitudinal or 1D) axis, effectively
realizing a quasi-1D gas. A digital micromirror device (DMD) enables the
creation of arbitrary optical potentials along the 1D axis [58]; for this
protocol, the atoms are initially trapped in a box potential of length
$L=80\>\mu\mathrm{m}$. By modulating the bottom of the box trap in the shape
of a cosine, a density perturbation in the shape of single $k$-mode is
imprinted in the condensate (see Fig. 4(a)). Once confined in the box, the
condensate density is 60-80 $\text{atoms}/\mu\mathrm{m}$, and the initial
state is well-described by a thermal state with temperature in the range
50-120 nK (depending on the degree of cooling performed). Here we consider
three quenches with dimensionless interaction strength and temperature:
1. (a)
$\gamma=1.8\cdot 10^{-3}$ and $\mathcal{T}=1.2\cdot 10^{3}$,
2. (b)
$\gamma=2.1\cdot 10^{-3}$ and $\mathcal{T}=2.9\cdot 10^{3}$,
3. (c)
$\gamma=1.6\cdot 10^{-3}$ and $\mathcal{T}=2.8\cdot 10^{3}$.
Comparing with figure 2, we find that the gas is realized in the quasi-
condensate phase near the thermal-quantum statistics boundary for the
fluctuations; in this parameter regime, diffusive effects are expected to be
weak.
To initiate dynamics, the confining potential is quenched to a flat-bottomed
box trap at time $t=0$ and the subsequent dynamics and relaxation is monitored
by recording the atomic density via absorption imaging. An example of the
measured density evolution is plotted in figure 4(b). Owing to a high degree
of control over the 1D potential, a single density mode of the condensate can
be excited, that is, the atomic density inside of the box trap is accurately
given by $n(z,t)=n_{0}+\delta n_{j}(t)\cos\left(k_{j}z\right)$, where $\delta
n_{j}(t)$ is the amplitude of the mode and $k_{j}=2\pi j/L$ is its momentum.
For the quenches analysed here, only the $j=1$ mode is excited, however, the
ability to also address higher modes was also demonstrated in Ref. [24]. In
the context of Bogoliubov theory, the excited density mode corresponds to an
eigenstate of the effective low-energy Hamiltonian [48]. However, as
previously mentioned, the phononic basis does not represent the true
eigenstates of the system [51]. Further, due to the high temperature of the
condensate, the applicability of the low-energy model is very limited; we will
discuss and demonstrate this in the following section.
((a))
((b))
((c))
Figure 4: (a) Illustration of the experimental protocol of Ref. [24]: A 1D
box trap with a cosine-modulated bottom is realized using a DMD. At time
$t=0$, dynamics is initiated by quenching the box bottom to flat. (b) Measured
evolution of density perturbation in quench (c). The dashed lines indicate the
position of the box walls. (c) Filling function of the initial thermal state
of quench (c), along with its decomposition into a stationary homogeneous
background and perturbation.
### 4.2 Analysis of quasi-particle propagation
#### 4.2.1 Comparing experimental observations with linearized GHD
((a))
((b))
((c))
((d))
((e))
Figure 5: Analysis of propagation in the single-mode quench experiment of
Ref. [24]. On the left, the experimentally measured evolution of the lowest
cosine mode (black dots) compared with linearized GHD simulations (colored
lines). The interaction strength and reduced temperatures of the background
state of the three quenches are: (a) $\gamma=1.8\cdot 10^{-3}$ and
$\mathcal{T}=1.2\cdot 10^{3}$, (b) $\gamma=2.1\cdot 10^{-3}$ and
$\mathcal{T}=2.9\cdot 10^{3}$, (c) $\gamma=1.6\cdot 10^{-3}$ and
$\mathcal{T}=2.8\cdot 10^{3}$. On the right, (d) the critical modes
$j_{\omega}^{\star}$ of eq. (18) for eigenstates of the propagation operator,
and (e) the coefficients (i.e. population) of the eigenstates for the single-
mode perturbation excited in the experiment. We label the eigenstates by the
imaginary part of the corresponding eigenvalue $\alpha_{\omega,j=1}$.
To assess whether a linearization of the dynamics is valid for the system, we
simulate the dynamics of the gas using linearized GHD [47] and compare with
experimental observations. First, to extract the filling function of the
background and perturbation, we spatially Fourier transform the filling of the
initial thermal states and identify the $j=0$ mode as the stationary
background. See figure 4(c) for an example. The perturbation
$\delta\vartheta(z,\theta)$ is comprised of all remaining modes; since density
is a non-linear function of the filling, an excitation of a single-density
mode generally features a perturbation perturbation $\delta\vartheta$
containing multiple modes. Indeed, after Fourier transforming
$\delta\vartheta(z,\theta)$ we find higher modes of the filling to be
occupied, however, their amplitude is relative low (less than 10% of the $j=1$
mode), whereby we will focus solely on the evolution of
$\delta\vartheta_{j=1}(t,\theta)$.
Next, given the background filling $\vartheta_{0}(\theta)$ we calculate and
diagonalize the propagation kernel, whereby the evolution of the perturbation
can be calculated efficiently using eq. (13). The results are plotted in
figure 5 for the three single mode quenches. We find that the linearized GHD
describes the measured dynamics to high accuracy, basically reproducing the
predictions of fully interacting (Euler-scale) GHD. The agreement remains very
accurate even for the higher amplitude quenches, where the density of the
perturbation $\delta n_{1}$ is around 20% of the background density $n_{0}$.
Note, the colored lines plotted in figures 5(a), 5(b), and 5(c) show the
results of the GHD simulation without accounting for diffusion; when
simulating the linearized GHD dynamics of the system with diffusion, we find
no discernible difference in the results. Other mechanisms of relaxation
through integrability-breaking processes can also be neglected: At such high
temperatures, excitations in the transverse trapping potential would normally
lead to thermalization of the system [59, 20], however an emergent Pauli
blocking of the associated scattering events protects the one-dimensionality
of the system [24]. By virtue of the high chemical potential of the
condensate, all low-rapidity states are completely filled, while at higher
rapidities the filling features large thermal tails. In transverse-exciting
scattering events, outgoing rapidities are much smaller than incoming due to
the large gain in transverse potential energy; thus, all outgoing scattering
states fall within rapidity states already occupied, and the amplitude of the
scattering process vanishes.
#### 4.2.2 Quasi-particle dephasing and applicability of phonon basis
With contributions of both diffusion and a dimensional crossover being
negligible, the apparent relaxation of the density mode seen in figure 5 must
be due to the ballistic dephasing of quasi-particle trajectories. In figure
5(e) we plot the coefficients $\eta_{\omega,j=1}$, representing the population
of each propagation eigenstate for the $j=1$ perturbation. The eigenstates are
ordered by the imaginary part (ballistic contribution) of their corresponding
eigenvalue $\alpha_{\omega,j=1}$. At low values of $\alpha_{\omega,j=1}$ the
population of the corresponding eigenstates is zero, as these states belongs
to the background $\vartheta_{0}(\theta)$. For all quenches, the population
$\eta_{\omega,j=1}$ is peaked around the edge of the background and features
long thermal tails. Having compared to other quenches of higher modes, we find
that $\eta_{\omega,j}$ is mostly independent of the mode number $j$; instead
its width/extend depends on the temperature, while its total area depends on
the initial mode amplitude. Thus, one can immediately understand the
temperature dependence of the relaxation rate: A larger spread in populated
ballistic eigenvalues $\alpha_{\omega,j=1}$ leads to a faster dephasing of the
eigenstates comprising the mode $\delta\vartheta_{j=1}$. In the context of
phonons, the extend of the populations $\eta_{\omega,j=1}$ is thus an
indication of the phonon lifetime. Given the fast relaxation of the density
mode (particularly in the hot realizations), a phonon basis is not
particularly suitable for describing the system. However, as the temperature
decreases, the population of large $\alpha_{\omega,j=1}$ vanishes,
substantially increasing the phonon lifetime. For this particular setup, a
condensate of 30 nK (half the temperature of quench (a)) would be reasonably
described by an effective low-energy Hamiltonian.
#### 4.2.3 Diffusive length scales
Finally, we calculate the critical mode $j_{\omega}^{\star}$, whose spectra
are shown in figure 5(d), also labelled by the imaginary part of the
corresponding eigenvalue. We find that $j_{\omega}^{\star}\gg 1$ for all
eigenstates, demonstrating that the GHD dynamics of the addressed mode is
entirely dominated by ballistic propagation of the quasi-particles. From the
spectrum of $j_{\omega}^{\star}$ we may identify the length scales at which
diffusion would become relevant. First, we note that the critical mode is
highly dependent on the eigenstate: Since $\alpha_{\omega,j=1}$ denotes the
ballistic contribution to quasi-particle propagation, we would expect
$j_{\omega}^{\star}$ to be minimal around low $\alpha$-values. This is indeed
the case, however, the corresponding states all belong to the stationary
background. Likewise, at large $\alpha$ the critical mode is very high.
Interestingly, at intermediate values of $\alpha_{\omega,j=1}$, the spectra of
$j_{\omega}^{\star}$ all feature a local minimum. The location of the minimum
is related to the states with maximal thermal thermal fluctuations (see eq.
(9)). Further, comparing with the eigenstate population of the perturbation of
figure 5(e), we find that the location of the minimum coincides with the most
populated eigenstates; the observed dynamics of the system is thus mainly
driven by the evolution of these particular eigenstates. From figure 5(d), we
find a minimum critical mode of $j_{\omega}^{\star}\sim 200$. Hence, given the
same experimental parameters, diffusive effects would become dominant for
quenches of the $j=200$ mode and higher. For an experimental box length of
$L=80$ µm, the corresponding diffusive length scale is around 0.4 µm, which is
close to the healing length of the condensate. At such length scales, higher
order (e.g. dispersive) corrections become relevant for dynamics [60, 61, 62,
63]. Further, from an experimental perspective, length scales on that order
are typically not resolvable by the available imaging techniques.
### 4.3 Single-mode quench in the maximal diffusive regime
((a))
((b))
((c))
Figure 6: Single-mode quench in a thermodynamic regime of maximal diffusion
(see text for parameters). (a) The critical modes $j_{\omega}^{\star}$ the
propagation operator and coefficients (i.e. population) of the eigenstates. We
label the eigenstates by the imaginary part of the corresponding eigenvalue
$\alpha_{\omega,j=1}$. (b) Evolution of addressed $j=1$ density mode
calculated using the linearized GHD with and without diffusion, labelled as
ballistic and diffusive, respectively. The area between the two curves has
been shaded in orange for improved visibility. (c) Filling function
$\vartheta$ after evolution time $t=t_{\mathrm{evol}}$.
To illustrate the difficulty in observing diffusive effects in the 1D Bose gas
from measuring its density, we consider a single-mode quench similar to the
experiment [24] but realized near the maximally diffusive regime, identified
via figure 2, here $\gamma=1$ and $\mathcal{T}=2$. The resulting parameters
are close to a system considered in Ref. [47]. For a gas of 87Rb atoms, this
would correspond to an extremely dilute system with a density of
$1\>\mathrm{atoms}/\mu\mathrm{m}$ and temperature $5.5$nK. In order to
simplify comparison to the experiment, we here also treat a quench of the
$j=1$ density mode, however, we scale the length of the system $L$ to near the
diffusive length scale, resulting in $L=2\>\mu\mathrm{m}$.
The resulting critical mode spectrum and eigenstate population are plotted in
figure 6(a). Compared with the spectrum of the experimental system (Fig.
5(d)), we find that the critical mode here is much lower for all populated
states. Further, the lack of a Fermi sea in the background state means that
the highly diffusive eigenstates (low $\alpha$) are also occupied by the
perturbation. Thus, on average, we would expect modes of length scale around
$L$ to exhibit diffusive behavior (following our choice of system length).
Next, in figure 6(b), we plot the density of the $j=1$ mode obtained from
linearized GHD simulations for a duration of
$t_{\mathrm{evol}}=3.5\,\mathrm{ms}$, both with and without diffusion. Despite
the system being realized around the maximally diffusive thermodynamic regime,
the influence of diffusion on the density dynamics is very limited. Indeed, we
find a hardly measurable difference in the relaxation of the density mode upon
accounting for diffusion. Comparing higher moments [64] does not yield a clear
indication of diffusion either.
Finally, figure 6(c) depicts the filling function $\vartheta$ at time
$t=t_{\mathrm{evol}}$. Here, the effect of adding diffusive corrections to the
ballistic propagation is evident; following ballistic propagation, the filling
function develops fine structures, which are washed out in the presence of
diffusion. Upon calculating expectation values of observables, the rapidity is
integrated over; functions, which are very sensitive to the exact shape of the
filling, could be used to detect the presence (or lack of) of fine structure
in the rapidity distribution. However, experimental setups are highly limited
in the measurable observables available. Measurement of the rapidity density
[65] integrates over space, thus similarly hiding the underlying structure.
Hence, for a 1D Bose gas confined to a box, is it very difficult to
differentiate a ballistically dephased system from one that has relaxed via
diffusion.
## 5 Conclusion
To summarize, we have quantified the effect of diffusion in 1D Bose gases by
calculating the diffusive spreading of quasi-particle trajectories relative to
their ballistic propagation velocity. Computing this measure for a number of
thermal states, we have found that diffusive effects are most prominent in the
transition regions between the different thermodynamic regimes of the Lieb-
Liniger model, particularly for interaction strengths and temperatures around
$\gamma\sim\mathcal{T}\sim 1$. Diffusion, which scales with thermal
fluctuations of the state, increases with temperature up to a certain point,
where the gas transitions into a state of free particles. The parameter regime
of maximal diffusion is accessible by existing experimental setups, which is
encouraging for future studies.
Next, by diagonalizing the linearized propagation operator, we have identified
diffusive length scales in an experimental quench of a single density mode in
a quasi-condensate. For this system, diffusive effects are found to be very
weak. Only upon approaching lengths scales around the condensate healing
length, do diffusive effects become dominent. Thus, on experimental time
scales, the effect of diffusion on the density evolution can be completely
neglected; the observed relaxation can instead be understood solely from the
dephasing of ballistic quasi-particle trajectories. Nevertheless, diffusive
effects are still important for the dynamics at longer time scales: Following
ballistic dephasing of the density mode, the system remains in a non-
equilibrium state where ever finer structures in the filling develop [66, 56,
67]; these structures are integrated over to obtain the density, and are
therefore not observed in the experiment. Once the length scale of said
structures approach the critical mode, diffusive effects wash out their
features and drive the system towards thermal equilibrium [14, 68]. Hence, in
order to observe diffusive dynamics in an experimental setup, one must realize
a system where expectation values of measurable observables for balistically
dephased states and thermalized states are distinguishable. One particular
setup, which fulfills this condition, is the quantum Newton’s cradle.
Finally, the analysis facilitates an evaluation of the applicability of a
phonon basis, which have been used extensively to describe quantum field
simulators realized with ultracold gases. For the given setup, the lifetime of
the excited mode is highly dependent on temperature: The particular quenches
analyzed here feature a rather short lifetime, however, for lower temperatures
(within experimental reach) the lifetime of the mode exceeds experimental time
scales.
## Acknowledgements
We thank Jacopo de Nardis and Andrew Urichuk for enlightening discussions and
for suggesting the measure $\Gamma$.
##### Funding information
This work is supported by the European Research Council: ERC-AdG ’Emergence in
Quantum Physics’ (EmQ) under Grant Agreement No. 101097858 and the DFG/FWF CRC
1225 ’ISOQUANT’, (Austrian Science Fund (FWF) P 36236) and Research Unit FOR
2724 ’Thermal machines in the thermal world’, (Austrian Science Fund (FWF) I
6047).-
## Appendix A Thermodynamic Bethe Ansatz
The Bethe Ansatz identifies the elementary excitations of the 1D Bose gas as
fermionic quasi-particles, uniquely labelled by their quasi-momentum, or
rapidity, $\theta$ [32, 33]. In the thermodynamics limit, the rapidity becomes
a continuous variable, with the density of occupied rapidities, i.e. the
density of quasi-particles, $\rho_{\mathrm{p}}(\theta)$ fully characterising
the thermodynamic properties of the local equilibrium macrostate. Similarly, a
density of holes $\rho_{\mathrm{h}}(\theta)$ can be introduced, which
describes the density of unoccupied rapidities; together the two densities
form the density of states $\rho_{\mathrm{s}}(\theta)$, which obeys the
relation
$\rho_{\mathrm{s}}(\theta)=\rho_{\mathrm{p}}(\theta)+\rho_{\mathrm{h}}(\theta)=\frac{1}{2\pi}-\frac{1}{2\pi}\int_{-\infty}^{\infty}d\theta^{\prime}\,\Delta(\theta,\theta^{\prime})\rho_{\mathrm{p}}(\theta^{\prime})\;,$
(20)
where
$\Delta(\theta,\theta^{\prime})=-\frac{2c}{c^{2}+(\theta-\theta^{\prime})^{2}}$
is the two-body scattering kernel of the Lieb-Liniger model. Given
$\rho_{\mathrm{p}}$ one can compute coarse-grained expectation values of local
observables, in particular the atomic density following
$n=\int_{-\infty}^{\infty}\mathrm{d}\theta\>\rho_{\mathrm{p}}(\theta)$.
Equivalently, one can encode the thermodynamic properties of the system in the
filling function
$\vartheta(\theta)=\rho_{\mathrm{p}}(\theta)/\rho_{\mathrm{s}}(\theta)$, which
describes the occupational fraction of allowed rapidity states. Quasi-
particles of the Lieb-Liniger model follow Fermionic statistics, whereby the
filling assumes values between 0 and 1. The ground (zero temperature) state is
given by a Fermi sea of rapidities, where $\vartheta(\theta)=1$ for rapidities
within the interval of two Fermi points (whose value is determined by
$\gamma$) and 0 everywhere else. Meanwhile, the filling of a thermal state is
given by
$\vartheta(\theta)=\frac{1}{1+e^{\varepsilon(\theta)\beta}}\;,$ (21)
where $\beta$ is the inverse temperature, and the pseudo-energy
$\varepsilon(\theta)$ is acquired from solving the Yang-Yang equation [36]
$\varepsilon(\theta)=\frac{\hbar^{2}\theta^{2}}{2m}-\mu+\frac{1}{2\pi\beta}\int_{-\infty}^{\infty}\mathrm{d}\theta^{\prime}\>\Delta(\theta,\theta^{\prime})\ln\left(1+e^{\varepsilon(\theta^{\prime})\beta}\right)\;,$
(22)
where $\mu$ is the local chemical potential.
## Appendix B Quasi-particle propagation versus energy
The thermodynamic regimes of the Lieb-Liniger model are identified by their
two-point correlation function $g^{(2)}(0)$ [35]. However, although the
minimum of the propagation measure $\Gamma$ (corresponding to a relative
maximal influence of diffusion on quasi-particle propagation) aligns with the
crossover between the different regimes, we find no clear relation between it
and the $g^{(2)}$-function. Instead, we study the relation between $\Gamma$
and the distribution of energy in the gas, which itself it related to the
$g^{(2)}$-function. In the asymptotic regimes of the Lieb-Liniger model, the
system energy is dominated either by kinetic energy (free particle regimes) or
interaction energy (quasi-condensate regime). To study the relation between
energy and diffusion, we plot $\Gamma/c$ as function of the mean interaction
and kinetic energy per atom of the system, which, respectively, are given by
[69]
$\displaystyle E_{int}/N$ $\displaystyle=\frac{\hbar^{2}}{2m}cng^{(2)}(0)\;,$
(23) $\displaystyle E_{kin}/N$ $\displaystyle=E/N-E_{int}/N\;,$ (24)
where the total energy per atom $E/N$ is
$E/N=\frac{1}{n}\int_{-\infty}^{\infty}\mathrm{d}\theta\>\epsilon^{\mathrm{dr}}\vartheta(\theta)\;.$
(25)
The result can be seen in figure 7, with the minimum of $\Gamma/c$ being
situated around the equipartition value $E_{int}/(Nk_{B}T)\sim
E_{kin}/(Nk_{B}T)\sim 0.5$.
Figure 7: Propagation measure $\Gamma$ of eq. (19) scaled by the coupling
strength $c$ as function of the mean kinetic and interaction energy per atom
in the gas. The dashed lines mark $E_{kin}/(Nk_{B}T)=0.5$ and
$E_{int}/(Nk_{B}T)=0.5$.
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|
# Active Covering
Heinrich Jiang Afshin Rostamizadeh
###### Abstract
We analyze the problem of active covering, where the learner is given an
unlabeled dataset and can sequentially label query examples. The objective is
to label query all of the positive examples in the fewest number of total
label queries. We show under standard non-parametric assumptions that a
classical support estimator can be repurposed as an offline algorithm
attaining an excess query cost of $\widetilde{\Theta}(n^{D/(D+1)})$ compared
to the optimal learner, where $n$ is the number of datapoints and $D$ is the
dimension. We then provide a simple active learning method that attains an
improved excess query cost of $\widetilde{O}(n^{(D-1)/D})$. Furthermore, the
proposed algorithms only require access to the positive labeled examples,
which in certain settings provides additional computational and privacy
benefits. Finally, we show that the active learning method consistently
outperforms offline methods as well as a variety of baselines on a wide range
of benchmark image-based datasets.
Active Learning
## 1 Introduction
Active learning is an increasingly important practical area of machine
learning as the amount of data grows faster than the resources to label these
datapoints, which can be costly. In this paper, we introduce a variant of the
active learning problem, called active covering, where the goal is to label
query all of the positive examples given an unlabeled dataset in as few total
queries as possible.
Active covering arises in many machine learning problems. In credit card fraud
detection, one goal is to find the instances of fraud with as few queries as
possible to the user asking if a transaction was fraudulent (Awoyemi et al.,
2017). In mineral exploration, the goal is to find all of the valuable
resources with as little exploration as necessary (Acosta et al., 2019). In
computational drug discovery, one goal is to discover all of the effective
drugs with as few suggested candidates as possible, as running trials on each
candidate can be costly (Ou-Yang et al., 2012). In bank loan applications, a
label query means granting the loan to an applicant (as the only way to know
if the applicant would default or not)– as such, negative label queries can be
costly (Khandani et al., 2010). Finally, many internet applications need to
moderate abusive content, and the goal is to quickly identify and remove all
of the abusive content with as few label queries as possible to human
operators (Nobata et al., 2016).
For our analysis, we assume that an unlabeled pool of $n$ datapoints is
generated by drawing i.i.d. samples from a mixture of two distributions, where
one of the distributions represents the positive examples and the other the
negative examples. Our goal is to retrieve all of the positive examples from
this pool with as few label queries as possible. Furthermore, we assume that
the support of the positive examples is compact and its density function is
lower bounded by a positive quantity. Finally, we leverage a few additional
standard and mild nonparametric regularity assumptions on the the curvature of
the support of positive examples. These assumptions allow for a wide range of
distributions (e.g. mixtures of truncated multivariate Gaussians), and in
particular the support of positives and negatives can be of any shape and can
arbitrarily overlap.
We first establish the optimal active covering algorithm, which has knowledge
of the support of positive examples. This algorithm, thus, will query all of
the examples that lie in the support of positive examples (which will contain
some negative examples where there is an overlap with support of negative
examples). Then, the performance of any active covering algorithm can be
compared to the optimal learner via the expected excess query cost that the
algorithm incurs in addition to what the optimal learner is expected to incur.
To begin with, we analyze an offline algorithm that is based on a classical
method in support estimation, providing both upper and lower bounds on the
excess query cost. We then provide an active learning procedure based on the
explore-then-commit strategy, where we first explore with an initial random
sample, and then exploit by querying the example closest to any of the
positive examples found thus far. We show that the active learning procedure
achieves a provably better excess query cost than the offline algorithm.
The presented methods also have the beneficial property of using only the
queried positive examples for learning. This not only has desirable
computational implications in that we don’t need to store the negative
examples (especially in highly label imbalanced datasets with few positives),
but also is practical in situations where there are strict privacy
requirements for the negative examples. For example, this arises in the
problem of fake account detection faced by social network platforms (García-
Recuero, 2016), where real account information is highly sensitive data and
may even not be allowed to use for training. Another such application is spam
email detection, where it may be desirable (or necessary) to avoid training
with non-spam emails (Li et al., 2008). While this privacy aspect is not a
focus of the paper, it may be a feature of independent interest.
We now summarize our contributions as follows.
* •
In Section 2, we introduce and formalize the active covering problem and
establish the notion of excess query cost.
* •
In Section 3, we analyze the offline learner and show matching upper and lower
bounds on the excess query cost of $\widetilde{\Theta}(n^{D/(D+1)})$.
* •
In Section 4, we introduce and analyze the Explore-then-Commit active
algorithm and show it has excess query cost $\widetilde{O}(n^{(D-1)/D})$.
* •
In Section 6, we show empirical results on a wide range of benchmark image-
based datasets (Letters, MNIST, Fashion MNIST, CIFAR10, CelebA) comparing the
Explore-then-Commit algorithm to a number of offline and active baselines.
## 2 Active Covering
In this section, we formulate the active covering problem. We are given an
unlabeled dataset of $n$ datapoints, $X$, and the goal is to minimize the
number of label queries necessary until all positive examples are labeled. We
provide the assumptions on the underlying distribution from which $X$ is drawn
from and establish the notion of excess query cost, which is the additional
label queries compared to the optimal procedure, which will be defined later.
### 2.1 Theoretical Setup
We make the following assumption on the data generating process, which says
that with probability $p$, we draw a positive example from distribution
$\mathcal{P}_{+}$ and with probability $1-p$, we draw a negative example from
distribution $\mathcal{P}_{-}$.
###### Assumption 1.
The dataset $X$ is drawn i.i.d. from a distribution
$\mathcal{P}:=p\cdot\mathcal{P}_{+}+(1-p)\cdot\mathcal{P}_{-}$, for some
$p\in(0,1)$ and $\mathcal{P}_{+}$ and $\mathcal{P}_{-}$ are distributions over
$\mathbb{R}^{D}$ of the positive and negative examples respectively.
We then require the following regularity assumption on the distribution of
positive examples. The first part ensures that the density of positive
examples is lower bounded in its support; otherwise, some positive examples
will only appear as outliers, making it impossible for any learner to find
them in a non-trivial manner. The second part ensures that the support of
positive examples does not become arbitrarily thin anywhere, otherwise there
may not be any samples drawn from these regions and it will be very difficult
to recover the entire support from a finite sample. This is a standard
assumption in a variety of density estimation scenarios e.g. (Cuevas et al.,
1997; Singh et al., 2009).
###### Assumption 2.
There exists density function $f_{+}:\mathbb{R}^{D}\rightarrow\mathbb{R}$
corresponding to $\mathcal{P}_{+}$ with a compact support $\mathcal{X}_{+}$
that can be decomposed into a finite number of connected components. There
exists $\lambda_{0},r_{0},C_{+}>0$ such that the following holds:
* •
$f_{+}(x)\geq\lambda_{0}$ for all $x\in\mathcal{X}_{+}$.
* •
For all $0<r<r_{0}$ and $x\in\mathcal{X}_{+}$, we have $\text{Vol}(B(x,r)\cap
X_{+})\geq C_{+}\cdot\text{Vol}(B(x,r))$, where
$B(x,r):=\\{x^{\prime}\in\mathbb{R}^{D}:|x-x^{\prime}|\leq r\\}$.
The final assumption ensures that the negative example’s density is upper
bounded to ensure that there are no possible arbitrarily dense clumps of
negative examples near the boundary of $\mathcal{X}_{+}$. Otherwise, querying
in such regions may yield too few positive examples and it may be difficult to
learn that such a region is part of the support of positive examples.
###### Assumption 3.
There exists density function $f_{-}:\mathbb{R}^{D}\rightarrow\mathbb{R}$
corresponding to $\mathcal{P}_{-}$ and $\lambda_{1}>0$ such that
$f_{-}(x)\leq\lambda_{1}$ for all $x\in\mathcal{X}$.
Our assumptions are quite mild as they allow for a wide range of
distributions. In particular, they are non-parametric so there are no
assumptions that the data is generated based on a parameterized model.
Moreover, the support of the positive examples (as well as the support of the
negative examples) can be of arbitrary bounded shape, need not be a single
connected component (i.e. can appear as a number of connected components), and
can intersect arbitrarily with the support of the opposite label. Such mild
assumptions can model the wide range of data distributions arising in
practice.
Perhaps the strongest of our assumptions is that the density on
$\mathcal{X}_{+}$ is lower bounded. We stress that if the density of the
positive examples on $\mathcal{X}_{+}$ can become arbitrarily small, then some
regions can become so low-density that the positive examples in those regions
can be considered outliers. For such outliers, it may be unrealistic to expect
a procedure to efficiently find them. Nonetheless, in practice, our
methodology can still be applied on datasets containing positive outliers, but
without guarantees that those outliers will be efficiently recovered. We made
this assumption to keep our theoretical analysis from becoming too involved;
however it’s worth noting that it is possible to relax this assumption and
allow the density to have smooth boundaries. To do so, we would assume
parameters that bound the rate of decay of the density as it goes to $0$ and
provide final guarantees that depend on the decay parameters. Assumptions used
in recent analysis on densities with smooth boundaries (Zhao & Lai, 2020) can
be adapted here, which is a future research direction.
### 2.2 Optimal Learner and Excess Query Cost
We will analyze a number of new learners for this problem. To do so, we
establish a metric called excess query cost, which compares the learner to
that of the the optimal learner, which takes the optimal strategy given
knowledge of $\mathcal{X}_{+}$. This learner is unattainable in practice, but
serves as a theoretical limit in which to quantify the excess query cost of an
algorithm with respect to, to be defined below.
The optimal learner has knowledge of $\mathcal{X}_{+}$ and therefore its
strategy is to label query every point in $\mathcal{X}_{+}$. Let
$Q_{\text{opt}}$ be the number of label queries incurred by the optimal
learner. Thus, the optimal learner attains an expected number of queries of:
$\displaystyle\mathbb{E}[Q_{\text{opt}}]=n\cdot\mathcal{P}(\mathcal{X}_{+}).$
We can then for any algorithm define the notion of excess query cost, which is
the additional label queries needed compared to the optimal learner. That is,
if the cost of an algorithm is $C$, then the excess query cost is
###### Definition 1 (Excess Query Cost).
Suppose that an algorithm needs to make $Q$ label queries before labeling all
of the positive examples. Then the excess query cost of the procedure is
defined as: $C:=Q-Q_{\text{opt}}$.
For the results in the paper, we analyze the expected excess query cost, where
the expectation is taken over the distribution from which sample $X$ is drawn
from.
### 2.3 Passive Learner
We first provide a result for the passive learning algorithm, which queries
labels uniformly at random until all positive examples have been retrieved and
serves as our most basic baseline. In the following theorem we show a lower
bound on the excess query cost that is linear in the pool size.
###### Theorem 1 (Lower bound for Passive Learner).
There exists a distribution satisfying Assumptions 1, 2, and 3 such that with
probability at least $\frac{1}{2}$, we have (letting $C_{\text{passive}}$ be
the excess query cost of the passive learner):
$\mathbb{E}[C_{\text{passive}}]\geq\frac{1}{2}\cdot n$.
Algorithm | Excess Query Cost
---|---
Passive | $\Theta(n)$
Offline | $\widetilde{\Theta}(n^{D/(D+1)})$
Active | $\widetilde{O}(n^{(D-1)/D})$
Table 1: Summary of algorithms and results.
## 3 Offline Learner
Algorithm 1 Offline Learner
Inputs: Dataset $X$, initial sample size $m$.
Let $X_{0}$ be $m$ examples sampled uniformly without replacement from $X$.
Label query $X_{0}$ and let $X_{0,+}$ be the positive examples.
Label query remaining examples in ascending order of minimum distance to
$X_{0,+}$ (i.e. $x\rightarrow\min_{x^{\prime}\in X_{0,+}}|x-x^{\prime}|$)
until all positive examples are label queried.
The offline learner (Algorithm 1) will first randomly sample $m$ points to
label query and then label queries the remaining examples in ascending order
of minimum distance to any of the initially sampled positive examples until
all positives are labeled. It’s worth noting that we may not know when all of
the positive examples are labeled– thus, in practice, we can terminate the
algorithm when enough positives are found depending on the task or when the
labeling budget runs out.
The offline learner is based on a classical technique in the support
estimation literature (Cuevas et al., 1997), where the support of a
distribution can be covered with a finite sample of $m$ points drawn from this
distribution by taking the $\epsilon$-neighborhood of the sample for
appropriate $\epsilon$. We apply this methodology to only the positive
examples in our initial sample of $m$ points. We show that the Algorithm 1
finishes when it label queries everything within
$\epsilon\approx(\log(m)/m)^{1/D}$ of the initial positive examples, and thus
it will cover all the examples in $\mathcal{X}_{+}$ and the excess cost will
be proportional to $\epsilon\cdot n$ (all details in the Appendix).
The formal guarantee for the excess query cost is a follows:
###### Theorem 2 (Excess Query Cost for Offline Learner).
Suppose that Assumptions 1, 2, and 3 hold. Let $0<\delta<1$. There exists
$C,M_{0}>0$ depending on $\mathcal{P}$ such that the following holds. In
Algorithm 1, suppose that $m$ is chosen sufficiently large such that
$m\geq\frac{2\log(2/\delta)}{p^{2}}$ and $\frac{m}{\log
m}\geq\log(4/\delta)\cdot M_{0}$. Then, with probability at least $1-\delta$,
Algorithm 1 has an expected excess query cost of:
$\displaystyle\mathbb{E}[C_{\text{offline}}]$ $\displaystyle\leq(1-p)\cdot
m+C\cdot\left(\frac{\log(4/\delta)\cdot\log(p\cdot m/2)}{m}\right)^{1/D}\cdot
n.$
We now have the following immediate result by optimizing $m$ as a function of
$n$, trading off the cost from the initial exploration (first term) and the
cost from the exploitation (second term):
###### Corollary 1.
Under the conditions of Theorem 2, setting $m\approx n^{D/(D+1)}$ results in
expected excess query cost bounded as follows:
$\displaystyle\mathbb{E}[C_{\text{offline}}]\leq\text{PolyLog}(n,1/\delta)\cdot
n^{D/(D+1)}.$
We also provide the following lower bound, which shows that the offline
learner cannot achieve a better rate (proof found in the appendix).
###### Theorem 3 (Lower Bound for Offline Learner).
There exists a distribution satisfying Assumptions 1, 2, and 3 such that with
probability at least $\frac{1}{4}$, we have for $n$ sufficiently large and
some constant $C>0$:
$\displaystyle\mathbb{E}[C_{\text{offline}}]\geq(1-p)\cdot
m+C\cdot\left(\frac{\log m}{m}\right)^{1/D}\cdot n.$
## 4 Active Explore-then-Commit Learner
We next show an active approach (Algorithm 2) inspired by Explore-then-Commit
strategy (Garivier et al., 2016) that proceeds by first exploring by randomly
sampling a set of examples, and then commits to a greedy approach of choosing
the closest unlabeled example to any positive example labeled thus far until
all of the positive examples are labeled.
Algorithm 2 Active Explore-then-Commit Learner
Inputs: Dataset $X$, initial sample size $m$.
Let $X_{0}$ be $m$ examples sampled uniformly without replacement from $X$.
Label query $X_{0}$ and let $X_{+,0}$ be the positive examples.
Initialize $X_{p}\leftarrow X_{+,0}$ and $X_{a}\leftarrow X_{0}$
while not all positives examples in $X$ are labeled do
Label query $x=\operatorname*{arg\,min}_{x\in X\backslash X_{a}}d(x,X_{p})$
if $x$ has a positive label then
$X_{p}\leftarrow X_{p}\cup\\{x\\}$.
end if
$X_{a}\leftarrow X_{a}\cup\\{x\\}$.
end while
To analyze the algorithm there are three key steps: first we show that in the
explore phase, we choose at least one example from each connected component
(CC) of $\mathcal{X}_{+}$ (Lemma 1). Next, we show that in any CC of
$\mathcal{X}_{+}$, all of the positive examples are in the same connected
component in the $\epsilon$-neighborhood graph for some $\epsilon$ specified
later (Lemma 2). The final step is combining these two results to show a bound
on the excess query cost.
We now give the following result which says that for $m$ sufficiently large,
depending on the probability mass distribution of the CCs of
$\mathcal{X}_{+}$, we will with high probability have a positive example from
each of the CCs in the initial sample.
###### Lemma 1.
Suppose Assumptions 1 and 2 hold and let $0<\delta<1$. Let the connected
components of $\mathcal{X}_{+}$ be $\mathcal{X}_{+,1},...,\mathcal{X}_{+,c}$.
Let $q:=\min_{i\in[c]}\mathcal{P}_{+}(\mathcal{X}_{+,i})$. If
$\displaystyle
m\geq\max\left\\{\frac{2\log(2c/\delta)}{p\cdot\log(1/(1-q))},\frac{2\log(2/\delta)}{p^{2}}\right\\},$
then with probability at least $1-\delta$, the initial $m$ examples will
contain a positive example in each of $\mathcal{X}_{+,i}$ for $i\in[c]$.
The next result shows that the positive examples in each CC of
$\mathcal{X}_{+}$ appear in the same CC of the $\epsilon$-neighborhood graph
of the positive example for appropriate choice of $\epsilon$. This will be
important in showing that after greedily sampling enough examples after the
explore phase, we will query all of the examples in $\mathcal{X}_{+}$ but not
query examples that are more than $\epsilon$ away from $\mathcal{X}_{+}$.
###### Lemma 2 (Connectedness).
Suppose Assumptions 1 and 2 hold. Let $0<\delta<1$ and
$\mathcal{X}_{+,1},...,\mathcal{X}_{+,c}$ be the connected components of
$\mathcal{X}_{+}$. The following holds with probability at least $1-\delta$.
For each $i\in[c]$, we have that $\mathcal{X}_{+,i}\cap X_{+}$ is connected in
the $\epsilon$-neighborhood graph of $X_{+}$, where
$\displaystyle\epsilon=3\left(\frac{C_{0}\cdot D\log(2/\delta)\cdot\log
n}{p\cdot C_{+}\cdot\lambda_{0}\cdot v_{D}\cdot n}\right)^{1/D},$
and $n$ is sufficiently large so that $\epsilon\leq r_{0}$.
We now combine the two results to obtain an excess query cost guarantee for
the active learner. Lemma 1 ensures that our initial sample of $m$ examples
contains an example from each CC of $\mathcal{X}_{+}$ and Lemma 2 ensures that
when we actively sample in a greedy manner, we eventually query all of the
positive examples and never query any example that is too far from
$\mathcal{X}_{+}$– this farness determines how much the active algorithm
samples outside of $\mathcal{X}_{+}$ and hence determines the expected excess
query cost.
###### Theorem 4 (Excess query cost for Algorithm 2).
Suppose Assumptions 1, 2, and 3 hold and let $0<\delta<1$. Let the connected
components of $\mathcal{X}_{+}$ be $\mathcal{X}_{+,1},...,\mathcal{X}_{+,c}$
and $q:=\min_{i\in[c]}\mathcal{P}_{+}(\mathcal{X}_{+,i})$. There exists
constants $C,N_{0}>0$ depending on $\mathcal{P}$ such that the following
holds. If
$\displaystyle m$
$\displaystyle\geq\max\left\\{\frac{2\log(4c/\delta)}{p\cdot\log(1/(1-q))},\frac{2\log(4/\delta)}{p^{2}}\right\\},$
and $\frac{n}{\log n}\geq N_{0}\log(4/\delta)$, then with probability at least
$1-\delta$, we have the following excess query cost guarantee for Algorithm 2:
$\displaystyle\mathbb{E}[C_{\text{exp-commit}}]\leq
m+C\cdot\left((\log(4/\delta)\cdot\log n\right)^{1/D}\cdot n^{(D-1)/D}.$
###### Remark 1.
Our requirement for $m$ is tight w.r.t. $q$ and $c$ in the case where
$q=\frac{1}{c}$ (i.e. equal probability of each CC). In this case, it reduces
down to the classic coupon collector problem (Boneh & Hofri, 1997): each CC is
a coupon and the expected number of times we draw a coupon with replacement
until we receive one example from each is $\Omega(c\log c)=\Omega(\log
c/\log(1/(1-q)))$ by Taylor expansion of $\log(1/(1-q))$.
###### Remark 2.
While our results all have a strong dependence on the dimension (commonly
referred to as the curse of dimensionality), it’s been shown that non-
parametric techniques such as these algorithms can automatically adapt to the
intrinsic dimension of the data and the convergence rates can be shown to
depend only on this dimension and not the ambient dimension (i.e. arguments
from Pelletier (2005); Kpotufe (2011); Jiang (2017, 2019) can be adapted
here).
## 5 Related Works
The problem of actively retrieving the positive labeled examples was studied
as active search by Garnett et al. (2012), where the goal is to label query as
many positive examples given a fixed budget and they propose a sequential
Bayesian approach that optimizes the expected number of positive labels across
timesteps; however their method is computationally expensive requiring
$O((2\cdot n)^{\ell})$ runtime where $\ell$ is the number of lookahead
timesteps to optimize over. Efficient approximations to this active search
technique have been proposed (Jiang et al., 2018, 2019). In our theoretical
setting, the goal is to label query all of the positive examples with as few
label queries as possible rather than having a fixed known labeling budget.
A recent work by Jain & Jamieson (2019) designs an algorithm to actively
identify the largest number of positive examples while minimizing or
constraining the false-discovery rate (i.e. false-negative rate). They propose
a bandit-style active elimination procedure which strategically label queries
datapoints (i.e. each datapoint can be seen as an arm and label querying can
be seen as pulling the arm) to find the best classification. In our work, we
leverage the structure of the data while Jain & Jamieson (2019) considers each
datapoint as its own bandit arm and doesn’t explicitly use information about
the location of these datapoints. The contributions of (Jain & Jamieson, 2019)
are primarily in the theoretical analysis of the setting and proposed
algorithm, while the practicality of the algorithm itself is limited due to
its computation complexity.
A related line of work is learning under one-sided feedback, first studied
under the name apple tasting by Helmbold et al. (2000), where the learner
receives the true labels for only examples it predicted positively on and the
goal is to have as high accuracy as possible. Recently, Jiang et al. (2020)
studied the one-sided feedback problem for generalized linear models and
attain regret guarantees using an adaptive UCB-based approach under their
proposed one-sided loss. This problem is similar to our proposed active
covering problem in that both cases we desire to label query the positive
examples; however, a key difference is that both Helmbold et al. (2000) and
Jiang et al. (2020) operate in the streaming setting, where predictions must
be made in real-time whereas here, the learner has access to the entire corpus
of unlabeled data.
It’s also worth mentioning the tangentially related set cover problem (Slavık,
1997), where the goal is identify the smallest sub-collection of sets whose
union equals the whole. The submodular set cover problem (Iwata & Nagano,
2009) involves finding a set that minimizes a modular cost function subject to
a submodular function constraint. Guillory & Bilmes (2010) propose an active
approach to solve the submodular set cover problem. Active covering however is
a different problem that tries to recover the set of datapoints rather than a
collection of subsets.
Our work is also related to the support estimation literature, which has a
long history. Some works include Geffroy (1964); Devroye & Wise (1980);
Korostelev & Tsybakov (1993); Cuevas et al. (1997); Biau et al. (2008). Our
offline algorithm applies the classical support estimator on the positive
samples found to find a covering for $\mathcal{X}_{+}$, which is the union of
the $\epsilon$-balls around the initial positive examples. Those works
established both upper and lower bounds on $\epsilon$ of order $(\log
m/m)^{1/D}$, which were key to the analysis for the offline algorithm.
More broadly, support estimation has also been studied under the name of one-
class classification, where the goal is identify the examples of a particular
class given only training examples from that class. There have been a wide
range of approaches proposed including using SVMs (Schölkopf et al., 2001;
Manevitz & Yousef, 2001), density functions (Hempstalk et al., 2008),
clustering (Ypma & Duin, 1998), optimization (Crammer & Chechik, 2004), and
deep learning (Ruff et al., 2018).
Figure 1: Plots of percentage of all the positive examples retrieved after
each batch. Top: Letters Recognition dataset for the first $4$ letters as the
positive classes. Middle: Various datasets using the label $4$ as the positive
class. Bottom: CelebA using various attributes as the label. We compare our
Explore-Commit method against the offline algorithm as well as the active
variants of the baselines we tested across. We see that in all these cases,
our method performs the best across batch sizes. Results averaged across $100$
runs.
## 6 Experiments
In this section, we describe the details of our experimental results. We test
the Explore-then-Commit algorithm (Algorithm 2) against a number of baselines
including the offline algorithm (Algorithm 1). We note that these algorithms
do not come with any additional hyperparameters.
### 6.1 Baselines
We propose a number of additional baselines based on the one-class
classification methods implemented in scikit-learn (Pedregosa et al., 2011)
that can score examples based on likelihood that they are in-class, namely the
One-Class SVM (Schölkopf et al., 2001), Isolation Forest (Liu et al., 2008),
and Robust Covariance (Rousseeuw & Driessen, 1999). For each of these
baselines, we have variants: offline and active. The offline version trains
the one-class classifier on the positive examples in the initial sample (see
next subsection) and scores all of the unlabeled examples and then samples in
order from most likely to least likely of being in-class. The active version
retrains the one-class classifier on the label queried examples found thus far
after each batch, and then scores the remaining examples to choose the next
batch. All methods have the property of only utilizing the positive queried
examples for learning.
We thus can list the baselines: 1. Offline (O); 2. Offline Linear SVM (O-LS);
3. Active Linear SVM (A-LS); 4. Offline RBF SVM (O-RS); 5. Active RBF SVM
(A-RS); 6. Offline Isolation Forest (O-IF); 7. Active Isolation Forest SVM
(A-IF); 8. Offline Robust Covariance (O-RC); 9. Active Robust Covariance
(A-RC).
### 6.2 Experiment Setup
For each of the datasets, we combine all of the data (i.e. any predetermined
train/test/validation splits) into one dataset and operate on this dataset. We
fix the initial sample size to a random stratified sample of $100$ datapoints.
We train a neural network on the initial sample and use the activations of the
second-last layer (i.e. the layer immediately before the logits) as an
embedding for the data and we fix the embedding throughout and all of the
methods will operate on this embedding instead of the original input. This is
because recent work has shown that it’s effective to use classical methods on
the intermediate embeddings of the neural network (Papernot & McDaniel, 2018;
Bahri et al., 2020; Bahri & Jiang, 2021); moreover, the original features may
have undesirable properties (i.e. relative scaling of the features, high-
dimensional pixel data, etc) which can hurt the performance of some of the
baselines.
For each dataset, we run experiments using each of the classes as the positive
class, as the datasets we used were all multiclass (with the exception of
CelebA, which came with a wide range of binary attributes which we used as
classes). Then for each of the datasets with the exception of SVHN and CelebA,
we let the batch size be $5\%$ of the remainder of the dataset (i.e. after
removing the initial sample) to obtain $20$ batches, and for SVHN and CelebA,
due to their size, we let the batch size be $1\%$ of the remainder of the
dataset and ran for $50$ batches for SVHN and $30$ batches for CelebA. For all
of the experimental results, we averaged across $100$ runs randomizing over
different initial samples and ran on a cluster of NVIDIATM TeslaTM V100 Tensor
Core GPUs.
### 6.3 Datasets and Embeddings
We tested on the following datasets:
1: UCI Letters Recognition (Dua & Graff, 2017), which has $20000$ datapoints
and $16$ features based on various statistics on the pixel intensities of the
original images of the letters, with $26$ classes – one for each letter. To
train the embedding, we used a fully-connected network with one hidden layer
of $100$ units and ReLU activations and trained for $20$ epochs.
2: MNIST, with 70000 28x28 pixel grayscale images of handwritten digits and
$10$ classes. We use the same model and epochs as Letters for the embeddings.
3: Fashion MNIST (Xiao et al., 2017), with same dimensions and embedding
training procedure as that of MNIST.
4: CIFAR10 with 60000 32x32 colour images in 10 classes. For the embeddings
use a simple VGG (Zhang et al., 2015) network and extract the second-last
layer which has $128$ dimensions and train for $100$ epochs.
5: SVHN (Netzer et al., 2011) with 99289 color images, cropped to 32x32
pixels. For the embeddings, we use LeNet5 (LeCun et al., 1998) and train for
$20$ epochs.
6\. CelebA (Liu et al., 2018) a large-scale face attributes dataset with more
than 162770; we resized the images to 28x28 celebrity images. The dataset has
40 attribute annotations which we use as separate binary classification tasks.
We use the same embedding procedure as that of SVHN.
### 6.4 Hyperparameters
Our method doesn’t come with any additional hyperparameters; however the
baselines do require the tuning of hyperparameters. For these methods, we
perform $5$-fold cross-validation on the initial sample using accuracy as the
metric (these methods as implemented in scikit-learn have predict methods
which classifies whether an example is an outlier relative to the positive
class). For the SVM methods, we tune the gamma parameter (kernel coefficient)
and nu (upper bound on the fraction of training errors and a lower bound of
the fraction of support vectors). For Isolation Forest, we tune the number of
estimators in the ensemble. For Robust Covariance, we tune the proportion of
contamination of the data set. For all of these aforementioned
hyperparameters, we search over a grid of powers of two.
### 6.5 Evaluation Metrics
We plot the percentage of positive examples label queried across batches for
each of the methods to illustrate the performance of each method. We also
compute the area under the curve for each method, defined as the average
number of positive examples retrieved across each batch, and use this as the
primary evaluation metric to compare the methods. Since we average over $100$
runs, we also compute an error band on the area under the curve metric. We do
this by computing the standard deviation of percentage of positive examples
retrieved for each of the $20$ (or $50$ and $30$ in the case of SVHN and
CelebA) batches. We average across these standard deviations and divide by
square root of the number of runs to obtain an estimate of the standard
deviation of the mean. We then form a $95\%$ confidence band based on this and
consider methods which have overlapping bands as statistical ties.
### 6.6 Results
We show the results for each baseline and dataset/label pair under our area
under the curve metric in Table 2. Due to space, we could only show partial
results for Letters and defer the rest along with the CelebA results to the
Appendix. We nonetheless summarize all of the results here:
1\. Letters. Our proposed method, Explore-then-Commit, outright outperforms
all the other baselines on all $26$ tasks.
2\. MNIST. Our method again outright outperforms all the other baselines on
all $10$ tasks.
3\. Fashion MNIST. Our method performs competitively on $9$ out of the $10$
tasks, with the next most competitive baseline (Active Isolation Forest) being
competitive on $7$ out of the $10$ tasks.
4\. CIFAR10. Here, our method performs competitively on $6$ of the $10$ tasks.
It’s worth noting that we only perform poorly when all of the methods perform
poorly suggesting that in such settings not much learning is possible (i.e.
from Table 2, we only perform non-competitively when the AUC metric is under
$55\%$. A passive learner that samples uniformly at random is expected to have
an AUC of $50\%$).
5\. SVHN. Our method is competitive for all tasks and outright wins for all
but one task.
6\. CelebA. Our method is competitive for 32 out of the 40 tasks. Due to
space, the results are shown in the Appendix. We again see a similar pattern
as in CIFAR10 where our method only performs poorly when all of the methods
perform poorly (i.e. we only perform non-competitively when the AUC metric is
under $20\%$. For comparison, a passive learner is expected to have an AUC of
$15\%$).
Dataset | Label | O | O-LS | A-LS | O-RS | A-RS | O-IF | A-IF | O-RC | A-RC | EC (Ours)
---|---|---|---|---|---|---|---|---|---|---|---
Letters | A | 91.14 | 52.52 | 52.49 | 84.81 | 89.02 | 59.52 | 84.69 | 64.27 | 86.87 | 97.12
B | 83.41 | 52.73 | 52.57 | 75.95 | 82.41 | 56.13 | 75.89 | 61.38 | 80.18 | 93.76
C | 84.48 | 56.19 | 56.08 | 75.92 | 83.78 | 55.78 | 78.68 | 59.21 | 81.92 | 94.2
D | 83.51 | 52.57 | 52.45 | 76.14 | 82.23 | 56.09 | 76.03 | 61.51 | 78.83 | 93.73
E | 78.6 | 52.85 | 52.99 | 74.84 | 81.41 | 55.77 | 73.7 | 60.17 | 77.27 | 89.5
F | 83.63 | 53.4 | 53.41 | 78.72 | 83.16 | 57.44 | 77.83 | 64.69 | 80.88 | 94.0
| G | 82.23 | 52.72 | 52.67 | 75.76 | 81.88 | 58.15 | 74.58 | 63.45 | 79.79 | 92.35
MNIST | 0 | 86.67 | 81.81 | 81.96 | 52.53 | 52.95 | 83.44 | 90.8 | 74.31 | 86.48 | 94.44
1 | 95.89 | 55.22 | 55.11 | 58.47 | 90.31 | 90.16 | 94.27 | 87.34 | 89.8 | 96.46
2 | 75.86 | 60.64 | 60.37 | 52.54 | 52.56 | 72.66 | 80.18 | 62.81 | 77.83 | 85.47
3 | 80.77 | 60.64 | 60.36 | 52.52 | 52.65 | 76.67 | 84.31 | 66.87 | 81.03 | 87.63
4 | 83.05 | 54.23 | 54.14 | 52.47 | 53.08 | 76.86 | 83.61 | 66.71 | 78.31 | 89.14
5 | 75.59 | 52.88 | 52.81 | 52.51 | 52.63 | 61.65 | 69.44 | 57.82 | 71.18 | 87.24
6 | 86.53 | 59.98 | 59.77 | 52.49 | 54.03 | 81.19 | 88.33 | 67.37 | 81.95 | 93.24
7 | 87.05 | 55.83 | 55.63 | 52.54 | 57.02 | 80.71 | 87.26 | 70.14 | 81.76 | 91.62
8 | 75.7 | 56.27 | 56.17 | 52.49 | 52.62 | 69.73 | 78.3 | 61.71 | 77.32 | 83.37
9 | 84.91 | 54.64 | 54.7 | 52.51 | 55.06 | 77.22 | 84.66 | 67.88 | 79.79 | 90.71
Fashion MNIST | 0 | 87.81 | 54.51 | 54.49 | 52.9 | 66.76 | 86.35 | 90.14 | 81.5 | 87.44 | 89.75
1 | 94.73 | 55.84 | 55.67 | 55.12 | 85.21 | 92.67 | 94.73 | 90.42 | 92.54 | 95.93
2 | 84.44 | 55.57 | 55.57 | 52.72 | 63.65 | 82.97 | 87.6 | 78.46 | 85.6 | 87.19
3 | 88.86 | 53.15 | 53.15 | 52.64 | 63.56 | 85.39 | 89.74 | 83.08 | 87.06 | 91.29
4 | 84.9 | 56.47 | 56.41 | 52.68 | 62.54 | 83.23 | 87.25 | 79.68 | 83.61 | 88.09
5 | 88.09 | 52.54 | 52.52 | 52.62 | 57.14 | 79.59 | 84.7 | 82.92 | 75.2 | 89.16
6 | 77.31 | 52.5 | 52.5 | 52.98 | 63.62 | 75.94 | 81.63 | 71.46 | 80.18 | 81.1
7 | 94.41 | 52.47 | 52.49 | 52.97 | 69.91 | 92.66 | 95.06 | 90.72 | 93.05 | 95.17
8 | 82.86 | 55.43 | 55.46 | 52.5 | 54.1 | 78.23 | 86.56 | 78.33 | 83.97 | 85.96
9 | 92.5 | 70.39 | 70.13 | 52.59 | 58.31 | 90.93 | 94.12 | 90.35 | 91.4 | 93.49
CIFAR10 | 0 | 67.06 | 54.33 | 54.27 | 64.84 | 64.8 | 64.6 | 68.03 | 65.49 | 69.36 | 70.69
1 | 57.24 | 52.57 | 52.54 | 55.67 | 55.01 | 54.25 | 53.68 | 57.68 | 50.3 | 54.06
2 | 65.43 | 52.48 | 52.51 | 59.75 | 59.92 | 62.09 | 64.61 | 63.42 | 65.15 | 68.71
3 | 53.98 | 52.5 | 52.51 | 53.21 | 53.28 | 53.34 | 52.8 | 54.13 | 50.09 | 52.2
4 | 70.94 | 52.5 | 52.55 | 66.68 | 67.52 | 68.54 | 71.52 | 67.95 | 68.29 | 73.97
5 | 57.59 | 52.53 | 52.48 | 56.13 | 56.09 | 56.55 | 56.7 | 58.83 | 53.12 | 53.9
6 | 72.16 | 52.48 | 52.49 | 67.67 | 67.89 | 67.79 | 70.87 | 70.78 | 65.26 | 74.73
7 | 58.51 | 52.54 | 52.53 | 56.0 | 55.93 | 56.48 | 57.7 | 58.07 | 53.92 | 58.36
8 | 70.25 | 52.48 | 52.47 | 66.67 | 66.86 | 67.92 | 71.57 | 68.13 | 69.91 | 71.42
9 | 62.79 | 52.54 | 52.49 | 61.8 | 61.74 | 63.51 | 66.18 | 63.97 | 62.65 | 55.63
SVHN | 0 | 31.11 | 25.47 | 25.49 | 28.54 | 29.89 | 28.0 | 30.12 | 31.57 | 39.59 | 39.67
1 | 28.23 | 25.53 | 25.52 | 25.63 | 25.25 | 25.08 | 25.44 | 32.81 | 35.19 | 37.32
2 | 28.66 | 25.53 | 25.52 | 26.28 | 26.49 | 26.32 | 26.69 | 30.86 | 32.68 | 34.31
3 | 28.19 | 25.57 | 25.56 | 26.11 | 26.47 | 26.34 | 26.79 | 29.37 | 31.0 | 33.81
4 | 28.01 | 25.51 | 25.49 | 25.54 | 25.16 | 25.21 | 26.66 | 27.31 | 33.4 | 36.31
5 | 29.02 | 25.56 | 25.53 | 26.55 | 26.98 | 26.77 | 27.75 | 29.67 | 32.81 | 34.44
6 | 28.8 | 25.49 | 25.5 | 26.37 | 26.6 | 26.24 | 27.72 | 28.7 | 32.34 | 35.29
7 | 29.36 | 25.52 | 25.48 | 26.46 | 26.18 | 25.7 | 27.22 | 27.79 | 35.14 | 37.62
8 | 28.04 | 25.51 | 25.48 | 26.29 | 27.03 | 26.41 | 27.43 | 27.29 | 31.43 | 33.37
9 | 28.48 | 25.49 | 25.49 | 26.82 | 27.5 | 26.28 | 28.04 | 27.62 | 32.83 | 34.36
Table 2: Area under the curve metric for various benchmark image-based
datasets. For each of the datasets and possible labels, we show the area under
the curve metric averaged across $100$ runs, with the top value bolded (any
methods whose $95\%$ confidence intervals overlap were considered statistical
ties). Due to space, we show the rest of the Letters results as well as the
CelebA results in the Appendix.
## 7 Conclusion
We have formalized the problem of active covering and introduced several
baselines, including a principled active approach that attains better
guarantees than the offline algorithm. We showed in experiments that our
proposed Explore-the-Commit algorithm has strong performance against a number
of baselines while having desirable properties including not having additional
hyperparameters, and not needing to store or use the queried negative
examples. Future work involves extending theoretical analysis relaxing the
hard boundary density assumption on $\mathcal{X}_{+}$, letting
$\mathcal{X}_{+}$ be a lower dimensional manifold embedded in the
$D$-dimensional space (rather than being full-dimensional) and attain excess
query cost guarantees that depend on this lower dimension, and investigating
the computational and privacy implications of such approaches.
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Appendix
## Appendix A Proofs
### A.1 Proofs for Section 2
###### Proof of Theorem 1.
Consider the distribution where there are two connected components, one for
$\mathcal{X}_{+}$ and one for $\mathcal{X}_{-}$, each with mixture probability
$p=\frac{1}{2}$. Thus, Assumption 1 holds and we are choose to free the other
parameters of the distribution in any way that satisfies Assumption 2 and 3
(e.g. a mixture of uniform density functions satisfies these assumptions). Now
note that with probability $\frac{1}{2}$, the final point that is label
queried by the passive learner will be positive and, thus, the passive
algorithm will need to query all of the points with probability $\frac{1}{2}$
in order to retrieve all positive points. In such an event, the excess query
cost is at least $\frac{1}{2}\cdot n$. ∎
### A.2 Proofs for Section 3
Much of our technical results require the following uniform high-probability
guarantee that balls of sufficient probability mass contain an example:
###### Lemma 3.
Let $0<\delta<1$ and $\mathcal{F}$ be some distribution over $\mathbb{R}^{D}$
and $X$ be a sample of size $n$ drawn i.i.d. from $\mathcal{F}$. There exists
universal constant $C_{0}$ such that the following holds with probability at
least $1-\delta$ uniformly over all balls $B\in\mathbb{R}^{D}$:
$\displaystyle\mathcal{F}(B)\geq\frac{C_{0}\cdot D\cdot\log(2/\delta)\log
n}{n}\Rightarrow|B\cap X|>0.$
###### Proof.
This follows by Lemma 7 of Chaudhuri & Dasgupta (2010). ∎
The following result bounds the volume of the $\epsilon$-neighborhood around
$\mathcal{X}_{+}$, which will be used later to bound the excess number of
points queried around $\mathcal{X}_{+}$. The result says that the volume of
the $\epsilon$-neighboorhood around $\mathcal{X}_{+}$ (and not including
$\mathcal{X}_{+}$) is linear in $\epsilon$.
###### Lemma 4.
Suppose Assumption 2 holds. Then there exists constants
$r_{1},C_{+}^{\prime}>0$ depending only on $\mathcal{X}_{+}$ such that for all
$0<\epsilon<r_{1}$, we have
$\displaystyle\text{Vol}(B(\mathcal{X}_{+},\epsilon)\backslash\mathcal{X}_{+})\leq
C_{+}^{\prime}\cdot\epsilon,$
where
$B(\mathcal{X}_{+},\epsilon):=\\{x\in\mathbb{R}^{D}:\inf_{x^{\prime}\in\mathcal{X}_{+}}|x-x^{\prime}|\leq\epsilon\\}$.
###### Proof of Lemma 4.
This follows from Gorin (1983). To see this, the equation on page 159 of Gorin
(1983) states that if $M$ and $N$ are respectively $d$-dimensional and
$(d+k)$-dimensional compact smooth Riemannian manifolds and $f:M\rightarrow N$
is a smooth isometric embedding, then we have
$\displaystyle\text{Vol}(B(f(M),\epsilon))=V_{k}\cdot\epsilon^{k}\cdot\text{Vol}(M)+O(\epsilon^{k+1}),$
where $V_{k}$ is the volume of a $k$-dimensional ball. Here, we take
$M=\mathcal{X}_{+}$ and $N=B(\mathcal{X}_{+},r_{1})$ for some $r_{1}>0$. Then,
we have $k=0$ and taking $f$ to be the identity function, we have
$\displaystyle\text{Vol}(B(\mathcal{X}_{+},\epsilon))=\text{Vol}(\mathcal{X}_{+})+O(\epsilon),$
and the result follows immediately. ∎
###### Proof of Theorem 2.
By Hoeffding’s inequality, out of the initial $m$ examples that Algorithm 1
label queries, we have with probability at least $1-\delta/2$ that at least
$p-\sqrt{\frac{1}{2m}\cdot\log(2/\delta)}$ fraction of them are positively
labeled, since the example being positive follows a Bernoulli distribution
with probability $p$. Then by the condition on $m$, we have that at least
$p/2$ fraction of the points are positively labeled.
Take
$\displaystyle\epsilon=\left(\frac{2\cdot C_{0}\cdot
D\cdot\log(4/\delta)\log(p\cdot m/2)}{p^{2}\cdot\lambda_{0}\cdot C_{+}\cdot
v_{D}\cdot m}\right)^{1/D},\hskip 14.22636ptM_{0}=\max\left\\{\frac{2\cdot
C_{0}\cdot D(\log(p\cdot/2)+1)}{p^{2}\cdot\lambda_{0}\cdot C_{+}\cdot
v_{D}\cdot\min\\{r_{0},r_{1}\\}^{D}},2e\right\\},$
where $v_{D}$ is the volume of a unit ball in $\mathbb{R}^{D}$. Then, we have
that the condition on $m$ and $M_{0}$ guarantees that
$\epsilon<\min\\{r_{0},r_{1}\\}$.
Let $x\in\mathcal{X}_{+}$. We have that the probability mass of positive
examples in $B(x,\epsilon)$ w.r.t. $\mathcal{P}$ is:
$\displaystyle p\cdot\mathcal{P}_{+}(B(x,\epsilon))$ $\displaystyle\geq
p\cdot\lambda_{0}\cdot\text{Vol}(B(x,\epsilon)\cap\mathcal{X}_{+})$
$\displaystyle\geq p\cdot\lambda_{0}\cdot C_{+}\cdot\text{Vol}(B(x,\epsilon))$
$\displaystyle\geq p\cdot\lambda_{0}\cdot C_{+}\cdot v_{D}\cdot\epsilon^{D}$
$\displaystyle\geq\frac{2\cdot C_{0}\cdot D\log(4/\delta)\log(p\cdot
m/2)}{p\cdot m}.$
Then by Lemma 3, we have with probability at least $1-\delta/2$ that all the
positive examples in $X$ are within $\epsilon$ of one of the positive examples
among the initially sampled $m$ examples. Therefore, Algorithm 1 retrieves all
of the positive examples.
Now we bound the expected regret:
$\displaystyle\mathbb{E}[C_{\text{offline}}]$ $\displaystyle\leq(1-p)\cdot
m+n\cdot(1-p)\cdot\mathcal{P}_{-}(B(\mathcal{X}_{+},\epsilon)\backslash\mathcal{X}_{+})$
$\displaystyle\leq(1-p)\cdot m+n\cdot(1-p)\cdot\lambda_{1}\cdot
C_{+}^{\prime}\cdot\epsilon.$
The result follows. ∎
###### Proof of Theorem 3.
Let $\mathcal{P}_{+}$ be the uniform distribution on the unit hypercube
$[0,1]^{D}$ and $\mathcal{P}_{-}$ be the uniform distribution on $[-1,2]^{D}$.
In the initial sampling phase of Algorithm 1, at most $m$ of the examples will
be positively labeled. Let $\widehat{\mathcal{X}_{+}}=X\cap\left(\cup_{x\in
X_{0,+}}B(x,\epsilon)\right)$, the set of points that Algorithm 1 labeled.
Then, Theorem 3b in (Cuevas et al., 1997) shows that for $n$ sufficiently
large, with probability at least $1/4$, we have
$\displaystyle
d_{H}(\widehat{\mathcal{X}_{+}},\mathcal{X}_{+})\geq\frac{1}{4}\left(\frac{\log
m}{m}\right)^{1/D}$
for any $\epsilon>0$, where $d_{H}(A,B):=\max\\{\sup_{x\in A}d(x,B),\sup_{x\in
B}d(x,A)\\}$ is the Hausdorff distance. Therefore, we have (in the case of
taking $\epsilon\rightarrow 0$):
$\displaystyle d_{H}(X_{0,+},\mathcal{X}_{+})\geq\frac{1}{4}\left(\frac{\log
m}{m}\right)^{1/D}.$
Since $X_{0,+}\subseteq\mathcal{X}_{+}$, it follows that
$d_{H}(X_{0,+},\mathcal{X}_{+})=\sup_{x\in\mathcal{X}_{+}}d(x,X_{0,+})$.
Therefore, we need $\epsilon\geq\frac{1}{4}\left(\frac{\log
m}{m}\right)^{1/D}$ in order for Algorithm 1 to recover all of the positive
examples. Thus, the expected regret is at least (for some $C>0$)
$\displaystyle\mathbb{E}[C_{\text{offline}}]\geq(1-p)\cdot
m+C\cdot\left(\frac{\log m}{m}\right)^{1/D}\cdot n,$
as desired. ∎
### A.3 Proofs for Section 4
###### Proof of Lemma 1.
By Hoeffding’s inequality, out of the initial $m$ examples that Algorithm 2
label queries, we have with probability at least $1-\delta/2$ that at least
$p-\sqrt{\frac{1}{2m}\cdot\log(2/\delta)}$ are fraction of them are positively
labeled, since the example being positive follows a Bernoulli distribution
with probability $p$. Then by the condition on $m$, we have that at least
$p/2$ fraction of the points are positively labeled and thus we have at least
$mp/2$ positive examples.
Then, we have that out of these $mp/2$ examples, the probability that none of
them are in $\mathcal{X}_{+,i}$ for each $i\in[c]$ is at most
$\displaystyle(1-\mathcal{P}_{+}(\mathcal{X}_{+,i}))^{mp/2}\leq(1-q)^{mp/2}\leq\frac{\delta}{2c}.$
The result follows by union bound. ∎
###### Proof of Lemma 2.
Let $x,x^{\prime}\in\mathcal{X}_{+,i}$. There exists a path $x=x_{1}\to
x_{2}\to\ldots\to x_{q}=x^{\prime}$ in $\mathcal{X}_{+,i}$ such that
$||x_{j}-x_{j+1}||\leq\epsilon/3$. We also have that the probability mass of
positive examples in $B(x_{j},\epsilon/3)$ w.r.t. $\mathcal{P}$ is:
$\displaystyle p\cdot\mathcal{P}_{+}(B(x_{j},\epsilon/2))$ $\displaystyle\geq
p\cdot C_{+}\cdot\lambda_{0}\cdot
v_{D}\cdot(\epsilon/3)^{D}\geq\frac{C_{0}\cdot D\log(2/\delta)\cdot\log
n}{n}.$
Therefore by Lemma 3, there exists $x_{j}^{\prime}\in B(x_{j},\epsilon/3)\cap
X_{+}$, where $X_{+}$ are the positive examples in $X$. Hence, by triangle
inequality, there exists path $x=x_{1}^{\prime}\to x_{2}^{\prime}\to...\to
x_{q}^{\prime}=x^{\prime}$ all in $X_{+}$ where
$||x_{j}^{\prime}-x_{j+1}^{\prime}||\leq||x_{j}^{\prime}-x_{j}||+||x_{j}-x_{j+1}||+||x_{j+1}-x_{j+1}^{\prime}||\leq\epsilon$
implying that $\mathcal{X}_{+,i}\cap X_{+}$ is connected in the
$\epsilon$-neighborhood graph of $X_{+}$. The result follows immediately. ∎
Finally, we combine these two results to show the final excess query cost
guarantee for Algorithm 2.
###### Proof of Theorem 4.
Take
$\displaystyle N_{0}=\frac{3^{D}\cdot C_{0}\cdot
D}{\min\\{r_{0},r_{1}\\}^{D}\cdot p\cdot C_{+}\cdot\lambda_{0}\cdot v_{D}}.$
By Lemma 1, there exists at least one positive example in the initial $m$
samples from each connected component of $\mathcal{X}_{+}$. Define
$\displaystyle\epsilon:=3\left(\frac{C_{0}\cdot D\log(4/\delta)\cdot\log
n}{p\cdot C_{+}\cdot\lambda_{0}\cdot v_{D}\cdot n}\right)^{1/D}.$
We have that the condition on $n$ implies that
$\epsilon\leq\min\\{r_{0},r_{1}\\}$. By Lemma 2, we have that all of the
positive examples of each connected component of $\mathcal{X}_{+}$ are in the
same CC of the $\epsilon$-neighborhood graph of the positive examples.
Therefore, when the algorithm terminates, the set of examples it will select
from will be contained in $B(\mathcal{X}_{+},\epsilon)$.
Therefore, we have
$\displaystyle\mathbb{E}[C_{\text{exp-commit}}]\leq(1-p)\cdot
m+C_{+}^{\prime}\cdot\epsilon\cdot n\leq(1-p)\cdot
m+C\cdot\left((\log(4/\delta)\cdot\log n\right)^{1/D}\cdot n^{(D-1)/D},$
for some $C$ depending on $\mathcal{P}$, as desired. ∎
## Appendix B Additional Experiment Plots
In Table 3, we show the full results for the Letters dataset for the area
under curve metrics. We see that in all cases, our method outperforms
outright. In Table 4, we show the full results for CelebA. We see that our
method is competitive for 32 out of the 40 tasks.
Dataset | Label | O | O-LS | A-LS | O-RS | A-RS | O-IF | A-IF | O-RC | A-RC | EC (Ours)
---|---|---|---|---|---|---|---|---|---|---|---
Letters | A | 91.14 | 52.52 | 52.49 | 84.81 | 89.02 | 59.52 | 84.69 | 64.27 | 86.87 | 97.12
B | 83.41 | 52.73 | 52.57 | 75.95 | 82.41 | 56.13 | 75.89 | 61.38 | 80.18 | 93.76
C | 84.48 | 56.19 | 56.08 | 75.92 | 83.78 | 55.78 | 78.68 | 59.21 | 81.92 | 94.2
D | 83.51 | 52.57 | 52.45 | 76.14 | 82.23 | 56.09 | 76.03 | 61.51 | 78.83 | 93.73
E | 78.6 | 52.85 | 52.99 | 74.84 | 81.41 | 55.77 | 73.7 | 60.17 | 77.27 | 89.5
F | 83.63 | 53.4 | 53.41 | 78.72 | 83.16 | 57.44 | 77.83 | 64.69 | 80.88 | 94.0
G | 82.23 | 52.72 | 52.67 | 75.76 | 81.88 | 58.15 | 74.58 | 63.45 | 79.79 | 92.35
H | 69.07 | 52.4 | 52.63 | 61.19 | 69.51 | 53.95 | 64.15 | 52.03 | 66.94 | 81.78
I | 84.26 | 52.37 | 52.59 | 73.96 | 80.72 | 55.95 | 77.56 | 59.11 | 83.83 | 93.89
J | 84.24 | 54.45 | 54.35 | 75.16 | 81.72 | 56.22 | 77.06 | 58.58 | 81.61 | 94.28
K | 72.7 | 52.57 | 52.62 | 65.15 | 72.03 | 55.46 | 68.12 | 51.45 | 74.38 | 89.4
L | 80.98 | 52.59 | 52.67 | 72.74 | 79.87 | 55.44 | 77.73 | 59.16 | 78.57 | 93.18
M | 81.83 | 54.41 | 54.25 | 77.48 | 83.23 | 59.55 | 78.61 | 60.62 | 81.67 | 93.07
N | 75.15 | 52.67 | 52.68 | 68.58 | 75.31 | 55.37 | 69.43 | 55.85 | 73.74 | 89.8
O | 87.71 | 52.51 | 52.62 | 80.88 | 86.08 | 57.88 | 78.08 | 67.62 | 82.71 | 94.69
P | 84.76 | 52.7 | 52.7 | 79.24 | 84.47 | 57.63 | 79.18 | 64.81 | 83.19 | 93.47
Q | 82.25 | 53.22 | 53.08 | 74.97 | 80.09 | 55.71 | 74.66 | 60.49 | 79.55 | 92.18
R | 82.68 | 52.59 | 52.48 | 76.72 | 82.47 | 56.32 | 75.3 | 61.37 | 79.26 | 92.77
S | 76.51 | 52.83 | 52.91 | 70.42 | 75.85 | 55.24 | 72.01 | 59.66 | 75.84 | 89.27
T | 83.47 | 55.32 | 55.35 | 76.38 | 82.95 | 56.87 | 79.06 | 62.87 | 81.96 | 93.34
U | 78.05 | 52.91 | 53.05 | 73.72 | 81.28 | 55.86 | 72.81 | 58.56 | 76.41 | 92.7
V | 89.82 | 54.61 | 54.59 | 82.3 | 87.65 | 59.74 | 82.18 | 61.01 | 85.18 | 96.19
W | 90.15 | 55.25 | 55.29 | 84.57 | 89.11 | 59.37 | 82.04 | 63.1 | 86.65 | 96.79
X | 80.18 | 52.63 | 52.6 | 74.84 | 80.68 | 55.92 | 73.04 | 62.48 | 78.1 | 92.93
Y | 81.73 | 54.65 | 54.7 | 72.3 | 79.67 | 57.49 | 76.38 | 53.48 | 79.33 | 92.46
Z | 83.78 | 54.6 | 54.54 | 76.89 | 84.18 | 56.46 | 78.56 | 60.14 | 82.64 | 93.35
Table 3: Letters: Area under curve metric. Label | O | O-LS | A-LS | O-RS | A-RS | O-IF | A-IF | O-RC | A-RC | EC (Ours)
---|---|---|---|---|---|---|---|---|---|---
5-o-Clock-Shadow | 20.17 | 15.54 | 15.55 | 17.31 | 18.28 | 17.89 | 18.85 | 20.25 | 22.49 | 23.85
Arched-Eyebrows | 19.29 | 15.52 | 15.51 | 17.54 | 17.69 | 18.34 | 18.65 | 19.87 | 19.76 | 20.93
Attractive | 18.4 | 19.19 | 18.32 | 18.6 | 18.7 | 18.86 | 18.89 | 18.03 | 17.93 | 18.61
Bags-Under-Eyes | 17.06 | 15.5 | 15.51 | 16.58 | 17.54 | 17.26 | 17.11 | 17.3 | 16.33 | 16.97
Bangs | 20.88 | 15.52 | 15.52 | 20.6 | 19.97 | 20.49 | 21.43 | 22.12 | 19.9 | 22.08
Bald | 18.52 | 15.48 | 15.49 | 16.6 | 16.98 | 15.48 | 17.92 | NA | NA | 28.34
Big-Lips | 15.81 | 15.51 | 15.5 | 15.44 | 14.91 | 15.26 | 15.46 | 15.61 | 15.68 | 16.35
Big-Nose | 16.58 | 15.53 | 15.53 | 15.92 | 16.58 | 15.95 | 16.23 | 16.88 | 16.12 | 17.23
Black-Hair | 22.87 | 15.49 | 15.5 | 20.8 | 19.43 | 21.35 | 22.52 | 21.23 | 20.5 | 24.6
Blond-Hair | 36.36 | 15.67 | 15.67 | 31.41 | 34.64 | 35.92 | 37.84 | 37.31 | 39.24 | 41.72
Blurry | 16.75 | 15.44 | 15.48 | 16.38 | 16.06 | 16.48 | 16.66 | 16.66 | 16.81 | 17.79
Brown-Hair | 20.88 | 15.48 | 15.48 | 18.82 | 20.45 | 20.64 | 21.34 | 20.94 | 20.77 | 21.57
Bushy-Eyebrows | 19.11 | 15.49 | 15.49 | 17.87 | 18.12 | 18.18 | 18.7 | 18.88 | 19.2 | 21.3
Chubby | 16.77 | 15.51 | 15.5 | 15.98 | 16.2 | 15.83 | 16.15 | 16.51 | 19.01 | 18.63
Double-Chin | 18.05 | 15.53 | 15.52 | 16.73 | 17.31 | 16.17 | 17.28 | 16.66 | 22.57 | 22.3
Eyeglasses | 16.48 | 15.51 | 15.5 | 16.14 | 17.12 | 16.48 | 16.78 | 17.74 | 15.88 | 15.44
Goatee | 17.57 | 15.49 | 15.51 | 16.86 | 16.69 | 16.22 | 16.93 | 16.98 | 17.98 | 19.81
Gray-Hair | 23.19 | 15.53 | 15.55 | 19.55 | 21.77 | 17.01 | 23.12 | 20.84 | 32.86 | 31.66
Heavy-Makeup | 21.42 | 15.85 | 15.58 | 20.49 | 20.5 | 21.24 | 21.94 | 21.29 | 20.4 | 22.45
High-Cheekbones | 19.17 | 15.38 | 15.17 | 18.59 | 18.79 | 18.91 | 18.89 | 19.78 | 18.77 | 20.07
Male | 18.64 | 15.49 | 15.57 | 20.09 | 19.36 | 19.14 | 18.8 | 18.43 | 16.5 | 19.24
Mouth-Slightly-Open | 17.37 | 15.66 | 15.56 | 17.43 | 17.34 | 17.42 | 17.16 | 17.79 | 17.2 | 17.77
Mustache | 17.16 | 15.5 | 15.48 | 16.77 | 16.46 | 15.98 | 16.99 | 16.64 | 17.05 | 18.13
Narrow-Eyes | 15.48 | 15.49 | 15.5 | 15.66 | 15.95 | 15.78 | 15.78 | 15.68 | 15.36 | 14.83
No-Beard | 15.82 | 16.38 | 16.37 | 15.79 | 15.93 | 16.0 | 15.99 | 16.08 | 15.98 | 15.83
Oval-Face | 18.02 | 15.5 | 15.5 | 16.76 | 17.11 | 17.18 | 17.37 | 18.56 | 18.37 | 19.22
Pale-Skin | 18.15 | 15.45 | 15.48 | 19.89 | 19.15 | 16.95 | 20.95 | 18.75 | 17.94 | 19.55
Pointy-Nose | 18.55 | 15.49 | 15.49 | 17.03 | 17.22 | 17.23 | 17.75 | 18.69 | 18.8 | 19.97
Receding-Hairline | 18.5 | 15.5 | 15.5 | 17.09 | 17.47 | 16.67 | 17.55 | 18.61 | 22.2 | 22.38
Rosy-Cheeks | 23.96 | 15.5 | 15.51 | 19.77 | 22.49 | 18.5 | 22.3 | 22.45 | 32.64 | 34.69
Sideburns | 18.32 | 15.51 | 15.52 | 17.23 | 17.19 | 16.9 | 17.39 | 17.41 | 19.36 | 21.77
Smiling | 19.46 | 16.01 | 15.52 | 18.97 | 19.05 | 19.31 | 19.15 | 19.89 | 18.81 | 20.15
Straight-Hair | 16.1 | 15.5 | 15.5 | 15.97 | 16.25 | 15.91 | 16.02 | 16.24 | 16.13 | 16.29
Wavy-Hair | 21.0 | 15.52 | 15.52 | 20.18 | 20.09 | 20.56 | 20.97 | 21.13 | 20.42 | 21.88
Wearing-Earrings | 18.75 | 15.47 | 15.47 | 16.93 | 17.87 | 17.75 | 18.37 | 19.28 | 20.0 | 20.62
Wearing-Hat | 18.32 | 15.48 | 15.49 | 18.33 | 18.27 | 17.44 | 19.95 | 20.12 | 15.16 | 16.82
Wearing-Lipstick | 20.42 | 19.0 | 17.34 | 19.64 | 19.65 | 20.49 | 20.8 | 20.21 | 19.51 | 21.06
Wearing-Necklace | 18.99 | 15.48 | 15.49 | 16.72 | 17.85 | 17.54 | 18.15 | 19.44 | 21.28 | 21.48
Wearing-Necktie | 19.56 | 15.52 | 15.51 | 17.51 | 18.13 | 16.53 | 18.36 | 18.45 | 23.79 | 24.46
Young | 15.51 | 16.22 | 16.21 | 15.54 | 15.63 | 15.55 | 15.55 | 15.56 | 15.51 | 15.57
Table 4: CelebA: Area under curve metric. We note that for Bald, there were no
results for the Robust Covariance metrics. This is because due to the low rate
of positive examples, it was not possible to tune Robust Covariance’s hyper-
parameters via cross-validation on the initial sample.
|
# The Spin-down of PSR J0821–4300 and PSR J1210–5226:
Confirmation of Central Compact Objects as Anti-Magnetars
E. V. Gotthelf, J. P. Halpern, and J. Alford Columbia Astrophysics
Laboratory, Columbia University, 550 West 120th Street, New York, NY 10027
###### Abstract
Using XMM–Newton and Chandra, we measure period derivatives for the second and
third known pulsars in the class of Central Compact Objects (CCOs) in
supernova remnants, proving that these young neutron stars have exceptionally
weak dipole magnetic field components. For the 112 ms PSR J0821$-$4300 in
Puppis A, $\dot{P}=(9.28\pm 0.36)\times 10^{-18}$. Its proper motion,
$\mu=61\pm 9$ mas yr-1, was also measured using Chandra. This contributes a
kinematic term to the period derivative via the Shklovskii effect, which is
subtracted from $\dot{P}$ to derive dipole $B_{s}=2.9\times 10^{10}$ G, a
value similar to that of first measured CCO PSR J1852$+$0040 in Kes 79, which
has $B_{s}=3.1\times 10^{10}$ G. Antipodal surface hot spots with different
temperatures and areas are deduced from the X-ray spectrum and pulse profiles.
Paradoxically, such nonuniform surface temperature appears to require strong
crustal magnetic fields, probably toroidal or quadrupolar components much
stronger than the external dipole. A spectral feature, consisting of either an
emission line at $\approx 0.75$ keV or absorption at $\approx 0.46$ keV, is
modulated in strength with the rotation. It may be due to a cyclotron process
in a magnetic field on the surface that is slightly stronger than the dipole
deduced from the spin-down. We also timed anew the 424 ms PSR J1210$-$5226,
resolving previous ambiguities about its spin-down rate. Its $\dot{P}=(2.22\pm
0.02)\times 10^{-17}$, corresponding to $B_{s}=9.8\times 10^{10}$ G. This is
compatible with a cyclotron resonance interpretation of its prominent
absorption line at 0.7 keV and harmonics. These results deepen the mystery of
the origin and evolution of CCOs: why are their numerous descendants not
evident?
###### Subject headings:
ISM: individual (Puppis A) — pulsars: individual (PSR J0821$-$4300, PSR
J1210$-$5226, PSR J1852$+$0040) — stars: neutron
## 1\. Introduction
The class of faint X-ray sources in supernova remnants (SNRs) known as central
compact objects (CCOs) are characterized by steady flux, predominantly surface
thermal X-ray emission, lack of a surrounding pulsar wind nebula, and absence
of detection at any other wavelength. Table 1 lists basic data on the well-
studied CCOs, as well as proposed candidates whose qualifications are not yet
well established. Of the eight most secure CCOs, three are known to be neutron
stars (NSs) with spin periods of 0.105, 0.424, and 0.112 s. Spin-down has been
detected for two of these, the 0.105 s pulsar PSR J1852$+$0040 in Kes 79 and
the 0.424 s pulsar PSR J1210$-$5226 in the SNR PKS 1209$-$51/52. For PSR
J1852$+$0040, the implied surface dipole field is only $B_{s}=3.1\times
10^{10}$ G (Halpern & Gotthelf, 2010a), smaller than that of any other young
known NS. In the case of PSR J1210$-$5226, archival data allow two alternative
timing solutions, with $B_{s}=9.9\times 10^{10}$ or $2.4\times 10^{11}$ G
(Halpern & Gotthelf, 2011).
It is natural to assume that CCOs that have not yet been seen to pulse are
isolated, weakly magnetized NSs of the same class as the CCO pulsars. Where
pulsar searches have been unsuccessful, it is possible that an even weaker
magnetic field, a more uniform surface temperature, or an unfavorable viewing
geometry, prevents detection of rotational modulation. The absence of
pulsations from the youngest known NS, the $\approx 330$ year old CCO in
Cassiopeia A, has been used, in combination with fitting of its X-ray
spectrum, to argue that it is covered with a uniform temperature, non-
magnetized atmosphere of carbon, the product of nuclear burning of H and He
(Ho & Heinke, 2009). Rapid cooling of the NS in Cas A, directly detected by
Chandra (Heinke & Ho, 2010), has been interpreted as evidence for neutron
superfluidity in the core (Page et al., 2011; Shternin et al., 2011).
Table 1Central Compact Objects in Supernova Remnants
CCO | SNR | Age | $d$ | $P$ | $f_{p}$aaUpper limits on pulsed fraction are for a search down to $P=12$ ms or smaller. | $B_{s}$ | $L_{x,\rm bol}$ | References
---|---|---|---|---|---|---|---|---
| | (kyr) | (kpc) | (s) | (%) | ($10^{10}$ G) | (erg s-1) |
RX J0822.0$-$4300 | Puppis A | 4.5 | 2.2 | 0.112 | 11 | 2.9 | $5.6\times 10^{33}$ | 1,2,3,4,5,6
CXOU J085201.4$-$461753 | G266.1$-$1.2 | 1 | 1 | … | $<7$ | … | $2.5\times 10^{32}$ | 7,8,9,10,11
1E 1207.4$-$5209 | PKS 1209$-$51/52 | 7 | 2.2 | 0.424 | 9 | 9.8 | $2.5\times 10^{33}$ | 6,12,13,14,15,16,17
CXOU J160103.1$-$513353 | G330.2$+$1.0 | $\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}3$ | 5 | … | $<40$ | … | $1.5\times 10^{33}$ | 18,19
1WGA J1713.4$-$3949 | G347.3$-$0.5 | 1.6 | 1.3 | … | $<7$ | … | $\sim 1\times 10^{33}$ | 11,20,21
XMMU J172054.5$-$372652 | G350.1$-$0.3 | 0.9 | 4.5 | … | … | … | $3.9\times 10^{33}$ | 22,23
CXOU J185238.6$+$004020 | Kes 79 | 7 | 7 | 0.105 | 64 | 3.1 | $5.3\times 10^{33}$ | 24,25,26,27
CXOU J232327.9$+$584842 | Cas A | 0.33 | 3.4 | … | $<12$ | … | $4.7\times 10^{33}$ | 27,28,29,30,31,32,33
2XMMi J115836.1$-$623516 | G296.8$-$0.3 | 10 | 9.6 | … | … | … | $1.1\times 10^{33}$ | 34
XMMU J173203.3$-$344518 | G353.6$-$0.7 | $\sim 27$ | 3.2 | … | $<9$ | … | $1.3\times 10^{34}$ | 35,36,37,38
CXOU J181852.0$-$150213 | G15.9$+$0.2 | $1-3$ | (8.5) | … | … | … | $\sim 1\times 10^{33}$ | 39
Note. — Above the line are eight well-established CCOs. Below the line are
three candidates.
References. — (1) Hui & Becker 2006a; (2) Gotthelf & Halpern 2009; (3)
Gotthelf et al. 2010; (4) De Luca et al. 2012; (5) Becker et al. 2012; (6)
this paper; (7) Slane et al. 2001; (8) Kargaltsev et al. 2002; (9) Bamba et
al. 2005; (10) Iyudin et al. 2005; (11) De Luca 2008; (12) Zavlin et al. 2000;
(13) Mereghetti et al. 2002; (14) Bignami et al. 2003; (15) De Luca et al.
2004; (16) Gotthelf & Halpern 2007; (17) Halpern & Gotthelf 2011; (18) Park et
al. 2006; (19) Park et al. 2009; (20) Lazendic et al. 2003; (21) Cassam-Chenaï
et al. 2004; (22) Gaensler et al. 2008; (23) Lovchinsky et al. 2011; (24)
Seward et al. 2003; (25) Gotthelf et al. 2005; (26) Halpern et al. 2007; (27)
Halpern & Gotthelf 2010a; (28) Pavlov et al. 2000; (29) Chakrabarty et al.
2001; (30) Mereghetti et al. 2002; (31) Pavlov & Luna 2009; (32) Ho & Heinke
2009; (33) Heinke & Ho 2010; (34) Sánchez-Ayaso et al. 2012; (35) Tian et al.
2008; (36) Abramowski et al. 2011; (37) Halpern & Gotthelf 2010b; (38) Halpern
& Gotthelf 2010c; (39) Reynolds et al. 2006.
The “anti-magnetar” explanation of CCOs, which is motivated by their weak
magnetic fields, absence of variability, and location on the $P-\dot{P}$
diagram, remains incomplete in detail. Specifically, it does not yet account
for the hot spots that are seen on the surfaces of CCO pulsars. Since the
spin-down power of a CCO pulsar is less than its X-ray luminosity, the latter
must be thermal emission from residual cooling, which can only be nonuniform
if there is anisotropic heat conduction. In the absence of strong magnetic
fields or magnetospheric activity, it is difficult to reproduce the light
curve and pulsed fraction of 64% from PSR J1852$+$0040 in Kes 79 (Halpern &
Gotthelf, 2010a; Shabaltas & Lai, 2012), or the two antipodal hot spots of
different temperatures and areas on RX J0822$-$4300 in Puppis A (Gotthelf &
Halpern, 2009; Gotthelf et al., 2010). The latter 0.112 s pulsar, hereafter
PSR J0821$-$4300, is a subject of this paper. Its spectrum is especially
puzzling in also displaying a phase-dependent emission feature at 0.7–0.8 keV
(Gotthelf & Halpern, 2009), which is reported to be variable in the long term
(De Luca et al., 2012).
Here we report the first spin-down measurement for PSR J0821$-$4300, based on
a dedicated program of phase-coherent timing jointly scheduled between
XMM–Newton and Chandra. It was also necessary to incorporate Chandra HRC
observations of the position and proper motion of PSR J0821$-$4300 in order to
determine its small period derivative accurately. The astrometric analysis is
described in Section 2. (The latter work was also carried out, with consistent
results, by Becker et al. 2012.) The results of the timing are presented in
Section 3. The X-ray flux and spectra are discussed in Section 4, with
particular attention paid to the spectral line and the question of its
possible variability. We also obtained new timing observations of PSR
J1210$-$5226 that resolve the prior ambiguity about its spin-down rate in
favor of the smaller value; this definitive result is presented in Section 5.
The nature of CCOs as anti-magnetars, and their possible evolutionary status,
are discussed in Section 6. Conclusions and proposals for future work follow
in Section 7.
## 2\. X-ray Position and Proper Motion
Evidence that PSR J0821$-$4300 has high proper motion from Chandra HRC images
over a 5 year baseline was reported by Hui & Becker (2006b) and Winkler &
Petre (2007), but with slightly disparate measurements of $\mu=107\pm 34$ mas
yr-1 and $\mu=165\pm 25$ mas yr-1, respectively, from the same data. Here, we
are concerned with timing this high-velocity pulsar over an extended period of
time with millisecond accuracy. When trying to measure a small $\dot{P}$ using
X-ray data, position and proper motion can contribute significant errors via
three effects. The first is an instrumental property of the Chandra CCDs when
used in continuous-clocking (CC) mode (see Section 3); the position of the
pulsar must be known a priori to $<0.\\!^{\prime\prime}5$ in order to
determine the time of arrival of each source photon. The second consideration
is the accuracy of the barycentric correction. The third effect is the
magnitude of the proper motion, which contributes a purely kinematic period
derivative via the “train whistle” effect (Shklovskii, 1970). The original
measurements of proper motion are not accurate enough to measure this effect,
which is crucial in the case of PSR J0821$-$4300.
Accordingly, we have reanalyzed the position and proper motion of PSR
J0821$-$4300 using the Chandra HRC-I data listed in Table 2, which now
includes a more recent pair of observations in 2010 August that extends the
baseline to 10.6 yr, enabling higher precision on both the contemporary
position for timing, and the proper motion. We will describe here any
differences between our method and previous work. For example, we did not use
an HRC-S observation (ObsID 1851) because of known systematic differences
between HRC-S and HRC-I. Ultimately, however, our results are consistent with
the recent analysis of the same data (including ObsID 1851) by Becker et al.
(2012).
Table 2Log of Chandra HRC-I Observations of PSR J0821$-$4300 ObsID | Date | Start Epoch | Exposure | Roll angle | Star A | PSR J0821$-$4300
---|---|---|---|---|---|---
| (UT) | (MJD) | (ks) | (∘) | (Counts)aaTotal counts collected in a $1.\\!^{\prime\prime}5$ radius aperture centered on the source. | (Counts)aaTotal counts collected in a $1.\\!^{\prime\prime}5$ radius aperture centered on the source.
749 | 1999 Dec 21 | 51533.95 | 18.0 | 338.7 | 47 | 3257
4612 | 2005 Apr 25 | 53485.31 | 40.2 | 261.9 | 123 | 7260
11819 | 2010 Aug 10 | 55418.72 | 33.7 | 163.4 | 101 | 5455
12201 | 2010 Aug 11 | 55419.13 | 38.7 | 162.9 | 117 | 6296
The data from all epochs were reprocessed and analyzed using the latest
calibration files and software (CIAO 4.4/CALDB 4.4.8). This processing
accounts for the HRC AMP_SF electronic ringing distortions discussed by Hui &
Becker (2006b). The HRC detector is well suited to astrometry, with its
processed pixel size of $0.\\!^{\prime\prime}1318$ that oversamples the on-
axis point spread function (PSF) by a factor of 5. For all observations, the
pulsar was placed close to the optical axis where the PSF is essentially
symmetric. In the following analysis we assume, as there is no evidence to the
contrary, that the HRC focal plane is linear and the aspect reconstruction
introduces no errors in roll angle. The two pointings on consecutive days in
2010 August are sufficiently different in their aspect reconstruction that we
analyze them individually.
The nominal uncertainty in aspect reconstruction for a typical Chandra
observation is $0.\\!^{\prime\prime}6$. It is often possible to remove most of
this systematic error by using nearby X-ray point sources with precisely
measured optical coordinates to correct the absolute astrometry. Hui & Becker
(2006b) used their X-ray detected “star A” as their sole fiducial point, and
fitted a model PSF interpolated from the CIAO library appropriate for its
position and estimated photon energy to determine its position. Winkler &
Petre (2007) used this star and two additional stars, and followed the updated
method outlined in the CIAO thread for generating a Monte Carlo PSF using the
CHaRT/MARX software for input into the CIAO/Sherpa spectral/image fitting
software package.
Figure 1.— Chandra HRC-I images around reference star A (3UC094-058669) used
to define the coordinate system for position and proper motion of PSR
J0821$-$4300\. Each panel shows the observed counts from Star A (left) and the
simulated PSF (right) for its location on the focal plane in native HRC-I
pixels. The plots cover $12^{\prime\prime}\times 12^{\prime\prime}$ in
celestial coordinates. The total counts in a $1.\\!^{\prime\prime}5$ radius
aperture are given in Table 3.
In our analysis, we also follow the CIAO thread to characterize the HRC-I PSF,
but we adopt a simpler approach to measuring source locations, one that is not
dependent on model fitting in the image domain. Our method is guided by the
following observations. First, the on-axis, symmetric image of the pulsar
contains enough counts that a simple centroid calculation is a sufficiently
accurate measurement of its position on the detector. Second, star A of Hui &
Becker (2006b), which lies $2\farcm 7$ from PSR J0821$-$4300, is the only
useful fiducial source for registering the X-ray image. The position and
proper motion of star A are taken from the UCAC3 (Zacharias et al., 2010),
where it is listed as 3UC094-058669 with coordinates (J2000.0) R.A.=$08^{\rm
h}21^{\rm m}46.\\!^{\rm s}2924(16)$,
decl.=$-43^{\circ}02^{\prime}03.\\!^{\prime\prime}640(49)$, and proper motion
$\mu_{\alpha}\,{\rm cos}\,\delta=-14.3(2.0)$, $\mu_{\delta}=-3.6(5.5)$ mas
yr-1. X-ray position measurements of the two weaker, off-axis stars used by
Winkler & Petre (2007) only add to the uncertainty (as quantified below) in
the absolute astrometry. Third, the few X-ray photons from star A, and its
broad off-axis PSF, do not warrant a sophisticated image fitting technique.
Instead, we use a “corrected centroid” method, as described below.
To determine the source location of star A in the X-ray images we start with
the CHaRT/MARX simulation of the PSF, as described in the CIAO user
webpages111http://cxc.harvard.edu/chart, for its respective locations on the
focal plane. Figure 1 shows the distribution of the counts from each HRC image
and corresponding Monte Carlo PSF. It is apparent that star A is poorly
sampled in the data, with total source counts in the range $47\leq N\leq 123$
(Table 2). The maximum number of counts per pixel is typically only $2-4$,
making forward fitting poorly constrained statistically, while sharp features
in the model can cause systematic offsets. Furthermore, the source is immersed
in a substantial diffuse background from the Puppis A SNR which, although it
only contributes a few photons over the source region, adds uncertainty to the
position measurement because of the small count statistics, especially when
fitting over a larger area.
Figure 2.— Four position measurements of PSR J0821$-$4300 spanning 10.6 yr,
after correction using the optical/X-ray star A (3UC094-058669) as a
reference. The two observations in 2010 nearly coincide in time. The positions
(diamonds) and their errors are fitted with a linear model (dashed lines). The
fitted parameters are listed in Table 4.
To sidestep these effects, the coordinates of star A are first found from its
centroid. Photons were extracted from circular aperture of radius
$1.\\!^{\prime\prime}5$, chosen to minimize the background counts. This
aperture encloses essentially all of the signal in that fraction of the PSF
with a finite probability of producing a single count during the observation.
This measurement was made using the CIAO tool dmstat and is iterated to
produce the final coordinates. However, while this results in a well-defined
and statistically meaningful measurement, it does not account for the shape of
the complex off-axis PSF, whose orientation depends on the spacecraft roll
angle (which differs for each observation; see Figure 1), or for the monotonic
off-axis deviations introduced by the flat focal plane. To quantify these
systematic effects we also measure the centroid of the simulated PSF of star A
in sky coordinates in each image and compare it to the coordinates that were
input to CHaRT/MARX for that PSF. The difference constitutes the small, but
critical correction to the centroid of star A.
We estimated the uncertainty in the derived coordinates of star A using a
Monte Carlo method. We generated 500 realizations of star A by sampling the
PSF using a random number generator to match the observed counts, and
accumulated the centroid measurements to build up a distribution in right
ascension and declination. To account for the observed background, we included
a random distribution of photons within the source aperture. The resulting
(Gaussian) width of the distribution of centroids is typically $\sigma\approx
0.\\!^{\prime\prime}06$ in each of the two coordinates (see Table 3). These
reproduce the expected “standard error” for a centroid,
$\approx{\sigma}/\sqrt{N}$. We also simulated the uncertainties for the two
other fiducial sources used in Winkler & Petre (2007) and find that their
inclusion in the position determination would only increase the uncertainty on
the final coordinates.
The pulsar itself is a strong source (see Table 2) whose coordinates are
precisely measured using a centroid calculation with uncertainty an order of
magnitude smaller than those measured for star A. Because of the symmetry of
its on-axis PSF, no systematic correction is required for the pulsar. Its
coordinates are then adjusted by the difference between the optical and X-ray
coordinates of star A at each epoch to produce its final astrometric position
in Table 3. Fitting the position of the pulsar as a function of time then
yields the proper motion. Figure 2 shows the $\chi^{2}$ fit with constant
velocity in each coordinate, and Table 4 lists the formal solution and
quantities derived from it. The derived total proper motion of $61\pm 9$ mas
yr-1 is in agreement with the value determined by Becker et al. (2012), $71\pm
12$ mas yr-1, and is smaller than previously published numbers for reasons
discussed in that paper.
Table 3Position Measurements for PSR J0821$-$4300
Epoch | Star A (Optical) | Star A (X-ray) | PSR J0821$-$4300 (corrected)
---|---|---|---
(year) | R.A. (h m s) | Decl. (${}^{\circ}\ {}^{\prime}\ {}^{\prime\prime}$) | R.A. (h m s) | Decl. (${}^{\circ}\ {}^{\prime}\ {}^{\prime\prime}$) | R.A. (h m s) | Decl. (${}^{\circ}\ {}^{\prime}\ {}^{\prime\prime}$)
1999.975 | 08 21 46.2928(16) | –43 02 03.637(49) | 08 21 46.2882(65) | -43 02 03.397(83) | 08 21 57.4024(67) | -43 00 16.894(96)
2005.316 | 08 21 46.2858(21) | –43 02 03.657(57) | 08 21 46.3011(47) | -43 02 03.756(41) | 08 21 57.3685(52) | -43 00 17.023(71)
2010.610 | 08 21 46.2789(31) | –43 02 03.676(76) | 08 21 46.2621(62) | -43 02 04.113(47) | 08 21 57.3488(69) | -43 00 17.185(89)
2010.611 | 08 21 46.2789(31) | –43 02 03.676(76) | 08 21 46.2575(57) | -43 02 04.068(51) | 08 21 57.3476(65) | -43 00 17.190(92)
Note. — All coordinates are equinox J2000. Optical coordinates for star A
(3UC094-058669) are corrected for the epoch of proper motion. The X-ray
position of star A is determined using the method described in the text. The
pulsar coordinates are corrected by the difference between the optical and
X-ray coordinates of star A. Uncertainties on the last digits are in
parentheses.
The tangential velocity of PSR J0821$-$4300 then depends on a distance
determination for Puppis A, which has ranged from 1 to 2.5 kpc according to
various methods, and is a matter of unresolved debate in the most recent
studies quoted here. One (Reynoso et al., 1995, 2003) is based on 21 cm H I
velocities, and a perceived morphological association of features in H I
surrounding the pulsar and the SNR at $v_{\rm lsr}=+16$ km s-1, the latter
corresponding to $d=2.2\pm 0.3$ kpc. The second method uses spectra of ground-
state hydroxyl lines (Woermann et al., 2000), which show absorption at $v_{\rm
lsr}<+7.6$ km s-1, and emission above this velocity, from which
$d=1.3^{+0.6}_{-0.8}$ kpc is derived. Both sets of authors employ assumptions
that are not mutually accepted, and are beyond the scope of this paper to
evaluate. We will adopt a fiducial distance of $2.2\pm 0.3$ kpc for purposes
of further calculations, while noting the implications of a possible smaller
distance where relevant.
Table 4Ephemeris of PSR J0821$-$4300 Parameter | Value
---|---
Position and Proper Motion
Epoch of position and $\mu$ (MJD) | 53964.0
R.A. (J2000) | $08^{\rm h}21^{\rm m}57.\\!^{\rm s}3653(31)$
Decl. (J2000) | $-43^{\circ}00^{\prime}17.\\!^{\prime\prime}074(43)$
R.A. proper motion, $\mu_{\alpha}\,{\rm cos}\,\delta$ | $-54.1\pm 8.3$ mas yr-1
Decl. proper motion, $\mu_{\delta}$ | $-28.1\pm 10.5$ mas yr-1
Total proper motion, $\mu$ | $61.0\pm 8.8$ mas yr-1
Position angle of proper motion | $242.\\!^{\circ}5\pm 9.\\!^{\circ}5$
Tangential velocityaaAssuming $d=2.2\pm 0.3$ kpc (Reynoso et al., 1995)., $v_{\perp,c}$ | $629\pm 126$ km s-1
Timing Solution
Epoch of ephemeris (MJD TDB)bbEpoch of fitted minima of the $1.5-4.5$ keV pulse profile; phase zero in Figure 5. | 55580.0000006
Span of ephemeris (MJD) | 55,182–56,027
Frequency, $f$ | 8.86529105448(32) Hz
Frequency derivative, $\dot{f}$ | $(-7.29\pm 0.28)\times 10^{-16}$ Hz s-1
Period, $P$ | 0.1127994550720(41) s
Period derivative, $\dot{P}$ | $(9.28\pm 0.36)\times 10^{-18}$
Kinematic period derivativeaaAssuming $d=2.2\pm 0.3$ kpc (Reynoso et al., 1995)., $\dot{P}_{k}$ | $(2.24\pm 0.72)\times 10^{-18}$
Intrinsic period derivativeaaAssuming $d=2.2\pm 0.3$ kpc (Reynoso et al., 1995)., $\dot{P}_{\rm int}$ | $(7.04\pm 0.80)\times 10^{-18}$
Surface dipole magnetic field, $B_{s}$ | $2.9\times 10^{10}$ G
Spin-down luminosity, $\dot{E}$ | $1.9\times 10^{32}$ erg s-1
Characteristic age, $\tau_{c}$ | 254 Myr
The tangential velocity of PSR J0821$-$4300 at $d=2.2\pm 0.3$ kpc is
$v_{\perp}=636\pm 126$ km s-1. This velocity is at the high end of the
distribution of two-dimensional velocities of pulsars measured by Hobbs et al.
(2005), who find mean values of $\bar{v}_{\perp}=246\pm 22$ km s-1 for 121
ordinary (non-recycled) pulsars, and $\bar{v}_{\perp}=307\pm 47$ km s-1 for 46
pulsars whose characteristic ages are $<3$ Myr. The individual pulsar
velocities are corrected for the solar motion with respect to the local
standard of rest and a flat Galactic rotation curve in order to express them
in the frame of the rotating Galactic disk. In the case of PSR J0821$-$4300
this correction is dominated by the solar motion for the range of plausible
distances, and is only $-7$ km s-1, resulting in a corrected tangential
velocity of $v_{\perp,c}=629$ km s-1. If the distance is as small as 1 kpc,
this reduces to an unexceptional 290 km s-1.
The Shklovskii (1970) effect, a purely kinematic contribution to the observed
period derivative, will be significant. For a source moving at constant
velocity the kinematic contribution is
$\dot{P}_{k}\,={\mu^{2}\,P\,d\over c}\ =\ {v^{2}_{\perp}\,P\over d\,c}\ .$
$None$
From the proper motion measurement of PSR J0821$-$4300, $\dot{P}_{k}=(2.24\pm
0.72)\times 10^{-18}$ is calculated, where we have propagated the
uncertainties on both $\mu$ and $d$.
## 3\. X-ray Timing
Previous observations of PSR J0821$-$4300 were only able to set upper limits
on its period derivative (Gotthelf & Halpern, 2009; Gotthelf et al., 2010; De
Luca et al., 2012). Evidently $\dot{P}$ is so small that it can only be
measured by phase-coherent timing. Accordingly, we designed a sequence of
observations coordinated between XMM–Newton and Chandra that would start and
maintain phase connection over a 2 year span, 2010 May – 2012 April. The
scheduling strategy is the same as was used for PSR J1852$+$0040 in Kes 79
(Halpern & Gotthelf, 2010a). By design, the resulting ephemeris was also
connected backward to archival observations that were obtained in 2009
December and 2010 April (De Luca et al., 2012), which extended the time span
to 2.3 years. All of the timing observations used in this analysis are listed
in Table 5. The discovery observations from 2001 (Gotthelf & Halpern, 2009)
are not included, as they are too far removed in time to be reliably
connected.
Table 5Log of X-ray Timing Observations of PSR J0821$-$4300 Mission | Instr/Mode | ObsID | Date | Elapsed time/ | Start Epoch | PeriodaaBarycentric period derived from a $Z^{2}_{1}$ test. The Leahy et al. (1983) uncertainty on the last digits is in parentheses. | $Z^{2}_{1}$
---|---|---|---|---|---|---|---
| | | (UT) | Livetime (ks) | (MJD) | (s) |
XMM | EPIC-pn/SW | 0606280101 | 2009 Dec 17,18 | 85.1/54.9 | 55182.820 | 0.112799488(12) | 173.0
XMM | EPIC-pn/SW | 0606280201 | 2010 Apr 05 | 42.2/29.4 | 55291.377 | 0.112799451(20) | 199.1
XMM | EPIC-pn/SW | 0650220201 | 2010 May 02 | 28.0/19.6 | 55318.782 | 0.112799390(41) | 135.5
Chandra | ACIS-S3/CC | 12108 | 2010 Aug 16 | 34.0/34.0 | 55424.625 | 0.112799470(21) | 192.3
XMM | EPIC-pn/SW | 0650220901 | 2010 Oct 15 | 23.5/16.4 | 55484.109 | 0.112799519(44) | 147.0
XMM | EPIC-pn/SW | 0650221001 | 2010 Oct 15 | 23.5/16.4 | 55484.987 | 0.112799462(39) | 156.7
XMM | EPIC-pn/SW | 0650221101 | 2010 Oct 19 | 26.5/18.6 | 55488.332 | 0.112799518(40) | 150.5
XMM | EPIC-pn/SW | 0650221201 | 2010 Oct 25 | 24.5/17.2 | 55494.228 | 0.112799486(35) | 162.1
XMM | EPIC-pn/SW | 0650221301 | 2010 Nov 12 | 23.5/16.5 | 55512.524 | 0.112799391(52) | 144.7
XMM | EPIC-pn/SW | 0650221401 | 2010 Dec 20 | 27.2/19.0 | 55550.159 | 0.112799450(35) | 163.3
Chandra | ACIS-S3/CC | 12109 | 2011 Feb 04 | 33.0/33.0 | 55596.837 | 0.112799445(27) | 173.4
XMM | EPIC-pn/SW | 0650221501 | 2011 Apr 12 | 30.0/21.0 | 55663.857 | 0.112799449(30) | 156.6
XMM | EPIC-pn/SW | 0657600101 | 2011 May 18 | 36.5/25.6 | 55699.925 | 0.112799480(17) | 195.6
Chandra | ACIS-S3/CC | 12541 | 2011 Aug 11 | 33.0/33.0 | 55784.655 | 0.112799412(22) | 180.4
XMM | EPIC-pn/SW | 0657600201 | 2011 Nov 08 | 37.2/26.1 | 55873.289 | 0.112799459(28) | 142.2
Chandra | ACIS-S3/CC | 12542,14395 | 2012 Feb 18,19 | 33.1/33.1 | 55975.446 | 0.112799481(35) | 165.8
XMM | EPIC-pn/SW | 0657600301 | 2012 Apr 10 | 35.3/24.7 | 56027.022 | 0.112799445(17) | 196.5
The Chandra observations used the Advanced Camera for Imaging and Spectroscopy
(ACIS-S3) in continuous-clocking (CC) mode to provide time resolution of 2.85
ms. All data were reprocessed from the level 1 event files with the
coordinates corrected for the proper motion of PSR J0821$-$4300 given in Table
4, and analyzed using the latest calibration files and software (CIAO
4.4/CALDB 4.4.8). Reprocessing with a source position that is accurate to $<1$
pixel ($<0.\\!^{\prime\prime}5$) ensures that the time assignment is precise
to $\lesssim 3$ ms. All of the XMM–Newton observations used the pn detector of
the European Photon Imaging Camera (EPIC-pn) in “small window” (SW) mode to
achieve 5.7 ms time resolution, and an absolute uncertainty of $\approx 3$ ms
on the arrival time of any photon. Each data set was examined and cleaned of
intervals of high particle background due to solar activity, as necessary. Two
pairs of data sets that were acquired on consecutive days were merged to
improve their statistics. The photon arrival times from all data were
transformed to Barycentric Dynamical Time (TDB) using the coordinates of the
pulsar corrected for proper motion.
For PSR J0821$-$4300, the pulsed signal strength is a strong function of
energy, not only because of its spectrum relative to the background but
because of the cancellation by emission from the opposite pole, as described
in Gotthelf et al. (2010). For each observation listed in Table 5 we extracted
source photons using an aperture centered on the source and optimized for the
signal strength in the hard $1.5-4.5$ keV energy band. For the XMM–Newton
observations we used an aperture of radius of $30^{\prime\prime}$. For the
Chandra CC-mode observations we selected five columns
($2.\\!^{\prime\prime}4$). We also examined the soft, phase shifted $0.5-1.0$
keV band. However, these data are noisier and their use did not in the end
improve the timing results significantly. The phase cancellation effect also
prevents pulsations from being detected by the Chandra HRC, which has
insufficient energy resolution.
As in our previous timing studies of CCOs (Halpern & Gotthelf, 2010a, 2011),
we employed two complementary approaches to fitting an ephemeris. First, we
used the $Z^{2}_{1}$ (Rayleigh) test (Strutt, 1880; Buccheri et al., 1983) in
a coherent analysis of the entire set of 17 observations. Beginning with the
closely spaced set spanning 2010 October 15–19, the $Z^{2}_{1}$ test
determined the pulse frequency with sufficient accuracy to connect in phase
uniquely to the next observation. This procedure was iterated by adding each
subsequent observation, and including a frequency derivative when it became
evident. We also worked backward in time, incorporating all 17 observations in
the resulting unique ephemeris. The fitted frequency derivative is
$\dot{f}=(-6.94\pm 0.28)\times 10^{-16}$ Hz s-1, where the $1\sigma$
uncertainty comes from the $\Delta Z^{2}_{1}=-2.3$ contour around the peak
power in ($f,\dot{f}$) space.
The second method also started with the $Z^{2}_{1}$ test statistic, this time
to find the period and pulse profile separately at each epoch. The 17 profiles
were cross-correlated, shifted, and summed to create a master pulse profile
template. The process was iterated to generate a more accurate template and a
set of time-of-arrival (TOA) measurements and their uncertainties for each
epoch. These TOAs were fitted with a quadratic model in frequency and
frequency derivative using a $\chi^{2}$ fitting routine to minimize their
phase residuals. We searched for a coherent phase-connected solution over a
grid of $f$ and $\dot{f}$ covering the range $f=8.8652906\pm 0.0000016$ Hz (at
epoch MJD 55,580) and $-3.1\times 10^{-14}<\dot{f}<1.9\times 10^{-14}$ Hz s-1,
with an oversampling factor of 10 for accuracy. This range corresponds to the
$3\sigma$ limits of an incoherent fit to all of the measured frequencies,
including the 2001 discovery observations. The resulting frequency derivative
from TOA fitting, $\dot{f}=(-7.29\pm 0.28)\times 10^{-16}$ Hz s-1, is
consistent with the value found above from the coherent $Z^{2}_{1}$ search. We
adopt the TOA result for the final timing solution listed in Table 4.
Figure 3.— $P-\dot{P}$ diagram of isolated pulsars (dots), binary radio
pulsars (circled dots), and other types of isolated X-ray pulsars (colored
symbols). The CCO pulsars (red stars) in Kes 79 and Puppis A have virtually
the same spin parameters. The upper limit on Calvera’s $\dot{P}$ is from
Halpern (2011). The radio pulsar death line $B/P^{2}=1.7\times 10^{11}$ G s-2
of Bhattacharya et al. (1992) is indicated. The van den Heuvel (1987) spin-up
limit for recycled pulsars corresponds to $P({\rm ms})=1.9\,(B/10^{9}\,{\rm
G})^{6/7}$. The exponent in this equation corrects a typographical error in
the caption to Figure 7 of Halpern & Gotthelf (2010a), although the
corresponding line in the Figure was correct.
The observed $\dot{P}=(9.28\pm 0.36)\times 10^{-18}$ can now be split into the
sum of its intrinsic and kinematic contributions, $\dot{P}=\dot{P}_{\rm
int}+\dot{P}_{k}$. Since we determined in Section 2 that $\dot{P}_{k}=(2.24\pm
0.72)\times 10^{-18}$, the intrinsic period derivative is $\dot{P}_{\rm
int}=(7.04\pm 0.80)\times 10^{-18}$. Parenthetically, we note that the small
observed period derivative is independent evidence that the proper motion of
the pulsar is not as large as the value originally quoted by Winkler & Petre
(2007), $\mu=165\pm 25$ mas yr-1. If so, and if $d=2.2$ kpc, $\dot{P}_{k}$
would be $(1.64\pm 0.55)\times 10^{-17}$, requiring $\dot{P}_{\rm int}$ to be
negative, i.e., the pulsar would be spinning up.
In the vacuum dipole spin-down formalism, the values of $P$ and $\dot{P}_{\rm
int}$ imply a surface magnetic field strength $B_{s}=3.2\times
10^{19}(P\dot{P})^{1/2}~{}G=2.9\times 10^{10}$ G, a spin-down luminosity
$\dot{E}=-I\Omega\dot{\Omega}=4\pi^{2}I\dot{P}/P^{3}=1.9\times 10^{32}$ erg
s-1, and characteristic age $\tau_{c}\equiv P/2\dot{P}=254$ Myr. PSR
J0821$-$4300 is nearly identical in its spin properties to PSR J1852$+$0040 in
Kes 79, as shown in Figure 3. The uncertainties in distance and proper motion
have only a small effect on the derived magnetic field. For a smaller distance
of 1 kpc, $\dot{P}_{k}$ is reduced to $1.02\times 10^{-17}$, and
$B_{s}=3.1\times 10^{10}$ G. An absolute upper limit regardless of distance
and proper motion is $B_{s}<3.3\times 10^{10}$ G.
The phase residuals from the best-fit solution are shown in Figure 4. The
weighted rms of the phase residuals is 5.1 ms, or 0.045 pulse cycles, which is
comparable to the individual measurement errors (average $\sigma=3.6$ ms). It
is not clear if there is any real timing noise and/or systematic errors in the
TOAs.
The light curves of PSR J0821$-$4300 in the soft and hard bands are shown in
Figure 5. These were derived by folding all the timing data on the best
fitting ephemeris given in Table 4. As revealed by a cross-correlation, the
soft and hard pulses are out of phase by $0.45\pm 0.02$ cycles, consistent
with that found by De Luca et al. (2012) using the 2009 December and 2010
April XMM–Newton data.
The energy dependence of the pulsar modulation provided an important
diagnostic for modeling the viewing geometry and surface emission of PSR
J0821$-$4300 (Gotthelf et al., 2010). By analyzing the lightcurve in narrower
energy bands than in Figure 5, we can resolve the signal modulation and phase
for PSR J0821$-$4300 over the $0.3-5$ keV range. The data were grouped into 23
energy bands that are at least $100$ eV in width and have a signal-to-noise
$N_{s}/\sqrt{N_{s}+N_{b}}>100$, where $N_{s},N_{b}$ are the source and
background counts, respectively. We used $Z_{1}^{2}$ to provide a model of the
unbinned lightcurve. The first Fourier component is a reasonable estimate as
the lightcurve is sinusoidal in each energy band to within counting
statistics. The error bar for the phase is calculated by cross-correlation,
with the profile of Figure 5 serving as a template.
The result is presented in Figure 6. As the energy dependent modulation
decreases, the phase becomes undefined in two energy bands; these two phase
points are not plotted. The modulation is qualitatively similar to that
predicted by the antipodal model of Gotthelf et al. (2010) (cf. their Figure
6), providing confirmation of the basic model. The prediction was based on
fitting the modulation in only three energy bands, using much less data, and
differs somewhat from the new, resolved data having an order-of-magnitude more
counts. In particular, the energy of the minimum modulation is lower (1.12 vs.
1.28 keV), and the form of the modulation is more complex than predicted.
Furthermore, the phase is seen to drift at lower energies and the transition
is not as sharp compared to the antipodal case, in which the phase was
statistically either 0.0 or 0.5. The observed characteristics likely imply
that the hotspots are offset from a strictly antipodal geometry. The high
quality data presented herein should allow a far more detailed modeling of the
surface emission of PSR J0821$-$4300.
Figure 4.— For the timing observations of PSR J0821$-$4300 listed in Table 5,
pulse-phase residuals from the linear term (dash-dot line) of the phase
ephemeris presented in Table 4. The quadratic term (solid line) contributes
$\approx\pm 0.5$ cycles to the ephemeris over the 2.3 years of timing.
Figure 5.— Summed pulse profiles of PSR J0821$-$4300 in the $1.5-4.5$ keV band
(top) and the $0.5-1.0$ keV band (bottom) using all of the observations listed
in Table 5, folded according to the ephemeris of Table 4. These hard and soft
pulse profiles are out of phase by $\phi=0.45\pm 0.02$ cycles. The intervals
between the vertical lines ($\Delta\phi=0.3$ cycles) correspond to the two
phase regions used in the phase resolved analysis of Section 4.3 and Table 8.
Figure 6.— Modulation and pulse phase as a function of energy for PSR
J0821$-$4300 using all of the observations listed in Table 5, folded according
to the ephemeris of Table 4. The modulation (top) and pulse phase (bottom)
reproduce the form predicted for the antipodal model of Gotthelf et al.
(2010). Two undefined phase points are omitted.
## 4\. Spectral Analysis
Previously, we modelled the original (year 2001) XMM–Newton spectra of PSR
J0821$-$4300 as surface blackbody emission from two antipodal spots of
different temperatures and areas. Crucially, this model is also able to
account for the observed energy-dependent pulse modulation and phase shift
(Gotthelf & Halpern, 2009; Gotthelf et al., 2010). Additionally, we found
evidence of a spectral line feature around 0.8 keV, which more recent data
obtained by De Luca et al. (2012) suggests has a time-variable centroid
energy. With the increased quantity of data now in hand on PSR J0821$-$4300,
we can re-examine this spectral feature, first by combining all 13 XMM–Newton
observations presented in Table 5 plus the two observations obtained in 2001,
and then by testing for any variability among the observations.
### 4.1. Summed XMM–Newton Spectrum
The spectral analysis presented here uses exclusively the data collected with
the EPIC pn. Although background is much reduced in the EPIC MOS, this
instrument is less sensitive at soft energies, collecting 4.7 times fewer
photons than the EPIC pn in the $0.3-1$ keV band. Furthermore, much of the
EPIC MOS data were lost as the high surface brightness of the Puppis A SNR
frequently triggered an automatic shutoff of that detector. We also neglect
the Chandra ACIS spectra here because of their even poorer low-energy
sensitivity and the increased background and other uncertainties involved in
the analyzing the Chandra CC-mode data. Lastly, one ACIS image taken in timed
exposure mode (ObsID 750) suffers from pileup, and was not used.
For the EPIC pn data, the main technical issue is that the SNR background
exceeds the point source counts below 1 keV. Surprisingly, the background
intensity and spectral shape are both strong functions of spacecraft roll
angle. This effect is made evident because the XMM–Newton data sets were
acquired in two narrow ranges of roll angle roughly 180∘ apart associated with
their respective visibility windows, with the time divided nearly equally
between the two. We checked all of our spectral results carefully for features
that might be dependent on roll angle due to systematic errors in background
subtraction. No such systematic effect were found, which indicates that the
background subtraction is reliable in general.
We extracted spectra for each observation from the EPIC pn detector using an
aperture of radius $0\farcm 3$ and a concentric background annulus of $0\farcm
5<r<0\farcm 6$, selecting only events with ${\tt PATTERN}\leq 4$ and FLAG$=0$.
The data were filtered to exclude time intervals of high background identified
by count rate $>0.1$ s-1 in the $10-12$ keV energy band. An inspection of the
pattern distribution of single and double events shows no evidence of pile-up
and suggests that a lower energy bound in the range $0.3-0.4$ keV is
acceptable. Response matrices and effective area files were generated for each
observation using the SAS software suite. We combined data from all
observations using the FTOOL addascaspec to produce a single source spectrum
and associated files. The combined spectrum was grouped to include at least
1000 counts per channel and was fitted using XSPEC v12.21 to a two blackbody
model with interstellar absorption over the energy range $0.3-5$ keV (see
Figure 7 and Table 6).
Table 6Models for the Summed XMM–Newton Spectrum of PSR J0821$-$4300
Model | Two Blackbody | $+$ Emis. line | $+$ Abs. line | $+$ Two Abs. lines | $+$ Cyclabs
---|---|---|---|---|---
$N{\rm{}_{H}}$ ($10^{21}$ cm-2) | $3.8\pm 0.1$ | $4.3\pm 0.3$ | $3.2\pm 0.2$ | $2.9\pm 0.4$ | … | $2.8\pm 1.01$ | …
$k{\rm T}_{w}$ (keV) | $0.26\pm 0.01$ | $0.25\pm 0.01$ | $0.29\pm 0.01$ | $0.29\pm 0.01$ | … | $0.28\pm 0.03$ | …
$k{\rm T}_{h}$ (keV) | $0.46\pm 0.01$ | $0.45\pm 0.01$ | $0.49\pm 0.02$ | $0.49\pm 0.03$ | … | $0.47\pm 0.02$ | …
$L_{w}({\rm bol})$ ($10^{33}$ erg s-1)aaBlackbody bolometric luminosity for a distance of 2.2 kpc. | $3.3\pm 0.1$ | $3.6\pm 0.2$ | $3.1\pm 0.2$ | $3.0\pm 0.2$ | … | $3.2\pm 0.8$ | …
$L_{h}({\rm bol})$ ($10^{33}$ erg s-1)aaBlackbody bolometric luminosity for a distance of 2.2 kpc. | $2.0\pm 0.2$ | $2.0\pm 0.2$ | $1.4\pm 0.3$ | $1.4\pm 0.3$ | … | $1.7\pm 0.4$ | …
$A_{w}$ (km2) | $72\pm 11$ | $89\pm 18$ | $44\pm 6$ | $40\pm 8$ | … | $53\pm 14$ | …
$A_{h}$ (km2) | $4.4\pm 0.9$ | $4.5\pm 0.9$ | $2.5\pm 0.8$ | $2.4\pm 1.0$ | … | $3.4\pm 0.8$ | …
$E_{0}$ (keV) | … | $0.75\pm 0.01$ | $0.46\pm 0.05$ | $0.46\pm 0.01$ | $2E_{o}$ | $0.46\pm 0.01$ | $2E_{o}$
WidthbbGaussian $\sigma$ for the emission or absorption lines, natural width $W$ for the cyclotron absorption model (Makishima et al., 1990; Mihara et al., 1990). (eV) | … | $75\pm 20$ | $85\pm 50$ | $106\pm 20$ | $34-62$ | $53-97$ | $35-290$
EW (eV) | … | $53\pm 10$ | … | … | … | … | …
$\tau_{o}$ccOptical depth at line center. | … | … | $0.1-0.9$ | $0.6-1.3$ | $<0.035$ | $0.9-1.7$ | $<0.14$
$\chi^{2}({\rm DoF})$ | $1.50(359)$ | 1.08(356) | $1.16(356)$ | $1.08(354)$ | $1.12(354)$
Note. — The $1\sigma$ uncertainties for three interesting parameters
($\Delta\chi^{2}=3.53$) are given.
The resulting high signal-to-noise ratio of the fitted spectrum reveals
significant features that are unaccounted for by the two blackbody model. The
best fit model, with $N_{\rm H}=(3.8\pm 0.01)\times 10^{21}$ cm-2,
$kT_{w}=0.26\pm 0.01$ keV, and $kT_{h}=0.46\pm 0.01$ keV, has reduced
$\chi^{2}_{\nu}=1.50$ for 359 degrees of freedom (DoF), which is formally
unacceptable. The deviations are evident in structure in the residuals in
Figure 7a. Adding a Gaussian emission line to the model as suggested by our
previous work improves the fit to $\chi^{2}_{\nu}=1.08$ for 358 DoF (Figure
7b). The centroid energy of the line is $0.75\pm 0.01$ keV, and its equivalent
width is $53\pm 10$ eV.
Figure 7.— (a) EPIC pn spectrum of the 16 summed XMM–Newton observations of
PSR J0821$-$4300 fitted to a double blackbody model. The residuals from the
fit are shown in the bottom panel. (b) The same spectrum fitted with a double
blackbody model plus Gaussian emission line. The parameters of these fits are
given in Table 6.
Considering the shape of the residuals from the two blackbody fit in Figure
7a, an alternative hypothesis is that an absorption feature, or features, is
responsible. Accordingly we applied a Gaussian absorption line, two Gaussian
absorption lines, and finally, the cyclotron absorption model of Makishima et
al. (1990) and Mihara et al. (1990), which is available in XSPEC as cyclabs.
As shown in Table 6 and Figure 8, all of these models gave acceptable fits.
Two absorption lines, when fitted independently, are separated by nearly a
factor of 2 in energy, which is suggestive of the fundamental and first
harmonic in a cyclotron model. Therefore we fixed the ratio of their centroid
energies at 2, with the result that their centroids are at 0.46 keV and 0.92
keV, bracketing the energy previously ascribed to an emission line. The
cyclabs model, which also fixes this ratio, yields the same centroid energies.
We conclude that, from the phase-averaged spectra alone, it is not possible to
distinguish an emission feature from one or two absorption lines, which leaves
the physical interpretation uncertain.
The fundamental energy of the electron cyclotron resonance falls at
$E_{0}=1.16\,(B/10^{11}\,{\rm G})/(1+z)\ {\rm keV},$ $None$
where $z$ is the gravitational redshift. Assuming a typical value of $z=0.3$,
and if $B\approx B_{s}=2.9\times 10^{10}$ G, the equatorial field from the
vacuum dipole spin-down result, then $E_{0}\approx 0.26$ keV is expected. When
compared with $E_{0}=0.46$ keV from the absorption-line fit, or $0.75$ keV
from the emission-line fit, this hints that the local surface field where the
line is formed is larger than the equatorial dipole field, but is perhaps more
compatible with the field at the pole, $B_{p}=2\,B_{s}$. Other factors
possibly affecting a comparison between the dipole spin-down $B$-field and the
surface $B$-field from the cyclotron energy will be explored in Section 6.2.
Figure 8.— Same as Figure 7 but showing the residuals from fits to a double
blackbody model plus, from top to bottom: no line, a Gaussian emission line, a
Gaussian absorption line, two Gaussian absorption lines, and a cyclotron
absorption line model. The parameters of the fits are given in Table 6.
The addition of emission or absorption lines to the model near the low-energy
end of the XMM–Newton spectrum affects the fitted value of the column density
$N_{\rm H}$, with values ranging from $(2.8-4.3)\times 10^{21}$ cm-2 for the
different models in Table 6. Therefore, it is possible that comparison with
independent measurements of $N_{\rm H}$ may suggest a preference for one or
another of these models. For example, the 21 cm H I emission in the foreground
of Puppis A amounts to $N_{\rm HI}=2.5\times 10^{21}$ cm-2 according to
Reynoso et al. (2003). To obtain this value, they integrated the H I line
emission in the radial velocity range $-10$ to $+16$ km s-1, the latter
velocity corresponding to their assumed 2.2 kpc distance of Puppis A.
Comparison with the values of $N_{\rm H}$ in Table 6 tends to favor the X-ray
spectral models that include absorption lines. While this agreement is
encouraging, it is subject to the caveat that the 21 cm column density is
formally a lower limit, assuming as it does that the line is optically thin.
Better support for a low column density comes from the XMM–Newton RGS spectra
of several regions of the Puppis A SNR fitted by Katsuda et al. (2012). These
require X-ray $N_{\rm H}$ in the range $(2.58-2.85)\times 10^{21}$ cm-2, which
also agrees closely with the $N_{\rm H}$ from our absorption-line models for
the pulsar spectrum.
Figure 9.— Blackbody temperatures and flux in the $0.5-5$ keV band for the 16
individual XMM–Newton observations of PSR J0821$-$4300, fitted to the two
blackbody model. Data are taken from Table 7. Errors bars are 1$\sigma$. The
weighted mean values are indicated by the dashed lines.
Figure 10.— Combined XMM–Newton EPIC pn spectrum of PSR J0821$-$4300 in two
pulse-phase intervals defined in Figure 5 ($\Delta\phi_{1}$, blue,
$\Delta\phi_{2}$, red), fitted to a double blackbody model plus Gaussian
emission line. The blackbody temperatures and line centroid energy are linked
between the two spectra. The fitted parameters are given in the last column of
Table 8. The residuals to the fit are shown in the bottom panel.
Finally, we note that the total luminosity and blackbody areas reported in
Table 6 differ from those presented in Gotthelf & Halpern (2009), the areas by
a factor of two or more. This is a consequence of the best fit values for each
model fit. The derived blackbody areas depend on blackbody temperature as
$T^{-4}$, which itself is strongly correlated with the fitted column density.
Future work, simultaneously modeling of the phase dependent spectra, should
better constrain the column density and temperatures, and consequently provide
a more accurate measurement of the blackbody areas.
### 4.2. Search for Variability
To test for long-term variability of PSR J0821$-$4300, we also fitted the
individual XMM–Newton spectra, searching for temperature variations on the
surface, for example. The spectra for each observation are well-fitted by the
two blackbody model, with parameters listed in Table 7; their statistics do
not warrant an additional line component. For this comparison, we held the
absorbing column fixed at the value in Table 6 derived from the fit of the
composite spectrum to the two blackbody model. Figure 9 displays the results.
No significant variation is evident in the flux or in either of the two
blackbody temperatures.
Table 7Individual XMM–Newton Spectra of PSR J0821$-$4300
Group | ObsID | Livetime | $kT_{w}$ | $kT_{h}$ | Flux ($10^{-12}$ | $\chi^{2}_{\nu}({\rm DoF})$
---|---|---|---|---|---|---
| | (ks) | (keV) | (keV) | erg s-1 cm-2) |
1 | 0113020101S | 15.2 | $0.27(0.25,0.29)$ | $0.47(0.44,0.52)$ | $4.23(4.14,4.30)$ | $1.06(154)$
| 0113020301S | 16.1 | $0.27(0.25,0.28)$ | $0.46(0.43,0.50)$ | $4.21(4.12,4.27)$ | $1.05(184)$
2 | 0606280101S | 25.7 | $0.27(0.26,0.28)$ | $0.47(0.45,0.51)$ | $4.15(4.09,4.20)$ | $1.19(282)$
| 0606280101U | 24.5 | $0.25(0.24,0.26)$ | $0.45(0.43,0.47)$ | $4.11(4.05,4.16)$ | $0.91(268)$
| 0606280201S | 24.6 | $0.27(0.26,0.28)$ | $0.47(0.45,0.50)$ | $4.14(4.09,4.20)$ | $1.02(261)$
| 0650220201S | 8.1 | $0.22(0.19,0.26)$ | $0.40(0.38,0.45)$ | $4.11(4.01,4.20)$ | $0.88(94)$
3 | 0650220901S | 16.4 | $0.25(0.24,0.27)$ | $0.45(0.42,0.48)$ | $4.13(4.06,4.19)$ | $1.01(191)$
| 0650221001S | 15.5 | $0.24(0.22,0.25)$ | $0.42(0.40,0.44)$ | $4.13(4.06,4.20)$ | $0.99(183)$
| 0650221101S | 18.6 | $0.26(0.25,0.28)$ | $0.48(0.45,0.51)$ | $4.19(4.12,4.25)$ | $0.97(222)$
| 0650221201S | 17.1 | $0.27(0.26,0.28)$ | $0.49(0.46,0.54)$ | $4.17(4.08,4.23)$ | $1.08(201)$
| 0650221301S | 16.4 | $0.28(0.27,0.29)$ | $0.52(0.48,0.57)$ | $4.06(3.98,4.12)$ | $1.21(182)$
| 0650221401S | 15.5 | $0.24(0.23,0.26)$ | $0.43(0.41,0.46)$ | $4.13(4.05,4.20)$ | $1.00(174)$
4 | 0650221501S | 20.3 | $0.27(0.25,0.28)$ | $0.45(0.43,0.48)$ | $4.26(4.20,4.32)$ | $0.96(225)$
| 0657600101S | 22.4 | $0.26(0.25,0.27)$ | $0.47(0.44,0.50)$ | $4.13(4.07,4.19)$ | $0.99(242)$
| 0657600201S | 26.1 | $0.26(0.25,0.27)$ | $0.45(0.43,0.47)$ | $4.24(4.19,4.30)$ | $1.07(285)$
| 0657600301S | 24.7 | $0.26(0.25,0.28)$ | $0.46(0.44,0.49)$ | $4.22(4.16,4.27)$ | $0.94(262)$
Note. — Results for a fit to an absorbed, two-blackbody model with the column
density held fixed at $N_{\rm H}=3.75\times 10^{21}$ cm-2, the value obtained
for the summed spectrum in Table 6. The absorbed flux in the $0.5-5$ keV range
is tabulated. The range of uncertainty of each value is the $1\sigma$
confidence interval for two interesting parameters.
We next test for pulse-phase dependence of the spectral line feature(s) by
generating spectra in two phase intervals, each of $\Delta\phi=0.3$ cycles in
width centered on the peaks of the soft and hard light curves, respectively
(see Figure 5.) This is motivated by the original finding in Gotthelf &
Halpern (2009) that an emission line is more strongly associated with the warm
region, i.e., the soft phase of the pulse. For simplicity, given the reduced
counts in the phased spectra, we used only the Gaussian emission-line model as
a representative for all of the possible line models. The two phase-resolved
spectra combined from all epochs are shown in Figure 10, where they are fitted
to the two blackbody plus Gaussian emission-line model. In the simultaneous
fit, the two temperatures and the line centroid energy are linked. As shown in
Table 8, the equivalent width of the Gaussian line is indeed about factor of 2
larger in the soft-phase spectrum than in the hard phase.
Figure 11.— XMM–Newton EPIC pn spectra of PSR J0821$-$4300, grouped into the
four sets as listed in Table 7, and fitted to a double blackbody model with
Gaussian emission line. As in Figure 10, red and blue represent the two pulse-
phase intervals defined in Figure 5. The residuals from the fits are shown in
the bottom panels. The fitted parameters are given in Table 8.
De Luca et al. (2012) presented evidence that the emission-line centroid had
decreased from 0.80 keV to 0.73 keV between 2001 and 2009. Here we extend the
examination for variability of the spectral feature, including its phase
dependence, by combining the XMM–Newton spectra into four groups of adjacent
observations as indicated Table 7. We fitted these four sets of spectra to the
two blackbody plus Gaussian emission-line model, with their temperatures and
line width (sigma) linked between the two phase intervals. The four fits are
displayed in Figure 11, and the fitted parameters for each set and their sum
are presented in Table 8. It is clear that the temperatures and their
contributed fluxes are steady in time, consistent with their phase-averaged
behavior and with no time dependence in their phase ratios. On the other hand,
the equivalent width of the fitted Gaussian emission line shows evidence for
having increased between 2001 and 2009, while the line centroid energy
decreased from $0.79^{+0.02}_{-0.03}$ keV to $0.71^{+0.02}_{-0.03}$ keV. While
these results are consistent with the De Luca et al. (2012) analysis of the
same data, there is little if any additional variability between 2009 and
2012.
Table 8XMM–Newton Phase-Resolved Spectra of PSR J0821$-$4300
Model | Group 1 | Group 2 | Group 3 | Group 4 | Sum
---|---|---|---|---|---
Parameter | 2001 Apr–Nov | 2009 Dec – 2010 May | 2010 Oct–Dec | 2011 Apr – 2012 Apr | 2001 Apr – 2012 Apr
$kT_{w}$ (keV) | $0.23(0.21,0.25)$ | $0.25(0.24,0.26)$ | $0.24(0.23,0.25)$ | $0.27(0.25,0.28)$ | $0.25(0.24,0.25)$
$kT_{h}$ (keV) | $0.41(0.39,0.44)$ | $0.45(0.43,0.47)$ | $0.44(0.42,0.46)$ | $0.47(0.44,0.52)$ | $0.44(0.43,0.45)$
$F[{\Delta\phi_{1}]}$aaAbsorbed flux quoted for the $0.5-5.0$ keV band in units of $10^{-12}$ erg s-1 cm-2. | $4.00(3.97,4.08)$ | $3.91(3.88,3.93)$ | $3.94(3.91,3.96)$ | $3.99(3.96,4.01)$ | $3.96(3.94,3.97)$
$F[{\Delta\phi_{2}]}$aaAbsorbed flux quoted for the $0.5-5.0$ keV band in units of $10^{-12}$ erg s-1 cm-2. | $4.33(4.28,4.38)$ | $4.37(4.35,4.40)$ | $4.37(4.34,4.39)$ | $4.45(4.42,4.47)$ | $4.40(4.38,4.41)$
$E_{0}$ (keV) | $0.79(0.76,0.81)$ | $0.71(0.68,0.73)$ | $0.72(0.69,0.74)$ | $0.69(0.63,0.73)$ | $0.72(0.70,0.73)$
$\sigma$ (eV) | $\leq 62$ | $44(21,70)$ | $68(45,95)$ | $133(78,195)$ | $69(52,89)$
$EW[{\Delta\phi_{1}]}$ (eV) | $40(24,51)$ | $77(59,89)$ | $58(46,76)$ | $86(67,107)$ | $61(53,74)$
$EW[{\Delta\phi_{2}]}$ (eV) | $14(3,22)$ | $34(23,42)$ | $42(32,57)$ | $39(26,52)$ | $31(26,39)$
$\chi^{2}_{\nu}{\rm(DoF)}$ | $1.01(87)$ | $1.19(229)$ | $1.05(269)$ | $0.99(250)$ | $1.20(212)$
Note. — Results from simultaneous fits to XMM–Newton spectra of PSR
J0821$-$4300 extracted from two pulse-phase intervals, $\Delta\phi_{1}$,
$\Delta\phi_{2}$, as defined in the Figure 5. Groups numbers are defined in
Table 7. The blackbody temperatures and Gaussian line energy are linked
between the two phases. The column density is fixed at $N_{\rm H}=4.28\times
10^{21}$ cm-2, the phase-averaged value for the summed spectrum in Table 6.
Quoted uncertainties are $1\sigma$ for three interesting parameters.
We also repeated this test for variability with the Gaussian absorption-line
model for the spectral feature (results not tabulated here). For either the
emission-line or the absorption-line model, the measured line centroids for
all epochs but 2001 cluster well within their 1$\sigma$ uncertainty (for two
interesting parameters, $\Delta\chi^{2}=2.3$). The 2001 set deviates from the
mean defined by the other three sets by $\approx 2\sigma$. In terms of
percentage deviation from the mean, the line centroid measured in 2001 was 14%
and 8% higher in the emission line and the absorption line model,
respectively. We conclude that, although the deviation seems large for the
emission line model, it is not inconsistent with the expected variance.
Further observations would be necessary to establish more definite evidence of
long-term variability.
## 5\. A Definitive Spin-Down Measurement for PSR J1210$-$5226
Archival timing observations of PSR J1210$-$5226 spanning the years 2000–2008
were too sparse to securely determine its spin-down rate, as described in
Halpern & Gotthelf (2011). Searching all possible parameter space for a phase-
coherent, quadratic ephemeris, we found two equally acceptable solutions, with
$\dot{f}=-1.243(22)\times 10^{-16}$ Hz s-1 and $\dot{f}=-7.084(22)\times
10^{-16}$ Hz s-1, corresponding to $B_{s}=9.9\times 10^{10}$ G (solution 1)
and $B_{s}=2.4\times 10^{11}$ G (solution 2), respectively. Since such low
$\dot{E}$ pulsars are generally very stable rotators with little timing noise
or glitch activity, it was deemed likely that one of these is the true
solution, and the other one is an alias with an incorrect cycle count. It is
also important that no solutions with smaller dipole $B_{s}$, and no spinning-
up solutions, were found.
Table 9Log of New X-ray Timing Observations of PSR J1210$-$5226 Mission | Instr/Mode | ObsID | Date | Exposure | Start Epoch | FrequencyaaBarycentric frequency derived from a $Z^{2}_{1}$ test. The given uncertainty is for the $1\sigma$ confidence interval. | $Z^{2}_{1}$
---|---|---|---|---|---|---|---
| | | (UT) | (ks) | (MJD) | (s) |
Chandra | ACIS-S3/CC | 14199 | 2011 Nov 25 | 31.0 | 55890.233 | 2.3577625(28) | 48.2
Chandra | ACIS-S3/CC | 14202 | 2012 Apr 10 | 33.0 | 56027.637 | 2.3577709(43) | 22.2
XMM | EPIC-pn/SW | 0679590101 | 2012 Jun 22 | 26.5 | 56100.537 | 2.3577687(25) | 68.3
XMM | EPIC-pn/SW | 0679590201 | 2012 Jun 24 | 22.3 | 56102.752 | 2.3577621(34) | 69.0
XMM | EPIC-pn/SW | 0679590301 | 2012 Jun 28 | 24.9 | 56106.490 | 2.3577636(23) | 109.6
XMM | EPIC-pn/SW | 0679590401 | 2012 Jul 02 | 24.5 | 56110.918 | 2.3577626(28) | 61.4
XMM | EPIC-pn/SW | 0679590501 | 2012 Jul 18 | 27.3 | 56126.553 | 2.3577640(27) | 51.9
XMM | EPIC-pn/SW | 0679590601 | 2012 Aug 11 | 27.3 | 56150.408 | 2.3577637(23) | 81.8
Chandra | ACIS-S3/CC | 14200 | 2012 Dec 01 | 31.1 | 56262.095 | 2.3577634(28) | 39.8
In 2011–2012 we started a new series of observations of PSR J1210$-$5226 with
Chandra and XMM–Newton that was designed to initiate and maintain a unique,
phase-connect timing solution of this pulsar for the first time and eliminate
the prior timing ambiguity. Table 9 is a log of the new observations. The
instrumental setups and analysis methods are the same as those described in
Section 3 for PSR J0821$-$4300, except that the XMM–Newton source photons were
extracted from a $20^{\prime\prime}$ radius aperture instead of
$30^{\prime\prime}$, and the $0.5-2.5$ keV band was selected to optimize the
pulsed signal.
Table 10Ephemeris of PSR J1210$-$5226 Parameter | Value
---|---
R.A. (J2000)aaMeasured from Chandra ACIS-S3 ObsID 3913. Typical uncertainty is $0.\\!^{\prime\prime}6$. | $12^{\rm h}10^{\rm m}00^{\rm s}\\!.91$
Decl. (J2000)aaMeasured from Chandra ACIS-S3 ObsID 3913. Typical uncertainty is $0.\\!^{\prime\prime}6$. | $-52^{\circ}26^{\prime}28^{\prime\prime}\\!.4$
Ephemeris Epoch (MJD TDB)bbEpoch of fitted minima of summed pulse profile; phase zero in Figure 13 | 53562.0000006
Ephemeris Span (MJD) | 51,549–56,262
Frequency, $f$ | 2.357763502865(65) Hz
Frequency derivative, $\dot{f}$ | $(-1.2363\pm 0.0091)\times 10^{-16}$ Hz s-1
Period, $P$ | 0.424130748816(12) s
Period derivative, $\dot{P}$ | $(2.224\pm 0.016)\times 10^{-17}$
Surface dipole dipole field, $B_{s}$ | $9.8\times 10^{10}$ G
Spin-down luminosity, $\dot{E}$ | $1.2\times 10^{31}$ erg s-1
Characteristic age, $\tau_{c}$ | 302 Myr
The coherently measured period from the new observations is precise enough to
reject previous timing solution 2, while solution 1 with the smaller $B_{s}$
extrapolates precisely to the new period. Furthermore, the new pulse phases
are aligned accurately with solution 1, while solution 2 gives random phases.
Thus a unique quadratic ephemeris fits all 12.9 years of Chandra and
XMM–Newton timing. Table 10 gives this global ephemeris, derived from a least-
square fit to the TOAs as described in Section 3, with the phase residuals
shown in Figure 12. The previous uncertainty on the $\dot{f}$ of solution 1 is
reduced by a factor of 2, with the result $\dot{f}=-1.2363(91)\times 10^{-16}$
Hz s-1; the corresponding dipole magnetic field is $B_{s}=9.8\times 10^{10}$
G. Figure 13 shows the pulse profile using data from all the observations
folded on the presented ephemeris.
Figure 12.— Pulse-phase residuals from the linear term (dash-dot line) of the
phase ephemeris of PSR J1210$-$5226 presented in Table 10. Included are the
new observations listed in Table 9 and the archival timing observations from
Table 1 of Halpern & Gotthelf (2011). The quadratic term (solid line)
corresponds to the uniquely determined period derivative spanning the years
2000–2012. The error bars are generally smaller than the symbol size. Figure
13.— Pulse profiles of PSR J1210$-$5226 in the $0.5-2.5$ keV band using data
from all timing observations, folded according to the ephemeris in Table 10.
Phase zero corresponds to the listed TDB epoch of the ephemeris. Included are
the new observations in Table 9 and the archival timing observations from
Table 1 of Halpern & Gotthelf (2011).
Strictly speaking, the derived period derivative is an upper limit to the
intrinsic one, as any proper motion has not been measured and taken into
account. PSR J1210$-$5226 and PSR J0821$-$4300 are at similar distances, and
the location of PSR J1210$-$5226 with respect to the geometric center of SNR
PKS 1209$-$51/52 allows (but does not require, since the kinematics of the SNR
are unknown) a proper motion of $\sim 70$ mas yr-1, or $v_{\perp}\sim 730$ km
s-1 (De Luca et al., 2011), similar to that of PSR J0821$-$4300\. If so, the
Shklovskii effect given by equation (1) would contribute $\dot{P}_{k}\sim
1.1\times 10^{-17}$, or half of the total $\dot{P}$, and the spin-down dipole
field would be reduced somewhat, to $B_{s}\approx 7\times 10^{10}$ G. Of
course, if these two CCOs both have velocities toward the high end of the
distribution of pulsars, it would be a tantalizing physical result in itself.
However, examination of another CCO, the NS in Cas A, doesn’t necessarily
support such high velocities for CCOs, in general; Thorstensen et al. (2001)
and Fesen et al. (2006) find $v_{\perp}\approx 350$ km s-1 with respect to the
explosion center of Cas A.
## 6\. Discussion
### 6.1. On the Age of Puppis A
Becker et al. (2012) discussed the impact of the revised proper motion of PSR
J0821$-$4300 on the inferred age of Puppis A. The age of the SNR had been
derived previously as $3700\pm 300$ yr from the proper motions of optical
filaments that point back to a common center, presumed to be the site of the
explosion, and assuming no deceleration. The motion of the NS also
extrapolates to the same center, but the distance traveled,
$371^{\prime\prime}\pm 31^{\prime\prime}$, corresponds to an age of $5200\pm
1000$ yr (or $6100\pm 1000$ yr for our proper motion of $61\pm 9$ mas yr-1).
Becker et al. (2012) refer to these marginally contradictory measurements as
independent, and choose to average them, giving the value 4.5 kyr listed in
Table 1. However, they are not truly independent, because they assume the same
starting location. Furthermore, the discrepancy is worse if the filaments have
decelerated. Just as the new Chandra observations of the NS have improved the
accuracy of its proper motion, we suggest that a contemporary observation of
the optical filaments of Puppis A may improve the precision of the original
Winkler et al. (1988) study on which the optical proper-motion age is based.
Such an investigation could lead to a more detailed understanding of dynamics
of the filaments, and a more accurate age for Puppis A.
### 6.2. PSR J0821–4300 and PSR J1210$-$5226 as Anti-Magnetars
The spin-down of power PSR J0821$-$4300, $\dot{E}=1.9\times 10^{32}$ erg s-1,
is consistent with being caused by magnetic braking of an isolated neutron
star with a weak dipole field of $B_{s}=2.9\times 10^{10}$ G. Its spin-down
power is much smaller than its observed thermal X-ray luminosity,
$L_{x}\approx 5.6\times 10^{33}\,d_{2.2}^{2}$ erg s-1, which rules out
rotation as a significant power source. As discussed in Halpern & Gotthelf
(2010a) for PSR J1852$+$0040, the very small $\dot{P}$ disfavors propeller
spin-down and accretion as a source of the X-rays. Thus, residual cooling
remains the most plausible source of the X-ray luminosity of PSR J0821$-$4300
and, by extension, of all CCOs. Given the meager spin-down power of PSR
J0821$-$4300, we can also discount the suggestion of Reynoso et al. (2003) and
Castelletti et al. (2006) that structure in the Puppis A SNR and associated H
I is caused by jets emitted by the pulsar, similar to the SS433/W50 system.
Parenthetically, if the H I morphology is not caused by the pulsar, then it
does not provide supporting evidence of its distance.
An unresolved question about PSR J0821$-$4300 is the origin of its phase-
dependent, possibly variable emission or absorption feature. In the absence of
accretion, it would be difficult to understand how an emission line is
generated, or why it would vary in a few years. The indication of variability
is not strong, and there is no other evidence of accretion. Therefore, we will
consider here an absorption-line interpretation, for which there is a good
precedent in PSR J1210$-$5226, the only isolated pulsar to show a series of
strong absorption lines in its spectrum (Sanwal et al., 2002; Mereghetti et
al., 2002; Bignami et al., 2003).
The spectral features in PSR J1210$-$5226 are now widely considered to
comprise the electron cyclotron fundamental at $E_{0}=0.7$ keV and its
harmonics. The surface magnetic field strength where the lines are formed is
inferred to be $B\approx 8\times 10^{10}$ G according to equation (2) and
assuming $z\approx 0.3$. The relative strength of the harmonics is explained
by treating them as resonances in the photospheric free-free opacity in the
presence of the magnetic field (Suleimanov et al., 2010, 2012). PSR
J1210$-$5226 is exceptional in having the largest known spin-down dipole
magnetic field among CCOs, now confirmed as $B_{s}\leq 9.8\times 10^{10}$ G.
This is only slightly larger than $B\approx 8\times 10^{10}$ G inferred from
the spectral features, and is the first case in which such a comparison has
been made between independent methods of measuring surface $B$-field on an
isolated pulsar. Effects that could eliminate the already minor discrepancy
include the unmeasured proper motion, which would decrease the inferred spin-
down dipole field, and the results of numerical models of a force-free
magnetosphere, which imply a different spin-down law from the standard vacuum
dipole expression
$B_{s}=\left(3c^{3}IP\dot{P}\over 8\pi^{2}R^{6}\,{\rm
sin}^{2}\alpha\right)^{1/2}=\ 3.2\times 10^{19}\,\left(P\dot{P}\over{\rm
sin}^{2}\alpha\right)^{1/2}\,{\rm G}$ $None$
such that, more accurately,
$B_{s}\approx\left(c^{3}IP\dot{P}\over 4\pi^{2}R^{6}\,[1+{\rm
sin}^{2}\alpha]\right)^{1/2}=\ 2.6\times 10^{19}\,\left(P\dot{P}\over 1+{\rm
sin}^{2}\alpha\right)^{1/2}\,{\rm G}$ $None$
(Spitkovsky, 2006). This means that a measured spin-down rate is obtained with
$B_{s}$ smaller by a factor $0.58-0.82$ than the conventional approximation
$B_{s}=3.2\times 10^{19}(P\dot{P})^{1/2}$ G.
In the case of PSR J0821$-$4300, by the same logic, its weaker inferred dipole
magnetic field of $B_{s}\leq 2.9\times 10^{10}$ G wouldn’t naturally account
for a spectral feature at $0.7-0.8$ keV, instead predicting $E_{0}\leq 0.26$
keV. The absorption model with the cyclotron fundamental at $E_{0}=0.46$ keV
comes closer to the prediction. However, additional variables can affect the
comparison of a dipole field inferred from spin-down with that from a
cyclotron line. First is the factor of 2 variation of dipole field strength
over the surface, with $B_{p}=2\,B_{s}$. Second, the actual field is not
likely to be a centered dipole, and may be larger than $B_{s}$ or $B_{p}$ in
places where the line is formed. The asymmetric distribution of surface
temperature on PSR J0821$-$4300 already appears to require a complex magnetic
field geometry, such as an off-center dipole or higher multipoles. Third, the
inferred spin-down field depends on the uncertain NS mass and radius as in
equations (3) and (4). $B_{s}$ scales as ${I^{1/2}\,R^{-3}}\propto
M^{1/2}R^{-2}(1+z)$ (Ravenhall & Pethick, 1994), where
$1+z=(1-2GM/Rc^{2})^{-1/2}\propto B/E_{0}$ from equation (2). For an
astrophysically likely $M=1.4\,M_{\odot}$, theoretical NS equations of state
allow $8<R<15$ km, therefore $1.18<1+z<1.44$. The spectroscopic $B$ is
therefore uncertain by $\pm 10\%$ when we adopt $z=0.3$, while the dipole
$B_{s}$ is uncertain by the much larger factor of $\sim 2$. (The gravitational
redshift cancels out in their ratio.) Allowing for these uncertainties, it is
reasonable to adopt the hypothesis that feature(s) in the spectrum of PSR
J0821$-$4300 are due to the cyclotron process.
The continuum X-ray spectrum and pulse profiles of PSR J0821$-$4300 are
indicative of antipodal hot spots of different areas and temperatures
(Gotthelf & Halpern, 2009; Gotthelf et al., 2010), which are difficult to
account for if the magnetic field is weak. A related problem is the high
pulsed fraction of 64% from PSR J1852$+$0040 (Halpern & Gotthelf, 2010a), a
timing twin of PSR J0821$-$4300\. Polar cap heating by any magnetospheric
accelerator must be negligible as a source of surface heating in CCOs, being
only a small fraction of the already insignificant spin-down power. Thermal
X-rays from residual cooling can be nonuniform if there is anisotropic heat
conduction in the star. The effect of different magnetic field configurations
on heat transport in the crust and envelope of NSs has been modelled, most
recently by Geppert et al. (2004, 2006), Pérez-Azorín et al. (2006a), and Pons
et al. (2009). A toroidal field is expected to be the initial configuration
generated by differential rotation in the proto-neutron star dynamo (Thompson
& Duncan, 1993). One of the effects of crustal toroidal field is to insulate
the magnetic equator from heat conduction, resulting in warm spots at the
poles. The warm regions can even be of different sizes due to the antisymmetry
of the poloidal component of the field (Geppert et al., 2006), which is
evocative of the antipodal thermal structure of PSR J0821$-$4300\. To have a
significant effect on the heat transport, the crustal toroidal field strength
required in these models is $\sim 10^{15}$ G, many orders of magnitude greater
than the poloidal field if the latter is measured by the spin-down. Purely
toroidal or poloidal fields are thought to be unstable in an initially fluid
NS (Tayler, 1973; Flowers & Ruderman, 1977), although the toroidal field may
be stabilized by a poloidal field that is several orders of magnitude weaker
(Braithwaite, 2009). We suggest the latter as a viable configuration for a
CCO.
Shabaltas & Lai (2012) tried to model the pulse profile of PSR J1852$+$0040
with anisotropic conduction, and concluded that they needed a toroidal crustal
field of $B_{\phi}>2\times 10^{14}$ G to produce its high pulsed fraction.
Even then, they observe, the shape of the modelled light curve doesn’t match
the observed one. Since the shape of the light curve is not reproduced, the
surface temperature distribution of PSR J1852$+$0040 and its physical origin
are still not known.
Page & Sarmiento (1996), Pérez-Azorín et al. (2006b), Zane & Turolla (2006),
and Zane (2007) investigated NS surface emission patterns using a combination
of star-centered dipole and quadrupole magnetic field components to model
asymmetric pulse profiles. The behavior of PSR J0821$-$4300 may ultimately be
explained by similar models. It remains to be shown that if CCOs can have
field configurations that are strong enough to affect heat transport to the
extent required, while not exceeding the spin-down limits on the external
dipole field.
### 6.3. Origin and Evolution of Anti-magnetars
PSR J0821$-$4300 is nearly a twin of PSR J1852$+$0040 in its spin properties,
and there are no other young neutron stars with measured magnetic fields this
weak. They fall in a region of the $P-\dot{P}$ diagram (Figure 3) that is
devoid of ordinary (non-recycled) radio pulsars (Manchester et al., 2005).
They overlap with the supposed mildly recycled pulsars in this area
(Belczynski et al., 2010). The characteristic age of 254 Myr for PSR
J0821$-$4300 is not meaningful because the pulsar was born spinning at its
current period, as is the case for the other CCO pulsars. If their magnetic
fields remain constant, they will not move in $P-\dot{P}$ space for $>10^{8}$
years (but see below). That CCOs are found in SNRs in comparable numbers to
other classes of NSs implies that they must represent a significant fraction
of NS births. If PSR J0821$-$4300 and PSR J1852$+$0040 are typical CCOs, the
area around them in the $P-\dot{P}$ diagram should be densely populated with
“orphaned CCOs” that remain after their SNRs dissipate in $\sim 10^{5}$ years.
Why there are few ordinary radio pulsars and no older X-ray pulsars near their
location is then a mystery, as emphasized by Kaspi (2010), which may indicate
that radio luminosity is a function of spin-down power.
There are not yet enough CCOs to know whether they are intrinsically radio-
quiet relative to ordinary and recycled pulsars, rather than unfavorably
beamed. It is also possible that some ordinary radio pulsars with
$B_{s}<10^{11}$ G are actually much younger than their characteristic ages,
and may be orphaned CCOs that could be recognized in X-rays. Whether or not
they are radio pulsars, nearby orphaned CCOs should be detectable as thermal
X-ray sources for $10^{5}-10^{6}$ years, similar to the seven ROSAT discovered
isolated NSs (INSs: Haberl, 2007) which, however, have strong magnetic fields
(Kaplan et al., 2009). It is likely that the INSs are kept hot for longer than
CCOs by continuing magnetic field decay (Pons et al., 2007), which would
explain their observed abundance relative to the elusive orphaned CCOs. It may
be difficult to detect and/or recognize orphaned CCOs if they cool faster than
ordinary neutron stars. One effect that can accelerate cooling is an accreted
light-element envelope, which has higher heat conduction than an iron surface
(Kaminker et al., 2006). The newly discovered 59 ms pulsar 1RXS
J141256.0$+$792204 (“Calvera”), originally detected in the ROSAT All-Sky
Survey, may be the first recognized example of an orphaned CCO (Zane et al.,
2011; Rutledge et al., 2008; Halpern, 2011), pending a measurement of its
period derivative. The isolated NS 2XMM J104608.7$-$594306 in the Carina
Nebula (Pires et al., 2012) has been suggested as another orphaned CCO.
It is possible that the weak magnetic field of CCOs is causally related to
slow rotation at birth through the turbulent dynamo (Thompson & Duncan, 1993)
that generates the magnetic field. (See Spruit 2008 for a review of possible
mechanisms for the origin of magnetic fields in neutron stars.) If the dipole
magnetic field $B_{s}$ is simply related to the initial spin period, and
assuming that the present period $P$ is in fact the birth period because the
spin-down time is so much longer than the true age, we may expect an anti-
correlation between $B_{s}$ and $P$. With only three data points to compare,
and with two of them having nearly identical values, there is not much
evidence to examine for a trend. In fact, the CCO with the longest period, PSR
J1210$-$5226, also has the strongest dipole field of the three, which would
not by itself support such a simple correlation. A population analysis of
radio pulsars by Faucher-Giguère & Kaspi (2006) concludes that there is a wide
distribution of birth periods, with a mean of $\sim 300$ ms and a dispersion
of $\sigma\sim 150$ ms. If this is true, the birth periods of CCOs are not in
fact long, and their weak dipole fields may be the effect of some as-yet
unknown parameter.
In an alternative theory for CCOs, a normal ($\sim 10^{12}$ G) magnetic field
is buried in the core or crust of a NS by prompt fall-back of supernova
debris, and takes thousands of years to diffuse back to the surface, during
which time the NS appears as an anti-magnetar. This is assuming that the
accreted matter is itself not magnetized. The timescale for diffusion is
highly dependent on the amount of matter accreted. According to Muslimov &
Page (1995), for accretion of $\sim 10^{-5}\,M_{\odot}$, the regrowth of the
surface field is largely complete after $\sim 10^{3}$ yr, but if
$>0.01\,M_{\odot}$ is accreted, then the diffusion time could be millions of
years. Chevalier (1989) calculated that the neutron star in SN 1987A could
have accreted $\sim 0.1\,M_{\odot}$ of fallback material in the hours after
the SN explosion, aided by a reverse shock from the helium layer of the
progenitor. If so, it may never emerge as a radio pulsar.
Interesting support for the theory of field burial and regrowth is the absence
of evidence for magnetic field strengths $<10^{11}$ G in accreting high-mass
X-ray binary pulsars, as inferred from their pulse periods and period
derivatives (Popov & Turolla, 2013). If intrinsic fields in the range
$10^{10}-10^{11}$ G were common, equilibrium spin periods of $0.1-1$ s should
be frequent in HMXBs, but they are not. If the NSs in these systems are born
in the same way as isolated pulsars, this could imply that birth fields are
never so small; instead, field regrowth has occurred in all cases after it was
initially buried. One caveat here is that $\sim 40\%$ of HMXBs do not have
measured spin periods, although there is no strong selection effect against
detecting $P<1$ s.
Bernal et al. (2010) has revisited the process of hypercritical accretion onto
a magnetized neutron star, while Ho (2011) made new calculations of the
subsequent diffusion to constrain the magnetic fields of CCOs at birth and the
accreted mass, finding $10^{-4}-10^{-5}\,M_{\odot}$ for the latter. These
models are difficult to test using CCOs, because it would involve measuring
the braking index or the change of the dipole magnetic field directly. Ho
(2011) implied that the isolation of CCOs in the $P-\dot{P}$ diagram may be
less problematic in this model because, as their dipole magnetic fields
increase, they will evolve to join the bulk of the pulsar population. However,
for the first $\sim 10^{5}$ yr, field growth can only move a CCO vertically
upward in the $P-\dot{P}$ diagram, as the braking index has a large, negative
value. Orphaned CCOs should still have periods of $\sim 0.1$ s and lie in a
sparse region. For this reason, an important test for an example of field
growth is to determine the $\dot{P}$ of Calvera, which will reveal if its
dipole field is greater than that of PSR J0821$-$4300 and PSR J1852$+$0040.
Previous calculations of field burial and re-emergence for CCOs were one
dimensional. The first two-dimensional calculations for this purpose were
recently reported by Viganò & Pons (2012). They find that the accretion must
be essentially isotropic to bury the field sufficiently to cause the required
orders-of-magnitude reduction of the external dipole. If the accretion were
instead confined to the equator or the magnetic poles, the reduction of the
dipole component would not be significant enough to produce a CCO. Finally, it
remains to be seen if the resulting surface thermal distribution during the
CCO phase can be made compatible with observed spectra and pulse profiles.
Viganò & Pons (2012) briefly presented the temperature distribution from one
of their models that produces hot polar caps at a time when the external
dipole field is $10^{10}$ G. But even in this case, the temperature does not
vary by as much as a factor of 2 over the surface, a range that would be
required to match the properties of PSR J0821$-$4300 and other CCOs.
## 7\. Conclusions and Future Work
Measurement of the spin-down rate of the 112 ms PSR J0821$-$4300 in Puppis A
was achieved using X-ray observations coordinated between XMM–Newton and
Chandra, the resulting phase-connected ephemeris spanning 2.3 yr. We also
measured the proper motion of the pulsar in Chandra HRC images over 10.6
years. The proper motion makes a non-negligible contribution to the period
derivative via the Shklovskii effect, and the uncertainty in proper motion and
distance limits the accuracy of the intrinsic period derivative to $\approx
16\%$. The straightforward interpretation of these results is dipole spin-down
due to a weak surface magnetic field of $B_{s}=2.9\times 10^{10}$ G, the
smallest measured for any young neutron star, and nearly identical to that of
the CCO pulsar PSR J1852$+$0040 in Kes 79.
A phase-dependent spectral feature in PSR J0821$-$4300 can be modelled either
as an emission line of energy $\approx 0.75$ keV, or as a cyclotron absorption
line and its harmonic, with $E_{0}\approx 0.46$ keV. For reasons that are not
clear, it is stronger during the pulse-phase interval in which the continuum
spectrum is softer. There is only marginal evidence for long-term variability
of this feature. The local magnetic field strength in the area where the
spectral feature is produced may have to be larger than the dipole spin-down
value.
The existing spin-down measurements for three CCOs, including a definitive new
result for PSR J1210$-$5226, are compelling evidence that a weak dipole field
component is the physical origin of the CCO class in general. It is reasonable
to assume that the CCOs which have not yet been seen to pulse have magnetic
fields that are similar to or weaker than those of PSR J0821$-$4300 and PSR
J1852$+$0040\. Otherwise, if their fields were stronger, their spectra should
show cyclotron lines similar to PSR J1210$-$5226 and (possibly) PSR
J0821$-$4300\. Deep X-ray timing searches of the remaining members of this
class, to smaller limits on pulsed fraction, could still discover new pulsars
and supply valuable data on their birth properties and evolution.
A remaining theoretical puzzle about CCOs is the origin of their surface
temperature anisotropies, in particular, the one or two warm/hot regions that
are smaller than the full neutron star surface. The measured spin-down power
is too small to contribute to this emission. Continuing accretion is unlikely
because of the small spin-down rates. It appears that any explanation will
have to involve stronger magnetic field components in the crust, with toroidal
or quadrupolar geometries, that do not contribute to the dipole spin-down
torque. A physical model of thermal and magnetic field structure that self-
consistently reproduces the spin-down rates, X-ray spectra, and pulse
profiles, is still needed. If it can be studied with better data, the details
of the phase-dependent and possibly variable spectral feature from PSR
J0821$-$4300 may contribute to a more definite model of its surface magnetic
field geometry.
The CCO pulsars PSR J1852$+$0040 and PSR J0821$-$4300 fall in a region of
$B-P$ space that overlaps with what are assumed to be moderately recycled
pulsars, but is otherwise empty. An understanding of the evolutionary status
of CCOs is critically dependent upon a search for their descendants, the
orphaned CCOs without SNRs. Some single radio pulsars with spin parameters
similar to CCOs may be orphaned CCOs rather than recycled pulsars, and they
may be much younger than their characteristic ages. X-ray observations of
pulsars in this region may find evidence of their relative youth via surface
thermal emission. The spin-down rates of orphaned CCOs would reveal whether
CCOs have intrinsically weak magnetic fields, or if field re-emerges on a
timescale of $\sim 10^{4}$ yr from a normal magnetic field that was buried by
prompt fallback of supernova debris, as is invoked in some theoretical
studies. Such rapid evolution would lead to orphaned CCOs that lie directly
above CCOs on the $P-\dot{P}$ diagram but are still detectable as thermal
X-ray sources. The radio-quiet pulsar Calvera is a possible such candidate.
Whether or not CCOs have buried fields, the paucity of radio pulsars with
similar spin parameters is real and requires an explanation.
We thank Dany Page, and an anonymous referee, for helpful comments. This
investigation is based on observations obtained with XMM–Newton, an ESA
science mission with instruments and contributions directly funded by ESA
Member States and NASA, and Chandra. The opportunity to propose joint,
coordinated observations between XMM–Newton and Chandra was crucial for the
success of this effort. Financial support was provided by NASA grants
NNX11AD19G and NNX12AD41G for the XMM–Newton observations, and by Chandra
awards GO1-12001X and SAO GO1-12071X, issued by the Chandra X-ray Observatory
Center, which is operated by the Smithsonian Astrophysical Observatory for and
on behalf of NASA under contract NAS8-03060.
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|
that $\mathbf{x}$ satisfies (2.14) and has nonzero entries.
Let
$\mathbf{x}\in(\operatorname{\mathbb{C}}^{*})^{\operatorname{\mathbb{Z}}^{3}}$
be a coherent solution of the Kashaev equation satisfying (2.14). Let
$A_{j}=[-j,j]^{3}\cap\operatorname{\mathbb{Z}}^{3}$ and $B_{j}=[-j,j]^{3}\cap
L$ for $j\in\operatorname{\mathbb{Z}}_{\geq 0}$. We claim that if, for all
$j$, there exist
$\mathbf{\tilde{x}}_{j}\in(\operatorname{\mathbb{C}}^{*})^{B_{j}}$ satisfying
the K-hexahedron equations that agree with $\mathbf{x}$ on $A_{j}$, then there
exists $\mathbf{\tilde{x}}\in(\operatorname{\mathbb{C}}^{*})^{L}$ satisfying
the K-hexahedron equations that agrees with $\mathbf{x}$ on
$\operatorname{\mathbb{Z}}^{3}$. Construct an infinite tree $T$ as follows:
* •
The vertices of $T$ are solutions of the K-hexahedron equation indexed by
$B_{j}$ that agree with $\mathbf{x}$ on $A_{j}$ (over
$j\in\operatorname{\mathbb{Z}}_{\geq 0}$).
* •
Add an edge between
$\mathbf{\tilde{x}}_{j}\in(\operatorname{\mathbb{C}}^{*})^{B_{j}}$ and
$\mathbf{\tilde{x}}_{j+1}\in(\operatorname{\mathbb{C}}^{*})^{B_{j+1}}$ if
$\mathbf{\tilde{x}}_{j+1}$ restricts to $\mathbf{\tilde{x}}_{j}$.
Thus, $T$ is an infinite tree in which every vertex has finite degree. By
König’s infinity lemma (see [11, Theorem 16.3]), there exists an infinite path
$\mathbf{\tilde{x}}_{0},\mathbf{\tilde{x}}_{1},\dots$ in $T$ with
$\mathbf{\tilde{x}}_{j}\in(\operatorname{\mathbb{C}}^{*})^{B_{j}}$. Thus,
there exists $\mathbf{\tilde{x}}\in(\operatorname{\mathbb{C}}^{*})^{L}$
restricting to $\mathbf{\tilde{x}}_{j}$ for all
$j\in\operatorname{\mathbb{Z}}_{\geq 0}$, so $\mathbf{\tilde{x}}$ is a
solution of the K-hexahedron equations that agrees with $\mathbf{x}$ on
$\operatorname{\mathbb{Z}}^{3}$.
Given $j\in\operatorname{\mathbb{Z}}_{\geq 0}$, we claim that there exists
$\mathbf{\tilde{x}}\in(\operatorname{\mathbb{C}}^{*})^{B_{j}}$ satisfying the
K-hexahedron equations that agree with $\mathbf{x}$ on $A_{j}$. It is
straightforward to show that there exists a sequence
$\mathbf{x}_{1},\mathbf{x}_{2},\ldots\in(\operatorname{\mathbb{C}}^{*})^{\operatorname{\mathbb{Z}}^{3}}$
of coherent solutions of the Kashaev equation that converge pointwise to
$\mathbf{x}$ whose restrictions to $\mathbb{Z}^{3}_{\textup{init}}$ are
generic. By Corollary 7.22, there exist
$\mathbf{\tilde{x}}_{1},\mathbf{\tilde{x}}_{2},\ldots\in(\operatorname{\mathbb{C}}^{*})^{L}$
satisfying the K-hexahedron equations such that $\mathbf{\tilde{x}}_{i}$
restricts to $\mathbf{x}_{i}$. However, the sequence
$\mathbf{\tilde{x}}_{1},\mathbf{\tilde{x}}_{2},\dots$ does not necessarily
converge (see Theorem 2.23). Let
$\mathbf{\tilde{x}}_{1}^{\prime},\mathbf{\tilde{x}}_{2}^{\prime},\dots\in(\operatorname{\mathbb{C}}^{*})^{B_{j}}$
be the restrictions of $\mathbf{\tilde{x}}_{1},\mathbf{\tilde{x}}_{2},\dots$
to $B_{j}$. There exists a subsequence of
$\mathbf{\tilde{x}}_{1}^{\prime},\mathbf{\tilde{x}}_{2}^{\prime},\dots$ that
converges to some
$\mathbf{\tilde{x}}\in(\operatorname{\mathbb{C}}^{*})^{B_{j}}$. (For each
$s\in B_{j}\setminus A_{j}$, we can partition the sequence
$\mathbf{\tilde{x}}_{1}^{\prime},\mathbf{\tilde{x}}_{2}^{\prime},\dots$ into
two sequences, each of which converges at $s$. Because $B_{j}$ is finite, the
claim follows.) The array $\mathbf{\tilde{x}}$ must satisfy the K-hexahedron
equations and agree with $\mathbf{x}$ on $A_{j}$, so we are done. ∎
We shall now work towards a proof of Theorem 2.23.
###### Lemma 7.23.
Let $\mathbf{\tilde{x}}\in(\operatorname{\mathbb{C}}^{*})^{L}$ be a solution
of the K-hexahedron equations. Let
$\mathbf{\tilde{x}}_{\textup{init}}\in(\operatorname{\mathbb{C}}^{*})^{L_{\textup{init}}}$
denote the restriction of $\mathbf{\tilde{x}}$ to $L_{\textup{init}}$. Let
$\mathbf{t}=(t_{s})\in\\{-1,1\\}^{\mathbb{Z}^{3}_{\boxdot}}$ be in the kernel
of $\psi$. Then $(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow
L}=(y_{s})_{s\in L}$, where
$\displaystyle y_{s}=\begin{cases}x_{s}&\text{if
}s\in\operatorname{\mathbb{Z}}^{3},\\\ t_{[s]}x_{s}&\text{if }s\in
L-\operatorname{\mathbb{Z}}^{3}.\end{cases}$
###### Proof.
This follows from Lemma 7.18 and Proposition 7.3. ∎
###### Lemma 7.24.
Let $\mathbf{\tilde{x}}\in(\operatorname{\mathbb{C}}^{*})^{L}$ be a solution
of the K-hexahedron equations. Let
$\mathbf{\tilde{x}}_{\textup{init}}\in(\operatorname{\mathbb{C}}^{*})^{L_{\textup{init}}}$
denote the restriction of $\mathbf{\tilde{x}}$ to $L_{\textup{init}}$. For
$\mathbf{t}\in\\{-1,1\\}^{\mathbb{Z}^{3}_{\boxdot}}$, the following are
equivalent:
* •
$\mathbf{\tilde{x}}$ and
$(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow L}$ agree on
$\operatorname{\mathbb{Z}}^{3}$,
* •
$\mathbf{t}$ is in the kernel of $\psi$ $($see Proposition 7.16).
###### Proof.
If $\mathbf{t}$ is in the kernel of $\psi$, then $\mathbf{\tilde{x}}$ and
$(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow L}$ agree on
$\operatorname{\mathbb{Z}}^{3}$ by Lemma 7.23.
If $\mathbf{t}$ is not in the kernel of $\psi$, let
$\mathbf{u}=(u_{s})_{s\in\operatorname{\mathbb{Z}}^{3}+\left(\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}}\right)}=\psi(\mathbf{t})$.
Write $\mathbf{\tilde{x}}=(x_{s})_{s\in L}$ and
$(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow
L}=(y_{s})_{s\in L}$. Choose
$v\in\operatorname{\mathbb{Z}}^{3}_{\\{3,4,5,\dots\\}}$ such that
$u_{v-\left(\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}}\right)}=-1$
and
$u_{w-\left(\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}}\right)}=1$
for all $w\in\operatorname{\mathbb{Z}}^{3}_{\\{3,4,5,\dots\\}}$ with $w<v$.
(Such a choice of $v$ exists because if
$u_{v-\left(\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}}\right)}=1$
for all $v\in\operatorname{\mathbb{Z}}^{3}_{\\{3,4,5,\dots,\\}}$, then
$u_{v-\left(\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}}\right)}=1$
for all $v\in\operatorname{\mathbb{Z}}^{3}$ because $\mathbf{u}$ must satisfy
equation (7.10) for all $v\in\operatorname{\mathbb{Z}}^{3}$.) Then by Lemma
7.18, $y_{v-s}=x_{v-s}$ for $s\in\\{0,1\\}^{3}-\\{(0,0,0)\\}$, and
$y_{v-s}=t_{[v-s]}x_{v-s}$ for
$s\in\left\\{\left(1,\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}}\right),\left(\operatorname{\frac{1}{2}},1,\operatorname{\frac{1}{2}}\right),\left(\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}},1\right)\right\\}$.
Hence,
$\displaystyle
y_{v}-x_{v}=-4\frac{x_{v-\left(1,\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}}\right)}x_{v-\left(\operatorname{\frac{1}{2}},1,\operatorname{\frac{1}{2}}\right)}x_{v-\left(\operatorname{\frac{1}{2}},\operatorname{\frac{1}{2}},1\right)}}{x_{v-\left(1,1,1\right)}^{2}}\not=0,$
so $y_{v}\not=x_{v}$. ∎
###### Proof of Theorem 2.23.
Let
$\mathbf{\tilde{x}}_{\textup{init}}\in(\operatorname{\mathbb{C}}^{*})^{L_{\textup{init}}}$
denote the restriction of $\mathbf{\tilde{x}}$ to $L_{\textup{init}}$.
Suppose that for some signs $\alpha_{i},\beta_{i},\gamma_{i}\in\\{-1,1\\}$,
$i\in\operatorname{\mathbb{Z}}$,
$\mathbf{\tilde{y}}\in(\operatorname{\mathbb{C}}^{*})^{L}$ satisfies equations
(2.19)–(2.22) for $\left(a,b,c\right)\in\operatorname{\mathbb{Z}}^{3}$. Define
$\mathbf{t}=(t_{s})\in\\{-1,1\\}^{\mathbb{Z}^{3}_{\boxdot}}$ by equations
(7.7)–(7.9), so $\mathbf{t}$ is in the kernel of $\psi$ by Proposition 7.16.
Hence, by Lemma 7.23,
$\mathbf{\tilde{y}}=(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow
L}$, so $\mathbf{\tilde{y}}$ satisfies the K-hexahedron equations, proving
part (a).
Next, if $\mathbf{\tilde{y}}\in(\operatorname{\mathbb{C}}^{*})^{L}$ satisfies
the K-hexahedron equations, then
$\mathbf{\tilde{y}}=(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow
L}$ for some $\mathbf{t}=(t_{s})\in\\{-1,1\\}^{\mathbb{Z}^{3}_{\boxdot}}$. By
Lemma 7.24, $\mathbf{t}$ is in the kernel of $\psi$. Hence, by Proposition
7.16, there exist signs $\alpha_{i},\beta_{i},\gamma_{i}\in\\{-1,1\\}$,
$i\in\operatorname{\mathbb{Z}}$, such that $\mathbf{t}$ is given by equations
(7.7)–(7.9). Hence, by Lemma 7.23, $\mathbf{\tilde{y}}$ satisfies equations
(2.19)–(2.22) for $\left(a,b,c\right)\in\operatorname{\mathbb{Z}}^{3}$,
proving part (b). ∎
## 8 Coherence for cubical complexes
In this section, we generalize Proposition 2.8 and Theorem 2.22 from
$\operatorname{\mathbb{Z}}^{3}$ to certain classes of $3$-dimensional cubical
complexes. Proposition 2.8 generalizes to arbitrary $3$-dimensional cubical
complexes embedded in $\operatorname{\mathbb{R}}^{3}$ (see Proposition 8.1),
while Theorem 2.22(b) generalizes to directed cubical complexes corresponding
to piles of quadrangulations of a polygon (see Proposition 8.3). Theorem
2.22(a) does not hold for arbitrary directed cubical complexes corresponding
to piles of quadrangulations of a polygon. It turns out that an additional
property of a cubical complex is required, which we call _comfortable-ness_.
This property is satisfied by the standard tiling of
$\operatorname{\mathbb{R}}^{3}$ with unit cubes, as well as by cubical
complexes corresponding to piles of $\Diamond$-tilings of $\mathbf{P}_{n}$
(see Proposition 8.8). Let $\varkappa$ be the directed cubical complex
corresponding to a pile of quadrangulations of a polygon. In Theorems
8.10–8.11, we show that Theorem 2.22(a) holds for $\varkappa$ if and only if
$\varkappa$ is comfortable. The proof of Theorem 8.10 is nearly identical to
the proof of Theorem 2.22(a) in Section 7.
First, we note that Proposition 2.8 generalizes to arbitrary $3$-dimensional
cubical complexes embedded in $\operatorname{\mathbb{R}}^{3}$ as follows:
###### Proposition 8.1.
Let $\varkappa$ be a $3$-dimensional cubical complex embedded in
$\operatorname{\mathbb{R}}^{3}$. Suppose that
$\mathbf{x}=(x_{s})_{s\in\varkappa^{0}}$ satisfies the Kashaev equation. Then
for any interior vertex $v\in\varkappa^{0}$ $($see Definition 3.2),
$\displaystyle\left(\prod_{C\ni
v}K_{v}^{C}(\mathbf{x})\right)^{2}=\left(\prod_{S\ni
v}(x_{v}x_{v_{2}}+x_{v_{1}}x_{v_{3}})\right)^{2},$
where
* •
the first product is over $3$-dimensional cubes $C$ incident to the vertex
$v$,
* •
the second product is over $2$-dimensional faces $S$ incident to the vertex
$v$, and
* •
$v$, $v_{1}$, $v_{2}$, $v_{3}$ are the vertices of such a face $S$ listed in
cyclic order.
###### Proof.
The proof is almost identical to the proof of Proposition 2.8 in Section 7. ∎
###### Remark 8.2.
With Proposition 8.1 in mind, we can think of the notion of coherence from
Definition 4.17 as follows. Let $\mathbf{T}=(T_{0},\dots,T_{\ell})$ be a pile
of quadrangulations of a polygon with $\varkappa=\varkappa(\mathbf{T})$. Start
with an arbitrary array $\mathbf{x}_{\textup{init}}$ indexed by
$\varkappa^{0}(T_{0})$ whose entries are “sufficiently generic”. We want to
extend $\mathbf{x}_{\textup{init}}$ to an array $\mathbf{x}$ indexed by
$\varkappa^{0}$ that is a coherent solution of the Kashaev equation. Building
$\mathbf{x}$ inductively, suppose we have defined the values of $\mathbf{x}$
at $\varkappa^{0}(T_{0},\dots,T_{i-1})$, and we need to define the value
$x_{w}$ of $\mathbf{x}$ at the new vertex $w$ in $T_{i}$. Let
$C\in\varkappa^{3}$ be the cube corresponding to the flip between $T_{i-1}$
and $T_{i}$, and let $v$ be the bottom vertex of $C$, i.e., let $v$ be the
unique vertex in $T_{i-1}$ but not $T_{i}$. In order that $\mathbf{x}$
continue to satisfy the Kashaev equation, there are $2$ possible values for
$x_{w}$, say $a$ and $b$, so that $K^{C}(\mathbf{x})=0$. If the vertex $v$ is
in $T_{0}$, i.e., $v$ is not an interior vertex of $\varkappa$, then we can
either set $x_{w}=a$ or $x_{w}=b$, and $\mathbf{x}$ will continue to be a
coherent solution of the Kashaev equation. Now, suppose $v$ is not in $T_{0}$,
i.e., $v$ is an interior vertex of $\varkappa$. Because we have chosen
$\mathbf{x}_{\textup{init}}$ to be “sufficiently generic”, the value of
$\prod_{C\ni v}K_{v}^{C}(\mathbf{x})$ depends on whether we set $x_{w}=a$ or
$x_{w}=b$. Proposition 8.1 tells us that for one of the $2$ possible values,
say $x_{w}=a$, equation (4.3) holds, while for the other value, $x_{w}=b$, the
following equation holds:
$\displaystyle\prod_{C\ni v}K_{v}^{C}(\mathbf{x})=-\prod_{S\ni
v}(x_{v}x_{v_{2}}+x_{v_{1}}x_{v_{3}}).$
Hence, the condition of coherence tells us which of the $2$ solutions is the
“correct” one when $v$ is an interior vertex of $\varkappa$.
We now prove the following generalization of Theorem 2.22(b).
###### Proposition 8.3.
Let $\mathbf{T}$ be a pile of quadrangulations of a polygon. Let
$\mathbf{\tilde{x}}=(x_{s})_{s\in\varkappa^{02}(\mathbf{T})}$ be an array
$($with $x_{s}\not=0$ for all $s\in\varkappa^{0}(\mathbf{T}))$ satisfying the
K-hexahedron equations. Then the restriction of $\mathbf{\tilde{x}}$ to
$\varkappa^{0}(\mathbf{T})$ is a coherent solution of the Kashaev equation.
###### Proof.
The proof follows almost exactly the same as the proof of Theorem 2.22(b). For
an interior vertex $v$ of $\varkappa$, there is exactly one cube $C$ for which
$v$ is the top vertex, and exactly one cube $C$ for which $v$ is the bottom
vertex. Let $\mathbf{x}$ be the restriction of $\mathbf{\tilde{x}}$ to
$\varkappa^{0}$. By Lemma 7.2, taking the product over the cubes incident to
$v$,
$\displaystyle\prod_{C\ni
v}K^{C}_{v}(\mathbf{x})=(-1)^{2}\prod_{S\in\varkappa^{2}\colon S\ni
v}x_{S}=\prod_{S\ni v}x_{S}^{2}=\prod_{S\ni
v}(x_{v}x_{v_{2}}+x_{v_{1}}x_{v_{3}}),$
so the restriction of $\mathbf{\tilde{x}}$ to $\mathbf{x}$ is a coherent
solution of the Kashaev equation. ∎
The following statement generalizes Theorem 2.9:
###### Corollary 8.4.
Let $\mathbf{T}$ be a pile of quadrangulations of a polygon. Let
$\mathbf{x}=(x_{s})_{s\in\varkappa^{0}(\mathbf{T})}$ be an array satisfying
the positive Kashaev recurrence. Then $\mathbf{x}$ is a coherent solution of
the Kashaev equation.
###### Proof.
This follows immediately from Proposition 8.3 because $\mathbf{x}$ can be
extended to an array indexed by $\varkappa^{02}(\mathbf{T})$ satisfying the
K-hexahedron equations by choosing the positive solutions from equation (2.9)
for $s\in\varkappa^{2}(\mathbf{T})$. ∎
###### Remark 8.5.
The converse of Proposition 8.3 (i.e., the counterpart of Theorems 2.22(a) and
4.19(a)) does not hold for an arbitrary choice of $\mathbf{T}$. In other
words, there exist piles $\mathbf{T}$ and arrays $\mathbf{x}$ indexed by
$\varkappa^{0}(\mathbf{T})$ with nonzero components that are coherent
solutions of the Kashaev equation, where $\mathbf{x}$ cannot be extended to an
array indexed by $\varkappa^{02}(\mathbf{T})$ satisfying the K-hexahedron
equations.
In order for a converse of Proposition 8.3 (equivalently, a generalization of
Theorems 2.22(a) and 4.19(a)) to hold, one must impose an additional condition
on the underlying cubical complexes; see Definition 8.6 below.
###### Definition 8.6.
Let $\varkappa$ be a three-dimensional cubical complex that can be embedded
into $\operatorname{\mathbb{R}}^{3}$, cf. Definition 3.2. (While this
embeddability condition can be relaxed, it is satisfied in all subsequent
applications. In fact, $\varkappa$ will always be the cubical complex
associated to a pile of quadrangulations.) Let $\sim$ be the equivalence
relation on $\varkappa^{2}$ generated by the equivalences $s_{1}\sim s_{2}$
for all pairs $(s_{1},s_{2})$ involving opposite faces of some $3$-dimensional
cube in $\varkappa^{3}$. Let $\varkappa_{\boxdot}$ denote the set of
equivalence classes under this equivalence relation. Denote by
$[s]\in\varkappa_{\boxdot}$ the equivalence class of $s\in\varkappa^{2}$. By
analogy with Definition 7.13, denote by
$\psi_{\varkappa}\colon\\{-1,1\\}^{\varkappa_{\boxdot}}\rightarrow\\{-1,1\\}^{\varkappa^{3}}$
the map sending an array $\mathbf{t}=(t_{[s]})_{[s]\in\varkappa_{\boxdot}}$ to
the array $\psi_{\varkappa}(\mathbf{t})=(u_{C})_{C\in\varkappa^{3}}$ defined
by $u_{C}=t_{[a]}t_{[b]}t_{[c]}$, where $a$, $b$, $c$ are representatives of
the three pairs of opposite $2$-dimensional faces of $C$. We say that the
cubical complex $\varkappa$ is _comfortable_ if the following statements are
equivalent for every $\mathbf{u}=(u_{C})\in\\{-1,1\\}^{\varkappa^{3}}$:
* (C1)
$\mathbf{u}$ is in the image of $\psi_{\varkappa}$,
* (C2)
for every interior vertex $v\in\varkappa^{0}$ (cf. Definition 3.2), we have
$\displaystyle\prod_{C\ni v}u_{C}=1,$
the product over $3$-dimensional cubes $C\in\varkappa^{3}$ containing $v$.
By Lemma 7.19, the standard tiling of $\operatorname{\mathbb{R}}^{3}$ by unit
cubes is comfortable.
###### Remark 8.7.
In Definition 8.6, the statement (C1) always implies (C2). Indeed, if
$\mathbf{u}=\psi_{\varkappa}(\mathbf{t})$ with
$\mathbf{t}=\left(t_{[s]}\right)_{[s]\in\varkappa_{\boxdot}}\in\\{-1,1\\}^{\varkappa_{\boxdot}}$,
then for any interior vertex $v\in\varkappa^{0}$,
$\displaystyle\prod_{v\in C\in\varkappa^{3}}u_{C}=\prod_{v\in
s\in\varkappa^{2}}t_{[s]}^{2}=1.$
Thus, in checking comfortableness, we simply must check that (C2) implies
(C1).
We next state four results (Propositions 8.8–8.9 and Theorems 8.10–8.11) which
the rest of this section is dedicated to proving. The reader may want to
review Definitions 3.6–3.7 before proceeding with the following proposition.
###### Proposition 8.8.
Let $\mathbf{T}$ be a pile of quadrangulations of a polygon. Suppose that the
divide associated to each quadrangulation in $\mathbf{T}$ is a pseudoline
arrangement. Then $\varkappa=\varkappa(\mathbf{T})$ is comfortable. In
particular, if $\mathbf{T}$ is a pile of $\Diamond$-tilings of the polygon
$\mathbf{P}_{n}$, then $\varkappa=\varkappa(\mathbf{T})$ is comfortable.
###### Proposition 8.9.
There exists a pile $\mathbf{T}$ of quadrangulations of some polygon such that
the cubical complex $\varkappa=\varkappa(\mathbf{T})$ is not comfortable.
We next state a generalization of Theorems 2.22 and 4.19.
###### Theorem 8.10.
Let $\mathbf{T}$ be a pile of quadrangulations of a polygon such that
$\varkappa=\varkappa(\mathbf{T})$ is comfortable. Any coherent solution of the
Kashaev equation $\mathbf{x}=(x_{s})_{s\in\varkappa^{0}}$ with nonzero
components satisfying condition (4.4) can be extended to an array
$\mathbf{\tilde{x}}=(x_{s})_{s\in\varkappa^{02}}$ satisfying the K-hexahedron
equations.
However, Theorems 2.22 and 4.19 don’t generalize to cubical complexes that are
not comfortable.
###### Theorem 8.11.
Let $\mathbf{T}$ be a pile of quadrangulations of a polygon such that
$\varkappa=\varkappa(\mathbf{T})$ is not comfortable. Then there exists a
coherent solution of the Kashaev equation $\mathbf{x}$ indexed by
$\varkappa^{0}$ which cannot be extended to an array indexed by
$\varkappa^{02}$ satisfying the K-hexahedron equations.
Note that Theorem 4.19(a) follows directly from Proposition 8.8 and Theorem
8.10. In Remark 8.12 below, we explain that Theorem 2.22(a) follows from
Proposition 8.8 and Theorem 8.10 as well.
###### Remark 8.12.
Together, Theorem 8.10 and Proposition 8.8 imply Theorem 2.22(a). For each
cube $[-j,j]^{3}\in\operatorname{\mathbb{R}}^{3}$, project the “bottom” faces
(i.e., $\\{-j\\}\times[-j,j]\times[-j,j]$, $[-j,j]\times\\{-j\\}\times[-j,j]$,
$[-j,j]\times[-j,j]\times\\{-j\\}$) onto $\operatorname{\mathbb{R}}^{2}$ to
obtain a quadrangulation $T_{j}$ of a region $R_{j}$, as shown in Fig. 22. The
divide associated to each quadrangulation $T_{i}$ is a pseudoline arrangement.
Hence, by Proposition 8.8, for any pile $\mathbf{T}_{i}$ including $T_{i}$,
$\varkappa(\mathbf{T}_{i})$ is comfortable. Choose $\mathbf{T}_{i}$, so that
we can associate the vertices of $\varkappa(\mathbf{T}_{i})$ with
$\\{-j,\dots,j\\}^{3}$, so that
$\bigcup_{j=1}^{\infty}\varkappa^{0}(\mathbf{T}_{i})=\operatorname{\mathbb{Z}}^{3}$.
Repeating the König’s infinity lemma argument from the end of the proof of
Theorem 2.22(a), Theorem 8.10 implies Theorem 2.22.
|
---|---
quadrangulation $T_{1}$ of $R_{1}$ | quadrangulation $T_{2}$ of $R_{2}$
Figure 22: The quadrangulations $T_{j}$ of regions $R_{j}$ described in Remark
8.12.
The rest of this section is dedicated to proving Propositions 8.8–8.9 and
Theorems 8.10–8.11.
We begin by proving Proposition 8.8.
###### Lemma 8.13.
Let $\mathbf{T}=(T_{0},\dots,T_{\ell})$ be a pile of quadrangulations of a
polygon such that $\varkappa(\mathbf{T})$ is comfortable. Given $0\leq i\leq
j\leq\ell$, let $\mathbf{T}^{\prime}=(T_{i},\dots,T_{j})$. Then
$\varkappa(\mathbf{T}^{\prime})$ is comfortable.
###### Proof.
It suffices to check that (C2) implies (C1) for
$\varkappa(\mathbf{T}^{\prime})$ (see Remark 8.7). Note that any
$\mathbf{u}=(u_{C})_{C\in\varkappa^{3}(\mathbf{T}^{\prime})}$ satisfying (C2)
can be extended to
$\mathbf{\tilde{u}}=(u_{C})_{C\in\varkappa^{3}(\mathbf{T})}$ satisfying (C2).
Identifying $\varkappa_{\boxdot}(\mathbf{T})$ and
$\varkappa_{\boxdot}(\mathbf{T}^{\prime})$, the fact that there exists
$\mathbf{t}$ such that
$\psi_{\varkappa(\mathbf{T})}(\mathbf{t})=\mathbf{\tilde{u}}$ implies that
$\psi_{\varkappa(\mathbf{T}^{\prime})}(\mathbf{t})=\mathbf{u}$, as desired. ∎
We can now prove Proposition 8.8 in the special case where $\mathbf{T}$ be a
pile of $\Diamond$-tilings of $\mathbf{P}_{n}$.
###### Lemma 8.14.
Let $\mathbf{T}$ be a pile of $\Diamond$-tilings of $\mathbf{P}_{n}$. Then
$\varkappa=\varkappa(\mathbf{T})$ is comfortable.
###### Proof.
Labeling the vertices of $\varkappa$ by subsets of $[n]$ (as in Section 4), we
can label the cubes in $\varkappa^{3}$ by $3$-element subsets of $[n]$ by
taking the symmetric difference of the labels of any opposite vertices in the
cube. Note that we can extend $\mathbf{T}$ to a longer pile
$\mathbf{T}^{\prime}$ so that for every $A\in\binom{[n]}{3}$, at least one
cube of $\varkappa(\mathbf{T}^{\prime})$ is labeled by $A$. Hence, by
Proposition 8.13, it suffices to prove the theorem under the additional
assumption that each set in $\binom{[n]}{3}$ labels at least one cube in
$\varkappa^{3}$.
Let $A_{1}$ be the set of $\mathbf{u}\in\\{-1,1\\}^{\varkappa^{3}}$ satisfying
(C1), and $A_{2}$ be the set of $\mathbf{u}$ satisfying (C2). Because
$A_{1}\subseteq A_{2}$, it suffices to show that
$\left|A_{1}\right|\geq\left|A_{2}\right|$ in order to prove that
$A_{1}=A_{2}$. We claim that both $A_{1}$ and $A_{2}$ have size
$2^{\binom{n-1}{2}}$.
First, we claim that $\left|A_{1}\right|\geq 2^{\binom{n-1}{2}}$. Identify
each element $S\in\varkappa_{\boxdot}$ with a $2$-element subset of $[n]$ by
taking the symmetric difference of the labels of any pair of opposite vertices
of any tile in $S$. Note that if $\mathbf{u}=\psi_{\varkappa}(\mathbf{t})$,
and a cube $C$ labeled by $\\{i,j,k\\}$, then
$u_{C}=t_{\\{i,j\\}}t_{\\{i,k\\}}t_{\\{j,k\\}}$. Define a map of vector spaces
$f\colon\\{-1,1\\}^{\binom{[n]}{2}}\rightarrow\\{-1,1\\}^{\binom{[n]}{3}}$
where
$f\big{(}(t_{S})_{S\in\binom{[n]}{2}}\big{)}=(u_{C})_{C\in\binom{[n]}{3}}$
with
$\displaystyle u_{\\{i,j,k\\}}=t_{\\{i,j\\}}t_{\\{i,k\\}}t_{\\{j,k\\}}.$
If we fix $t_{\\{1,2\\}}=\cdots=t_{\\{1,n\\}}=1$, then
$u_{\\{1,j,k\\}}=t_{\\{j,k\\}}$, so the rank of $f$ is at least the number of
$2$-element subsets of $\\{2,\dots,n\\}$, i.e., $\binom{n-1}{2}$. Hence, it
follows that $\left|A_{1}\right|\geq 2^{\binom{n-1}{2}}$.
Thus, in order to prove the proposition, we must show that
$\left|A_{2}\right|\leq 2^{\binom{n-1}{2}}$. Note that there are
$\binom{n-1}{2}$ vertices in the interior of any $\Diamond$-tiling of
$\mathbf{P}_{n}$. In choosing $\mathbf{u}$ satisfying (C2), we can make an
arbitrary choice of sign for any cube that shares its bottom vertex with
$T_{0}$, but the signs of the remaining cubes is determined by condition (C2).
Hence, because at most $\binom{n-1}{2}$ cubes can share their bottom vertices
with $T_{0}$ (the bottom of a cube cannot be on the boundary of $T_{0}$),
there are at most $2^{\binom{n-1}{2}}$ such $\mathbf{u}$ satisfying condition
(C2), proving our claim. ∎
We can now prove Proposition 8.8 in its full generality.
###### Proof of Proposition 8.8.
Let $\mathbf{T}=(T_{0},\dots,T_{\ell})$. We claim that we can “embed” the
quadrangulations $T_{0},\dots,T_{\ell}$ in $\Diamond$-tilings of
$\mathbf{P}_{n}$. Let $D_{0},\dots,D_{\ell}$ be the divides associated to
$T_{0},\dots,T_{\ell}$. Because $D_{0},\dots,D_{\ell}$ are pseudoline
arrangements connected by braid moves, we can extend $D_{0},\dots,D_{\ell}$ to
pseudoline arrangements $\tilde{D}_{0},\dots,\tilde{D}_{\ell}$, still
connected by braid moves, in which every pair of branches intersects exactly
once. By Proposition 3.8, there exists a pile
$\mathbf{\tilde{T}}=(\tilde{T}_{0},\dots,\tilde{T}_{\ell})$ of
$\Diamond$-tilings of $\mathbf{P}_{n}$, for which the divides associated to
$\tilde{T}_{0},\dots,\tilde{T}_{\ell}$ are
$\tilde{D}_{0},\dots,\tilde{D}_{\ell}$.
By Lemma 8.14, $\varkappa(\mathbf{\tilde{T}})$ is comfortable. The cubical
complex $\varkappa(\mathbf{\tilde{T}})$ consists of
$\varkappa=\varkappa(\mathbf{T})$, unioned with $2$-dimensional faces that are
not part of any $3$-dimensional cube. Hence, it follows that $\varkappa$ is
comfortable as well. ∎
###### Proof of Proposition 8.9.
We describe a pile $\mathbf{T}=(T_{0},\dots,T_{8})$ of quadrangulations of a
square such that $\varkappa=\varkappa(\mathbf{T})$ is not comfortable. Let
$T_{0}$ be as in Fig. 23. It is easier to understand this example by looking
at the divides associated to $T_{0},\dots,T_{8}$, displayed in Fig. 24. Note
that the divides associated to these quadrangulations are not pseudoline
arrangements. Note that $\varkappa$ has no interior vertices. Hence, every
$\mathbf{u}\in\\{-1,1\\}^{\varkappa^{3}}$ satisfies (C2). However, it is not
difficult to check that if $\mathbf{u}$ satisfies (C1), then the sign on a
given cube is determined by the sign on the other $7$. Hence, $\varkappa$ is
not comfortable. ∎
Figure 23: The quadrangulation $T_{0}$ from the proof of Proposition 8.9, with the associated divide drawn on top in blue. | |
---|---|---
$T_{0}$ | $T_{1}$ | $T_{2}$
| |
$T_{3}$ | $T_{4}$ | $T_{5}$
| |
$T_{6}$ | $T_{7}$ | $T_{8}$
Figure 24: The divides associated to the quadrangulations $T_{0},\dots,T_{8}$
from the proof of Proposition 8.9.
The rest of this section is dedicated to the proofs of Theorems 8.10–8.11.
###### Definition 8.15.
Let $\mathbf{T}=(T_{0},\dots,T_{\ell})$ be a pile of quadrangulations of a
polygon, with $\varkappa=\varkappa(\mathbf{T})$, and
$\mathbf{x}=(x_{s})_{s\in\varkappa^{0}}$. We say that $\mathbf{x}$ is
_generic_ if for all extensions of $\mathbf{x}_{\textup{init}}$ (the
restriction of $\mathbf{x}$ to $\varkappa^{0}(T_{0})$) to an array
$\mathbf{\tilde{x}}$ indexed by $\varkappa^{02}(\mathbf{T})$ satisfying the
K-hexahedron equations, the entries of $\mathbf{\tilde{x}}$ are all nonzero.
###### Definition 8.16.
Let $\mathbf{T}=(T_{0},\dots,T_{\ell})$ be a pile of quadrangulations of a
polygon, with $\varkappa=\varkappa(\mathbf{T})$. Given an array
$\mathbf{\tilde{x}}_{\textup{init}}=(x_{s})_{s\in\varkappa^{02}(T_{0})}$ and
$\mathbf{t}=\left(t_{[s]}\right)_{[s]\in\varkappa_{\boxdot}}$, set
$\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}}=(y_{s})_{s\in\varkappa^{02}(T_{0})}$,
where
$\displaystyle y_{s}=\begin{cases}x_{s}&\text{if }s\in\varkappa^{0}(T_{0}),\\\
t_{[s]}x_{s}&\text{if }s\in\varkappa^{2}(T_{0}).\end{cases}$
Given a generic array $\mathbf{\tilde{x}}_{\textup{init}}$ indexed by
$\varkappa^{02}(T_{0})$, define
$(\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\varkappa^{02}}$ to be the
unique extension of $\mathbf{\tilde{x}}_{\textup{init}}$ to an array indexed
by $\varkappa^{02}$ satisfying the K-hexahedron equations. Define
$(\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\varkappa^{0}}$ to be the
restriction of $(\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\varkappa^{02}}$
to $\varkappa^{0}$.
###### Lemma 8.17.
Let $\mathbf{T}=(T_{0},\dots,T_{\ell})$ be a pile of quadrangulations of a
polygon, with $\varkappa=\varkappa(\mathbf{T})$. Fix a generic array
$\mathbf{\tilde{x}}_{\textup{init}}$ indexed by $\varkappa^{02}(T_{0})$
satisfying equation (2.9) for $s\in\varkappa^{2}(T_{0})$, and
$\mathbf{t}\in\\{-1,1\\}^{\varkappa_{\boxdot}}$. Then the following are
equivalent:
* •
$\psi_{\varkappa}(\mathbf{t})$ has value $1$ on $C_{1},\dots,C_{i-1}$, and
value $-1$ on $C_{i}$,
* •
$(\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\varkappa^{0}}$ and
$(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\varkappa^{0}}$
agree at $\varkappa^{0}(T_{0}),\dots,\varkappa^{0}(T_{i-1})$ but not at
$\varkappa^{0}(T_{i})$.
###### Proof.
The proof follows directly from Lemma 7.11. ∎
###### Lemma 8.18.
Let $\mathbf{T}=(T_{0},\dots,T_{\ell})$ be a pile of quadrangulations of a
polygon, with $\varkappa=\varkappa(\mathbf{T})$. Let $\mathbf{x}$ and
$\mathbf{x}^{\prime}$ be generic and distinct coherent solutions of the
Kashaev equation, both indexed by $\varkappa^{0}$, such that $\mathbf{x}$ and
$\mathbf{x}^{\prime}$ agree at $\varkappa^{0}(T_{0})$. Let $i$ be the minimum
value such that $\mathbf{x}$ and $\mathbf{x}^{\prime}$ do not agree at
$\varkappa^{0}(T_{i})$. Then the cube $C_{i}$ shares its bottom vertex with
$T_{0}$.
###### Proof.
Assume (for contradiction) that $C_{i}$ doesn’t share its bottom vertex with
$T_{0}$. Then the bottom vertex of $C_{i}$ must be an interior vertex of
$\varkappa$. Hence, by the coherence and genericity of $\mathbf{x}$ and
$\mathbf{x}^{\prime}$, the values of $\mathbf{x}$ and $\mathbf{x}^{\prime}$
are uniquely determined by their values at
$\varkappa^{0}(T_{0}),\dots,\varkappa^{0}(T_{i-1})$, which are the same for
$\mathbf{x}$ and $\mathbf{x}^{\prime}$. Hence, $\mathbf{x}$ and
$\mathbf{x}^{\prime}$ agree at the top vertex of $C_{i}$, and thus agree at
$\varkappa^{0}(T_{i})$, a contradiction. ∎
We can now prove a weaker version of Theorem 8.10, under the additional
constraint of genericity.
###### Corollary 8.19.
Let $\mathbf{T}$ be a pile of quadrangulations of a polygon such that
$\varkappa=\varkappa(\mathbf{T})$ is comfortable. Any generic, coherent
solution of the Kashaev equation $\mathbf{x}=(x_{s})_{s\in\varkappa^{0}}$ can
be extended to $\mathbf{\tilde{x}}=(x_{s})_{s\in\varkappa^{02}}$ satisfying
the K-hexahedron equations.
###### Proof.
Choose an arbitrary extension of $\mathbf{x}_{\textup{init}}$, the restriction
of $\mathbf{x}$ to $\varkappa^{0}(T_{0})$ to an array
$\mathbf{\tilde{x}}_{\textup{init}}^{\prime}$ indexed by
$\varkappa^{02}(T_{0})$ satisfying equation (2.9) for
$s\in\varkappa^{2}(T_{0})$. The result follows once we can show that there
exists $\mathbf{t}\in\\{-1,1\\}^{\varkappa_{\boxdot}}$ such that
$(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}}^{\prime})^{\uparrow\varkappa^{02}}$
agrees with $\mathbf{x}$ on $\varkappa^{0}$.
We proceed by induction, and assume that there exists $\mathbf{t}$ such that
$(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}}^{\prime})^{\uparrow\varkappa^{02}}$
agrees with $\mathbf{x}$ on $\varkappa^{0}(T_{j})$ for $j=0,\dots,i-1$. If
$(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}}^{\prime})^{\uparrow\varkappa^{02}}$
agrees with $\mathbf{x}$ on $\varkappa^{0}(T_{j})$ for $j=0,\dots,i$, we are
done. Suppose that
$(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}}^{\prime})^{\uparrow\varkappa^{02}}$
does not agree with $\mathbf{x}$ on $\varkappa^{0}(T_{i})$. We need to find
$\mathbf{t}_{i}$ such that
$(\mathbf{t}_{i}\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}}^{\prime})^{\uparrow\varkappa^{02}}$
agrees with $\mathbf{x}$ on $\varkappa^{0}(T_{j})$ for $j=0,\dots,i$. By Lemma
8.17, this is equivalent to finding $\mathbf{t}_{i}$ such that
$\psi_{\varkappa}(\mathbf{t}_{i})$ is $1$ on $C_{1},\dots,C_{i-1}$, and $-1$
on $C_{i}$. By Lemma 8.18, the cube $C_{i}$ shares its bottom vertex with
$T_{0}$. Hence, there exists $\mathbf{u}=(u_{s})\in\\{-1,1\\}^{\varkappa^{3}}$
satisfying (C2) such that $u_{C_{1}}=\cdots=u_{C_{i-1}}=1$ and $u_{C_{i}}=-1$.
(For example, choose $\mathbf{u}$ so that $u_{C_{i}}=-1$, and $u_{C}=1$ for
all other cubes $C$ that share a bottom vertex with $T_{0}$. Then the
remaining values are determined by condition (C2).) Because $\varkappa$ is
comfortable, there exists $\mathbf{t}_{i}$ such that
$\psi_{\varkappa}(\mathbf{t}_{i})=\mathbf{u}$, as desired. ∎
###### Proof of Theorem 8.10..
We need to loosen the condition that $\mathbf{x}$ is generic from Corollary
8.19 to the conditions that $\mathbf{x}$ has nonzero components and satisfies
condition (4.4).
Let $\mathbf{x}\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{0}}$ be a
coherent solution of the Kashaev equation with nonzero components that
satisfies condition (4.4). It is straightforward to show that there exists a
sequence
$\mathbf{x}_{1},\mathbf{x}_{2},\ldots\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{0}}$
of generic, coherent solutions of the Kashaev equation that converge pointwise
to $\mathbf{x}$. By Corollary 8.19, there exist
$\mathbf{\tilde{x}}_{1},\mathbf{\tilde{x}}_{2},\ldots\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{02}}$
satisfying the K-hexahedron equations such that $\mathbf{\tilde{x}}_{i}$
restricts to $\mathbf{x}_{i}$. There exists a subsequence of
$\mathbf{\tilde{x}}_{1},\mathbf{\tilde{x}}_{2},\dots$ that converges to an
array $\mathbf{\tilde{x}}$. (For each $s\in\varkappa^{2}(\mathbf{T})$, we can
partition the sequence $\mathbf{\tilde{x}}_{1},\mathbf{\tilde{x}}_{2},\dots$
into two sequences, each of which converges at $s$. Because
$\varkappa^{2}(\mathbf{T})$ is finite, the claim follows.) The array
$\mathbf{\tilde{x}}$ must satisfy the K-hexahedron equations and restrict to
$\mathbf{x}$, so we are done. ∎
In order to complete the proof of Theorem 8.11, we will need the following
technical lemma.
###### Lemma 8.20.
Let $\mathbf{T}=(T_{0},\dots,T_{\ell})$ be a pile of quadrangulations of a
polygon such that $\varkappa(\mathbf{T})$ is not comfortable, but
$\varkappa(T_{0},\dots,T_{\ell-1})$ is comfortable. Let $C_{\ell}$ be the cube
of $\varkappa$ corresponding to the flip from $T_{\ell-1}$ to $T_{\ell}$.
1. $(a)$
Let $v$ be the bottom vertex of the cube $C_{\ell}$, i.e., let $v$ be the
vertex of $T_{\ell-1}$ not in $T_{\ell}$. Then $v$ is in $T_{0}$.
2. $(b)$
Let $\mathbf{w}=(w_{C})_{C\in\varkappa^{3}}$ where $w_{C_{\ell}}=-1$, and
$w_{C}=1$ for $C\not=C_{\ell}$. Then $\mathbf{w}$ is not in the image of
$\psi_{\varkappa}$.
###### Proof.
Let $\varkappa^{\prime}=\varkappa(T_{0},\dots,T_{\ell-1})$. Let
* •
$a_{1}$ be the number of $\mathbf{u}\in\\{-1,1\\}^{(\varkappa^{\prime})^{3}}$
satisfying (C1),
* •
$a_{2}$ be the number of $\mathbf{u}\in\\{-1,1\\}^{(\varkappa^{\prime})^{3}}$
satisfying (C2),
* •
$b_{1}$ be the number of $\mathbf{u}\in\\{-1,1\\}^{\varkappa^{3}}$ satisfying
(C1), and
* •
$b_{2}$ be the number of $\mathbf{u}\in\\{-1,1\\}^{\varkappa^{3}}$ satisfying
(C2).
Because $a_{1}$, $a_{2}$, $b_{1}$, $b_{2}$ enumerate the elements of vector
fields over $\mathbb{F}_{2}$, all four quantities must be powers of $2$.
Because $\varkappa^{\prime}$ is comfortable, $a_{1}=a_{2}$. Because
$\varkappa$ is not comfortable, $b_{1}<b_{2}$. It is clear that $b_{1}\leq
2a_{1}$ and $b_{2}\leq 2a_{2}$. Hence, it follows that
$a_{1}=a_{2}=b_{1}=b_{2}/2$.
Assume (for contradiction) that $v$ is not in $T_{0}$, so $v$ is in the
interior of $\varkappa$. But then if $\mathbf{u}=(u_{C})_{C\in\varkappa^{3}}$
satisfies (C2),
$\displaystyle u_{C_{\ell}}=\prod_{C\in\varkappa^{3}\colon w\in
C\not=C_{\ell}}u_{C},$
so $a_{2}=b_{2}$, a contradiction. Hence, we have proved (a).
Because $a_{1}=b_{1}$, it follows that for each
$\mathbf{u}^{\prime}\in\\{-1,1\\}^{(\varkappa^{\prime})^{3}}$ satisfying (C1),
there exists exactly one $\mathbf{u}\in\\{-1,1\\}^{\varkappa^{3}}$ satisfying
(C1) that restricts to $\mathbf{u}^{\prime}$. Because
$\mathbf{u}=(u_{C})_{C\in\varkappa^{3}}$ and
$\mathbf{u}^{\prime}=(u_{C})_{C\in(\varkappa^{\prime})^{3}}$ where $u_{C}=1$
for all $C$ satisfy (C1), $\mathbf{w}$ cannot satisfy (C1), proving (b). ∎
###### Proof of Theorem 8.11.
Without loss of generality, we assume that $\varkappa(T_{0},\dots,T_{\ell-1})$
is comfortable. (If not, let $m$ be minimum so that
$\varkappa(T_{0},{\dots},T_{m})$ is not comfortable, but
$\varkappa(T_{0},{\dots},T_{m{-}1})$ is comfortable. If we can prove the
theorem for $\varkappa(T_{0},\dots,T_{m})$, it follows that it holds for
$\varkappa(\mathbf{T})$.)
We now construct an array $\mathbf{x}$ satisfying the desired conditions. Let
$C$ be the cube of $\varkappa$ corresponding to the flip from $T_{\ell-1}$ to
$T_{\ell}$, and let $v$ be the top vertex of $C$ (i.e., $v$ is the new vertex
in $T_{i}$). Choose arbitrary positive values for
$\mathbf{x}_{\textup{init}}$. Extend $\mathbf{x}_{\textup{init}}$ to
$\mathbf{x}$ by the positive Kashaev recurrence until we reach $v$, where we
choose the other value such that $K^{C}(\mathbf{x})=0$.
By construction, $\mathbf{x}$ restricted to
$\varkappa(T_{0},\dots,T_{\ell-1})$ satisfies the positive Kashaev recurrence,
and hence is a coherent solution of the Kashaev equation. By Lemma 8.20(a), no
vertices of $C$ are in the interior of $\varkappa$. Hence, $\mathbf{x}$ is a
coherent solution of the Kashaev equation.
Next, we show that $\mathbf{x}$ cannot be extended to an array indexed by
$\varkappa^{02}$ satisfying the K-hexahedron equations. Let
$\mathbf{x}_{\text{pK}}$ be the array satisfying the positive Kashaev
recurrence that restricts to $\mathbf{x}_{\textup{init}}$ at $T_{0}$ (so
$\mathbf{x}_{\text{pK}}$ agrees with $\mathbf{x}$ everywhere except $v$). Let
$\mathbf{\tilde{x}}_{\text{pK}}$ be an extension of $\mathbf{x}_{\text{pK}}$
to $\varkappa^{02}$ satisfying the K-hexahedron equations. Assume (for
contradiction) that there exists
$\mathbf{t}\in\\{-1,1\\}^{\tilde{\varkappa}_{2}}$ such that
$\mathbf{\tilde{x}}(\mathbf{t}\cdot(\mathbf{\tilde{x}}_{\text{pK}})_{0})$
restricts to $\mathbf{x}$. Hence, by Lemma 8.17,
$\psi_{\varkappa}(\mathbf{t})$ has value $-1$ at $C$, and value $1$ everywhere
else. But Lemma 8.20(b) says that array is not in the image of
$\psi_{\varkappa}$, a contradiction. Hence, no such $\mathbf{t}$ exists, so
$\mathbf{x}$ cannot be extended to an array indexed by $\varkappa^{02}$
satisfying the K-hexahedron equations. ∎
## 9 Proofs of Corollary 4.23 and Theorem 4.26
This section contains the proofs of Corollary 4.23 and Theorem 4.26.
We use the following lemma in proving Corollary 4.23.
###### Lemma 9.1.
Let $\mathbf{T}$ be a pile of $\Diamond$-tilings of $\mathbf{P}_{n}$, with
$\varkappa=\varkappa(\mathbf{T})$. Let
$\mathbf{x}=(x_{s})\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{0}}$ be a
coherent solution of the Kashaev equation satisfying condition (4.2). Suppose
$s_{1},s_{2}\in\varkappa^{0}$ are labeled by the same subset of $[n]$. Then
$x_{s_{1}}=x_{s_{2}}$.
###### Proof.
Note that due to the homogeneity of the Kashaev equation and the coherence
equations (equation (4.17)), we can rescale the components of $\mathbf{x}$ to
obtain a standard array. Hence, we can assume that $\mathbf{x}$ is standard.
By Theorem 4.19(a), we can extend $\mathbf{x}$ to an array
$\mathbf{\tilde{x}}$ indexed by $\varkappa^{02}$ satisfying the K-hexahedron
equations. Note that we can choose a sequence
$\mathbf{\tilde{x}}_{1},\mathbf{\tilde{x}}_{2},\dots$ of standard arrays
indexed by $\varkappa^{02}$ satisfying the K-hexahedron equations converging
to $\mathbf{\tilde{x}}$ such that the restriction of $\mathbf{\tilde{x}}_{i}$
to $\varkappa^{02}(T)$ for any tiling $T$ in $\mathbf{T}$ is generic. By
Theorems 4.9 and 4.12, there exist symmetric $n\times n$ matrices such that
$\mathbf{\tilde{x}}_{i}=\mathbf{\tilde{x}}_{\varkappa(\mathbf{T})}(M_{i})$.
Hence, the components of $\mathbf{\tilde{x}}_{i}$ at $s_{1}$ and $s_{2}$ must
agree, so the components of $\mathbf{\tilde{x}}$ at $s_{1}$ and $s_{2}$ must
agree. ∎
###### Proof of Corollary 4.23..
The first bullet point implies the second two by Corollary 4.14, and it is
obvious that the third implies the second. Thus, we need to show that the
second bullet point implies the first.
Next, suppose $\mathbf{T}=(T_{0},\dots,T_{\ell})$ is a pile of
$\Diamond$-tilings of $\mathbf{P}_{n}$ in which every $I\subseteq[n]$ labels
at least one vertex of $\varkappa(\mathbf{T})$, and
$\mathbf{x}=\mathbf{x}_{\varkappa(\mathbf{T})}(\mathbf{\bar{x}})$ is a
coherent solution of the Kashaev equation. Let
$\mathbf{T}^{\prime}=(T_{0},\dots,T_{\ell},\dots,T_{\ell^{\prime}})$ be an
extension of $\mathbf{T}$ where $\mathbf{T}^{\prime}$ contains the tiling
$T_{\textup{min},n}$. By Lemma 9.1,
$\mathbf{x}_{\varkappa(\mathbf{T}^{\prime})}(\mathbf{\bar{x}})$ is the unique
extension of $\mathbf{x}$ to $\varkappa^{0}(\mathbf{T}^{\prime})$. By Theorem
4.19(a), there exists an array $\mathbf{\tilde{x}}$ indexed by
$\varkappa^{02}(\mathbf{T}^{\prime})$ extending
$\mathbf{x}_{\varkappa(\mathbf{T}^{\prime})}(\mathbf{\bar{x}})$ that satisfies
the K-hexahedron equations. By Theorem 4.10, Proposition 4.11, and Corollary
4.15 (all of which are due to Kenyon and Pemantle [5]), there exists a unique
symmetric matrix $M$ such that
$\mathbf{\tilde{x}}=\mathbf{\tilde{x}}_{\varkappa(\mathbf{T}^{\prime})}(M)$,
so $M$ satisfies condition (4.5). ∎
Next, we shall work towards a proof of Theorem 4.26.
###### Proposition 9.2.
Let $M$ be an $n\times n$ symmetric matrix, and let
$\mathbf{\bar{x}}=\mathbf{\bar{x}}(M)$. Then for all $I\subseteq[n]$ and
$A\in\binom{[n]}{4}$, equation (4.10) holds.
###### Proof.
Note that it suffices to prove Proposition 9.2 for generic, symmetric $M$,
because any symmetric matrix can be written as a limit of generic, symmetric
matrices. Fix a generic, symmetric $n\times n$ matrix $M$ for the rest of the
proof.
For $I\subset[n]$ and distinct $i,j\in[n]$, let
$\displaystyle
x_{I,\\{i,j\\}}=(-1)^{\lfloor(\left|I^{\prime}\right|+1)/2\rfloor}M_{I^{\prime}\cup\\{i\\}}^{I^{\prime}\cup\\{j\\}},$
where $I^{\prime}=I\setminus\\{i,j\\}$. Note that if $\mathbf{T}$ is a pile of
$\Diamond$-tilings of $\mathbf{P}_{n}$, a cube of $\varkappa(\mathbf{T})$
containing vertices labeled by $I$ and $I\cup\\{i,j,k\\}$ for $i,j,k\not\in I$
has top/bottom vertices labeled by $I\cup\\{j\\}$ and $I\cup\\{i,k\\}$. Hence,
by Lemma 7.2 and the fact that $\mathbf{\tilde{x}}_{\varkappa(\mathbf{T})}(M)$
satisfies the K-hexahedron equations, where $\mathbf{T}$ is any pile of
$\Diamond$-tilings of $\mathbf{P}_{n}$, it follows that
$\displaystyle K_{I,\\{i,j,k\\}}(\mathbf{\bar{x}})=\pm
x_{I,\\{i,j\\}}x_{I,\\{\\{i,k\\}}x_{I,\\{j,k\\}},$ (9.1)
where the plus sign appears on the right-hand side of equation (9.1) if either
* •
$i,k\in I$ and $j\not\in I$, or
* •
$j\in I$ and $i,k\not\in I$,
and the minus sign appears otherwise.
Let $I\subseteq[n]$ and $A\in\binom{[n]}{4}$. It is straightforward to check
that an even number of $\\{i<j<k\\}\in\binom{A}{3}$ satisfy neither of the
bullet points above. Hence, by equation (9.1), it follows that
$\displaystyle\prod_{J\in\binom{A}{3}}K_{I,J}(\mathbf{\bar{x}})=\prod_{J\in\binom{A}{2}}x_{I,J}^{2}=\prod_{J\in\binom{A}{2}}L_{I,J}(\mathbf{\bar{x}}),$
as desired. ∎
The reader may want to review Example 3.17 before proceeding with the
following lemma.
###### Lemma 9.3.
Fix an array $\mathbf{\bar{x}}=(x_{I})_{I\subseteq[4]}$, satisfying the
conditions that
* •
$x_{I}\not=0$ for all $I\subseteq[4]$,
* •
for any $I\subseteq[4]$ and distinct $i,j\in[4]$, $L_{I,\\{i,j\\}}\not=0$,
* •
for all $I\subseteq[4]$ and distinct $i,j,k\in[4]$, equation (4.8) holds,
* •
for all $I\subseteq[4]$, equation (4.9) holds.
Let
$\mathbf{T}_{1}=(T_{1,0},\dots,T_{1,4}),\mathbf{T}_{2}=(T_{2,0},\dots,T_{2,4})\in\mathcal{C}(4)$
be the two distinct piles in $\mathcal{C}(4)$. Let
$\mathbf{\tilde{x}}_{1}\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{02}(\mathbf{T}_{1})}$
be an extension of $\mathbf{x}_{\varkappa(\mathbf{T}_{1})}(\mathbf{\bar{x}})$
satisfying the K-hexahedron equations. Then there exists an extension
$\mathbf{\tilde{x}}_{2}\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{02}(\mathbf{T}_{2})}$
of $\mathbf{x}_{\varkappa(\mathbf{T}_{2})}(\mathbf{\bar{x}})$ satisfying the
K-hexahedron equations that agrees with $\mathbf{\tilde{x}}_{1}$ on
$\varkappa^{02}(T_{1,0})=\varkappa^{02}(T_{2,0})$.
###### Proof.
By the homogeneity of equations (4.8)–(4.9), we can rescale the components of
$\mathbf{\bar{x}}$ so that $x_{\varnothing}=1$. Hence, it follows from
Corollary 4.25 that there exists an extension
$\mathbf{\tilde{x}}_{1}^{\prime}\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{02}(\mathbf{T}_{1})}$
of $\mathbf{x}_{\varkappa(\mathbf{T}_{1})}(\mathbf{\bar{x}})$ and an extension
$\mathbf{\tilde{x}}_{2}^{\prime}\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{02}(\mathbf{T}_{2})}$
of $\mathbf{x}_{\varkappa(\mathbf{T}_{2})}(\mathbf{\bar{x}})$ that agree on
$\varkappa^{02}(T_{1,0})=\varkappa^{02}(T_{2,0})$.
Let $\mathbf{\tilde{x}}_{\textup{init}}^{\prime}$ be the restriction of
$\mathbf{\tilde{x}}_{1}^{\prime}$ to $\varkappa^{02}(T_{1,0})$. Let
$\mathbf{t}\in\\{-1,1\\}^{\varkappa^{2}(T_{1,0})}$ (where we associate
$\varkappa^{2}(T_{1,0})$ with $\tilde{\varkappa}_{2}(\mathbf{T}_{1})$) so that
$\mathbf{\tilde{x}}_{1}=(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}}^{\prime})^{\uparrow\varkappa^{02}(\mathbf{T}_{1})}$.
By Lemma 8.17, because $\mathbf{\tilde{x}}_{1}^{\prime}$ and
$\mathbf{\tilde{x}}_{1}$ agree on $\varkappa^{0}(\mathbf{T}_{1})$,
$\psi_{\varkappa(\mathbf{T}_{1})}(\mathbf{t})$ has value $1$ at every cube of
$\varkappa(\mathbf{T}_{1})$. Because
$\psi_{\varkappa(\mathbf{T}_{1})}(\mathbf{t})$ has value $1$ at every cube of
$\varkappa(\mathbf{T}_{1})$, $\psi_{\varkappa(\mathbf{T}_{2})}(\mathbf{t})$
has value $1$ at every cube of $\varkappa(\mathbf{T}_{2})$. Hence,
$(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}}^{\prime})^{\uparrow\varkappa^{2}(\mathbf{T}_{2})}$
agrees with $\mathbf{\tilde{x}}_{1}$ on
$\varkappa^{02}(T_{1,0})=\varkappa^{02}(T_{2,0})$ and restricts to
$\mathbf{x}_{\varkappa(\mathbf{T}_{2})}(\mathbf{\bar{x}})$. ∎
The reader may want to review Definition 3.21 before proceeding with the
following definition.
###### Definition 9.4.
Let $\mathbf{T}_{1}=(T_{1,0},\dots,T_{1,\ell})$ and
$\mathbf{T}_{2}=(T_{2,0},\dots,T_{2,\ell})$ be two piles, such that the
directed cubical complexes $\varkappa(\mathbf{T}_{1})$ and
$\varkappa(\mathbf{T}_{2})$ are related by a flip. Label the vertices of
$\varkappa(\mathbf{T}_{1})$ and $\varkappa(\mathbf{T}_{2})$ involved in the
$3$-flip by subsets of $[4]$, as in Fig. 25. Let
$\mathbf{x}_{1}\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{0}(\mathbf{T}_{1})}$
and
$\mathbf{x}_{2}\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{0}(\mathbf{T}_{2})}$
be arrays satisfying condition (4.2). We say that the pair
$(\mathbf{x}_{1},\mathbf{x}_{2})$ is _K-flipped_ when
* •
$\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ agree everywhere, except at the vertex
at which $\varkappa(\mathbf{T}_{1})$ and $\varkappa(\mathbf{T}_{2})$ differ,
* •
writing $\mathbf{\bar{x}}=(x_{I})_{I\subseteq[4]}$, where $x_{I}$ is the
component of $\mathbf{x}_{1}$ and/or $\mathbf{x}_{2}$ at the vertex labeled by
$I$, $\mathbf{\bar{x}}$ satisfies equation (4.8) for all $I\subseteq[4]$ and
distinct $i,j,k\in[4]$ and equation (4.9) for all $I\subseteq[4]$.
In
$\varkappa(\mathbf{T}_{1})$:$\bm{\varnothing}$$\mathbf{1}$$\mathbf{12}$$\mathbf{123}$$\mathbf{1234}$$\mathbf{234}$$\mathbf{34}$$\mathbf{4}$$\mathbf{2}$$\mathbf{3}$$\mathbf{23}$$\bm{\varnothing}$$\mathbf{1}$$\mathbf{12}$$\mathbf{123}$$\mathbf{1234}$$\mathbf{234}$$\mathbf{34}$$\mathbf{4}$$\mathbf{13}$$\mathbf{3}$$\mathbf{23}$$\bm{\varnothing}$$\mathbf{1}$$\mathbf{12}$$\mathbf{123}$$\mathbf{1234}$$\mathbf{234}$$\mathbf{34}$$\mathbf{4}$$\mathbf{13}$$\mathbf{3}$$\mathbf{134}$$\bm{\varnothing}$$\mathbf{1}$$\mathbf{12}$$\mathbf{123}$$\mathbf{1234}$$\mathbf{234}$$\mathbf{34}$$\mathbf{4}$$\mathbf{13}$$\mathbf{14}$$\mathbf{134}$$\bm{\varnothing}$$\mathbf{1}$$\mathbf{12}$$\mathbf{123}$$\mathbf{1234}$$\mathbf{234}$$\mathbf{34}$$\mathbf{4}$$\mathbf{124}$$\mathbf{14}$$\mathbf{134}$
---
In
$\varkappa(\mathbf{T}_{2})$:$\bm{\varnothing}$$\mathbf{1}$$\mathbf{12}$$\mathbf{123}$$\mathbf{1234}$$\mathbf{234}$$\mathbf{34}$$\mathbf{4}$$\mathbf{2}$$\mathbf{3}$$\mathbf{23}$$\bm{\varnothing}$$\mathbf{1}$$\mathbf{12}$$\mathbf{123}$$\mathbf{1234}$$\mathbf{234}$$\mathbf{34}$$\mathbf{4}$$\mathbf{23}$$\mathbf{2}$$\mathbf{24}$$\bm{\varnothing}$$\mathbf{1}$$\mathbf{12}$$\mathbf{123}$$\mathbf{1234}$$\mathbf{234}$$\mathbf{34}$$\mathbf{4}$$\mathbf{124}$$\mathbf{2}$$\mathbf{24}$$\bm{\varnothing}$$\mathbf{1}$$\mathbf{12}$$\mathbf{123}$$\mathbf{1234}$$\mathbf{234}$$\mathbf{34}$$\mathbf{4}$$\mathbf{124}$$\mathbf{14}$$\mathbf{24}$$\bm{\varnothing}$$\mathbf{1}$$\mathbf{12}$$\mathbf{123}$$\mathbf{1234}$$\mathbf{234}$$\mathbf{34}$$\mathbf{4}$$\mathbf{124}$$\mathbf{14}$$\mathbf{134}$
Figure 25: Labeling the vertices involved in a flip between
$\varkappa(\mathbf{T}_{1})$ and $\varkappa(\mathbf{T}_{2})$ in Lemma 9.5 with
subsets of $[4]$ (in blue).
###### Lemma 9.5.
Let $\mathbf{T}_{1}=(T_{1,0},\dots,T_{1,\ell})$ and
$\mathbf{T}_{2}=(T_{2,0},\dots,T_{2,\ell})$ be two piles, such that the
directed cubical complexes $\varkappa(\mathbf{T}_{1})$ and
$\varkappa(\mathbf{T}_{2})$ are related by a flip. Let
$\mathbf{x}_{1}\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{0}(\mathbf{T}_{1})}$
and
$\mathbf{x}_{2}\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{0}(\mathbf{T}_{2})}$
be arrays satisfying condition (4.2), such that the pair
$(\mathbf{x}_{1},\mathbf{x}_{2})$ is K-flipped.
1. $(a)$
Then $\mathbf{x}_{1}$ is a coherent solution of the Kashaev equation if and
only if $\mathbf{x}_{2}$ is a coherent solution of the Kashaev equation.
2. $(b)$
Suppose $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ are both coherent solutions of
the Kashaev equation. Let
$\mathbf{\tilde{x}}_{1}\\!\in\\!(\operatorname{\mathbb{C}}^{*})^{\varkappa^{02}(\mathbf{T}_{1})}$
be an extension of $\mathbf{x}_{1}$ to $\varkappa^{02}(\mathbf{T}_{1})$, and
let $\mathbf{\tilde{x}}_{\textup{init}}$ be the restriction of
$\mathbf{\tilde{x}}_{1}$ to $\varkappa^{02}(T_{1,0})$. Then, identifying
$\varkappa^{02}(T_{1,0})$ and $\varkappa^{02}(T_{2,0})$,
$(\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\varkappa^{02}(\mathbf{T}_{2})}$
restricts to $\mathbf{x}_{2}$.
###### Proof.
This result follows from Lemma 9.3. ∎
###### Lemma 9.6.
There exists a sequence
$\mathbf{T}_{0},\dots,\mathbf{T}_{\binom{n}{3}-\binom{n-1}{2}}\in\mathcal{C}(n)$
of piles, where we write $\mathbf{T}_{i}=(T_{i,0},\dots,T_{i,\binom{n}{3}})$
for $i=0,\dots,\binom{n}{3}-\binom{n-1}{2}$, such that
* •
the directed cubical complexes $\varkappa(\mathbf{T}_{i-1})$ and
$\varkappa(\mathbf{T}_{i})$ are related by a flip for $i=1,\dots,\ell$,
* •
the directed cubes of $\varkappa(\mathbf{T}_{0})$ corresponding to the flips
between $T_{0,i-1}$ and $T_{0,i}$ for $i=1,\dots,\binom{n-1}{2}$ share their
bottom vertex with $T_{0,0}$,
* •
for $i=1,\dots,\binom{n}{3}-\binom{n-1}{2}$, $T_{0,j}=\cdots=T_{i-1,j}$ for
$j=i+\binom{n-1}{2},\dots,\binom{n}{3}$,
* •
for $i=1,\dots,\binom{n}{3}-\binom{n-1}{2}$, the directed cube of
$\varkappa(\mathbf{T}_{i-1})$ corresponding to the flip between
$T_{i-1,i-1+\binom{n}{3}-\binom{n-1}{2}}$ and
$T_{i-1,i-1+\binom{n}{3}-\binom{n-1}{2}}$ is the top of the four cubes of
$\varkappa(\mathbf{T}_{i-1})$ involved in the flip between
$\varkappa(\mathbf{T}_{i-1})$ and $\varkappa(\mathbf{T}_{i})$.
###### Remark 9.7.
The idea behind Lemma 9.6 is as follows. Let $C_{i}$ be the cube of
$\varkappa(\mathbf{T}_{0})$ corresponding to the flip between $T_{0,i-1}$ and
$T_{0,i}$. The second bullet point states that the cubes
$C_{1},\dots,C_{\binom{n-1}{2}}$ share their bottom vertex with $T_{0,0}$, and
hence do not have their bottom vertices in the interior of
$\varkappa(\mathbf{T}_{0})$. As a consequence of the remaining bullet points,
there is a sequence of $\binom{n}{3}-\binom{n-1}{2}$ flips on the directed
cubical complex $\varkappa(\mathbf{T}_{0})$ in which
$C_{\binom{n-1}{2}+1},\dots,C_{\binom{n}{3}}$ (in that order) are the top
cubes involved in the flips.
###### Proof of Lemma 9.6.
Define a total order $<_{\textup{lex}}$ on $\binom{[n]}{k}$, where
$\\{i_{1}<\cdots<i_{k}\\}<_{\textup{lex}}\\{i_{1}^{\prime}<\cdots<i_{k}^{\prime}\\}$
when there exists $j$ such that $i_{\ell}=i_{\ell}^{\prime}$ for $\ell<j$, and
$i_{j}<i_{j}^{\prime}$. Set
$\\{\alpha_{1}<_{\textup{lex}}\cdots<_{\textup{lex}}\alpha_{\binom{n}{3}}\\}=\binom{[n]}{3}$.
Note that the permutation
$\sigma_{0}=(\alpha_{1},\dots,\alpha_{\binom{n}{3}})$ of $\binom{[n]}{3}$ is
admissible (see Definition 3.23). Let $\mathbf{T}_{0}$ be the pile
corresponding to $(\alpha_{1},\dots,\alpha_{\binom{n}{3}})$ (see Theorem
3.25). Note that $1\in\alpha_{i}$ for $i=1,\dots,\binom{n-1}{2}$, so the
second bullet point holds.
We now construct the piles
$\mathbf{T}_{1},\dots,\mathbf{T}_{\binom{n}{3}-\binom{n-1}{2}}$ inductively as
follows. For $i=1,\dots,\binom{n}{3}-\binom{n-1}{2}$, the admissible
permutation $\sigma_{i}$ corresponding to $\mathbf{T}_{i}$ should have the
following properties:
* •
The inversion set of $\sigma_{i}$ is
$\\{\\{1\\}\cup\alpha_{\binom{n-1}{2}+1},\dots,\\{1\\}\cup\alpha_{\binom{n-1}{2}+i}\\}$.
Hence, the inversion sets of $\sigma_{i-1}$ and $\sigma_{i}$ differ by the
element $\\{1\\}\cup\alpha_{\binom{n-1}{2}+i}$, so
$\varkappa(\mathbf{T}_{i-1})$ and $\varkappa(\mathbf{T}_{i})$ are related by a
flip. Hence, the first bullet point holds.
* •
Writing $\sigma_{i-1}=(\beta_{1},\dots,\beta_{\binom{n}{3}})$,
$\beta_{j}=\alpha_{j}$ for $j=i+\binom{n-1}{2},\dots,\binom{n}{3}$. Hence, the
third bullet point holds. Because
$\beta_{i+\binom{n-1}{2}}=\alpha_{i+\binom{n-1}{2}}$ and the flip between
$\varkappa(\mathbf{T}_{i-1})$ and $\varkappa(\mathbf{T}_{i})$ consists of the
inclusion of $\\{1\\}\cup\alpha_{i+\binom{n-1}{2}}$ to the inversion set, the
fourth bullet point follows.
For $i=1,\dots,\binom{n}{3}-\binom{n-1}{2}$, write
$\sigma_{i-1}=(\beta_{1},\dots,\beta_{\binom{n}{3}})$. We want to obtain
$\sigma_{i}$. Write
$\alpha_{i+\binom{n-1}{2}}=\beta_{i+\binom{n-1}{2}}=\\{i_{1}<i_{2}<i_{3}\\}$.
Let $(\gamma_{1},\dots,\gamma_{i+\binom{n-1}{2}-4})$ be the subsequence of
$(\beta_{1},\dots,\beta_{i+\binom{n-1}{2}-4})$ excluding
$\\{1,i_{1},i_{2}\\}$, $\\{1,i_{1},i_{3}\\}$, and $\\{1,i_{2},i_{3}\\}$.
Setting
$\displaystyle\sigma_{i}=\big{(}\gamma_{1},\dots,\gamma_{i+\binom{n-1}{2}-4},\\{i_{1},i_{2},i_{3}\\},\\{1,i_{2},i_{3}\\},\\{1,i_{1},i_{3}\\},\\{1,i_{1},i_{2}\\},\beta_{i+\binom{n-1}{2}},\dots,\beta_{\binom{n}{3}}\big{)},$
it is straightforward to check that $\sigma_{i}$ is an admissible permutation
with the desired properties. ∎
###### Remark 9.8.
The pile $\mathbf{T}_{0}$ constructed in the proof of Lemma 9.6 is a
representative for the smallest element of the third higher Bruhat order. The
sequence
$\big{(}\varkappa(\mathbf{T}_{0}),\dots,\varkappa\big{(}\mathbf{T}_{\binom{n}{3}-\binom{n-1}{2}}\big{)}\big{)}$,
where $\mathbf{T}_{0},\dots,\mathbf{T}_{\binom{n}{3}-\binom{n-1}{2}}$ are the
piles constructed in the proof of Lemma 9.6, are the first
$\binom{n}{3}-\binom{n-1}{2}+1$ elements for a representative for the smallest
element of the fourth higher Bruhat order. See [9] or [12] for further
discussion of higher Bruhat orders.
###### Lemma 9.9.
Let $\mathbf{\bar{x}}=(x_{I})_{I\subseteq[n]}$ be an array satisfying the
conditions that $L_{I,\\{i,j\\}}\not=0$ for any $I\subseteq[n]$ and distinct
$i,j\in[n]$, and $x_{\varnothing}=1$. Suppose that for all $I\subseteq[n]$ and
distinct $i,j,k\in[n]$, equation (4.8) holds, and for all $I\subseteq[n]$ and
$A\in\binom{[n]}{4}$, equation (4.10) holds. Then there exists
$\mathbf{T}\in\mathcal{C}(n)$ such that
$\mathbf{x}_{\varkappa(\mathbf{T})}(\mathbf{\bar{x}})$ is a coherent solution
of the Kashaev equation.
###### Proof.
Let
$\mathbf{T}_{0},\dots,\mathbf{T}_{\binom{n}{3}-\binom{n-1}{2}}\in\mathcal{C}(n)$
be a sequence of piles satisfying the conditions of Lemma 9.6, where we write
$\mathbf{T}_{i}=(T_{i,0},\dots,T_{i,\binom{n}{3}})$ for
$i=0,\dots,\binom{n}{3}-\binom{n-1}{2}$. We will show that
$\mathbf{x}_{\varkappa(\mathbf{T}_{0})}(\mathbf{\bar{x}})$ is a coherent
solution of the Kashaev equation.
We claim that
$\mathbf{x}_{\varkappa(T_{0,0},\dots,T_{0,j})}(\mathbf{\bar{x}})$ is a
coherent solution of the Kashaev equation for $j=1,\dots,\binom{n}{3}$ and
proceed by induction. Because equation (4.8) holds for all $I\subseteq[n]$ and
distinct $i,j,k\in[n]$,
$\mathbf{x}_{\varkappa(T_{0,0},\dots,T_{0,j})}(\mathbf{\bar{x}})$ satisfies
the Kashaev equation. Hence, we only have to check coherence, i.e., we need to
check that equation (4.3) holds for every interior vertex of
$\varkappa(T_{0,0},\dots,T_{0,j})$. For $j=1,\dots,\binom{n-1}{2}$, none of
the vertices $\varkappa^{0}(T_{0,0},\dots,T_{0,j})$ are interior vertices of
$\varkappa(T_{0,0},\dots,T_{0,j})$, so
$\mathbf{x}_{\varkappa(T_{0,0},\dots,T_{0,j})}(\mathbf{\bar{x}})$ is a
coherent solution of the Kashaev equation. By our inductive hypothesis,
$\mathbf{x}_{\varkappa(T_{0,0},\dots,T_{0,j-1})}(\mathbf{\bar{x}})$ is a
coherent solution of the Kashaev equation. By construction,
$\varkappa(T_{i-1,0},\dots,T_{i-1,j-1})$ and
$\varkappa(T_{i,0},\dots,T_{i,j-1})$ are related by a flip for
$i=1,\dots,j-\binom{n-1}{2}-1$. Hence, the pairs
$\big{(}\mathbf{x}_{\varkappa(T_{i-1,0},\dots,T_{i-1,j-1})}(\mathbf{\bar{x}}),\mathbf{x}_{\varkappa(T_{i,0},\dots,T_{i,j-1})}(\mathbf{\bar{x}})\big{)}$
are K-flipped for $i=1,\dots,j-\binom{n-1}{2}-1$ by the conditions of the
lemma. By repeated applications of Lemma 9.5(a), it follows that
$\mathbf{x}_{\varkappa\big{(}T_{j-\binom{n-1}{2}-1,0},\dots,T_{j-\binom{n-1}{2}-1,j-1}\big{)}}(\mathbf{\bar{x}})$
is a coherent solution of the Kashaev equation. By construction, the cube of
$\varkappa\big{(}T_{j-\binom{n-1}{2}-1,0},\dots,T_{j-\binom{n-1}{2}-1,j}\big{)}$
corresponding to the flip between $T_{j-\binom{n-1}{2}-1,j-1}$ and
$T_{j-\binom{n-1}{2}-1,j}$ is the top of four cubes where a flip can take
place. Hence, because equation (4.10) holds for all $I\subseteq[n]$ and
$A\in\binom{[n]}{4}$,
$\mathbf{x}_{\varkappa\big{(}T_{j-\binom{n-1}{2}-1,0},\dots,T_{j-\binom{n-1}{2}-1,j}\big{)}}(\mathbf{\bar{x}})$
is a coherent solution of the Kashaev equation. By construction,
$\varkappa(T_{i-1,0},\dots,T_{i-1,j})$ and
$\varkappa(T_{i-1,0},\dots,T_{i-1,j})$ are related by a flip for
$i=1,\dots,j-\binom{n-1}{2}-1$, so the pairs
$\big{(}\mathbf{x}_{\varkappa(T_{i-1,0},\dots,T_{i-1,j})}(\mathbf{\bar{x}}),\mathbf{x}_{\varkappa(T_{i,0},\dots,T_{i,j})}(\mathbf{\bar{x}})\big{)}$
are K-flipped for $i=1,\dots,j-\binom{n-1}{2}-1$. Thus, by repeated
applications of Lemma 9.5(a), it follows that
$\mathbf{x}_{\varkappa(T_{0,0},\dots,T_{0,j})}(\mathbf{\bar{x}})$ is a
coherent solution of the Kashaev equation. ∎
###### Proof of Theorem 4.26..
By Corollary 4.23, the first two bullet points are equivalent, and by
Corollary 4.23 and Proposition 9.2, the first bullet point implies the third
bullet point. Hence, we just need to show that the third bullet point implies
the first.
Suppose that the third bullet point holds. By Lemma 9.9, there exists
$\mathbf{T}\in\mathcal{C}(n)$ such that
$\mathbf{x}_{\varkappa(\mathbf{T})}(\mathbf{\bar{x}})$ is a coherent solution
of the Kashaev equation. Hence, by Theorem 4.19, there exists an extension
$\mathbf{\tilde{x}}\in(\operatorname{\mathbb{C}}^{*})^{\varkappa^{02}(\mathbf{T})}$
of $\mathbf{x}_{\varkappa(\mathbf{T})}(\mathbf{\bar{x}})$ to
$\varkappa^{02}(\mathbf{T})$ that satisfies the K-hexahedron equations. Hence,
there exists a symmetric matrix $M$ such that
$\mathbf{\tilde{x}}=\mathbf{\tilde{x}}_{\varkappa(\mathbf{T})}(M)$. Let
$\mathbf{\tilde{x}}_{\textup{init}}$ be the restriction of
$\mathbf{\tilde{x}}$ to $T_{\min,n}$.
Given any $\mathbf{T}^{\prime}\in\mathcal{C}(n)$, by Proposition 3.22, there
exists a sequence of piles $\mathbf{T}_{0},\dots,\mathbf{T}_{\ell}$ with
$\mathbf{T}=\mathbf{T}_{0}$ and $\mathbf{T}^{\prime}=\mathbf{T}_{\ell}$ such
that $\varkappa(\mathbf{T}_{i-1})$ and $\varkappa(\mathbf{T}_{i})$ are related
by a flip for $i=1,\dots,\ell$. Hence,
$(\mathbf{x}_{\varkappa(\mathbf{T}_{i-1})}(\mathbf{\bar{x}}),\mathbf{x}_{\varkappa(\mathbf{T}_{i})}(\mathbf{\bar{x}}))$
is K-flipped, so by repeated applications of Lemma 9.5(a) and (b),
$(\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\varkappa^{02}(\mathbf{T}^{\prime})}$
restricts to $\mathbf{x}_{\varkappa(\mathbf{T}^{\prime})}(\mathbf{\bar{x}})$.
Because every $I\subseteq[n]$ labels a vertex in
$\varkappa(\mathbf{T}^{\prime})$ for some
$\mathbf{T}^{\prime}\in\mathcal{C}(n)$,
$\mathbf{\bar{x}}=\mathbf{\bar{x}}(M)$, as desired. ∎
## 10 Generalizations of the Kashaev equation
In this section, we describe an axiomatic setup for equations similar to the
Kashaev equation and the examples from Sections 5–6. This allows us to prove
all of the results from Sections 5–6. This section is organized as follows:
* •
Proposition 10.2 and Lemma 10.8 generalize Propositions 2.8, 5.4, 6.1, and
6.7.
* •
In Definition 10.11, given a polynomial equation resembling the Kashaev
equation, we describe how to obtain a set of equations with the same
properties as the K-hexahedron equations.
* •
In Definition 10.18, we define certain signs that appear in our generalized
“coherence” equation (10.15).
* •
Theorem 10.24, the main result in this section, generalizes Theorems 2.22,
5.9, 6.4, and 6.10. The proof of Theorem 10.24 is nearly identical to the
proof of Theorem 2.22 from Section 7.
###### Definition 10.1.
For $d\geq 1$ and
$\mathbf{a}=(a_{1},\dots,a_{d})\in\operatorname{\mathbb{Z}}^{d}$, we denote by
$\displaystyle[\mathbf{a}]=\big{\\{}(b_{1},\dots,b_{d})\in\operatorname{\mathbb{Z}}^{d}\colon
0\leq|b_{i}|\leq|a_{i}|\text{ and }a_{i}b_{i}\geq 0\text{ for all
$i$}\big{\\}}$
the set of integer points in the
$\left|a_{1}\right|\times\cdots\times\left|a_{d}\right|$ box with opposite
vertices $(0,\dots,0)$ and $(a_{1},\dots,a_{d})$. For
$\mathbf{b}=(b_{1},\dots,b_{d}),\mathbf{c}=(c_{1},\dots,c_{d})\in\operatorname{\mathbb{Z}}^{d}$,
we write
$\mathbf{b}\odot\mathbf{c}=(b_{1}c_{1},\dots,b_{d}c_{d})\in\operatorname{\mathbb{Z}}^{d}$.
Denote by $\mathbf{1}=(1,\dots,1)\in\operatorname{\mathbb{Z}}^{d}$ the all
$1$’s vector, and set
$\mathbf{1}_{i}=(0,\dots,0,1,0,\dots,0)\in\operatorname{\mathbb{Z}}^{d}$, with
$1$ in the $i$th place, for $i=1,\dots,d$. Let
$\displaystyle\mathbf{z}_{[\mathbf{a}]}=\\{z_{\mathbf{i}}\colon\mathbf{i}\in[\mathbf{a}]\\}$
be a set of indeterminates. For $i=1,\dots,d$, let
$\pi_{\mathbf{a},i}\colon\mathbf{z}_{[\mathbf{a}]}\rightarrow\mathbf{z}_{[\mathbf{a}]}$
be the involution defined by
$\displaystyle z_{(j_{1},\dots,j_{d})}\mapsto
z_{(j_{1},\dots,a_{i}-j_{i},\dots,j_{d})},$
i.e., we “flip” the index of each variable in its $i$th coordinate. The action
of $\pi_{\mathbf{a},i}$ extends from $\mathbf{z}_{[\mathbf{a}]}$ to the
polynomial ring $\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$. Given
an array
$\mathbf{x}=(x_{s})\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}}$
and integer vectors $v\in\operatorname{\mathbb{Z}}^{d}$,
$\mathbf{a}\in\operatorname{\mathbb{Z}}_{\geq 0}^{d}$, and
$\bm{\alpha}\in\\{-1,1\\}^{d}$, we denote by
$\mathbf{x}_{v+[\mathbf{a}\odot\bm{\alpha}]}\in\operatorname{\mathbb{C}}^{[\mathbf{a}]}$
the array whose entries are
$\displaystyle(\mathbf{x}_{v+[\mathbf{a}\odot\bm{\alpha}]})_{\mathbf{i}}=x_{v+\mathbf{i}\odot\bm{\alpha}},\qquad\text{for}\quad\mathbf{i}\in[\mathbf{a}].$
In particular,
$\displaystyle(\mathbf{x}_{v+[\mathbf{a}]})_{\mathbf{i}}=x_{v+\mathbf{i}},\qquad\text{for}\quad\mathbf{i}\in[\mathbf{a}].$
Thus, given a polynomial
$f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$, the number
$f(\mathbf{x}_{v+[\bm{\alpha}\odot\mathbf{a}]})\in\operatorname{\mathbb{C}}$
is obtained by setting $z_{\mathbf{i}}=x_{v+\bm{\alpha}\odot\mathbf{i}}$ for
each variable $z_{\mathbf{i}}$ for $\mathbf{i}\in[\mathbf{a}]$. We say that
$\mathbf{x}\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}}$
_satisfies $f$_ if $f(\mathbf{x}_{v+[\mathbf{a}]})=0$ for all
$v\in\operatorname{\mathbb{Z}}^{d}$.
###### Proposition 10.2.
Let $\mathbf{a}=(a_{1},\dots,a_{d})\in\operatorname{\mathbb{Z}}^{d}_{\geq 1}$,
and a polynomial $f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$
satisfy the following conditions:
1. $(1)$
$f$ is invariant under the action of $\pi_{\mathbf{a},i}$ for $i=1,\dots,d$,
2. $(2)$
$f$ has degree $2$ with respect to the variable $z_{\mathbf{a}}$; as a
quadratic polynomial in $z_{\mathbf{a}}$, $f$ has discriminant $D$ which
factors as a product $D=f_{1}\cdots f_{d}$, where each polynomial
$f_{i}\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}-\mathbf{1}_{i}]}]$
is invariant under the action of $\pi_{\mathbf{a}-\mathbf{1}_{i},j}$ for
$j=1,\dots,d$.
Then for any
$\mathbf{x}=(x_{s})\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}}$
satisfying $f$, we have, for all $v\in\operatorname{\mathbb{Z}}^{d}$:
$\displaystyle\left(\prod_{\bm{\alpha}\in\\{-1,0\\}^{d}}\\!\frac{\partial
f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]})\right)^{2}=\left(\prod_{i=1}^{d}\prod_{\begin{subarray}{c}\bm{\beta}=(\beta_{1},\dots,\beta_{d})\in\\{-1,0\\}^{d}\\\
\beta_{i}=0\end{subarray}}\\!\\!f_{i}(\mathbf{x}_{v+\bm{\beta}+[\mathbf{a}]})\right)^{2}\\!\\!.\\!\\!\\!$
(10.1)
Moreover, for all $v\in Z^{d}$, we have
$\displaystyle\left(\prod_{\begin{subarray}{c}\bm{\alpha}=(\alpha_{1},\dots,\alpha_{d})\\\
\alpha_{1},\dots,\alpha_{d}\in\\{-1,0\\}\\\ \alpha_{1}+\cdots+\alpha_{d}\text{
even}\end{subarray}}\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]})\right)^{2}$
$\displaystyle\qquad{}=\left(\prod_{\begin{subarray}{c}\bm{\alpha}=(\alpha_{1},\dots,\alpha_{d})\\\
\alpha_{1},\dots,\alpha_{d}\in\\{-1,0\\}\\\ \alpha_{1}+\cdots+\alpha_{d}\text{
odd}\end{subarray}}\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]})\right)^{2}$
$\displaystyle\qquad{}=\prod_{i=1}^{d}\prod_{\begin{subarray}{c}\bm{\beta}=(\beta_{1},\dots,\beta_{d})\in\\{-1,0\\}^{d}\\\
\beta_{i}=0\end{subarray}}f_{i}(\mathbf{x}_{v+\bm{\beta}+[\mathbf{a}]}).$
(10.2)
###### Remark 10.3.
The subscripts
$v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]$
appearing on the left-hand side of (10.1) run over all boxes of size
$a_{1}\times\cdots\times a_{d}$ containing $v+[\mathbf{a}-\mathbf{1}]$. The
subscripts
$v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]$
appearing on the right-hand side of (10.1) run over $i=1,\dots,d$ and boxes of
size $a_{1}\times\cdots\times a_{i-1}\times(a_{i}-1)\times
a_{i+1}\times\cdots\times a_{d}$ containing $v+[\mathbf{a}-\mathbf{1}]$. In
particular, when $a_{1}=\cdots=a_{d}=1$, all of these products are over boxes
of a certain size containing the vertex $v$. For example, in the case
$\mathbf{a}=(1,2)$ (like in Proposition 6.7), the boxes we are considering on
the left-hand side of (10.1) are given in the top row of Fig. 19, while the
boxes we are considering on the right-hand side of (10.1) are given in the
bottom row of Fig. 19.
Before proving Proposition 10.2, we give several examples of polynomials
discussed in previous sections that satisfy conditions (1)–(2) from
Proposition 10.2. In the examples below, we write $z_{i_{1}\cdots
i_{d}}=z_{(i_{1},\dots,i_{d})}$ for
$(i_{1},\dots,i_{d})\in\operatorname{\mathbb{Z}}^{d}$.
###### Example 10.4.
Our first example is the Kashaev equation. Let
$\mathbf{a}=(1,1,1)\in\operatorname{\mathbb{Z}}^{3}$, and let
$\displaystyle
f=2\big{(}a^{2}+b^{2}+c^{2}+d^{2}\big{)}-(a+b+c+d)^{2}-4(s+t)\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}],$
where
$\displaystyle a=z_{000}z_{111},\qquad b=z_{100}z_{011},\qquad
c=z_{010}z_{101},\qquad d=z_{001}z_{110},$ $\displaystyle
s=z_{000}z_{011}z_{101}z_{110},\qquad t=z_{100}z_{010}z_{001}z_{111}.$
The polynomial $f$ is invariant not only under the action of the
$\pi_{\mathbf{a},i}$, but under all symmetries of the cube. Its discriminant
(as a polynomial in $z_{111}$) $D$ factors as a product $D=f_{1}f_{2}f_{3}$,
with
$\displaystyle f_{1}=16(z_{000}z_{011}+z_{010}z_{001}),\qquad
f_{2}=z_{000}z_{101}+z_{100}z_{001},\qquad
f_{3}=z_{000}z_{110}+z_{100}z_{010}.\ $
Hence, $f$ satisfies conditions (1)–(2) from Proposition 10.2. Therefore,
Proposition 2.8 is a special case of Proposition 10.2.
###### Example 10.5.
Let $\mathbf{a}=(1,1)\in\operatorname{\mathbb{Z}}^{2}$, and let
$\displaystyle
f=z_{00}^{2}+z_{10}^{2}+z_{01}^{2}+z_{11}^{2}-2(z_{00}z_{10}+z_{10}z_{11}+z_{11}z_{01}+z_{01}z_{00})$
$\displaystyle\hphantom{f=}{}-6(z_{00}z_{11}+z_{10}z_{01})\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}].$
The polynomial $f$ is invariant not only under the action of the
$\pi_{\mathbf{a},i}$, but under all symmetries of the square. Its discriminant
(as a polynomial in $z_{11}$) $D$ factors as a product $D=f_{1}f_{2}$, with
$\displaystyle f_{1}=32(z_{00}+z_{01}),\qquad f_{2}=z_{00}+z_{10}.$
Hence, $f$ satisfies conditions (1)–(2) from Proposition 10.2. Therefore,
Proposition 5.4 is a special case of Proposition 10.2.
###### Example 10.6.
Fix $\alpha_{1},\alpha_{2},\alpha_{3}\in\operatorname{\mathbb{C}}$. Let
$\mathbf{a}=(3)\in\operatorname{\mathbb{Z}}^{1}$, and let
$\displaystyle
f=z_{0}^{2}z_{3}^{2}+\alpha_{1}z_{1}^{2}z_{2}^{2}+\alpha_{2}z_{0}z_{1}z_{2}z_{3}+\alpha_{3}\big{(}z_{0}z_{2}^{3}+z_{1}^{3}z_{3}\big{)}\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}].$
The polynomial $f$ is invariant under the action of $\pi_{\mathbf{a},1}$,
i.e., replacing $z_{i}$ by $z_{3-i}$. Its discriminant (as a polynomial in
$z_{11}$) $D=f_{1}$, with
$\displaystyle
f_{1}=\alpha_{3}^{2}z_{1}^{6}+2\alpha_{2}\alpha_{3}z_{0}z_{1}^{4}z_{2}+\big{(}\alpha_{2}^{2}-4\alpha_{1}\big{)}z_{0}^{2}z_{1}^{2}z_{2}^{2}-4\alpha_{3}z_{0}^{3}z_{2}^{3}.$
Hence, $f$ satisfies conditions (1)–(2) from Proposition 10.2. Therefore,
Proposition 6.1 is a special case of Proposition 10.2.
###### Example 10.7.
Fix $\alpha_{1},\alpha_{2}\in\operatorname{\mathbb{C}}$. Let
$\mathbf{a}=(1,2)\in\operatorname{\mathbb{Z}}^{2}$, and let
$\displaystyle
f=z_{00}^{2}z_{12}^{2}+z_{10}^{2}z_{02}^{2}+\frac{\alpha_{2}^{2}-\alpha_{1}^{2}}{4}z_{01}^{2}z_{11}^{2}-\alpha_{1}\big{(}z_{00}z_{02}z_{11}^{2}+z_{10}z_{12}z_{01}^{2}\big{)}$
$\displaystyle\hphantom{f=}{}-2z_{00}z_{10}z_{02}z_{12}-\alpha_{2}(z_{00}z_{12}z_{01}z_{11}+z_{10}z_{02}z_{01}z_{11})\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}].$
The polynomial $f$ is invariant under the action of $\pi_{\mathbf{a},1}$ and
$\pi_{\mathbf{a},2}$. Its discriminant (as a polynomial in $z_{12}$) factors
as a product $D=f_{1}f_{2}$, with
$\displaystyle f_{1}=\alpha_{1}z_{01}^{2}+4z_{00}z_{02},\qquad
f_{2}=\alpha_{1}\big{(}z_{00}^{2}z_{11}^{2}+z_{01}^{2}z_{10}^{2}\big{)}+2\alpha_{2}z_{00}z_{01}z_{10}z_{11}.$
Hence, $f$ satisfies conditions (1)–(2) from Proposition 10.2. Therefore,
Proposition 6.7 is a special case of Proposition 10.2.
###### Proof of Proposition 10.2..
Let $g$ denote the coefficient of $z_{\mathbf{a}}^{2}$ in $f$ (viewed as a
polynomial in $z_{\mathbf{a}}$). It is easy to check that
$\displaystyle D=f_{1}\cdots f_{d}=\left(\frac{\partial f}{\partial
z_{\mathbf{a}}}\right)^{2}-4fg.$ (10.3)
Because $\mathbf{x}$ satisfies $f$, we have
$\displaystyle\left(\frac{\partial f}{\partial
z_{\mathbf{a}}}\right)^{2}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(1+2\bm{\alpha})\odot\mathbf{a}]})=(f_{1}\cdots
f_{d})(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(1+2\bm{\alpha})\odot\mathbf{a}]}).$
Because each $f_{i}$ is invariant under the action of
$\pi_{\mathbf{a}-\mathbf{1}_{i},j}$ for $j=1,\dots,d$, we have
$\displaystyle\left(\frac{\partial f}{\partial
z_{\mathbf{a}}}\right)^{2}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(1+2\bm{\alpha})\odot\mathbf{a}]})=\prod_{i=1}^{d}f_{i}(\mathbf{x}_{v+\bm{\alpha}\odot(\mathbf{1}-\mathbf{1}_{i})+[\mathbf{a}]}).$
(10.4)
Given $i\in\\{1,\dots,d\\}$ and
$\bm{\beta}=(\beta_{1},\dots,\beta_{d})\in\\{-1,0\\}^{d}$ with $\beta_{i}=0$,
there exist exactly two
$\bm{\alpha}=(\alpha_{1},\dots,\alpha_{d})\in\\{-1,0\\}^{d}$ with
$(\mathbf{1}-\mathbf{1}_{i})\odot\bm{\alpha}=\bm{\beta}$: one with
$\alpha_{i}=0$ and the other with $\alpha_{i}=1$. Hence,
$\alpha_{1}+\cdots+\alpha_{d}$ is even for one such choice of $\bm{\alpha}$,
and odd for the other. Taking the product over
$\bm{\alpha}=(\alpha_{1},\dots,\alpha_{d})\in\\{-1,0\\}^{d}$ with
$\alpha_{1}+\cdots+\alpha_{d}$ odd (or even) in (10.4), we obtain (10.2).
Equation (10.1) follows. ∎
###### Lemma 10.8.
Let $\mathbf{a}=(a_{1},\dots,a_{d})\in\operatorname{\mathbb{Z}}^{d}_{\geq 1}$,
and let $f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$ be a
polynomial satisfying conditions $(1)$–$(2)$ from Proposition 10.2. Let
$\mathbf{x}=(x_{s})\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}}$
be an array satisfying $f$. Fix $v\in\operatorname{\mathbb{Z}}^{d}$ and
$\gamma\in\\{-1,1\\}$. Then
$\displaystyle\prod_{\bm{\alpha}\in\\{-1,0\\}^{d}}\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]})=\gamma\prod_{i=1}^{d}\prod_{\begin{subarray}{c}\bm{\beta}=(\beta_{1},\dots,\beta_{d})\in\\{-1,0\\}^{d}\\\
\beta_{i}=0\end{subarray}}f_{i}(\mathbf{x}_{v+\bm{\beta}+[\mathbf{a}]})$
if and only if
$\displaystyle\prod_{\begin{subarray}{c}\bm{\alpha}=(\alpha_{1},\dots,\alpha_{d})\\\
\alpha_{1},\dots,\alpha_{d}\in\\{-1,0\\}\\\ \alpha_{1}+\cdots+\alpha_{d}\text{
even}\end{subarray}}\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]})=\gamma\prod_{\begin{subarray}{c}\bm{\alpha}=(\alpha_{1},\dots,\alpha_{d})\\\
\alpha_{1},\dots,\alpha_{d}\in\\{-1,0\\}\\\ \alpha_{1}+\cdots+\alpha_{d}\text{
odd}\end{subarray}}\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]}).$
###### Proof.
This follows immediately from Proposition 10.2. ∎
###### Definition 10.9.
Given $\mathbf{a}=(a_{1},\dots,a_{d})\in\operatorname{\mathbb{Z}}^{d}_{\geq
1}$ and $1\leq i\leq d$, let
$\displaystyle F^{\mathbf{a}}_{i}=\big{\\{}v+[\mathbf{a}-\mathbf{1}_{i}]\colon
v\in\operatorname{\mathbb{Z}}^{d}\big{\\}}$
denote the set of boxes of size $a_{1}\times\cdots\times
a_{i-1}\times(a_{i}-1)\times a_{i+1}\times\cdots\times a_{d}$ in
$\operatorname{\mathbb{Z}}^{d}$. Set
$\displaystyle F^{\mathbf{a}}=\bigcup_{i=1}^{d}F^{\mathbf{a}}_{i}.$
Given an array
$\mathbf{\tilde{x}}=(x_{s})_{s\in\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ with
$\mathbf{x}=(x_{s})_{s\in\operatorname{\mathbb{Z}}^{d}}$,
$i\in\\{1,\dots,d\\}$, and $v\in\operatorname{\mathbb{Z}}^{d}$, we remark that
$\mathbf{x}_{v+[\mathbf{a}-\mathbf{1}_{i}]}$ (with $\mathbf{x}$ bold) refers
to the array defined in Definition 10.1, whereas
$x_{v+[\mathbf{a}-\mathbf{1}_{i}]}$ (with $x$ not bold) refers to the
component of $\mathbf{\tilde{x}}$ indexed by $v+[\mathbf{a}-\mathbf{1}_{i}]\in
F^{\mathbf{a}}$.
###### Definition 10.10.
For $\mathbf{a}\in\operatorname{\mathbb{Z}}_{\geq 1}^{d}$, define
$[\mathbf{a}^{*}]$ by
$\displaystyle[\mathbf{a}^{*}]=\left([\mathbf{a}]\setminus\\{\mathbf{a}\\}\right)\cup\bigcup_{i=1}^{d}\\{[\mathbf{a}-\mathbf{1}_{i}]\\}.$
In other words, the set $[\mathbf{a}^{*}]$ consists of
$[\mathbf{a}]\setminus\\{\mathbf{a}\\}\subset\operatorname{\mathbb{Z}}^{d}$,
along with the sets $[\mathbf{a}-\mathbf{1}_{i}]\in F^{\mathbf{a}}$ for
$i=1,\dots,d$.
We want to develop a generalization of the K-hexahedron equations for arrays
indexed by $\operatorname{\mathbb{Z}}^{d}\cup F^{\mathbf{a}}$. Suppose that
$f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$ is a polynomial
satisfying conditions (1)–(2) from Proposition 10.2, with the polynomials
$f_{1},\dots,f_{d}$ from condition (2) fixed. Let $g$ and $h$ be the
coefficients of $z_{\mathbf{a}}^{2}$ and $z_{\mathbf{a}}$ in $f$, viewed as a
polynomial in $z_{\mathbf{a}}$. We consider arrays
$\mathbf{\tilde{x}}=(x_{s})\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ such that
$\displaystyle
x_{v+\mathbf{a}}=\frac{-h(\mathbf{x}_{v+[\mathbf{a}]})+\prod\limits_{i=1}^{d}x_{v+[\mathbf{a}-\mathbf{1}_{i}]}}{2g(\mathbf{x}_{v+[\mathbf{a}]})}\qquad\text{for
all $v\in\operatorname{\mathbb{Z}}^{d}$},$ (10.5) $\displaystyle
x_{v+\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}=r_{i}(x_{v+s}\colon
s\in[\mathbf{a}^{*}])\qquad\text{for $i=1,\dots,d$}\text{ and for all
$v\in\operatorname{\mathbb{Z}}^{d}$},$ (10.6) $\displaystyle
x_{v+[\mathbf{a}-\mathbf{1}_{i}]}^{2}=f_{i}(\mathbf{x}_{v+[\mathbf{a}-\mathbf{1}_{i}]})\qquad\text{for
$i=1,\dots,d$}\text{ and for all $v\in\operatorname{\mathbb{Z}}^{d}$},$ (10.7)
where $r_{1},\dots,r_{d}$ are some rational functions in the variables $z_{s}$
for $s\in[\mathbf{a}^{*}]$. Note that if $\mathbf{\tilde{x}}$ satisfies
conditions (10.5) and (10.7), then by the quadratic formula, its restriction
$\mathbf{x}=(x_{s})_{s\in\operatorname{\mathbb{Z}}^{d}}$ satisfies $f$. In the
following definition, we formulate the properties that our tuple of rational
functions $(r_{1},\dots,r_{d})$ should have in order for the subsequent
developments to follow.
###### Definition 10.11.
Let $\mathbf{a}=(a_{1},\dots,a_{d})\in\operatorname{\mathbb{Z}}^{d}_{\geq 1}$,
and let $f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$ be a
polynomial satisfying conditions (1)–(2) from Proposition 10.2. Fix the
polynomials $f_{1},\dots,f_{d}$ from condition (2) of Proposition 10.2. Let
$g$ be the coefficient of $z_{\mathbf{a}}^{2}$ in $f$, viewed as a polynomial
in $z_{\mathbf{a}}$. For $i=1,\dots,d$, let $r_{i}=\frac{p_{i}}{q_{i}}$ be
rational functions in the variables $z_{s}$ for $s\in[\mathbf{a}^{*}]$, with
$p_{i}$, $q_{i}$ polynomials in these variables. We say that
$(r_{1},\dots,r_{d})$ is _adapted to $(f;f_{1},\dots,f_{d})$_ if there exist
signs $\beta_{1},\dots,\beta_{d}\in\\{-1,1\\}$ such that the following
properties hold for $i=1,\dots,d$:
* •
the denominator $q_{i}$ of $r_{i}$ is of the form
$\displaystyle
q_{i}=g^{b_{i}}\prod_{j\in\\{1,\dots,d\\}\setminus\\{i\\}}z_{[\mathbf{a}-\mathbf{1}_{j}]}^{b_{ij}},$
where $b_{i}\in\operatorname{\mathbb{Z}}_{\geq 0}$ and $b_{ij}\in\\{0,1\\}$,
* •
for all arrays
$\mathbf{\tilde{x}}=(x_{s})\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ satisfying (10.5), (10.7), and
$\displaystyle q_{i}(x_{v+s}\colon s\in[\mathbf{a}^{*}])\not=0\qquad\text{for
all $v\in\operatorname{\mathbb{Z}}^{d}$},$ (10.8) $\displaystyle
g(\mathbf{x}_{v+[\mathbf{a}]})\not=0\qquad\text{for all
$v\in\operatorname{\mathbb{Z}}^{d}$},$ (10.9)
the following condition holds:
$\displaystyle r_{i}(x_{v+s}\colon
s\in[\mathbf{a}^{*}])=\beta_{i}\frac{\left(\frac{\partial f}{\partial
z_{\mathbf{a}}}\right)(\mathbf{x}_{v+a_{i}\mathbf{1}_{i}+[\mathbf{a}\odot(\mathbf{1}-2\mathbf{1}_{i})]})}{\prod\limits_{j\in\\{1,\dots,d\\}-\\{i\\}}x_{v+[\mathbf{a}-\mathbf{1}_{j}]}}.$
(10.10)
Note that one can obtain a tuple $(r_{1},\dots,r_{d})$ adapted to
$(f;f_{1},\dots,f_{d})$ by choosing the signs
$\beta_{1},\dots,\beta_{d}\in\\{-1,1\\}$ and using condition (10.5) to replace
all instances of $x_{v+\mathbf{a}}$ in (10.10).
In the following proposition, we show that with $(r_{1},\dots,r_{d})$ adapted
to $(f;f_{1},\dots,f_{d})$, the recurrence (10.5)–(10.6) “propagates” the
condition (10.7).
###### Proposition 10.12.
Let $\mathbf{a}=(a_{1},\dots,a_{d})\in\operatorname{\mathbb{Z}}^{d}_{\geq 1}$,
and let $f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$ be a
polynomial satisfying conditions $(1)$–$(2)$ from Proposition 10.2. Fix the
polynomials $f_{1},\dots,f_{d}$ from condition $(2)$ of Proposition 10.2. Let
$g$ be the coefficient of $z_{\mathbf{a}}^{2}$ in $f$, viewed as a polynomial
in $z_{\mathbf{a}}$. Let $(r_{1},\dots,r_{d})$ be a $d$-tuple of rational
functions in the variables $z_{s}$ for $s\in[\mathbf{a}^{*}]$ adapted to
$(f;f_{1},\dots,f_{d})$. Fix $v\in\operatorname{\mathbb{Z}}^{d}$. Let
$\mathbf{\tilde{x}}=(x_{s})\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ be an array satisfying conditions (10.5)–(10.6),
(10.8)–(10.9), and
$\displaystyle
x_{v+[\mathbf{a}-\mathbf{1}_{i}]}^{2}=f_{i}(\mathbf{x}_{v+[\mathbf{a}-\mathbf{1}_{i}]})\qquad\text{for
$i=1,\dots,d$}.$
Then
$\displaystyle
x_{v+\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}^{2}=f_{i}(\mathbf{x}_{v+\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]})\qquad\text{for
$i=1,\dots,d$}.$
###### Proof.
By identity (10.3),
$\displaystyle\left(\frac{\partial f}{\partial
z_{\mathbf{a}}}\right)^{2}(\mathbf{x}_{v+a_{i}\mathbf{1}_{i}+[\mathbf{a}\odot(\mathbf{1}-2\mathbf{1}_{i})]})=(f_{1}\cdots
f_{d})(\mathbf{x}_{v+a_{i}\mathbf{1}_{i}+[\mathbf{a}\odot(\mathbf{1}-2\mathbf{1}_{i})]})$
$\displaystyle\hphantom{\left(\frac{\partial f}{\partial
z_{\mathbf{a}}}\right)^{2}(\mathbf{x}_{v+a_{i}\mathbf{1}_{i}+[\mathbf{a}\odot(\mathbf{1}-2\mathbf{1}_{i})]})}{}=f_{i}(\mathbf{x}_{v+\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]})\prod_{j\in\\{1,\dots,d\\}-\\{i\\}}f_{j}(\mathbf{x}_{v+[\mathbf{a}-\mathbf{1}_{j}]}).$
Hence,
$\displaystyle
x_{v+\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}^{2}=\beta_{i}^{2}\frac{\left(\frac{\partial
f}{\partial
z_{\mathbf{a}}}\right)^{2}(\mathbf{x}_{v+a_{i}\mathbf{1}_{i}+[\mathbf{a}\odot(\mathbf{1}-2\mathbf{1}_{i})]})}{\prod\limits_{j\in\\{1,\dots,d\\}-\\{i\\}}x_{v+[\mathbf{a}-\mathbf{1}_{j}]}^{2}}$
$\displaystyle\hphantom{x_{v+\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}^{2}}{}=\frac{f_{i}(\mathbf{x}_{v+\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]})\prod\limits_{j\in\\{1,\dots,d\\}-\\{i\\}}f_{j}(\mathbf{x}_{v+[\mathbf{a}-\mathbf{1}_{j}]})}{\prod\limits_{j\in\\{1,\dots,d\\}-\\{i\\}}f_{j}(\mathbf{x}_{v+[\mathbf{a}-\mathbf{1}_{j}]})}=f_{i}(\mathbf{x}_{v+\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}),$
as desired. ∎
We now describe $d$-tuples of rational functions adapted to
$(f;f_{1},\dots,f_{d})$ for the four polynomials $f$ in the Examples
10.4–10.7.
###### Example 10.13.
Continuing with Example 10.4, let us write
$\displaystyle
z_{i_{1}\left(i_{2}+\operatorname{\frac{1}{2}}\right)\left(i_{3}+\operatorname{\frac{1}{2}}\right)}=z_{(i_{1},i_{2},i_{3})+[\mathbf{a}-\mathbf{1}_{1}]},\qquad
z_{\left(i_{1}+\operatorname{\frac{1}{2}}\right)i_{2}\left(i_{3}+\operatorname{\frac{1}{2}}\right)}=z_{(i_{1},i_{2},i_{3})+[\mathbf{a}-\mathbf{1}_{2}]},$
$\displaystyle
z_{\left(i_{1}+\operatorname{\frac{1}{2}}\right)\left(i_{2}+\operatorname{\frac{1}{2}}\right)i_{3}}=z_{(i_{1},i_{2},i_{3})+[\mathbf{a}-\mathbf{1}_{3}]}.$
Set
$\displaystyle r_{1}(z_{s}\colon
s\in[\mathbf{a}^{*}])=\frac{4z_{\frac{1}{2}0\frac{1}{2}}z_{\frac{1}{2}\frac{1}{2}0}+z_{0\frac{1}{2}\frac{1}{2}}z_{100}}{z_{000}},\qquad
r_{2}(z_{s}\colon
s\in[\mathbf{a}^{*}])=\frac{z_{0\frac{1}{2}\frac{1}{2}}z_{\frac{1}{2}\frac{1}{2}0}+4z_{\frac{1}{2}0\frac{1}{2}}z_{010}}{4z_{000}},$
$\displaystyle r_{3}(z_{s}\colon
s\in[\mathbf{a}^{*}])=\frac{z_{0\frac{1}{2}\frac{1}{2}}z_{\frac{1}{2}0\frac{1}{2}}+4z_{\frac{1}{2}\frac{1}{2}0}z_{001}}{4z_{000}}.$
It can be checked that $(r_{1},r_{2},r_{3})$ is adapted to
$(f;f_{1},f_{2},f_{3})$ by following the construction at the end of Definition
10.11 with $\beta_{1}=\beta_{2}=\beta_{3}=-1$ and using condition (10.7). Note
that $r_{1}$, $r_{2}$, $r_{3}$ matches the right-hand-sides of (2.15)–(2.17)
and the K-hexahedron equations (2.15)–(2.18) and (2.9) are the same as
conditions (10.5)–(10.7) if each $z_{v+[\mathbf{a}-\mathbf{1}_{1}]}$ for
$v\in\operatorname{\mathbb{Z}}^{3}$ is rescaled by a factor of $4$.
###### Example 10.14.
Continuing with Example 10.5, let us write
$\displaystyle
z_{i_{1}\left(i_{2}+\operatorname{\frac{1}{2}}\right)}=z_{(i_{1},i_{2})+[\mathbf{a}-\mathbf{1}_{1}]},\qquad
z_{\left(i_{1}+\operatorname{\frac{1}{2}}\right)i_{2}}=z_{(i_{1},i_{2})+[\mathbf{a}-\mathbf{1}_{2}]}.$
Set
$\displaystyle r_{1}(z_{s}\colon
s\in[\mathbf{a}^{*}])=z_{0\frac{1}{2}}+8z_{\frac{1}{2}0},\qquad
r_{2}(z_{s}\colon
s\in[\mathbf{a}^{*}])=z_{\frac{1}{2}0}+\frac{1}{4}z_{0\frac{1}{2}}.$
It can be checked that $(r_{1},r_{2})$ is adapted to $(f;f_{1},f_{2})$ by
following the construction at the end of Definition 10.11 with
$\beta_{1}=\beta_{2}=-1$ and using condition (10.7). Note that $r_{1}$,
$r_{2}$ matches the right-hand-sides of (5.8)–(5.9) and the conditions
(5.7)–(5.11) are the same as conditions (10.5)–(10.7) if each
$z_{v+[\mathbf{a}-\mathbf{1}_{1}]}$ for $v\in\operatorname{\mathbb{Z}}^{2}$ is
rescaled by a factor of $4\sqrt{2}$.
###### Example 10.15.
Continuing with Example 10.6, let us write
$\displaystyle w_{i+1}=z_{i+[\mathbf{a}-\mathbf{1}_{1}]}.$
Set
$\displaystyle r_{1}(z_{s}\colon
s\in[\mathbf{a}^{*}])=\frac{\alpha_{3}^{2}z_{1}^{6}+\alpha_{2}\alpha_{3}z_{0}z_{1}^{4}z_{2}+2\alpha_{3}z_{0}^{3}z_{2}^{3}+w_{1}^{2}+\big{(}{-}2\alpha_{3}z_{1}^{3}-\alpha_{2}z_{0}z_{1}z_{2}\big{)}w_{1}}{2z_{0}^{3}}.$
It can be checked that $(r_{1})$ is adapted to $(f;f_{1})$ by following the
construction at the end of Definition 10.11 with $\beta_{1}=1$ and using
condition (10.7). Note that $r_{1}$ matches the right-hand side of equation
(6.6), and conditions (6.5)–(6.7) are the same as conditions (10.5)–(10.7).
###### Example 10.16.
Continuing with Example 10.7, let us write
$\displaystyle
w_{i_{1}(i_{2}+1)}=z_{(i_{1},i_{2})+[\mathbf{a}-\mathbf{1}_{1}]},\qquad
w_{\left(i_{1}+\operatorname{\frac{1}{2}}\right)\left(i_{2}+\operatorname{\frac{1}{2}}\right)}=z_{(i_{1},i_{2})+[\mathbf{a}-\mathbf{1}_{2}]}.$
Set
$\displaystyle r_{1}(z_{s}\colon
s\in[\mathbf{a}^{*}])=\frac{z_{10}w_{01}+w_{\operatorname{\frac{1}{2}}\operatorname{\frac{1}{2}}}}{z_{00}},$
$\displaystyle r_{2}(z_{s}\colon
s\in[\mathbf{a}^{*}])=\frac{z_{01}(\alpha_{1}z_{01}z_{10}+\alpha_{2}z_{00}z_{11})w_{01}+\big{(}\alpha_{1}z_{01}^{2}+2z_{00}z_{02}\big{)}w_{\operatorname{\frac{1}{2}}\operatorname{\frac{1}{2}}}}{2z_{00}^{2}}.$
It can be checked that $(r_{1},r_{2})$ is adapted to $(f;f_{1},f_{2})$ by
following the construction at the end of Definition 10.11 with
$\beta_{1}=\beta_{2}=-1$ and using condition (10.7). Note that $r_{1}$,
$r_{2}$ matches the right-hand side of equation (6.15)–(6.16), and conditions
(6.14)–(6.18) are the same as conditions (10.5)–(10.7).
###### Lemma 10.17.
Let $\mathbf{a}=(a_{1},\dots,a_{d})\in\operatorname{\mathbb{Z}}^{d}_{\geq 1}$.
Let $f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$ be a polynomial
that is irreducible over $\operatorname{\mathbb{C}}$ and satisfies conditions
$(1)$–$(2)$ from Proposition 10.2. Fix the polynomials $f_{1},\dots,f_{d}$
from condition $(2)$ of Proposition 10.2. Let $(r_{1},\dots,r_{d})$ be a tuple
of rational functions in the variables $z_{s}$ for $s\in[\mathbf{a}^{*}]$ that
is adapted to $(f;f_{1},\dots,f_{d})$. Then for all
$\bm{\alpha}\in\\{-1,1\\}^{d}$, there exists a unique sign
$\gamma_{\bm{\alpha}}\in\\{-1,1\\}$ such the following condition holds for all
arrays
$\mathbf{\tilde{x}}=(x_{s})\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ satisfying (10.5)–(10.7) and (10.8)–(10.9):
$\displaystyle\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v+[\bm{\alpha}\odot\mathbf{a}]})=\gamma_{\bm{\alpha}}\prod_{i=1}^{d}x_{v+[\bm{\alpha}\odot(\mathbf{a}-\mathbf{1}_{i})]}\qquad\text{for
all $v\in\operatorname{\mathbb{Z}}^{d}$}.$ (10.11)
###### Definition 10.18.
For $f$ and $(r_{1},\dots,r_{d})$ as in Lemma 10.17, we call the signs
$(\gamma_{\bm{\alpha}})_{\bm{\alpha}\in\\{-1,1\\}^{d}}$ given in Lemma 10.17
the _propagation signs_ corresponding to
$(f;f_{1},\dots,f_{d};r_{1},\dots,r_{d})$.
The proof of Lemma 10.17 relies on the following lemma.
###### Lemma 10.19.
Let $\mathbf{a}=(a_{1},\dots,a_{d})\in\operatorname{\mathbb{Z}}^{d}_{\geq 1}$.
Let $f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$ be a polynomial
that is irreducible over $\operatorname{\mathbb{C}}$ and satisfies conditions
$(1)$–$(2)$ from Proposition 10.2. Let $j\not=k\in\\{1,\dots,d\\}$. Then there
exists $\alpha_{jk}\in\\{-1,1\\}$ such that for all
$\bm{\alpha}\in\\{0,1\\}^{d}$,
$\displaystyle\frac{\partial f}{\partial
z_{\mathbf{a}\odot\bm{\alpha}}}\frac{\partial f}{\partial
z_{\mathbf{a}\odot\bm{\alpha}+(\mathbf{a}-2\mathbf{a}\odot\bm{\alpha})\odot(\mathbf{1}_{j}+\mathbf{1}_{k})}}-\alpha_{jk}\frac{\partial
f}{\partial
z_{\mathbf{a}\odot\bm{\alpha}+(\mathbf{a}-2\mathbf{a}\odot\bm{\alpha})\odot\mathbf{1}_{j}}}\frac{\partial
f}{\partial
z_{\mathbf{a}\odot\bm{\alpha}+(\mathbf{a}-2\mathbf{a}\odot\bm{\alpha})\odot\mathbf{1}_{k}}}$
is a multiple of $f$.
###### Proof.
Because $f$ satisfies conditions (1)–(2) from Proposition 10.2,
$\displaystyle\left(\frac{\partial f}{\partial
z_{\mathbf{a}}}\right)^{2}\left(\frac{\partial f}{\partial
z_{\mathbf{a}\odot(\mathbf{1}-\mathbf{1}_{j}-\mathbf{1}_{k})}}\right)^{2}-\left(\frac{\partial
f}{\partial
z_{\mathbf{a}\odot(\mathbf{1}-\mathbf{1}_{j})}}\right)^{2}\left(\frac{\partial
f}{\partial z_{\mathbf{a}\odot(\mathbf{1}-\mathbf{1}_{k})}}\right)^{2}$
$\displaystyle\qquad{}\equiv(f_{1}\cdots
f_{d})(\pi_{\mathbf{a},j}\pi_{\mathbf{a},k}(f_{1}\cdots
f_{d}))-(\pi_{\mathbf{a},j}(f_{1}\cdots f_{d}))(\pi_{\mathbf{a},k}(f_{1}\cdots
f_{d}))$
$\displaystyle\qquad{}=(1-1)f_{j}f_{k}(\pi_{\mathbf{a},j}(f_{j}))(\pi_{\mathbf{a},k}(f_{k}))\prod_{i\in\\{1,\dots,d\\}\setminus\\{j,k\\}}f_{j}^{2}=0$
mod $f$. Hence, by the irreducibility of $f$, there exists
$\alpha_{jk}\in\\{-1,1\\}$ such that
$\displaystyle\frac{\partial f}{\partial z_{\mathbf{a}}}\frac{\partial
f}{\partial
z_{\mathbf{a}\odot(\mathbf{1}-\mathbf{1}_{j}-\mathbf{1}_{k})}}-\alpha_{jk}\frac{\partial
f}{\partial z_{\mathbf{a}\odot(\mathbf{1}-\mathbf{1}_{j})}}\frac{\partial
f}{\partial z_{\mathbf{a}\odot(\mathbf{1}-\mathbf{1}_{k})}}$
is a multiple of $f$. The full lemma follows from condition (1) of Proposition
10.2. ∎
###### Proof of Lemma 10.17.
We proceed by induction on the number of $-1$s in $\bm{\alpha}$. When
$\bm{\alpha}=\mathbf{1}$, then $\gamma_{\bm{\alpha}}=1$ by condition (10.5).
If $\bm{\alpha}$ contains one $-1$, say
$\bm{\alpha}=\mathbf{1}-\mathbf{1}_{i}$, then
$\gamma_{\bm{\alpha}}=\beta_{i}$, where $\beta_{i}$ is the sign from
Definition 10.11.
Suppose $\ell\geq 2$, and $\bm{\alpha}$ has $\ell$ $-1$s, including $-1$s at
positions $j$ and $k$. Then by Lemma 10.19 and our inductive hypothesis,
$\displaystyle\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v+[\bm{\alpha}\odot\mathbf{a}]})=\frac{\alpha_{jk}\frac{\partial
f}{\partial
z_{\mathbf{a}\odot(\mathbf{1}-\mathbf{1}_{j})}}(\mathbf{x}_{v+[\bm{\alpha}\odot\mathbf{a}]})\frac{\partial
f}{\partial
z_{\mathbf{a}\odot(\mathbf{1}-\mathbf{1}_{k})}}(\mathbf{x}_{v+[\bm{\alpha}\odot\mathbf{a}]})}{\frac{\partial
f}{\partial
z_{\mathbf{a}\odot(\mathbf{1}-\mathbf{1}_{j}-\mathbf{1}_{k})}}(\mathbf{x}_{v+[\bm{\alpha}\odot\mathbf{a}]})}$
$\displaystyle\hphantom{\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v+[\bm{\alpha}\odot\mathbf{a}]})}{}=\alpha_{jk}\gamma_{\bm{\alpha}+2(\mathbf{1}_{j}+\mathbf{1}_{k})}\gamma_{\bm{\alpha}+2\mathbf{1}_{j}}\gamma_{\bm{\alpha}+2\mathbf{1}_{k}}\prod_{i=1}^{d}x_{v+[\bm{\alpha}\odot(\mathbf{a}-\mathbf{1}_{i})]},$
so setting
$\gamma_{\bm{\alpha}}=\alpha_{jk}\gamma_{\bm{\alpha}+2(\mathbf{1}_{j}+\mathbf{1}_{k})}\gamma_{\bm{\alpha}+2\mathbf{1}_{j}}\gamma_{\bm{\alpha}+2\mathbf{1}_{k}}\in\\{-1,1\\}$,
we obtain the desired result. ∎
###### Example 10.20.
Let us continue with Examples 10.4 and 10.13. Following the argument in the
proof of Lemma 10.17, it can be shown that $\gamma_{\bm{\alpha}}=1$ if
$\bm{\alpha}=\pm\mathbf{1}$, and $\gamma_{\bm{\alpha}}=-1$ otherwise. Note
that this fact is equivalent to Lemma 7.2.
###### Example 10.21.
Let us continue with Examples 10.5 and 10.14. Following the argument in the
proof of Lemma 10.17, it can be shown that $\gamma_{\bm{\alpha}}=1$ if
$\bm{\alpha}=\mathbf{1}$, and $\gamma_{\bm{\alpha}}=-1$ otherwise. In
particular, $\gamma_{-\mathbf{1}}=1$. Hence, if
$\mathbf{\tilde{x}}=(x_{s})\in(\operatorname{\mathbb{C}}^{*})^{\operatorname{\mathbb{Z}}^{2}\cup
F_{\mathbf{a}}}$ satisfies conditions (5.7)–(5.11), it follows that
$(x_{-s})_{s\in\operatorname{\mathbb{Z}}^{2}\cup F_{\mathbf{a}}}$ cannot
satisfy conditions (5.7)–(5.11).
###### Example 10.22.
Let us continue with Examples 10.6 and 10.15. It is straightforward to show
that $\gamma_{(1)}=\gamma_{(-1)}=1$.
###### Example 10.23.
Let us continue with Examples 10.7 and 10.16. Following the argument in the
proof of Lemma 10.17, it can be shown that $\gamma_{\bm{\alpha}}=1$ if
$\bm{\alpha}=\pm\mathbf{1}$, and $\gamma_{\bm{\alpha}}=-1$ otherwise.
We now state the main theorem of this section.
###### Theorem 10.24.
Let $\mathbf{a}=(a_{1},\dots,a_{d})\in\operatorname{\mathbb{Z}}^{d}_{\geq 1}$.
Let $f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$ be a polynomial
that is irreducible over $\operatorname{\mathbb{C}}$ and satisfies conditions
$(1)$–$(2)$ from Proposition 10.2. Fix the polynomials $f_{1},\dots,f_{d}$
from condition $(2)$ of Proposition 10.2. Let $g$ and $h$ be the coefficients
of $z_{\mathbf{a}}^{2}$ and $z_{\mathbf{a}}$ in $f$, viewed as a polynomial in
$z_{\mathbf{a}}$. Let $(r_{1},\dots,r_{d})$ be a tuple of rational functions
in the variables $z_{s}$ for $s\in[\mathbf{a}^{*}]$ that is adapted to
$(f;f_{1},\dots,f_{d})$. Let
$(\gamma_{\bm{\alpha}})_{\bm{\alpha}\in\\{-1,1\\}^{d}}$ be the propagation
signs corresponding to $(f;f_{1},\dots,f_{d};r_{1},\dots,r_{d})$.
1. $(a)$
Let $\mathbf{x}=(x_{s})_{s\in\operatorname{\mathbb{Z}}^{d}}$ be an array such
that
$\displaystyle\mathbf{x}\text{ satisfies }f,$ (10.12)
$\displaystyle\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v+[\mathbf{a}]})\not=0\qquad\text{for all
$v\in\operatorname{\mathbb{Z}}^{d}$ if $d>1$},$ (10.13) $\displaystyle
g(\mathbf{x}_{v+[\mathbf{a}]})\not=0\qquad\text{for all
$v\in\operatorname{\mathbb{Z}}^{d}$},$ (10.14)
$\displaystyle\prod_{\bm{\alpha}\in\\{-1,0\\}^{d}}\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]})$
(10.15)
$\displaystyle\qquad{}=\left(\prod_{\bm{\alpha}\in\\{-1,1\\}^{d}}\gamma_{\bm{\alpha}}\right)\prod_{i=1}^{d}\prod_{\bm{\beta}=(\beta_{1},\dots,\beta_{d})\in\\{-1,0\\}^{d}\colon\beta_{i}=0}f_{i}(\mathbf{x}_{v+\bm{\beta}+[\mathbf{a}]})\qquad\text{for
all $v\in\operatorname{\mathbb{Z}}^{d}$}.$
Then $\mathbf{x}$ can be extended to an array
$\mathbf{\tilde{x}}=(x_{s})\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ satisfying (10.5)–(10.7).
2. $(b)$
Conversely, if
$\mathbf{\tilde{x}}=(x_{s})\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ satisfies conditions (10.5)–(10.7) and (10.8)–(10.9), then
the restriction of $\mathbf{\tilde{x}}$ to $\operatorname{\mathbb{Z}}^{d}$
satisfies $f$ and the condition (10.15).
###### Remark 10.25.
The equation (10.15) is a generalization of the coherence condition (equation
(2.7)) for the Kashaev equation.
The following proposition states that the sign
$\prod\limits_{\bm{\alpha}\in\\{-1,1\\}^{d}}\gamma_{\bm{\alpha}}$ in (10.15)
is independent of the choice of $(r_{1},\dots,r_{d})$ if $d\geq 2$.
###### Proposition 10.26.
Let $\mathbf{a}=(a_{1},\dots,a_{d})\in\operatorname{\mathbb{Z}}^{d}_{\geq 1}$
with $d\geq 2$. Let $f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$
be a polynomial that is irreducible over $\operatorname{\mathbb{C}}$ and
satisfies conditions $(1)$–$(2)$ from Proposition 10.2. Then there exists a
sign $\gamma\in\\{-1,1\\}$ such that for any tuple of rational functions
$(r_{1},\dots,r_{d})$ in the variables $z_{s}$ for $s\in[\mathbf{a}^{*}]$ that
is adapted to $(f;f_{1},\dots,f_{d})$, we have
$\displaystyle\gamma=\prod_{\bm{\alpha}\in\\{-1,1\\}^{d}}\gamma_{\bm{\alpha}},$
where $(\gamma_{\bm{\alpha}})_{\bm{\alpha}\in\\{-1,1\\}^{d}}$ are the
propagation signs corresponding to $(f;f_{1},\dots,f_{d};r_{1},\dots,r_{d})$.
###### Remark 10.27.
By Lemma 10.8, given
$f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$ satisfying
conditions (1)–(2) from Proposition 10.2 and an array
$\mathbf{x}=(x_{s})_{s\in\operatorname{\mathbb{Z}}^{d}}$ satisfying $f$, the
following are equivalent:
* •
$\mathbf{x}$ satisfies condition (10.15),
* •
$\mathbf{x}$ satisfies
$\displaystyle\prod_{\begin{subarray}{c}\bm{\alpha}=(\alpha_{1},\dots,\alpha_{d})\\\
\alpha_{1},\dots,\alpha_{d}\in\\{-1,0\\}\\\ \alpha_{1}+\cdots+\alpha_{d}\text{
even}\end{subarray}}\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]})$
$\displaystyle\qquad{}=\gamma\prod_{\begin{subarray}{c}\bm{\alpha}=(\alpha_{1},\dots,\alpha_{d})\\\
\alpha_{1},\dots,\alpha_{d}\in\\{-1,0\\}\\\ \alpha_{1}+\cdots+\alpha_{d}\text{
odd}\end{subarray}}\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]})\qquad\text{for
all $v\in\operatorname{\mathbb{Z}}^{d}$}.$ (10.16)
Hence, one can replace condition (10.15) in Theorem 10.24 by condition
(10.16).
###### Example 10.28.
Continuing with Examples 10.4, 10.13, and 10.20, Theorem 2.22 is a special
case of Theorem 10.24. Theorem 2.9 is a special case of Theorem 10.24(b),
where we require all values of $\mathbf{\tilde{x}}$, including the values
indexed by $F^{\mathbf{a}}$, to be positive.
###### Example 10.29.
Continuing with Examples 10.5, 10.14, and 10.21, Theorem 5.9 is a special case
of Theorem 10.24. Theorem 5.7 is a special case of Theorem 10.24(b), where we
require $x_{s}>0$ for $s\in\operatorname{\mathbb{Z}}^{2}_{\\{0,1,2,\dots\\}}$,
$s\in\operatorname{\mathbb{Z}}^{2}_{\\{0,1,2,\dots\\}}+[\mathbf{a}-\mathbf{1}_{1}]$,
and
$s\in\operatorname{\mathbb{Z}}^{2}_{\\{0,1,2,\dots\\}}+[\mathbf{a}-\mathbf{1}_{2}]$.
###### Example 10.30.
Continuing with Examples 10.6, 10.15, and 10.22, Theorem 6.4 is a special case
of Theorem 10.24. Theorem 6.3 is a special case of Theorem 10.24(b), where we
require all values of $\mathbf{\tilde{x}}$, including the values indexed by
$F^{\mathbf{a}}$, to be positive.
###### Example 10.31.
Continuing with Examples 10.7, 10.16, and 10.23, Theorem 6.10 is a special
case of Theorem 10.24. Theorem 6.9 is a special case of Theorem 10.24(b),
where we require all values of $\mathbf{\tilde{x}}$, including the values
indexed by $F^{\mathbf{a}}$, to be positive.
Before we prove Theorem 10.24, we first prove Proposition 10.26. Proposition
10.26 follows from the lemma below.
###### Lemma 10.32.
Let $\mathbf{a}=(a_{1},\dots,a_{d})\in\operatorname{\mathbb{Z}}^{d}_{\geq 1}$.
Let $f\in\operatorname{\mathbb{C}}[\mathbf{z}_{[\mathbf{a}]}]$ be a polynomial
that is irreducible over $\operatorname{\mathbb{C}}$ and satisfies conditions
$(1)$–$(2)$ from Proposition 10.2. Fix $j\in\\{1,\dots,d\\}$. Let
$(r_{1},\dots,r_{d})$ and $(\tilde{r}_{1},\dots,\tilde{r}_{d})$ be tuples of
rational functions in the variables $z_{s}$ for $s\in[\mathbf{a}^{*}]$ that
are adapted to $(f;f_{1},\dots,f_{d})$, such that $\tilde{r}_{j}=-r_{j}$ and
$\tilde{r}_{i}=r_{i}$ for $i\not=j$. Let
$(\gamma_{\bm{\alpha}})_{\bm{\alpha}\in\\{-1,1\\}^{d}}$ and
$(\tilde{\gamma}_{\bm{\alpha}}\in\\{-1,1\\})_{\bm{\alpha}\in\\{-1,1\\}^{d}}$
be the propagation signs corresponding to
$(f;f_{1},\dots,f_{d};r_{1},\dots,r_{d})$ and
$(f;f_{1},\dots,f_{d};\tilde{r}_{1},\dots,\tilde{r}_{d})$, respectively. Given
$\bm{\alpha}=(\alpha_{1},\dots,\alpha_{d})\in\\{-1,1\\}^{d}$, the following
are equivalent:
* •
$\gamma_{\bm{\alpha}}=\tilde{\gamma}_{\bm{\alpha}}$,
* •
$\alpha_{j}=1$.
###### Proof.
We proceed by induction on the number of $-1$s in $\bm{\alpha}$. Note that
$\tilde{\gamma}_{\mathbf{1}}=\gamma_{\mathbf{1}}=1$,
$\tilde{\gamma}_{\mathbf{1}-2\mathbf{1}_{j}}=-\gamma_{\mathbf{1}-2\mathbf{1}_{j}}$,
and
$\tilde{\gamma}_{\mathbf{1}-2\mathbf{1}_{i}}=\gamma_{\mathbf{1}-2\mathbf{1}_{i}}$
for $i\not=j$. Suppose $\bm{\alpha}$ has at least two $-1$s. As we showed in
the proof to Lemma 10.17, if $k_{1}$ and $k_{2}$ are distinct values such that
$\alpha_{k_{1}}=\alpha_{k_{2}}=1$, then
$\gamma_{\bm{\alpha}}=\alpha_{k_{1}k_{2}}\gamma_{\bm{\alpha}+2(\mathbf{1}_{k_{1}}+\mathbf{1}_{k_{2}})}\gamma_{\bm{\alpha}+2\mathbf{1}_{k_{1}}}\gamma_{\bm{\alpha}+2\mathbf{1}_{k_{2}}}$
and
$\tilde{\gamma}_{\bm{\alpha}}=\alpha_{k_{1}k_{2}}\tilde{\gamma}_{\bm{\alpha}+2(\mathbf{1}_{k_{1}}+\mathbf{1}_{k_{2}})}\tilde{\gamma}_{\bm{\alpha}+2\mathbf{1}_{k_{1}}}\tilde{\gamma}_{\bm{\alpha}+2\mathbf{1}_{k_{2}}}$.
If $\alpha_{j}=1$, let $k_{1}$, $k_{2}$ be distinct values such that
$\alpha_{k_{1}}=\alpha_{k_{2}}=-1$, so
$\displaystyle\tilde{\gamma}_{\bm{\alpha}}=\alpha_{k_{1}k_{2}}\tilde{\gamma}_{\bm{\alpha}+2(\mathbf{1}_{k_{1}}+\mathbf{1}_{k_{2}})}\tilde{\gamma}_{\bm{\alpha}+2\mathbf{1}_{k_{1}}}\tilde{\gamma}_{\bm{\alpha}+2\mathbf{1}_{k_{2}}}=\alpha_{k_{1}k_{2}}\gamma_{\bm{\alpha}+2(\mathbf{1}_{k_{1}}+\mathbf{1}_{k_{2}})}\gamma_{\bm{\alpha}+2\mathbf{1}_{k_{1}}}\gamma_{\bm{\alpha}+2\mathbf{1}_{k_{2}}}=\gamma_{\bm{\alpha}}.$
If $\alpha_{j}=-1$, let $k\not=j$ be a value such that $\alpha_{k}=-1$, so
$\displaystyle\tilde{\gamma}_{\bm{\alpha}}=\alpha_{jk}\tilde{\gamma}_{\bm{\alpha}+2(\mathbf{1}_{j}+\mathbf{1}_{k})}\tilde{\gamma}_{\bm{\alpha}+2\mathbf{1}_{j}}\tilde{\gamma}_{\bm{\alpha}+2\mathbf{1}_{k}}=-\alpha_{jk}\gamma_{\bm{\alpha}+2(\mathbf{1}_{j}+\mathbf{1}_{k})}\gamma_{\bm{\alpha}+2\mathbf{1}_{j}}\gamma_{\bm{\alpha}+2\mathbf{1}_{k}}=-\gamma_{\bm{\alpha}}.$
∎
###### Proof of Proposition 10.26.
It suffices to prove the proposition for tuples of rational functions
$(r_{1},\dots,r_{d})$ and $(\tilde{r}_{1},\dots,\tilde{r}_{d})$ satisfying the
conditions of Lemma 10.32. Let
$(\gamma_{\bm{\alpha}})_{\bm{\alpha}\in\\{-1,1\\}^{d}}$ and
$(\tilde{\gamma}_{\bm{\alpha}}\in\\{-1,1\\})_{\bm{\alpha}\in\\{-1,1\\}^{d}}$
be the propagation signs corresponding to $(f;f_{1},\dots,f_{d};\allowbreak
r_{1},\dots,r_{d})$ and
$(f;f_{1},\dots,f_{d};\tilde{r}_{1},\dots,\tilde{r}_{d})$, respectively. Then
by Lemma 10.32,
$\displaystyle\prod_{\bm{\alpha}\in\\{-1,1\\}^{d}}\tilde{\gamma}_{\bm{\alpha}}=(-1)^{2^{d-1}}\prod_{\bm{\alpha}\in\\{-1,1\\}^{d}}\gamma_{\bm{\alpha}}=\prod_{\bm{\alpha}\in\\{-1,1\\}^{d}}\gamma_{\bm{\alpha}},$
because $d\geq 2$. ∎
The remainder of this section is dedicated to the proof of Theorem 10.24. For
the rest of this section, we fix all quantities given in Theorem 10.24, and
set
$\displaystyle a=a_{1}+\cdots+a_{d}.$
###### Proof of Theorem 10.24(b)..
Let $\mathbf{x}$ be the restriction of $\mathbf{\tilde{x}}$ to
$\operatorname{\mathbb{Z}}^{d}$. It is clear that $\mathbf{x}$ must satisfy
$f$. By Lemma 10.17,
$\displaystyle\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]})=\gamma_{\mathbf{1}+2\bm{\alpha}}\prod_{i=1}^{d}x_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(1+2\bm{\alpha})\odot(\mathbf{a}-\mathbf{1}_{i})]}$
$\displaystyle\hphantom{\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]})}{}=\gamma_{\mathbf{1}+2\bm{\alpha}}\prod_{i=1}^{d}x_{v+\bm{\alpha}\odot(\mathbf{1}-\mathbf{1}_{i})+[\mathbf{a}-\mathbf{1}_{i}]}.$
Taking the product over $\bm{\alpha}\in\\{-1,0\\}^{d}$, we get
$\displaystyle\prod_{\bm{\alpha}\in\\{-1,0\\}^{d}}\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{v-(\mathbf{a}-\mathbf{1})\odot\bm{\alpha}+[(\mathbf{1}+2\bm{\alpha})\odot\mathbf{a}]})$
$\displaystyle\qquad{}=\left(\prod_{\bm{\alpha}\in\\{-1,1\\}^{d}}\gamma_{\bm{\alpha}}\right)\prod_{i=1}^{d}\prod_{\bm{\beta}=(\beta_{1},\dots,\beta_{d})\in\\{-1,0\\}^{d}\colon\beta_{i}=0}x_{v+\bm{\beta}+[\mathbf{a}-\mathbf{1}_{i}]}^{2}$
$\displaystyle\qquad{}=\left(\prod_{\bm{\alpha}\in\\{-1,1\\}^{d}}\gamma_{\bm{\alpha}}\right)\prod_{i=1}^{d}\prod_{\bm{\beta}=(\beta_{1},\dots,\beta_{d})\in\\{-1,0\\}^{d}\colon\beta_{i}=0}f_{i}(\mathbf{x}_{v+\bm{\beta}+[\mathbf{a}]}).$
∎
The proof of Theorem 10.24(a) below is nearly identical to the proof of
Theorem 2.22(a).
###### Definition 10.33.
For $U\subseteq\operatorname{\mathbb{Z}}$, let
$\operatorname{\mathbb{Z}}^{d}_{U}$ denote the set
$\displaystyle\operatorname{\mathbb{Z}}^{d}_{U}=\big{\\{}(i_{1},\dots,i_{d})\in\operatorname{\mathbb{Z}}^{d}\colon
i_{1}+\cdots i_{d}\in U\big{\\}}.$
For $U\subseteq\operatorname{\mathbb{Z}}$, we will also use the notation
$\displaystyle F_{\mathbf{a},U}=\\{v+[\mathbf{a}-\mathbf{1}_{i}]\colon
v\in\operatorname{\mathbb{Z}}_{U},i\in\\{1,\dots,d\\}\\}.$
In particular, we will be interested in
$\mathbb{Z}^{d}_{a,\textup{init}}=\operatorname{\mathbb{Z}}^{d}_{\\{0,\dots,a-1\\}}$
and $F^{\mathbf{a}}_{\textup{init}}=F_{\mathbf{a},\\{0\\}}$.
###### Definition 10.34.
We say that an array $\mathbf{\tilde{x}}_{\textup{init}}$ indexed by
$\mathbb{Z}^{d}_{a,\textup{init}}\cup F^{\mathbf{a}}_{\textup{init}}$
satisfying condition (10.7) is _generic_ if there exists an extension of
$\mathbf{\tilde{x}}_{\textup{init}}$ to an array $\mathbf{\tilde{x}}$ indexed
by $\operatorname{\mathbb{Z}}^{d}\cup F^{\mathbf{a}}$ satisfying equations
(10.5)–(10.7) where the restriction of $\mathbf{\tilde{x}}$ to
$\operatorname{\mathbb{Z}}^{d}$ satisfies conditions (10.13)–(10.14).
Similarly, we say that an array $\mathbf{x}_{\textup{init}}$ indexed by
$\mathbb{Z}^{d}_{a,\textup{init}}$ is generic if every extension of
$\mathbf{x}_{\textup{init}}$ to an array $\mathbf{\tilde{x}}_{\textup{init}}$
indexed by $\mathbb{Z}^{d}_{a,\textup{init}}\cup
F^{\mathbf{a}}_{\textup{init}}$ satisfying condition (10.7) is generic.
###### Definition 10.35.
Let $\mathbf{\tilde{x}}_{\textup{init}}$ be a generic array indexed by
$\mathbb{Z}^{d}_{a,\textup{init}}\cup F^{\mathbf{a}}_{\textup{init}}$
satisfying condition (10.7). We denote by
$(\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\operatorname{\mathbb{Z}}^{d}\\!{\cup}F^{\mathbf{a}}}$
the unique extension of $\mathbf{\tilde{x}}_{\textup{init}}$ to
$\operatorname{\mathbb{Z}}^{d}\\!{\cup}F^{\mathbf{a}}$ where
$(\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\operatorname{\mathbb{Z}}^{d}\\!{\cup}F^{\mathbf{a}}}$
satisfies equations (10.5)–(10.7).
The next lemma generalizes Lemma 7.11.
###### Lemma 10.36.
Let $S=[\mathbf{a}]\cup\\{b\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]\colon
b\in\\{0,1\\},i\in\\{1,\dots,d\\}\\}$, i.e., $S$ is the set of vertices of
$[\mathbf{a}]\subset\operatorname{\mathbb{Z}}^{d}$ and boxes of
$F^{\mathbf{a}}$ completely contained in $[\mathbf{a}]$. Fix values
$t_{i}\in\\{-1,1\\}$ for $i=1,\dots,d$. Suppose
$\mathbf{\tilde{x}}=(x_{s})_{s\in S}$ and $\mathbf{\tilde{y}}=(y_{s})_{s\in
S}$ are arrays of complex numbers such that
* •
$\mathbf{\tilde{x}}$ and $\mathbf{\tilde{y}}$ both satisfy equations
(10.5)–(10.7), with the denominators in equations (10.5)–(10.6) non-vanishing,
* •
$y_{s}=x_{s}$ for $s\in[\mathbf{a}]-\\{\mathbf{a}\\}$,
* •
$y_{[\mathbf{a}-\mathbf{1}_{i}]}=t_{i}x_{[\mathbf{a}-\mathbf{1}_{i}]}$ for
$i=1,\dots,d$, and
* •
$\prod\limits_{i=1}^{d}t_{i}=1$.
Then the following equations hold:
$\displaystyle
y_{\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}=t_{i}x_{\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}\qquad\text{for
$i=1,\dots,d$},$ $\displaystyle y_{\mathbf{a}}=x_{\mathbf{a}}.$
###### Proof.
Note that
$\displaystyle
y_{\mathbf{a}}-x_{\mathbf{a}}=\left(\prod_{i=1}^{d}t_{i}-1\right)\frac{\prod\limits_{i=1}^{d}x_{[\mathbf{a}-\mathbf{1}_{i}]}}{2g(\mathbf{x}_{[\mathbf{a}-\mathbf{1}_{i}]})}=0,$
so $y_{\mathbf{a}}=x_{\mathbf{a}}$. By (10.11),
$\displaystyle\begin{split}&\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{a_{i}\mathbf{1}_{i}+[(\mathbf{1}-2\mathbf{1}_{i})\odot\mathbf{a}]})=\gamma_{(\mathbf{1}-2\mathbf{1}_{i})}\prod_{j=1}^{d}x_{a_{i}\mathbf{1}_{i}+[((\mathbf{1}-2\mathbf{1}_{i}))\odot(\mathbf{a}-\mathbf{1}_{j})]}\\\
&\hphantom{\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{a_{i}\mathbf{1}_{i}+[(\mathbf{1}-2\mathbf{1}_{i})\odot\mathbf{a}]})}{}=\gamma_{(\mathbf{1}-2\mathbf{1}_{i})}x_{\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}\prod_{j\in\\{1,\dots,d\\}\setminus\\{i\\}}x_{[\mathbf{a}-\mathbf{1}_{j}]}\end{split}$
and
$\displaystyle\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{y}_{a_{i}\mathbf{1}_{i}+[(\mathbf{1}-2\mathbf{1}_{i})\odot\mathbf{a}]})=\gamma_{(\mathbf{1}-2\mathbf{1}_{i})}\prod_{j=1}^{d}y_{a_{i}\mathbf{1}_{i}+[((\mathbf{1}-2\mathbf{1}_{i}))\odot(\mathbf{a}-\mathbf{1}_{j})]}$
$\displaystyle\hphantom{\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{y}_{a_{i}\mathbf{1}_{i}+[(\mathbf{1}-2\mathbf{1}_{i})\odot\mathbf{a}]})}{}=\gamma_{(\mathbf{1}-2\mathbf{1}_{i})}y_{\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}\prod_{j\in\\{1,\dots,d\\}\setminus\\{i\\}}y_{[\mathbf{a}-\mathbf{1}_{j}]}$
$\displaystyle\hphantom{\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{y}_{a_{i}\mathbf{1}_{i}+[(\mathbf{1}-2\mathbf{1}_{i})\odot\mathbf{a}]})}{}=\gamma_{(\mathbf{1}-2\mathbf{1}_{i})}y_{\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}\prod_{j\in\\{1,\dots,d\\}\setminus\\{i\\}}t_{j}x_{[\mathbf{a}-\mathbf{1}_{j}]}$
$\displaystyle\hphantom{\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{y}_{a_{i}\mathbf{1}_{i}+[(\mathbf{1}-2\mathbf{1}_{i})\odot\mathbf{a}]})}{}=\gamma_{(\mathbf{1}-2\mathbf{1}_{i})}t_{i}y_{\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}\prod_{j\in\\{1,\dots,d\\}\setminus\\{i\\}}x_{[\mathbf{a}-\mathbf{1}_{j}]}.$
Because
$\displaystyle\frac{\partial f}{\partial
z_{\mathbf{a}}}(\mathbf{x}_{a_{i}\mathbf{1}_{i}+[(\mathbf{1}-2\mathbf{1}_{i})\odot\mathbf{a}]})=\frac{\partial
f}{\partial
z_{\mathbf{a}}}(\mathbf{y}_{a_{i}\mathbf{1}_{i}+[(\mathbf{1}-2\mathbf{1}_{i})\odot\mathbf{a}]}),$
it follows that
$x_{\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}=t_{i}y_{\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}$.
∎
###### Definition 10.37.
Define an equivalence relation on $F^{\mathbf{a}}$ by setting $s_{1}\sim
s_{2}$ if and only if $s_{1}=v+[\mathbf{a}-\mathbf{1}_{i}]$ and
$s_{2}=v+\beta\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]$ for some $1\leq
i\leq d$, $v\in\operatorname{\mathbb{Z}}^{d}$, and
$\beta\in\operatorname{\mathbb{Z}}$. Let $F^{\mathbf{a}}_{\boxdot}$ denote the
set of equivalence classes under this equivalence relation. Denote by $[s]\in
F^{\mathbf{a}}_{\boxdot}$ the equivalence class of $s\in F^{\mathbf{a}}$.
###### Definition 10.38.
Define an action of $\\{-1,1\\}^{F^{\mathbf{a}}_{\boxdot}}$ on arrays indexed
by $\mathbb{Z}^{d}_{a,\textup{init}}\cup F^{\mathbf{a}}_{\textup{init}}$ as
follows: given $\mathbf{t}=(t_{s})_{s\in
F^{\mathbf{a}}_{\boxdot}}\in\\{-1,1\\}^{F^{\mathbf{a}}_{\boxdot}}$ and
$\mathbf{\tilde{x}}_{\textup{init}}=(x_{s})_{s\in\mathbb{Z}^{d}_{a,\textup{init}}\cup
F^{\mathbf{a}}_{\textup{init}}}$, define
$\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}}=(\tilde{x}_{s})_{\mathbb{Z}^{d}_{a,\textup{init}}\cup
F^{\mathbf{a}}_{\textup{init}}}$, where
$\displaystyle\tilde{x}_{s}=\begin{cases}x_{s}&\text{if
$s\in\mathbb{Z}^{d}_{a,\textup{init}}$},\\\ t_{[s]}x_{s}&\text{if $s\in
F^{\mathbf{a}}_{\textup{init}}$}.\end{cases}$
###### Definition 10.39.
For $\mathbf{t}=(t_{s})\in\\{-1,1\\}^{F^{\mathbf{a}}_{\boxdot}}$, define
$\psi(\mathbf{t})=(u_{s})\in\\{-1,1\\}^{\operatorname{\mathbb{Z}}^{d}+\mathbf{a}/2}$
by
$\displaystyle
u_{s+\mathbf{a}/2}=\prod_{i=1}^{d}t_{[s+[\mathbf{a}-\mathbf{1}_{i}]]}$
for $s\in\operatorname{\mathbb{Z}}^{d}$.
The next lemma generalizes Lemma 7.18.
###### Lemma 10.40.
Let $\mathbf{\tilde{x}}_{\textup{init}}$ be a generic array indexed by
$\mathbb{Z}^{d}_{a,\textup{init}}\cup F^{\mathbf{a}}_{\textup{init}}$
satisfying condition (10.7). Let
$\mathbf{t}\in\\{-1,1\\}^{F^{\mathbf{a}}_{\boxdot}}$, and
$\mathbf{u}=(u_{s})_{s\in\operatorname{\mathbb{Z}}^{d}+\mathbf{a}/2}=\psi(\mathbf{t})$.
Let
$(\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}=(x_{s})_{s\in\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$, and
$(\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}=(y_{s})_{s\in\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$. Suppose
$v\in\operatorname{\mathbb{Z}}^{d}_{\\{a,a+1,\dots\\}}$ satisfies the
condition that $u_{w-\mathbf{a}/2}=1$ for all
$w\in\operatorname{\mathbb{Z}}^{d}_{\\{a,a+1,\dots\\}}$ with $w\leq v$. Then:
1. $(a)$
$y_{v}=x_{v}$,
2. $(b)$
$y_{v-\mathbf{a}+\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}=t_{[v-\mathbf{a}+\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]]}x_{v-\mathbf{a}+\mathbf{1}_{i}+[\mathbf{a}-\mathbf{1}_{i}]}$
for $i=1,\dots,d$.
###### Proof.
We prove parts (a) and (b) together by induction. Assume that we have proved
parts (a) and (b) for all
$w\in\operatorname{\mathbb{Z}}^{d}_{\\{a,a+1,\dots\\}}$ with $w<v$. By
construction, $x_{w}=y_{w}$ for all $w\in\mathbb{Z}^{d}_{a,\textup{init}}$ and
statement (b) holds for all $w\in\operatorname{\mathbb{Z}}^{d}_{\\{a-1\\}}$.
Hence, $y_{v-s^{\prime}}=x_{v-s^{\prime}}$ for
$s\in[\mathbf{a}]-\\{\mathbf{0}\\}$, and
$y_{v-\mathbf{a}+[\mathbf{a}-\mathbf{1}_{i}]}=t_{[v-\mathbf{a}+[\mathbf{a}-\mathbf{1}_{i}]]}x_{v-\mathbf{a}+[\mathbf{a}-\mathbf{1}_{i}]}$
for $i=1,\dots,d$. Because $u_{v-\mathbf{a}/2}=1$, statements (a) and (b)
follow from Lemma 10.36. ∎
The next lemma generalizes Lemma 7.19.
###### Lemma 10.41.
An array
$\mathbf{u}=(u_{s})\in\\{-1,1\\}^{\operatorname{\mathbb{Z}}^{d}+\mathbf{a}/2}$
is in the image of $\psi$ $($see Definition 10.39) if and only if for every
$v\in\operatorname{\mathbb{Z}}^{3}$,
$\displaystyle\prod_{\bm{\alpha}\in\\{0,1\\}^{d}}u_{v+\mathbf{a}/2+\bm{\alpha}}=1.$
(10.17)
###### Proof.
First, suppose $\mathbf{u}=\psi(\mathbf{t})$, where
$\mathbf{t}=(t_{s})\in\\{-1,1\\}^{F^{\mathbf{a}}_{\boxdot}}$. Then for any
$v\in\operatorname{\mathbb{Z}}^{d}$,
$\displaystyle\prod_{\bm{\alpha}\in\\{0,1\\}^{d}}u_{v+\mathbf{a}/2+\bm{\alpha}}=\prod_{\bm{\alpha}\in\\{0,1\\}^{d}}\prod_{i=1}^{d}t_{[v+[\mathbf{a}-\mathbf{1}_{i}]]}=\prod_{i=1}^{d}\prod_{\begin{subarray}{c}\bm{\alpha}=(\alpha_{1},\dots,\alpha_{d})\in\\{0,1\\}^{d}\\\
\alpha_{i}=0\end{subarray}}t_{[v+\bm{\alpha}+[\mathbf{a}-\mathbf{1}_{i}]]}^{2}=1.$
Next, suppose that condition (10.17) holds. It is clear that $\mathbf{u}$ is
uniquely determined by its components at
$S=\left\\{(v_{1},\dots,v_{d})+\mathbf{a}/2\colon v_{1}\cdots
v_{d}=0\right\\}$ and condition (10.17). For
$v=(v_{1},\dots,v_{d})\in\operatorname{\mathbb{Z}}^{d}_{\\{0\\}}$ and
$i\in\\{1,\dots,d\\}$, set
$\displaystyle
t_{[v+[\mathbf{a}-\mathbf{1}_{i}]]}=\begin{cases}\prod\limits_{j=1}^{i}u_{(v_{1},\dots,v_{j-1},0,v_{j+1},\dots,v_{d})+\mathbf{a}/2}&\text{if
}v_{1},\dots,v_{i-1}\not=0,\\\ 1&\text{otherwise}.\end{cases}$
Set $\mathbf{t}=(t_{s})\in\\{-1,1\\}^{F^{\mathbf{a}}_{\boxdot}}$. It is
straightforward to check that $\psi(\mathbf{t})$ agrees with $\mathbf{u}$ at
$S$. Hence, because $\psi(\mathbf{t})$ and $\mathbf{u}$ both satisfy condition
(10.17), it follows that $\mathbf{u}=\psi(\mathbf{t})$. ∎
The next lemma generalizes Lemma 7.21.
###### Lemma 10.42.
Let $\mathbf{\hat{x}}$ be an array indexed by
$\operatorname{\mathbb{Z}}^{d}_{\\{0,1,\dots,a+d-1\\}}$. Assume that
$\mathbf{\hat{x}}$ satisfying $f$, and, moreover, its restriction to
$\mathbb{Z}^{d}_{a,\textup{init}}$ is generic. Then there exists an array
$\mathbf{\tilde{x}}$ indexed by $\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}$ satisfying equations (10.5)–(10.7) and extending
$\mathbf{\hat{x}}$.
###### Proof.
For $i=a,\dots,a+d-1$, we will show by induction on $i$ that there exists an
array $\mathbf{\tilde{x}}_{\textup{init}}$ indexed by
$\mathbb{Z}^{d}_{a,\textup{init}}\cup F^{\mathbf{a}}_{\textup{init}}$
satisfying (10.7) such that
$(\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ agrees with
$\mathbf{\hat{x}}=(x_{s})_{s\in\operatorname{\mathbb{Z}}^{d}_{\\{0,\dots,a+d-1\\}}}$
on $\operatorname{\mathbb{Z}}^{d}_{\\{0,\dots,i\\}}$. Let
$\mathbf{\tilde{x}}_{\textup{init}}^{\prime}$ be an array indexed by
$\mathbb{Z}^{d}_{a,\textup{init}}\cup F^{\mathbf{a}}_{\textup{init}}$
satisfying (10.7) such that
$(\mathbf{\tilde{x}}_{\textup{init}}^{\prime})^{\uparrow\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}=(y_{s})_{s\in\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ agrees with $\mathbf{\hat{x}}$ on
$\operatorname{\mathbb{Z}}^{d}_{\\{0,\dots,i-1\\}}$. (For $i=a$, we can obtain
$\mathbf{\tilde{x}}_{\textup{init}}^{\prime}$ be taking an arbitrary extension
of $\mathbf{x}_{\textup{init}}$ to $\mathbb{Z}^{d}_{a,\textup{init}}\cup
F^{\mathbf{a}}_{\textup{init}}$ satisfying condition (10.7). For $i>a$, we
have shown that $\mathbf{\tilde{x}}_{\textup{init}}^{\prime}$ exists by
induction.) Choose
$\mathbf{\tilde{u}}=(u_{s})\in\\{-1,1\\}^{\operatorname{\mathbb{Z}}^{d}_{\\{a,\dots,a+d-1\\}}-\mathbf{a}/2}$
so that
* •
$u_{s-\mathbf{a}/2}=1$ if $s\in\operatorname{\mathbb{Z}}^{d}_{\\{i\\}}$ and
$x_{s}=y_{s}$,
* •
$u_{s-\mathbf{a}/2}=-1$ if $s\in\operatorname{\mathbb{Z}}^{d}_{\\{i\\}}$ and
$x_{s}=y_{s}$,
* •
$u_{s-\mathbf{a}/2}=1$ if $s\in\operatorname{\mathbb{Z}}^{d}_{\\{j\\}}$ for
$0\leq j<i$.
Extend $\mathbf{\tilde{u}}$ to
$\mathbf{u}=(u_{s})_{s\in\operatorname{\mathbb{Z}}^{d}+\mathbf{a}/2}$ by
condition (10.7). By Lemma 10.41, there exists
$\mathbf{t}\in\\{-1,1\\}^{F^{\mathbf{a}}_{\boxdot}}$ such that
$\mathbf{u}=\psi(\mathbf{t})$. Set
$\mathbf{\tilde{x}}_{\textup{init}}=\mathbf{t}\cdot\mathbf{\tilde{x}}_{\textup{init}}^{\prime}$.
Then by Lemma 10.40,
$(\mathbf{\tilde{x}}_{\textup{init}})^{\uparrow\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ agrees with $\mathbf{\hat{x}}$ on
$\operatorname{\mathbb{Z}}^{d}_{\\{0,\dots,i\\}}$, as desired. ∎
We can now prove a weaker version of Theorem 10.24(a), under the additional
constraint of genericity.
###### Corollary 10.43.
Let $\mathbf{x}=(x_{s})_{s\in\operatorname{\mathbb{Z}}^{d}}$ be an array that
satisfies $f$ and condition (10.15), and whose restriction to
$\mathbb{Z}^{d}_{a,\textup{init}}$ is generic. Then $\mathbf{x}$ can be
extended to an array $\mathbf{\tilde{x}}$ indexed by
$\operatorname{\mathbb{Z}}^{d}\cup F^{\mathbf{a}}$ satisfying equations
(10.5)–(10.7).
###### Proof.
Let $\mathbf{x}=(x_{s})_{s\in\operatorname{\mathbb{Z}}^{d}}$ be an array that
satisfies $f$ and condition (10.15), and whose restriction to
$\mathbb{Z}^{d}_{a,\textup{init}}$ is generic. By Lemma 10.42, there exists an
array $\mathbf{\tilde{x}}$ indexed by $\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}$ satisfying equations (10.5)–(10.7) that agrees with
$\mathbf{x}$ on $\operatorname{\mathbb{Z}}^{d}_{\\{0,\dots,a+d-1\\}}$. Let
$\mathbf{x}^{\prime}$ be the restriction of $\mathbf{\tilde{x}}$ to
$\operatorname{\mathbb{Z}}^{d}$. By Theorem 10.24(b), $\mathbf{x}^{\prime}$
satisfies $f$ and (10.15). There is a unique solution of $f$ satisfying
condition (10.15) agreeing with $\mathbf{x}$ at
$\operatorname{\mathbb{Z}}^{d}_{\\{0,\dots,a+d-1\\}}$, as condition (10.15)
gives the remaining values as rational expressions in the values at
$\operatorname{\mathbb{Z}}^{d}_{\\{0,\dots,a+d-1\\}}$, where the denominators
do not vanish because conditions (10.13)–(10.14) hold for $\mathbf{x}$ (as
$\mathbf{x}$ is generic). Hence, $\mathbf{x}^{\prime}=\mathbf{x}$, as desired.
∎
###### Proof of Theorem 10.24(a)..
We need to loosen the genericity condition in Corollary 10.43 to the condition
that $\mathbf{x}$ satisfies (10.13)–(10.14).
Let $\mathbf{x}$ satisfy $f$ along with conditions (10.13)–(10.14) and
condition (10.15). Let $A_{j}=[-j,j]^{d}\cap\operatorname{\mathbb{Z}}^{d}$,
and let $B_{j}=\left\\{s\in F^{\mathbf{a}}\colon
s\subseteq[-j,j]^{d}\right\\}$. We claim that if there exist
$\mathbf{\tilde{x}}_{j}\in\operatorname{\mathbb{C}}^{A_{j}\cup B_{j}}$
satisfying equations (10.5)–(10.7) that agree with $\mathbf{x}$ on $A_{j}$ for
all $j$, then there exists
$\mathbf{\tilde{x}}\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ satisfying equations (10.5)–(10.7) that agrees with
$\mathbf{x}$ on $\operatorname{\mathbb{Z}}^{d}$. Construct an infinite tree
$T$ as follows:
* •
The vertices of $T$ are arrays indexed by $A_{j}\cup B_{j}$ satisfying
equations (10.5)–(10.7) that agree with $\mathbf{x}$ on $A_{j}$ (over
$j\in\operatorname{\mathbb{Z}}_{\geq 0}$).
* •
Add an edge between
$\mathbf{\tilde{x}}_{j}\in\operatorname{\mathbb{C}}^{A_{j}\cup B_{j}}$ and
$\mathbf{\tilde{x}}_{j+1}\in\operatorname{\mathbb{C}}^{A_{j+1}\cup B_{j+1}}$
if $\mathbf{\tilde{x}}_{j+1}$ restricts to $\mathbf{\tilde{x}}_{j}$.
Thus, $T$ is an infinite tree in which every vertex has finite degree. By
König’s infinity lemma, there exists an infinite path
$\mathbf{\tilde{x}}_{0},\mathbf{\tilde{x}}_{1},\dots$ in $T$ with
$\mathbf{\tilde{x}}_{j}\in\operatorname{\mathbb{C}}^{A_{j}\cup B_{j}}$. Thus,
there exists
$\mathbf{\tilde{x}}\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ restricting to $\mathbf{\tilde{x}}_{j}$ for all
$j\in\operatorname{\mathbb{Z}}_{\geq 0}$, so $\mathbf{\tilde{x}}$ satisfies
equations (10.5)–(10.7) and agrees with $\mathbf{x}$ on
$\operatorname{\mathbb{Z}}^{d}$.
Given $j\in\operatorname{\mathbb{Z}}_{\geq 0}$, we claim that there exists
$\mathbf{\tilde{x}}\in\operatorname{\mathbb{C}}^{A_{j}\cup B_{j}}$ satisfying
equations (10.5)–(10.7) that agrees with $\mathbf{x}$ on $A_{j}$. Because
$\mathbf{x}$ satisfies conditions (10.13)–(10.14), there exists a sequence
$\mathbf{x}_{1},\mathbf{x}_{2},\dots$ of arrays satisfying $f$ along with
conditions (10.13)–(10.14) and condition (10.15), whose restrictions to
$\mathbb{Z}^{d}_{a,\textup{init}}$ are generic. By Corollary 10.43, there
exist
$\mathbf{\tilde{x}}_{1},\mathbf{\tilde{x}}_{2},\ldots\in\operatorname{\mathbb{C}}^{\operatorname{\mathbb{Z}}^{d}\cup
F^{\mathbf{a}}}$ satisfying equations (10.5)–(10.7) such that
$\mathbf{\tilde{x}}_{i}$ restricts to $\mathbf{x}_{i}$. However, the sequence
$\mathbf{\tilde{x}}_{1},\mathbf{\tilde{x}}_{2},\dots$ does not necessarily
converge. Let
$\mathbf{\tilde{x}}_{1}^{\prime},\mathbf{\tilde{x}}_{2}^{\prime},\ldots\in\operatorname{\mathbb{C}}^{A_{j}\cup
B_{j}}$ be the restrictions of
$\mathbf{\tilde{x}}_{1},\mathbf{\tilde{x}}_{2},\dots$ to $A_{j}\cup B_{j}$.
There exists a subsequence of
$\mathbf{\tilde{x}}_{1}^{\prime},\mathbf{\tilde{x}}_{2}^{\prime},\dots$ that
converges to some $\mathbf{\tilde{x}}\in\operatorname{\mathbb{C}}^{A_{j}\cup
B_{j}}$. (For each $s\in B_{j}$, we can partition the sequence
$\mathbf{\tilde{x}}_{1}^{\prime},\mathbf{\tilde{x}}_{2}^{\prime},\dots$ into
two sequences, each of which converges at $s$. Because $B_{j}$ is finite, the
claim follows.) The array $\mathbf{\tilde{x}}$ must satisfy equations
(10.5)–(10.7) and agree with $\mathbf{x}$ on $A_{j}$, so we are done. ∎
### Acknowledgements
I would like to thank my Ph.D. advisor, Sergey Fomin, for his invaluable
mathematical insights and the countless hours he dedicated to our meetings
while I was writing this paper. I am also grateful to Dmitry Chelkak for
pointing out the connection with s-holomorphicity, and to Thomas Lam and John
Stembridge for helpful discussions and editorial suggestions. Finally, I would
like to thank the anonymous referees for their thorough reading of this
manuscript, and for their suggestions that improved the quality of this paper.
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|
# Entropy properties of mostly expanding partially hyperbolic diffeomorphisms
Jinhua Zhang
###### Abstract
The statistical properties of mostly expanding partially hyperbolic
diffeomorphisms have been substantially studied. In this paper, we would like
to address the entropy properties of mostly expanding partially hyperbolic
diffeomorphisms. We prove that for mostly expanding partially hyperbolic
diffeomorphisms with minimal strong stable foliation and one-dimensional
center bundle, there exists a $C^{1}$-open neighborhood of them, in which the
topological entropy varies continuously and the intermediate entropy property
holds. To prove that, we show that each non-hyperbolic ergodic measure is
approached by horseshoes in entropy and in weak$*$-topology.
###### Contents
1. 1 Introduction
1. 1.1 Statements of our main results
2. 2 Preliminaries
1. 2.1 Topological entropy and metric entropy
2. 2.2 Liao’s shadowing lemma and Pliss lemma
3. 2.3 Plaque families and uniform size of invariant manifolds
4. 2.4 Constructions of horseshoes from hyperbolic periodic orbits
5. 2.5 Unstable entropy and entropy formulas
6. 2.6 Mostly expanding partially hyperbolic diffeomorphisms
3. 3 Existence of periodic orbits with prescribed behavior
1. 3.1 Generating orbit segments with weak hyperbolicity
2. 3.2 Proof of Theorem 3.1
4. 4 Approximation of ergodic measures by horseshoes: Proofs of our main results
## 1 Introduction
Entropy (topological entropy and metric entropy) is an important topological
invariant and plays a central role in describing the complexity of a dynamical
system. The dependence of entropy on the systems is an interesting topic and
has been substantially studied by many mathematicians. Smoothness of a system
is relevant to the upper semi-continuity of entropy. For
$C^{\infty}$-diffeomorphisms, Newhouse [N] proved that the metric entropy
varies upper semi-continuously and Yomdin [Yo] proved that the topological
entropy also varies upper semi-continuously. For systems with lower
regularity, a bifurcation phenomena called _homoclinic tangency_ (i.e. the
existence of non-transverse homoclinic intersections between the stable and
unstable manifolds of a hyperbolic saddle) seems to be an obstruction to the
upper semi-continuity of topological entropy. Misiurewicz [Mi] constructed
$C^{r}$-diffeomorphisms with homoclinic tangencies at which the entropy fails
to be upper semi-continuous. For $C^{1}$-diffeomorphisms far away from
homoclinic tangencies, one can recover the upper semi-continuity of entropy
(see [LVY, DFPV]). As for the lower semi-continuity of the topological
entropy, the hyperbolicity is involved in. For $C^{1}$-uniformly hyperbolic
diffeomorphisms, the topological entropy is locally constant due to Smale’s
structural stability theorem. Using Pesin’s theory, Katok [Ka] proved that
hyperbolic ergodic measures of $C^{1+\alpha}$ surface diffeomorphisms are
approached by hyperbolic horseshoes and thus the topological entropy is lower
semi-continuous for $C^{1+\alpha}$ surface diffeomorphisms. As a consequence,
the topological entropy varies continuously among $C^{\infty}$ surface
diffeomorphisms.
Then it is natural to ask: among which classes of differentiable systems, does
the topological entropy vary continuously? The examples in [Mi] and the
results in [LVY, DFPV] tell us that one should consider the systems away from
homoclinic tangencies. Such diffeomorphisms are partially hyperbolic with
multi-one dimensional center bundles due to [CSY]. Therefore, it is reasonable
to firstly consider the continuity of topological entropy for partially
hyperbolic diffeomorphisms with one-dimensional center bundle. There are three
types of classical examples of partially hyperbolic diffeomorphisms with one
dimensional center bundle: skew-products with circle fiber, derived from
Anosov diffeomorphisms and time one maps of Anosov flows. The topological
entropy of these classical examples varies continuously. See for instance
[BFSV, U, SY]. In recent years, some new anomalous examples appeared. The
continuity of the topological entropy of the anomalous examples in [BGP, BGHP]
was proved in [YZ]. Saghin and Yang [SY] conjectured that _the topological
entropy varies continuously among the set of partially hyperbolic
diffeomorphisms with one-dimensional center bundle_.
The classical variational principle gives the relation between metric entropy
and topological entropy: topological entropy equals the supremum of the metric
entropy. For partially hyperbolic diffeomorphisms with one-dimensional center
bundle, the usual approach to prove the continuity of the topological entropy
is to find horseshoes approximating ergodic measures in weak$*$-topology and
in entropy. This makes the problem of the continuity of topological entropy
closely related to the intermediate entropy property, since horseshoes satisfy
the intermediate entropy property.
Recall that a diffeomorphism $f$ satisfies the _intermediate entropy property_
if for any $h\in[0,h_{top}(f))$, there exists an ergodic measure $\nu$ such
that $h_{\nu}(f)=h.$ In general, one can not expect that $h$ is chosen as
$h_{top}(f)$, since the ergodic measures whose metric entropies obtain the
topological entropy do not always exist [Mi]. Katok proposed a conjecture that
_$C^{r}$ ($r\geq 1$)-diffeomorphisms satisfy the intermediate entropy
property_. It is classical that hyperbolic systems satisfy the intermediate
entropy property. Katok’s result [Ka] implies that $C^{r}$ ($r>1$) non-
uniformly hyperbolic systems have the intermediate entropy property. In
general, for dynamics beyond uniform hyperbolicity, this conjecture is widely
open and there are not so many results. See for instance [S1, S2, S3, GSW, YZ,
LSWW]. Even for partially hyperbolic diffeomorphisms with one-dimensional
center bundle, Katok’s conjecture is still open. See [S1, S2, YZ] for some
partial results.
There is a special class of partially hyperbolic diffeomorphisms called mostly
expanding diffeomorphisms whose study was initiated in [ABV]111The notion in
[ABV] nowadays is called partially hyperbolic diffeomorphisms with non-
uniformly expanding center and it is slightly different from the notion of
mostly expanding that we use. See the discussions and examples in [AnV1]. It
was proved in [AnV1] that mostly expanding partially hyperbolic
diffeomorphisms form an open set. There are many works on studying such kind
of diffeomorphisms, and people aim to show the statistical properties of such
systems, for instance the existence and finiteness of physical measures or SRB
measures, statistical stability of the set of SRB measures and the mixing
properties of SRB measures. See for instance [An, AnV1, AnV2, ADLP, ALi, Y]
and references therein. But there are few results on the entropy properties of
such systems. In this paper, we are interested in studying the continuity of
topological entropy and intermediate entropy properties of such systems. The
method we use is finding horseshoes approximating ergodic measures in
weak$*$-topology and in entropy.
### 1.1 Statements of our main results
A diffeomorphism $f\in\operatorname{Diff}^{1}(M)$ is _partially hyperbolic_ if
there exist a $Df$-invariant continuous splitting $TM=E^{s}\oplus E^{c}\oplus
E^{u}$ and some constants $C>1,\lambda\in(0,1)$ such that for any $x\in M$ and
$n\in\mathbb{N}$, one has
* •
$\|Df^{n}|_{E^{s}(x)}\|<C\lambda^{n}\textrm{~{}~{}and~{}~{}}\|Df^{-n}|_{E^{u}(x)}\|<C\lambda^{n}.$
* •
$\|Df^{n}|_{E^{s}(x)}\|\cdot\|Df^{-n}|_{E^{c}(f^{n}(x))}\|<C\lambda^{n}\textrm{
and }\|Df^{n}|_{E^{c}(x)}\|\cdot\|Df^{-n}|_{E^{u}(f^{n}(x))}\|<C\lambda^{n}.$
By [Go], up to changing a metric, one can assume that $C=1$, and we will do so
throughout this paper. Due to [HPS], the bundles $E^{s}$ and $E^{u}$ are
uniquely integrable to $f$-invariant foliations, called strong stable and
strong unstable foliations and denoted by ${\cal F}^{s}$ and ${\cal F}^{u}$
respectively. We denote by ${\cal F}^{*}(x)$ the ${\cal F}^{*}$-leaf through
the point $x$, for $*=s,u.$
Let $\mu$ be an invariant measure of a partially hyperbolic diffeomorphism $f$
with one dimensional center bundle (i.e. $\operatorname{dim}(E^{c})=1$), then
we define
$\chi^{c}(\mu,f):=\int\log\|Df|_{E^{c}}\|\operatorname{d}\mu.$
When $\mu$ is ergodic, by Oseledec’s theorem and Birkhoff’s ergodic theorem,
$\chi^{c}(\mu,f)$ coincides with the Lyapunov exponent of $\mu$ along the
center bundle $E^{c}$ (called _center Lyapunov exponent_). When there is no
ambiguity, we will simply write $\chi^{c}(\mu)$.
For a partially hyperbolic diffeomorphism $f$, we denote by $G^{u}(f)$ the set
of invariant measures satisfying the entropy formula for unstable entropy (see
Equation (1)). We remark that for $f\in\operatorname{Diff}^{1+\alpha}(M)$, the
set $G^{u}(f)$ coincides with the set of u-Gibbs states (see Theorem 2.12),
and moreover, $f$ is called _mostly expanding_ if all the center Lyapunov
exponents of each $u$-Gibbs state are positive (see Section 2.6 for more
information).
Recall that a foliation is _minimal_ if every leaf is dense in the manifold.
###### Theorem A.
Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism
with $\operatorname{dim}(E^{c})=1$. Assume that
* •
$\chi^{c}(\mu)>0$ for any $\mu\in G^{u}(f);$
* •
the strong stable foliation is minimal.
Then there exists a $C^{1}$-open neighborhood
$\mathcal{U}\subset\operatorname{Diff}^{1}(M)$ of $f$ such that
* –
each $g\in{\cal U}$ satisfies the intermediate entropy property;
* –
the entropy function $g\in{\cal U}\mapsto h_{top}(g)$ is continuous.
###### Remark 1.1.
In fact, each $g\in{\cal U}$ also satisfies our assumptions. The first
assumption is an open condition due to the upper semi-continuity of the
compact set $G^{u}(f)$ (see Lemma 2.13). In general, the second assumption is
not an open condition, but combining with the first assumption, one can deduce
that the strong stable foliation is $C^{1}$-robustly minimal (see Theorem
2.20).
We can apply our result to some conservative partially hyperbolic
diffeomorphisms, and obtain the continuity of topological entropy and
intermediate entropy property.
###### Corollary B.
Let $f\in\operatorname{Diff}_{\operatorname{m}}^{1+\alpha}(M)$ be a partially
hyperbolic diffeomorphism preserving a smooth volume $\operatorname{m}$ and
with $\operatorname{dim}(E^{c})=1.$ Assume that
* •
$\int\log\|Df|_{E^{c}}\|\operatorname{d}\operatorname{m}>0$;
* •
the strong stable foliation is minimal.
Then there exists a $C^{1}$-neighborhood
$\mathcal{U}\subset\operatorname{Diff}^{1}(M)$ of $f$ such that
* –
each $g\in{\cal U}$ satisfies the intermediate entropy property;
* –
the entropy function $g\in{\cal U}\mapsto h_{top}(g)$ is continuous.
###### Remark 1.2.
One can deduce that $f$ is $C^{1}$-stably ergodic using the minimality of
strong stable foliation and the center Lyapunov exponent being positive (see
Theorem 2.22).
For the classical examples of partially hyperbolic diffeomorphisms (skew-
products with circle fiber, derived from Anosov diffeomorphisms and time one
maps of Anosov flows), the results in [BF, TY, CPo] show that under some mild
open conditions, ergodic measures with high entropy are hyperbolic. However,
for the systems that we consider, it is not clear for us if there exist
hyperbolic ergodic measures of maximal entropy (i.e. ergodic measures whose
metric entropies equal the topological entropy), and we are not able to
exclude the existence of non-hyperbolic ergodic measures of maximal entropy.
To show the continuity of topological entropy and intermediate entropy
property, we prove that non-hyperbolic ergodic measures are approached by
hyperbolic horseshoes in entropy and in weak$*$-topology.
Given an invariant compact set $K\subset M$ of
$f\in\operatorname{Diff}^{1}(M)$, we denote by $\mathcal{M}_{inv}(f,K)$ and
$\mathcal{M}_{erg}(f,K)$ the sets of $f$-invariant and $f$-ergodic measures on
$K$ respectively. When $K=M$, we will simply write $\mathcal{M}_{inv}(f)$ and
$\mathcal{M}_{erg}(f)$. Recall that an $f$-invariant compact set $\Lambda$ is
called a _hyperbolic basic set_ if it is a hyperbolic transitive set and there
exists a neighborhood $U$ of $\Lambda$ such that
$\cap_{n\in{\mathbb{Z}}}f^{n}(\overline{U})=\Lambda.$
###### Theorem C.
Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism
with $\operatorname{dim}(E^{c})=1.$ Assume that
* •
$\chi^{c}(\mu)>0$ for any $\mu\in G^{u}(f);$
* •
the strong stable foliation is minimal.
Then there exist a $C^{1}$-neighborhood ${\cal
U}\subset\operatorname{Diff}^{1}(M)$ of $f$, and constants $\kappa>0$ and
$\chi_{1}>0$ such that for any $g\in{\cal U}$, any ergodic measure
$\nu\in\mathcal{M}_{\rm erg}(g)$ with $-\chi_{1}\leq\chi^{c}(\nu)\leq 0$ and
any ${\varepsilon}>0$, there exists a hyperbolic basic set
$\Lambda_{\varepsilon}$ of $g$ whose center bundle is uniformly expanding such
that
* –
$h_{top}(g,\Lambda_{{\varepsilon}})>\frac{h_{\nu}(g)-{\varepsilon}}{1+\kappa\cdot(|\chi^{c}(\nu)|+{\varepsilon})};$
* –
the set ${\cal M}_{\rm inv}(g,{\Lambda_{\varepsilon}})$ is contained in the
$\kappa\cdot(|\chi^{c}(\nu)|+{\varepsilon})$-neighborhood of $\nu.$
Furthermore, the set $\big{\\{}\nu\in\mathcal{M}_{erg}(g)|~{}\chi^{c}(\nu)\geq
0\big{\\}}$ is path connected.
###### Remark 1.3.
1. 1.
In Theorem C, if $\nu$ is non-hyperbolic (i.e. $\chi^{c}(\nu)=0$), then $\nu$
is approached by horseshoes in weak$*$-topology and in entropy;
2. 2.
For any $g\in{\cal U}$, all the periodic points with positive center Lyapunov
exponent are homoclinically related222Recall that two hyperbolic periodic
orbits are homoclinically related, if the stable manifold of one periodic
orbit intersects the unstable manifold of the other transversely, and vice
versa. due to Theorem 2.20.
3. 3.
Comparing with the results in [BZ, DGS, YZ], we assume neither the minimality
of strong unstable foliation nor the existence of blenders.
Theorem C is not only used to prove Theorem A and has its own interest. The
approximation of ergodic measures by hyperbolic sets in various way has been
studied. See for instance the results in $C^{1+\alpha}$-setting by [Ka, SW,
Ge] and in $C^{1}$-setting with domination by [G2, C, Ge] for hyperbolic
ergodic measures, and the results in $C^{1}$-generic setting without
domination by [BCF] for ergodic measures. The approximation of non-hyperbolic
ergodic measures came into sight in recent years, and many results have been
obtained. See for instance [DGR, BZ, YZ, DGS], and one can refer to [DG] for
more results. The main novelty here is that we neither assume the existence of
blenders nor the minimality of strong unstable foliation. We use mostly
expanding property to replace the existence of blenders and minimality of
strong unstable foliation, and in some sense, mostly expanding property which
is a non-uniformly hyperbolic property can play the same role in these
problems.
Besides, one can get more information on the set of points with vanishing
center Lyapunov exponent, which in general is a larger set than the union of
the generic points of non-hyperbolic ergodic measures. Let $f$ be a partially
hyperbolic diffeomorphism with one-dimensional center bundle, and let us
define the $0$-level of the center Lyapunov regular set:
$\mathcal{R}_{f}(0):=\big{\\{}x\in
M|\lim_{|n|\rightarrow\infty}\frac{1}{n}\log\|Df^{n}|_{E^{c}(x)}\|=0\big{\\}}.$
###### Theorem D.
Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism
with $\operatorname{dim}(E^{c})=1$. Assume that
* •
$\chi^{c}(\mu)>0$ for any $\mu\in G^{u}(f);$
* •
the strong stable foliation is minimal.
Then there exists a $C^{1}$-neighborhood ${\cal
U}\subset\operatorname{Diff}^{1}(M)$ of $f$ such that for any $g\in{\cal U}$,
and any $h\leq h_{top}(g,\mathcal{R}_{g}(0))$ and ${\varepsilon}>0$, there
exists a hyperbolic basic set $\Lambda_{\varepsilon}$ of $g$ whose center is
uniformly expanding such that
* –
$h_{top}(g,\Lambda_{{\varepsilon}})>h-{\varepsilon},$
* –
for each $\mu\in\mathcal{M}_{erg}(g,\Lambda_{\varepsilon})$, one has
$0<\chi^{c}(\mu,g)<{\varepsilon}.$
###### Remark 1.4.
For any non-hyperbolic ergodic measure $\nu\in\mathcal{M}_{erg}(g)$, one has
$\nu(\mathcal{R}_{g}(0))=1$. By Theorem 3 in [Bow] and the monotonicity of
entropy, one has $h_{\nu}(g)\leq h_{top}(g,\mathcal{R}_{g}(0))$. Thus, Theorem
D also implies that non-hyperbolic ergodic measures are approached by
horseshoes in entropy.
Acknowledgments. The author would like to thank C. Bonatti, L. Díaz, S.
Crovisier, K. Gelfert and R. Saghin for helpful comments. The author benefits
a lot from the discussions with L. Díaz, S. Crovisier, K. Gelfert, D. Yang and
J. Yang. The author is partially supported by National Key R&D Program of
China (2022YFA1005801), National Key R$\&$D Program of China (2021YFA1001900),
NSFC 12001027 and the Fundamental Research Funds for the Central Universities.
## 2 Preliminaries
In this section, we collect the results and notions that involved in this
paper.
### 2.1 Topological entropy and metric entropy
In this section, let $f:X\to X$ be a homeomorphism on a compact metric space
$(X,\operatorname{d})$.
Given $n\in\mathbb{N}$ and ${\varepsilon}>0$, let us recall that
* •
a _$(n,{\varepsilon})$ -Bowen ball_ centered at a point $x\in X$ is defined as
$B_{n}(x,{\varepsilon})=\big{\\{}y\in
X:\operatorname{d}(f^{i}(x),f^{i}(y))<{\varepsilon}\textrm{~{}for any $0\leq
i<n$}\big{\\}}.$
* •
a subset $S\subset X$ is called a _$(n,{\varepsilon})$ -separated set_, if for
any two different points $x,y\in S$, there exists $j\in\\{0,\cdots,n-1\\}$
such that $\operatorname{d}(f^{j}(x),f^{j}(y))>{\varepsilon}.$
Given an invariant compact subset $K\subset X$, let $s(n,{\varepsilon},K)$ be
the maximal cardinality of $(n,{\varepsilon})$-separated sets contained in
$K$. Then _the topological entropy of $f$ on $K$_ is defined as
$h_{top}(f,K)=\lim_{{\varepsilon}\rightarrow
0}\limsup_{n\rightarrow+\infty}\frac{1}{n}\log s(n,{\varepsilon},K).$
When $K=X$, we simply denote $h_{top}(f):=h_{top}(f,X).$
In analogy with the definition of Hausdorff dimension, Bowen [Bow] introduced
topological entropy for non-compact sets via open covers and here we present
an equivalent one via Bowen balls (see [Pe]). Let $Y\subset X$. For any
${\varepsilon}>0$ and $h\in\mathbb{R}$, let us define
${\rm
m}_{{\varepsilon},h}(Y)=\lim_{n\rightarrow+\infty}\inf\big{\\{}\sum_{i\in\mathbb{N}}e^{-hn_{i}}|~{}\textrm{$Y\subset\cup_{i\in\mathbb{N}}B_{n_{i}}(x_{i},{\varepsilon})$
and $n_{i}\geq n$}\big{\\}}.$
Define
$h_{top}(f,Y,{\varepsilon})=\inf\big{\\{}h|~{}{\rm
m}_{{\varepsilon},h}(Y)=0\big{\\}}=\sup\big{\\{}h|~{}{\rm
m}_{{\varepsilon},h}(Y)=+\infty\big{\\}}.$
Now, the topological entropy of $f$ on $Y$ is defined as
$h_{top}(f,Y)=\lim_{{\varepsilon}\rightarrow 0}h_{top}(f,Y,{\varepsilon}).$
###### Remark 2.1.
Bowen showed that if $Y\subset X$ is compact, then the dimension-like
definition of topological entropy coincides with the canonical definition
using separated sets (see [Bow, Proposition 1]).
One can also define the entropy for the non-compact set $Y$ by counting the
cardinality of separated sets. Let $s(n,{\varepsilon},Y)$ be the maximal
cardinality of $(n,{\varepsilon})$-separated sets in $Y$, then one can define
the _lower capacity entropy_ and _upper capacity entropy_ of $f$ on $Y$ as
follows:
$\underline{Ch}_{top}(f,Y)=\lim_{{\varepsilon}\rightarrow
0}\liminf_{n\rightarrow+\infty}\frac{1}{n}\log{s(n,{\varepsilon},Y)}$
and
$\overline{Ch}_{top}(f,Y)=\lim_{{\varepsilon}\rightarrow
0}\limsup_{n\rightarrow+\infty}\frac{1}{n}\log{s(n,{\varepsilon},Y)}$
Now, we recall some basic properties for topological entropy.
###### Proposition 2.2 (Proposition 2 in [Bow] and Section 11 in [Pe]).
Let $f\in\operatorname{Homeo}(X)$ be a homeomorphism on a compact metric space
$(X,\operatorname{d})$. Then one has
* •
for any subsets $Y_{1}\subset Y_{2}\subset X$, one has $h_{top}(f,Y_{1})\leq
h_{top}(f,Y_{2}).$
* •
$h_{top}(f,Y)=\sup_{n}h_{top}(f,Y_{n})$, where
$Y=\cup_{n\in\mathbb{N}}Y_{n}\subset X.$
* •
$\overline{Ch}_{top}(f,Y)\geq\underline{Ch}_{top}(f,Y)\geq h_{top}(f,Y).$
At last, we recall the definition of metric entropy of an $f$-ergodic measure
$\mu$ given by Katok [Ka]. Given $\delta\in(0,1)$, $n\in\mathbb{N}$ and
${\varepsilon}>0$, let $s(n,{\varepsilon},\delta)$ be the minimal cardinality
of $(n,{\varepsilon})$-Bowen balls whose union covers a set with $\mu$-measure
no less than $\delta.$ Then the _metric entropy of $f$ with respect to $\mu$_
is defined as
$h_{\mu}(f)=\lim_{{\varepsilon}\rightarrow
0}\limsup_{n\rightarrow+\infty}\frac{1}{n}\log
s(n,{\varepsilon},\delta)=\lim_{{\varepsilon}\rightarrow
0}\liminf_{n\rightarrow+\infty}\frac{1}{n}\log s(n,{\varepsilon},\delta).$
### 2.2 Liao’s shadowing lemma and Pliss lemma
A $Df$-invariant splitting $T_{\Lambda}M=E\oplus F$ over an invariant compact
set $\Lambda$ of $f\in\operatorname{Diff}^{1}(M)$ is a _dominated splitting_ ,
if there exists $N\in\mathbb{N}$ such that
$\|Df^{N}|_{E(x)}\|\cdot\|Df^{-N}|_{F(f^{N}(x))}\|\leq\frac{1}{2}\textrm{~{}for
any $x\in\Lambda$.}$
Now, we recall Liao’s shadowing lemma which was improved by S. Gan. This
shadowing lemma allows us to find periodic orbits chasing (long) orbit
segments with ”weak” center Lyapunov exponent for a large proportion of time.
###### Theorem 2.3 ([Li, G1]).
Let $f\in\operatorname{Diff}^{1}(M)$ and $\Lambda$ be an $f$-invariant compact
set. Assume that $\Lambda$ admits a dominated splitting of the form
$T_{\Lambda}M=E\oplus F$. Then for any $\lambda\in(0,1)$, there exist
constants $L>1$ and $d_{0}>0$ such that for any $d\in(0,d_{0})$, any
$x\in\Lambda$ and any $n\in\mathbb{N}$ satisfying
* •
$\operatorname{d}(f^{n}(x),x)<d;$
* •
$\|Df^{j}|_{E(x)}\|<\lambda^{j}\textrm{~{}and~{}}\|Df^{-j}|_{F(f^{n}(x))}\|<\lambda^{j},\textrm{~{}for
any $1\leq j\leq n$};$
there exists a periodic point $p$ of period $n$ such that
$\operatorname{d}(f^{i}(x),f^{i}(p))<L\cdot d\textrm{~{} for any $0\leq i\leq
n-1$}.$
###### Remark 2.4.
We will apply this shadowing lemma to the splitting $E^{s}\oplus(E^{c}\oplus
E^{u})$ and in this case, one only needs to consider the norm along the bundle
$E^{c}\oplus E^{u}.$
Now, we recall the Pliss lemma which can be used for finding points with
uniform size of stable or unstable manifolds.
###### Lemma 2.5 (Pliss lemma [Pl]).
Let $a_{1},\cdots,a_{k}$ be some real numbers and assume that $\max_{i\leq
k}a_{i}\leq C$ for some $C\in{\mathbb{R}}$. Suppose that
$\sum_{i=1}^{k}a_{i}\geq k\chi_{1}$ for some $\chi_{1}$. Then for any
$\chi_{2}<\chi_{1}$, there exist $1\leq j_{1}<j_{2}<\cdots<j_{l}\leq k$ such
that
* •
$\rho:=\frac{l}{k}\geq\frac{\chi_{1}-\chi_{2}}{C-\chi_{2}}$;
* •
for each $1\leq n\leq l$ and each $1\leq m\leq j_{n}$, one has
$\sum_{i=m}^{j_{n}}a_{i}\geq(j_{n}-m+1)\chi_{2}.$
### 2.3 Plaque families and uniform size of invariant manifolds
The existence of plaque family was given by Theorem 5.5 in [HPS] for a single
diffeomorphism and was extended to a neighborhood of a diffeomorphism. See
Lemma 3.5 in [CPu]. Given a diffeomorphism $f\in\operatorname{Diff}^{1}(M)$,
let $\Lambda$ be an invariant compact set and $E$ be a vector bundle over
$\Lambda$. For $x\in\Lambda$ and $r>0$, let us denote
$E(x,r):=\big{\\{}v\in
E(x)|~{}\|v\|<r\big{\\}}\textrm{~{}and~{}}E(r)=\cup_{x\in\Lambda}E(x,r).$
###### Theorem 2.6 (Plaque Family Theorem).
Let $f\in\operatorname{Diff}^{1}(M)$ and $\Lambda$ be an invariant compact set
with the dominated splitting of the form $T_{\Lambda}M=E\oplus F.$
Then there exist a $C^{1}$-neighborhood ${\cal U}$ of $f$ and a neighborhood
$U$ of $\Lambda$ such that for any $g\in{\cal U}$, the maximal invariant
compact set $\Lambda_{g}$ of $g$ in $U$ admits a dominated splitting
$T_{\Lambda_{g}}M=E_{g}\oplus F_{g},$ and there exist two continuous families
of maps ${\cal W}^{cs}_{g}:E_{g}(1)\to M$ and ${\cal W}^{cu}_{g}:F_{g}(1)\to
M$ satisfying the following properties:
* •
for each $x\in\Lambda_{g}$, the map ${\cal W}^{cs}_{x,g}:E_{g}(x,1)\to M$
(resp. ${\cal W}^{cu}_{x,g}:F_{g}(x,1)\to M$) is a $C^{1}$-embedding, $x={\cal
W}^{cs}_{x,g}(0_{x})$ (resp. $x={\cal W}^{cu}_{x,g}(0_{x})$ ) and its graph is
tangent to $E_{g}(x)$ (resp. $F_{g}(x)$) at the point $x$;
* •
the families $\\{{\cal W}^{cs}_{x,g}\\}$ and $\\{{\cal W}^{cu}_{x,g}\\}$ of
$C^{1}$-embedding maps are continuous with respect to $x,g$ in
$C^{1}$-topology;
* •
for any $\delta\in(0,1)$, there exists $\delta^{\prime}>0$ such that
$g({\cal W}^{cs}_{x,g}(E_{g}(x,\delta^{\prime})))\subset{\cal
W}^{cs}_{g(x),g}(E_{g}(g(x),\delta))$
and
$g^{-1}({\cal W}^{cu}_{x,g}(F_{g}(x,\delta^{\prime})))\subset{\cal
W}^{cu}_{g^{-1}(x),g}(F_{g}(g^{-1}(x),\delta)).$
For $g\in{\cal U}$, $x\in\Lambda_{g}$ and $\delta\in(0,1)$, we denote by
${\cal W}^{cs}_{\delta}(x,g):={\cal W}^{cs}_{x,g}(E_{g}(x,\delta)))$ and
${\cal W}^{cu}_{\delta}(x,g):={\cal W}^{cu}_{x,g}(F_{g}(x,\delta)))$.
When there is no ambiguity, we will drop the index $g$ for simplicity. The
$C^{1}$-submanifolds ${\cal W}^{cs}_{\delta}(x,g)$ (resp. ${\cal
W}^{cu}_{\delta}(x,g)$) is called the $cs$-plaque (resp. $cu$-plaque) centered
at $x$ and of radius $\delta$. The last item in Theorem 2.6 tells us that the
$cu$-plaques and $cs$-plaques are locally $g$-invariant.
The following result gives the uniformity on the size of unstable manifolds
for a single diffeomorphism, whose proof can be found in [ABC, Section 8.2].
###### Lemma 2.7.
Let $f\in\operatorname{Diff}^{1}(M)$ and $\Lambda$ be an invariant compact set
with the dominated splitting of the form $T_{\Lambda}M=E\oplus F.$ Consider a
$cu$-plaque family ${\cal W}^{cu}$ corresponding to the bundle $F$. For any
$\chi>0$, there exists $\delta>0$ such that if $x\in\Lambda$ satisfies that
$\prod_{i=0}^{n-1}\|Df|_{F(f^{-i}(x))}\|\leq e^{-n\chi}\textrm{~{}for any
$n\in\mathbb{N}$},$
then ${\cal W}^{cu}_{\delta}(x)$ is contained in the unstable manifold of $x$.
Using the uniform continuity of the plaque families among nearby systems and
Lemma 2.7, one can obtain the similar result for diffeomorphisms close to $f$
and for our purpose we state it in partially hyperbolic setting.
###### Lemma 2.8.
Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism
with $\operatorname{dim}(E^{c})=1$, and $\widetilde{{\cal U}}$ be a
$C^{1}$-neighborhood of $f$ given by Lemma 2.6 together with a plaque family
${\cal W}^{cu}_{g}$ with $g\in\widetilde{\cal U}$, which corresponds to the
bundle $E_{g}^{c}\oplus E_{g}^{u}$ and depends continuously on $g$.
Then for any $\chi>0$, there exist a $C^{1}$-small neighborhood ${\cal
U}\subset\widetilde{{\cal U}}$ of $f$, and a small constant
${\varepsilon}_{0}>0$ such that
* •
for any $x\in M$, one has
* –
$g^{-1}({\cal W}^{cu}_{{\varepsilon}_{0}}(x,g))\subset{\cal
W}^{cu}_{1/2}(g^{-1}(x),g)$;
* –
$\big{|}\log\|Dg^{-1}|_{E_{g}^{c}(x)}\|-\log\|Dg^{-1}|_{T_{y}{\cal
W}^{cu}_{{\varepsilon}_{0}}(x,g)}\|\big{|}<\chi/4\,\textrm{~{}for any
$y\in{\cal W}^{cu}_{{\varepsilon}_{0}}(x,g)$};$
* •
if a point $x\in M$ satisfies that $\|Dg^{-n}|_{E_{g}^{c}(x)}\|<e^{-n\chi}$
for any $n\in{\mathbb{N}}$, then
* –
${\cal W}^{cu}_{{\varepsilon}_{0}}(x,g)$ is contained in the unstable manifold
of $x$ and is tangent everywhere to $E_{g}^{c}\oplus E_{g}^{u};$
* –
$\|Dg^{-n}|_{E_{g}^{c}(y)}\|<e^{-3n\chi/4}$ for any $n\in{\mathbb{N}}$ and any
$y\in{\cal W}^{cu}_{{\varepsilon}_{0}}(x,g)$.
Given ${\varepsilon}>0$, $\ell>0$ and a plaque family ${\cal W}^{cu}$
corresponding to the bundle $E^{c}\oplus E^{u}$, one says that the strong
stable foliation $\mathcal{F}^{s}$ is _$(\ell,{\varepsilon})$ -dense with
respect to ${\cal W}^{cu}$_, if for any $x,y\in M$ the local strong stable
manifold $\mathcal{F}^{s}_{\ell}(x)$ has non-empty transverse intersection
with ${\cal W}^{cu}_{\varepsilon}(y)$, where $\mathcal{F}^{s}_{\ell}(x)$ is
the $\ell$-neighborhood of $x$ in $\mathcal{F}^{s}(x)$ under its intrinsic
topology.
The following result comes from the uniform continuity of strong stable
manifolds and the plaque families with respect to the bundle $E^{c}\oplus
E^{u}$.
###### Lemma 2.9.
Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism
and $\widetilde{{\cal U}}$ be a $C^{1}$ neighborhood of $f$ given by Lemma 2.6
together with a plaque family ${\cal W}^{cu}_{g}$ with $g\in\widetilde{\cal
U}$, which corresponds to the bundle $E_{g}^{c}\oplus E_{g}^{u}$ and depends
continuously on $g$. Assume, in addition, that strong stable foliation of $f$
is minimal.
Then for any ${\varepsilon}>0$, there exist a $C^{1}$-neighborhood ${\cal
U}_{\varepsilon}\subset\widetilde{\cal U}$ of $f$ and a constant $\ell>0$ such
that the strong stable foliation of $g\in{\cal U}_{\varepsilon}$ is
$(\ell,{\varepsilon})$-dense with respect to ${\cal W}_{g}^{cu}$.
### 2.4 Constructions of horseshoes from hyperbolic periodic orbits
Katok and Mendoza [KM] gave a way to construct horseshoes approaching
hyperbolic ergodic measures on surfaces in $C^{1+\alpha}$-setting and this has
been generalized to higher dimension in [Ge]. Here we state a mechanism using
a collection of periodic orbits to find horseshoes in $C^{1}$-setting with
domination.
Given an invariant compact set $\Lambda$ of $f\in\operatorname{Diff}^{1}(M)$
exhibiting a dominated splitting of the form $T_{\Lambda}M=E\oplus F$ and
given $\chi>0$, let us define
${\rm
NUH}_{\chi}=\big{\\{}x\in\Lambda|\,\prod_{i=0}^{k-1}\|Df|_{E(f^{i}(x))}\|\leq
e^{-k\chi},~{}\prod_{i=0}^{k-1}\|Df^{-1}|_{F(f^{-i}(x))}\|\leq
e^{-k\chi}\textrm{~{}for any $k\in{\mathbb{N}}$}\big{\\}}.$
###### Theorem 2.10 (Theorem 4.1 in [YZ]).
Let $f\in\operatorname{Diff}^{1}(M)$ and $\Lambda$ be an invariant compact set
admitting a dominated splitting of the form $T_{\Lambda}M=E\oplus F$. For any
$\chi>0$ and any ${\varepsilon}>0$, there exists $\xi_{0}>0$ satisfying the
following properties. For any $n\in{\mathbb{N}}$ and any $\xi\in(0,\xi_{0})$,
if there exist periodic points $p_{1},\cdots,p_{m}\in{\rm NUH}_{\chi}$ of the
same period $l$ such that
* •
$d(p_{i},p_{j})<\xi/16$ for $1\leq i<j\leq m$;
* •
$\\{p_{1},\cdots,p_{m}\\}$ is a $(l,\xi)$-separated set;
then there exists a hyperbolic basic set $K$ whose stable bundle has dimension
$\operatorname{dim}(E)$ such that
* –
$h_{top}(f,K)\geq\frac{\log m}{l}$;
* –
the set ${\cal M}_{\rm inv}(f,K)$ is contained in the
${\varepsilon}$-neighborhood of $\big{\\{}\sum_{i=1}^{m}t_{i}\delta_{{\cal
O}_{p_{i}}}|\,t_{i}\geq 0,~{}\sum_{i=1}^{m}t_{i}=1\big{\\}}$, where
$\delta_{{\cal O}_{p_{i}}}=\frac{1}{l}\sum_{j=0}^{l-1}\delta_{f^{j}(p_{i})}.$
### 2.5 Unstable entropy and entropy formulas
In this section, we firstly recall the definition of the metric entropy along
unstable manifolds, which was defined by F. Ledrappier and L. Young [LeYo1,
LeYo2]. Then we recall its applications based on some entropy formulas.
Given a Borel probability measure $\mu$ and a foliation ${\cal F}$ on the
manifold $M$, a measurable partition ${\cal A}$ is called _$\mu$ -subordinate
to ${\cal F}$_ if for $\mu$ a.e. $x\in M$, one has
* •
${\cal A}(x)\subset{\cal F}(x)$, where ${\cal A}(x)$ denotes the element of
${\cal A}$ containing $x$;
* •
there exists $r_{x}>0$ such that ${\cal F}_{r_{x}}(x)\subset{\cal A}(x)$,
where ${\cal F}_{r_{x}}(x)$ denotes the $r_{x}$-neighborhood of $x$ in ${\cal
F}(x)$ under its intrinsic topology.
Given $f\in\operatorname{Diff}^{1}(M)$, a measurable partition ${\cal A}$ is
_increasing_ if $f({\cal A})\prec{\cal A}$. If
$f\in\operatorname{Diff}^{1}(M)$ is partially hyperbolic and $\mu$ is an
$f$-invariant measures, then there exists an increasing measurable partition
which is $\mu$-subordinate to the strong unstable foliation due to [LeS] (see
also [Y]). Given two increasing measurable partitions ${\cal A}_{1},{\cal
A}_{2}$ which are $\mu$-subordinate to the unstable foliation, Ledrappier and
Young [LeYo1] proved that $H_{\mu}({\cal A}_{1}|f({\cal A}_{1}))=H_{\mu}({\cal
A}_{2}|f({\cal A}_{2}))$, and this yields the following definition.
###### Definition 2.11.
Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism
and $\mu\in{\cal M}_{inv}(f)$. The _unstable (metric)-entropy_ of $\mu$ is
defined as
$h_{\mu}(f,{\cal F}^{u})=H_{\mu}\big{(}{\cal A}|f({\cal A})\big{)},$
where ${\cal A}$ is an increasing measurable partition $\mu$-subordinate to
${\cal F}^{u}.$
Recall that an invariant measure of a $C^{1+\alpha}$-partially hyperbolic
diffeomorphism is called a _$u$ -Gibbs state_, if its conditional measures
along the strong unstable manifolds are absolutely continuous with respect to
the Lebesgue measure on the strong unstable manifolds. The existence of
$u$-Gibbs states was firstly proved by Y. Pesin and Y. Sinai [PeSi]. It has
been shown by Ledrappier that u-Gibbs property is equivalent to the unstable
entropy formula.
###### Theorem 2.12 (Théorème in [Le]).
Let $f\in\operatorname{Diff}^{1+\alpha}(M)$ be partially hyperbolic and
$\mu\in{\cal M}_{inv}(f)$. Then $\mu$ is a $u$-Gibbs state if and only if
$h_{\mu}(f,\mathcal{F}^{u})=\int\log|\operatorname{det}(Df|_{E^{u}})|\operatorname{d}\mu.$
For a $C^{1}$-partially hyperbolic diffeomorphism $f$, let us denote
$G^{u}(f):=\big{\\{}\mu\in\mathcal{M}_{inv}(f)|h_{\mu}(f,\mathcal{F}^{u})=\int\log|\operatorname{det}(Df|_{E^{u}})|\operatorname{d}\mu\big{\\}}.$
(1)
The following result describes the set $G^{u}(f)$.
###### Lemma 2.13 ([CYZ, HYY, Y]).
Let $f$ be a $C^{1}$-partially hyperbolic diffeomorphism. Then
* •
$G^{u}(f)$ is a non-empty convex compact set;
* •
$G^{u}(f)$ varies upper semi-continuously with respect to $f$;
* •
if $\mu$ belongs to $G^{u}(f)$, so do its ergodic components.
In $C^{1}$-setting, the large deviation property for the set $G^{u}(f)$ holds
due to [CYZ]. Recall that for the partially hyperbolic splitting
$TM=E^{s}\oplus E^{c}\oplus E^{u}$, one can associate a continuous cone field
of size $\beta>0$ to the strong unstable bundle $E^{u}$:
$\mathcal{C}^{u}_{\beta}:=\big{\\{}v\in TM|~{}v=v^{cs}+v^{u},~{}v^{cs}\in
E^{s}\oplus E^{c},~{}v^{u}\in
E^{u},~{}\|v^{cs}\|\leq\beta\cdot\|v^{u}\|\big{\\}}.$
For $\beta>0$ small, the cone field $\mathcal{C}^{u}_{\beta}$ is strictly
$Df$-invariant, that is, there exists $\beta^{\prime}\in(0,\beta)$ such that
$Df(\mathcal{C}^{u}_{\beta}(x))\subset\mathcal{C}^{u}_{\beta^{\prime}}(f(x))$
for any $x\in M$. When there is no ambiguity, we would drop the index $\beta$
for simplicity.
###### Theorem 2.14 (Theorem D′ in [CYZ]).
Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism.
Then for any continuous function $\varphi:M\to\mathbb{R}$ and any $\kappa>0$,
there exist positive constants $r_{0},a_{\kappa},b_{\kappa}>0$ and a
$Df$-invariant cone field $\mathcal{C}^{u}$ for the strong unstable bundle
such that for any disc $D$ tangent to the cone field $\mathcal{C}^{u}$ and of
diameter smaller than $r_{0}$, one has
$\operatorname{Leb}_{D}\bigg{(}\big{\\{}x\in
D:~{}~{}\operatorname{d}\big{(}\frac{1}{n}S_{n}\varphi(x),I(\varphi)\big{)}>\kappa\big{\\}}\bigg{)}<a_{\kappa}\cdot
e^{-nb_{\kappa}}\textrm{~{} for any $n\in\mathbb{N}$},$
where $\operatorname{Leb}_{D}$ denotes the Lebesgue measure on the submanifold
$D$, $S_{n}\varphi=\sum_{i=0}^{n-1}\varphi\circ f^{i}$ and
$I(\varphi)=\big{\\{}\int\varphi\operatorname{d}\nu:\nu\in G^{u}(f)\big{\\}}.$
In $C^{1+\alpha}$-setting, $u$-Gibbs states are the candidate for SRB
measures. Recall that an invariant measure is called an _SRB_ if it admits
positive Lyapunov exponents and its conditional measures along the Pesin’s
unstable manifolds are absolutely continuous with respect to the Lebesgue
measure.
###### Theorem 2.15 (Theorem A in [LeYo1]).
Let $f\in\operatorname{Diff}^{1+\alpha}(M)$ and $\mu$ be an invariant measure
with positive Lyapunov exponents. Then $\mu$ is an SRB if and only if $\mu$
satisfies Pesin’s entropy formula, that is,
$h_{\mu}(f)=\int\sum\lambda^{+}(x)\operatorname{d}\mu,$
where $\sum\lambda^{+}(x)$ is the sum of positive Lyapunov exponents of $x$
with multiplicities counted.
###### Remark 2.16.
Ledrappier and Young [LeYo1] proved this result in $C^{2}$-setting and Brown
[Br] generalized it to $C^{1+\alpha}$-setting.
In [LeYo2], Ledrappier and Young also gave an entropy formula in terms of
Lyapunov exponents and transverse dimensions. The following result is a direct
consequence of Corollary 7.2.2 in [LeYo2] which assumes that $f$ is $C^{2}$,
and it has been relaxed to $C^{1+\alpha}$-setting in [Br].
###### Theorem 2.17.
Let $f\in\operatorname{Diff}^{1+\alpha}(M)$ be a partially hyperbolic
diffeomorphism and $\mu\in\mathcal{M}_{inv}(f)$. If all the center Lyapunov
exponents of $\mu$ are non-positive, then $h_{\mu}(f)=h_{\mu}(f,{\cal
F}^{u}).$
### 2.6 Mostly expanding partially hyperbolic diffeomorphisms
Recall that a partially hyperbolic diffeomorphism
$f\in\operatorname{Diff}^{1+\alpha}(M)$ is _mostly expanding_ , if all the
center Lyapunov exponents of each $u$-Gibbs state of $f$ are positive.
The following result comes from Lemma 4.1 and Theorem B in [AnV1], and one can
also refer to the proof of Theorem C in [Y]. It tells us that mostly expanding
is a $C^{1}$-open property.
###### Theorem 2.18.
Let $f\in\operatorname{Diff}^{1+\alpha}(M)$ be a mostly expanding partially
hyperbolic diffeomorphism. Then there exist a constant $\chi>0$, an integer
$n_{0}\in\mathbb{N}$ and a $C^{1}$-neighborhood ${\cal V}$ of $f$ such that
for each $g\in{\cal V}\cap\operatorname{Diff}^{1+\alpha}(M)$ and any $u$-Gibbs
state $\mu$ of $g$, one has
$\int\log\|Dg^{-n_{0}}|_{E^{c}}\|\operatorname{d}\mu<-\chi.$
As the set of $u$-Gibbs states of $f\in\operatorname{Diff}^{1+\alpha}(M)$
coincides with the compact set $G^{u}(f)$ (due to Theorem 2.12) and $G^{u}(f)$
varies upper semi-continuously with respect to $f$ (due to Lemma 2.13), one
immediately obtains the following.
###### Corollary 2.19.
Let $f\in\operatorname{Diff}^{1+\alpha}(M)$ be a mostly expanding partially
hyperbolic diffeomorphism. Then there exist a constant $\chi>0$, an integer
$n_{0}\in\mathbb{N}$ and a $C^{1}$-neighborhood ${\cal V}$ of $f$ such that
for each $g\in{\cal V}$ and any $\mu\in G^{u}(g)$, one has
$\int\log\|Dg^{-{n_{0}}}|_{E^{c}}\|\operatorname{d}\mu<-\chi.$
In general the minimality of strong stable foliation is not an open property.
Nevertheless, the following result tells us that under the hypothesis of
mostly expanding property, the minimality of strong stable foliation is a
$C^{1}$-open property.
###### Theorem 2.20.
Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism
with $\operatorname{dim}(E^{c})=1.$ Assume that
* •
$\int\log\|Df|_{E^{c}}\|\operatorname{d}\mu>0$ for any $\mu\in G^{u}(f)$;
* •
the strong stable foliation is minimal.
Then there exists a $C^{1}$-neighborhood
$\mathcal{U}\subset\operatorname{Diff}^{1}(M)$ of $f$ such that for each
$g\in{\cal U}$, one has
* –
$\int\log\|Dg|_{E_{g}^{c}}\|\operatorname{d}\mu>0$ for any $\mu\in G^{u}(g)$;
* –
the strong stable foliation of $g$ is minimal.
###### Proof.
As $\operatorname{dim}(E^{c})=1$, by Corollary 2.19, there exist a
$C^{1}$-neighborhood ${\cal V}_{1}$ of $f$ and a constant $\chi>0$ such that
for any $g\in{\cal V}_{1}$, each invariant measure $\mu\in G^{u}(g)$ satisfies
$\int\log\|Dg|_{E^{c}_{g}}\|\operatorname{d}\mu>\chi.$ For the constant
$\chi>0$, by Lemma 2.8, there exist a $C^{1}$-neighborhood $\widetilde{\cal
V}_{2}\subset\operatorname{Diff}^{1}(M)$ of $f$, a plaque family ${\cal
W}^{cu}_{g}$ with $g\in\widetilde{\cal V}_{2}$ and a constant
${\varepsilon}_{0}>0$ satisfying the posited properties. As the strong stable
foliation of $f$ is minimal, by Lemma 2.9, there exist a $C^{1}$-neighborhood
${\cal V}_{2}\subset\widetilde{\cal V}_{2}$ of $f$ and $\ell_{0}>0$ such that
for any $g\in{\cal V}_{2}$ and any $x,y\in M$, the local strong stable
manifold $\mathcal{F}^{s}_{\ell_{0}}(x)$ of $g$ has non-empty transverse
intersection with ${\cal W}_{{\varepsilon}_{0}}^{cu}(y,g).$
Let ${\cal U}={\cal V}_{1}\cap{\cal V}_{2}$, and fix $g\in{\cal U}$. Let us
denote by $\chi^{u}:=\inf_{\mu\in
G^{u}(g)}\int\log\|Dg|_{E^{c}_{g}}\|\operatorname{d}\mu>\chi,$
$\varphi(x)=\log\|Dg|_{E^{c}_{g}(x)}\|$ and $\kappa=\frac{\chi^{u}-\chi}{2}.$
Now, apply Theorem 2.14 to the continuous function $\varphi$ and $\kappa$, and
one obtains positive constants $r_{0},a_{\kappa},b_{\kappa}$ such that for any
$w\in M$, one has
$\operatorname{Leb}_{\mathcal{F}_{r_{0}}^{u}(w)}\bigg{(}\big{\\{}y\in\mathcal{F}_{r_{0}}^{u}(w):~{}~{}\operatorname{d}\big{(}\frac{1}{n}S_{n}\varphi(y),I(\varphi)\big{)}>\kappa\big{\\}}\bigg{)}<a_{\kappa}\cdot
e^{-nb_{\kappa}}\,\textrm{~{}for any $n\in\mathbb{N}.$}$ (2)
Let $\chi_{\rm max}(g)=\sup_{x\in M}|\varphi(x)|.$ Apply $C=\chi_{\rm
max}(g)$, $\chi_{1}=(\chi^{u}+\chi)/2$ and $\chi_{2}=\chi$ to Lemma 2.5, and
one obtains $\rho\in(0,1)$ with the posited properties
Fix $x,z\in M$. We are going to show that $\mathcal{F}^{s}(z)\cap
B_{2{\varepsilon}}(x)\neq\emptyset$ for any ${\varepsilon}>0$. For any
${\varepsilon}>0$, by the uniform expansion of $g$ along the strong unstable
foliation, there exists an integer $k=k({\varepsilon})$ such that
$\mathcal{F}^{u}_{r_{0}}(g^{k}(x))\subset
g^{k}(\mathcal{F}_{\varepsilon}^{u}(x)).$ Take $N_{1}$ large enough such that
for any $j\geq N_{1},$ one has
* •
$a_{\kappa}\cdot
e^{-jb_{\kappa}}<\operatorname{Leb}_{\mathcal{F}_{r_{0}}^{u}(g^{k}(x))}(\mathcal{F}_{r_{0}}^{u}(g^{k}(x))/2,$
* •
$e^{-j\chi/2}\cdot{\varepsilon}_{0}\cdot\|Dg^{-1}\|^{k}<{\varepsilon}$, where
$\|Dg^{-1}\|=\sup_{w\in M}\|Dg^{-1}(w)\|$.
For $n>\rho^{-1}N_{1}+1$, by Equation (2), one has
$\operatorname{Leb}_{\mathcal{F}_{r_{0}}^{u}(g^{k}(x))}\bigg{(}\big{\\{}y\in\mathcal{F}_{r_{0}}^{u}(g^{k}(x)):~{}~{}\operatorname{d}\big{(}\frac{1}{n}S_{n}\varphi(y),I(\varphi)\big{)}>\kappa\big{\\}}\bigg{)}<\operatorname{Leb}_{\mathcal{F}_{r_{0}}^{u}(g^{k}(x))}(\mathcal{F}_{r_{0}}^{u}(g^{k}(x))/2.$
Therefore there exists a point $y\in\mathcal{F}^{u}_{r_{0}}(g^{k}(x))$ such
that
$\operatorname{d}\big{(}\frac{1}{n}S_{n}\varphi(y),I(\varphi)\big{)}\leq\kappa$,
and thus
$\frac{1}{n}S_{n}\varphi(y)\geq\chi^{u}-\kappa=\frac{\chi^{u}+\chi}{2}$. By
Pliss lemma, there exists $n^{\prime}\geq n\rho>N_{1}$ such that
$\|Dg^{-j}|_{E^{c}_{g}(g^{n^{\prime}}(y))}\|\leq e^{-j\chi}\,\textrm{~{}for
any $j\in[1,n^{\prime}].$}$ (3)
As $g\in{\cal U}\subset{\cal V}_{2}$, the local stable manifold
$\mathcal{F}^{s}_{\ell_{0}}(g^{k+n^{\prime}}(z))$ has non-empty transverse
intersection with ${\cal W}^{cu}_{{\varepsilon}_{0}}(g^{n^{\prime}}(y),g).$ As
the strong stable foliation is $g$-invariant, $\mathcal{F}^{s}(g^{k}(z))$ has
non-empty transverse intersection with $g^{-n^{\prime}}\big{(}{\cal
W}^{cu}_{{\varepsilon}_{0}}(g^{n^{\prime}}(y),g)\big{)}$. As $g\in{\cal
V}_{2}\subset\widetilde{\cal V}_{2}$, by Equation (3), the first item in Lemma
2.8 and the locally invariant property of the plaque family, one has
$g^{-n^{\prime}}\big{(}{\cal
W}^{cu}_{{\varepsilon}_{0}}(g^{n^{\prime}}(y),g)\big{)}\subset{\cal
W}^{cu}_{e^{-n^{\prime}\chi/2}{\varepsilon}_{0}}(y,g).$
Note that
$e^{-n^{\prime}\chi/2}\cdot{\varepsilon}_{0}\cdot\|Dg^{-1}\|^{k}<{\varepsilon}$
due to $n^{\prime}>N_{1}$, then the strong stable manifold
$\mathcal{F}^{s}(z)$ intersects ${\cal W}^{cu}_{{\varepsilon}}(g^{-k}(y),g)$
which is contained in $B_{\varepsilon}(g^{-k}(y))$. Since $g^{-k}(y)\in{\cal
F}^{u}_{\varepsilon}(x)$, one has $\mathcal{F}^{s}(z)\cap
B_{2{\varepsilon}}(x)\neq\emptyset.$ By the arbitrariness of $x,z$ and
${\varepsilon}$, one deduces that the strong stable foliation of $g$ is
minimal. ∎
It has been shown in [Y] that there are plenty of mostly expanding partially
hyperbolic diffeomorphisms among conservative partially hyperbolic
diffeomorphisms with positive center Lyapunov exponents. Recall that a
partially hyperbolic diffeomorphism is _accessible_ , if any pair of points
can be connected by a path which is a concatenation of paths lying entirely in
a strong stable or a strong unstable manifold.
###### Theorem 2.21 (Theorem D in [Y]).
Let $f\in\operatorname{Diff}_{\rm m}^{1+\alpha}(M)$ be a volume preserving
partially hyperbolic diffeomorphism with $\operatorname{dim}(E^{c})=1.$ Assume
that $f$ is accessible and the volume measure has positive center Lyapunov
exponent. Then $f$ is mostly expanding.
Instead of assuming accessibility, one can assume the minimality of strong
stable foliation and obtain the similar result. It will be used to prove
Corollary B.
###### Theorem 2.22.
Let $f\in\operatorname{Diff}_{\rm m}^{1+\alpha}(M)$ be a partially hyperbolic
diffeomorphism preserving a smooth volume ${\rm m}$. Assume that
* •
the strong stable foliation of $f$ is minimal;
* •
$\operatorname{dim}(E^{c})=1$ and
$\int\log\|Df|_{E^{c}}\|\operatorname{d}\operatorname{m}>0$.
Then one has that
* –
$f$ is mostly expanding;
* –
$(f,\operatorname{m})$ is $C^{1}$-stably ergodic, that is, there exists a
$C^{1}$-neighborhood $\mathcal{U}\subset\operatorname{Diff}^{1}(M)$ of $f$
such that each $g\in{\cal U}\cap\operatorname{Diff}_{\rm m}^{1+\alpha}(M)$ is
ergodic with respect to the volume measure $\operatorname{m}.$
###### Proof.
By the absolute continuity of the strong stable and unstable foliations, each
ergodic component of the volume measure $\operatorname{m}$ is both a $u$-Gibbs
state and an $s$-Gibbs state (i.e. a $u$-Gibbs state for $f^{-1}$).
By assumption, there exists an ergodic component $\operatorname{m}_{0}$ of
$\operatorname{m}$ such that $\chi^{c}(\operatorname{m}_{0},f)>0.$ Note that
$\operatorname{m}_{0}$ is a $u$-Gibbs state for $f$ as well as a $u$-Gibbs
state for $f^{-1}$. As $\operatorname{m}_{0}$ has negative center Lyapunov
exponent for $f^{-1}$, thus $\operatorname{m}_{0}$ is also an SRB for
$f^{-1}$.
###### Claim 2.23.
$f^{-1}$ has a unique $u$-Gibbs state $\operatorname{m}_{0}$.
###### Proof.
Let $\mu$ be an ergodic $u$-Gibbs state for $f^{-1}$. Note that the
conditional measures of $\operatorname{m}_{0}$ along strong unstable manifolds
of $f^{-1}$ are absolutely continuous with respect to the Lebesgue measure,
and $\operatorname{m}_{0}$ has negative center Lyapunov exponent for $f^{-1}$,
by the absolute continuity of the Pesin stable manifold for
$(\operatorname{m}_{0},f^{-1})$, the minimality of the strong unstable
foliation for $f^{-1}$ and the ergodicity of the measures
$\operatorname{m}_{0}$ and $\mu$, one deduces that $\operatorname{m}_{0}=\mu$.
∎
Claim 2.23 implies that $\operatorname{m}$ has only one ergodic component
$\operatorname{m}_{0}$ proving the ergodicity of $\operatorname{m}.$
Consider an ergodic $u$-Gibbs state $\mu$ for $f$, and we need to show that
$\mu$ has positive center Lyapunov exponent for $f$. By Theorem 2.12, one has
$h_{\mu}(f,\mathcal{F}^{u})=\int\log\operatorname{det}(Df|_{E^{u}})\operatorname{d}\mu$.
If $\chi^{c}(\mu,f)\leq 0$, by Theorem 2.17, one has
$h_{\mu}(f)=h_{\mu}(f,\mathcal{F}^{u})=\int\log\operatorname{det}(Df|_{E^{u}})\operatorname{d}\mu.$
As $f$ is volume preserving, one has
$\int\log\operatorname{det}(Df|_{E^{u}})\operatorname{d}\mu=-\int\log\operatorname{det}(Df|_{E^{s}\oplus
E^{c}})\operatorname{d}\mu.$ This gives that
$h_{\mu}(f)=-\int\log\operatorname{det}(Df|_{E^{s}\oplus
E^{c}})\operatorname{d}\mu$. By Theorem 2.15, $\mu$ is an SRB measure for
$f^{-1}$. By the absolute continuity of the strong unstable manifolds of
$f^{-1}$, $\mu$ is also a $u$-Gibbs state for $f^{-1}$. Then Claim 2.23 gives
that $\mu=\operatorname{m}_{0}$ which contradicts the fact that
$\chi^{c}(\mu,f)\leq 0$ and $\chi^{c}(\operatorname{m}_{0},f)>0$. This proves
that all the ergodic $u$-Gibbs states of $f$ have positive center Lyapunov
exponent, and hence $f$ is mostly expanding.
Let ${\cal U}\subset\operatorname{Diff}^{1}(M)$ be the $C^{1}$-open
neighborhood of $f$ given by Theorem 2.20. For each $g\in{\cal
U}\cap\operatorname{Diff}^{1+\alpha}_{\operatorname{m}}(M),$ by the absolute
continuity of the strong unstable foliation, one has $\operatorname{m}\in
G^{u}(g)$, and thus
$\int\log\|Dg|_{E^{c}_{g}}\|\operatorname{d}\operatorname{m}>0$ due to the
first item of Theorem 2.20. Now apply the arguments above to
$(g,\operatorname{m})$ with $g\in{\cal
U}\cap\operatorname{Diff}^{1+\alpha}_{\operatorname{m}}(M)$, and one deduces
that $(g,\operatorname{m})$ is ergodic. ∎
## 3 Existence of periodic orbits with prescribed behavior
This section gives the main ingredient for proving our main results. We will
first find some periodic orbits which shadow the generic points of an ergodic
measure with non-positive center Lyapunov exponent for some proportion of
time.
Throughout this section, given a foliation $\mathcal{F}$ on $M$ with
$C^{1}$-leaves, $x\in M$ and $\ell>0$, we denote by $\mathcal{F}_{\ell}(x)$
the $\ell$-neighborhood of $x$ in the leaf $\mathcal{F}(x)$ under its
intrinsic topology. Given a partially hyperbolic diffeomorphism, we simply
call a plaque family corresponding to the splitting $E^{c}\oplus E^{u}$ as a
_$cu$ -plaque family_.
Given a partially hyperbolic diffeomorphism $f$ with
$\operatorname{dim}(E^{c})=1$ and $\chi>0$, define the Pesin block of the
following form:
${\rm NUH}_{\chi}=\big{\\{}x\in M:\|Df^{-n}|_{E^{c}(x)}\|\leq
e^{-n\chi}\textrm{~{}for any $n\in\mathbb{N}$}\big{\\}}.$
###### Theorem 3.1.
Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism
with $\operatorname{dim}(E^{c})=1$. Suppose that
* •
there exists $\chi_{0}>0$ such that
$\int\log\|Df|_{E^{c}}\|\operatorname{d}\mu>\chi_{0}$ for any $\mu\in
G^{u}(f)$;
* •
let ${\cal W}^{cu}$ be a $cu$-plaque family and ${\varepsilon}_{0}>0$ be given
by Lemma 2.8 corresponding to $\chi_{0}$, then there exists a constant
$\ell_{0}>0$ such that the strong stable foliation is
$(\ell_{0},{\varepsilon}_{0}/2)$-dense with respect to ${\cal W}^{cu}$.
Then there exist a constant $\rho_{1}>0$ and a hyperbolic periodic point $q$
with positive center Lyapunov exponent and whose unstable manifold has inner
radius ${\varepsilon}_{0}$ such that the following properties hold. For any
ergodic measure $\nu$ with $\chi_{0}/24<\chi^{c}(\nu)\leq 0$ and any
${\varepsilon}>0$, there exists $\xi_{1}>0$ such that for any
$\xi\in(0,\xi_{1})$, there exist periodic points $p_{1},\cdots,p_{m}\in{\rm
NUH}_{2(|\chi^{c}(\nu)|+{\varepsilon})}$ of the same period $l\in\mathbb{N}$
satisfying that
* •
each $p_{i}$ is homoclinically related to $q$;
* •
$\operatorname{d}(p_{i},p_{j})<\xi/16$ for $1\leq i,j\leq m$;
* •
$\\{p_{1},\cdots,p_{m}\\}$ is a $(l,\xi)$-separated set;
* •
$\frac{\log
m}{l}>\frac{h_{\nu}(f)-{\varepsilon}}{1+\rho_{1}(|\chi^{c}(\nu)|+{\varepsilon})};$
* •
$\operatorname{d}(\delta_{{\cal
O}_{p_{j}}},\nu)<\rho_{1}(|\chi^{c}(\nu)|+{\varepsilon})$ for any $1\leq j\leq
m$.
The periodic orbits obtained in Theorem 3.1 would be used to find horseshoes
approaching the ergodic measure $\nu$ with $\chi^{c}(\nu)\leq 0$ in
weak$*$-topology and in entropy. To find periodic orbits, we first need to
find some periodic pseudo-orbit which are close to the ergodic measure in
weak$*$-topology and have weak (but not too weak) center Lyapunov exponent,
then Liao’s shadowing lemma is applied. Some similar ideas can be found in
[BZ, YZ], which assume both minimality of strong foliations and existence of
$cs$-blenders and $cu$-blenders (one can refer to [BD] for the definition of
blenders).
### 3.1 Generating orbit segments with weak hyperbolicity
The following is a key result to find pseudo orbits with some weak expansion
along the center and satisfying the assumption in Liao’s shadowing lemma.
###### Proposition 3.2.
Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism
with $\operatorname{dim}(E^{c})=1$. Suppose that
* •
there exists $\chi_{0}>0$ such that
$\int\log\|Df|_{E^{c}}\|\operatorname{d}\mu>\chi_{0}$ for any $\mu\in
G^{u}(f)$;
* •
let ${\cal W}^{cu}$ be a $cu$-plaque family and ${\varepsilon}_{0}>0$ be given
by Lemma 2.8 corresponding to $\chi_{0}$, then there exists a constant
$\ell_{0}>0$ such that the strong stable foliation is
$(\ell_{0},{\varepsilon}_{0}/2)$-dense with respect to ${\cal W}^{cu}$.
Then there exist a hyperbolic periodic point $q$ and a constant $\rho_{0}>0$
with the following properties:
* •
$\|Df^{-i}|_{E^{c}(q)}\|<e^{-i\chi_{0}}$ for any $i\in{\mathbb{N}}$;
* •
for any $C>1$, $\chi\in(-\chi_{0}/24,\chi_{0}/24)$,
${\varepsilon}\in(0,\chi_{0}/24)$ and $d>0$, there exist integers
$\tau=\tau({\varepsilon},d)<T=T(C,\chi,{\varepsilon},d)$ such that for any
$n\geq T$ and any $x\in M$ satisfying
$C^{-1}\cdot e^{k(\chi-{\varepsilon})}\leq\|Df^{k}|_{E^{c}(x)}\|\leq C\cdot
e^{k(\chi+{\varepsilon})}\>\textrm{~{}for any $0\leq k\leq n,$}$
there exist a point $w_{x}\in W^{u}_{d/2}(q)$ and an integer
$l_{x}\in\big{(}n,\>n+\rho_{0}\cdot(|\chi|+{\varepsilon})\cdot n\big{)}$ such
that
* –
$f^{l_{x}}(w_{x})\in{\cal F}^{s}_{d/2}(q)$;
* –
$\operatorname{d}(f^{\tau+i}(w_{x}),f^{i}(x))<d$ for any $i\in[\tau,n]$;
* –
$\|Df^{-i}|_{E^{c}(f^{l_{x}}(w_{x}))}\|\leq e^{-i2(|\chi|+{\varepsilon})}$ for
any $i\in[1,l_{x}]$.
The idea of proving Proposition 3.2 is that we first replace $x$ by a point
$x^{s}$ on the local unstable manifold of a hyperbolic periodic saddle $q$,
then we choose a thin $cu$-strip at $x^{s}$ which is $C^{1}$-foliated by discs
tangent to a strong unstable cone field and whose size in the center direction
is exponentially small; then following the forward orbit of $x^{s}$, we
iterate this strip for a long time $n$ and we are interested in those points
in the $cu$-strip whose forward orbits stay close to the one of $x^{s}$ until
time $n$. Then we iterate the strip at $f^{n}(x^{s})$ by some uniform finite
time $\tau$ to make the size in the strong unstable direction at some uniform
scale so that we can apply the large deviation property for the set $G^{u}(f)$
given by Theorem 2.14. Combining with Pliss lemma, we can find a point has
”Pliss-property” which guarantees that after iterating the strip at
$f^{n+\tau}(x^{s})$ by $m$-times, the $cu$-strip has size at scale
${\varepsilon}_{0}$, and thus it intersects the local strong stable manifold
of $q$. Our argument can guarantee that $m/n$ is at
scale-$\rho_{0}\cdot(|\chi|+{\varepsilon})$ for some uniform constant
$\rho_{0}.$
In this section, a _center curve_ means a $C^{1}$-curve everywhere tangent to
the center bundle.
###### Proof of Proposition 3.2.
Define
$\chi_{\rm max}=\sup_{y\in M}|\log\|Df|_{E^{c}(y)}\||>0.$
By Lemma 2.13, the set $G^{u}(f)$ is non-empty and compact. Let us define
$\chi^{u}=\inf\big{\\{}\int\log\|Df|_{E^{c}}\|\operatorname{d}\mu\,|\,~{}\mu\in
G^{u}(f)\big{\\}}.$
By assumption, one has $0<\chi_{0}<\chi^{u}\leq\chi_{\rm max}.$ By the local
invariant property of the plaque family ${\cal W}^{cu}$, for
${\varepsilon}_{0}$ given in the assumption, there exists $\delta_{0}>0$ such
that
$f^{-1}({\cal W}_{\delta_{0}}^{cu}(y))\subset{\cal
W}^{cu}_{{\varepsilon}_{0}}(f^{-1}(y))\,\textrm{~{} for any $y\in M.$}$ (4)
Since hyperbolic ergodic measures are approximated by periodic measures [C,
G2, Ge], by the continuity of the function $\log\|Df|_{E^{c}}\|$, there exists
a periodic point $q$ with $\chi^{c}(\delta_{{\cal O}_{q}})>\chi_{0}.$ By Pliss
lemma (Lemma 2.5), up to replacing $q$ by some $f^{i}(q)$, one can assume that
$\|Df^{-i}|_{E^{c}(q)}\|<e^{-i\chi_{0}}\textrm{~{} for any $i\in\mathbb{N}.$}$
(5)
In the following, for simplicity, we shall assume that $q$ is a fixed point.
Up to changing a metric, we shall assume that $E^{c}$ is orthogonal to
$E^{u}$. By Lemma 2.8, the point $q$ has unstable manifold
$W^{u}_{{\varepsilon}_{0}}(q)={\cal W}^{cu}_{{\varepsilon}_{0}}(q)$ of size
${\varepsilon}_{0}$ and tangent everywhere to $E^{c}\oplus E^{u}$.
###### Claim 3.3.
For any $z\in W^{u}_{{\varepsilon}_{0}}(q)$, one has that ${\cal
W}^{cu}_{{\varepsilon}_{0}}(z)\subset W^{u}(q)$.
###### Proof.
Take $z\in W^{u}_{{\varepsilon}_{0}}(q)={\cal W}^{cu}_{{\varepsilon}_{0}}(q)$.
By Equation (5) and the second item of Lemma 2.8, one has
$\|Df^{-i}|_{E^{c}(z)}\|<e^{-3i\chi_{0}/4}\textrm{~{} for any
$i\in\mathbb{N}.$}$
Recall that ${\varepsilon}_{0}$ is given by Lemma 2.8 corresponding to
$\chi_{0}$. By the first item of Lemma 2.8, for any $w\in{\cal
W}^{cu}_{{\varepsilon}_{0}}(z)$, one has
$\|Df^{-1}|_{T_{w}{\cal W}^{cu}_{{\varepsilon}_{0}}(z)}\|<e^{-\chi_{0}/2}.$
By the local invariance of the plaque family ${\cal W}^{cu}$, one has
$f^{-1}({\cal W}^{cu}_{{\varepsilon}_{0}}(z))\subset{\cal
W}^{cu}_{{\varepsilon}_{0}\cdot e^{-\chi_{0}/2}}(f^{-1}(z)).$ Once again using
the first item of Lemma 2.8, one has
$\|Df^{-2}|_{T_{w}{\cal
W}^{cu}_{{\varepsilon}_{0}}(z)}\|<e^{\chi_{0}/2}\cdot\|Df^{-2}|_{E^{c}(z)}\|<e^{-\chi_{0}}.$
By the local invariance of the plaque familly, one has $f^{-2}({\cal
W}^{cu}_{{\varepsilon}_{0}}(z))\in{\cal W}^{cu}_{{\varepsilon}_{0}\cdot
e^{-\chi_{0}}}(f^{-2}(z)).$ Inductively using the arguments above, one deduces
that $f^{-i}({\cal W}^{cu}_{{\varepsilon}_{0}}(z))\subset{\cal
W}^{cu}_{{\varepsilon}_{0}\cdot e^{-i\chi_{0}/2}}(f^{-i}(z))$ for any
$i\in\mathbb{N}.$ Since $z\in W^{u}_{{\varepsilon}_{0}}(q)$, one deduces that
${\cal W}^{cu}_{{\varepsilon}_{0}}(z)\subset W^{u}(q)$. ∎
Now apply $\varphi(x)=\log\|Df|_{E^{c}(x)}\|$ and
$\kappa=(\chi^{u}-\chi_{0})/2$ to Theorem 2.14, and one obtains positive
constants $r_{0},a,b>0$ and a $Df$-invariant cone field
$\widetilde{\mathcal{C}}^{u}$ with posited properties. We take a smaller
$Df$-invariant cone field $\mathcal{C}^{u}\subset\widetilde{\mathcal{C}}^{u}$
such that $f$ is uniformly expanding along $\mathcal{C}^{u}$, that is, there
exists $\lambda^{u}>1$ such that
$\|Df(v)\|\geq\lambda^{u},\,\textrm{~{}for any $x\in M$ and any unit vector
$v\in\mathcal{C}^{u}(x)$}.$
Fix a $C^{1}$-foliation $\widehat{F}^{u}$ on the local unstable manifold
$W^{u}_{{\varepsilon}_{0}}(q)$, whose leaves are discs tangent to the cone
field $\mathcal{C}^{u}$ (of course, this foliation is not $f$-invariant and
not unique, and we just fix one). Let $C_{u}>1$ be an upper bound of the
Lipschitz constant for the holonomy maps (from center curves to center curves)
of this foliation $\widehat{F}^{u}$.
By the uniform continuity of the center bundle, for ${\varepsilon}>0$, there
exists $\delta\in(0,\min\\{{\varepsilon}_{0}/2,\delta_{0}\\})$ such that
$\big{|}\log\|Df|_{E^{c}(y_{1})}\|-\log\|Df|_{E^{c}(y_{2})}\|\big{|}<{\varepsilon}/4,\textrm{~{}for
any $y_{1},y_{2}\in M$ with $\operatorname{d}(y_{1},y_{2})<\delta$.}$ (6)
For any $d>0$, up to shrinking $\delta$, one can assume that $\delta<d/2$. As
$f$ is uniformly contracting and expanding along $E^{s}$ and $\mathcal{C}^{u}$
respectively, and as $q$ has local unstable manifold of size
${\varepsilon}_{0}$, there exists an integer $\tau=\tau({\varepsilon},d)>0$
such that for any $k\geq\tau$ one has
* •
$f^{k}({\cal F}^{s}_{\ell_{0}}(y))\subset{\cal F}^{s}_{\delta}(f^{k}(y))$ for
any $y\in M$;
* •
for disc $D$ tangent to $\mathcal{C}^{u}$ and of inner radius $\delta/2$, the
disc $f^{k}(D)$ is tangent to $\mathcal{C}^{u}$ and contains a disc of inner
radius $r_{0}$;
* •
$f^{-k}(W^{u}_{{\varepsilon}_{0}}(q))\subset W^{u}_{\delta}(q)$.
Take $n>\tau$ large (which will be precised later). Now for every $x\in M$
satisfying
$C^{-1}\cdot e^{k(\chi-{\varepsilon})}\leq\|Df^{k}|_{E^{c}(x)}\|\leq C\cdot
e^{k(\chi+{\varepsilon})}\>\textrm{~{}for any $0\leq k\leq n,$}$
let $x^{s}\in{\cal F}^{s}_{\ell_{0}}(x)\cap W^{u}_{{\varepsilon}_{0}/2}(q)$,
then one has
$C^{-1}\cdot e^{-3\tau\chi_{\rm max}}\cdot
e^{i(\chi-5{\varepsilon}/4)}\leq\|Df^{i}|_{E^{c}(x^{s})}\|\leq C\cdot
e^{3\tau\chi_{\rm max}}\cdot e^{i(\chi+5{\varepsilon}/4)}\>\textrm{~{}for any
$0\leq i\leq n$}.$ (7)
Now, consider a center curve $\sigma_{n}\subset W^{u}_{{\varepsilon}_{0}}(q)$
centered at $x^{s}$ of length $\delta\cdot e^{-2n(|\chi|+{\varepsilon})}$ and
the $C^{1}$-foliation $\widehat{F}^{u}$ on $W^{u}_{{\varepsilon}_{0}}(q)$
whose leaves are disks tangent to the cone field $\mathcal{C}^{u}$. Let us
denote by
$\mathfrak{F}_{n}^{cu}(x^{s}):=\cup_{z\in\sigma_{n}}\widehat{F}^{u}_{\delta}(z)$
which is a $C^{1}$-submanifold of $W^{u}_{{\varepsilon}_{0}}(q)$. By Claim
3.3, one also has $\mathfrak{F}_{n}^{cu}(x^{s})\subset{\cal
W}^{cu}_{{\varepsilon}_{0}}(x^{s})\subset W^{u}(q).$ We call
$\cup_{z\in\partial\sigma_{n}}\widehat{F}^{u}_{\delta}(z)$ as the u-boundary
of $\mathfrak{F}_{n}^{cu}(x^{s})$ which has two connected components. One
defines the center-size of $\mathfrak{F}_{n}^{cu}(x^{s})$ by the infimum of
the length of the center curves in $\mathfrak{F}_{n}^{cu}(x^{s})$ which joins
the two connected components of the u-boundary of
$\mathfrak{F}_{n}^{cu}(x^{s})$. Then the center-size of
$\mathfrak{F}_{n}^{cu}(x^{s})$ is bounded from below by
$C_{u}^{-1}\cdot\delta\cdot e^{-2n(|\chi|+{\varepsilon}))}$ and from above by
$C_{u}\cdot\delta\cdot e^{-2n(|\chi|+{\varepsilon})}$. For each $i\in[0,n]$,
let $\mathfrak{F}_{n}^{cu}(f^{i}(x^{s}))$ be the connected component of
$f^{i}(\mathfrak{F}_{n}^{cu}(x^{s}))\cap B(f^{i}(x^{s}),\delta)$ containing
$f^{i}(x^{s})$. By the locally invariant property of the plaque family, one
deduces that $\mathfrak{F}_{n}^{cu}(f^{i}(x^{s}))\subset{\cal
W}^{cu}_{\delta}(f^{i}(x^{s})).$ Besides, one can analogously define the
u-boundary and the center-size of $\mathfrak{F}_{n}^{cu}(f^{i}(x^{s}))$
respectively.
Let $N_{1}=N_{1}({\varepsilon},d,C)$ be an integer such that
$C_{u}\cdot C\cdot e^{5\tau\chi_{\rm
max}}<e^{k{\varepsilon}/4}\,\textrm{~{}and~{}}\,C\cdot e^{6\tau\chi_{\rm
max}}<e^{k{\varepsilon}^{2}/(2\chi_{\rm max})}\,\textrm{~{}for any $k\geq
N_{1}$}.$ (8)
Therefore for $n\geq N_{1}$, the center size of $\mathfrak{F}_{n}^{cu}(x^{s})$
is much smaller than $\delta$.
###### Claim 3.4.
For any $n\geq N_{1}$ and each $i\in[0,n]$, the disc
$\mathfrak{F}_{n}^{cu}(f^{i}(x^{s}))$ has its center-size in between
$\big{(}\delta\cdot e^{-4n(|\chi|+{\varepsilon})},\>\delta\cdot
e^{-n(|\chi|+{\varepsilon}/4)}\big{)}$.
###### Proof.
By Equation (6), for each $y\in\mathfrak{F}_{n}^{cu}(f^{i}(x^{s}))$ and any
$j\leq i+1$, one has
$\|Df^{j}|_{E^{c}(x^{s})}\|\cdot
e^{-j{\varepsilon}/4}\leq\|Df^{j}|_{E^{c}(y)}\|\leq\|Df^{j}|_{E^{c}(x^{s})}\|\cdot
e^{j{\varepsilon}/4},$
and thus by Equation (7), one has
$C^{-1}\cdot e^{-3\tau\chi_{\rm max}}\cdot
e^{j(\chi-3{\varepsilon}/2)}\leq\|Df^{j}|_{E^{c}(y)}\|\leq C\cdot
e^{3\tau\chi_{\rm max}}\cdot e^{j(\chi+3{\varepsilon}/2)}.$ (9)
We inductively prove this claim. Assume that the conclusion holds for $i<n$,
then consider a center curve $\gamma\subset
f(\mathfrak{F}_{n}^{cu}(f^{i}(x^{s})))$ which joins the two u-boundary
components of the $cu$-strip $f(\mathfrak{F}_{n}^{cu}(f^{i}(x^{s})))$, then by
the uniform expansion of $f$ along the cone field $\mathcal{C}^{u}$ and the
invariance of the center bundle, $f^{-j}(\gamma)$ is a center curve joining
the two u-boundary components of $\mathfrak{F}_{n}^{cu}(f^{i+1-j}(x^{s}))$.
Let $\tilde{\gamma}:[0,1]\to M$ denote the center curve $f^{-(i+1)}(\gamma)$,
then the length $\ell(\tilde{\gamma})$ of $\tilde{\gamma}$ has the following
bounds
$C_{u}^{-1}\cdot\delta\cdot
e^{-2n(|\chi|+{\varepsilon})}\leq\ell(\tilde{\gamma})\leq
C_{u}\cdot\delta\cdot e^{-2n(|\chi|+{\varepsilon})},$
and the length $\ell(\gamma)$ of the center curve $\gamma$ is given by
$\ell(\gamma)=\int_{0}^{1}\|Df^{i+1}\frac{\operatorname{d}\tilde{\gamma}(t)}{\operatorname{d}t}\|\operatorname{d}t=\int_{0}^{1}\|Df^{i+1}|_{E^{c}(\tilde{\gamma}(t))}\|\cdot\|\frac{\operatorname{d}\tilde{\gamma}(t)}{\operatorname{d}t}\|\operatorname{d}t.$
By Equation (9), one deduces that
$C^{-1}\cdot e^{-3\tau\chi_{\rm max}}\cdot
e^{{(i+1)}(\chi-3{\varepsilon}/2)}\cdot\ell(\tilde{\gamma})\leq\ell(\gamma)\leq
C\cdot e^{3\tau\chi_{\rm max}}\cdot
e^{{(i+1)}(\chi+3{\varepsilon}/2)}\cdot\ell(\tilde{\gamma}).$
Combining with Equation (8), one has
$\displaystyle\ell(\gamma)$ $\displaystyle\leq\delta\cdot C_{u}\cdot C\cdot
e^{3\tau\chi_{\rm max}}\cdot e^{{(i+1)}(\chi+3{\varepsilon}/2)}\cdot
e^{-2n(|\chi|+{\varepsilon})}$ $\displaystyle<\delta\cdot
e^{n{\varepsilon}/4}\cdot e^{n(|\chi|+3{\varepsilon}/2)}\cdot
e^{-2n(|\chi|+{\varepsilon})}$ $\displaystyle=\delta\cdot
e^{-n(|\chi|+{\varepsilon}/4)}$
and
$\displaystyle\ell(\gamma)$ $\displaystyle\geq\delta\cdot C_{u}^{-1}\cdot
C^{-1}\cdot e^{-3\tau\chi_{\rm max}}\cdot
e^{{(i+1)}(\chi-3{\varepsilon}/2)}\cdot e^{-2n(|\chi|+{\varepsilon})}$
$\displaystyle>\delta\cdot e^{-n{\varepsilon}/4}\cdot
e^{-n(|\chi|+3{\varepsilon}/2)}\cdot e^{-2n(|\chi|+{\varepsilon})}$
$\displaystyle>\delta\cdot e^{-4n(|\chi|+{\varepsilon})}.$
∎
By the choice of $\tau$, the cu-disc
$f^{\tau}(\mathfrak{F}_{n}^{cu}(f^{n}(x^{s})))$ is $C^{1}$-foliated by discs
tangent to the cone field $\mathcal{C}^{u}$ of radius no less than $r_{0}$ and
its center size is bounded from below by $\delta\cdot
e^{-4n(|\chi|+{\varepsilon})}\cdot e^{-\tau\chi_{\rm max}}$. Let $D\subset
f^{\tau}(\mathfrak{F}_{n}^{cu}(f^{n}(x^{s})))$ be the $C^{1}$-disc through the
point $f^{n+\tau}(x^{s})$ of radius $r_{0}$, then there exists a small
neighborhood $\widehat{D}$ of $f^{n+\tau}(x^{s})$ in $D$ such that for each
point $y\in\widehat{D}$, the disc
$f^{\tau}(\mathfrak{F}_{n}^{cu}(f^{n}(x^{s})))$ contains a $cu$-disc centered
at $y$ of radius $\delta_{n}$, where $\delta_{n}:=2^{-1}\cdot\delta\cdot
e^{-4n(|\chi|+{\varepsilon})}\cdot e^{-\tau\chi_{\rm max}}$. Now we show that
the $cu$-plaque of size $\delta_{n}$ centered at $y\in\widehat{D}$ is
contained in $f^{\tau}(\mathfrak{F}_{n}^{cu}(f^{n}(x^{s})))$, which is
essentially due to Claim 3.3 and the locally invariant property of
$cu$-plaques.
###### Claim 3.5.
For any $y\in\widehat{D}$, one has ${\cal W}^{cu}_{\delta_{n}}(y)\subset
f^{\tau}(\mathfrak{F}_{n}^{cu}(f^{n}(x^{s}))).$
###### Proof.
By the choice of $\delta_{0}$ in Equation (4), one has
$f^{-1}({\cal W}^{cu}_{\delta_{0}}(f^{-n-\tau+1}(y)))\subset{\cal
W}^{cu}_{{\varepsilon}_{0}}(f^{-n-\tau}(y)).$
Since $\mathfrak{F}_{n}^{cu}(x^{s})\subset W_{{\varepsilon}_{0}}^{u}(q)$, one
has that $f^{-n-\tau}(y)\in\mathfrak{F}_{n}^{cu}(x^{s})\subset
W_{{\varepsilon}_{0}}^{u}(q)$. By Claim 3.3, one has that ${\cal
W}^{cu}_{{\varepsilon}_{0}}(f^{-n-\tau}(y))\subset W^{u}(q)$. As
$\mathfrak{F}_{n}^{cu}(f(x^{s}))$ is the connected component of
$f(\mathfrak{F}_{n}^{cu}(x^{s}))\cap B_{\delta}(f(x^{s}))$ containing
$f(x^{s})$, combining with Claim 3.4 and $\delta<\delta_{0}$, one deduces that
${\cal
W}^{cu}_{\delta_{n}}(f^{-n-\tau+1}(y)))\subset\mathfrak{F}_{n}^{cu}(f(x^{s})).$
By applying the argument above inductively, one can conclude. ∎
Apply the disc $D$ to Theorem 2.14, and one deduces
$\operatorname{Leb}_{D}\big{(}\big{\\{}y\in
D:\operatorname{d}\big{(}\frac{1}{k}\log\|Df^{k}|_{E^{c}(y)}\|,I(\varphi)\big{)}\leq(\chi^{u}+\chi_{0})/2\big{\\}}\big{)}>\operatorname{Leb}_{D}(D)-a\cdot
e^{-kb}\,\textrm{~{} for any $k\in{\mathbb{N}}$},$
where
$I(\varphi)=\big{[}\chi^{u},\,\sup\\{\int\log\|Df|_{E^{c}}\|\operatorname{d}\mu\,|~{}\mu\in
G^{u}(f)\\}\big{]}$ is an interval. As $b>0$, there exists
$k_{0}\in{\mathbb{N}}$ such that
$\operatorname{Leb}_{D}\big{(}\big{\\{}y\in\widehat{D}:\|Df^{k}|_{E^{c}(y)}\|\geq
e^{k(\chi^{u}+\chi_{0})/2}\big{\\}}\big{)}>0\,\textrm{~{}for any $k\geq
k_{0}$.}$
Let $N_{2}=N_{2}(\chi,{\varepsilon})\in\mathbb{N}$ be an integer such that for
any $n\geq N_{2}$, one has
$\frac{24n(|\chi|+{\varepsilon})}{\chi_{0}\cdot\rho}>k_{0},$
where $\rho=\frac{\chi^{u}-\chi_{0}}{2\chi_{\rm max}-\chi^{u}-\chi_{0}}$ is
given by Pliss Lemma (Lemma 2.5) with $C=\chi_{\rm
max},~{}\chi_{1}=(\chi^{u}+\chi_{0})/2~{},\chi_{2}=\chi_{0}.$ Take
$~{}\widetilde{m}=\big{[}\frac{12n(|\chi|+{\varepsilon})}{\chi_{0}\cdot\rho}\big{]}+1>k_{0}.$
(10)
Then there exists a point $y_{0}\in\widehat{D}$ such that
$\|Df^{\widetilde{m}}|_{E^{c}(y_{0})}\|\geq
e^{\widetilde{m}(\chi^{u}+\chi_{0})/2}$. By Pliss Lemma, there exists an
integer $m\in[\widetilde{m}\rho,\widetilde{m}]$ such that
$\|Df^{-i}|_{E^{c}(f^{m}(y_{0}))}\|\leq e^{-i\chi_{0}}\,\textrm{~{} for every
$1\leq i\leq m$}.$ (11)
###### Claim 3.6.
The $cu$-plaque ${\cal W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0}))$ satisfies
that
* •
$f^{-m}({\cal W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0})))\subset{\cal
W}^{cu}_{{\varepsilon}_{0}\cdot e^{-3m\chi_{0}/4}}(y_{0})$;
* •
$\|Df^{-i}|_{T_{w}{\cal
W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0}))}\|<e^{-3i\chi_{0}/4}$ for every
$w\in{\cal W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0}))$ and $1\leq i\leq m$.
###### Proof.
By the first item in Lemma 2.8 and Equation (11), for any point $w\in{\cal
W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0}))$, one has
$\|Df^{-1}|_{T_{w}{\cal
W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0}))}\|<e^{\chi_{0}/4}\cdot\|Df^{-1}|_{E^{c}(f^{m}(y_{0}))}\|\leq
e^{-3\chi_{0}/4}.$
By the local invariance of the plaque family, one has $f^{-1}({\cal
W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0})))\subset{\cal
W}^{cu}_{{\varepsilon}_{0}\cdot e^{-3\chi_{0}/4}}(f^{m-1}(y_{0}))$.
Inductively apply the above arguments, and one deduces that
* •
$f^{-m}({\cal W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0})))\subset{\cal
W}^{cu}_{{\varepsilon}_{0}\cdot e^{-3m\chi_{0}/4}}(y_{0})$;
* •
$\|Df^{-i}|_{T_{w}{\cal
W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0}))}\|<e^{-3i\chi_{0}/4}$ for every
$w\in{\cal W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0}))$ and $1\leq i\leq m$.
∎
Recall that $\delta$ is determined by ${\varepsilon}$ (by Equation (6)). There
exists $N_{3}=N_{3}({\varepsilon})\in{\mathbb{N}}$ such that
$2\cdot{\varepsilon}_{0}<\delta\cdot e^{k{\varepsilon}}\,\textrm{~{}for any
$k\geq N_{3}.$ }$
Thus for $n>\max\\{N_{1},N_{3}\\}$, as $m\geq\widetilde{m}\rho$, by Equations
(8) and (10), one has
$\displaystyle{\varepsilon}_{0}\cdot e^{-3m\chi_{0}/4}<{\varepsilon}_{0}\cdot
e^{-3\widetilde{m}\rho\chi_{0}/4}<2^{-1}\cdot\delta\cdot
e^{n{\varepsilon}-9n(|\chi|+{\varepsilon})}<2^{-1}\cdot\delta\cdot
e^{-4n(|\chi|+{\varepsilon})}\cdot e^{-\tau\chi_{\rm max}}=\delta_{n}.$
By Claims 3.5 and 3.6, one gets that
$f^{-m}({\cal W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0})))\subset{\cal
W}^{cu}_{{\varepsilon}_{0}\cdot e^{-3m\chi_{0}/4}}(y_{0})\subset{\cal
W}^{cu}_{\delta_{n}}(y_{0})\subset
f^{\tau}(\mathfrak{F}_{n}^{cu}(f^{n}(x^{s}))).$
Since the strong stable foliation is $(\ell_{0},{\varepsilon}_{0}/2)$-dense
with respect to the plaque family ${\cal W}^{cu}$, there exists a point
$z_{0}\in{\cal F}^{s}_{\ell_{0}}(q)\cap{\cal
W}^{cu}_{{\varepsilon}_{0}/2}(f^{m}(y_{0}))\subset{\cal
F}^{s}_{\ell_{0}}(q)\cap f^{m+\tau}(\mathfrak{F}_{n}^{cu}(f^{n}(x^{s}))).$
Since $\mathfrak{F}_{n}^{cu}(f^{n}(x^{s})))$ is tangent to $E^{c}\oplus
E^{u}$, so is ${\cal W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0}))$. As $E^{c}$ is
dominated by $E^{u}$, by the second item of Claim 3.6, one has that
$\|Df^{-i}|_{E^{c}(z_{0})}\|<-3i\chi_{0}/4\textrm{~{}for any $1\leq i\leq
m$.}$ (12)
By the choices of $z_{0}$ and $\tau$, one has $f^{\tau}(z_{0})\subset{\cal
F}^{s}_{\delta}(q)$. By the uniform continuity of the center bundle and
Equation (5), there exists an integer $t=t({\varepsilon},d)>\tau$ such that
$\|Df^{-i}|_{E^{c}(f^{\tau+t}(z_{0}))}\|<e^{-3i\chi_{0}/4}\textrm{~{}for any
$i\leq t+\tau$}.$
Thus one has
$\|Df^{-i}|_{E^{c}(f^{\tau+t}(z_{0}))}\|<e^{-3i\chi_{0}/4}\textrm{~{} for
every $i\leq t+\tau+m$}.$ (13)
Since $t$ and $\tau$ depend only on ${\varepsilon}$ and $d$, there exists
$N_{4}=N_{4}({\varepsilon},d)\in\mathbb{N}$ such that
$(3\tau+t)\cdot\chi_{\rm max}<k\cdot{\varepsilon}\textrm{~{} for any $k\geq
N_{4}$}.$ (14)
Let
$T=T(C,{\varepsilon},d,\chi)=\max\big{\\{}\tau,N_{1},N_{2},N_{3},N_{4}\big{\\}}.$
From now on, we require that $n>T.$
Let $w_{x}=f^{-m-2\tau-n}(z_{0})$ and $l_{x}=m+n+3\tau+t$. Note that $w_{x}\in
f^{-n-\tau}(\mathfrak{F}_{n}^{cu}(f^{n}(x^{s})))\subset
f^{-\tau}(W^{u}_{{\varepsilon}_{0}}(q))\subset W^{u}_{\delta}(q)\subset
W^{u}_{d/2}(q)$ and
$f^{l_{x}}(w_{x})=f^{\tau+t}(z_{0})\in\mathcal{F}^{s}_{\delta}(q)\subset\mathcal{F}^{s}_{d/2}(q)$
since $\delta<d/2$, proving the first item for $w_{x}$.
Using Equations (10) and (14), and the fact that $m\leq\widetilde{m}$, one has
$l_{x}<\frac{12n(|\chi|+{\varepsilon})}{\chi_{0}\cdot\rho}+n+\frac{n{\varepsilon}}{\chi_{\rm
max}}\leq n+n\cdot(\frac{1}{\chi_{\rm
max}}+\frac{12}{\chi_{0}\cdot\rho})\cdot(|\chi|+{\varepsilon}).$
It suffices to take
$\rho_{0}=\frac{1}{\chi_{\rm max}}+\frac{12}{\chi_{0}\cdot\rho}.$
As $z_{0}\in{\cal W}^{cu}_{{\varepsilon}_{0}}(f^{m}(y_{0}))\subset
f^{m+\tau}(\mathfrak{F}_{n}^{cu}(f^{n}(x^{s})))$, then for every $0\leq i\leq
n$ one deduces that $d(f^{i+\tau}(w_{x}),f^{i}(x^{s}))<\delta.$ Since
$\operatorname{d}(f^{i}(x),f^{i}(x^{s}))<\delta$ for every $\tau\leq i\leq n$
and $\delta/2<d$, one gets that $d(f^{i+\tau}(w_{x}),f^{i}(x))<2\delta<d$ for
every $\tau\leq i\leq n$, proving the second item for $w_{x}$.
It remains to show the last item for $w_{x}$, that is,
$\|Df^{j}|_{E^{c}(f^{l_{x}-j}(w_{x}))}\|>e^{2j\cdot(|\chi|+{\varepsilon})}\,\textrm{~{}for
any $1\leq j\leq l_{x}$.}$
By Equations (6) and (7), for each $0\leq i\leq n$ one has
$\|Df^{i}|_{E^{c}(f^{\tau}(w_{x}))}\|\leq
e^{i{\varepsilon}/4}\cdot\|Df^{i}|_{E^{c}(x^{s})}\|\leq C\cdot
e^{3\tau\chi_{\rm max}}\cdot e^{i(\chi+3{\varepsilon}/2)}$ (15)
and
$\|Df^{i}|_{E^{c}(f^{\tau}(w_{x}))}\|\geq
e^{-i{\varepsilon}/4}\cdot\|Df^{i}|_{E^{c}(x^{s})}\|\geq C^{-1}\cdot
e^{-3\tau\chi_{\rm max}}\cdot e^{i(\chi-3{\varepsilon}/2)}.$
Combining with Equation (13) and the fact that
$f^{\tau+t}(z_{0})=f^{l_{x}}(w_{x})$, one has
$\displaystyle\|Df^{l_{x}}|_{E^{c}(f(w_{x}))}\|=$
$\displaystyle\|Df^{\tau}|_{E^{c}(w_{x})}\|\cdot\|Df^{n}|_{E^{c}(f^{\tau}(w_{x}))}\|\cdot\|Df^{\tau}|_{E^{c}(f^{n+\tau}(w_{x}))}\|\cdot\|Df^{m+t+\tau}|_{E^{c}(f^{n+2\tau}(w_{x}))}\|$
$\displaystyle\geq$ $\displaystyle e^{-\tau\chi_{\rm max}}\cdot C^{-1}\cdot
e^{-3\tau\chi_{\rm max}}\cdot e^{n(\chi-3{\varepsilon}/2)}\cdot
e^{-\tau\chi_{\rm max}}\cdot e^{3(t+\tau+m)\chi_{0}/4}$ $\displaystyle=$
$\displaystyle C^{-1}\cdot e^{-5\tau\chi_{\rm max}}\cdot
e^{n(\chi-3{\varepsilon}/2)}\cdot e^{3(t+\tau+m)\chi_{0}/4}$
$\displaystyle\text{\tiny by Equation~{}\eqref{eq:choice-of-N-bounding-C}}>$
$\displaystyle e^{n(\chi-2{\varepsilon})}\cdot e^{3(t+\tau+m)\chi_{0}/4}.$
To show that $3(t+\tau+m)\chi_{0}/4+n(\chi-2{\varepsilon})\geq
3(m+3\tau+t+n)(|\chi|+{\varepsilon})$, it suffices to prove
$3m\chi_{0}/4-n(|\chi|+2{\varepsilon})\geq
3(m+3\tau+t+n)(|\chi|+{\varepsilon})$. As
$|\chi|+{\varepsilon}<\chi_{0}/12<\chi_{\rm max}$ and $n>T\geq N_{4}$, by
Equation (14), it suffices to show that $3m\chi_{0}/4\geq
n(|\chi|+3{\varepsilon})+3(m+n)(|\chi|+{\varepsilon})$. Since
$|\chi|+{\varepsilon}<\chi_{0}/12$, one only needs to show that
$m\chi_{0}/2\geq n(4|\chi|+6{\varepsilon})$, and this inequality holds due to
the fact that
$m\geq\widetilde{m}\rho\geq\frac{12n(|\chi|+{\varepsilon})}{\chi_{0}}.$
This proves that
$\|Df^{l_{x}}|_{E^{c}(w_{x})}\|>e^{3(m+3\tau+t+n)\cdot(|\chi|+{\varepsilon})}.$
Let $i_{0}\in[1,l_{x}]$ be the biggest integer satisfying that
$\|Df^{j}|_{E^{c}(f^{i_{0}-j}(w_{x}))}\|>e^{2j\cdot(|\chi|+{\varepsilon})}\,\textrm{~{}for
any $1\leq j\leq i_{0}$}.$ (16)
Then we claim that $i_{0}=l_{x}.$ By Pliss lemma, one has
$i_{0}\geq l_{x}\cdot\frac{|\chi|+{\varepsilon}}{\chi_{\rm
max}-2(|\chi|+{\varepsilon})}>l_{x}\cdot\frac{|\chi|+{\varepsilon}}{\chi_{\rm
max}}>\frac{n{\varepsilon}}{\chi_{\rm max}}.$
Assume, on the contrary, that $i_{0}\in(n{\varepsilon}/\chi_{\rm
max},n+2\tau]$. For each $i\in[\tau,n]$, by Equation (15), one has
$\|Df^{i}|_{E^{c}(w_{x})}\|=\|Df^{\tau}|_{E^{c}(w_{x})}\|\cdot\|Df^{i-\tau}|_{E^{c}(f^{\tau}(w_{x}))}\|\leq
C\cdot e^{4\tau\chi_{\rm max}}\cdot e^{(i-\tau)(\chi+3{\varepsilon}/2)},$
then combining with Equations (8) and (15), one has
$\displaystyle\|Df^{i_{0}}|_{E^{c}(w_{x})}\|$ $\displaystyle\leq C\cdot
e^{4\tau\chi_{\rm max}}\cdot e^{(i_{0}-\tau)(\chi+3{\varepsilon}/2)}\cdot
e^{2\tau\chi_{\rm max}}$ $\displaystyle<e^{n{\varepsilon}^{2}/(2\chi_{\rm
max})}\cdot e^{i_{0}(|\chi|+3{\varepsilon}/2)}$
$\displaystyle<e^{i_{0}{\varepsilon}/2}\cdot
e^{i_{0}(|\chi|+3{\varepsilon}/2)}$ $\displaystyle\leq
e^{i_{0}(|\chi|+2{\varepsilon})}$
which contradicts with
$\|Df^{i_{0}}|_{E^{c}(w_{x})}\|>e^{2i_{0}(|\chi|+{\varepsilon})}.$ Thus
$i_{0}\in(n+2\tau,l_{x}]$. By Equation (13) and
$f^{\tau+t}(z_{0})=f^{l_{x}}(w_{x})$, one has
$\|Df^{j}|_{E^{c}(f^{l_{x}-j}(w_{x}))}\|>e^{2j\cdot(|\chi|+{\varepsilon})}\,\textrm{~{}for
any $n+2\tau\leq j\leq l_{x}$,}$
which implies that $i_{0}=l_{x}$, since $i_{0}$ is the largest integer in
$[1,l_{x}]$ satisfying Equation (16). The proof of Proposition 3.2 is now
completed.
∎
According to our proof the constant $\rho_{0}$ is given by
$\rho_{0}=\frac{1}{\chi_{\rm max}}+\frac{12(2\chi_{\rm
max}-\chi^{u}-\chi_{0})}{\chi_{0}\cdot(\chi^{u}-\chi_{0})},$
where $\chi_{\rm max}=\sup_{y\in M}|\log\|Df|_{E^{c}(y)}\||$ and
$\chi^{u}=\inf\big{\\{}\int\log\|Df|_{E^{c}}\|\operatorname{d}\mu~{}|\,~{}\mu\in
G^{u}(f)\big{\\}}.$
### 3.2 Proof of Theorem 3.1
Given a partially hyperbolic diffeomorphism $f$ with
$\operatorname{dim}(E^{c})=1$, and constants $C>1$, $\chi\in\mathbb{R}$ and
${\varepsilon}>0$, let us define the following type of Pesin block:
${\rm\mathcal{L}}_{C,\chi,{\varepsilon}}=\big{\\{}x\in M|~{}~{}C^{-1}\cdot
e^{k(\chi-{\varepsilon})}\leq\|Df^{k}|_{E^{c}(x)}\|\leq C\cdot
e^{k(\chi+{\varepsilon})}\,\textrm{~{}for any $k\in\mathbb{N}$}\big{\\}}.$
Thus ${\rm\mathcal{L}}_{C,\chi,{\varepsilon}}$ concerns about the set of
points whose center Lyapunov exponents are ${\varepsilon}$-close to $\chi$.
Given some separated sets, the following result allows us to find a collection
of periodic points which satisfy the assumptions of Theorem 2.10. We will
apply Proposition 3.2 to these separated sets to find some periodic pseudo-
orbits with some weak expansion along the center bundle, then we apply Liao’s
shadowing lemma to get periodic points.
###### Lemma 3.7.
Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism
with $\operatorname{dim}(E^{c})=1$. Suppose that
* •
there exists $\chi_{0}>0$ such that
$\int\log\|Df|_{E^{c}}\|\operatorname{d}\mu>\chi_{0}$ for any $\mu\in
G^{u}(f)$;
* •
let ${\cal W}^{cu}$ be a $cu$-plaque family and ${\varepsilon}_{0}>0$ be given
by Lemma 2.8 corresponding to $\chi_{0}$, then there exists a constant
$\ell_{0}>0$ such that the strong stable foliation is
$(\ell_{0},{\varepsilon}_{0}/2)$-dense with respect to ${\cal W}^{cu}$.
Then there exist $\rho_{0}>0$ and a hyperbolic periodic point $q$ with
positive center Lyapunov exponent such that for any continuous functions
$\varphi_{1},\cdots,\varphi_{m}$ on $M$, any
${\varepsilon}\in(0,\chi_{0}/24)$, any $\chi\in(-\chi_{0}/24,0]$, any $C>1$
and any $\xi>0$, there exists $N\in\mathbb{N}$ such that for any $n>N$, any
$h\geq 0$ and any $(n,2\xi)$-separated set
$S\subset{\rm\mathcal{L}}_{C,\chi,{\varepsilon}}$ with cardinality $\\#S\geq
e^{n(h-{\varepsilon}/2)}$, there exists a collection $P\subset{\rm
NUH}_{|\chi|+{\varepsilon}}$ of periodic points of the same period
$l\in\big{(}n,n+\rho_{0}(|\chi|+{\varepsilon})n\big{)}$ such that
* –
$\\#P>e^{n(h-{\varepsilon})};$
* –
$P$ is a $(l,\xi)$-separated set;
* –
$\operatorname{d}(p,p^{\prime})<\xi/16$ for any $p,p^{\prime}\in P$;
* –
each $p\in P$ is homoclinically related to $q$;
* –
for any $p\in P$, there exists $x\in S$ such that for each $1\leq j\leq m$,
one has
$\big{|}\int\varphi_{j}\operatorname{d}\delta_{{\cal
O}_{p}}-\int\varphi_{j}\operatorname{d}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x)}\big{|}<\big{(}2\rho_{0}(|\chi|+{\varepsilon})+{\varepsilon}\big{)}\|\varphi_{j}\|_{C^{0}},$
where $\|\varphi_{j}\|_{C^{0}}=\sup_{x\in M}|\varphi_{j}(x)|.$
###### Proof.
Let $\rho_{0}>0$ be the constant and $q$ be the hyperbolic periodic point
given by Proposition 3.2. Let us fix ${\varepsilon}\in(0,\chi_{0}/24)$,
$\chi\in(-\chi_{0}/24,0]$, $C>1$, $\xi>0$ and continuous functions
$\varphi_{1},\cdots,\varphi_{m}$ on $M$.
For ${\varepsilon}>0$, by the uniform continuity of the center bundle and the
uniform continuity of the functions $(\varphi_{j})_{1\leq j\leq m}$, there
exists $\eta_{1}>0$ such that for any $x,y\in M$ with
$\operatorname{d}(x,y)<\eta_{1}$, one has
$\big{|}\log\|Df|_{E^{c}(x)}\|-\log\|Df|_{E^{c}(y)}\|\big{|}<{\varepsilon}/2\,\textrm{~{}and~{}}\,\max_{1\leq
j\leq
m}\frac{|\varphi_{j}(x)-\varphi_{j}(y)|}{\|\varphi_{j}\|_{C^{0}}}<{\varepsilon}/2.$
(17)
Apply $\lambda=e^{-2(|\chi|+{\varepsilon})}$ and the dominated splitting
$TM=E^{s}\oplus(E^{c}\oplus E^{u})$ to Theorem 2.3, and one obtains the
constants $L>1$ and $d_{0}>0$ with the posited properties.
By Lemma 2.7, there exists $\delta_{0}>0$ such that if a point $x\in M$
satisfies that
$\|Df^{-n}|_{E^{c}(x)}\|\leq e^{-n(|\chi|+{\varepsilon})}\,\textrm{~{}for any
$n\in\mathbb{N}$},$
then $x$ has unstable manifold $W^{u}_{\delta_{0}}(x)$ of size $\delta_{0}$
and tangent everywhere to $E^{c}\oplus E^{u}.$ As the bundles $E^{s}$ and
$E^{c}\oplus E^{u}$ have uniform angle, there exists $\eta_{0}>0$ such that
for any $x,y\in M$ with $\operatorname{d}(x,y)<\eta_{0}$, if $D$ is a disc
which is centered at $y$, of size $\delta_{0}$ and tangent everywhere to
$E^{c}\oplus E^{u}$, then ${\cal F}_{\delta_{0}}^{s}(x)$ has non-empty
transverse intersection with $D$.
Apply $C>1$, $\chi$, ${\varepsilon}>0$ and
$d=\min\\{\,d_{0},\,\eta_{0}\cdot(L+1)^{-1},\,\eta_{1}\cdot(L+1)^{-1},\,32^{-1}\cdot(L+1)^{-1}\xi\,\\}$
to Proposition 3.2, and one obtains integers $\tau=\tau({\varepsilon},d)$ and
$T=T(C,\chi,{\varepsilon},d)$ with the posited properties.
Let $N_{1}=N_{1}({\varepsilon},d,\xi)$ be an integer such that for any $n\geq
N_{1}$, one has
* •
$\rho_{0}(|\chi|+{\varepsilon})n<e^{n{\varepsilon}/4}$
* •
${4\tau}<n{\varepsilon}$;
* •
the minimal number of $(\tau,\xi)$-Bowen balls covering the whole manifold is
less than $e^{n{\varepsilon}/4}$.
Let $N=\max\\{T,N_{1}\\}$. Let $n>N$ and $S$ be a $(n,2\xi)$-separated set in
${\rm\mathcal{L}}_{C,\chi,{\varepsilon}}$ with cardinality
$\\#S>e^{n(h-{\varepsilon}/2)}$. Using the pigeonhole principle, one gets a
subset $\widetilde{S}\subset S$ such that
* •
$\widetilde{S}$ is contained in a Bowen ball $B_{\tau}(z_{0},\xi)$ for some
$z_{0}\in M$;
* •
$\\#\widetilde{S}>e^{n(h-3{\varepsilon}/4)}.$
By Proposition 3.2, for any $x\in\widetilde{S}$, there exist $w_{x}\in
W^{u}_{d/2}(q)$ and an integer
$l_{x}\in\big{(}n,\>n+\rho_{0}\cdot(|\chi|+{\varepsilon})\cdot n\big{)}$ such
that
* •
$f^{l_{x}}(w_{x})\in{\cal F}^{s}_{d/2}(q)$;
* •
$\operatorname{d}(f^{\tau+i}(w_{x}),f^{i}(x))<d$ for any $i\in[\tau,n]$;
* •
$\|Df^{-i}|_{E^{c}(f^{l_{x}}(w_{x}))}\|\leq e^{-2i(|\chi|+{\varepsilon})}$ for
any $i\in[1,l_{x}]$.
As $\rho_{0}(|\chi|+{\varepsilon})n<e^{n{\varepsilon}/4}$, by the pigeonhole
principle, there exists a subset $\widehat{S}\subset\widetilde{S}$ such that
* •
$l_{x}$ is a constant on $\widehat{S}$ and we denote it by
$l\in\big{(}n,\>n+\rho_{0}\cdot(\chi+{\varepsilon})\cdot n\big{)}$;
* •
$\\#\widehat{S}>e^{n(h-{\varepsilon})}.$
For each $x\in\widehat{S}$, since
$\operatorname{d}(w_{x},f^{l}(w_{x}))\leq\operatorname{d}(w_{x},q)+\operatorname{d}(q,f^{l}(w_{x}))\leq
d<d_{0}$, by Theorem 2.3, there exists a periodic point $p_{x}$ of period $l$
such that
$\operatorname{d}(f^{i}(p_{x}),f^{i}(w_{x}))<L\cdot d\textrm{ for any $0\leq
i\leq l-1$}.$ (18)
This gives a collection of periodic points
$P:=\\{\,p_{x}|~{}x\in\widehat{S}\;\\}$ of the same period
$l\in\big{(}n,n+\rho_{0}(|\chi|+{\varepsilon})\big{)}$. For each
$x\in\widehat{S}$, one has
$\operatorname{d}(p_{x},q)\leq\operatorname{d}(p_{x},w_{x})+\operatorname{d}(w_{x},q)<(L+1)d$.
As $(L+1)d\leq\xi/32$, one gets that $P\subset B_{\xi/32}(q)$ which implies
the third item of Lemma 3.7. It also holds that $(L+1)d<\eta_{0}$, then by the
choice of $\eta_{0}$, the periodic point $p_{x}$ is homoclinically related to
$q$ which proves the fourth item of Lemma 3.7. Besides, for each
$x\in\widehat{S}$, one has
$\operatorname{d}(f^{\tau+i}(p_{x}),f^{i}(x))\leq\operatorname{d}(f^{\tau+i}(p_{x}),f^{i}(w_{x}))+\operatorname{d}(f^{\tau+i}(w_{x}),f^{i}(x))<(L+1)d\textrm{~{}
for any $i\in[\tau,n]$. }$ (19)
Since $(L+1)d<\eta_{1},$ for each $x\in\widehat{S}$, by Equations (17) and
(18), one has
$\|Df^{-i}|_{E^{c}(p_{x})}\|\leq
e^{i{\varepsilon}/2}\cdot\|Df^{-i}|_{E^{c}(f^{l}(w_{x}))}\|<e^{-i(|\chi|+{\varepsilon})}\textrm{
for any $i\in[1,l]$}$ (20)
which implies that $P\subset{\rm NUH}_{|\chi|+{\varepsilon}}$.
Recall that $4\tau<n{\varepsilon}$. Now, for each $1\leq j\leq m$, and
$x\in\widehat{S},$ one has
$\displaystyle\big{|}\int\varphi_{j}\operatorname{d}\delta_{{\cal
O}_{p_{x}}}-\int\varphi_{j}\operatorname{d}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x)}\big{|}$
$\displaystyle=\big{|}\int\varphi_{j}\operatorname{d}\frac{1}{l}\sum_{i=0}^{l-1}\delta_{f^{i}(p_{x})}-\int\varphi_{j}\operatorname{d}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x)}\big{|}$
$\displaystyle\leq\big{|}\frac{1}{l}\big{(}\sum_{i=0}^{l-1}\varphi_{j}(f^{i}(p_{x}))-\sum_{i=0}^{n-1}\varphi_{j}(f^{i}(x))\big{)}+(\frac{1}{l}-\frac{1}{n})\sum_{i=0}^{n-1}\varphi_{j}(f^{i}(x))\big{|}$
$\displaystyle\leq\frac{1}{l}\sum_{i=0}^{2\tau-1}\big{|}\varphi_{j}(f^{i}(p_{x}))\big{|}+\frac{1}{l}\sum_{i=\tau}^{n-1}\big{|}\varphi_{j}(f^{i+\tau}(p_{x}))-\varphi_{j}(f^{i}(x))\big{|}+\frac{1}{l}\sum_{i=n}^{l-1}\big{|}\varphi_{j}(f^{i}(p_{x}))\big{|}$
$\displaystyle\hskip 14.22636pt+\frac{l-n}{n\cdot
l}\sum_{i=0}^{n-1}\big{|}\varphi_{j}(f^{i}(x))\big{|}$ by Equations (19) and
(17)
$\displaystyle\leq\frac{2\tau}{l}\|\varphi_{j}\|_{C^{0}}+\frac{n-\tau}{l}\cdot\frac{{\varepsilon}}{2}\cdot\|\varphi_{j}\|_{C^{0}}+\frac{2(l-n)}{l}\cdot\|\varphi_{j}\|_{C^{0}}$
$\displaystyle\leq\big{(}\frac{{\varepsilon}}{2}+\frac{{\varepsilon}}{2}+2\rho_{0}(|\chi|+{\varepsilon})\big{)}\cdot\|\varphi_{j}\|_{C^{0}}=\big{(}{\varepsilon}+2\rho_{0}(|\chi|+{\varepsilon})\big{)}\cdot\|\varphi_{j}\|_{C^{0}},$
which gives the last item of Lemma 3.7.
It remains to show that $P$ is $(l,\xi)$-separated and
$\\#P>e^{n(h-{\varepsilon})}$.
###### Claim 3.8.
For two different points $x,x^{\prime}\in\widehat{S}$, there exists $2\tau\leq
k<n+\tau$ such that
$\operatorname{d}(f^{k}(p_{x}),f^{k}(p_{x^{\prime}}))>\xi.$
###### Proof.
Since $\widehat{S}\subset S$ is also a $(n,2\xi)$-separated and is contained
in $B_{\tau}(z_{0},\xi)$, there exists $\tau\leq i<n$ such that
$\operatorname{d}(f^{i}(x),f^{i}(x^{\prime}))>2\xi$. As $2Ld<\xi$, by Equation
(19), one deduces that
$\operatorname{d}(f^{i+\tau}(p_{x}),f^{i+\tau}(p_{x^{\prime}}))\geq\operatorname{d}(f^{i}(x),f^{i}(x^{\prime}))-\operatorname{d}(f^{i+\tau}(p_{x}),f^{i}(x))-\operatorname{d}(f^{i+\tau}(p_{x^{\prime}}),f^{i}(x^{\prime}))>\xi.$
∎
By Claim 3.8, the set of periodic points
$P=\\{\,p_{x}|~{}x\in\widehat{S}\;\\}$ is $(n+\tau,\xi)$-separated and has
cardinality $\\#P=\\#\widehat{S}>e^{n(h-{\varepsilon})}$, proving the first
and the second items of Lemma 3.7. ∎
Now, we use Lemma 3.7 to prove Theorem 3.1. Fix a sequence of continuous
functions $(\varphi_{n})$ on the manifold $M$ which forms a dense subset of
the space of continuous functions on $M$ under $C^{0}$-topology. Let us define
the metric on the space of Borel probability measures. The distance of two
Borel probability measures $\mu_{1}$ and $\mu_{2}$ is defined as
$\operatorname{d}(\mu_{1},\mu_{2})=\sum_{n\in{\mathbb{N}}}\frac{|\int\varphi_{n}\operatorname{d}\mu_{1}-\int\varphi_{n}\operatorname{d}\mu_{2}|}{2^{n}(1+\|\varphi_{n}\|_{C^{0}})},$
where $\|\varphi\|_{C^{0}}=\sup_{x\in M}|\varphi(x)|$ for any continuous
function $\varphi:M\to{\mathbb{R}}$.
###### Proof of Theorem 3.1.
Let $\rho_{0}>0$ be the constant and $q$ be the hyperbolic periodic point
given by Lemma 3.7.
Given $\nu\in{\cal M}_{erg}(f)$ with $\chi_{0}/24<\chi^{c}(\nu)\leq 0$ and
${\varepsilon}\in(0,\chi_{0}/24)$, take $k_{0}\geq 1$ such that the set
$\displaystyle{\rm Basin}_{k_{0},{\varepsilon}}(\nu):=\big{\\{}x\in M|~{}~{}$
$\displaystyle
e^{k(\chi^{c}(\nu)-{\varepsilon})}\leq\|Df^{k}|_{E^{c}(x)}\|\leq
e^{k(\chi^{c}(\nu)+{\varepsilon})}\textrm{~{}and~{}}$ $\displaystyle\hskip
14.22636pt\operatorname{d}(\nu,\frac{1}{k}\sum_{i=0}^{k-1}\delta_{f^{i}(x)})<{\varepsilon}/2,\>\textrm{~{}for
any $k\geq k_{0}$}\big{\\}}$
has $\nu$-measure no less than $1/2.$ Furthermore, there exists $C>1$ such
that for any point $x\in{\rm Basin}_{k_{0},{\varepsilon}}(\nu)$, one has
$C^{-1}\cdot e^{k(\chi^{c}(\nu)-{\varepsilon})}\leq\|Df^{k}|_{E^{c}(x)}\|\leq
C\cdot e^{k(\chi^{c}(\nu)+{\varepsilon})}\,\textrm{~{}for any
$k\in\mathbb{N}$},$
in formula, ${\rm
Basin}_{k_{0},{\varepsilon}}(\nu)\subset{\rm\mathcal{L}}_{C,\chi^{c}(\nu),{\varepsilon}}.$
For any $\xi>0$, denote by $S(n,2\xi)$ a $(n,2\xi)$-separated set of ${\rm
Basin}_{k_{0},{\varepsilon}}(\nu)$ with maximal cardinality. By Katok’s
definition of metric entropy (see Section 2.1), one has
$\lim_{\xi\rightarrow
0}\liminf_{n\rightarrow+\infty}\frac{1}{n}\log\\#S(n,2\xi)\geq h_{\nu}(f).$
Thus there exists $\xi_{1}>0$ such that for any $\xi\leq\xi_{1}$, one has
$\liminf_{n\rightarrow+\infty}\frac{1}{n}\log\\#S(n,2\xi)>h_{\nu}(f)-{\varepsilon}/4.$
For any $\xi<\xi_{1}$, there exists an integer $N_{1}>k_{0}$ such that
$\\#S(n,2\xi)>e^{n(h_{\nu}(f)-{\varepsilon}/2)}\textrm{~{} for any $n\geq
N_{1}$.}$
For ${\varepsilon}>0$, consider an integer $m=m({\varepsilon})$ such that
$2^{1-m}<{\varepsilon}.$ Apply the continuous functions
$\varphi_{1},\cdots,\varphi_{m}$, ${\varepsilon}\in(0,\chi_{0}/24)$,
$\chi=\chi^{c}(\nu)$, $C>1$ and $\xi<\xi_{1}$ to Lemma 3.7, and one obtains an
integer $N$. For any $n>\max\\{N,N_{1}\\}$, using the $(n,2\xi)$-separated set
$S(n,2\xi)$, one gets a collection $P\subset{\rm NUH}_{|\chi|+{\varepsilon}}$
of periodic points of the same period
$l\in\big{(}n,n+\rho_{0}(|\chi^{c}(\nu)|+{\varepsilon})n\big{)}$ such that
* •
$\\#P>e^{n(h_{\nu}(f)-{\varepsilon})};$
* •
$P$ is a $(l,\xi)$-separated set;
* •
$\operatorname{d}(p,p^{\prime})<\xi/16$ for any $p,p^{\prime}\in P$;
* •
for any $p\in P$, there exists a point $x_{p}\in S(n,2\xi)$ such that for each
$1\leq j\leq m$, one has
$\big{|}\int\varphi_{j}\operatorname{d}\delta_{{\cal
O}_{p}}-\int\varphi_{j}\operatorname{d}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x_{p})}\big{|}<\big{(}2\rho_{0}(|\chi^{c}(\nu)|+{\varepsilon})+{\varepsilon}\big{)}\|\varphi_{j}\|_{C^{0}}.$
As $S(n,2\xi)\subset{\rm Basin}_{k_{0},{\varepsilon}}(\nu)$ and $n\geq
N_{1}>k_{0}$, one has the following estimate
$\displaystyle\operatorname{d}\big{(}\delta_{{\cal O}_{p}},\nu\big{)}$
$\displaystyle\leq\operatorname{d}\big{(}\delta_{{\cal
O}_{p}},\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x_{p})}\big{)}+\operatorname{d}\big{(}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x_{p})},\nu\big{)}$
$\displaystyle<\sum_{j=1}^{\infty}\frac{\big{|}\int\varphi_{j}\operatorname{d}\delta_{{\cal
O}_{p}}-\int\varphi_{j}\operatorname{d}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x_{p})}\big{|}}{2^{j}\|\varphi_{j}\|_{C^{0}}}+\frac{{\varepsilon}}{2}$
$\displaystyle\leq
2\rho_{0}(|\chi^{c}(\nu)|+{\varepsilon})+{\varepsilon}+\sum_{j=m+1}^{\infty}\frac{1}{2^{j}}+\frac{{\varepsilon}}{2}$
$\displaystyle<2\rho_{0}(|\chi^{c}(\nu)|+{\varepsilon})+2{\varepsilon}.$
Besides, one has
$\frac{\log\\#P}{l}>\frac{n(h_{\nu}(f)-{\varepsilon})}{n+n\rho_{0}(|\chi^{c}(\nu)|+{\varepsilon})}=\frac{h_{\nu}(f)-{\varepsilon}}{1+\rho_{0}(|\chi^{c}(\nu)|+{\varepsilon})}.$
Now, it suffices to take $\rho_{1}=2\rho_{0}+2,$ ending the proof of Theorem
3.1. ∎
## 4 Approximation of ergodic measures by horseshoes: Proofs of our main
results
In this section, we will first show that non-hyperbolic ergodic measures are
approached by horseshoes in entropy and in weak$*$-topology (i.e., proving
Theorem C). Then we prove the continuity of topological entropy and the
intermediate entropy property (i.e., proving Theorem A). At last, we give the
proof of Theorem D and Corollary B.
###### Proof of Theorem C.
Let $f$ be as in the assumption. Recall that $G^{u}(f)$ is compact and varies
upper semi-continuously with respect to $f$ (see Lemma 2.13), and thus there
exist a $C^{1}$-open neighborhood ${\cal U}_{1}$ of $f$ and a constant
$\chi_{0}>0$ such that
$\int\log\|Dg|_{E_{g}^{c}}\|\operatorname{d}\mu>\chi_{0}\,\textrm{~{}for any
$g\in{\cal U}_{1}$ and $\mu\in G^{u}(g)$.}$
By Lemma 2.8, for $\chi_{0}>0$, there exist a $C^{1}$-neighborhood
$\widetilde{\cal U}_{2}$ of $f$, a $cu$-plaque family ${\cal W}^{cu}_{g}$
relying on $g\in\widetilde{\cal U}_{2}$ continuously and ${\varepsilon}_{0}>0$
satisfying the posited properties. As the strong stable foliation of $f$ is
minimal, by Lemma 2.9, there exist a $C^{1}$-neighborhood ${\cal
U}_{2}\subset\widetilde{\cal U}_{2}$ of $f$ and $\ell_{0}>0$ such that the
strong stable foliation of $g\in{\cal U}_{2}$ is
$(\ell_{0},{\varepsilon}_{0}/2)$-dense with respect to ${\cal W}^{cu}_{g}$.
Now let ${\cal U}={\cal U}_{1}\cap{\cal U}_{2}$, then each $g\in{\cal U}$
satisfies the assumption of Proposition 3.2. Let $\chi_{1}=\chi_{0}/24$.
Up to shrinking ${\cal U}$, one can assume that
$\inf\big{\\{}\int\log\|Dg|_{E^{c}_{g}}\|\operatorname{d}\mu\;|\;g\in{\cal
U}\,\textrm{~{}and~{}}\mu\in G^{u}(g)\big{\\}}>\chi_{0}.$
Fix $g\in{\cal U}$, then Theorem 3.1 gives a constant $\rho_{1}=\rho_{1}(g)>0$
for $g.$
Let $\nu$ be an ergodic measure of $g$ with $\chi_{0}/24<\chi^{c}(\nu)\leq 0.$
Take ${\varepsilon}>0$ small, apply
$\lambda=e^{-(|\chi^{c}(\nu)|+{\varepsilon})}$ and the dominated splitting
$TM=E_{g}^{s}\oplus(E_{g}^{c}\oplus E_{g}^{u})$ to Theorem 2.10, and one
obtains $\xi_{0}>0$ with the posited properties. Apply $\nu$ and
${\varepsilon}>0$ to Theorem 3.1, and one obtains $\xi_{1}>0$ with posited
properties. For $\xi<\min\\{\xi_{0},\xi_{1}\\}$, by Theorem 3.1, there exist
finitely many periodic points $p_{1},\cdots,p_{m}\in{\rm
NUH}_{|\chi^{c}(\nu)|+{\varepsilon}}$ of the same period $l$ such that
* •
$\operatorname{d}(p_{i},p_{j})<\xi/16$ for any $i,j\in\\{1,\cdots,m\\}$;
* •
$\\{p_{1},\cdots,p_{m}\\}$ is a $(l,\xi)$-separated set;
* •
$\frac{\log
m}{l}\geq\frac{h_{\nu}(g)-{\varepsilon}}{1+\rho_{1}(|\chi^{c}(\nu)|+{\varepsilon})};$
* •
$\operatorname{d}(\delta_{{\cal
O}_{p_{i}}},\nu)<\rho_{1}(|\chi^{c}(\nu)|+{\varepsilon})$ for any
$i\in\\{1,\cdots,m\\}$;
Now, we apply Theorem 2.10 to the set of period points $p_{1},\cdots,p_{m}$
and obtain a horseshoe $\Lambda$ whose center is uniformly expanding such that
* •
$h_{top}(g,{\Lambda})\geq\frac{\log
m}{l}>\frac{h_{\nu}(g))-{\varepsilon}}{1+\rho_{1}(|\chi^{c}(\nu)|+{\varepsilon})};$
* •
${\cal M}_{inv}(g,\Lambda)$ is contained in the ${\varepsilon}$-neighborhood
of
$\big{\\{}\sum_{i=1}^{m}t_{i}\cdot\delta_{{\cal O}_{p_{i}}}|~{}t_{i}\geq
0\textrm{~{}and~{}}\sum_{i=1}^{m}t_{i}=1\big{\\}},$
and thus in the
$\rho_{1}(|\chi^{c}(\nu)|+{\varepsilon})+{\varepsilon}$-neighborhood of $\nu.$
Let us denote $\chi_{\rm max}(g)=\sup_{x\in M}|\log\|Dg|_{E_{g}^{c}(x)}\||$
and $\chi^{u}(g)=\inf_{\mu\in
G^{u}(g)}\int\log\|Dg|_{E_{g}^{c}}\|\operatorname{d}\mu.$ By the proofs of
Theorem 3.1 and Proposition 3.2, it suffices to take
$\kappa=\sup_{g\in{\cal U}}\rho_{1}(g)+1=\sup_{g\in{\cal
U}}2\big{(}\frac{1}{\chi_{\rm max}(g)}+\frac{12(2\chi_{\rm
max}(g)-\chi^{u}(g)-\chi_{0})}{\chi_{0}\cdot(\chi^{u}(g)-\chi_{0})}\big{)}+3.$
Now, we need the robustly minimality of the strong stable foliation which is
guaranteed by Theorem 2.20. Up to shrinking ${\cal U}$, one can assume that
the strong stable foliation of $g\in{\cal U}$ is minimal, and we will show
that $\\{\nu\in\mathcal{M}_{erg}(g)|\chi^{c}(\nu,g)\geq 0\\}$ is path
connected for any $g\in{\cal U}$. Consider
$\nu_{1},\nu_{2}\in\mathcal{M}_{erg}(g)$ with non-negative center Lyapunov
exponents, then there exists a sequences of periodic orbits $\\{{\cal
O}_{p_{n}}\\}_{n\in{\mathbb{Z}}}$ with positive center Lyapunov exponents such
that $\lim_{n\rightarrow+\infty}\delta_{{\cal O}_{p_{n}}}=\nu_{1}$ and
$\lim_{n\rightarrow-\infty}\delta_{{\cal O}_{p_{n}}}=\nu_{2}.$ For each
$n\in{\mathbb{Z}}$, by the minimality of the strong stable foliation, ${\cal
O}_{p_{n}}$ is homoclinically related to ${\cal O}_{p_{n+1}}$ and thus by the
Birkhoff-Smale’s theorem, there exists a hyperbolic horseshoe $\Lambda_{n}$ of
$g$ containing ${\cal O}_{p_{n}}$ and ${\cal O}_{p_{n+1}}$. By Theorem B in
[Sig], there exists a continuous path $\alpha_{n}$ on
$\mathcal{M}_{erg}(g,\Lambda_{n})$, which is defined on
$[1-2^{-n},1-2^{-(n+1)}]$ provided that $n\geq 0$ or defined on
$[2^{n}-1,2^{n+1}-1]$ provided that $n<0$, such that the two endpoints of
$\alpha_{n}$ are $\delta_{{\cal O}_{p_{n}}}$ and $\delta_{{\cal O}_{p_{n+1}}}$
respectively; furthermore, one can require that the path $\alpha_{n}$ is
contained in the $2\operatorname{d}(\delta_{{\cal O}_{p_{n}}},\delta_{{\cal
O}_{p_{n+1}}})$-neighborhood of $\delta_{{\cal O}_{p_{n}}}$ (see the comments
after Theorem B in [Sig]). As
$\lim_{|n|\rightarrow+\infty}\operatorname{d}(\delta_{{\cal
O}_{p_{n}}},\delta_{{\cal O}_{p_{n+1}}})=0$, one gets a continuous path
$\alpha:[-1,1]\to\\{\nu\in\mathcal{M}_{erg}(f)|\chi^{c}(\nu)\geq 0\\}$ defined
as
$\alpha(t)=\begin{cases}\nu_{2}&\text{if }t=-1,\\\ \alpha_{n}(t)&\text{if
}t\in[2^{n}-1,2^{n+1}-1]\text{~{}and~{}}n<0,\\\ \alpha_{n}(t)&\text{if
}t\in[1-2^{n},1-2^{n+1}]\text{~{}and~{}}n\geq 0,\\\ \nu_{1}&\text{if
}t=1.\end{cases}$
∎
Now, we give the proof of Theorem D which gives some descriptions on the set
of points with vanishing center Lyapunov exponents. The idea is that the
entropy of $h_{top}(\mathcal{R}_{g}(0))$ is bounded by lower capacity entropy
which can give some separated sets, then one can apply Lemma 3.7 to the
separated sets and gets some periodic orbits satisfying Theorem 2.10.
###### Proof of Theorem D.
We take the $C^{1}$-open neighborhood ${\cal U}$ of $f$ as in the proof of
Theorem C. Hence each $g\in{\cal U}$ satisfies the hypothesis of Lemma 3.7 and
one obtains a constant $\rho_{0}>0$ and a hyperbolic periodic point $q$ with
the posited properties.
Now, we fix $g\in{\cal U}.$ Denote by
$\chi_{\rm max}(g)=\sup_{x\in M}\big{|}\log\|Dg|_{E^{c}_{g}(x)}\|\big{|}$
Let ${\varepsilon}>0$ and $h\in[0,h_{top}(g,\mathcal{R}_{g}(0))].$ Take
${\varepsilon}^{\prime}\in(0,\chi_{0}/24)$ such that
$\frac{h-{\varepsilon}^{\prime}}{1+\rho_{0}{\varepsilon}^{\prime}}>h-{\varepsilon}\,\textrm{~{}and~{}}\,2{\varepsilon}^{\prime}+\big{(}2\rho_{0}{\varepsilon}^{\prime}+{\varepsilon}^{\prime}\big{)}\chi_{\rm
max}(g)<{\varepsilon}.$
By the uniform continuity of the center bundle, there exists $\eta_{1}>0$ such
that
$|\log\|Dg|_{E^{c}_{g}(x)}\|-\log\|Dg|_{E^{c}_{g}(y)}\||<{\varepsilon}^{\prime},\textrm{
for any $x,y\in M$ with $d(x,y)<\eta_{1}.$}$ (21)
By the compactness of the set of Borel probability measures on $M$, there
exists $\eta_{2}>0$ such that for any Borel probability measures
$\nu_{1},\nu_{2}$ on $M$ with $\operatorname{d}(\nu_{1},\nu_{2})<\eta_{2}$,
one has
$\big{|}\int\log\|Dg|_{E^{c}_{g}}\|\operatorname{d}\nu_{1}-\int\log\|Dg|_{E^{c}_{g}}\|\operatorname{d}\nu_{2}\big{|}<{\varepsilon}^{\prime}.$
(22)
Apply $\lambda=e^{-{\varepsilon}^{\prime}}$, $\eta_{2}>0$ and the dominated
splitting $TM=E^{s}_{g}\oplus(E^{c}_{g}\oplus E^{u}_{g})$ to Theorem 2.10, and
one gets a constant $\xi_{0}>0$ with the posited properties.
For any $k\in\mathbb{N}$, define
$\mathcal{R}_{g}^{k}(0)=\big{\\{}x\in\mathcal{R}_{g}(0)\;|~{}e^{-|n|{\varepsilon}^{\prime}}\leq\|Dg^{n}|_{E^{c}_{g}(x)}\|\leq
e^{|n|{\varepsilon}^{\prime}},\textrm{~{}for any $n\in\mathbb{Z}$ with
$|n|>k$}\big{\\}}.$
Then $\mathcal{R}_{g}^{k}(0)\subset\mathcal{R}_{g}^{k+1}(0)$ for any
$k\in\mathbb{N}$ and
$\mathcal{R}_{g}(0)=\cup_{k\in\mathbb{N}}\mathcal{R}_{g}^{k}(0).$ By the
second item in Proposition 2.2, one has
$h_{top}(g,\mathcal{R}_{g}(0))=\lim_{k\rightarrow\infty}h_{top}(g,\mathcal{R}^{k}_{g}(0))$.
As $0\leq h\leq h_{top}(g,\mathcal{R}_{g}(0))$, there exists
$k_{0}\in\mathbb{N}$ such that $h_{top}(g,\mathcal{R}_{g}^{k_{0}}(0))\geq
h-{\varepsilon}^{\prime}/4.$ Let $C>1$ be a constant such that for any
$x\in\mathcal{R}_{g}^{k_{0}}(0)$, one has
$C^{-1}\cdot e^{-|n|{\varepsilon}^{\prime}}\leq\|Dg^{n}|_{E_{g}^{c}(x)}\|\leq
C\cdot e^{|n|{\varepsilon}^{\prime}}\textrm{~{}for any $n\in\mathbb{Z}$ .}$
For $\xi>0$ and $n\in\mathbb{N}$, fix a $(n,2\xi)$-separated set $S(n,2\xi)$
of $\mathcal{R}_{g}^{k_{0}}(0)$ with maximal cardinality. By the third item of
Proposition 2.2, one has
$\underline{Ch}_{top}(g,\mathcal{R}_{g}^{k_{0}}(0))\geq
h_{top}(g,\mathcal{R}_{g}^{k_{0}}(0))\geq h-{\varepsilon}^{\prime}/4.$ Then
there exists $\xi_{1}>0$ such that for any $\xi\leq\xi_{1}$, there exists
$k_{0}<\widehat{N}=\widehat{N}(\xi,{\varepsilon}^{\prime})\in\mathbb{N}$ such
that $\\#S(n,2\xi)\geq e^{n(h-{\varepsilon}^{\prime}/2)}$ for any
$n>\widehat{N}$.
Now, apply the continuous function $\varphi(x)=\log\|Dg|_{E_{g}^{c}(x)}\|$,
$\chi=0$, ${\varepsilon}^{\prime}\in(0,\chi_{0}/24)$, $C>1$, and
$0<\xi<\min\\{\xi_{0},\xi_{1}\\}$ to Lemma 3.7, and one gets an integer
$N\in\mathbb{N}$. For $n>\max\\{N,\widehat{N}\\}$, apply Lemma 3.7 to the
$(n,2\xi)$-separated set $S(n,2\xi)$ and one obtains a collection
$P\subset{\rm NUH}_{{\varepsilon}^{\prime}}$ of periodic points with the same
period $l\in(n,n+\rho_{0}{\varepsilon}^{\prime}n)$ such that
* •
$\\#P>e^{n(h-{\varepsilon}^{\prime})};$
* •
$P$ is a $(l,\xi)$-separated set;
* •
$\operatorname{d}(p,p^{\prime})<\xi/16$ for any $p,p^{\prime}\in P$;
* •
for any $p\in P$, there exists $x_{p}\in S(n,2\xi)$ such that
$\big{|}\int\log\|Dg|_{E^{c}_{g}}\|\operatorname{d}\delta_{{\cal
O}_{p}}-\int\log\|Dg|_{E^{c}_{g}}\|\operatorname{d}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x_{p})}\big{|}<\big{(}2\rho_{0}{\varepsilon}^{\prime}+{\varepsilon}^{\prime}\big{)}\chi_{\rm
max}(g).$
For any $p\in P$, as $S(n,2\xi)\subset\mathcal{R}^{k_{0}}_{g}(0)$ and
$n>k_{0}$, one has
$\begin{split}\int\log\|Dg|_{E^{c}_{g}}\|\operatorname{d}\delta_{{\cal
O}_{p}}&<\int\log\|Dg|_{E^{c}_{g}}\|\operatorname{d}\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{i}(x)}+\big{(}2\rho_{0}{\varepsilon}^{\prime}+{\varepsilon}^{\prime}\big{)}\chi_{\rm
max}(g)\\\
&\leq{\varepsilon}^{\prime}+\big{(}2\rho_{0}{\varepsilon}^{\prime}+{\varepsilon}^{\prime}\big{)}\chi_{\rm
max}(g).\end{split}$ (23)
As $\xi<\xi_{0}$, apply $P$ to Theorem 2.10, and one gets a horseshoe
$\Lambda$ whose center is uniformly expanding such that
* •
$h_{top}(g,\Lambda)\geq\frac{\log\\#P}{l}>\frac{h-{\varepsilon}^{\prime}}{1+\rho_{0}{\varepsilon}^{\prime}}>h-{\varepsilon};$
* •
each $\mu\in\mathcal{M}_{inv}(g,\Lambda)$ belongs to the
$\eta_{2}$-neighborhood of $\\{\sum_{p\in P}t_{p}\delta_{{\cal
O}_{p}}|\,t_{p}\geq 0\,,\sum t_{p}=1\\}.$
By Equations (22) and (23), for each $\mu\in{\cal M}_{inv}(g,\Lambda)$, one
has
$\int\|Dg|_{E_{g}^{c}}\|\operatorname{d}\mu\leq
2{\varepsilon}^{\prime}+\big{(}2\rho_{0}{\varepsilon}^{\prime}+{\varepsilon}^{\prime}\big{)}\chi_{\rm
max}(g)<{\varepsilon}.$
Since $\Lambda$ has expanding center, one has
$\int\|Dg|_{E_{g}^{c}}\|\operatorname{d}\mu>0.$ This ends the proof of Theorem
D.
∎
Now, we turn to prove the continuity of topological entropy and the
intermediate entropy property.
###### Proof of Theorem A.
Let ${\cal U}$ be the $C^{1}$-open neighborhood of $f$ given by Theorem C. For
partially hyperbolic diffeomorphisms with one-dimensional center, by Corollary
C and Theorem D in [LVY] (see also Theorem 1.1 in [DFPV]), one has
* •
the metric entropy varies upper semi-continuously, and in particular, there
exist measures of maximal entropy;
* •
the function $g\in{\cal U}\mapsto h_{top}(g)$ is upper semi-continuous.
For each $g\in{\cal U}$, let $\nu$ be an ergodic measure of maximal entropy.
By Theorem C and Katok’s result [Ka] (see also [Ge]), $\nu$ is approximated by
horseshoes in weak$*$-topology and in entropy. By the structural stability of
horseshoes, one deduces that the topological entropy is lower semi-continuous
at $g$, and hence is continuous at $g$, proving that $g\in{\cal U}\mapsto
h_{top}(g)$ is continuous. It is classical that horseshoes satisfy the
intermediate entropy property implying that $g$ also satisfies the
intermediate entropy property. ∎
At last, we give the proof of Corollary B.
###### Proof of Corollary B.
By Theorem 2.22, $f$ is mostly expanding, and thus all the ergodic measures in
$G^{u}(f)$ have positive center Lyapunov exponents. Then Corollary B follows
directly from Theorem A. ∎
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_Jinhua Zhang_
---
School of Mathematical Sciences, Beihang University, Beijing, 100191, P. R.
China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
|
$|y_{i_{0}}\rangle,|y_{i_{1}}\rangle$ for $i_{0}\neq i_{1}$, such that
$|y_{i_{0}}\rangle=|y_{i_{1}}\rangle=0$. Then, the vectors
$|x_{i_{0}}\rangle,|x_{i_{1}}\rangle$ are orthonormal. We simply define
$|S_{0}\rangle=|i_{0}\rangle,|S_{1}\rangle=|i_{1}\rangle$,
$|R_{0}\rangle=|x_{i_{0}}\rangle$ and $|R_{1}\rangle=|x_{i_{1}}\rangle$. One
can calculate that $R_{*}E_{0}S_{*}={\rm 1l}_{\mathbb{C}^{2}}$ and
$R_{*}E_{1}S_{*}=0$.
In the third case, for all $i\in\\{0,\ldots,3\\}$ vectors
$|x_{i}\rangle,|y_{i}\rangle$ are not linearly independent and there is at
most one zero vector $|y_{i_{3}}\rangle$ for some $i_{3}\in\\{0,\ldots,3\\}$.
Define indices $i_{0},i_{1},i_{2}\in\\{0,\ldots,3\\}$ as the remaining labels,
such that $\\{i_{0},\ldots,i_{3}\\}$ covers the whole set $\\{0,\ldots,3\\}$.
Define the matrix
$M=\left[\begin{array}[]{ccc}\langle{y_{i_{0}}}|{x_{i_{0}}}\rangle&\langle{y_{i_{1}}}|{x_{i_{1}}}\rangle&\langle{y_{i_{2}}}|{x_{i_{2}}}\rangle\\\
\langle{y_{i_{0}}}|{y_{i_{0}}}\rangle&\langle{y_{i_{1}}}|{y_{i_{1}}}\rangle&\langle{y_{i_{2}}}|{y_{i_{2}}}\rangle\end{array}\right].$
(111)
In the first sub-case we assume that $\mathrm{rank}(M)=1$. Define
$b=\frac{\langle{y_{i_{1}}}|{y_{i_{1}}}\rangle}{\langle{y_{i_{0}}}|{y_{i_{0}}}\rangle}$.
We can take $|S_{0}\rangle=|i_{0}\rangle$, $|S_{1}\rangle=|i_{1}\rangle$,
$|R_{0}\rangle=|y_{i_{0}}\rangle$ and
$|R_{1}\rangle=\frac{1}{b}|y_{i_{1}}\rangle$. One can calculate that
$R_{*}E_{0}S_{*}=\langle{y_{i_{0}}}|{x_{i_{0}}}\rangle{\rm
1l}_{\mathbb{C}^{2}}$ and
$R_{*}E_{1}S_{*}=\langle{y_{i_{0}}}|{y_{i_{0}}}\rangle{\rm
1l}_{\mathbb{C}^{2}}$.
In the second sub-case we assume that $\mathrm{rank}(M)=2$. Define indices
$j_{1},j_{2}\in\\{0,1,2\\}$, such that
$\mathrm{rank}\left(\left[\begin{array}[]{cc}M_{0,j_{1}}&M_{0,j_{2}}\\\
M_{1,j_{1}}&M_{1,j_{2}}\end{array}\right]\right)=2.$ (112)
Define $j_{0}\in\\{0,1,2\\}$ as the remaining label, such that
$\\{j_{0},j_{1},j_{2}\\}$ covers the whole set $\\{0,1,2\\}$. Take
$|S_{0}\rangle=|i_{j_{0}}\rangle$, $|R_{0}\rangle=|y_{i_{j_{0}}}\rangle$ and
define
$(b_{1},b_{2})^{\top}\coloneqq\left[\begin{array}[]{cc}\langle{y_{i_{j_{1}}}}|{x_{i_{j_{1}}}}\rangle&\langle{y_{i_{j_{2}}}}|{x_{i_{j_{2}}}}\rangle\\\
\langle{y_{i_{j_{1}}}}|{y_{i_{j_{1}}}}\rangle&\langle{y_{i_{j_{2}}}}|{y_{i_{j_{2}}}}\rangle\end{array}\right]^{-1}(\langle{y_{i_{j_{0}}}}|{x_{i_{j_{0}}}}\rangle,\langle{y_{i_{j_{0}}}}|{y_{i_{j_{0}}}}\rangle)^{\top}.$
(113)
We may take $|S_{1}\rangle=|i_{j_{1}}\rangle+|i_{j_{2}}\rangle$ and
$|R_{1}\rangle=\bar{b}_{1}|y_{i_{j_{1}}}\rangle+\bar{b}_{2}|y_{i_{j_{2}}}\rangle$.
Direct calculations reveal that
$R_{*}E_{0}S_{*}=\langle{y_{i_{j_{0}}}}|{x_{i_{j_{0}}}}\rangle{\rm
1l}_{\mathbb{C}^{2}}$ and
$R_{*}E_{1}S_{*}=\langle{y_{i_{j_{0}}}}|{y_{i_{j_{0}}}}\rangle{\rm
1l}_{\mathbb{C}^{2}}$. ∎
### A.15 Proof of Theorem 16
Theorem 16. Let $\mathcal{E}_{r}\in\mathcal{C}(\mathcal{Y})$ be a random
quantum channel defined according to Eq. (32). Then, the following two
implications hold
$\begin{split}r<\frac{\dim(\mathcal{X})\dim(\mathcal{Y})}{\dim(\mathcal{X})^{2}-1}&\implies\mathcal{P}\left(\mathcal{E}_{r}\in\xi(\mathcal{X},\mathcal{Y})\right)=1,\\\
\mathcal{P}\left(\mathcal{E}_{r}\in\xi_{1}(\mathcal{X},\mathcal{Y})\right)=1&\implies
r<\sqrt{\frac{\dim(\mathcal{Y})}{\dim(\mathcal{X})-1}}.\end{split}$ (114)
###### Proof.
For $r\in\mathbb{N}$ satisfying
$r<\frac{\dim(\mathcal{X})\dim(\mathcal{Y})}{\dim(\mathcal{X})^{2}-1}$, let
$(G_{i})_{i=1}^{r}\subset\mathcal{M}(\mathcal{Y})$ be a tuple of random and
independent Ginibre matrices and $Q=\sum_{i=1}^{r}G_{i}^{\dagger}G_{i}$.
Define the projector $\Pi=\sum_{i=0}^{\dim(\mathcal{X})-1}|i\rangle\\!\langle
i|$ and consider the set
$A=\left\\{(G_{i})_{i=1}^{r}:\quad\mathrm{rank}(Q)=\dim(\mathcal{Y}),\mathrm{rank}\left(\sum_{i=1}^{r}G_{i}^{\dagger}\Pi
G_{i}\right)=\min\\{r\dim(\mathcal{X}),\dim(\mathcal{Y})\\}\right\\}.$ (115)
One can observe that $\mathcal{P}((G_{i})_{i=1}^{r}\in A)=1$. Let
$\mathcal{E}_{r}\in\mathcal{C}(\mathcal{Y})$ be a random channel defined
according to Eq. (32) for $(G_{i})_{i=1}^{r}\in A$, that is
$\mathcal{E}_{r}(Y)=\sum_{i=1}^{r}\left(G_{i}Q^{-1/2}\right)Y\left(G_{i}Q^{-1/2}\right)^{\dagger}.$
Define $S=Q^{1/2}\tilde{S}$ for
$\tilde{S}\in\mathcal{M}(\mathcal{X},\mathcal{Y})$ and $R=\tilde{R}\Pi$ for
$\tilde{R}\in\mathcal{M}(\mathcal{Y},\mathcal{X})$. We obtain
$RG_{i}Q^{-1/2}S=\tilde{R}\Pi G_{i}\tilde{S}.$ Utilizing Lemma 9, Proposition
12 and Theorem 1 $(D)$ for $\tilde{\mathcal{E}}=\mathcal{K}\left((\Pi
G_{i})_{i=1}^{r}\right)\in s\mathcal{C}(\mathcal{Y})$, there exist
$\tilde{S},\tilde{R}$, such that $\tilde{R}\Pi G_{i}\tilde{S}\propto{\rm
1l}_{\mathcal{X}}$ and $\tilde{R}\Pi G_{i_{0}}\tilde{S}\neq 0$ for some
$i_{0}$. Eventually, $\mathcal{E}_{r}\in\xi(\mathcal{X},\mathcal{Y})$.
Now, for a given $r\in\mathbb{N}$ let us define
$B=\\{\mathcal{E}_{r}:\,\,\mathcal{E}_{r}\in\xi_{1}(\mathcal{X},\mathcal{Y})\\}$.
From the assumption $\mathcal{P}(B)=1$, we obtain that $B$ is a dense subset
of
$\\{\mathcal{E}\in\mathcal{C}(\mathcal{Y}):\,\,\mathrm{rank}(J(\mathcal{E}))\leq
r\\}$. Imitating the proof of Theorem 7, we get that if
$\mathcal{E}\in\mathcal{C}(\mathcal{Y})$ and
$\mathrm{rank}(J(\mathcal{E}))\leq r$, then
$\mathcal{E}\in\xi_{1}(\mathcal{X},\mathcal{Y})$. That implies $r\leq
r_{1}(\mathcal{X},\mathcal{Y})$. By using Lemma 10 we obtain the desired
inequality. ∎
### A.16 Proof of Proposition 17
Proposition 17. Let $\Upsilon\subset\mathcal{C}(\mathcal{Y})$ be a nonempty
and convex family of noise channels. Define $\mu$ to be a probability measure
defined on $\Upsilon$ and assume that the support of $\mu$ is equal to
$\Upsilon$. Let
$\bar{\mathcal{E}}=\int_{\Upsilon}\mathcal{E}\mu(d\mathcal{E})\in\mathcal{C}(\mathcal{Y})$
and fix $(\SS,\mathcal{R})\in s\mathcal{C}(\mathcal{X},\mathcal{Y})\times
s\mathcal{C}(\mathcal{Y},\mathcal{X})$. The following conditions are
equivalent:
1. (A)
For each $\mathcal{E}\in\Upsilon$ there exists $p_{\mathcal{E}}\geq 0$ such
that $\mathcal{R}\mathcal{E}\SS=p_{\mathcal{E}}\mathcal{I}_{\mathcal{X}}$ and
$\int_{\Upsilon}p_{\mathcal{E}}\mu(d\mathcal{E})>0.$
2. (B)
It holds that
$0\neq\mathcal{R}\bar{\mathcal{E}}\SS\propto\mathcal{I}_{\mathcal{X}}$.
###### Proof.
$(B)\implies(A)$
Let us assume that
$\mathcal{R}\bar{\mathcal{E}}\SS=p\mathcal{I}_{\mathcal{X}}$ for $p>0$. There
exists a $k$ dimensional affine subspace $\mathcal{L}$ such that
$\Upsilon\subset\mathcal{L}$ and
$\mathrm{int}_{\mathcal{L}}(\Upsilon)\neq\emptyset$. Take arbitrary
$\mathcal{E}_{0}\in\Upsilon$. There exist
$\mathcal{E}_{1},\ldots,\mathcal{E}_{k}\in\Upsilon$ such that convex hull of
points $\mathcal{E}_{0},\ldots,\mathcal{E}_{k}$ is a $k$-dimensional simplex
$\Delta_{k}$. For any state
$|\psi\rangle\\!\langle\psi|\in\mathcal{D}(\mathcal{X})$ it holds
$p|\psi\rangle\\!\langle\psi|=\mathcal{R}\bar{\mathcal{E}}\SS(|\psi\rangle\\!\langle\psi|)=\int_{\Upsilon}\mathcal{R}\mathcal{E}\SS(|\psi\rangle\\!\langle\psi|)\mu(d\mathcal{E})\geq\int_{\Delta_{k}}\mathcal{R}\mathcal{E}\SS(|\psi\rangle\\!\langle\psi|)\mu(d\mathcal{E}).$
(116)
Inside $\Delta_{k}$ each $\mathcal{E}$ can be uniquely represented as
$\sum_{i=0}^{k}q_{i}(\mathcal{E})\mathcal{E}_{i}$, where
$(q_{i}(\mathcal{E}))_{i=0}^{k}$ is a probability vector which depends on
$\mathcal{E}$. Hence,
$p|\psi\rangle\\!\langle\psi|\geq\sum_{i=0}^{k}\int_{\Delta_{k}}q_{i}(\mathcal{E})\mathcal{R}\mathcal{E}_{i}\SS(|\psi\rangle\\!\langle\psi|)\mu(d\mathcal{E})\geq\left(\int_{\Delta_{k}}q_{0}(\mathcal{E})\mu(d\mathcal{E})\right)\mathcal{R}\mathcal{E}_{0}\SS(|\psi\rangle\\!\langle\psi|).$
(117)
There exists $\epsilon$ small ball $B_{\epsilon}$ around $\mathcal{E}_{0}$,
such that for each channel $\mathcal{E}\in B_{\epsilon}\cap\Delta_{k}$ it
holds $q_{0}(\mathcal{E})\geq\frac{1}{2}$. Hence,
$\int_{\Delta_{k}}q_{0}(\mathcal{E})\mu(d\mathcal{E})\geq\frac{1}{2}\mu\left(B_{\epsilon}\cap\Delta_{k}\right)>0,$
where in the last inequality we used the fact that the support of $\mu$ is
equal to $\Upsilon$. Therefore, it holds that for any
$|\psi\rangle\\!\langle\psi|\in\mathcal{D}(\mathcal{X})$ we have
$\mathcal{R}\mathcal{E}_{0}\SS(|\psi\rangle\\!\langle\psi|)\propto|\psi\rangle\\!\langle\psi|$
and from Lemma 18 there exists $p_{\mathcal{E}_{0}}\geq 0$ such that
$\mathcal{R}\mathcal{E}_{0}\SS=p_{\mathcal{E}_{0}}\mathcal{I}_{\mathcal{X}}$.
The instant relation $\int_{\Upsilon}p_{\mathcal{E}}\mu(d\mathcal{E})=p>0$
ends the proof. ∎
|
Comparing Session Type Systems derived from Linear Logict1
Work partially supported by the Dutch Research Council (NWO) under the VIDI Project No. 016.Vidi.189.046 (Unifying Correctness for Communicating Software).
Bas van den Heuvel
Jorge A. Pérez
University of Groningen, The Netherlands
Session types are a typed approach to message-passing concurrency, where types describe sequences of intended exchanges over channels.
Session type systems
have been given strong logical foundations via Curry-Howard correspondences with linear logic, a resource-aware logic that naturally captures structured interactions.
These logical foundations provide an elegant framework to specify and (statically) verify message-passing processes.
In this paper, we rigorously compare different type systems for concurrency derived from the Curry-Howard correspondence between linear logic and session types.
We address the main divide between these type systems: the classical and intuitionistic presentations of linear logic.
Along the years, these presentations have given rise to separate research strands on logical foundations for concurrency; the differences between their derived type systems have only been addressed informally.
To formally assess these differences, we develop
, a session type system that encompasses type systems derived from classical and intuitionistic interpretations of linear logic.
Based on a fragment of Girard's Logic of Unity, provides a basic reference framework:
we compare existing session type systems by characterizing fragments of that coincide with classical and intuitionistic formulations.
We analyze the significance of our characterizations by considering the locality principle (enforced by intuitionistic interpretations but not by classical ones) and forms of process composition induced by the interpretations.
Session types are an approach to correct message-passing concurrency in which communication over channels is abstracted by types as sequences of exchanges.
Session type systems have been given logical foundations via Curry-Howard correspondences with linear logic, both in intuitionistic and classical presentations.
The type systems derived from these two logics enforce communication correctness on classes of $\pi$-calculus processes that largely coincide.
However, there are significant differences between these classes.
It has been informally observed that, unlike the classical type system, the intuitionistic type system enforces locality for shared channels (i.e. received channels cannot be used for replicated receives).
In this paper, we revisit this observation from a formal angle.
We develop United Linear Logic (), a logic encompassing both classical and intuitionistic linear logic based on Girard's Logic of Unity.
Simultaneously, we follow the Curry-Howard correspondences for session types to define a session type system for the $\pi$-calculus based on , denoted .
We then use to formally assess the differences between the intuitionistic and classical type systems and their classes of typable processes, and justify the role of locality and duality therein.
We confirm the informal observation concerning locality, but also discuss a newly discovered difference concerning non-local empty send actions.
We further corroborate the usefulness of by considering extensions with rules for more expressive parallel composition and channel connection, and find that such extensions do not fit in type systems derived from intuitionistic linear logic.
Concurrency, linear logic, $\pi$-calculus, session types.
§ INTRODUCTION
Establishing the correctness of message-passing programs is a central but challenging problem.
Within formal methods for concurrency, session types [27, 49, 31] are now a consolidated approach to statically verifying safety and liveness properties of communicating programs.
Session types specify communication over channels as sequences of exchanges.
This way, e.g., the session type $!\mathsf{int}.?\mathsf{bool}.\tend$ describes a channel's intended protocol:
send an integer, receive a boolean, and close the channel.
Due to its simplicity and expressivity, the $\pi$-calculus [39, 47]—the paradigmatic model of concurrency and interaction—is a widely used specification language for developing session types and establishing their basic theory.
In this paper, we are interested in developing further the logical foundations of session type systems for the $\pi$-calculus.
Context: Linear Logic and Session Types
In a line of work developed by Caires, Pfenning, Wadler, and several others, the theory of session types has been given firm logical foundations in the form of Curry-Howard-style correspondences.
Caires and Pfenning [6] discovered a correspondence between session types for the $\pi$-calculus and Girard's linear logic [22]:
session types
logical propositions
$\pi$-calculus processes
process communication
cut elimination
Based on this logical bridge, the resulting type systems simultaneously ensure important properties for processes, such as session fidelity (processes respect session types), communication safety (absence of communication errors), and progress/deadlock-freedom (processes never reach stuck states).
There are two presentations of linear logic, classical and intuitionistic, and session type systems derived from linear logic have inherited this dichotomy.
Under Curry-Howard interpretations, typing judgments and typing rules correspond to the logical sequents and inference rules of the underlying linear logic.
In both classical and intuitionistic cases, judgments assign independent session protocols to the channels of a process.
Because these judgments differ between the presentations of linear logic, there are consequences for their respective interpretations as type systems:
* Caires and Pfenning's correspondence [6] uses an intuitionistic linear logic where judgments for process are two-sided, with zero or many channels on the left and exactly one channel on the right.
Such judgments have a convenient rely-guarantee reading: the process relies on the behaviors described by the channels on the left to guarantee the behavior of the one channel on the right.
In this interpretation, each logical connective thus requires two rules: the right rule specifies how to offer a behavior, while the left rule specifies how to use a behavior.
* Wadler's correspondence [52] uses classical linear logic, where judgments are single-sided (all channels appear on the right) and lack a rely-guarantee reading.
In this interpretation, there is only a single rule per logical connective, which makes such type systems direct and economical.
Although the differences and relationship between intuitionistic and classical linear logic are relevant and well-known (see, e.g., [46, 5, 36]), the differences between their derived session type systems have only been addressed informally.
In particular, Caires [11] observe that, unlike classical interpretations, an intuitionistic interpretation guarantees locality for shared channels.
In the meantime, both interpretations have been extended in multiple, exciting directions (see, e.g., [8, 1, 12, 48, 37, 3, 7, 13, 50, 9, 32, 44, 18, 30]).
This emergence of families of type systems (one classical, one intuitionistic) has deepened the original dichotomy and somewhat obscured our understanding of logical foundations of concurrency as a whole.
This state of affairs calls for a formal comparison between the session type systems derived from classical and intuitionistic linear logics that goes beyond their superficial differences. This seems to us as an indispensable step in consolidating the logical foundations of message-passing concurrency.
Goal of the Paper
In this paper, we aim at formally comparing the session type systems derived from interpretations of intuitionistic and classical linear logic.
We are concerned with two families of type systems for the $\pi$-calculus, which naturally induce two classes of typable processes, namely those typable in intuitionistic and classical interpretations, respectively.
These two classes are known to largely overlap with each other;
informal observations suggest that some processes are typable in a classical setting but not in an intuitionistic setting.
We are interested in determining precisely how these classes of processes relate to each other, but especially in uncovering why their logical underpinnings justify such relationships.
The key step in our approach is to define a basic framework of reference in which both type systems, intuitionistic and classical, can be objectively compared.
Girard's Logic of Unity (LU) [23] has a similar goal: to objectively compare classical logic, intuitionistic logic, and (classical and intuitionistic) linear logic in a single system, abstracting away from syntactical differences.
It is then only natural to use LU as a reference for our comparison.
We develop , a session type system for the $\pi$-calculus that subsumes classical and intuitionistic interpretations of linear logic as session types.
Based on a fragment of LU that suits our purposes (dubbed ), the type system allows us to define a class of typable processes that contains processes typable both in the intuitionistic type system, as well as in the classical type system.
In our approach, the type system provides an effective framework and a yardstick for a formal comparison.
Because its typing rules encompass both classical and intuitionistic formulations, we can readily characterize their two (sub-)classes of typable processes by considering the different typing rules that apply in each case.
Analyzing these different portions of rules allows us to then characterize the differences (in expressiveness/typability) between the intuitionistic and classical type systems.
Moreover, these characterizations allow use to determine how extensions to the type systems might be supported by the intuitionistic and classical interpretations.
Contributions and Outline
Summing up, this paper makes the following contributions:
* The session type system , which is derived from a concurrent interpretation of (a linear fragment of LU), together with its soundness results (<Ref>, sec:ull);
* A formal comparison between (i) the classes of processes typable under interpretations of intuitionistic and classical linear logic and (ii) those typable in (sec:comparison).
We prove that
(1) precisely captures the class of processes typable under the classical interpretation (<Ref>), and that
(2) the class of processes typable under the intuitionistic interpretation is strictly included in (<Ref>).
Together, these two results confirm the informal observation that the two interpretations induce different classes of processes.
* An analysis of the significance of our characterizations in terms of two different aspects: the locality principle enforced by intuitionistic interpretations, and process composition usually limited in type systems derived from linear logic in comparison to session type systems not based on logical foundations (sec:discussion).
This analysis corroborates the observation made in [11] concerning locality, and suitably extends it to show that intuitionistic interpretations
cannot type empty sends on previously received channels.
Finally, we draw some conclusions in <Ref>.
Related Work
Our work can be seen as an expressiveness study on the classes of processes induced by different type systems.
Concretely, our results can be seen as providing formal evidence that session type systems derived from classical linear logic are more expressive than those based on intuitionistic linear logic, based on the fact that the former are more permissive than the latter.
The study of the relative expressiveness of process calculi has a long, fruitful history (see, e.g. [25, 42, 40]).
The aim is to compare two different process languages by defining an encoding (a translation up to some correctness criteria) between them or by proving its non-existence.
Even though our aims are similar in spirit to expressiveness studies, at a technical level there are substantial differences, because our focus is on assessing the influence that different type systems have on the same process language—as such, our comparisons do not rely on encodings.
Although most studies on relative expressiveness have addressed untyped process languages, some works have studied the influence of (behavioral) types on the expressiveness of process languages—see, e.g., [15, 21, 14, 35, 41].
Salient examples are the works [17, 43], which compare type systems that enforce the deadlock-freedom and termination properties, respectively.
In particular, a main result in [17] is that (classical) linear logic induces a strict subclass of deadlock-free processes with respect to the class of typable processes in non-logically motivated type systems.
Unlike our work, the focus in [17] is on different process languages, each with a different type system.
This requires the definition of encodings, on processes but also on types, that abstract away from the syntactic differences between the classes of processes induced by each typed framework.
This paper is an extended, revised version of the workshop paper [28].
We present several improvements and developments with respect to that work.
First, here we have significantly improved the correctness results for (<Ref>).
and considered branching/selection types (not studied in [28]).
Also, we have refined the presentation of in [28] with a more explicit treatment of duality, presented in <Ref>. Moreover, we have broadened the analysis of our comparison results in <Ref> by considering extensions to the type system with more expressive forms of parallel composition and restriction.
§ A SESSION TYPE SYSTEM BASED ON LU
Girard [23] developed Logic of Unity (LU) to study and compare classical, intuitionistic, and linear logic, without having to “change the rules of the game by, e.g., passing from one style of sequent to the other.”
The idea is simple: there is one form of sequent with an abundance of inference rules.
The several logics subsumed by LU are then characterized by (possibly overlapping) subsets of those rules.
Clearly, we find ourselves in a similar situation:
we want to compare intuitionistic and classical linear logic as session type systems, by abstracting away from typing judgments and rules of different forms.
To this end, in this section we introduce United Linear Logic (), a logic based on the linear fragment of LU, and present the Curry-Howard interpretation of as a session type system for the $\pi$-calculus, dubbed , following [6, 52] (ssec:proccal).
As we will see in <Ref>, we can then characterize the session type interpretations of intuitionistic and classical linear logic as subsets of the typing rules of .
We also present correctness properties common to (logically motivated) session type systems (ssec:procprop).
Finally, we discuss an alternative presentation of that has a more explicit account of duality (ssec:duality).
§.§ The Process Calculus and Type System
A session type represents a sequence of exchanges that should be performed along a channel.
Propositions in —interpreted as session types in —are defined as follows:
propositions/types are generated by the following grammar:
\begin{align*}
A, B ::= \1 \bnf \bot \bnf A \tensor B \bnf A \lolli B \bnf \oplus\{i:A\}_{i \in I} \bnf \&\{i:A\}_{i \in I} \bnf \bang A \bnf \whynot A
\end{align*}
<Ref> gives the intuitive reading of the interpretation of propositions as session types.
Some details are worth noting.
First, receiving in is defined using the intuitionistic connective ;
we will see later that can be used to define the classical connective —also interpreted as receiving—which is not supported by intuitionistic linear logic.
Second, supports labeled $n$-ary branching constructs (following, e.g., Caires and Pérez in [7]).
In linear logic, $\oplus$ and $\&$ are binary connectives [22, 24, 23].
In [6, 52], Caires, Pfenning, and Wadler originally interpret them as a choice between a left and a right option.
However, in session-typed $\pi$-calculi (see, e.g., [31]) $n$-ary choice is standard and more convenient.
$\1$ and $\bot$
Close the channel
$A \tensor B$
Send a channel of type $A$ and continue as $B$
$A \lolli B$
Receive a channel of type $A$ and continue as $B$
$\oplus\{i:A_i\}_{i \in I}$
Send a label $i \in I$ and continue as $A_i$
$\&\{i:A_i\}_{i \in I}$
Receive a label $i \in I$ and continue as $A_i$
$\bang A$
Repeatedly provide a service of type $A$
$\whynot A$
Connect to a service of type $A$
Interpretation of propositions as session types.
The duality of propositions is defined as follows:
Duality is a binary relation on propositions/types, denoted $\dual{A}$, defined as follows:
\begin{align*}
\dual{\1}
&:= \bot
& \dual{(A \tensor B)}
&:= A \lolli \dual{B}
& \dual{(\oplus\{i:A_i\}_{i \in I})}
&:= \&\{i:\dual{A_i}\}_{i \in I}
& \dual{(\bang A)}
&:= \whynot \dual{A} \\
\dual{\bot}
&:= \1
& \dual{(A \lolli B)}
&:= A \tensor \dual{B}
& \dual{(\&\{i:A_i\}_{i \in I})}
&:= \oplus\{i:\dual{A_i}\}_{i \in I}
& \dual{(\whynot A)}
&:= \bang \dual{A}
\end{align*}
Duality in reflects the intended reciprocity of protocols between two parties:
when a process on one side of a channel sends, the process on the opposite side must receive, and vice versa.
It is easy to see that duality is an involution:
$\dual{(\dual{A})} = A$.
As mentioned before, we can use duality to define in terms of : $A \parr B := \dual{A} \lolli B$;
as we will see, the rules for of classical linear logic can be recovered from the rules for in using duality.
Duality also shows that and are De Morgan-style duals in classical linear logic:
\begin{align*}
\dual{(A \tensor B)}
&= A \lolli \dual{B} = \dual{(\dual{A})} \lolli \dual{B} = \dual{A} \parr \dual{B} \\
\dual{(A \parr B)}
&= \dual{(\dual{A} \lolli B)} = \dual{A} \tensor \dual{B}
\end{align*}
The discipline of binary session types deals with concurrent processes that communicate through point-to-point channels.
The $\pi$-calculus [39, 47] offers a rigorous yet expressive framework for defining this discipline and establishing its fundamental properties.
is a type system for $\pi$-calculus processes defined as follows:
Process terms are generated by the following grammar:
\begin{align*}
P, Q ::=
\0 \bnf \nu{x}P \bnf P \| Q \bnf \send{x}{y}.P \bnf \recv{x}{y}.P \bnf x \triangleleft \ell.P \bnf x \triangleright \{i:P_i\}_{i \in I} \\
\bnf
\serv{x}{y}.P \bnf \fwd{x}{y} \bnf \send{x}{}.P \bnf \recv{x}{}.P
\end{align*}
Process constructs for inaction $\0$, channel restriction $\nu{x}P$, and parallel composition $P \| Q$ have standard readings.
The same applies to constructs for sending, receiving, selection, branching, and replicated receive prefixes, which are respectively denoted $\send{x}{y}.P$, $\recv{x}{y}.P$, $x \triangleleft \ell.P$, $x \triangleright \{i:P_i\}_{i \in I}$, and $\serv{x}{y}.P$.
Note that all these prefixes are blocking, which means that our process calculus implements synchronous communication (see, e.g., [12, 29] for interpretations of linear logic as session type systems in the asynchronous setting).
Process $\fwd{x}{y}$ denotes a forwarder that “fuses” channels $x$ and $y$;
it is akin to the link processes used in encodings of name passing using internal mobility [2].
We consider also constructs $\send{x}{}.P$ and $\recv{x}{}.P$, which specify the explicit closing of channels:
their synchronization represents the explicit de-allocation of linear resources.
We use these constructs to give a non-silent interpretation of $\1$ and $\bot$ (see ssec:types).
In $\nu{y}{P}$, $\recv{x}{y}.P$, and $\serv{x}{y}.P$ the occurrence of $y$ is binding, with scope $P$.
The set of free names of a process $P$ is denoted $\fn(P)$ and is the complement of $\bn(P)$, the set of bound names of $P$.
We identify processes up to consistent renaming of bound names, writing $\equiv_\alpha$ for this congruence.
We write $P\subst{y/x}$ for the capture-avoiding substitution of free occurrences of $y$ for $x$ in $P$.
Structural Congruence
The following is an important notation of syntactical equivalence for processes.
Structural congruence is a binary relation on process terms, denoted $P \equiv Q$.
It is defined as the least congruence on processes (i.e., closed under arbitrary process contexts) that satisfies the axioms in <Ref> (top).
The above formulation of structural congruence internalizes the conditions induced by typing (i.e., proof transformations in ), necessary for the correctness properties proven in <Ref>.
As is usual for Curry-Howard correspondences, computation is related to cut reduction.
Cuts are used in logic to combine two proofs that contain dual propositions.
As we will see, a cut in session type interpretations of linear logic entails the parallel composition and connection of two processes.
Generally, cut reduction transforms cuts into cuts on smaller types.
In correspondences between linear logic and session types, cut reduction corresponds to communication—the notion of computation in the $\pi$-calculus, which expresses internal behavior of processes.
The definition of the reduction relation follows.
Note that, for a simpler presentation, it includes a set of rules for commuting conversions ($\kappa$-rules);
an alternative presentation could treat commuting conversions as a behavioral equivalence.
νx ( P Q ) ≡νx ( Q P )
x ∉(Q)
y ∉(P)
νx ( P νy ( Q R ) ) ≡νy ( Q νx ( P R ) )
x ∉(R)
y ∉(P)
νx ( P νy ( Q R ) ) ≡νy ( νx ( P Q ) R )
x ≠y
νx(P xy) Py/x
νx ( x. x.Q ) Q
νx ( νy xy . ( P_1 P_2 ) xz.Q ) νx ( P_2 νy ( P_1 Qy/z ) )
j ∈I
νx ( x ◃j.P x ▹{i:Q_i}_i ∈I ) νx ( P Q_j )
νx ( νy xy.P xz.Q ) νx ( νy ( P Qy/z ) xz.Q )
u ∉(P)
νu ( P xz.Q ) P
νy ( P x.Q ) x.νy ( P Q )
y ∈(Q_2)
νy ( P νz xz.( Q_1 Q_2 ) ) νz xz.( Q_1 νy ( P Q_2 ) )
y ∈(Q_1)
νy ( P νz xz.( Q_1 Q_2 ) ) νz xz.( νy ( P Q_1 ) Q_2 )
νy ( P xz.Q ) xz.νy ( P Q )
νy ( P x ◃ℓ.Q ) x ◃ℓ.νy ( P Q )
νy ( x ▹{i:P_i}_i ∈I Q ) x ▹{i:νy ( P_i Q )}_i ∈I
Structural congruence and reduction for .
Reduction is a binary relation on process terms, denoted $P \red Q$, defined in <Ref>.
It is closed under the following rules:
Q Q'
P Q P Q'
P Q
νyP νyQ
P ≡P'
P' Q'
Q' ≡Q
P Q
We write $\red_\beta$ to denote reductions that follow from $\beta$-rules.
In this definition, Rule $\beta$id replaces a channel $x$ with a channel $y$ which $x$ is forwarded to.
Rule $\beta$close formalizes the explicit channel de-allocation mentioned above: the synchronization between an empty send and an empty receive effectively closes channel $x$.
Rule $\beta$send formalizes the synchronization between a send and a corresponding receive, substituting the sent name for the received name in the continuation of the receive.
Rule $\beta$sel formalizes the synchronization between a selection and a branch.
Rule $\beta$serv allows to synchronize a client and a service; a copy of the service $Q$ is spawned with the sent name substituted for the received name while the replicated receive remains.
Rule $\beta$weaken allows to clean up a service if it has no clients.
The $\kappa$-rules reflect commuting conversions in linear logic, which in the process calculus allow to pull prefixes on free names out of series of consecutive restrictions;
as we will see, we use them such that we can state a general progress result.
Rules par, res, and sc close reduction under parallel composition, restriction and structural congruence, respectively (<Ref>).
Type Checking
The inference system of is a sequent calculus with sequents of the form $\Gamma; \Delta \vdash \Lambda$.
Here, $\Gamma$, $\Delta$ and $\Lambda$ denote regions which collect propositions and obey different structural criteria.
$\Gamma$ is the unrestricted region, which contains propositions that can be indefinitely used.
$\Delta$ and $\Lambda$ are the linear regions, which contain propositions that must be used exactly once.
We write `$\emptyset$' to denote an empty region.
Also, we extend duality to regions:
$\dual{(\Delta)}$ contains exactly the duals of the propositions in $\Delta$.
The type system for is an extension of 's inference system with process and channel name annotations on sequents, such that judgments are of the form $\Gamma; \Delta \vdash P :: \Lambda$.
Then, the regions $\Gamma$, $\Delta$ and $\Lambda$ denote the unrestricted resp. linear contexts of $P$, consisting of assignments $x:A$ where $x$ is a channel name and $A$ is a proposition/type.
Γ; x:A ⊢xy :: y:A
Γ; x:A, y:A ⊢xy :: ∅
Γ; ∅⊢x. :: x:
Γ; Δ⊢P :: Λ
Γ; Δ, x: ⊢x.P :: Λ
Γ; Δ⊢P :: Λ
Γ; Δ⊢x.P :: Λ, x:
Γ; x:⊢x. :: ∅
Γ; Δ⊢P :: Λ, y:A
Γ; Δ' ⊢Q :: Λ', x:B
Γ; Δ, Δ' ⊢νy xy.(P Q) :: Λ, Λ', x:A B
Γ; Δ, y:A, x:B ⊢P :: Λ
Γ; Δ, x:A B ⊢xy.P :: Λ
Γ; Δ⊢P :: Λ, y:A, x:B
Γ; Δ⊢xy.P :: Λ, x:A B
Γ; Δ, y:A ⊢P :: Λ
Γ; Δ', x:B ⊢Q :: Λ'
Γ; Δ, Δ', x:A B ⊢νy xy.(P Q) :: Λ, Λ'
Γ; Δ, y:A ⊢P :: Λ, x:B
Γ; Δ⊢xy.P :: Λ, x:A B
Γ; Δ⊢P :: Λ, y:A
Γ; Δ', x:B ⊢Q :: Λ'
Γ; Δ, Δ', x:A B ⊢νy
xy.(P Q) :: Λ, Λ'
Γ; Δ⊢P :: Λ, x:A_j
j ∈I
Γ; Δ⊢x ◃j .P :: Λ, x:⊕{i:A_i}_i ∈I
∀i ∈I. Γ; Δ, x:A_i ⊢P_i :: Λ
Γ; Δ, x:⊕{i:A_i}_i ∈I ⊢x ▹{i:P_i}_i ∈I :: Λ
∀i ∈I. Γ; Δ⊢P_i :: Λ, x:A_i
Γ; Δ⊢x ▹{i:P_i}_i ∈I :: Λ, x:&{i:A_i}_i ∈I
Γ; Δ, x:A_j ⊢P :: Λ
j ∈I
Γ; Δ, x:&{i:A_i}_i ∈I ⊢x ◃j .P :: Λ
Γ, u:A; Δ⊢P :: Λ, x:A
Γ, u:A; Δ⊢νx ux.P :: Λ
Γ, u:A; Δ, x:A ⊢P :: Λ
Γ, u:A; Δ⊢νx ux.P :: Λ
Γ; ∅⊢P :: y:A
Γ; ∅⊢xy.P :: x:A
Γ, u:A; Δ⊢P :: Λ
Γ; Δ, x:A ⊢Px/u :: Λ
Γ, u:A; Δ⊢P :: Λ
Γ; Δ⊢Px/u :: Λ, x:A
Γ; y:A ⊢P :: ∅
Γ; x:A ⊢xy.P :: ∅
The $\pull$ type system.
Γ; Δ⊢P :: Λ, x:A
Γ; Δ', x:A ⊢Q :: Λ'
Γ; Δ, Δ' ⊢νx(P Q) :: Λ, Λ'
Γ; Δ, x:A ⊢P :: Λ
Γ; Δ' ⊢Q :: Λ', x:A
Γ; Δ, Δ' ⊢νx(P Q) :: Λ, Λ'
Γ; Δ⊢P :: Λ, x:A
Γ; Δ' ⊢Q :: Λ', x:A
Γ; Δ, Δ' ⊢νx(P Q) :: Λ, Λ'
Γ; Δ, x:A ⊢P :: Λ
Γ; Δ', x:A ⊢Q :: Λ'
Γ; Δ, Δ' ⊢νx(P Q) :: Λ, Λ'
Γ, u:A; Δ⊢P :: Λ
Γ; ∅⊢Q :: x:A
Γ; Δ⊢νu ( P ux.Q ) :: Λ
Γ; ∅⊢P :: x:A
Γ, u:A; Δ⊢Q :: Λ
Γ; Δ⊢νu ( ux.P Q ) :: Λ
Γ, u:A; Δ⊢P :: Λ
Γ; x:A ⊢Q :: ∅
Γ; Δ⊢νu ( P ux.Q ) :: Λ
Γ; x:A ⊢P :: ∅
Γ, u:A; Δ⊢Q :: Λ
Γ; Δ⊢νu ( ux.P Q ) :: Λ
Cut-rules of the type system.
<Ref> give the typing rules of , which are based directly on the linear fragment of LU in [23] (some rules are marked with $\ast$, which we will refer to later).
We comment on the rules in <Ref>.
Axioms idR and idL type forwarding constructs which connect two channels of dual type.
Axioms $\1$R and $\bot$L type processes that close a session with an empty send after which they become inactive.
Rules $\bot$R and $\1$L type processes that close a session with an empty receive.
These four rules define a non-silent interpretation for $\1$ and $\bot$ that entails process communication (cf. <Ref>), which corresponds to cut reductions in proofs.
(An alternative silent interpretation of $\1$ is discussed in <Ref>.)
The typing system elegantly induces processes under the internal mobility discipline, whereby only fresh channels are exchanged in communications [45, 2].
Rules $\tensor$R, $\parr$L, and $\lolli$L type bound sends, where one process provides the sent channel independently from another process which provides the continuation channel.
Rules $\tensor$L, $\parr$R, and $\lolli$R type receive-prefixed processes.
Rules $\oplus$R and $\&$L type selection and rules $\oplus$L and $\&$R type branching.
Our interpretation of $\bang A$ and $\whynot A$ as server and client behaviors follows the interpretation of classical linear logic in [11].
Rules copyR and copyL type clients that connect to a service by sending a fresh channel.
Rules $\bang$R and $\whynot$L allow the typing of unused services and Rules $\bang$L and $\whynot$R allow to add unused services to the unrestricted context.
<Ref> gives a series of so-called cut-rules, that type channel connections.
The number and shape of these rules is a difference with respect to previous presentations.
Rules cutRL, cutLR, cutLL, and cutRR type pairs of processes that have a channel of dual type in common by composing them in parallel and immediately binding their common channel.
The four similar rules provide for all possible sides the cut channel can appear on.
This way, constructs for restriction and parallel composition are jointly treated.
Rules cut$\bang$R, cut$\bang$L, cut$\whynot$R, and cut$\whynot$R type the connection of a service provider with potential clients; a process $Q$ with potential clients has a channel $u$ in its unrestricted context, so the rules create a service from a process $P$ that has a single channel $x$ of type dual to $u$'s type by prefixing it with replicated reception on $u$ (forming $\serv{u}{x}.P$) and then composing this process in parallel with $Q$ and binding $u$.
This abundance of cut-rules is derived from the generality of 's judgments and necessary for proving the correctness results presented in <Ref>.
In <Ref> we shall consider an alternative presentation of , which allows moving channels between the left and side regions of typing judgments using duality; as we will see, in such a presentation we will be able to drastically cut down the number of cut-rules.
Differences with LU
As already mentioned, for the purposes of our formal comparison we consider a linear logic derived from LU [23] restricted to linear connectives.
The following are notable differences between our linear logic and the linear fragment of LU:
* we include a Rule idL which is complementary to idR;
* we include Rules $\1$L and $\bot$R which are lacking in LU;
* we omit rules for $\top$ and 0 (the units of $\&$ and $\oplus$, resp.), which are usually disregarded in session type interpretations of linear logic (an exception is [30], which uses $\top$ and 0 to give a local account of subtyping);
* we omit rules that move propositions between the left and right linear regions using duality (in <Ref> we will return to these rules).
* because the order of assumptions in typing rules makes a practical difference, we include additional symmetric cut-rules.
§.§ Correctness Properties
Session type systems for the $\pi$-calculus derived from the Curry-Howard correspondence enforce strong correctness properties for processes, which follow directly from properties of the logic, in particular from cut elimination.
This is no different for .
Our first result is the safety property of subject congruence and reduction (<Ref>), which says that typability is consistent across structural congruence and reductions.
If $\Gamma ; \Delta \vdash P :: \Lambda$ and $P \equiv Q$, then $\Gamma ; \Delta \vdash Q :: \Lambda$.
By induction on the derivation of the structural congruence.
The only inductive case is the closure under arbitrary process contexts, which follows from the IH directly.
The base cases correspond to a rule in <Ref> (top).
In each case, we infer the typing of $P$ and $Q$ from the shapes of the processes in the rule, and show that these typing inferences have identical assumptions and conclusion.
The cases of Rules cutAssocL and cutAssocR are straightforward as usual.
The analysis of the more interesting case of Rule cutSymm depends on the last-applied cut-rule.
If the left-hand side uses, e.g., Rule cutLR, then the right-hand side should use Rule cutRL.
If $\Gamma; \Delta \vdash P :: \Lambda$ and $P \red Q$, then $\Gamma; \Delta \vdash Q :: \Lambda$.
By induction on the derivation of the reduction.
The cases correspond to the rules in <Ref> (bottom), as well as the closure rules in <Ref>.
In each case, we infer the typing of $P$ and construct on for $Q$ from the shapes of the processes in the rule, and show that these typing inferences have identical assumptions and conclusions.
We detail two cases.
The case of Rule $\beta$serv serves to illustrate the need for multiple symmetric cut-rules.
Suppose, for example, that the last-applied rule is Rule cut$\bang$R, and that the client request is derived using Rule copyL:
\[
\begin{bussproof}
\bussAssume{
\Gamma , x:A ; \Delta , y:A \vdash P :: \Lambda
\bussUn[\ruleLabel{copyL}]{
\Gamma , x:A ; \Delta \vdash \nu{y} \send{x}{y}.P :: \Lambda
\bussAssume{
\Gamma ; \emptyset \vdash Q :: z:A
\bussBin[\ruleLabel{cut$\bang$R}]{
\Gamma ; \Delta \vdash \nu{x} ( \nu{y} \send{x}{y}.P \| \serv{x}{z}.Q ) :: \Lambda
\end{bussproof}
\]
As per Rule $\beta$serv, we need to identically type $\nu{x} ( \nu{y} ( P \| Q\subst{y/z} ) \| \serv{x}{z}.Q )$ using the same assumptions as above.
Had we only had, e.g., Rules cutRL and cutLL, this would not be possible.
However, with the rules in <Ref> it is no problem.
We first have to add $x:A$ into the persistent regions of the proof of the typing of $Q$, and substitute $y$ for $z$, after which we derive the following:
\[
\begin{bussproof}
\bussAssume{
\Gamma , x:A ; \Delta , y:A \vdash P :: \Lambda
\bussAssume{
\Gamma , x:A ; \emptyset \vdash Q\subst{y/z} :: y:A
\bussBin[\ruleLabel{cutLR}]{
\Gamma , x:A ; \Delta \vdash \nu{y} ( P \| Q\subst{y/z} ) :: \Lambda
\bussAssume{
\Gamma ; \emptyset \vdash Q :: z:A
\bussBin[\ruleLabel{cut$\bang$R}]{
\Gamma ; \Delta \vdash \nu{x} ( \nu{y} ( P \| Q\subst{y/z} ) \| \serv{x}{z}.Q ) :: \Lambda
\end{bussproof}
\]
As another representative case, we consider Rule $\beta{\tensor}{\parr}$.
We have $P = \nu{x} ( \nu{y}\send{x}{y}.(R \| S) \| x(z).T) \red \nu{x} ( S \| \nu{y} ( R \| T\subst{y/z} ) ) = Q$.
There are multiple ways to type $P$, depending on the cut-rule applied.
Here, we give the example of Rule cutR.
The proof of $\Gamma; \Delta \vdash P :: \Lambda$ looks as follows:
\[
\begin{bussproof}
\bussAssume{
\Gamma; \Delta_1 \vdash R :: \Lambda_1, y:A
\bussAssume{
\Gamma; \Delta_2 \vdash S :: \Lambda_2, x:B
\bussBin[\ruleLabel{$\tensor$R}]{
\Gamma; \Delta_1, \Delta_2 \vdash \nu{y}\send{x}{y}.(R \| S) :: \Lambda_1, \Lambda_2, x:A \tensor B
\bussAssume{
\Gamma; \Delta_3, z:A, x:B \vdash T :: \Lambda_3
\bussUn[\ruleLabel{$\tensor$L}]{
\Gamma; \Delta_3, x:A \tensor B \vdash x(z).T :: \Lambda_3
\bussBin[\ruleLabel{cutR}]{
\Gamma; \underbrace{\Delta_1, \Delta_2, \Delta_3}_{\Delta} \vdash \underbrace{\nu{x}(\nu{y}\send{x}{y}(R \| S) \| T)}_{P} :: \underbrace{\Lambda_1, \Lambda_2, \Lambda_3}_{\Lambda}
\end{bussproof}
\]
We can then construct a proof of $\Gamma; \Delta \vdash Q :: \Lambda$ using the assumptions in the above proof.
\[
\begin{bussproof}
\bussAssume{
\Gamma; \Delta_2 \vdash S :: \Lambda_2, x:B
\bussAssume{
\Gamma; \Delta_1 \vdash R :: \Lambda_1, y:A
\bussAssume{
\Gamma; \Delta_3, y:A, x:B \vdash T \subst{y/z} :: \Lambda_3
\bussBin[\ruleLabel{cutR}]{
\Gamma; \Delta_1, \Delta_3, x:B \vdash \nu{y}(R \| T \subst{y/z} ) :: \Lambda_1, \Lambda_3
\bussBin[\ruleLabel{cutR}]{
\Gamma; \Delta \vdash \underbrace{\nu{x}( S \| \nu{y}(R \| T \subst{y/z}) )}_{Q} :: \Lambda
\end{bussproof}
\]
Our second result is the liveness property of progress, which says that the specific form of composition and restriction in following from the cut-rule enables communication, and that processes never get stuck waiting for each other:
If $\Gamma; \Delta \vdash P :: \Lambda$ and $P \equiv \nu{x}(Q \| R)$, then there exists $P'$ such that $P \red P'$.
By induction on the size of the proof of $\Gamma; \Delta \vdash P :: \Lambda$.
By <Ref>, $\Gamma ; \Delta \vdash \nu{x} ( Q \| R ) :: \Lambda$.
By assumption, the last inference of the derivation thereof is either a linear cut or an unrestricted cut.
(Case linear cut)
The last-applied rule can be cutR or cutL.
W.l.o.g. assume the former.
By inversion of cutR, we have a proof $\pi_Q$ of $\Gamma; \Delta_Q \vdash Q :: \Lambda_Q, x:A$ and a proof $\pi_R$ of $\Gamma; \Delta_R, x:A \vdash R :: \Lambda_R$ where $\Delta_Q, \Delta_R = \Delta$ and $\Lambda_Q, \Lambda_R = \Lambda$.
If the last-applied rules in $\pi_Q$ and $\pi_R$ are both on $x$, then we apply a $\beta$-reduction depending on $A$.
For example, assume $A = B \tensor C$.
Then the last-applied rules in $\pi_Q$ and $\pi_R$ are $\tensor$R and $\tensor$L, respectively.
Hence, by Rule $\beta{\tensor}{\parr}$, $P \equiv \nu{x}(\nu{y}\send{x}{y}.(Q_y \| Q_x) \| \recv{x}{y}.R') \red \nu{x}(Q_x \| \nu{y}(Q_y \| R'))$.
Otherwise, w.l.o.g. assume the last-applied rule not on $x$ is in $\pi_Q$.
Then, if $Q$ is a cut, by the induction hypothesis, $Q \red Q'$, and hence $P \equiv \nu{x}(Q \| R) \red \nu{x}(Q' \| R)$.
Otherwise, $Q$ is prefixed by an action on some free channel $y$ which is not a free channel of $R$.
Hence, we apply a $\kappa$-conversion depending on the type of the channel the last-applied rule in $\pi_Q$ works on.
For example, if this rule introduces $y:B \tensor C$ on the right, then, by Rule $\kappa{\tensor}$, $P \equiv \nu{x}(\nu{z}\send{y}{z}.(Q_z \| Q_y) \| R) \red \nu{z}\send{y}{z}.\nu{x}(Q_z \| Q_y \| R)$.
(Case unrestricted cut)
The last-applied rule can be cut$\bang$ or cut$\whynot$.
W.l.o.g. assume the former.
Then $Q \equiv \serv{x}{y}.Q'$, and, by inversion of this rule, we have a proof $\pi_{Q'}$ of $\Gamma; \emptyset \vdash Q' :: y:A$ and a proof $\pi_R$ of $\Gamma, x:A; \Delta \vdash R :: \Lambda$.
If $x \notin \fn(R)$, then, by Rule $\beta$weakenL, $P \equiv \nu{x}(\serv{x}{y}.Q' \| R) \red R$.
Otherwise, the next step depends on the last-applied rule in $\pi_R$.
If the last-applied rule in $\pi_R$ is on $x$, then it must be copyR or copyL.
W.l.o.g. assume the former.
Then $R \equiv \nu{y}\send{x}{y}.R'$, so, by Rule $\beta\bang\whynot$, $P \equiv \nu{x}(\serv{x}{y}.Q' \| \nu{y}\send{x}{y}.R') \red \nu{x}(\serv{x}{y}.Q' \| \nu{y}(Q' \| R'))$.
Otherwise, the proof proceeds as in the last part of the case of linear cut.
An important corollary of subject reduction and progress is that closed processes are deadlock-free.
The corresponding result from linear logic is cut-elimination, but this can be misleading: most reductions actually increase the number of cuts.
Therefore, to prove deadlock-freedom we need another notion than the size of a proof.
Here, we use the cost of a proof (following, e.g., [10, 13]), which is determined by the sum of the costs of its cuts.
The cost of a cut is the sum of the cut propositions/types, which depends on the amount of connectives it has.
For example, the cost associated to type $\1 \tensor \bot \parr \1$ is five.
This way, the cost of a proof decreases upon reduction, because the new cuts are on propositions/types that cost less than before.
Let us write $\red_\beta^\ast$ to denote the reflexive, transitive closure of $\red_\beta$. We have:
If $\emptyset; \emptyset \vdash P :: z:\1$ or $\emptyset; z:\bot \vdash P :: \emptyset$, then $P \red_\beta^\ast \send{z}{}.\0$.
By induction on the number of client requests (1) and the cut cost of the derivation of the typing of $P$ (2).
In the ultimate base case, there are no client requests, and the cut cost is zero, where well-typedness gives us that $P = \send{z}{}.\0$.
Otherwise, the derivation of the typing of $P$ must end with a cut-rule.
By <Ref> (progress), there is $Q$ s.t. $P \red Q$.
Hence, by <Ref> (subject reduction), we have $\emptyset; \emptyset \vdash Q :: z:\1$ (resp. $\emptyset; z:\bot \vdash Q :: \emptyset$).
By its type, the free channel $z$ cannot guard any actions on bound channels, and there are no other free channels.
Therefore, following the proof of <Ref>, the reduction $P \red Q$ can only be the result of a $\beta$-reduction, i.e. $P \red_\beta Q$.
The analysis depends on whether the reduction involves a client/server or not.
If the reduction involves a client/server, the number of clients reduces, so the thesis follows from 1.
Otherwise, the reduction replaces a cut with cuts on smaller types, so the cut cost of the typing of $Q$ is less than that of $P$, and the thesis follows from 2.
§.§ On Duality
It may seem that there is an extensive redundancy in the system of rules in <Ref>, caused by the two-sidedness of 's judgments:
every connective can be inferred on either side of judgments.
For example, Rules $\tensor$R, $\parr$L, and $\lolli$L all type the send of a channel;
which rule to use depends on the side of the judgment the involved channels are on.
However, there is no actual redundancy, for if we were to omit rules for, e.g., $\parr$ and $\lolli$, it would be impossible to type a send on a previously received channel.
This abundance of typing rules in can be explained by its full support for duality:
for every rule inferring a connective on one side of a judgment, there is rule for inferring the connective's dual on the other side of a judgment.
To make this duality explicit, we define an alternative type system by restricting 's rules to a specific fragment and adding LU's rules for moving propositions between sides of judgments:
The type system , with judgments $\Gamma; \Delta \vpull P :: \Lambda$, is defined on the process calculus as defined in <Ref>.
Its rules are the $\ast$-marked rules in <Ref> plus the following rules:
Γ; ΔP :: Λ, x:A
Γ; Δ, x:A P :: Λ
Γ; Δ, x:A P :: Λ
Γ; ΔP :: Λ, x:A
Fortunately, in the presence of these two rules, a number of other rules become truly redundant:
all rules in <Ref> not marked with $\ast$ are admissible or derivable in .
Dually, Rules $\mleft$ and $\mright$ are admissible in vanilla .
The following theorem formalizes these facts:
* The rules in <Ref> not marked with $\ast$ are admissible or derivable in , and
* Rules $\mleft$ and $\mright$, as given in <Ref>, are admissible in .
(Item 1)
Suppose given a proof of $\Gamma; \Delta \vpull P :: \Lambda$.
By applying induction on the structure of this proof we show that any applications of non-$\ast$-marked rules can be replaced with applications of $\ast$-marked rules in combination with uses of Rules $\mleft$ or $\mright$.
We discuss every possible last-applied rule, omitting cases of $\ast$-marked rules as they follow directly from the induction hypothesis.
\begin{align*}
& \bullet \ruleLabel{idL}
&& 1\quad
&& \Gamma; x:A, y:\dual{A} \vpull \fwd{x}{y} :: \emptyset
\tag{assumption} \\
& && 2\quad
&& \Gamma; x:A \vpull \fwd{x}{y} :: y:A
\tag{\ruleLabel{idR}} \\
& && 3\quad
&& \Gamma; x:A, y:\dual{A} \vpull \fwd{x}{y} :: \emptyset
\tag{\ruleLabel{$\mleft$} on 2}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\bot$R}
&& 1\quad
&& \Gamma; \Delta \vpull \recv{x}{}.P :: \Lambda, x:\bot
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta \vpull P :: \Lambda
\tag{inversion on 1} \\
& && 3\quad
&& \Gamma; \Delta \vpull P :: \Lambda ~\text{with only $\ast$ rules}
\tag{IH on 2} \\
& && 4\quad
&& \Gamma; \Delta, x:\1 \vpull P :: \Lambda
\tag{\ruleLabel{$\1$L} on 3} \\
& && 5\quad
&& \Gamma; \Delta \vpull P :: \Lambda, x:\bot
\tag{\ruleLabel{$\mright$} on 4}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\bot$L}
&& 1\quad
&& \Gamma; x:\bot \vpull \send{x}{}.\0 :: \emptyset
\tag{assumption} \\
& && 2\quad
&& \Gamma; \emptyset \vpull \send{x}{}.\0 :: x:\1
\tag{\ruleLabel{$\1$R}} \\
& && 3\quad
&& \Gamma; x:\bot \vpull \send{x}{}.\0 :: \emptyset
\tag{\ruleLabel{$\mleft$} on 2}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\parr$R}
&& 1\quad
&& \Gamma; \Delta \vpull \recv{x}{y}.P :: \Lambda, x:A \parr B
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta \vpull P :: \Lambda, y:A, x:B
\tag{inversion on 1} \\
& && 3\quad
&& \Gamma; \Delta \vpull P :: \Lambda, y:A, x:B ~\text{with only $\ast$ rules}
\tag{IH on 2} \\
& && 4\quad
&& \Gamma; \Delta, y:\dual{A}, x:\dual{B} \vpull P :: \Lambda
\tag{\ruleLabel{$\mleft$} twice on 3} \\
& && 5\quad
&& \Gamma; \Delta, x:\dual{A} \tensor \dual{B} \vpull \recv{x}{y}.P :: \Lambda
\tag{\ruleLabel{$\tensor$L} on 4} \\
& && 6\quad
&& \Gamma; \Delta \vpull \recv{x}{y}.P :: \Lambda, x:A \parr B
\tag{\ruleLabel{$\mright$} on 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\parr$L}
&& 1\quad
&& \Gamma; \Delta, \Delta', x:A \parr B \vpull \nu{y}\send{x}{y}.(P \| Q) :: \Lambda, \Lambda'
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta, y:A \vpull P :: \Lambda \\
& && 3\quad
&& \Gamma; \Delta', x:B \vpull Q :: \Lambda'
\tag{inversion on 1} \\
& && 4\quad
&& \Gamma; \Delta, y:A \vpull P :: \Lambda ~\text{with only $\ast$ rules}
\tag{IH on 2} \\
& && 5\quad
&& \Gamma; \Delta', x:B \vpull Q :: \Lambda' ~\text{with only $\ast$ rules}
\tag{IH on 3} \\
& && 6\quad
&& \Gamma; \Delta \vpull P :: \Lambda, y:\dual{A}
\tag{\ruleLabel{$\mright$} on 4} \\
& && 7\quad
&& \Gamma; \Delta' \vpull Q :: \Lambda', x:\dual{B}
\tag{\ruleLabel{$\mright$} on 5} \\
& && 8\quad
&& \Gamma; \Delta, \Delta' \vpull \nu{y}\send{x}{y}.(P \| Q) :: \Lambda, \Lambda', x:\dual{A} \tensor \dual{B}
\tag{\ruleLabel{$\tensor$R} on 6 and 7} \\
& && 9\quad
&& \Gamma; \Delta, \Delta', x:A \parr B \vpull \nu{y}\send{x}{y}.(P \| Q) :: \Lambda, \Lambda'
\tag{\ruleLabel{$\mleft$} on 8}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{copyR}
&& 1\quad
&& \Gamma, u:A; \Delta \vpull \nu{x}\send{u}{x}.P :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma, u:A; \Delta \vpull P :: \Lambda, x:\dual{A}
\tag{inversion on 1} \\
& && 3\quad
&& \Gamma, u:A; \Delta \vpull P :: \Lambda, x:\dual{A} ~\text{with only $\ast$ rules}
\tag{IH on 2} \\
& && 4\quad
&& \Gamma, u:A; \Delta, x:A \vpull P :: \Lambda
\tag{\ruleLabel{$\mleft$} on 3} \\
& && 5\quad
&& \Gamma, u:A; \Delta \vpull \nu{x}\send{u}{x}.P :: \Lambda
\tag{\ruleLabel{copyL} on 4}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\whynot$R}
&& 1\quad
&& \Gamma; \Delta \vpull P\subst{x/u} :: \Lambda, x:\whynot \dual{A}
\tag{assumption} \\
& && 2\quad
&& \Gamma, u:A; \Delta \vpull P :: \Lambda
\tag{inversion on 1} \\
& && 3\quad
&& \Gamma, u:A; \Delta \vpull P :: \Lambda ~\text{with only $\ast$ rules}
\tag{IH on 2} \\
& && 4\quad
&& \Gamma; \Delta, x:\bang A \vpull P\subst{x/u} :: \Lambda
\tag{\ruleLabel{$\bang$L} on 3} \\
& && 5\quad
&& \Gamma; \Delta \vpull P\subst{x/u} :: \Lambda, x:\whynot \dual{A}
\tag{\ruleLabel{$\mright$} on 4}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\whynot$L}
&& 1\quad
&& \Gamma; x:\whynot A \vpull \serv{x}{y}.P :: \emptyset
\tag{assumption} \\
& && 2\quad
&& \Gamma; y:A \vpull P :: \emptyset
\tag{inversion on 1} \\
& && 3\quad
&& \Gamma; y:A \vpull P :: \emptyset ~\text{with only $\ast$ rules}
\tag{IH on 2} \\
& && 4\quad
&& \Gamma; \emptyset \vpull P :: y:\dual{A}
\tag{\ruleLabel{$\mright$} on 3} \\
& && 5 \quad
&& \Gamma; \emptyset \vpull \serv{x}{y}.P :: x:\bang \dual{A}
\tag{\ruleLabel{$\bang$R} on 4} \\
& && 6 \quad
&& \Gamma; x:\whynot A \vpull \serv{x}{y}.P :: \emptyset
\tag{\ruleLabel{$\mleft$} on 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cutRR}
&& 1\quad
&& \Gamma ; \Delta , \Delta' \vpull \nu{x} ( P \| Q ) :: \Lambda , \Lambda'
\tag{assumption} \\
& && 2\quad
&& \Gamma ; \Delta \vpull P :: \Lambda , x:A \\
& && 3\quad
&& \Gamma ; \Delta' \vpull Q :: \Lambda' , x:\dual{A}
\tag{inversion on 1} \\
& && 4\quad
&& \Gamma ; \Delta \vpull P :: \Lambda , x:A ~\text{with only $\ast$ rules}
\tag{IH on 2} \\
& && 5\quad
&& \Gamma ; \Delta' \vpull Q :: \Lambda' , x:\dual{A} ~\text{with only $\ast$ rules}
\tag{IH on 3} \\
& && 6\quad
&& \Gamma ; \Delta' , x:A \vpull Q :: \Lambda'
\tag{\ruleLabel{$\mleft$} on 5} \\
& && 7\quad
&& \Gamma; \Delta, \Delta' \vpull \nu{x} ( P \| Q ) :: \Lambda, \Lambda'
\tag{\ruleLabel{cutRL} on 4 and 6}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cutLL}
&& 1\quad
&& \Gamma; \Delta, \Delta' \vpull \nu{x}(P \| Q) :: \Lambda, \Lambda'
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta, x:A \vpull P :: \Lambda \\
& && 3\quad
&& \Gamma; \Delta', x:\dual{A} \vpull Q :: \Lambda'
\tag{inversion on 1} \\
& && 4\quad
&& \Gamma; \Delta, x:A \vpull P :: \Lambda ~\text{with only $\ast$ rules}
\tag{IH on 2} \\
& && 5\quad
&& \Gamma; \Delta', x:\dual{A} \vpull Q :: \Lambda' ~\text{with only $\ast$ rules}
\tag{IH on 3} \\
& && 6\quad
&& \Gamma; \Delta \vpull P :: \Lambda, x:\dual{A}
\tag{\ruleLabel{$\mright$} on 4} \\
& && 7\quad
&& \Gamma; \Delta, \Delta' \vpull \nu{x}(P \| Q) :: \Lambda, \Lambda'
\tag{\ruleLabel{cutRL} on 6 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cut$\whynot$R}
&& 1\quad
&& \Gamma; \Delta \vpull \nu{u} ( P \| \serv{u}{x}.Q ) :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma, u:A; \Delta \vpull P :: \Lambda \\
& && 3\quad
&& \Gamma; x:\dual{A} \vpull Q :: \emptyset
\tag{inversion on 1} \\
& && 4\quad
&& \Gamma, u:A; \Delta \vpull P :: \Lambda ~\text{with only $\ast$ rules}
\tag{IH on 2} \\
& && 5\quad
&& \Gamma; x:\dual{A} \vpull Q :: \emptyset ~\text{with only $\ast$ rules}
\tag{IH on 3} \\
& && 6\quad
&& \Gamma; \emptyset \vpull Q :: x:A
\tag{\ruleLabel{$\mright$} on 5} \\
& && 7\quad
&& \Gamma; \Delta \vpull \nu{u} ( P \| \serv{u}{x}.Q ) :: \Lambda
\tag{\ruleLabel{cut$\bang$R} on 4 and 6}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cut$\whynot$L}
&& 1\quad
&& \Gamma; \Delta \vpull \nu{u}(\serv{u}{x}.P \| Q) :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma; x:\dual{A} \vpull P :: \emptyset \\
& && 3\quad
&& \Gamma, u:A; \Delta \vpull Q :: \Lambda
\tag{inversion on 1} \\
& && 4\quad
&& \Gamma; x:\dual{A} \vpull P :: \emptyset ~\text{with only $\ast$ rules}
\tag{IH on 2} \\
& && 5\quad
&& \Gamma, u:A; \Delta \vpull Q :: \Lambda ~\text{with only $\ast$ rules}
\tag{IH on 3} \\
& && 6\quad
&& \Gamma; \emptyset \vpull P :: x:A
\tag{\ruleLabel{$\mright$} on 4} \\
& && 7\quad
&& \Gamma; \Delta \vpull \nu{u}(\serv{u}{x}.P \| Q) :: \Lambda
\tag{\ruleLabel{cut$\bang$L} on 6 and 5}
\end{align*}
(Item 2)
Suppose given a proof of $\Gamma; \Delta \vdash P :: \Lambda$, possibly with applications of Rules $\mleft$ and $\mright$.
By applying induction on the structure of this proof we show that it can be transformed to not contain any applications of $\mleft$ and $\mright$.
We discuss every possible last-applied rule.
However, all cases except $\mleft$ and $\mright$ follow directly from the induction hypothesis.
Therefore, we only detail the case of $\mleft$—the case of $\mright$ is analogous.
Since $\mleft$ is the last-applied rule, we know the assumption is of the form $\Gamma; \Delta, x:\dual{A} \vdash P :: \Lambda$.
By inversion, we have $\Gamma; \Delta \vdash P :: \Lambda, x:A$.
The idea is to move the application of $\mleft$ up the proof tree, applied to a subtype of $A$.
We apply the induction hypothesis to find a proof of $\Gamma; \Delta \vdash P :: \Lambda, x:A$ without applications of $\mleft$ and $\mright$.
Typing rules leave all channels/types untouched except the ones they work on.
Therefore, we can traverse up the proof tree—remembering which steps were taken—until we encounter the rule that introduces $x:A$.
Note that these steps do not include applications of $\mleft$ and $\mright$.
The consequence of this rule looks like $\Gamma'; \Delta' \vdash P' :: \Lambda', x:A$, for some $\Gamma'$, $\Delta'$, $P'$ and $\Lambda'$.
Now, we apply induction on the size of $A$ (with induction hypothesis denoted 2) to prove $\Gamma'; \Delta', x:\dual{A} \vdash P' :: \Lambda'$.
We discuss every possible last-applied rule that introduces $x:A$:
\begin{align*}
& \bullet \ruleLabel{idR}
&& 1\quad
&& \Gamma'; y:A \vdash \fwd{y}{x} :: x:A
\tag{assumption} \\
& && 2\quad
&& \Gamma'; y:A, x:\dual{A} \vdash \fwd{y}{x} :: \emptyset
\tag{\ruleLabel{idL}}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\1$R}
&& 1\quad
&& \Gamma'; \emptyset \vdash \send{x}{}.\0 :: x:\1
\tag{assumption} \\
& && 2\quad
&& \Gamma'; x:\bot \vdash \send{x}{}.\0 :: \emptyset
\tag{\ruleLabel{$\bot$L}}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\bot$R}
&& 1\quad
&& \Gamma'; \Delta' \vdash \recv{x}{}.P' :: \Lambda', x:\bot ~\text{without \ruleLabel{$\mleft$}/\ruleLabel{$\mright$}}
\tag{assumption} \\
& && 2\quad
&& \Gamma'; \Delta' \vdash P' :: \Lambda'
\tag{inversion on 1} \\
& && 3\quad
&& \Gamma'; \Delta', x:\1 \vdash \recv{x}{}.P' :: \Lambda'
\tag{\ruleLabel{$\1$L} on 2}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\tensor$R}
&& 1\quad
&& \Gamma'; \Delta', \Delta'' \vdash \nu{y}\send{x}{y}.(P' \| Q') :: \Lambda', \Lambda'', x:B \tensor C \\
& &&
&& ~\text{without \ruleLabel{$\mleft$}/\ruleLabel{$\mright$}}
\tag{assumption} \\
& && 2\quad
&& \Gamma'; \Delta' \vdash P' :: \Lambda', y:B \\
& && 3\quad
&& \Gamma'; \Delta'' \vdash Q' :: \Lambda'', x:C
\tag{inversion on 1} \\
& && 4\quad
&& \Gamma'; \Delta', y:\dual{B} \vdash P' :: \Lambda'
\tag{\ruleLabel{$\mleft$} on 2} \\
& && 5\quad
&& \Gamma'; \Delta'', x:\dual{C} \vdash Q' :: \Lambda''
\tag{\ruleLabel{$\mleft$} on 3} \\
& && 6\quad
&& \Gamma'; \Delta', y:\dual{B} \vdash P' :: \Lambda' ~\text{without \ruleLabel{$\mleft$}/\ruleLabel{$\mright$}}
\tag{\ih2 on 4} \\
& && 7\quad
&& \Gamma'; \Delta'', x:\dual{C} \vdash Q' :: \Lambda'' ~\text{without \ruleLabel{$\mleft$}/\ruleLabel{$\mright$}}
\tag{\ih2 on 5} \\
& && 8\quad
&& \Gamma'; \Delta', \Delta'', x:\dual{B} \parr \dual{C} \vdash \nu{y}\send{x}{y}.(P' \| Q') :: \Lambda', \Lambda''
\tag{\ruleLabel{$\parr$L} on 8 and 9}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\lolli$R}
&& 1\quad
&& \Gamma'; \Delta' \vdash \recv{x}{y}.P' :: \Lambda', x:B \lolli C ~\text{without \ruleLabel{$\mleft$}/\ruleLabel{$\mright$}}
\tag{assumption} \\
& && 2\quad
&& \Gamma'; \Delta', y:B \vdash P' :: \Lambda', x:C
\tag{inversion on 1} \\
& && 3\quad
&& \Gamma'; \Delta', y:B, x:\dual{C} \vdash P' :: \Lambda'
\tag{\ruleLabel{$\mleft$} on 2} \\
& && 4\quad
&& \Gamma'; \Delta', y:B, x:\dual{C} \vdash P' :: \Lambda' ~\text{without \ruleLabel{$\mleft$}/\ruleLabel{$\mright$}}
\tag{\ih2 on 3} \\
& && 5\quad
&& \Gamma'; \Delta', x:B \tensor \dual{C} \vdash \recv{x}{y}.P' :; \Lambda'
\tag{\ruleLabel{$\tensor$L} on 4}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\oplus$R}
&& 1\quad
&& \Gamma'; \Delta' \vdash x \triangleleft j.P' :: \Lambda', x:\oplus\{i:A_i\}_{i \in I} ~\text{without \ruleLabel{$\mleft$}/\ruleLabel{$\mright$}}
\tag{assumption} \\
& && 2\quad
&& \Gamma'; \Delta' \vdash P' :: \Lambda', x:A_j \\
& && 3\quad
&& j \in I
\tag{inversion on 1} \\
& && 4\quad
&& \Gamma'; \Delta', x:\dual{A_j} \vdash P' :: \Lambda'
\tag{\ruleLabel{$\mleft$} on 3} \\
& && 5\quad
&& \Gamma'; \Delta', x:\dual{A_j} \vdash P' :: \Lambda'
\tag{\ih2 on 4} \\
& && 6\quad
&& \Gamma'; \Delta', x:\&\{i:\dual{A_i}\}_{i \in I} \vdash x \triangleleft j.P' :: \Lambda'
\tag{\ruleLabel{$\&$L} on 5 and 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\&$R}
&& 1\quad
&& \Gamma'; \Delta' \vdash x \triangleright \{i:P'_i\}_{i \in I} :: \Lambda', x:\&\{i:A_i\}_{i \in I} ~\text{without \ruleLabel{$\mleft$}/\ruleLabel{$\mright$}}
\tag{assumption} \\
& && 2\quad
&& \forall i \in I.~ \Gamma'; \Delta' \vdash P'_i :: \Lambda', x:A_i
\tag{inversion on 1} \\
& && 3\quad
&& \forall i \in I.~ \Gamma'; \Delta', x:\dual{A_i} \vdash P'_i :: \Lambda'
\tag{\ruleLabel{$\mleft$} on 2} \\
& && 4\quad
&& \forall i \in I.~ \Gamma'; \Delta', x:\dual{A_i} \vdash P'_i :: \Lambda' ~\text{without \ruleLabel{$\mleft$}/\ruleLabel{$\mright$}}
\tag{\ih2 on 3} \\
& && 5\quad
&& \Gamma'; \Delta', x:\oplus\{i:\dual{A_i}\}_{i \in I} \vdash x \triangleright \{i:P'_i\}_{i \in I} :: \Lambda'
\tag{\ruleLabel{$\oplus$L} on 4}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\bang$R}
&& 1\quad
&& \Gamma'; \emptyset \vdash \serv{x}{y}.P' :: x:\bang B ~\text{without \ruleLabel{$\mleft$}/\ruleLabel{$\mright$}}
\tag{assumption} \\
& && 2\quad
&& \Gamma'; \emptyset \vdash P' :: y:B
\tag{inversion on 1} \\
& && 3\quad
&& \Gamma'; y:\dual{B} \vdash P' :: \emptyset
\tag{\ruleLabel{$\mleft$} on 2} \\
& && 4\quad
&& \Gamma'; y:\dual{B} \vdash P' :: \emptyset ~\text{without \ruleLabel{$\mleft$}/\ruleLabel{$\mright$}}
\tag{\ih2 on 3} \\
& && 5\quad
&& \Gamma'; x:\whynot \dual{B} \vdash \serv{x}{y}.P' :: \emptyset
\tag{\ruleLabel{$\whynot$L} on 4}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\whynot$R}
&& 1\quad
&& \Gamma'; \Delta' \vdash P'\subst{x/u} :: \Lambda', x:\whynot B
\tag{assumption} \\
& && 2\quad
&& \Gamma', u:\dual{A}; \Delta' \vdash P' :: \Lambda'
\tag{inversion on 1} \\
& && 3\quad
&& \Gamma'; \Delta', x:\bang \dual{B} \vdash P'\subst{x/u} :: \Lambda'
\tag{\ruleLabel{$\bang$L} on 2}
\end{align*}
Finally, we recall the steps we have traversed up the tree and remember that they do not include applications of $\mleft$ and $\mright$.
We re-apply them on $\Gamma'; \Delta', x:\dual{A} \vdash P' :: \Lambda'$,
without affecting $x:\dual{A}$.
Instead, they only affect $\Gamma'$, $\Delta'$ and $\Lambda'$ to give us a proof of $\Gamma; \Delta, x:\dual{A} \vdash P :: \Lambda$ without applications of $\mleft$ and $\mright$.
This concludes the proof of <Ref>.
§ COMPARING INTUITIONISTIC AND CLASSICAL INTERPRETATIONS
In this section, we rigorously compare the class of -typable processes to the classes of processes typable in session type interpretations of linear logic, in classical [11, 52] and intuitionistic [6] settings.
is an independent yardstick for this comparison, because it is derived from which subsumes both linear logics by design.
We have discussed in <Ref> several design choices we made when defining and .
Besides differences stemming from the dichotomy between classical and intuitionistic linear logic, logically-motivated session type systems also present features induced by certain design choices.
For a fair comparison, we want to make sure that the differences come only from typing.
This means that we need to make the same design choices for both interpretations: we require explicit closing, a separate unrestricted context, and identity as forwarding.
§.§ Session Type Systems Derived from Intuitionistic and Classical Linear Logic
Γ; x:A xy :: y:A
Γ; ∅x. :: x:
Γ; ΔP :: z:C
Γ; Δ, x: x.P :: z:Z
Γ; ΔP :: y:A
Γ; Δ' Q :: x:B
Γ; Δ, Δ' νyxy.(P Q) :: x:A B
Γ; Δ, y:A, x:B P :: z:C
Γ; Δ, x:A B xy.P :: z:C
Γ; Δ, y:A P :: x:B
Γ; Δxy.P :: x:A B
Γ; ΔP :: y:A
Γ; Δ', x:B Q :: z:C
Γ; Δ, Δ', x:A B νyxy.(P Q) :: z:C
Γ; ΔP :: x:A_j
j ∈I
Γ; Δx ◃j .P :: x:⊕{i:A_i}_i ∈I
∀i ∈I. Γ; Δ, x:A_i P_i :: z:C
Γ; Δ, x:⊕{i:A_i}_i ∈I x ▹{i:P_i}_i ∈I :: z:C
∀i ∈I. Γ; ΔP_i :: x:A_i
Γ; Δx ▹{i:P_i}_i ∈I :: x:&{i:A_i}_i ∈I
Γ; Δ, x:A_j P :: z:C
j ∈I
Γ; Δ, x:&{i:A_i}_i ∈I x ◃j .P :: z:C
Γ, u:A; Δ, x:A P :: z:C
Γ, u:A; Δνxux.P :: z:C
Γ; ∅P :: y:A
Γ; ∅xy.P :: x:A
Γ, u:A; ΔP :: z:C
Γ; Δ, x:A Px/u :: z:C
Γ; ΔP :: x:A
Γ; Δ', x:A Q :: z:C
Γ; Δ, Δ' νx(P Q) :: z:C
Γ; Δ, x:A P :: z:C
Γ; Δ' Q :: x:A
Γ; Δ, Δ' νx(P Q) :: z:C
Γ, u:A; ΔP :: z:C
Γ; ∅Q :: x:A
Γ; Δνu ( P ux.Q ) :: z:C
Γ; ∅P :: x:A
Γ, u:A; ΔQ :: z:C
Γ; Δνu(ux.P Q) :: z:C
The type system.
xy Γ; x:A, y:A
P Γ; Δ
x.P Γ; Δ, x:
x. Γ; x:
P Γ; Δ, y:A
Q Γ; Δ', x:B
νyxy.(P Q) Γ; Δ, Δ', x:A B
P Γ; Δ, y:A, x:B
xy.P Γ; Δ, x:A B
P Γ; Δ, x:A_j
j ∈I
x ◃j.P Γ; Δ, x:⊕{i:A_i}_i ∈I
∀i ∈I. P_i Γ; Δ, x:A_i
x ▹{i:P_i}_i ∈I Γ; Δ, x:&{i:A_i}_i ∈I
P Γ, u:A; Δ, y:A
νyuy.P Γ, u:A; Δ
P Γ, u:A; Δ
Px/u Γ; Δ, x:A
P Γ; y:A
xy.P Γ; x:A
P Γ; Δ, x:A
Q Γ; Δ', x:A
νx(P Q) Γ; Δ, Δ'
P Γ, u:A; Δ
Q Γ; x:A
νu ( P ux.Q ) Γ; Δ
P Γ; x:A
Q Γ, u:A; Δ
νu(ux.P Q) Γ; Δ
The type system.
We want to derive session type systems from intuitionistic and classical linear logic for the same process syntax as uses (cf. <Ref>).
<Ref> gives the inference rules for the type system derived from intuitionistic linear logic, denoted .
This system is based on the presentation by Caires, Pfenning, and Toninho in [10]; the judgment is denoted as follows:
\Gamma; \Delta \vdill P :: z:C
With respect to the intuitionistic interpretation introduced in [6, 11], features a non-silent interpretation of $\1$ and $\bot$, based on explicit closure of sessions (cf. Rules $\1$R and $\1$L).
We adopt this interpretation because, as explained in [10], it leads to a Curry-Howard correspondence that is tighter than correspondences with silent interpretations (such as those in [6, 11]).
A more superficial difference is that follows standard presentations of session type systems by supporting $n$-ary labeled choices; in contrast, the systems in [6, 11] support binary labeled choices.
We also include symmetric variants of the cut-rules, as they are necessary for type preservation under Rule CutSymm of structural congruence.
<Ref> gives the inference rules for the type system derived from classical linear logic.
It is based on a combination of features from Caires, Pfenning, and Toninho's in [11] and Wadler's in [52];
in the following, it is denoted .
The corresponding judgment is as follows:
P \vdcll \Gamma; \Delta
<Ref> summarizes the differences in the design choices between , , and ; these differences are merely superficial:
* As we have seen, an explicit closing of sessions (as in ) concerns a non-silent interpretation of the atomic propositions $\1$ and $\bot$.
In contrast, realizes an implicit (silent) closing of sessions.
* Sequents with a separate unrestricted context (as in ) are of the form $P \vdash \Gamma; \Delta$, which can also be written as $P \vdash \Delta, \Gamma'$ where $\Gamma'$ contains only types of the form $!A$.
* The identity axiom can be interpreted as the forwarding process, which enables to account for behavioral polymorphism (i.e., universal and existential quantification over propositions/session types) [52, 8].
As already mentioned, forwarding is not a typical process construct in session $\pi$-calculi.
Note that, since typing judgments are one-sided there is no need for symmetric cut-rules.
§.§ Formal Comparison
Explicit closing
Separate context
Identity as
[11] No Yes No
[52] Yes No Yes
(this paper) Yes Yes Yes
Feature comparison of three session type interpretations of classical linear logic.
Now that we have presented all three systems, we start our comparison by contrasting the shape of their typing judgments:
\[
\Gamma; \Delta \vdash P :: \Lambda
\]
\[
\Gamma; \Delta \vdill P :: x:A
\]
\[
P \vdcll \Gamma; \Delta
\]
In and judgments are similar, but in they have exactly one channel/type pair on the right.
We will see that the difference between and can be characterized by this fact alone.
Judgments in are different from those in and :
they have only one linear context and both the linear and the unrestricted contexts are on the right.
As we will see, our results reflect this with a duality relation between the contexts of and .
Our formal results rely on classes of processes typable in the three typing systems:
Let $\mathbb{P}$ denote the set of all processes induced by <Ref>.
\begin{align*}
\U &= \{P \in \mathbb{P} \mid \exists \Gamma, \Delta, \Lambda ~\text{such that}~ \Gamma; \Delta \vdash P :: \Lambda\}, \\
\C &= \{P \in \mathbb{P} \mid \exists \Gamma, \Delta ~\text{such that}~ P \vdcll \Gamma; \Delta\}, \\
\I &= \{P \in \mathbb{P} \mid \exists \Gamma, \Delta, x, A ~\text{such that}~ \Gamma; \Delta \vdill P :: x:A\}.
\end{align*}
Our first result is that $\U = \C$, i.e.,
is merely a two-sided representation of .
$\U = \C$.
We briefly discuss how we prove this result.
On the one hand (from left to right), if $P \in \U$, there is a proof of $\Gamma; \Delta \vdash P :: \Lambda$.
By backtracking on this proof, we can use rules in analogous to the -rules used to generate an equivalent proof of $P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda$ thus showing $P \in \C$.
Note how the single sidedness of judgments requires us to move $\Gamma$ and $\Delta$ to the right-hand side using duality.
On the other hand (from right to left), if $P \in \C$, there is a proof of $P \vdcll \Gamma; \Delta$.
Again, by backtracking on this proof, we can use a combination of rules similar to those used in in combination with Rules $\mleft$ and $\mright$ from <Ref> to prove $\dual{(\Gamma)}; \emptyset \vpull \Delta$ in .
Going through simplifies this process, since using $\mleft$ and $\mright$ enables us to guarantee that all of $\Delta$ ends up on the right-hand side of the typing judgment instead of dividing unpredictably between left and right.
Note that we do have to use duality to move $\Gamma$ to the left.
Since $\mleft$ and $\mright$ are admissible in (cf. <Ref>) and all the other rules in are also present in , this means we also have a proof of $\dual{(\Gamma)}; \emptyset \vdash \Delta$ in .
Hence, $P \in \U$.
($\,\U \subseteq \C$)
Take any $P \in \U$.
Then, by <Ref>, there are $\Gamma$, $\Delta$, $\Lambda$ s.t. $\Gamma; \Delta \vdash P :: \Lambda$.
By showing that this implies $P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda$, we have $P \in \C$.
We show this by induction on the structure of the proof of $\Gamma; \Delta \vdash P :: \Lambda$.
\begin{align*}
& \bullet \ruleLabel{$\id$R}
&& 1\quad
&& \Gamma; x:A \vdash \fwd{x}{y} :: y:A
\tag{assumption} \\
& && 2\quad
&& \fwd{x}{y} \vdcll \dual{(\Gamma)}; x:\dual{A}, y:A
\tag{\ruleLabel{id}}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{idL}
&& 1\quad
&& \Gamma; x:A, y:\dual{A} \vdash \fwd{x}{y} :: \emptyset
\tag{assumption} \\
& && 2\quad
&& \fwd{x}{y} \vdcll \dual{(\Gamma)}; x:\dual{A}, y:A
\tag{\ruleLabel{id}}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\1$R}
&& 1\quad
&& \Gamma; \emptyset \vdash \send{x}{}.\0 :: x:\1
\tag{assumption} \\
& && 2\quad
&& \send{x}{}.\0 \vdcll \dual{(\Gamma)}; x:\1
\tag{\ruleLabel{$\1$}}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\1$L}
&& 1\quad
&& \Gamma; \Delta, x:\1 \vdash \recv{x}{}.P :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta \vdash P :: \Lambda
\tag{inversion on 1} \\
& && 3\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda
\tag{IH on 2} \\
& && 4\quad
&& \recv{x}{}.P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\bot
\tag{\ruleLabel{$\bot$} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\bot$R}
&& 1\quad
&& \Gamma; \Delta \vdash \recv{x}{}.P :: \Lambda, x:\bot
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta \vdash P :: \Lambda
\tag{inversion on 1} \\
& && 3\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda
\tag{IH on 2} \\
& && 4\quad
&& \recv{x}{}.P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\bot
\tag{\ruleLabel{$\bot$} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\bot$L}
&& 1\quad
&& \Gamma; x:\bot \vdash \send{x}{}.\0 :: \emptyset
\tag{assumption} \\
& && 2\quad
&& \send{x}{}.\0 \vdcll \dual{(\Gamma)}; x:\1
\tag{\ruleLabel{$\1$}}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\tensor$R}
&& 1\quad
&& \Gamma; \Delta, \Delta' \vdash \nu{y}\send{x}{y}.(P \| Q) :: \Lambda, \Lambda', x:A \tensor B
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta \vdash P :: \Lambda, y:A \\
& && 3\quad
&& \Gamma; \Delta' \vdash Q :: \Lambda', x:B
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, y:A
\tag{IH on 2} \\
& && 5\quad
&& Q \vdcll \dual{(\Gamma)}; \dual{(\Delta')}, \Lambda', x:B
\tag{IH on 3} \\
& && 6\quad
&& \nu{y}\send{x}{y}.(P \| Q) \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \dual{(\Delta')}, \Lambda, \Lambda', x:A \tensor B
\tag{\ruleLabel{$\tensor$} on 4 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\tensor$L}
&& 1\quad
&& \Gamma; \Delta, x:A \tensor B \vdash \recv{x}{y}.P :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta, y:A, x:B \vdash P :: \Lambda
\tag{inversion on 1} \\
& && 3\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, y:\dual{A}, x:\dual{B}
\tag{IH on 2} \\
& && 4\quad
&& \recv{x}{y}.P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\dual{A} \parr \dual{B}
\tag{\ruleLabel{$\parr$} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\parr$R}
&& 1\quad
&& \Gamma; \Delta \vdash \recv{x}{y}.P :: \Lambda, x:A \parr B
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta \vdash P :: \Lambda, y:A, x:B
\tag{inversion on 1} \\
& && 3\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, y:A, x:B
\tag{IH on 2} \\
& && 4\quad
&& \recv{x}{y}.P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:A \parr B
\tag{\ruleLabel{$\parr$} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\parr$L}
&& 1\quad
&& \Gamma; \Delta, \Delta', x:A \parr B \vdash \nu{y}\send{x}{y}.(P \| Q) :: \Lambda, \Lambda'
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta, y:A \vdash P :: \Lambda \\
& && 3\quad
&& \Gamma; \Delta', x:B \vdash Q :: \Lambda'
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, y:\dual{A}
\tag{IH on 2} \\
& && 5\quad
&& Q \vdcll \dual{(\Gamma)}; \dual{(\Delta')}, \Lambda', x:\dual{B}
\tag{IH on 3} \\
& && 6\quad
&& \nu{y}\send{x}{y}.(P \| Q) \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \dual{(\Delta')}, \Lambda, \Lambda', x:\dual{A} \tensor \dual{B}
\tag{\ruleLabel{$\tensor$} on 4 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\lolli$R}
&& 1\quad
&& \Gamma; \Delta \vdash \recv{x}{y}.P :: \Lambda, x:A \lolli B
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta, y:A \vdash P :: \Lambda, x:B
\tag{inversion on 1} \\
& && 3\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, y:\dual{A}, x:B
\tag{IH on 2} \\
& && 4\quad
&& \recv{x}{y}.P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\dual{A} \parr B
\tag{\ruleLabel{$\parr$} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\lolli$L}
&& 1\quad
&& \Gamma; \Delta, \Delta', x:A \lolli B \vdash \nu{y}\send{x}{y}.(P \| Q) :: \Lambda, \Lambda'
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta \vdash P :: \Lambda, y:A \\
& && 3\quad
&& \Gamma; \Delta', x:B \vdash Q :: \Lambda'
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, y:A
\tag{IH on 2} \\
& && 5\quad
&& Q \vdcll \dual{(\Gamma)}; \dual{(\Delta')}, \Lambda', x:\dual{B}
\tag{IH on 3} \\
& && 6\quad
&& \nu{y}\send{x}{y}.(P \| Q) \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \dual{(\Delta')}, \Lambda, \Lambda', x:A \tensor \dual{B}
\tag{\ruleLabel{$\tensor$} on 4 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\&$R}
&& 1\quad
&& \Gamma; \Delta \vdash x \triangleright \{i:P_i\}_{i \in I} :: \Lambda, x:\&\{i:A_i\}_{i \in I}
\tag{assumption} \\
& && 2\quad
&& \forall i \in I.~ \Gamma; \Delta \vdash P_i :: \Lambda, x:A_i
\tag{inversion on 1} \\
& && 3\quad
&& \forall i \in I.~ P_i \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:A_i
\tag{IH on 2} \\
& && 4\quad
&& x \triangleright \{i:P_i\}_{i \in I} \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\&\{i:A_i\}_{i \in I}
\tag{\ruleLabel{$\&$} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\&$L}
&& 1\quad
&& \Gamma; \Delta, x:\&\{i:A_i\} \vdash x \triangleleft j.P :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta, x:A_j \vdash P :: \Lambda \\
& && 3\quad
&& j \in I
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\dual{A_j}
\tag{IH on 2} \\
& && 5\quad
&& x \triangleright j.P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\oplus\{i:\dual{A_i}\}_{i \in I}
\tag{\ruleLabel{$\oplus$} on 4 and 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\oplus$R}
&& 1\quad
&& \Gamma; \Delta \vdash x \triangleleft j.P :: \Lambda, x:\oplus\{i:A_i\}_{i \in I}
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta \vdash P :: \Lambda, x:A_j \\
& && 3\quad
&& j \in I
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:A_j
\tag{IH on 2} \\
& && 5\quad
&& x \triangleleft j.P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\oplus\{i:A_i\}_{i \in I}
\tag{\ruleLabel{$\oplus$} on 4 and 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\oplus$L}
&& 1\quad
&& \Gamma; \Delta, x:\oplus\{i:A_i\}_{i \in I} \vdash x \triangleright \{i:P_i\}_{i \in I} :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \forall i \in I.~ \Gamma; \Delta, x:A_i \vdash P_i :: \Lambda
\tag{inversion on 1} \\
& && 3\quad
&& \forall i \in I.~ P_i \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\dual{A_i}
\tag{IH on 2} \\
& && 4\quad
&& x \triangleright \{i:P_i\}_{i \in I} \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\&\{i:\dual{A_i}\}_{i \in I}
\tag{\ruleLabel{$\&$} on 4}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{copyR}
&& 1\quad
&& \Gamma, u:A; \Delta \vdash \nu{x}\send{u}{x}.P :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma, u:A; \Delta \vdash P :: \Lambda, x:\dual{A}
\tag{inversion on 1} \\
& && 3\quad
&& P \vdcll \dual{(\Gamma)}, u:\dual{A}; \dual{(\Delta)}, \Lambda, x:\dual{A}
\tag{IH on 2} \\
& && 4\quad
&& \nu{x}\send{u}{x}.P \vdcll \dual{(\Gamma)}, u:\dual{A}; \dual{(\Delta}), \Lambda
\tag{\ruleLabel{copy} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{copyL}
&& 1\quad
&& \Gamma, u:A; \Delta \vdash \nu{x}\send{u}{x}.P :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma, u:A; \Delta, x:A \vdash P :: \Lambda
\tag{inversion on 1} \\
& && 3\quad
&& P \vdcll \dual{(\Gamma)}, u:\dual{A}; \dual{(\Delta)}, \Lambda, x:\dual{A}
\tag{IH on 2} \\
& && 4\quad
&& \nu{x}\send{u}{x}.P :: \dual{(\Gamma)}, u:\dual{A}; \dual{(\Delta)}, \Lambda
\tag{\ruleLabel{copy} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\bang$R}
&& 1\quad
&& \Gamma; \emptyset \vdash \serv{x}{y}.P :: x:\bang A
\tag{assumption} \\
& && 2\quad
&& \Gamma; \emptyset \vdash P :: y:A
\tag{inversion on 1} \\
& && 3\quad
&& P \vdcll \dual{(\Gamma)}; y:A
\tag{IH on 2} \\
& && 4\quad
&& \serv{x}{y}.P \vdcll \dual{(\Gamma)}; y:\bang A
\tag{\ruleLabel{$\bang$} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\bang$L}
&& 1\quad
&& \Gamma; \Delta, x:\bang A \vdash P\subst{x/u} :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma, u:A; \Delta \vdash P :: \Lambda
\tag{inversion on 1} \\
& && 3\quad
&& P \vdcll \dual{(\Gamma)}, u:\dual{A}; \dual{(\Delta)}, \Lambda
\tag{IH on 2} \\
& && 4\quad
&& P\subst{x/u} \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\whynot \dual{A}
\tag{\ruleLabel{$\whynot$} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\whynot$R}
&& 1\quad
&& \Gamma; \Delta \vdash P\subst{x/u} :: \Lambda, x:\whynot \dual{A}
\tag{assumption} \\
& && 2\quad
&& \Gamma, u:A; \Delta \vdash P :: \Lambda
\tag{inversion on 1} \\
& && 3\quad
&& P \vdcll \dual{(\Gamma)}, u:\dual{A}; \dual{(\Delta)}, \Lambda
\tag{IH on 2} \\
& && 4\quad
&& P\subst{x/u} \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\whynot \dual{A}
\tag{\ruleLabel{$\whynot$} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\whynot$L}
&& 1\quad
&& \Gamma; x:\whynot A \vdash \serv{x}{y}.P :: \emptyset
\tag{assumption} \\
& && 2\quad
&& \Gamma; y:A \vdash P :: \emptyset
\tag{inversion on 1} \\
& && 3\quad
&& P \vdcll \dual{(\Gamma)}; y:\dual{A}
\tag{IH on 2} \\
& && 4\quad
&& \serv{x}{y}.P \vdcll \dual{(\Gamma)}; x:\bang \dual{A}
\tag{\ruleLabel{$\bang$} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cutRL}
&& 1\quad
&& \Gamma; \Delta, \Delta' \vdash \nu{x}(P \| Q) :: \Lambda, \Lambda'
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta \vdash P :: \Lambda, x:A \\
& && 3\quad
&& \Gamma; \Delta', x:A \vdash Q :: \Lambda'
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:A
\tag{IH on 2} \\
& && 5\quad
&& Q \vdcll \dual{(\Gamma)}; \dual{(\Delta')}, \Lambda', x:\dual{A}
\tag{IH on 3} \\
& && 6\quad
&& \nu{x}(P \| Q) \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \dual{(\Delta')}, \Lambda, \Lambda'
\tag{\ruleLabel{cut} on 4 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cutLR}
&& 1\quad
&& \Gamma; \Delta, \Delta' \vdash \nu{x}(P \| Q) :: \Lambda, \Lambda'
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta, x:A \vdash P :: \Lambda \\
& && 3\quad
&& \Gamma; \Delta' \vdash Q :: \Lambda', x:A
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\dual{A}
\tag{IH on 2} \\
& && 5\quad
&& Q \vdcll \dual{(\Gamma)}; \dual{(\Delta')}, \Lambda', x:A
\tag{IH on 3} \\
& && 6\quad
&& \nu{x}(P \| Q) \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \dual{(\Delta')}, \Lambda, \Lambda'
\tag{\ruleLabel{cut} on 4 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cutRR}
&& 1\quad
&& \Gamma; \Delta, \Delta' \vdash \nu{x}(P \| Q) :: \Lambda, \Lambda'
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta \vdash P :: \Lambda, x:A \\
& && 3\quad
&& \Gamma; \Delta' \vdash Q :: \Lambda', x:\dual{A}
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:A
\tag{IH on 2} \\
& && 5\quad
&& Q \vdcll \dual{(\Gamma)}; \dual{(\Delta')}, \Lambda', x:\dual{A}
\tag{IH on 3} \\
& && 6\quad
&& \nu{x}(P \| Q) \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \dual{(\Delta')}, \Lambda, \Lambda'
\tag{\ruleLabel{cut} on 4 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cutLL}
&& 1\quad
&& \Gamma; \Delta, \Delta' \vdash \nu{x}(P \| Q) :: \Lambda, \Lambda'
\tag{assumption} \\
& && 2\quad
&& \Gamma; \Delta, x:A \vdash P :: \Lambda \\
& && 3\quad
&& \Gamma; \Delta', x:\dual{A} \vdash Q :: \Lambda'
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda, x:\dual{A}
\tag{IH on 2} \\
& && 5\quad
&& Q \vdcll \dual{(\Gamma)}; \dual{(\Delta')}, \Lambda', x:A
\tag{IH on 3} \\
& && 6\quad
&& \nu{x}(P \| Q) \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \dual{(\Delta')}, \Lambda, \Lambda'
\tag{\ruleLabel{cut} on 4 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cut$\bang$R}
&& 1\quad
&& \Gamma; \Delta \vdash \nu{u}(P \| \serv{u}{x}.Q) :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma, u:A; \Delta \vdash P :: \Lambda \\
& && 3\quad
&& \Gamma; \emptyset \vdash Q :: x:A
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}, u:\dual{A}; \dual{(\Delta)}, \Lambda
\tag{IH on 2} \\
& && 5\quad
&& Q \vdcll \dual{(\Gamma)}; x:A
\tag{IH on 3} \\
& && 6\quad
&& \nu{u} ( P \| \serv{u}{x}.Q ) \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda
\tag{\ruleLabel{cut$\whynot$R} on 4 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cut$\bang$L}
&& 1\quad
&& \Gamma; \Delta \vdash \nu{u}(\serv{u}{x}.P \| Q) :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma; \emptyset \vdash P :: x:A \\
& && 3\quad
&& \Gamma, u:A; \Delta \vdash Q :: \Lambda
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}; x:A
\tag{IH on 2} \\
& && 5\quad
&& Q \vdcll \dual{(\Gamma)}, u:\dual{A}; \dual{(\Delta)}, \Lambda
\tag{IH on 3} \\
& && 6\quad
&& \nu{u}(\serv{u}{x}.P \| Q) \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda
\tag{\ruleLabel{cut$\whynot$L} on 4 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cut$\whynot$R}
&& 1\quad
&& \Gamma; \Delta \vdash \nu{u} ( P \| \serv{u}{x}.Q ) :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma, u:A; \Delta \vdash P :: \Lambda \\
& && 3\quad
&& \Gamma; x:\dual{A} \vdash Q :: \emptyset
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}, u:\dual{A}; \dual{(\Delta)}, \Lambda
\tag{IH on 2} \\
& && 5\quad
&& Q \vdcll \dual{(\Gamma)}; x:A
\tag{IH on 3} \\
& && 6\quad
&& \nu{u} ( P \| \serv{u}{x}.Q ) \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda
\tag{\ruleLabel{cut$\whynot$R} on 4 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cut$\whynot$L}
&& 1\quad
&& \Gamma; \Delta \vdash \nu{u}(\serv{u}{x}.P \| Q) :: \Lambda
\tag{assumption} \\
& && 2\quad
&& \Gamma; x:\dual{A} \vdash P :: \emptyset \\
& && 3\quad
&& \Gamma, u:A; \Delta \vdash Q :: \Lambda
\tag{inversion on 1} \\
& && 4\quad
&& P \vdcll \dual{(\Gamma)}; x:A
\tag{IH on 2} \\
& && 5\quad
&& Q \vdcll \dual{(\Gamma)}, u:\dual{A}; \dual{(\Delta)}, \Lambda
\tag{IH on 3} \\
& && 6\quad
&& \nu{u}(\serv{u}{x}.P \| Q) \vdcll \dual{(\Gamma)}; \dual{(\Delta)}, \Lambda
\tag{\ruleLabel{cut$\whynot$L} on 4 and 5}
\end{align*}
($\,\C \subseteq \U$)
Take any $P \in \C$.
Then, there are $\Gamma$, $\Delta$ s.t. $P \vdcll \Gamma; \Delta$.
By showing that this implies $\dual{(\Gamma)}; \emptyset \vdash P :: \Delta$, we have $P \in \U$.
As per the second item of <Ref>, Rules $\mleft$ and $\mright$ from <Ref> are admissible in .
Therefore, it suffices to show $\dual{(\Gamma)}; \emptyset \vpull P :: \Delta$.
We do so by induction on the structure of the proof of $P \vdcll \Gamma; \Delta$.
\begin{align*}
& \bullet \ruleLabel{id}
&& 1\quad
&& \fwd{x}{y} \vdcll \Gamma; x:A, y:\dual{A}
\tag{assumption} \\
& && 2\quad
&& \dual{(\Gamma)}; x:\dual{A} \vpull \fwd{x}{y} :: y:\dual{A}
\tag{\ruleLabel{idR}} \\
& && 3\quad
&& \dual{(\Gamma)} ; \emptyset \vpull \fwd{x}{y} :: x:A , y:\dual{A}
\tag{\ruleLabel{$\mright$}}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\1$}
&& 1\quad
&& \send{x}{}.\0 \vdcll \Gamma; x:\1
\tag{assumption} \\
& && 2\quad
&& \dual{(\Gamma)}; \emptyset \vpull \send{x}{}.\0 :: x:\1
\tag{\ruleLabel{$\1$R}}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\bot$}
&& 1\quad
&& \recv{x}{}.P \vdcll \Gamma; \Delta, x:\bot
\tag{assumption} \\
& && 2\quad
&& P \vdcll \Gamma; \Delta
\tag{inversion 1} \\
& && 3\quad
&& \dual{(\Gamma)}; \emptyset \vpull P :: \Delta
\tag{IH on 2} \\
& && 4\quad
&& \dual{(\Gamma)}; x:\1 \vpull \recv{x}{}.P :: \Delta
\tag{\ruleLabel{$\1$L} on 3} \\
& && 5\quad
&& \dual{(\Gamma)}; \emptyset \vpull \recv{x}{}.P :: \Delta, x:\bot
\tag{\ruleLabel{$\mright$} on 4}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\tensor$}
&& 1\quad
&& \nu{y}\send{x}{y}.(P \| Q) \vdcll \Gamma; \Delta, \Delta', x:A \tensor B
\tag{assumption} \\
& && 2\quad
&& P \vdcll \Gamma; \Delta, y:A \\
& && 3\quad
&& Q \vdcll \Gamma; \Delta', x:B
\tag{inversion on 1} \\
& && 4\quad
&& \dual{(\Gamma)}; \emptyset \vpull P :: \Delta, y:A
\tag{IH on 2} \\
& && 5\quad
&& \dual{(\Gamma)}; \emptyset \vpull Q :: \Delta', x:B
\tag{IH on 3} \\
& && 6\quad
&& \dual{(\Gamma)}; \emptyset \vpull \nu{y}\send{x}{y}(P \| Q) :: \Delta, \Delta', x:A \tensor B
\tag{\ruleLabel{$\tensor$R} on 4 and 5}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\parr$}
&& 1\quad
&& \recv{x}{y}.P \vdcll \Gamma; \Delta, x:A \parr B
\tag{assumption} \\
& && 2\quad
&& P \vdcll \Gamma; \Delta, y:A, x:B
\tag{inversion on 1} \\
& && 3\quad
&& \dual{(\Gamma)}; \emptyset \vpull P :: \Delta, y:A, x:B
\tag{IH on 2} \\
& && 4\quad
&& \dual{(\Gamma)}; y:\dual{A} \vpull P :: \Delta, x:B
\tag{\ruleLabel{$\mleft$} on 3} \\
& && 5\quad
&& \dual{(\Gamma)}; \emptyset \vpull \recv{x}{y}.P :: \Delta, x:\underbrace{\dual{A} \lolli B}_{A \parr B}
\tag{\ruleLabel{$\lolli$R} on 4}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\oplus$}
&& 1\quad
&& x \triangleleft j.P \vdcll \Gamma; \Delta, x:\oplus\{i:A_i\}_{i \in I}
\tag{assumption} \\
& && 2\quad
&& P \vdcll \Gamma; \Delta, x:A_j \\
& && 3\quad
&& j \in I
\tag{inversion on 1} \\
& && 4\quad
&& \dual{(\Gamma)}; \emptyset \vpull P :: \Delta, x:A_j
\tag{IH on 2} \\
& && 5\quad
&& \dual{(\Gamma)}; \emptyset \vpull x \triangleleft j.P :: \Delta, x:\oplus\{i:A_i\}_{i \in I}
\tag{\ruleLabel{$\oplus$R} on 4 and 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\&$}
&& 1\quad
&& x \triangleright \{i:P_i\}_{i \in I} \vdcll \Gamma; \Delta, x:\&\{i:A_i\}_{i \in I}
\tag{assumption} \\
& && 2\quad
&& \forall i \in I.~ P_i \vdcll \Gamma; \Delta, x:A_i
\tag{inversion on 1} \\
& && 3\quad
&& \forall i \in I.~ \dual{(\Gamma)}; \emptyset \vpull P_i :: \Delta, x:A_i
\tag{IH on 2} \\
& && 4\quad
&& \dual{(\Gamma)}; \emptyset \vpull x \triangleright \{i:P_i\}_{i \in I} :: \Delta, x:\&\{i:A_i\}_{i \in I}
\tag{\ruleLabel{$\&$R} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{copy}
&& 1\quad
&& \nu{x}\send{u}{x}.P \vdcll \Gamma, u:A; \Delta
\tag{assumption} \\
& && 2\quad
&& P \vdcll \Gamma, u:A; \Delta, x:A
\tag{inversion on 1} \\
& && 3\quad
&& \dual{(\Gamma)}, u:\dual{A}; \emptyset \vpull P :: \Delta, x:A
\tag{IH on 2} \\
& && 4\quad
&& \dual{(\Gamma)}, u:\dual{A}; x:\dual{A} \vpull P :: \Delta
\tag{\ruleLabel{$\mleft$} on 3} \\
& && 5\quad
&& \dual{(\Gamma)}, u:\dual{A}; \emptyset \vpull \nu{x}\send{u}{x}.P :: \Delta
\tag{\ruleLabel{copyL} on 4}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\bang$}
&& 1\quad
&& \serv{x}{y}.P \vdcll \Gamma; x:\bang A
\tag{assumption} \\
& && 2\quad
&& P \vdcll \Gamma; y:A
\tag{inversion on 1} \\
& && 3\quad
&& \dual{(\Gamma)}; \emptyset \vpull P :: y:A
\tag{IH on 2} \\
& && 4\quad
&& \dual{(\Gamma)}; \emptyset \vpull \serv{x}{y}.P :: x:\bang A
\tag{\ruleLabel{$\bang$R} on 3}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{$\whynot$}
&& 1\quad
&& P\subst{x/u} \vdcll \Gamma; \Delta, x:\whynot A
\tag{assumption} \\
& && 2\quad
&& P \vdcll \Gamma, u:A; \Delta
\tag{inversion on 1} \\
& && 3\quad
&& \dual{(\Gamma)}, u:\dual{A}; \emptyset \vpull P :: \Delta
\tag{IH on 2} \\
& && 4\quad
&& \dual{(\Gamma)}; x:\bang \dual{A} \vpull P\subst{x/u} :: \Delta
\tag{\ruleLabel{$\bang$L} on 3} \\
& && 5\quad
&& \dual{(\Gamma)}; \emptyset \vpull P\subst{x/u} :: \Delta, x:\whynot A
\tag{\ruleLabel{$\mright$} on 4}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cut}
&& 1\quad
&& \nu{x}(P \| Q) \vdcll \Gamma; \Delta, \Delta'
\tag{assumption} \\
& && 2\quad
&& P \vdcll \Gamma; \Delta, x:A \\
& && 3\quad
&& Q \vdcll \Gamma; \Delta', x:\dual{A}
\tag{inversion on 1} \\
& && 4\quad
&& \dual{(\Gamma)}; \emptyset \vpull P :: \Delta, x:A
\tag{IH on 2} \\
& && 5\quad
&& \dual{(\Gamma)}; \emptyset \vpull Q :: \Delta', x:\dual{A}
\tag{IH on 3} \\
& && 6\quad
&& \dual{(\Gamma)}; x:A \vpull Q :: \Delta'
\tag{\ruleLabel{$\mleft$} on 5} \\
& && 7\quad
&& \dual{(\Gamma)}; \emptyset \vpull \nu{x}(P \| Q) :: \Delta, \Delta'
\tag{\ruleLabel{cutRL} on 4 and 6}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cut$\whynot$R}
&& 1\quad
&& \nu{u} ( P \| \serv{u}{x}.Q ) \vdcll \Gamma; \Delta
\tag{assumption} \\
& && 2\quad
&& P \vdcll \Gamma, u:A; \Delta \\
& && 3\quad
&& Q \vdcll \Gamma; x:\dual{A}
\tag{inversion on 1} \\
& && 4\quad
&& \dual{(\Gamma)}, u:\dual{A}; \emptyset \vpull P :: \Delta
\tag{IH on 2} \\
& && 5\quad
&& \dual{(\Gamma)}; \emptyset \vpull Q :: x:\dual{A}
\tag{IH on 3} \\
& && 6\quad
&& \dual{(\Gamma)}; \emptyset \vpull \nu{u} ( P \| \serv{u}{x}.Q ) :: \Delta
\tag{\ruleLabel{cut$\bang$R}}
\displaybreak[1] \\[5pt]
& \bullet \ruleLabel{cut$\whynot$L}
&& 1\quad
&& \nu{u}(\serv{u}{x}.P \| Q) \vdcll \Gamma; \Delta
\tag{assumption} \\
& && 2\quad
&& P \vdcll \Gamma; x:\dual{A} \\
& && 3\quad
&& Q \vdcll \Gamma, u:A; \Delta
\tag{inversion on 1} \\
& && 4\quad
&& \dual{(\Gamma)}; \emptyset \vpull P :: x:\dual{A}
\tag{IH on 2} \\
& && 5\quad
&& \dual{(\Gamma)}, u:\dual{A}; \emptyset \vpull Q :: \Delta
\tag{IH on 3} \\
& && 6\quad
&& \dual{(\Gamma)}; \emptyset \vpull \nu{u}(\serv{u}{x}.P \| Q) :: \Delta
\tag{\ruleLabel{cut$\bang$L}}
\end{align*}
This concludes the proof of <Ref>.
We now turn our attention to , the class of processes typable under the intuitionistic interpretation.
The observation by Caires, Pfenning and Toninho in [11] entails that (-typable processes) should be a strict subset of (-typable processes).
We formalize this fact by characterizing the intuitionistic fragment of (which coincides with ) by limiting its typing rules and by showing that the class of processes typable in this fragment coincides with (<Ref>).
It then follows that this class of processes is strictly contained in (-typable processes; <Ref>).
Since we have just shown that $\U = \C$, this formalizes the fact that is a strict subset of .
<Ref> below formalizes two equivalent characterizations of the intuitionistic fragment of .
One characterization is based on the limited form of duality of :
the Rules $\mleft$ and $\mright$ in <Ref> may not be used.
The other characterization is based on the restricted two-sided form of sequents:
the rules in <Ref> are limited to have exactly one channel/type pair on the right.
This requires the following auxiliary definition:
The of a sequent $\Gamma; \Delta \vdash P :: \Lambda$ is the size of $\Lambda$.
We say this sequent has $|\Lambda|$.
We now have:
Given a process $P$, the following are equivalent:
* there are $\Gamma$, $\Delta$, $x$ and $A$ such that $\Gamma; \Delta \vdill P :: x:A$;
* there are $\Gamma$, $\Delta$ and $\Lambda$ such that $\Gamma; \Delta \vdash P :: \Lambda$ where all sequents in its proof have 1;
* there are $\Gamma$, $\Delta$ and $\Lambda$ such that $\Gamma; \Delta \vpull P :: \Lambda$ where its proof never uses Rules $\mleft$ and $\mright$.
We first argue that items 1 and 2 are equivalent; then, we argue that items 2 and 3 imply each other:
(Equivalence of items 1 and 2)
We show that restricting the sequents in to an of 1 entails the limitation of its rules to a strict subset: those marked with $\ast$ in <Ref>.
This set of rules coincides with the set of rules for in <Ref>.
Hence, a proof of typability in (item 1) can be replicated in with 1 (item 2) and vice versa.
Because the proof for each rule follows the same pattern, we only detail two cases: one without $\ast$ and one with $\ast$.
* Rule $\parr$L is not $\ast$-marked.
Suppose we have a proof of typability in with 1 and suppose this includes an application of $\parr$L.
By assumption, this application's antecedents have 1, so its consequence must have 2.
This contradicts the assumption that all sequents have 1, showing that Rule $\parr$L is not usable in this fragment of .
* Rule $\lolli$L is $\ast$-marked.
By assumption, in any application of this rule its antecedents would have 1.
Then, its consequence also has 1, so this rule can be used without problems.
(Item 2 implies item 3)
By <Ref>, the set of rules of consists of only $\mleft$, $\mright$, and the $\ast$-marked rules in <Ref>.
In , the only rules that alter the s in judgments are $\mleft$ and $\mright$.
Hence, if these rules may not be used, proofs of typability in have a constant .
Since all axioms of (the $\ast$-marked Axioms $\id$R and $\1$R in <Ref>) have 1, this constant is 1.
Therefore, a proof of typability in without using $\mleft$ and $\mright$ (item 3) can be replicated exactly in with restricted to 1 (item 2).
(Item 3 implies item 2)
The proof of equivalence of items 1 and 2 above shows that with 1 the only usable rules in are those marked with $\ast$ in <Ref>.
By <Ref>, without Rules $\mleft$ and $\mright$, consists only of these $\ast$-marked rules.
Hence, a proof of typability in with restricted to 1 (item 2) can be replicated exactly in without using $\mleft$ and $\mright$ (item 3).
We may now state our final result:
$\I \subsetneq \U$.
<Ref> ensures that $\I \subseteq \U$.
To show that this inclusion is strict, we give a process $P \in \U$ such that $P \notin \I$.
Assuming a proof for $u:B; \emptyset \vdash P' :: z:A$, let $P = \recv{x}{y}.\serv{y}{z}.P' \subst{x/u}$;
there are precisely three ways to prove $P \in \U$: they follow from the three ways to infer a receive in ($\parr$R, $\lolli$R, and $\tensor$L).
The proofs are as follows:
u:B; ∅⊢P' :: z:A
u:B; ∅⊢yz.P' :: y:A
∅; ∅⊢yz.P' x/u :: y:A, x:B
∅; ∅⊢xy.yz.P' x/u :: x:(A) (B)
u:B; ∅⊢P' :: z:A
u:B; z:A ⊢P' :: ∅
u:B; y:A ⊢yz.P' :: ∅
∅; y:A ⊢yz.P' x/u :: x:B
∅; ∅⊢xy.yz.P' x/u :: x:(A) (B)
u:B; ∅⊢P' :: z:A
u:B; z:A ⊢P' :: ∅
u:B; y:A ⊢yz.P' :: ∅
∅; y:A, x:B ⊢yz.P' x/u :: ∅
∅; x:(A) (B) ⊢xy.yz.P' x/u :: ∅
Clearly, all these proofs contain judgments of different from 1 (first case) or require using $\mleft$/$\mright$ (second and third cases).
Hence, by <Ref>, $P \notin \I$.
§ ANALYSIS
Now that we have established characterizations for the intuitionistic and classical fragments of , we analyze the meaning of these results and discuss possible extensions.
Our analysis is twofold.
First, we consider the informal observation by Caires, Pfenning, and Toninho [11] that intuitionistic type systems enforce the locality principle (ssec:locality);
unlike the classical formulation, the intuitionistic fragment of (and thus ) has a partial form of duality, which ensures that typability guarantees locality.
Next, we discuss how can support alternative, more expressive forms of parallel composition and restriction, and how such extensions transfer to classical or intuitionistic type systems (ssec:mixcycle);
we will see that the complete duality of classical type systems allows for such extensions, while the rely-guarantee reading of intuitionistic type systems cannot account for them.
§.§ Locality
Locality is a well-known principle in concurrency research [38].
The idea is that freshly created channels are local.
Local channels are mutable, in the sense that they can be used for receives.
Once a channel has been transmitted to another location, it becomes non-local, and thus immutable:
it can only be used for sends—receives are no longer allowed.
This makes locality particularly relevant for giving formal semantics to distributed programming languages;
a prime example is the join calculus [20], whose theory relies on (and is deeply influenced by) the locality principle [19].
Locality also makes an appearance in other contexts.
Honda and Laurent's [26] correspondence between a typed $\pi$-calculus and polarized proof-nets enforces locality due to receives having negative polarity while received names may only be of positive polarity.
Also, Dal Lago 's [16] typed $\pi$-calculus with intersection types goes further:
processes are hyperlocalized, meaning that there may be no receives on free channels after a prior receive at all.
Neither the intuitionistic or the classical interpretation guarantee full-fledged locality through typing:
both systems allow receives on previously received channels.
The exception is replicated receive, which is used to define a shared channel that can continuously receive linear channels over which to perform a service.
The intuitionistic interpretation guarantees locality for shared channels;
in other words, cannot type replicated receives on non-local channels.
The following example, taken from the work by Caires, Pfenning and Toninho [11], is typable in but not in :
\[
\nu{x}(\recv{x}{y}.\serv{y}{z}.P \| \nu{q}\send{x}{q}.Q)
\]
Consider the left process in the parallel composition, $\recv{x}{y}.\serv{y}{z}.P$.
It first receives a channel $y$ over channel $x$; then, it uses $y$ for replicated receive, thus defining it as a shared channel.
In , channel $x$ has type $\bang A \parr B$.
The intuitionistic variant of this type is $\dual{(\bang A)} \lolli B = \whynot \dual{A} \lolli B$ on the right of the typing judgment, and $\whynot \dual{A} \tensor \dual{B}$ on the left.
It is impossible to type a process with a channel of such a type in , because it lacks rules to type `' channels.
The fact that cannot type non-local shared channels—due to the absence of a dual for `'—suggests that there should be another kind of non-local channels that cannot type:
there is no dual for .
Indeed, cannot type empty sends on previously received channels, a feature that is not addressed in [11].
The following example is typable in but not in :
\[
\recv{x}{y}.\recv{x}{}.\send{y}{}.\0
\]
In , the type of $x$ in this process is $\1 \parr \bot$.
The intuitionistic variant of this type is $\dual{\1} \lolli \bot = \bot \lolli \bot$ on the right of the typing judgment, and $\bot \tensor \1$ on the left.
This process is not typable in , because it has no rules to type channels.
We make this more precise by giving an alternative proof to <Ref> ($\I \subsetneq \U$) based on this observation:
By <Ref>, it is sufficient to give $P \in \U$ such that $P \notin \I$.
Let $P := \recv{x}{y} . \recv{x}{} . \send{y}{} . \0$.
There are three ways to prove that $P \in \U$:
Γ; y:⊢y . :: ∅
Γ; y:, x: ⊢x . y . :: ∅
Γ; x: ⊢xy . x . y . :: ∅
Γ; ∅⊢y . :: y:
Γ; ∅⊢x . y . :: y: , x:
Γ; ∅⊢xy . x . y . :: x: x:
Γ; y:⊢y . :: ∅
Γ; y:⊢x . y . :: x:
Γ; ∅⊢xy . x . y . :: x:
Clearly, all these proofs contain judgments of different from 1.
Hence, by <Ref>, $P \notin \I$.
Based on these two proofs for <Ref>, we have corroborated Caires 's observation on locality, and extended it to the case of empty sends and receives.
Judgments, Revisited
The absence of rules for and in is not a design choice, but an inherent consequence of the form of its judgments.
As shown by <Ref>, rules for and are an impossibility when judgments are required to have exactly one assignment (channel/proposition) on the right.
<Ref> also shows this is closely related to duality, which is not a complete relation in the intuitionistic interpretation.
In the classical case, the type system is symmetrical (as shown by a support for $\mleft$/$\mright$-rules), while the intuitionistic type system is asymmetrical.
§.§ Parallel Composition and Restriction
The type systems we have discussed so far are rather restrictive in terms of how processes can be composed and connected:
there is only the cut-rule which composes and connects two processes that have exactly one channel of dual type in common.
Indeed, the cut-rule jointly handles constructs for parallel composition and restriction, which contrasts with many non-logical type systems for the $\pi$-calculus (e.g., [34, 33, 51]) where parallel composition and restriction typically have a dedicated rule each.
In this section we discuss extensions to the type systems that decompose cut into two separate rules:
mix for parallel composition, and cycle for channel connection (restriction).
As we will see, these notions form another clear distinction point between Curry-Howard interpretations of classical and intuitionistic linear logic.
§.§.§ Independent Parallel Composition
Caires [11] discuss an alternative form of parallel composition. In the intuitionistic interpretation, the so-called independent parallel composition
connects (i) an arbitrary process $Q$ with (ii) a process $P$ with a channel of type on the right; it is derivable in the silent interpretation of , in which
Axioms $\1$R and $\bot$L type the inactive process $\0$ and Rules $\bot$R and $\1$L leave processes untouched.
This way, the rules for $\1$ would be the following (the rules for $\bot$ are similar):
Γ; ∅⊢ :: x:
Γ; Δ⊢P :: Λ
Γ; Δ, x: ⊢P :: Λ
In this case, the correspondence relies on structural congruences in processes and proofs (suitably extended),
rather than on reduction.
To obtain independent parallel composition, one uses Rule cut to connect the right channel of $P$ with a corresponding (dual) channel in $Q$, which is exposed on the left using Rule $\1$L:
\[
\begin{bussproof}
\bussAssume{
\Gamma; \Delta \vdill P :: z:\1
\bussAssume{
\Gamma; \Delta' \vdill Q :: x:C
\bussUn[\ruleLabel{$\1$L}]{
\Gamma; \Delta', z:\1 \vdill Q :: x:C
\bussBin[\ruleLabel{cutRL}]{
\Gamma; \Delta, \Delta' \vdill \nu{z}(P \| Q) :: x:C
\end{bussproof}
\]
The requirement that process $P$ above has a channel of type on the right is rather restrictive:
only left-rules can be used for typing, so $P$ must be a process that relies on multiple behaviors but can offer a behavior of type $\1$.
Related to this constraint,
Caires show that in the silent interpretation
$\Gamma; \Delta \vdill P :: x:A$ implies
$\Gamma; \Delta,x:\dual{A} \vdill P :: z:\1$, for some fresh $z$, provided that $A$ is exponential-free <cit.>.
However, this “movement” from the right to the left is only possible under the absence of the identity axiom, which is in turn necessary when considering, e.g., behavioral polymorphism [8].
Indeed, given any type $A$ we can prove $\emptyset; x:A \vdill \fwd{x}{y} :: y:A$ using the identity axiom, but there is no way to prove $\emptyset; x:A, y:\dual{A} \vdill \fwd{x}{y} :: z:\1$.
§.§.§ Parallel Composition: Mix
Girard [22] discusses an extension to linear logic which allows the combination of two independent proofs.
Wadler gives a Curry-Howard interpretation of this rule as the parallel composition of two processes that have no channels in common [52].
In [4], Caires complements the extension with an axiom that types the inactive process with an empty context.
We now define , which is extended with this form of parallel composition.
The only change is the addition of the following rules:
P Γ; Δ
Q Γ; Δ'
P Q Γ; Δ, Δ'
Γ; ∅
We straightforwardly define as a similar extension of by adding the following rules:
Γ; Δ⊢P :: Λ
Γ; Δ' ⊢Q :: Λ'
Γ; Δ, Δ' ⊢P Q :: Λ, Λ'
Γ; ∅⊢ :: ∅
It is easy to see that <Ref>—the equivalence of and —still holds for and .
Following the reasoning of <Ref>—the characterization of in terms of —, Rules mix and empty do not belong to the intuitionistic fragment of :
Rule empty has 0, and if the antecedents of Rule mix both have 1 then its consequence would have 2.
This leads us to conclude that the more flexible ways of composition induced by mix cannot be supported by intuitionistic systems.
The Equivalence of and
Caires [4] notes that, in the classical setting, Rules mix and empty make it possible to prove $\bot \lolli \1$ and $\1 \lolli \bot$ (where denotes linear implication).
Hence, it is possible to consider and equivalent, writing a single symbol (say, `$\bullet$') for either—very similar to the singular, self-dual type in standard session types.
The work of Atkey [1] is also relevant, as it develops a detailed treatment of the conflation of $\bot$ and $\1$.
Due to the absence of and the impossibility of mix and empty in the intuitionistic interpretation, it is not possible to prove and equivalent.
This reinforces the fact that in the intuitionistic setting there is a significant difference between channels/propositions in the left and the right contexts.
§.§.§ Channel Connection: Cycle
The cut-rule only allows us to connect two processes on a single channel.
Although this neatly guarantees deadlock-freedom, it prevents many deadlock-free processes from being typable (see [17] for comparisons between these classes of processes).
For example, we can construct a process $\nu{x}\nu{y}(P \| Q)$ where
\begin{align*}
&:= \nu{u}\send{x}{u}.(\recv{y}{v}.(\recv{u}{}.\recv{v}{}.\0 \| \send{y}{}.\0) \| \send{x}{}.\0) ~\text{and} \\
&:= \recv{x}{w}.\nu{z}\send{y}{z}.(\recv{x}{}.\send{z}{}.\0 \| \recv{y}{}.\send{w}{}.\0)
\end{align*}
This process is deadlock-free, but not typable using cut.
Connecting on multiple channels at once does have the danger of introducing the circular dependencies between sessions that are at the heart of deadlocked processes.
For example, suppose that we replace $P$ with $P'$, where we swap the send on $x$ and the receive on $y$:
\[
P' := \recv{y}{v}.\nu{u}\send{x}{u}.(\recv{u}{}.\recv{v}{}.\0 \| \send{y}{}.\0 \| \send{x}{}.\0).
\]
Now, the composed and connected process would be stuck with $P'$ waiting for a receive on $y$ and $Q$ for a receive on $x$.
Dardha and Gay [13] present a session type system based on classical linear logic in which they replace the cut-rule with a rule called cycle, which connects two channels of dual types in the same judgment.
This new rule has a side-condition that allows their type system to maintain deadlock-freedom.
We can define as an extension of with the cycle-rule.
Since we are specifically interested in this form of channel restriction, below we abstract away from Dardha and Gay's side-condition by simply writing $\phi$.
Note that we first require some modifications to restriction and reduction, based on [51].
Restriction in should involve two channel names, i.e. writing $\nu{x y}P$ instead of $\nu{x}P$, because the involved channels appear in the same judgment and thus need to be uniquely named.
Indeed, $\nu{x y}P$ says that $x$ and $y$ are the two endpoints of the same channel in $P$.
\[
\begin{bussproof}[cycle]
\bussAssume{
P \vdcll \Gamma; \Delta, x:A, y:\dual{A}
\bussAssume{
\phi
\bussBin{
\nu{x y}P \vdcll \Gamma; \Delta
\end{bussproof}
\]
Defining as a similar extension of is again straightforward.
Similarly to the cut-rules in <Ref>, we add four rules that operate on different sides of the typing judgment:
Γ; Δ, x:A ⊢P :: Λ, y:A
Γ; Δ⊢νy xP :: Λ
Γ; Δ, x:A ⊢P :: Λ, y:A
Γ; Δ⊢νx yP :: Λ
Γ; Δ⊢P :: Λ, x:A, y:A
Γ; Δ⊢νx yP :: Λ
Γ; Δ, x:A, y:A ⊢P :: Λ
Γ; Δ⊢νx yP :: Λ
Again, it is easy to see that <Ref> still holds for and .
Interestingly, if we follow <Ref>, then Rule cycleLL actually is in 's intuitionistic fragment.
Unfortunately, this rule is useless for .
Every type has to contain at least one atomic type, which in the case of can only be .
Since cycleLL requires two channels of dual types, one of the channels then must contain the atomic type which we have seen earlier to be impossible in the intuitionistic setting.
We again conclude that the intuitionistic setting imposes restrictions that make more flexible ways of connecting processes than through cut an impossibility.
Adding mix- and cycle-rules enables a more flexible formulation of structural congruence, as well as more flexible typing of sends.
For one, e.g., Rule $\tensor$R would only require a single continuation that provides both the payload and the continuation of the send.
We can then add structural congruence rules that are found in traditional, untyped settings, such as $P \| Q \equiv Q \| P$ and $P \| \0 \equiv P$.
Preventing Circular Dependencies
We briefly discuss some solutions to the side-condition for deadlock-freedom $\phi$.
Dardha and Gay annotate types with priorities and include side-conditions on these priorities in other rules [13].
This approach is based on Kobayashi's type systems for deadlock-freedom <cit.>.
A completely orthogonal approach is by Toninho and Yoshida in [50].
It is based on multiparty session type theory in which session types describe interactions between two or more participants (as opposed to the binary session types we study in this paper, which always involve exactly two participants) <cit.>.
Toninho and Yoshida generate partial multiparty session types from binary typings of processes.
Processes can then be connected on multiple channels at once if their partial multiparty types are compatible, in which case deadlock-freedom results transfer from the multiparty theory to the binary system.
§ CONCLUSION
Curry-Howard correspondences between linear logic and session types explain how
linear logic can provide an ample, principled framework to conceive different type disciplines for message-passing processes specified in the $\pi$-calculus.
There is no single canonical interpretation, as there are multiple interpretations, depending on design choices involving, e.g., the logical connectives considered and their respective process operators.
In this context, this paper has pursued a very concrete goal: to formally compare the interpretations of classical and intuitionistic linear logic as session types. The comparison results reported in <Ref> are an indispensable step to consolidating the logical foundations of message-passing concurrency; they also have a number of relevant ramifications, as reported in <Ref>.
Our technical approach to this goal relies on a fragment of Girard's Logic of Unity (LU) [23] as a basis to develop , a new session type system that can type all processes typable in both and , which correspond to classical and intuitionistic interpretations, respectively.
The linear logic , on which the type system stands, is an admittedly modest fragment of Girard's LU. Still, we emphasize that is sufficient for our purposes, as it gives us a fair and rigorous basis for addressing our declared goal of comparing type systems based on different linear logics.
The development of computational interpretations for LU, including but going beyond and , is surely an interesting direction but one that lies outside the scope of our work.
In judgments have a particular reading in which a process relies on several channels (on the left of judgments) and guarantees a behavior along a single designated channel (on the right).
uses two-sided judgments as well, similar to LU.
However, does not retain 's rely-guarantee reading, because it does not distinguish between the sides of its sequents.
The consequence is that supports a full duality relation, as opposed to .
This allows to mimic the explicit duality in the single-sided .
On the other hand, restricting the right side of 's judgments to exactly one channel—thus limiting support for duality—characterizes a fragment of 's typing rules that precisely coincides with .
Our results confirm the informal observation by Caires [11] that the difference between session type systems based on classical and intuitionistic linear logic is in the enforcement of locality of shared names.
We have not only confirmed that cannot type processes that do not respect locality for shared channels:
a new insight obtained in this paper is that also forbids empty sends on received channels.
Our results show that these constraints are a consequence of the strict requirement that 's typing judgments must have exactly one channel/type on the right.
and do not have such constraints and thus fully support duality.
Our work can be seen as providing a technical answer to long-suspected but fuzzily understood differences between and , concerning the role of duality (implicit in , explicit in ) and locality of shared channels (enforced by but not by ). We believe it is important to give a formal footing to this kind of informal observations. Formally stating the required comparisons is not obvious, and insightful in itself, as implicit assumptions may emerge in the process. In this respect, we adopted as it provides an effective yardstick for comparisons, independent from both and , and based on an already existing framework (Girard's LU). In future work, it would be interesting to explore other approaches towards our technical results (<Ref>).
If one adopts the stance that a permissive type system is also an expressive one, then the results from our comparisons (notably, the strict inclusion of in ) indicate that classical linear logic induces a more expressive class of typed processes than intuitionistic linear logic.
There is an alternative stance, which considers little permissive type systems as being more precise than other, more permissive type system.
The locality property for shared channels, as enforced by intuitionistic interpretations, is a case in point here.
In our view, the connection between permissiveness, expressiveness, and precision is an interesting question, which largely depends on the value of the intended properties.
Indeed, the precision of type systems derived from intuitionistic interpretations may not be meaningful in settings/applications where locality is simply not a relevant property.
On the other hand, the conditions that make intuitionistic interpretations less permissive can always be imposed on more permissive interpretations as side-conditions, making them more precise.
Finally, as observed by Caires [11], it is surely remarkable that intuitionistic interpretations of session types precisely capture a principle such as locality of shared names, which was known and exploited in different contexts [38] long before the Curry-Howard interpretations were first spelled out [6].
To conclude, we believe that LU and have other useful applications besides the comparison of type systems based on vanilla linear logic;
our discussion of more expressive parallel composition and channel connection in <Ref> corroborates this.
We have been able to rely on to study the effects of mix- and cycle-rules in the intuitionistic interpretation by transferring extensions of the classical interpretation to and following the methodology of our comparison results.
In this case, extensions of intuitionistic interpretations with such rules do not make sense due to the constraints of typing judgments.
This makes sense from a logical perspective:
intuitionistic argumentation is based on a constructivist philosophy in which putting unrelated things together (mix) and then relating them later (cycle) may not be acceptable.
We are grateful to
Juan Jaramillo,
Joseph Paulus,
Revantha Ramanayake
for helpful discussions.
We would also like to thank the anonymous reviewers of PLACES'20 for their suggestions, which were helpful to improve the presentation.
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## 0.1 Feature Transformation for Style Transfer
Once we have the feature grid representation of a scene, we can tackle the
task of stylizing 3D scenes. Given a reference style image, our goal is to
render stylized novel views of the 3D scene with multi-view consistency. To
achieve this, we apply transformations to the features of the grid.
One plausible solution to this task is to apply style transfer to the feature
grid directly. This solution is efficient in evaluations as it can render any
stylized views with a single style transfer process only. However, it is
impractical to train such transformation as it needs to stylize the whole
feature grid in every iteration. Another solution is to apply an off-the-shelf
zero-shot style transfer method to the features of the sampled 3D points.
While this solution can reduce computational cost through decreasing the size
of training patch and the number of sampled points, it has two problems: 1)
vanilla zero-shot style transformation is conditioned on holistic statistics
of the sampled point batch [li2019learning, huang2017arbitrary,
liu2021adaattn], which violates multi-view consistency in volume rendering as
the feature transformation of a specific 3D point will vary across different
sampled points; 2) volume rendering requires sampling hundreds of points along
a single ray, which makes transformation on the point batch memory-intensive.
Motivated by the observation that style transformation is conditioned on both
content information and style information, we decompose the style
transformation into sampling-invariant content transformation (SICT) and
deferred style transformation (DST). After the decomposition, SICT will be
conditioned solely on the content information while DST conditioned solely on
the style information, more details to be elaborated in the ensuing
subsections.
### 0.1.1 Sampling-invariant Content Transformation
[b]0.4 [b]0.4
Figure 1: Vanilla IN. Figure 2: Volume-adaptive IN. Figure 3: Comparison
between vanilla instance normalization (IN) in (a) and volume-adaptive IN in
(b). During evaluation, volume-adaptive IN uses learned mean and standard-
deviation, discarding dependency over the sampled point batch’s holistic
statistics (indicated by the redred arrows in the left graph).
Given a batch of sampled points, we can get their corresponding features
$F_{i}\in\mathbb{R}^{C},i\in[1,2,...,N]$ from the feature grid, where $N$ is
the number of the sampled points along a ray and $C$ is the number of the
feature channels. The goal of SICT is to transform the extracted features
$F_{i}$ so that they can be better stylized. We formulate SICT as a channel-
wise self-attention operation to the features after instance normalization
(IN) [ulyanov2016instance]. Specifically, we formulate $Q$(query), $K$(key),
and $V$(value) as:
$\displaystyle Q$ $\displaystyle=q(Norm(F_{i})),$ (1) $\displaystyle K$
$\displaystyle=k(Norm(F_{i})),$ (2) $\displaystyle V$
$\displaystyle=v(Norm(F_{i})),$ (3)
where $q,k,v$ are $1\times 1$ convolution layers which reduce the channel
number from $C$ to $C^{\prime}$ for computational efficiency, and $Norm$
denotes the IN. However, as shown in fig:IN, vanilla IN calculates per-
dimension mean and standard-deviation of the batch of sampled points, which
varies with different sampled points and incurs multi-view inconsistency
accordingly. Thus we design volume-adaptive IN which, during training, keeps
running estimates of the computed mean and standard-deviation, and uses them
for normalization during evaluations (instead of computing from the sampled
point batch). Through volume-adaptive IN, we can ensure that the content
transformation is consistent regardless of the sampled point batch’s holistic
statistics.
Channel-wise self-attention can thus be implemented by:
$\bar{F_{i}}=V\otimes\mathrm{Softmax}\left(\widetilde{cov}(Q,K)\right),$ (4)
where $\otimes$ denotes matrix multiplication and
$\widetilde{cov}(Q,K)\in\mathbb{R}^{N\times C^{\prime}\times C^{\prime}}$
denotes the covariance matrix in the channel dimension.
### 0.1.2 Deferred Style Transformation
Figure 4: Deferred style transformation. We apply the style transformation to
the volume-rendered feature maps $\bar{F_{c}}$ according to the style feature
maps $F_{s}$. To ensure multi-view consistency, we modulate the bias (e.g. the
mean value of the style feature maps $\mu(F_{s})$) with the sum weight of
sampled points along each ray $w_{\textbf{r}}$.
After applying SICT to the features of each 3D point, we apply DST to the
volume-rendered 2D feature maps $\bar{F_{c}}$ rather than 3D point features
$\bar{F_{i}}$. To ensure multi-view consistency, we formulate the
transformation as matrix multiplication and adaptive bias addition as
illustrated in fig:DST.
Specifically, we first extract feature maps $F_{s}$ of the reference style $S$
using a pre-trained VGG[simonyan2014very], and then generate the style
transformation matrix $T\in\mathbb{R}^{C^{\prime}\times C^{\prime}}$ using
feature covariance $cov(F_{s})$ following [li2019learning]. Next, we apply
matrix multiplication with $T$ to the feature maps $\bar{F_{c}}$ and use a
$1\times 1$ convolution layer $conv$ without bias to restore the channel
number from $C^{\prime}$ to $C$. Though these operations can partially instill
style information, they are not expressive enough without bias addition
containing style information [wu2021styleformer]. Thus following
[huang2017arbitrary], we multiply the feature maps with the standard-deviation
value $\sigma(F_{s})$ and add the mean value $\mu(F_{s})$. To ensure it is
equivalent when applying the transformation to either 3D point features or 2D
feature maps, we adaptively modulate the mean value $\mu(F_{s})$ with the sum
weight of sampled points along each ray $w_{\textbf{r}}$. DST can be
mathematically formulated by:
$F_{cs}=conv\left(T\otimes\bar{F_{c}}\right)\times\sigma(F_{s})+w_{\textbf{r}}\times\mu(F_{s}),$
(5)
$\text{where}\quad\bar{F_{c}}=\sum_{i=1}^{N}w_{i}\bar{F_{i}},w_{\textbf{r}}=\sum_{i=1}^{N}w_{i},\textbf{r}\in\mathcal{R}$
(6)
where $w_{i}$ denotes the weight of sampled point $i$ (eq:weight),
$\bar{F_{i}}$ denotes the feature of sample $i$ after SICT, and $\mathcal{R}$
is the set of rays in each training batch.
Note $conv$ is a $1\times 1$ convolution layer without bias, so it is
basically a matrix multiplication operation. And $\sigma(S),\mu(S)$ are
scalars. Together with the adaptive bias modulation $w_{\textbf{r}}$,
eq:styletrans can be reformulated by:
$F_{cs}=\sum_{i=1}^{N}w_{i}\left(\underbrace{conv\left(T\otimes\bar{F_{i}}\right)\times\sigma(F_{s})+\mu(F_{s})}_{\mbox{(i)}}\right),$
(7)
where part (i) can be seen as applying style transformation on every 3D point
feature independently before volume rendering. This proves that applying DST
on 2D feature maps is equivalent to applying the transformation on 3D points’
features, maintaining multi-view consistency. The full derivation of
eq:styletrans2 is provided in the appendix.
Finally, we adopt a 2D CNN decoder to project the stylized feature maps
$F_{cs}$ to RGB space to generate the final stylized novel view images.
|
# The n-th root of sequential effect algebras††thanks: This project is
supported by Natural Science Foundation of China (10771191 and 10471124) and
Natural Science Foundation of Zhejiang Province of China (Y6090105).
Shen Jun1,2, Wu Junde1 Tel: 86-571-87951609-8111, E-mail<EMAIL_ADDRESS>
###### Abstract
In 2005, Professor Gudder presented 25 open problems of sequential effect
algebras, the 20th problem asked: In a sequential effect algebra, if the
square root of some element exists, is it unique ? We can strengthen the
problem as following: For each given positive integer $n>1$, is there a
sequential effect algebra such that the n-th root of its some element $c$ is
not unique and the n-th root of $c$ is not the k-th root of $c$ ($k<n$) ? In
this paper, we answer the strengthened problem affirmatively.
1Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
2Department of Mathematics, Anhui Normal University, Wuhu 241003, P. R. China
Keywords. Effect algebra, sequential effect algebra, root.
PACS numbers: 02.10-v, 02.30.Tb, 03.65.Ta.
Let $H$ be a complex Hilbert space and ${\cal D}(H)$ the set of density
operators on $H$, i.e., the trace class positive operators on $H$ of unit
trace, which represent the states of quantum system. A self-adjoint operator
$A$ on $H$ such that $0\leq A\leq I$ is called a quantum effect ([1, 2]), the
set of quantum effects on $H$ is denoted by ${\cal E}(H)$. The set of
orthogonal projection operators on $H$ is denoted by ${\cal P}(H)$. For each
$P\in{\cal P}(H)$ is associated a so-called Lüders transformation
$\Phi_{L}^{P}:{\cal D}(H)\rightarrow{\cal D}(H)$ such that for each $T\in{\cal
D}(H)$, $\Phi_{L}^{P}(T)=PTP$. Moreover, each quantum effect $B\in{\cal E}(H)$
gives also to a general Lüders transformation $\Phi_{L}^{B}$ such that for
each $T\in{\cal D}(H)$, $\Phi_{L}^{B}(T)=B^{\frac{1}{2}}TB^{\frac{1}{2}}$
([3-4]).
Let $B,C\in{\cal E}(H)$ be two quantum effects. It is easy to prove that the
composition $\Phi_{L}^{B}\circ\Phi_{L}^{C}$ satisfies that for each $T\in{\cal
D}(H)$,
($\Phi_{L}^{B}\circ\Phi_{L}^{C})(T)=(B^{\frac{1}{2}}CB^{\frac{1}{2}})^{\frac{1}{2}}T(B^{\frac{1}{2}}CB^{\frac{1}{2}})^{\frac{1}{2}}$
([4]). Professor Gudder called $B^{\frac{1}{2}}CB^{\frac{1}{2}}$ the
sequential product of $B$ and $C$, and denoted it by $B\circ C$ ([5-7]). This
sequential product has been generalized to an algebraic structure called a
sequential effect algebra ([8]). Now, we state the basic definitions and
results of sequential effect algebras.
An effect algebra is a system $(E,0,1,\oplus)$, where 0 and 1 are distinct
elements of $E$ and $\oplus$ is a partial binary operation on $E$ satisfying
that [9]:
(EA1). If $a\oplus b$ is defined, then $b\oplus a$ is defined and $b\oplus
a=a\oplus b$.
(EA2). If $a\oplus(b\oplus c)$ is defined, then $(a\oplus b)\oplus c$ is
defined and
$(a\oplus b)\oplus c=a\oplus(b\oplus c).$
(EA3). For each $a\in E$, there exists a unique element $b\in E$ such that
$a\oplus b=1$.
(EA4). If $a\oplus 1$ is defined, then $a=0$.
In an effect algebra $(E,0,1,\oplus)$, if $a\oplus b$ is defined, we write
$a\bot b$. For each $a\in(E,0,1,\oplus)$, it follows from (EA3) that there
exists a unique element $b\in E$ such that $a\oplus b=1$, we denote $b$ by
$a^{\prime}$. Let $a,b\in(E,0,1,\oplus)$, if there exists a $c\in E$ such that
$a\bot c$ and $a\oplus c=b$, then we say that $a\leq b$, if in addition,
$a\neq b$, then we write $a<b$. It follows from [9] that $\leq$ is a partial
order of $(E,0,1,\oplus)$ and satisfies that for each $a\in E$, $0\leq a\leq
1$, $a\bot b$ if and only if $a\leq b^{\prime}$.
A sequential effect algebra is an effect algebra $(E,0,1,\oplus)$ and another
binary operation $\circ$ defined on $(E,0,1,\oplus)$ satisfying that [8]:
(SEA1). The map $b\mapsto a\circ b$ is additive for each $a\in E$, that is, if
$b\bot c$, then $a\circ b\bot a\circ c$ and $a\circ(b\oplus c)=a\circ b\oplus
a\circ c$.
(SEA2). $1\circ a=a$ for each $a\in E$.
(SEA3). If $a\circ b=0$, then $a\circ b=b\circ a$.
(SEA4). If $a\circ b=b\circ a$, then $a\circ b^{\prime}=b^{\prime}\circ a$ and
$a\circ(b\circ c)=(a\circ b)\circ c$ for each $c\in E$.
(SEA5). If $c\circ a=a\circ c$ and $c\circ b=b\circ c$, then $c\circ(a\circ
b)=(a\circ b)\circ c$ and $c\circ(a\oplus b)=(a\oplus b)\circ c$ whenever
$a\bot b$.
Let $(E,0,1,\oplus,\circ)$ be a sequential effect algebra. Then the operation
$\circ$ is said to be a sequential product on $(E,0,1,\oplus,\circ)$. If
$a,b\in(E,0,1,\oplus,\circ)$ and $a\circ b=b\circ a$, then $a$ and $b$ is said
to be sequentially independent and write $a|b$ ([8]). Let
$a\in(E,0,1,\oplus,\circ)$. If there exists an element
$b\in(E,0,1,\oplus,\circ)$ such that $\underbrace{b\circ b\circ\cdots\circ
b}\limits_{the\ number\ is\ n}=a$, then we write $b^{n}=a$ and $b$ is said to
be a n-th root of $a$. Note that $b$ is a n-th root of $a$ implies that $a$
can be obtained by measuring $b$ n-times repeatedly.
The sequential effect algebra is an important and interesting mathematical
model for studying the quantum measurement theory [5-8]. In [10], Professor
Gudder presented 25 open problems to motivate the study of sequential effect
algebra theory. The 20th problem asked:
Problem 1 ([10]). In a sequential effect algebra $(E,0,1,\oplus,\circ)$, if
the square root of some element exists, is it unique ?
Now, we can strengthen Problem 1 as following:
Problem 2. For each given positive integer $n>1$, is there a sequential effect
algebra $(E,0,1,\oplus,\circ)$ such that the n-th root of its some element $c$
is not unique and the n-th root of $c$ is not the k-th root of $c$ ($k<n$) ?
i.e., are there $a,b\in E$, such that $a\neq b$, $a^{n}=c=b^{n}$ and
$a^{k}\neq c$, $b^{k}\neq c$ for $k<n$ ?
In this paper, we present an example to answer Problem 2 affirmatively.
Actually, we will construct a sequential effect algebra $E_{0}$, such that
there are elements $a,b,c\in E_{0}$ having the relations
$a>a^{2}>\cdots>a^{n},$ $b>b^{2}>\cdots>b^{n},$ $a^{k}\neq b^{k}\ for\ k<n\ ,\
a^{n}=b^{n}=c\neq 0.$
In order to construct our example, we need some preliminary steps:
Suppose $Z$ be the integer set, $n>1$ be a given positive integer.
Let $p(x)=\sum\limits_{i=1}^{n-1}k_{i}x^{i}$, where $k_{i}\in Z$, $k_{i}\equiv
0$ or the first nonzero $k_{i}>0$, we denote all the polynomials characterized
above by $I_{0}$ .
Suppose $p_{1},p_{2}\in I_{0}$ and
$p_{1}(x)=\sum\limits_{i=1}^{n-1}k_{1,i}x^{i}$ ,
$p_{2}(x)=\sum\limits_{i=1}^{n-1}k_{2,i}x^{i}$ , let
$F(p_{1},p_{2})(x)=\sum\limits_{i+j\leq n-1}k_{1,i}k_{2,j}x^{i+j}$ ,
$G(p_{1},p_{2})=\sum\limits_{i+j=n}k_{1,i}k_{2,j}$ . Then it is easy to see
that $F(p_{1},p_{2})\in I_{0}$ and $G(p_{1},p_{2})\in Z$ .
Thus we defined mappings
$F:I_{0}\times I_{0}\longrightarrow I_{0}$ and $G:I_{0}\times
I_{0}\longrightarrow Z$ .
Moreover, suppose $p_{1},p_{2},p_{3}\in I_{0}$ and
$p_{1}(x)=\sum\limits_{i=1}^{n-1}k_{1,i}x^{i}$ ,
$p_{2}(x)=\sum\limits_{i=1}^{n-1}k_{2,i}x^{i}$ ,
$p_{3}(x)=\sum\limits_{i=1}^{n-1}k_{3,i}x^{i}$ , let
$\overline{F}(p_{1},p_{2},p_{3})(x)=\sum\limits_{i+j+m\leq
n-1}k_{1,i}k_{2,j}k_{3,m}x^{i+j+m}$ ,
$\overline{G}(p_{1},p_{2},p_{3})=\sum\limits_{i+j+m=n}k_{1,i}k_{2,j}k_{3,m}$ .
Then it is also easy to see that $\overline{F}(p_{1},p_{2},p_{3})\in I_{0}$
and $\overline{G}(p_{1},p_{2},p_{3})\in Z$ . Thus we defined mappings
$\overline{F}:I_{0}\times I_{0}\times I_{0}\longrightarrow I_{0}$ and
$\overline{G}:I_{0}\times I_{0}\times I_{0}\longrightarrow Z$ .
Lemma 1. Suppose $p,p_{1},p_{2},p_{3}\in I_{0}$, we have
(1). $F(p_{1},p_{2})=F(p_{2},p_{1})$, $G(p_{1},p_{2})=G(p_{2},p_{1})$;
(2). $F(p_{1},p_{2}+p_{3})=F(p_{1},p_{2})+F(p_{1},p_{3})$,
$G(p_{1},p_{2}+p_{3})=G(p_{1},p_{2})+G(p_{1},p_{3})$;
(3). $F(0,p)=0$, $G(0,p)=0$;
(4). if $F(p_{1},p_{2})=0$, then $G(p_{1},p_{2})\geq 0$;
(5). $p_{1}-F(p_{1},p_{2})\in I_{0}$, and
$p_{1}=F(p_{1},p_{2})\Longleftrightarrow p_{1}=0$;
(6). $F(F(p_{1},p_{2}),p_{3})=\overline{F}(p_{1},p_{2},p_{3})$,
$G(F(p_{1},p_{2}),p_{3})=\overline{G}(p_{1},p_{2},p_{3})$;
(7). $p_{1}+p_{2}\in I_{0}$, and $p_{1}+p_{2}=0\Longleftrightarrow
p_{1}=p_{2}=0$.
Proof. (1),(2),(3),(6) and (7) are trivial.
(4). Except for the trivial cases, we may suppose
$p_{1}(x)=\sum\limits_{i=n_{1}}^{n-1}k_{1,i}x^{i}$,
$p_{2}(x)=\sum\limits_{i=n_{2}}^{n-1}k_{2,i}x^{i}$, with $k_{1,n_{1}}>0$ and
$k_{2,n_{2}}>0$. Then from $F(p_{1},p_{2})=0$ we have $n_{1}+n_{2}\geq n$. If
$n_{1}+n_{2}=n$, then $G(p_{1},p_{2})=k_{1,n_{1}}k_{2,n_{2}}>0$; otherwise
$n_{1}+n_{2}>n$ and $G(p_{1},p_{2})=0$.
(5). Except for the trivial cases, we may suppose
$p_{1}(x)=\sum\limits_{i=n_{1}}^{n-1}k_{1,i}x^{i}$,
$p_{2}(x)=\sum\limits_{i=n_{2}}^{n-1}k_{2,i}x^{i}$, with $k_{1,n_{1}}>0$ and
$k_{2,n_{2}}>0$. Then the first item of $p_{1}-F(p_{1},p_{2})$ is
$k_{1,n_{1}}x^{n_{1}}$, so $p_{1}-F(p_{1},p_{2})\in I_{0}$. If $p_{1}\neq 0$,
then from the above reason we know that $p_{1}-F(p_{1},p_{2})\neq 0$. Thus,
the lemma is proved.
Now, we take two infinite sets $U$ and $V$ such that $U\cap V=\emptyset$. Let
$f:I_{0}\times I_{0}\times Z\rightarrow U$ and $g:I_{0}\times I_{0}\times
Z\rightarrow V$ be two one to one maps. Then, we construct our example as
following:
Let $E_{0}=\\{f(p,q,m),g(p,q,m)|p,q\in I_{0},m\in Z\ and\ satisfy\ that\ m\geq
0\ whenever\ p=q=0\\}$.
First, we define a partial binary operation $\oplus$ on $E_{0}$ as follows
(when we write $x\oplus y=z$, we always mean that $x\oplus y=z=y\oplus x$):
(i). $f(p_{1},q_{1},m_{1})\oplus
f(p_{2},q_{2},m_{2})=f(p_{1}+p_{2},q_{1}+q_{2},m_{1}+m_{2})$ (the right side
is well-defined, see Lemma 1(7));
(ii). for $p_{2}-p_{1}\in I_{0}$, $q_{2}-q_{1}\in I_{0}$, and satisfy that
$m_{2}\geq m_{1}$ when $p_{2}=p_{1}$ and $q_{2}=q_{1}$,
$f(p_{1},q_{1},m_{1})\oplus
g(p_{2},q_{2},m_{2})=g(p_{2}-p_{1},q_{2}-q_{1},m_{2}-m_{1})$ .
No other $\oplus$ operation is defined.
Next, we define a binary operation $\circ$ on $E_{0}$ as follows (when we
write $x\circ y=z$, we always mean that $x\circ y=z=y\circ x$):
(i). $f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})=f\Big{(}F(p_{1},p_{2}),F(q_{1},q_{2}),G(p_{1},p_{2})+G(q_{1},q_{2})\Big{)}$
(the right side is well-defined, see Lemma 1(4));
(ii). $f(p_{1},q_{1},m_{1})\circ
g(p_{2},q_{2},m_{2})=f\Big{(}p_{1}-F(p_{1},p_{2}),q_{1}-F(q_{1},q_{2}),m_{1}-G(p_{1},p_{2})-G(q_{1},q_{2})\Big{)}$
(the right side is well-defined, see Lemma 1(3), (5));
(iii). $g(p_{1},q_{1},m_{1})\circ
g(p_{2},q_{2},m_{2})=g\Big{(}p_{1}+p_{2}-F(p_{1},p_{2}),q_{1}+q_{2}-F(q_{1},q_{2}),m_{1}+m_{2}-G(p_{1},p_{2})-G(q_{1},q_{2})\Big{)}$
(the right side is well-defined, see Lemma 1(3), (5), (7)).
We denote $f(0,0,0)$ by $0$, $g(0,0,0)$ by $1$.
Proposition 1. $(E_{0},0,1,\oplus,\circ)$ is a sequential effect algebra.
Proof. In the proof below, we will use Lemma 1 frequently without annotation.
First, we verify that $(E_{0},0,1,\oplus)$ is an effect algebra.
(EA1) is obvious. We verify (EA2) as follows:
(i). $f(p_{1},q_{1},m_{1})\oplus\Big{(}f(p_{2},q_{2},m_{2})\oplus
f(p_{3},q_{3},m_{3})\Big{)}=\Big{(}f(p_{1},q_{1},m_{1})\oplus
f(p_{2},q_{2},m_{2})\Big{)}\oplus
f(p_{3},q_{3},m_{3})=f(p_{1}+p_{2}+p_{3},q_{1}+q_{2}+q_{3},m_{1}+m_{2}+m_{3})$;
(ii). $f(p_{1},q_{1},m_{1})\oplus\Big{(}f(p_{2},q_{2},m_{2})\oplus
g(p_{3},q_{3},m_{3})\Big{)}$ or $\Big{(}f(p_{1},q_{1},m_{1})\oplus
f(p_{2},q_{2},m_{2})\Big{)}\oplus g(p_{3},q_{3},m_{3})$ is defined iff
$p_{3}-p_{1}-p_{2}\in I_{0}$, $q_{3}-q_{1}-q_{2}\in I_{0}$ and satisfy that
$m_{3}\geq m_{1}+m_{2}$ when $p_{3}=p_{1}+p_{2}$ and $q_{3}=q_{1}+q_{2}$, at
this point, they all equal to
$g(p_{3}-p_{1}-p_{2},q_{3}-q_{1}-q_{2},m_{3}-m_{1}-m_{2})$.
Note that $f(p,q,m)\oplus g(p,q,m)=g(0,0,0)=1$, we verified (EA3).
For (EA4), we note from our construction that the unique element orthogonal to
$g(0,0,0)(=1)$ is $f(0,0,0)(=0)$, that is, $f(0,0,0)\bot g(0,0,0)$ and
$f(0,0,0)\oplus g(0,0,0)=g(0,0,0)$.
So far, we have proved that $(E_{0},0,1,\oplus)$ is an effect algebra.
Next, we verify that $(E_{0},0,1,\oplus,\circ)$ is a sequential effect
algebra.
(SEA3) and (SEA5) are obvious.
We verify (SEA1) as follows:
(i). $f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\oplus
f(p_{3},q_{3},m_{3})\Big{)}=f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\oplus f(p_{1},q_{1},m_{1})\circ
f(p_{3},q_{3},m_{3})=f\Big{(}F(p_{1},p_{2}+p_{3}),F(q_{1},q_{2}+q_{3}),G(p_{1},p_{2}+p_{3})+G(q_{1},q_{2}+q_{3})\Big{)}$,
$g(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\oplus
f(p_{3},q_{3},m_{3})\Big{)}=g(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\oplus g(p_{1},q_{1},m_{1})\circ
f(p_{3},q_{3},m_{3})=f\Big{(}p_{2}+p_{3}-F(p_{1},p_{2}+p_{3}),q_{2}+q_{3}-F(q_{1},q_{2}+q_{3}),m_{2}+m_{3}-G(p_{1},p_{2}+p_{3})-G(q_{1},q_{2}+q_{3})\Big{)}$;
(ii). when $f(p_{2},q_{2},m_{2})\oplus g(p_{3},q_{3},m_{3})$ is defined, i.e.,
when $p_{3}-p_{2}\in I_{0}$, $q_{3}-q_{2}\in I_{0}$, and satisfy that
$m_{3}\geq m_{2}$ if $p_{3}=p_{2}$ and $q_{3}=q_{2}$ ,
$f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\oplus
g(p_{3},q_{3},m_{3})\Big{)}=f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\oplus f(p_{1},q_{1},m_{1})\circ
g(p_{3},q_{3},m_{3})=f\Big{(}p_{1}-F(p_{1},p_{3}-p_{2}),q_{1}-F(q_{1},q_{3}-q_{2}),m_{1}-G(p_{1},p_{3}-p_{2})-G(q_{1},q_{3}-q_{2})\Big{)}$,
$g(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\oplus
g(p_{3},q_{3},m_{3})\Big{)}=g(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\oplus g(p_{1},q_{1},m_{1})\circ
g(p_{3},q_{3},m_{3})=g\Big{(}p_{1}+p_{3}-p_{2}-F(p_{1},p_{3}-p_{2}),q_{1}+q_{3}-q_{2}-F(q_{1},q_{3}-q_{2}),m_{1}+m_{3}-m_{2}-G(p_{1},p_{3}-p_{2})-G(q_{1},q_{3}-q_{2})\Big{)}$.
We verify (SEA2) as follows:
$1\circ f(p,q,m)=g(0,0,0)\circ f(p,q,m)=f(p,q,m);$ $1\circ
g(p,q,m)=g(0,0,0)\circ g(p,q,m)=g(p,q,m).$
We verify (SEA4) as follows:
(i). $f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\circ
f(p_{3},q_{3},m_{3})\Big{)}$
$=f(p_{1},q_{1},m_{1})\circ
f\Big{(}F(p_{2},p_{3}),F(q_{2},q_{3}),G(p_{2},p_{3})+G(q_{2},q_{3})\Big{)}$
$=f\Big{(}F(p_{1},F(p_{2},p_{3})),F(q_{1},F(q_{2},q_{3})),G(p_{1},F(p_{2},p_{3}))+G(q_{1},F(q_{2},q_{3}))\Big{)}$
$=f\Big{(}\overline{F}(p_{1},p_{2},p_{3}),\overline{F}(q_{1},q_{2},q_{3}),\overline{G}(p_{1},p_{2},p_{3})+\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
by symmetry,
$\Big{(}f(p_{1},q_{1},m_{1})\circ f(p_{2},q_{2},m_{2})\Big{)}\circ
f(p_{3},q_{3},m_{3})$
$=f(p_{3},q_{3},m_{3})\circ\Big{(}f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\Big{)}$
$=f\Big{(}\overline{F}(p_{1},p_{2},p_{3}),\overline{F}(q_{1},q_{2},q_{3}),\overline{G}(p_{1},p_{2},p_{3})+\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
so we have
$f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\circ
f(p_{3},q_{3},m_{3})\Big{)}=\Big{(}f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\Big{)}\circ f(p_{3},q_{3},m_{3})$.
(ii). $f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}$
$=f(p_{1},q_{1},m_{1})\circ
f\Big{(}p_{2}-F(p_{2},p_{3}),q_{2}-F(q_{2},q_{3}),m_{2}-G(p_{2},p_{3})-G(q_{2},q_{3})\Big{)}$
$=f\Big{(}F(p_{1},p_{2}-F(p_{2},p_{3})),F(q_{1},q_{2}-F(q_{2},q_{3})),G(p_{1},p_{2}-F(p_{2},p_{3}))+G(q_{1},q_{2}-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}F(q_{2},q_{3}))\Big{)}$
$=f\Big{(}F(p_{1},p_{2})-F(p_{1},F(p_{2},p_{3})),F(q_{1},q_{2})-F(q_{1},F(q_{2},q_{3})),G(p_{1},p_{2})-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(p_{1},F(p_{2},p_{3}))+G(q_{1},q_{2})-G(q_{1},F(q_{2},q_{3}))\Big{)}$
$=f\Big{(}F(p_{1},p_{2})-\overline{F}(p_{1},p_{2},p_{3}),F(q_{1},q_{2})-\overline{F}(q_{1},q_{2},q_{3}),G(p_{1},p_{2})-\overline{G}(p_{1},p_{2},p_{3})+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(q_{1},q_{2})-\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
$\Big{(}f(p_{1},q_{1},m_{1})\circ f(p_{2},q_{2},m_{2})\Big{)}\circ
g(p_{3},q_{3},m_{3})$
$=f\Big{(}F(p_{1},p_{2}),F(q_{1},q_{2}),G(p_{1},p_{2})+G(q_{1},q_{2})\Big{)}\circ
g(p_{3},q_{3},m_{3})$
$=f\Big{(}F(p_{1},p_{2})-F(F(p_{1},p_{2}),p_{3}),F(q_{1},q_{2})-F(F(q_{1},q_{2}),q_{3}),G(p_{1},p_{2})+G(q_{1},q_{2})-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(F(p_{1},p_{2}),p_{3})-G(F(q_{1},q_{2}),q_{3})\Big{)}$
$=f\Big{(}F(p_{1},p_{2})-\overline{F}(p_{1},p_{2},p_{3}),F(q_{1},q_{2})-\overline{F}(q_{1},q_{2},q_{3}),G(p_{1},p_{2})-\overline{G}(p_{1},p_{2},p_{3})+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(q_{1},q_{2})-\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
so we have
$f(p_{1},q_{1},m_{1})\circ\Big{(}f(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}=\Big{(}f(p_{1},q_{1},m_{1})\circ
f(p_{2},q_{2},m_{2})\Big{)}\circ g(p_{3},q_{3},m_{3})$ .
(iii). $f(p_{1},q_{1},m_{1})\circ\Big{(}g(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}$
$=f(p_{1},q_{1},m_{1})\circ
g\Big{(}p_{2}+p_{3}-F(p_{2},p_{3}),q_{2}+q_{3}-F(q_{2},q_{3}),m_{2}+m_{3}-G(p_{2},p_{3})-G(q_{2},q_{3})\Big{)}$
$=f\Big{(}p_{1}-F(p_{1},p_{2}+p_{3}-F(p_{2},p_{3})),q_{1}-F(q_{1},q_{2}+q_{3}-F(q_{2},q_{3})),m_{1}-G(p_{1},p_{2}+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p_{3}-F(p_{2},p_{3}))-G(q_{1},q_{2}+q_{3}-F(q_{2},q_{3}))\Big{)}$
$=f\Big{(}p_{1}-F(p_{1},p_{2}+p_{3})+\overline{F}(p_{1},p_{2},p_{3}),q_{1}-F(q_{1},q_{2}+q_{3})+\overline{F}(q_{1},q_{2},q_{3}),m_{1}-G(p_{1},p_{2}+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p_{3})+\overline{G}(p_{1},p_{2},p_{3})-G(q_{1},q_{2}+q_{3})+\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
$\Big{(}f(p_{1},q_{1},m_{1})\circ g(p_{2},q_{2},m_{2})\Big{)}\circ
g(p_{3},q_{3},m_{3})$
$=f\Big{(}p_{1}-F(p_{1},p_{2}),q_{1}-F(q_{1},q_{2}),m_{1}-G(p_{1},p_{2})-G(q_{1},q_{2})\Big{)}\circ
g(p_{3},q_{3},m_{3})$
$=f\Big{(}p_{1}-F(p_{1},p_{2})-F(p_{1}-F(p_{1},p_{2}),p_{3}),q_{1}-F(q_{1},q_{2})-F(q_{1}-F(q_{1},q_{2}),q_{3}),m_{1}-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(p_{1},p_{2})-G(q_{1},q_{2})-G(p_{1}-F(p_{1},p_{2}),p_{3})-G(q_{1}-F(q_{1},q_{2}),q_{3})\Big{)}$
$=f\Big{(}p_{1}-F(p_{1},p_{2}+p_{3})+\overline{F}(p_{1},p_{2},p_{3}),q_{1}-F(q_{1},q_{2}+q_{3})+\overline{F}(q_{1},q_{2},q_{3}),m_{1}-G(p_{1},p_{2}+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p_{3})+\overline{G}(p_{1},p_{2},p_{3})-G(q_{1},q_{2}+q_{3})+\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
so we have
$f(p_{1},q_{1},m_{1})\circ\Big{(}g(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}=\Big{(}f(p_{1},q_{1},m_{1})\circ
g(p_{2},q_{2},m_{2})\Big{)}\circ g(p_{3},q_{3},m_{3})$.
(iv). $g(p_{1},q_{1},m_{1})\circ\Big{(}g(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}$
$=g(p_{1},q_{1},m_{1})\circ
g\Big{(}p_{2}+p_{3}-F(p_{2},p_{3}),q_{2}+q_{3}-F(q_{2},q_{3}),m_{2}+m_{3}-G(p_{2},p_{3})-G(q_{2},q_{3})\Big{)}$
$=g\Big{(}p_{1}+p_{2}+p_{3}-F(p_{2},p_{3})-F(p_{1},p_{2}+p_{3}-F(p_{2},p_{3})),q_{1}+q_{2}+q_{3}-F(q_{2},q_{3})-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}F(q_{1},q_{2}+q_{3}-F(q_{2},q_{3})),m_{1}+m_{2}+m_{3}-G(p_{2},p_{3})-G(q_{2},q_{3})-G(p_{1},p_{2}+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p_{3}-F(p_{2},p_{3}))-G(q_{1},q_{2}+q_{3}-F(q_{2},q_{3}))\Big{)}$
$=g\Big{(}p_{1}+p_{2}+p_{3}-F(p_{2},p_{3})-F(p_{1},p_{2})-F(p_{1},p_{3})+\overline{F}(p_{1},p_{2},p_{3}),q_{1}+q_{2}+q_{3}-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}F(q_{2},q_{3})-F(q_{1},q_{2})-F(q_{1},q_{3})+\overline{F}(q_{1},q_{2},q_{3}),m_{1}+m_{2}+m_{3}-G(p_{2},p_{3})-\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}G(p_{1},p_{2})-G(p_{1},p_{3})+\overline{G}(p_{1},p_{2},p_{3})-G(q_{2},q_{3})-G(q_{1},q_{2})-G(q_{1},q_{3})+\linebreak~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\overline{G}(q_{1},q_{2},q_{3})\Big{)}$,
by symmetry, we have
$g(p_{1},q_{1},m_{1})\circ\Big{(}g(p_{2},q_{2},m_{2})\circ
g(p_{3},q_{3},m_{3})\Big{)}=\Big{(}g(p_{1},q_{1},m_{1})\circ
g(p_{2},q_{2},m_{2})\Big{)}\circ g(p_{3},q_{3},m_{3})$.
Thus, we proved that $(E_{0},0,1,\oplus,\circ)$ is a sequential effect algebra
and the theorem is proved.
Now, let $P_{i}(x)=x^{i}$. Then it is easy to see that
$F(P_{1},P_{j})=\left\\{\begin{array}[]{ll}P_{1+j}\ ,&\hbox{$if\ j<n-1$;}\\\
0\ ,&\hbox{$if\
j=n-1$.}\end{array}\right.~{}and~{}~{}G(P_{1},P_{j})=\left\\{\begin{array}[]{ll}0\
,&\hbox{$if\ j<n-1$;}\\\ 1\ ,&\hbox{$if\ j=n-1$.}\end{array}\right.$
Thus we have
$[f(P_{1},0,0)]^{k}=f(P_{1},0,0)\circ f(P_{k-1},0,0)=f(P_{k},0,0)$ for $k<n$,
$[f(P_{1},0,0)]^{n}=f(P_{1},0,0)\circ f(P_{n-1},0,0)=f(0,0,1)$,
$[f(P_{1},0,0)]^{n+1}=f(P_{1},0,0)\circ f(0,0,1)=0$,
and
$[f(0,P_{1},0)]^{k}=f(0,P_{1},0)\circ f(0,P_{k-1},0)=f(0,P_{k},0)$ for $k<n$,
$[f(0,P_{1},0)]^{n}=f(0,P_{1},0)\circ f(0,P_{n-1},0)=f(0,0,1)$,
$[f(0,P_{1},0)]^{n+1}=f(0,P_{1},0)\circ f(0,0,1)=0$.
If we denote $f(P_{1},0,0)$ by $a$, $f(0,P_{1},0)$ by $b$, $f(0,0,1)$ by $c$,
then it is easy to get the relations
$a>a^{2}>\cdots>a^{n}>a^{n+1},$ $b>b^{2}>\cdots>b^{n}>b^{n+1},$ $a^{k}\neq
b^{k}\ for\ k<n\ ,\ a^{n}=b^{n}=c\neq 0\ and\ a^{n+1}=b^{n+1}=0.$
That is, $a,b$ are the n-th root of $c$, but $a,b$ are not the k-th root of
$c$, where $k=2,3,\cdots,n-1$, moreover, $a,b$ are also the n+1-th root of
$0$, so, the Problem 2 is answered affirmatively.
Finally, we would like to point out that for the advances of sequential effect
algebras, see [11-16].
Acknowledgement
The authors wish to express their thanks to the referee for his valuable
comments and suggestions.
References
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1983.
[2]. Ludwig, G. An Axiomatic Basis for Quantum Mechanics (II), Springer, New
York, 1986.
[3]. Davies, E. B. Quantum Theory of Open Systems, Academic Press, London,
1976.
[4]. Busch, P, Grabowski, M and Lahti P. J, Operational Quantum Physics,
Springer-Verlag, Beijing Word Publishing Corporation, 1999.
[5]. Gudder, S, Nagy, G. Sequential quantum measurements. J. Math. Phys.
42(2001), 5212-5222.
[6]. Gheondea, A, Gudder, S. Sequential product of quantum effects. Proc.
Amer. Math. Soc. 132 (2004), 503-512.
[7]. Gudder, S, Latr moli re, F. Characterization of the sequential product on
quantum effects. J. Math. Phys. 49 (2008), 052106-052112.
[8]. Gudder, S, Greechie, R. Sequential products on effect algebras. Rep.
Math. Phys. 49(2002), 87-111.
[9]. Foulis, D J, Bennett, M K. Effect algebras and unsharp quantum logics.
Found Phys 24 (1994), 1331-1352.
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Theory. Physi. 44 (2005), 2219-2230.
[11] Shen Jun and Wu Junde. Not each sequential effect algebra is sharply
dominating. Phys. Letter A. 373, 1708-1712, (2009)
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Math. Phys. 63, 441-446, (2009)
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${\cal E}(H)$. J. Phys. A: Math. Theor. 44, 345203-345214, (2009)
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185206-185215, (2009)
[16] Liu Weihua and Wu Junde. On fixed points of Lüders operation. J. Math.
Phys. 50, 103531-103532, (2009)
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By (10.15), $\left(\int_{A}\phi_{x_{0}}^{2}dm\right)^{\frac{1}{2}}\leq
Cn^{\frac{\psi}{2}\left(\frac{2}{\alpha}-1\right)}$ and by Proposition 2.6,
$\left(\int f_{n}^{2}dm\right)^{\frac{1}{2}}\leq
Cn^{\frac{1}{\alpha}-\frac{1}{2}}$. Hence we may bound (10.16) by
$Cn^{\left(1+\psi\right)\left(\frac{1}{\alpha}-\frac{1}{2}\right)}$.
To bound (II), let $B=U_{n}\cap(T_{\sigma^{i}\omega}^{j-i})^{-1}(U_{n}^{c})$.
Then,
(10.18)
$\displaystyle\int_{U_{n}\cap(T_{\sigma^{i}\omega}^{j-i})^{-1}(U_{n}^{c})}f_{n}\cdot
f_{n}\circ T_{\sigma^{i}\omega}^{j-i}dm\leq\int_{B}f_{n}\cdot\phi_{x_{0}}\circ
T_{\sigma^{i}\omega}^{j-i}dm\leq\left(\int
f_{n}^{2}dm\right)^{\frac{1}{2}}\left(\int_{B}\phi_{x_{0}}^{2}\circ
T_{\sigma^{i}\omega}^{j-i}dm\right)^{\frac{1}{2}}.$
As before $\left(\int f_{n}^{2}dm\right)^{\frac{1}{2}}\leq
Cn^{\frac{1}{\alpha}-\frac{1}{2}}$ and
$\displaystyle\left(\int_{B}\phi_{x_{0}}^{2}\circ
T_{\sigma^{i}\omega}^{j-i}dm\right)^{\frac{1}{2}}$
$\displaystyle\leq\left(\int\phi_{x_{0}}^{2}\circ
T_{\sigma^{i}\omega}^{j-i}\mathbf{1}_{(T_{\sigma^{i}\omega}^{j-i})^{-1}(U_{n}^{c})}dm\right)^{\frac{1}{2}}\leq
C\left(\int_{U_{n}^{c}}\phi_{x_{0}}^{2}dm\right)^{\frac{1}{2}}\leq
Cn^{\frac{\psi}{2}\left(\frac{2}{\alpha}-1\right)}$
by (10.15), and so (10.18) is bounded by
$Cn^{\left(1+\psi\right)\left(\frac{1}{\alpha}-\frac{1}{2}\right)}$.
It follows that ${\rm(II)}+{\rm(III)}\leq C(\log
n)n^{1+\left(1+\psi\right)\left(\frac{1}{\alpha}-\frac{1}{2}\right)}=o(n^{\frac{2}{\alpha}})$,
since $\psi<1$. This proves that (10.14) is a $o(b_{n}^{2})$ and concludes the
proof of (10.13).
Finally, from (10.11), (10.12) and (10.13), we obtain
(10.19) $\displaystyle\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\frac{1}{b_{n}^{2}}{\mathbb{E}}_{\nu^{\omega}}[(S_{\omega,n,n})^{2}]=0,$
which gives the result by taking the limit first in $n$ and then in
$\varepsilon$ in (10.10). ∎
### 10.2. Intermittent maps
We prove convergence to a stable law in the setting of Example 5.12 when
$\alpha\in(0,1)$.
###### Proof of Theorem 6.9.
We apply Proposition 5.8. By Theorem 6.6, it remains to prove (5.7), since
$\alpha\in(0,1)$. We will need an estimate for
${\mathbb{E}}_{\nu^{\omega}}(|\phi_{x_{0}}|\mathbf{1}_{\left\\{\phi_{x_{0}}\leq\varepsilon
b_{n}\right\\}})$ which is independent of $\omega$. For this purpose, we
introduce the absolutely continuous probability measure $\nu_{\rm max}$ whose
density is given by $h_{\rm max}(x)=\kappa x^{-\gamma_{\rm max}}$. Since all
densities $h_{\omega}$ belong to the cone $L$, we have that
$h_{\omega}\leq\frac{a}{\kappa}h_{\rm max}$ for all $\omega$. Thus,
$\frac{1}{b_{n}}\sum_{j=0}^{n-1}{\mathbb{E}}_{\nu^{\sigma^{j}\omega}}(\phi_{x_{0}}\mathbf{1}_{\left\\{|\phi_{x_{0}}|\leq\varepsilon
b_{n}\right\\}})\leq\frac{n}{b_{n}}\frac{a}{\kappa}{\mathbb{E}}_{\nu_{\rm
max}}(\phi_{x_{0}}{\mathbf{1}}_{\left\\{|\phi_{x_{0}}|\leq\varepsilon
b_{n}\right\\}}).$
We can easily verify that $\phi_{x_{0}}$ is regularly varying of index
$\alpha$ with respect to $\nu_{\rm max}$, with scaling sequence equal to
$(b_{n})_{n\geq 1}$ up to a multiplicative constant factor. Consequently, by
Proposition 2.6, we have that, for some constant $c>0$,
${\mathbb{E}}_{\nu_{\rm
max}}(\phi_{x_{0}}\mathbf{1}_{\left\\{|\phi_{x_{0}}|\leq\varepsilon
b_{n}\right\\}})\sim c\varepsilon^{1-\alpha}n^{\frac{1}{\alpha}-1},$
which implies (5.7). ∎
## 11\. The annealed case
In this section, we consider the annealed counterparts of our results. Even
though the annealed versions do not seem to follow immediately from the
quenched version, it is easy to obtain them from our proofs in the quenched
case. We take $\phi_{x_{0}}(x)=d(x,x_{0})^{-\frac{1}{\alpha}}$ as before we
consider the convergence on the measure space $\Omega\times[0,1]$ with respect
to $\nu_{F}(d\omega,dx)={\mathbb{P}}(d\omega)\nu^{\omega}(dx)$. We give
precise annealed results in the case of Theorems 6.7 and 6.9, where we
consider
$X^{a}_{n}(\omega,x)(t):=\frac{1}{b_{n}}\sum_{j=0}^{\lfloor
nt\rfloor-1}\phi_{x_{0}}(T_{\omega}^{j}x)-tc_{n},\>t\geq 0,$
viewed as a random process defined on the probability space
$(\Omega\times[0,1],\nu_{F})$.
###### Theorem 11.1.
Under the same assumptions as Theorem 6.7, the random process $X^{a}_{n}(t)$
converges in the $J_{1}$ topology to the Lévy $\alpha$-stable process
$X_{(\alpha)}(t)$ under the probability measure $\nu_{F}$.
###### Proof.
We apply [TK10b, Theorem 1.2] to the skew-product system
$(\Omega\times[0,1],F,\nu_{F})$ and the observable $\phi_{x_{0}}$ naturally
extended to $\Omega\times[0,1]$. Recall that $\nu_{F}$ is given by the
disintegration $\nu_{F}(d\omega,dx)={\mathbb{P}}(d\omega)\nu^{\omega}(dx)$.
We have to prove that
1. (a)
$N_{n}\,{\stackrel{{\scriptstyle d}}{{\to}}\,}N_{(\alpha)}$,
2. (b)
if $\alpha\in[1,2)$, for all $\delta>0$,
$\lim_{\varepsilon\to
0}\limsup_{n\to\infty}\nu_{F}\left((\omega,x)\,:\,\max_{1\leq k\leq
n}\left|\frac{1}{b_{n}}\sum_{j=0}^{k-1}\left[\phi_{x_{0}}(T_{\omega}^{j}x){\mathbf{1}}_{\left\\{|\phi_{x_{0}}\circ
T_{\omega}^{j}|\leq\varepsilon
b_{n}\right\\}}(x)-{\mathbb{E}}_{\nu}(\phi_{x_{0}}{\mathbf{1}}_{\left\\{|\phi_{x_{0}}|\leq\varepsilon
b_{n}\right\\}})\right]\right|\geq\delta\right)=0,$
where
$N_{n}(\omega,x)(B):=N_{n}^{\omega}(x)(B)=\\#\left\\{j\geq
1\,:\,\left(\frac{j}{n},\frac{\phi_{x_{0}}(T_{\omega}^{j-1}(x))}{b_{n}}\right)\in
B\right\\},\;n\geq 1.$
To prove (a), we take $f\in
C_{K}^{+}((0,\infty)\times({\mathbb{R}}\setminus\\{0\\}))$ arbitrary. Then, by
Theorem 6.5, we have for ${\mathbb{P}}$-a.e. $\omega$
$\lim_{n\to\infty}{\mathbb{E}}_{\nu^{\omega}}(e^{-N_{n}^{\omega}(f)})={\mathbb{E}}(e^{-N(f)}).$
Integrating with respect to ${\mathbb{P}}$ and using the dominated convergence
theorem yields
$\lim_{n\to\infty}{\mathbb{E}}_{\nu_{F}}(e^{-N_{n}(f)})={\mathbb{E}}(e^{-N(f)}),$
which proves (a).
To prove (b), we simply have to integrate with respect to ${\mathbb{P}}$ in
the estimates in the proof of Theorem 6.7, which hold uniformly in
$\omega\in\Omega$, and then to take the limits as $n\to\infty$ and
$\varepsilon\to 0$. ∎
Similarly, we have:
###### Theorem 11.2.
Under the same assumptions as Theorem 6.9,
$X_{n}^{a}(1)\,{\stackrel{{\scriptstyle d}}{{\to}}\,}X_{(\alpha)}(1)$ under
the probability measure $\nu_{F}$.
###### Proof.
We can proceed as for Theorem 11.1 in order to check the assumptions of
[TK10b, Theorem 1.3] for the skew-product system
$(\Omega\times[0,1],F,\nu_{F})$ and the observable $\phi_{x_{0}}$. ∎
## 12\. Appendix
The observation that our distributional limit theorems hold for any measures
$\mu\ll\nu^{\omega}$ follows from Theorem 1, Corollary 1 and Corollary 3 of
Zweimüller’s work [Zwe07].
Let
$S_{n}(x)=\frac{1}{b_{n}}[\sum_{j=0}^{n-1}\phi\circ T_{\omega}^{j}(x)-a_{n}].$
and suppose
$S_{n}\rightarrow_{\nu_{\omega}}Y$
where $Y$ is a Lévy random variable.
We consider first the setup of Example 5.12. We will show that for any measure
$\nu$ with density $h$ i.e. $d\nu=hdm$ in the cone $L$ of Example 5.12, in
particular Lebesgue measure $m$ with $h=1$,
$S_{n}\rightarrow_{\nu}Y$
We focus on $m$. According to [Zwe07, Theorem 1] it is enough to show that
$\int\psi(S_{n})d\nu_{\omega}-\int\psi(S_{n})dm\to 0.$
for any $\psi:{\mathbb{R}}\rightarrow{\mathbb{R}}$ which is bounded and
uniformly Lipschitz.
Fix such a $\psi$ and consider
$\int\psi(\frac{1}{b_{n}}[\sum_{j=0}^{n-1}\phi\circ
T_{\omega}^{j}(x)-a_{n}])(h_{\omega}-1)dm$
$\leq\int\psi(\frac{1}{b_{n}}[\sum_{j=0}^{n-1}\phi\circ
T_{\sigma^{k}\omega}^{j}(x)-a_{n}])P_{\omega}^{k}(h_{\omega}-1)dm$
$\leq\|\psi\|_{\infty}\|P_{\omega}^{k}(h_{\omega}-1)\|_{L^{1}(m)}.$
Since $\|P_{\omega}^{k}(h_{\omega}-1)\|_{L^{1}_{m}}\to 0$ in case of Example
5.12 and maps satisfying (LY), (Dec) and (Min) the assertion is proved. By
[Zwe07, Corollary 3], the proof for continuous time distributional limits
follows immediately.
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|
where $D_{t}=(\overline{\nabla f(y_{t})}^{\top}p_{t})^{2}$ and $y_{t}$ depends
on $\theta_{t}$. This corresponds to finding the fixed-point of $g$, so we
apply the fixed-point iteration method. Specifically, we first let
$\theta_{t}=0$, then $y_{t}=x_{t}$, and let $\theta_{t}\leftarrow
g(\theta_{t})$ (the above corresponding to Line 7 of Algorithm 4); then we
calculate $y_{t}$ again using the new value of $\theta_{t}$ (corresponding to
Line 8), and let $\theta_{t}\leftarrow g(\theta_{t})$ (corresponding to Line
9). We find that two iterations are able to lead to satisfactory performance.
Note that two additional queries are required to obtain $\nabla
f(y_{t}^{(0)})^{\top}p_{t}$ and $\nabla f(y_{t}^{(1)})^{\top}p_{t}$ used in
Line 7 and Line 9.
Since $D_{t}=(\overline{\nabla f(y_{t})}^{\top}p_{t})^{2}=\frac{(\nabla
f(y_{t})^{\top}p_{t})^{2}}{\|\nabla f(y_{t})\|^{2}}$, we need to estimate
$\|\nabla f(y_{t})\|^{2}$ as introduced in Section C.3. However, $y_{t}^{(0)}$
and $y_{t}^{(1)}$ in Algorithm 4 are different from both $y_{t}$ and
$y_{t-1}$, so to estimate $\|\nabla f(y_{t}^{(0)})\|^{2}$ and $\|\nabla
f(y_{t}^{(1)})\|^{2}$ as in Section C.3, many additional queries are required
(since the query results of the directional derivative at $y_{t-1}$ or $y_{t}$
cannot be reused). Therefore, we introduce one additional approximation: we
use the estimate of $\|\nabla f(y_{t-1})\|^{2}$ as the approximation of
$\|\nabla f(y_{t}^{(0)})\|^{2}$ and $\|\nabla f(y_{t}^{(1)})\|^{2}$. Since the
gradient norm itself is relatively large (compared with e.g. directional
derivatives) and in zeroth-order optimization, the single-step update is
relatively small, we expect that $\|\nabla f(y_{t}^{(0)})\|^{2}$ and $\|\nabla
f(y_{t}^{(1)})\|^{2}$ are closed to $\|\nabla f(y_{t-1})\|^{2}$. In Algorithm
4, Line 14 estimates $\|\nabla f(y_{t})\|^{2}$ by Eq. (153), and the estimator
is denoted $\|\hat{\nabla}f_{t}\|^{2}$; in the next iteration, it is used to
approximate $\|\nabla f(y_{t+1}^{(0)})\|^{2}$ and $\|\nabla
f(y_{t+1}^{(1)})\|^{2}$.
Finally we note that in the experiments, we find that when using Algorithm 4,
the error brought by approximation of $\|\nabla f(y_{t}^{(0)})\|^{2}$ and
$\|\nabla f(y_{t}^{(1)})\|^{2}$ sometimes makes the performance of the
algorithm not robust, especially when $q$ is small (e.g. $q=10$), which could
lead the algorithm to divergence. Therefore, we propose two tricks to suppress
the influence of approximation error (we note that in practice, the second
trick is more important, while the first trick is often not necessary given
the application of the second trick):
* •
To reduce the variance of $\|\hat{\nabla}f_{t}\|$ when $q$ is small, we let
$\displaystyle\|\hat{\nabla}f_{t}^{\mathrm{avg}}\|^{2}=\frac{1}{k}\sum_{s=t-k+1}^{t}\|\hat{\nabla}f_{s}\|^{2},$
(167)
and use $\|\hat{\nabla}f_{t-1}^{\mathrm{avg}}\|^{2}$ to replace
$\|\hat{\nabla}f_{t-1}\|^{2}$ in Line 7 and Line 9. In our experiments we
choose $k=10$. Compared with $\|\hat{\nabla}f_{t-1}\|^{2}$, using
$\|\hat{\nabla}f_{t-1}^{\mathrm{avg}}\|^{2}$ to estimate $\|\nabla
f(y_{t}^{(0)})\|^{2}$ and $\|\nabla f(y_{t}^{(1)})\|^{2}$ could reduce the
variance at the cost of increased bias.
* •
Although $D_{t}\leq 1$, It is possible that $\hat{D}_{t}$ in Line 7 and Line 9
is larger than $1$, which could lead to a negative $\theta_{t}$. Therefore, a
clipping of $\hat{D}_{t}$ is required. In our experiments, we observe that a
$\hat{D}_{t}$ which is less than but very close to $1$ (when caused by the
accidental large approximation error) could also lead to instability of
optimization, perhaps because that it leads to a too large value of
$\theta_{t}$ used to determine $y_{t}$ and to update $m_{t}$. Therefore, we
let $\hat{D}_{t}\leftarrow\min\\{\hat{D}_{t},B_{\mathrm{ub}}\\}$ in Line 7 and
Line 9 before calculating $\theta_{t}$, where $0<B_{\mathrm{ub}}\leq 1$ is
fixed. In our experiments we set $B_{\mathrm{ub}}$ to $0.6$.
We leave a more systematic study of the approximation error as future work.
### C.5 Implementation of History-PARS in practice (History-PARS-Impl)
In PARS, when using a specific prior instead of the prior from a general
source, we can utilize some properties of the prior. When using the historical
prior, we find that $D_{t}$ is usually similar to $D_{t-1}$, and intuitively
it happens when the smoothness of the objective function does not change
quickly along the optimization trajectory. Therefore, the best value of
$\theta_{t}$ should also be similar to the best value of $\theta_{t-1}$. Based
on this observation, we can directly use $\theta_{t-1}$ as the value of
$\theta_{t}$ in step $t$, and the value of $\theta_{t-1}$ is obtained with
$y_{t-1}$ in step $t-1$. Following this thread, we present our implementation
of History-PARS, i.e. History-PARS-Impl, in Algorithm 5.
Algorithm 5 History-PARS in implementation (History-PARS-Impl)
1:The $L$-smooth black-box function $f$; initialization $x_{0}$; $\hat{L}$ as
an upper bound of $L$ ($\hat{L}\geq L$); Query count per iteration $q$ (cannot
be too small); total number of iterations $T$; $\gamma_{0}>0$.
2:$x_{T}$ as the approximate minimizer of $f$.
3:$m_{0}\leftarrow x_{0}$;
4:$\theta_{-1}\leftarrow 0$;
5:$v_{-1}\sim\mathcal{U}(\mathbb{S}_{d-1})$;
6:for $t=0$ to $T-1$ do
7: $y_{t}\leftarrow(1-\alpha_{t})x_{t}+\alpha_{t}m_{t}$, where $\alpha_{t}\geq
0$ is a positive root of the equation
$\alpha_{t}^{2}=\theta_{t-1}(1-\alpha_{t})\gamma_{t}$;
$\gamma_{t+1}\leftarrow(1-\alpha_{t})\gamma_{t}$;
8: Sample an orthonormal set $\\{u_{i}\\}_{i=1}^{q}$ in the subspace
perpendicular to $v_{t-1}$;
9: $g_{1}(y_{t})\leftarrow\sum_{i=1}^{q}\nabla f(y_{t})^{\top}u_{i}\cdot
u_{i}+\nabla f(y_{t})^{\top}v_{t-1}\cdot v_{t-1}$;
10: $g_{2}(y_{t})\leftarrow\frac{d-1}{q}\sum_{i=1}^{q}\nabla
f(y_{t})^{\top}u_{i}\cdot u_{i}+\nabla f(y_{t})^{\top}v_{t-1}\cdot v_{t-1}$;
11:
$\theta_{t}\leftarrow\frac{D_{t}+\frac{q}{d-1}(1-D_{t})}{\hat{L}\left(D_{t}+\frac{d-1}{q}(1-D_{t})\right)}$,
where $D_{t}$ is estimated using Eq. (154) with $p_{t}=v_{t-1}$;
12: $x_{t+1}\leftarrow y_{t}-\frac{1}{\hat{L}}g_{1}(y_{t})$,
$m_{t+1}\leftarrow m_{t}-\frac{\theta_{t-1}}{\alpha_{t}}g_{2}(y_{t})$;
13: $v_{t}\leftarrow g_{1}(y_{t})$;
14:end for
15:return $x_{T}$.
### C.6 Full version of Algorithm 8 considering the strong convexity
parameter and its convergence theorem
Algorithm 6 Extended accelerated random search framework for $\tau\geq 0$
1:The $L$-smooth and $\tau$-strongly convex black-box function $f$;
initialization $x_{0}$; $\hat{L}$ as an upper bound of $L$ ($\hat{L}\geq L$);
$\hat{\tau}$ as a lower bound of $\tau$ ($0\leq\hat{\tau}\leq\tau$); iteration
number $T$; $\gamma_{0}\geq\hat{\tau}$.
2:$x_{T}$ as the approximate minimizer of $f$.
3:$m_{0}\leftarrow x_{0}$;
4:for $t=0$ to $T-1$ do
5: Find a $\theta_{t}$ such that
$\theta_{t}\leq\frac{\mathbb{E}_{t}\left[\left(\nabla
f(y_{t})^{\top}v_{t}\right)^{2}\right]}{\hat{L}\cdot\mathbb{E}_{t}[\|g_{2}(y_{t})\|^{2}]}$
in which $\theta_{t}$, $y_{t}$ and $g_{2}(y_{t})$ is defined in the following
3 steps:
6: Step 1: $y_{t}\leftarrow(1-\beta_{t})x_{t}+\beta_{t}m_{t}$, where
$\beta_{t}:=\frac{\alpha_{t}\gamma_{t}}{\gamma_{t}+\alpha_{t}\hat{\tau}}$,
$\alpha_{t}\geq 0$ is a positive root of the equation
$\alpha_{t}^{2}=\theta_{t}((1-\alpha_{t})\gamma_{t}+\alpha_{t}\hat{\tau})$;
$\gamma_{t+1}\leftarrow(1-\alpha_{t})\gamma_{t}+\alpha_{t}\hat{\tau}$;
7: Step 2: Let $v_{t}$ be a random vector s.t. $\|v_{t}\|=1$;
$g_{1}(y_{t})\leftarrow\nabla f(y_{t})^{\top}v_{t}\cdot v_{t}$;
8: Step 3: Let $g_{2}(y_{t})$ be an unbiased estimator of $\nabla f(y_{t})$,
i.e. $\mathbb{E}_{t}[g_{2}(y_{t})]=\nabla f(y_{t})$;
9: $\lambda_{t}\leftarrow\frac{\alpha_{t}}{\gamma_{t+1}}\hat{\tau}$;
10: $x_{t+1}\leftarrow y_{t}-\frac{1}{\hat{L}}g_{1}(y_{t})$,
$m_{t+1}\leftarrow(1-\lambda_{t})m_{t}+\lambda_{t}y_{t}-\frac{\theta_{t}}{\alpha_{t}}g_{2}(y_{t})$;
11:end for
12:return $x_{T}$.
In fact, the ARS algorithm proposed in [26] requires knowledge of the strong
convexity parameter $\tau$ of the objective function, and the original
algorithm depends on $\tau$. The ARS algorithm has a convergence rate for
general smooth convex functions, and also have another potentially better
convergence rate if $\tau>0$. In previous sections, for simplicity, we suppose
$\tau=0$ and illustrate the corresponding extension in Algorithm 8. In fact,
for the general case $\tau\geq 0$, the original ARS can also be extended to
allow for incorporation of prior information. We present the extension to ARS
with $\tau\geq 0$ in Algorithm 6. Note that our modification is similar to
that in Algorithm 8. For Algorithm 6, we can also provide its convergence
guarantee as shown in Theorem 7. Note that after considering the strong
convexity parameter in the algorithm, we have an additional convergence
guarantee, i.e. Eq. (169). In the corresponding PARS algorithm, we have
$\theta_{t}\geq\frac{q^{2}}{\hat{L}d^{2}}$, so the convergence rate of PARS is
not worse than that of ARS and admits improvement given a good prior.
###### Theorem 7.
Let $x^{*}:=\mathrm{argmin}_{x}f(x)$. Then in Algorithm 6, if $f$ is convex,
then we have
$\displaystyle\mathbb{E}\left[(f(x_{T})-f(x^{*}))\left(1+\frac{\sqrt{\gamma_{0}}}{2}\sum_{t=0}^{T-1}\sqrt{\theta_{t}}\right)^{2}\right]\leq
f(x_{0})-f(x^{*})+\frac{\gamma_{0}}{2}\|x_{0}-x^{*}\|^{2}.$ (168)
and
$\displaystyle\mathbb{E}\left[(f(x_{T})-f(x^{*}))\exp\left(\sqrt{\hat{\tau}}\sum_{t=0}^{T-1}\sqrt{\theta_{t}}\right)\right]\leq
f(x_{0})-f(x^{*})+\frac{\gamma_{0}}{2}\|x_{0}-x^{*}\|^{2}.$ (169)
###### Proof.
Let $L_{e}:=\frac{\hat{L}}{2\hat{L}-L}\cdot\hat{L}$. We still have Eq. (18),
so
$\displaystyle\mathbb{E}_{t}[f(x_{t+1})]$ $\displaystyle\leq
f(y_{t})-\frac{\mathbb{E}_{t}\left[\left(\nabla
f(y_{t})^{\top}v_{t}\right)^{2}\right]}{2L_{e}}$ (170) $\displaystyle\leq
f(y_{t})-\frac{\mathbb{E}_{t}\left[\left(\nabla
f(y_{t})^{\top}v_{t}\right)^{2}\right]}{2\hat{L}}.$ (171)
For any $x$, define
$\delta_{t}(x):=\frac{\gamma_{t}}{2}\|m_{t}-x\|^{2}+f(x_{t})-f(x)$. Let
$p_{t}:=(1-\lambda_{t})m_{t}+\lambda_{t}y_{t}$. We first prove a lemma.
Since
$(1-\beta_{t})x_{t}+\beta_{t}m_{t}=y_{t}=(1-\beta_{t})y_{t}+\beta_{t}y_{t}$,
we have $m_{t}-y_{t}=\frac{1-\beta_{t}}{\beta_{t}}(y_{t}-x_{t})$. So
$\displaystyle p_{t}$
$\displaystyle=(1-\lambda_{t})m_{t}+\lambda_{t}y_{t}=y_{t}+(1-\lambda_{t})(m_{t}-y_{t})=y_{t}+(1-\lambda_{t})\frac{1-\beta_{t}}{\beta_{t}}(y_{t}-x_{t}).$
(172)
By $\beta_{t}=\frac{\alpha_{t}\gamma_{t}}{\gamma_{t}+\alpha_{t}\hat{\tau}}$,
$\gamma_{t+1}=(1-\alpha_{t})\gamma_{t}+\alpha_{t}\hat{\tau}$ and
$\lambda_{t}=\frac{\alpha_{t}}{\gamma_{t+1}}\hat{\tau}$, after eliminating
$\gamma_{t}$ and $\gamma_{t+1}$, we have
$(1-\lambda_{t})\frac{1-\beta_{t}}{\beta_{t}}=\frac{1-\alpha_{t}}{\alpha_{t}}$.
Hence $p_{t}=y_{t}+\frac{1-\alpha_{t}}{\alpha_{t}}(y_{t}-x_{t})$, which means
$\displaystyle y_{t}=(1-\alpha_{t})x_{t}+\alpha_{t}p_{t}.$ (173)
Now we start the main proof.
$\displaystyle\delta_{t+1}(x)$
$\displaystyle=\frac{\gamma_{t+1}}{2}\|m_{t+1}-x\|^{2}+f(x_{t+1})-f(x)$ (174)
$\displaystyle=\frac{\gamma_{t+1}}{2}\|p_{t}-x\|^{2}-\frac{\gamma_{t+1}\theta_{t}}{\alpha_{t}}g_{2}(y_{t})^{\top}(p_{t}-x)+\frac{\gamma_{t+1}\theta_{t}^{2}}{2\alpha_{t}^{2}}\|g_{2}(y_{t})\|^{2}+f(x_{t+1})-f(x)$
(175)
$\displaystyle=\frac{\gamma_{t+1}}{2}\|p_{t}-x\|^{2}-\alpha_{t}g_{2}(y_{t})^{\top}(p_{t}-x)+\frac{\theta_{t}}{2}\|g_{2}(y_{t})\|^{2}+f(x_{t+1})-f(x).$
(176)
Hence
$\displaystyle\mathbb{E}_{t}[\delta_{t+1}(x)]$
$\displaystyle=\frac{\gamma_{t+1}}{2}\|p_{t}-x\|^{2}-\alpha_{t}\nabla
f(y_{t})^{\top}(p_{t}-x)+\frac{\theta_{t}}{2}\mathbb{E}_{t}[\|g_{2}(y_{t})\|^{2}]+\mathbb{E}_{t}[f(x_{t+1})]-f(x)$
(177) $\displaystyle\leq\frac{\gamma_{t+1}}{2}\|p_{t}-x\|^{2}-\alpha_{t}\nabla
f(y_{t})^{\top}(p_{t}-x)+\frac{\mathbb{E}_{t}\left[\left(\nabla
f(y_{t})^{\top}v_{t}\right)^{2}\right]}{2\hat{L}}+\mathbb{E}_{t}[f(x_{t+1})]-f(x)$
(178) $\displaystyle\leq\frac{\gamma_{t+1}}{2}\|p_{t}-x\|^{2}-\alpha_{t}\nabla
f(y_{t})^{\top}(p_{t}-x)+f(y_{t})-f(x)$ (179)
$\displaystyle=\frac{\gamma_{t+1}}{2}\|p_{t}-x\|^{2}-\nabla
f(y_{t})^{\top}(\alpha_{t}p_{t}-\alpha_{t}x)+f(y_{t})-f(x)$ (180)
$\displaystyle=\frac{\gamma_{t+1}}{2}\|p_{t}-x\|^{2}+\nabla
f(y_{t})^{\top}(-y_{t}+(1-\alpha_{t})x_{t}+\alpha_{t}x)+f(y_{t})-f(x)$ (181)
$\displaystyle=\frac{\gamma_{t+1}}{2}\|p_{t}-x\|^{2}+\alpha_{t}\left(f(y_{t})+\nabla
f(y_{t})^{\top}(x-y_{t})\right)$ (182)
$\displaystyle\quad+(1-\alpha_{t})\left(f(y_{t})+\nabla
f(y_{t})^{\top}(x_{t}-y_{t})\right)-f(x)$ (183)
$\displaystyle\leq\frac{\gamma_{t+1}}{2}\|p_{t}-x\|^{2}+(1-\alpha_{t})f(x_{t})-(1-\alpha_{t})f(x)-\frac{\alpha_{t}\tau}{2}\|x-y_{t}\|^{2}.$
(184)
We also have
$\displaystyle\frac{\gamma_{t+1}}{2}\|p_{t}-x\|^{2}$
$\displaystyle=\frac{\gamma_{t+1}}{2}\|(1-\lambda_{t})m_{t}+\lambda_{t}y_{t}-x\|^{2}$
(185)
$\displaystyle=\frac{\gamma_{t+1}}{2}\|(1-\lambda_{t})(m_{t}-x)+\lambda_{t}(y_{t}-x)\|^{2}$
(186)
$\displaystyle\leq\frac{\gamma_{t+1}(1-\lambda_{t})}{2}\|m_{t}-x\|^{2}+\frac{\gamma_{t+1}\lambda_{t}}{2}\|y_{t}-x\|^{2}$
(187)
$\displaystyle=\frac{\gamma_{t+1}(1-\lambda_{t})}{2}\|m_{t}-x\|^{2}+\frac{\alpha_{t}\hat{\tau}}{2}\|y_{t}-x\|^{2}$
(188)
$\displaystyle=(1-\alpha_{t})\frac{\gamma_{t}}{2}\|m_{t}-x\|^{2}+\frac{\alpha_{t}\hat{\tau}}{2}\|x-y_{t}\|^{2},$
(189)
where the inequality is due to Jensen’s inequality applied to the convex
function $\|\cdot\|^{2}$, and the third equality is obtained after
substituting $\lambda_{t}\gamma_{t+1}=\alpha_{t}\hat{\tau}$ by the definition
of $\lambda_{t}$. Since
$\gamma_{t+1}=(1-\alpha_{t})\gamma_{t}+\alpha_{t}\hat{\tau}=(1-\alpha_{t})\gamma_{t}+\lambda_{t}\gamma_{t+1}$,
we have $\gamma_{t+1}(1-\lambda_{t})=(1-\alpha_{t})\gamma_{t}$, which leads to
the last equality.
Hence
$\displaystyle\mathbb{E}_{t}[\delta_{t+1}(x)]$
$\displaystyle\leq\frac{\gamma_{t+1}}{2}\|p_{t}-x\|^{2}+(1-\alpha_{t})f(x_{t})-(1-\alpha_{t})f(x)-\frac{\alpha_{t}\tau}{2}\|x-y_{t}\|^{2}$
(190)
$\displaystyle=(1-\alpha_{t})\delta_{t}(x)+\frac{\alpha_{t}(\hat{\tau}-\tau)}{2}\|x-y_{t}\|^{2}$
(191) $\displaystyle\leq(1-\alpha_{t})\delta_{t}(x).$ (192)
Therefore,
$\displaystyle\delta_{0}(x)$
$\displaystyle\geq\frac{1}{1-\alpha_{0}}\mathbb{E}[\delta_{1}(x)]=\mathbb{E}\left[\frac{\delta_{1}(x)}{1-\alpha_{0}}\right]$
$\displaystyle\geq\mathbb{E}\left[\frac{\mathbb{E}_{1}[\delta_{2}(x)]}{(1-\alpha_{0})(1-\alpha_{1})}\right]=\mathbb{E}\left[\mathbb{E}_{1}\left[\frac{\delta_{2}(x)}{(1-\alpha_{0})(1-\alpha_{1})}\right]\right]=\mathbb{E}\left[\frac{\delta_{2}(x)}{(1-\alpha_{0})(1-\alpha_{1})}\right]$
$\displaystyle\geq\ldots$
$\displaystyle\geq\mathbb{E}\left[\frac{\delta_{T}(x)}{\prod_{t=0}^{T-1}(1-\alpha_{t})}\right].$
We have $\delta_{T}(x)\geq f(x_{T})-f(x)$. To prove the theorem, let
$x=x^{*}$. The remaining is to give an upper bound of
$\prod_{t=0}^{T-1}(1-\alpha_{t})$. Let
$\psi_{k}:=\prod_{t=0}^{k-1}(1-\alpha_{t})$ and
$a_{k}:=\frac{1}{\sqrt{\psi_{k}}}$, we have
$\displaystyle a_{k+1}-a_{k}$
$\displaystyle=\frac{1}{\sqrt{\psi_{k+1}}}-\frac{1}{\sqrt{\psi_{k}}}=\frac{\sqrt{\psi_{k}}-\sqrt{\psi_{k+1}}}{\sqrt{\psi_{k}\psi_{k+1}}}=\frac{\psi_{k}-\psi_{k+1}}{\sqrt{\psi_{k}\psi_{k+1}}(\sqrt{\psi_{k}}+\sqrt{\psi_{k+1}})}$
(193)
$\displaystyle\geq\frac{\psi_{k}-\psi_{k+1}}{\sqrt{\psi_{k}\psi_{k+1}}(\sqrt{\psi_{k}})}$
(194)
$\displaystyle=\frac{\psi_{k}-(1-\alpha_{k})\psi_{k}}{2\psi_{k}\sqrt{\psi_{k+1}}}=\frac{\alpha_{k}}{2\sqrt{\psi_{k+1}}}=\frac{\sqrt{\gamma_{k+1}\theta_{k}}}{2\sqrt{\psi_{k+1}}}=\frac{\sqrt{\theta_{k}}}{2}\sqrt{\frac{\gamma_{k+1}}{\psi_{k+1}}}$
(195) $\displaystyle\geq\frac{\sqrt{\gamma_{0}\theta_{k}}}{2}.$ (196)
The last step is because $\gamma_{t+1}\geq(1-\alpha_{t})\gamma_{t}$, so
$\frac{\gamma_{k+1}}{\gamma_{0}}\geq\prod_{t=0}^{k}(1-\alpha_{t})=\psi_{k+1}$.
Since $\psi_{0}=1$, $a_{0}=1$. Hence $a_{T}\geq
1+\frac{\sqrt{\gamma_{0}}}{2}\sum_{t=0}^{T-1}\sqrt{\theta_{t}}$. Therefore,
$\displaystyle\psi_{T}\leq\frac{1}{\left(1+\frac{\sqrt{\gamma_{0}}}{2}\sum_{t=0}^{T-1}\sqrt{\theta_{t}}\right)^{2}}.$
(197)
Meanwhile, since $\gamma_{0}\geq\hat{\tau}$ and
$\gamma_{t+1}=(1-\alpha_{t})\gamma_{t}+\alpha_{t}\hat{\tau}$, we have that
$\forall t,\gamma_{t}\geq\hat{\tau}$. Then
$\alpha_{t}^{2}=\theta_{t}((1-\alpha_{t})\gamma_{t}+\alpha_{t}\hat{\tau})\geq\theta_{t}\hat{\tau}$,
then we have that $\alpha_{t}\geq\sqrt{\hat{\tau}\theta_{t}}$. Therefore,
$\displaystyle\psi_{T}\leq\prod_{t=0}^{T-1}\left(1-\sqrt{\hat{\tau}\theta_{t}}\right)\leq\exp\left(-\sqrt{\hat{\tau}}\sum_{t=0}^{T-1}\sqrt{\theta_{t}}\right).$
(198)
The proof is completed. ∎
## Appendix D Supplemental materials for Section 5
### D.1 More experimental settings in Section 5.1
In experiments in this section, we set the step size $\mu$ used in finite
differences (Eq. (1)) to $10^{-6}$.
#### D.1.1 Experimental settings for Figure 1
##### Prior
We adopt the setting in Section 4.1 of [20] to mimic the case that the prior
is a biased version of the true gradient. Specifically, we let
$p_{t}=\overline{\overline{\nabla f(x_{t})}+(b+n)}$, where $b$ is a fixed
vector and $n$ is a random vector uniformly sampled each iteration, $\|b\|=1$
and $\|n\|=1.5$.
##### Test functions
Our test functions are as follows. We choose $f_{1}$ as the "worst-case smooth
convex function" used to construct the lower bound complexity of first-order
optimization, as used in [26]:
$\displaystyle
f_{1}(x)=\frac{1}{2}(x^{(1)})^{2}+\frac{1}{2}\sum_{i=1}^{d-1}(x^{(i+1)}-x^{(i)})^{2}+\frac{1}{2}(x^{(d)})^{2}-x^{(1)},\text{
where }x_{0}=\mathbf{0}.$ (199)
We choose $f_{2}$ as a simple smooth and strongly-convex function with a
worst-case initialization:
$\displaystyle
f_{2}(x)=\sum_{i=1}^{d}\left(\frac{i}{d}\cdot(x^{(i)})^{2}\right),\text{ where
}x_{0}^{(1)}=d,x_{0}^{(i)}=0\text{ for }i\geq 2.$ (200)
We choose $f_{3}$ as the Rosenbrock function ($f_{8}$ in [13]) which is a
well-known non-convex function used to test the performance of optimization
problems:
$\displaystyle
f_{3}(x)=\sum_{i=1}^{d-1}\left(100\left((x^{(i)})^{2}-x^{(i+1)}\right)^{2}+(x^{(i)}-1)^{2}\right),\text{
where }x_{0}=\mathbf{0}.$ (201)
We note that ARS, PARS-Naive and PARS could depend on a strong convexity
parameter (see Section C.6) when applied to a strongly convex function.
Therefore, for $f_{2}$ we set this parameter to the ground truth value. For
$f_{1}$ and $f_{3}$ we set it to zero, i.e. we use Algorithm 8.
#### D.1.2 Experimental settings for Figure 2
In this part we set $d=500$ and set $q$ such that each iteration of each
algorithm costs $11$ queries. Since when using the historical prior, we aim to
build algorithms agnostic to parameters of the objective function, we set the
strong convexity parameter in ARS-based methods to $0$ even though we know
that e.g. $f_{2}$ is strongly-convex. Correspondingly, we adopt adaptive
restart of function scheme [27] to reach the ideal performance. We introduce
our implementation here. In each iteration (suppose that currently it is step
$t$) of Algorithm 5, we check whether $f(y_{t})\leq f(y_{t-1})$. If not, we
set $m_{t+1}\leftarrow x_{t+1}$ and $\gamma_{t+1}\leftarrow\gamma_{0}$ as the
restart.
### D.2 More experimental settings in Section 5.2
We perform attacks under the $\ell_{2}$ norm with the perturbation bound set
to $3.514$ ($=32/255\times\sqrt{784}$) if each pixel value has the range
$[0,1]$. To deal with the constraints in optimization, in each iteration we
perform projection after the update to ensure that the constraints are
satisfied. The objective function to maximize is the C&W loss [7], i.e.
$f(x)=Z(x)_{t}-\max_{i\neq t}Z(x)_{i}$, where $Z(x)$ is the logits given the
input $x$. The network architecture is from the PyTorch example
(https://github.com/pytorch/examples/tree/master/mnist). We set the step size
$\mu$ used in finite differences (Eq. (1)) to $10^{-4}$.
## Appendix E Potential negative societal impacts
As a theoretical work, we think this paper can provide valuable insights on
understanding existing algorithms and may inspire new algorithms for zeroth-
order optimization, while having no significant potential negative societal
impacts. One may pay attention to its application for query-based black-box
adversarial attacks.
|
capbtabboxtable[][]
# Training Semantic Segmentation
on Heterogeneous Datasets
Panagiotis Meletis and Gijs Dubbelman P. Meletis<EMAIL_ADDRESS>and G.
Dubbelman<EMAIL_ADDRESS>are with the Department of Electrical
Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands.
0000-0001-8054-1760 0000-0001-6635-3245
###### Abstract
We explore semantic segmentation beyond the conventional, single-dataset
homogeneous training and bring forward the problem of Heterogeneous Training
of Semantic Segmentation (HTSS). HTSS involves simultaneous training on
multiple heterogeneous datasets, _i.e_. datasets with conflicting label
spaces and different (weak) annotation types from the perspective of semantic
segmentation. The HTSS formulation exposes deep networks to a larger and
previously unexplored aggregation of information that can potentially enhance
semantic segmentation in three directions: i) performance: increased
segmentation metrics on seen datasets, ii) generalization: improved
segmentation metrics on unseen datasets, and iii) knowledgeability: increased
number of recognizable semantic concepts. To research these benefits of HTSS,
we propose a unified framework, that incorporates heterogeneous datasets in a
single-network training pipeline following the established FCN standard. Our
framework first curates heterogeneous datasets to bring them in a common
format and then trains a single-backbone FCN on all of them simultaneously. To
achieve this, it transforms weak annotations, which are incompatible with
semantic segmentation, to per-pixel labels, and hierarchizes their label
spaces into a universal taxonomy. The trained HTSS models demonstrate
performance and generalization gains over a wide range of datasets and extend
the inference label space entailing hundreds of semantic classes.
###### Index Terms:
semantic segmentation, heterogeneous datasets, multi-dataset/domain training,
weakly/semi-supervised training
## I Introduction
Semantic Segmentation [1, 2, 3] is a indispensable building block of upstream
systems in various domains, such as automated driving [4, 5], biomedical image
analysis [6], virtual/augmented reality, and surveillance [3]. Semantic
segmentation is part of the bigger family of image recognition tasks, which
include, among others, image classification and object detection. The success
of supervised Convolutional Neural Networks (CNNs) in image classification [7,
8, 9] set them as the de-facto solution for related image recognition tasks,
which is in a large part attributed to the successful utilization of very
large datasets. Unlike CNNs trained for classification or detection, where
data collection and labeling is easier, CNNs for semantic segmentation face
two fundamental data challenges that are analyzed in the next paragraphs.
Limited size of existing datasets. Datasets for semantic segmentation [10, 11,
12] contain typically 100 to 1000 times less annotated images than for image
classification and object detection, as visualized in Fig. 1. The lack of rich
datasets causes CNNs for semantic segmentation to exhibit limited performance
when evaluated on seen datasets and poor generalization capabilities on unseen
datasets [13, 14, 15]. This challenge becomes particularly acute in data-
scarce areas, such as street-scene understanding. The main reason for the
difference in dataset sizes is the required level-of-detail of the annotations
for the task at hand. For example, COCO creators [16] report that annotating
pixels for semantic segmentation of general scenes is 15x slower than drawing
bounding boxes, while according to [17] annotating pixels is 78x slower than
choosing image-level tags.
Figure 1: Comparison of various image understanding datasets with respect to
their: i) annotation type, ii) number of semantic classes, and iii) number of
images (visualized by the area of the circles). Networks for semantic
segmentation are typically trained using a single per-pixel (fine) labeled
dataset.
Low diversity of represented semantic concepts. The second fundamental
challenge is related to the number of recognizable semantic concepts a CNNs
can predict. Typical a CNN trained with segmentation datasets can recognize a
few dozens concepts, since fine-grained semantic classes are absent from these
datasets. The complexity of (manual) per-pixel labeling practically constrains
the annotated semantic classes to represent an order of 100 different scene
concepts, while datasets with less spatially-detailed annotations (bounding
boxes or tags) [18, 19, 20] can reach up to $1,000-10,000$ unique semantic
classes (see vert. axis of Fig. 1).
The straightforward way to address the aforementioned two challenges, is to
annotate more images at the pixel level, or refine existing annotations with
finer-grained (sub-)classes by manual or semi-automated means. This is a
natural yet costly approach, since manual labeling is laborious (_e.g_. 90
min. per image for Cityscapes and Vistas [11, 10]) and semi-automated
procedures result in insufficient quality of annotations when not complemented
with human quality control.
These two practical challenges motivate us to research an alternative
approach. Having analyzed the annotation quality and semantic diversity of
existing image understanding datasets (Fig. 1) we observed that, as a whole,
they cover thousands of semantic concepts and contain millions of images.
Thus, we contemplate a solution that can combine many existing datasets and
solve both aforementioned challenges. However, combining datasets is not an
easy task, due to their structural difference, but when successful, it allows
for training more capable CNN models with minimal extra manual effort.
In this work, we investigate the challenges and benefits of combined dataset
training in three directions: segmentation performance on seen datasets
(validation/testing splits of datasets on which the CNN is trained),
generalization on unseen datasets (splits of datasets not used during
training), and Knowledgeability, _i.e_., the number of recognizable semantic
concepts with sufficient segmentation quality. Instead of combining only
datasets for semantic segmentation, _i.e_. pixel-labeled datasets, a larger
candidate pool of heterogeneous datasets from various image understanding
tasks is admitted. These datasets, can significantly increase the available
training data and the number of semantic classes leading to improvements in
all three directions.
Combining different heterogeneous datasets brings up new challenges, since
these datasets are created for different tasks or application domains, and
thus they contain conflicting or unrelated label spaces and incompatible
(weak) annotations. As a consequence, it is impossible to employ established
training strategies for semantic segmentation, _e.g_. a fully convolutional
network pipeline (FCN) [21]. To advance the state-of-the-art in multi-dataset
training, we generalize semantic segmentation training over heterogeneous
datasets and analyze the related challenges. Subsequently, we propose a
unified methodology that decouples dataset specifics (structure, annotation
types, label spaces) from the task formulation of semantic segmentation. In
this way, a plethora of existing image understanding datasets can be leveraged
under the same consistent and robust FCN-based framework. The contributions of
this work can be summarized as follows.
* •
The formulation of the heterogeneous datasets training problem for semantic
segmentation (HTSS) and characterization of the challenges.
* •
A methodology for combining label spaces with different semantic granularity
(level-of-detail) and with different semantic concepts, thus enabling
simultaneous training on datasets with disjoint or conflicting label spaces.
* •
A methodology for consolidating strong (pixel) and weak (bounding-box or
image-tag) supervision, and thus enabling simultaneous training with mixed
supervision.
* •
The novel Knowledgeability metric, which quantifies the number of recognizable
semantic classes by a network wrt. achievable performance for these classes,
and that can be used to compare the performance of a network across datasets
irrespective of the number of classes.
## II Related work
Multi-dataset training is gaining traction in various areas, _e.g_. in object
detection [22, 23], depth estimation [24], and domain adaptation [25, 26],
since it improves model robustness and generalization capabilities. This work
focuses on semantic segmentation and relaxation of the requirements that a
dataset has to comply to, in order to be suited for multi-dataset training.
The proposed work generalizes related approaches in literature for semantic
segmentation [27, 28] and complements recent work [29], [30], [31], [32].
Figure 2: Motivation and overview of the proposed framework. HTSS aims at
using a wide range of heterogeneous image understanding datasets with
incompatible annotation formats and conflicting label spaces (1). Our
methodology derives a unified label space (2) and consolidates supervision (3)
so they can be used to simultaneously train an FCN (4) on all datasets. The
trained network (5) has better performance on the training datasets,
generalizes better to unseen images, and recognizes finer-grained semantic
classes.
### II-A Multi-dataset semantic segmentation
The majority of previous works focus on using multiple datasets with possibly
different label spaces, but a single type of supervision, _i.e_. pixel-level
labels. Most of the works solve the challenges that arise from conflicts in
label semantics through dataset-based solutions [33, 27], architecture-based
solutions [34, 35, 36, 37, 38, 39], or loss-based solutions [40, 35, 38]. In
these works, all label spaces of the employed datasets are combined into a
common taxonomy, by merging, splitting, or ignoring semantic concepts or by
manual re-labeling when needed. Early works extend the conventional FCN
architecture with multiple heads/decoders up to one for each dataset [37], or
multiple (hierarchical) classifiers [35], thereby effectively approaching the
problem from the multi-task learning perspective. The authors of [33] combine
six datasets, while in [27] thirteen datasets are combined to create a large-
scale training and testing platform.
Contrary to existing works, the proposed Heterogeneous Training of Semantic
Segmentation (HTSS) framework does not require any dataset relabeling and does
not ignore classes for simultaneous training of an FCN with multiple datasets.
Moreover, it maintains the network architecture, since it solves all label
space conflicts at the stage of loss calculation, and therefore the method can
be applied to any semantic segmentation network without requiring
architectural changes.
### II-B Semantic Segmentation with weak supervision
Semantic segmentation is by definition a pixel-based task and it is
conventionally realized by training a CNN with per-pixel homogeneous
supervision. Previous works have used a diverse set of less detailed (weak)
heterogeneous supervision, either to accommodate strong supervision, or
independently in a semi/weakly-supervised setting [41, 42, 43, 44, 36].
Several methods generate candidate masks from bounding-box supervision [45,
43, 46, 42, 35] using external modules, internal network predictions, or
heuristics to refine weak annotations. These masks are used to train networks
alone or together with strong supervision. Even weaker forms of supervision
have been employed and examples of this include point-level [17] and image-
level [47, 48, 49] annotations, mainly within a multiple instance learning
formulation. Finally, methods that use a combination of multiple weaker types
of supervision have been proposed, such as bounding boxes and image-level tags
[50, 51, 44, 52, 53].
Inspired by earlier works, the proposed framework achieves pixel-accurate
training using weak supervision by a pre-processing step that generates
pseudo-labels and a refinement process during training. Moreover, unlike
previous methods, the HTSS framework treats all types of weakly-labeled
datasets uniformly and uses them in combination with strongly-labeled
datasets.
### II-C Other related tasks
Two related semantic segmentation tasks that encapsulate multiple datasets in
their formulation are transfer learning [54] and domain adaptation [26, 25].
These tasks aim at transferring rich knowledge from a source dataset/domain to
a target dataset/domain, where knowledge is scarce or even non-existing. They
mainly concentrate on the performance in the target domain, which may be
available during training, in some limited form. Recently, variations of these
tasks also track performance in the source domain and investigate multiple-
source versions of the problems [55, 56]. The HTSS formulation considers
performance on all employed datasets and during training it does not depend on
information from the testing datasets explicitly, as in domain adaptation.
The following four tasks are briefly addressed for their relevance with
aspects of HTSS. First, multi-dataset semantic segmentation has been tackled
in literature using multi-task learning [24, 57, 58, 59, 60], where a network
head/branch is devoted to each dataset independently, _i.e_. segmentation for
each dataset is modeled as a separate “task”. Second, continual learning is
relevant to the concept of knowledgeability, since in this task new classes
are discovered or added during training or inference [61, 62] and old data may
not be available. Third, self-training or pseudo-label approaches [63, 64, 65,
41, 66] have addressed the absence of labels during training for some
datasets/domains. Finally, learning with partial labels [67, 68] is related to
conflicting labels spaces. The partial-label formulation associates a training
sample with a set of candidate labels among which at most one is the correct
label.
This plethora of research shows that training with multiple and heterogeneous
datasets is a desired capability of modern training pipelines.
### II-D Conventional semantic segmentation
We recap here the formulation for the conventional image semantic
segmentation, which we use in the following section to define Heterogeneous
Training of Semantic Segmentation. The task of Semantic Segmentation [1, 21,
2, 10, 3] involves the per-pixel classification of an image into a
predetermined set of mutually-exclusive semantic classes. A semantic
segmentation system has a 2-D image $\mathbf{x}$ as input and uses a given
label space $\mathcal{L}^{\text{pred}}$ of semantic classes. The aim of the
system is to predict a 2-D matrix $\mathbf{y}^{\text{pred}}$, where each
element corresponds to an image pixel with a semantic class from
$\mathcal{L}^{\text{pred}}$ assigned to it.
In the conventional supervised learning setting, the task entails a dataset
$\mathcal{S}=(\mathcal{D},\mathcal{L})$, which consists of $N$ image-label
pairs
$\mathcal{D}=\\{\left(\mathbf{x}_{i},\mathbf{y}_{i}\right),~{}i=1,\dots,N\\}$
and a label space $\mathcal{L}=\left\\{l_{j},~{}j=0,\dots,L\right\\}$ of $L$
semantic classes. Every label $\mathbf{y}\in\mathcal{L}^{H\times W}$ is a 2-D
matrix with spatial size $H\times W$ and every position corresponds to a
single pixel in the image $\mathbf{x}$. It is common that a semantic class
(_e.g_. the $l_{0}$) corresponds to a special label that denotes unlabeled or
void pixels. The semantic classes represent semantic entities of a scene,
_e.g_. vehicle, person, tree, or sky. It is essential that all $l_{j}$ have
unambiguous and mutually-exclusive semantic definitions $def(l_{j})$ within a
dataset,_e.g_. all possible types of cars in a dataset should not resemble any
of the trucks and vice versa. If this does not hold, _i.e_. annotations are
noisy or concepts overlap between classes, then a classifier trained on this
dataset may be “confused” and the evaluation is inaccurate.
Although in literature, existing datasets strive to define unambiguous and
non-overlapping classes within their label space, in practice some ambiguity
exists, mainly due to defining semantic classes using a single-word. In this
work, we assume that such noise in labels is negligible.
## III Heterogeneous Training for Semantic Segmentation
In this section we describe the problem and challenges of Heterogeneous
Training of Semantic Segmentation (HTSS), prior to presenting the proposed
framework in Sec. IV. The terminology follows the conventional semantic
segmentation task described in Sec. II-D.
In the conventional setting described in Section II-D, the formats of the task
definition and the given dataset are in full agreement with each other. In
this case, the output label space $\mathcal{L}^{\text{pred}}$ can be set to be
the dataset label space $\mathcal{L}^{\text{pred}}\equiv\mathcal{L}$ and the
predictions are congruent to the per-pixel annotations
$\mathbf{y}^{\text{pred}}\cong\mathbf{y}$. The symmetry between the training
data and the task goal is advantageous, however it limits the available
information that a system can be exposed to, _e.g_. only one dataset from the
first column of Fig. 1. These datasets are limited in size and semantic
concepts, which renders them inadequate for large-scale, in-the-wild semantic
segmentation.
We consider two generalizations to the traditional semantic segmentation
problem and encapsulate them in a unified formulation. The first aims at
enriching the output semantic space of the task using multiple heterogeneous
label spaces. The second intends to increase the amount of available
supervision by generalizing the task input to more types of supervision. We
incorporate these generalization within a unified formulation by maintaining
the task output identical, while relaxing the requirements for the given
datasets. This enables potential inclusion of datasets, which are originally
created for other scene understanding tasks, _e.g_. multiple datasets from
all columns of Fig. 1, when training a network for semantic segmentation.
Heterogeneous semantic segmentation enables trained networks to aggregate
information from diverse datasets and it could demonstrate potential
improvements in the following three aspects. First, multi-dataset training
increases examples for underrepresented classes and provides diversity in
recognizable semantics, which could be advantageous for performance, _i.e_.
segmentation accuracy on seen (training) datasets. Second, the diversity of
employed datasets should bring benefits to generalization, _i.e_.
segmentation accuracy on unseen (testing) datasets. We evaluate this aspect
under the cross-dataset zero-shot setting [27]. Third, network predictions are
more fine-grained and semantically rich by incorporating semantics from
multiple datasets. As can be observed from Fig. 1, label spaces of pixel-
labeled datasets are generally smaller than weakly-labeled datasets, _i.e_.
$\mathcal{L}^{\text{pixel}}\ll\mathcal{L}^{\text{bbox}}\ll\mathcal{L}^{\text{tag}}$.
We propose a metric, detailed in Sec. IV, to quantify the semantic richness of
the predicted classes w.r.t. to the segmentation performance for these
classes.
Label type | Definition
---|---
Pixel (dense) | $\mathbf{y}\in\mathcal{L}^{H\times W},~{}\left|y_{k}=l_{0}\right|\ll\left|y_{k}\neq l_{0}\right|,~{}k\in H\times W$
Pixel (coarse) | $\mathbf{y}\in\mathcal{L}^{H\times W},~{}\left|y_{k}=l_{0}\right|\gg\left|y_{k}\neq l_{0}\right|,~{}k\in H\times W$
Bound. boxes | $\mathbf{y}=\left\\{\left(l_{k},~{}\text{bbox-coords}_{k}\right),~{}k=1,\dots,B\right\\}$
Image tags | $\mathbf{y}=\left\\{l_{k},~{}k=1,\dots,T\right\\}$
TABLE I: A variety of annotation formats are considered. The dataset indexing
superscript $(i)$ and the dataset sample subscript $j$ are omitted for
clarity. $l_{0}$ is the void class.
### III-A Problem formulation
Similarly to the conventional segmentation task definition, heterogeneous
semantic segmentation aims at predicting a 2-D matrix
$\mathbf{y}^{\text{pred}}$ with semantic classes, given a 2-D image
$\mathbf{x}$ and a label space $\mathcal{L}^{\text{pred}}$. Contrary to the
traditional single-dataset formulation, we assume that a collection
$\mathbb{S}$ of $D$ heterogeneous datasets is available, where each dataset
$\mathcal{S}^{(i)}$ includes $N$ image-label pairs $\mathcal{D}^{(i)}$ and the
corresponding label space $\mathcal{L}^{(i)}$ with $L^{(i)}$ semantic classes:
$\displaystyle\mathbb{S}=\left\\{\mathcal{S}^{(i)},~{}i=1,\dots,D\right\\}~{},$
(1)
$\displaystyle\mathcal{S}^{(i)}=\left(\mathcal{D}^{(i)},\mathcal{L}^{(i)}\right)~{},$
(2)
$\displaystyle\mathcal{D}^{(i)}=\left\\{\left(\mathbf{x}^{(i)}_{j},\mathbf{y}^{(i)}_{j}\right),~{}j=1,\dots,N^{(i)}\right\\}~{},$
(3)
$\displaystyle\mathcal{L}^{(i)}=\left\\{l^{(i)}_{m},~{}m=0,\dots,L^{(i)}\right\\}~{}.$
(4)
The type of labels $\mathbf{y}$ considered in this work are provided in Tab.
I.
The goal is to train a system for semantic segmentation, such that it utilizes
information from all heterogeneous datasets in $\mathbb{S}$. The system should
have a consistent label space and recognize the semantic concepts from all
considered label spaces
$\mathcal{L}^{\text{pred}}=\uplus_{i=1}^{D}\mathcal{L}^{(i)}$. Within the
researched problem formulation, the following conditions should hold for the
employed datasets.
1\. Intra-dataset label space consistency. Each label space
$\mathcal{L}^{(i)}$ should include consistent and mutually-exclusive semantic
classes, as explained in Sec. II-D:
$def\left(l^{(i)}_{m}\right)\cap
def\left(l^{(i)}_{n}\right)=\emptyset,~{}\forall~{}i=1,\dots,D~{}.$ (5)
where $def(l)$ denotes the proper definition of the semantic class $l$.
2\. Condition for weakly-labeled classes. Any semantic class from a weakly-
labeled dataset $\mathcal{S}^{(W)}$ should either be identical to, or contain
partially semantics from, a class in a strongly-labeled dataset
$\mathcal{S}^{(S)}$:
$\exists~{}l^{(S)}~{}\text{so that}~{}def\left(l^{(W)}\right)\subseteq
def\left(l^{(S)}\right)~{}.$ (6)
Note that: i) cond. 2 implies that there must be at least one pixel-labeled
dataset available for training, and ii) cond. 1 does not imply inter-dataset
label space consistency, which is one of the challenges addressed by our HTSS
framework.
Challenges | HTSS components
---|---
Label spaces | Label type | Preparation (once) | Supervision during training (each step)
no conflicts | strong | - | standard cross-entropy (CE)
weak (Sec. IV-B) | create pseudo-labels (Sec. IV-B1, Fig. 5) | conditional CE, refine pseu- do-labels (Sec. IV-B2, Fig. 6)
with conflicts | strong (Sec. IV-A) | build taxonomy (Sec. IV-A1, Fig. 3) | supervise semantic atoms (Sec. IV-A2, Fig. 4)
weak (Sec. IV-C) | all above | all above
TABLE II: Overview of the HTSS components developed in this work. Each row
includes methods for combining a strongly-labeled dataset and any other
dataset(s) with the type of supervision denoted by the second column.
### III-B Challenges
Heterogeneous semantic segmentation brings forward incompatibilities in the
annotation formats and the conflicting label spaces among datasets. The
following two paragraphs analyze the related challenges.
Label space conflicts. Datasets are annotated over different label spaces on a
vast spectrum of semantic detail, as they are collected to serve different
purposes, which leads to conflicting or overlapping definitions of classes
between datasets. If the class definitions for all labels is matching between
datasets then a simple union of the label spaces is feasible. However, this is
usually not doable because of potential conflicts. The main source of
conflicts stems from partial overlapping semantic class definitions between
two arbitrary datasets $\mathcal{S}^{(X)}$ and $\mathcal{S}^{(Y)}$, which is
specified by:
$def\left(l^{(X)}\right)\cap def\left(l^{(Y)}\right)\neq\emptyset~{}.$ (7)
Since, the class definitions can overlap only partially, merging them or
including them both in the combined label space will introduce ambiguity to
the output label space of a trained network. A special common case occurs when
conflicts arise from differences in the semantic level-of-detail between
classes. For example, a class $l^{(X)}$ from dataset $\mathcal{S}^{(X)}$
describes a high-level concept which contains many more fine-grained classes
for dataset $\mathcal{S}^{(Y)}$, giving:
$def\left(l^{(X)}\right)=\bigcup_{m}def\left(l_{m}^{(Y)}\right)~{}.$ (8)
The inclusion of all classes $l^{(X)}$, $l^{(Y)}_{m},\forall m$ in a single
label space would also imply introducing conflicts.
Annotation format incompatibilities. A plethora of diverse datasets for scene
understanding cannot be used in semantic segmentation when using standard
training schemes, due to their incompatible annotation formats. The most
common of them are shown in Tab. I. Semantic segmentation is by definition a
pixel-wise task, thus it is convenient that training datasets to provide
annotations at the same pixel-level format. The spatial localizability of
labels from other datasets of Fig. 1 is not adequate to train a network,
because they introduce spatial localization uncertainties during pixel-wise
training for segmentation. The incompatibilities in annotation formats are
even more pronounced in a multi-dataset training scenario, where a variety of
incompatible annotation formats can exist. Heterogeneous semantic segmentation
thus requires to extract useful supervision at the pixel level from a much
coarser source of information.
## IV Methodology
The development of our methodology for heterogeneous multi-dataset training
abides to the design principle of maintaining the established single-backbone
FCN pipeline. This desideratum enables straightforward applicability of the
proposed methods to current or future FCN-based architectures, and scalability
to an arbitrary number of datasets.
Figure 3: Combined taxonomy of 118 semantic atoms (void atom is not shown)
merging a total of 174 semantic classes from Cityscapes (27), Vistas (65), IDD
(33), and MTS (50) datasets. Each dataset section corresponds to the grouping
of the combined taxonomy labels in order to apply strong supervision from the
respective dataset. Figure 4: The HTSS classifier is supervised using the
standard cross-entropy (CE) loss between the semantic atoms vector and a
different ground truth vector per dataset (in boxes). This is achieved by
combining the CNN output onto a new multinomial distribution vector that
matches each dataset’s labels. A selection of classes from Fig. 3 are shown in
order to explain the procedure of probabilities accumulation.
The proposed Heterogeneous Training of Semantic Segmentation (HTSS) framework
addresses the challenges of Sec. III-B by introducing a methodology for
combing disjoint or conflicting label spaces (Sec. IV-A), training a single-
backbone network with strong and weak supervision simultaneously (Sec. IV-B,
Sec. IV-C). An overview of the methodology is depicted in Fig. 2. Tab. II
associates the components of our methodology with the challenges they address.
### IV-A Combine datasets with different label spaces
This subsection describes our approach for training a single-backbone, single-
classifier FCN on multiple pixel-labeled datasets, which have disjoint or
conflicting semantic label spaces. The approach enables training on an
arbitrary number of datasets and it consists of two steps.
Before training. First, the label spaces of the training datasets have to be
consolidated into a unified taxonomy (Sec. IV-A1). This procedure can be
automated if the label spaces are part of a semantic ontology or hierarchical
lexical database (_e.g_. WordNet [69] or BabelNet [70]). For example, the
classes of ImageNet [19] are organized according to the WordNet hierarchy. An
ontology [71] contains lexico-semantic relations (_e.g_. hypernymy/hyponymy,
holonymy/meronymy) and the semantic relatedness (is-a, has-a) of concepts,
which can be used to group classes together or split them into finer semantic
concepts. In this work, the label spaces of the employed datasets are not
constructed according to an ontology, thus we used lexico-semantic relations
from WordNet, BabelNet, thesaurus.com, and merriam-webster.com to generate a
unified taxonomy of semantic concepts and minimize manual effort.
During training. A single classifier makes predictions over the consistent,
unified taxonomy of labels obtained from the previous step. Only during
training, these predictions are mapped back to the label space of each dataset
with specific per-dataset converters (Sec. IV-A2), in which case the
segmentation loss can be directly applied.
#### IV-A1 Generate unified taxonomy of label spaces
When combining multiple datasets it is highly probable that classes between
datasets have conflicting definitions, as described in Sec. III-B. In order to
solve these conflicts and generate a unified taxonomy, we introduce the
concept of the semantic atoms. A semantic atom $\alpha$ is a fine-level
semantic primitive (class), whose definition coincide either fully or
partially to a definition of a semantic class from a dataset in $\mathbb{S}$.
A set of properly chosen semantic atoms
$\mathcal{A}=\left\\{\alpha_{m},~{}m=1,\dots,A\right\\}$ fully covers the
semantics of all employed datasets without any conflicts. Using the ontologies
from the aforementioned online sources we automate the construction of set
$\mathcal{A}$. The set is subsequently validated by a human merely for
inconsistencies due to ambiguities in their (natural language) description
(Algorithm 1).
Data: label spaces from all datasets $\mathcal{L}^{(i)}$, $i=1,\dots,D$
Result: the set of semantic atoms $\mathcal{A}$
/* Initialize semantic atoms with the multi-set of all labels, then remove
ones not complying to conditions. */
all_labels $\leftarrow$ $\uplus_{i=1}^{D}\mathcal{L}^{(i)}$
repeat
/* generate combinations of labels pairs */
pairs = generate_combinations(all_labels)
for _pair in pairs_ do
ls, lo = pair
if _(synonym(ls, lo) or hypernym(ls, lo) or holonym(ls, lo))_ then
all_labels.remove(ls)
break
until _no label is removed from all_labels_
$\mathcal{A}$ $\leftarrow$ all_labels
Algorithm 1 Combine the labels from existing datasets to create the set of
semantic atoms using the lexico-semantic relations from WordNet, BabelNet,
thesaurus.com, and merriam-webster.com.
The following three properties hold for all semantic atoms. First, each
semantic atom should have a concise and unique semantic definition that does
not overlap with any of the other semantic atoms:
$\text{def}\left(\alpha_{k}\right)\cap\text{def}\left(\alpha_{m}\right)=\emptyset,~{}\forall~{}k\neq
m~{}.$ (9)
Second, its definition matches fully or partially to a definition of a
semantic class from a dataset and thus every semantic atom corresponds to
(is-a, hyponym, meronym) at most one semantic class:
$\text{def}(\alpha_{m})\subseteq\text{def}(l_{n}),~{}\forall\alpha_{m}\in\mathcal{A},~{}l_{n}\in\mathfrak{L}~{},$
(10)
where $\mathfrak{L}=\cup_{i}\mathcal{L}^{(i)},~{}i=1,\dots,D$ is the set of
labels from all label spaces of the datasets to be combined. Third, the set of
all semantic atoms should completely describe the semantics of all datasets,
which yields that every semantic class $l_{n}$ consists of (has-a, hypernym,
holonym) at least one semantic atom:
$\text{def}(l_{n})=\bigcup_{m\in M_{n}}\text{def}(\alpha_{m}),~{}\forall
l_{n}\in\mathfrak{L}~{},$ (11)
where $M_{n}$ is a set of indices corresponding to class $l_{n}$.
As long as the manual process for the extraction of semantic atoms is
completed, the taxonomy of semantic classes from all datasets can be generated
(see Figure 3). Then, we can train a single-classifier FCN using the semantic
atoms as output classes and supervise this CNN using the original label spaces
from each dataset, by combing the atoms using the generated unified taxonomy
as described in the next section.
#### IV-A2 Supervise semantic atoms with the original label spaces
Having extracted the set of semantic atoms $\mathcal{A}$ that fully covers the
semantics of the employed datasets $\mathbb{S}$, we can train a single-
backbone, single-classifier FCN with output label space $\mathcal{A}$. This
procedure is shown in Fig. 4 for a selection of semantic atoms from Fig. 3.
The output of the classifier for spatial position (pixel) $p$ and dataset $i$
is the categorical probability vector
$\bm{\sigma}^{(i)}_{p}\in\left[0,1\right]^{A}$, of cardinality
$A=\mathcal{A}$, where each element corresponds to the probability of a
semantic atom in $\mathcal{A}$. Since $\bm{\sigma}^{(i)}_{p}$ represents a
categorical probability it holds that $\sum_{m}\sigma^{(i)}_{p,m}=1$. In the
following, we describe how $\bm{\sigma}^{(i)}_{p}$ is transformed to be
compatible with the original label space $\mathcal{L}^{(i)}$ of each dataset
of the taxonomy, in order to train the classifier using the conventional
cross-entropy loss.
Conceptually, for each supervising dataset $i$, we are going to map the
categorical output $\bm{\sigma}_{p}^{(i)}$ to the categorical labels. Via this
mapping the labels of the original dataset can supervise (in)directly the
training of the semantic atoms. The extraction of semantic atoms induces a
collection of sets $\\{G^{(i)}_{m},~{}i=1,\dots,D,~{}m=0,\dots,L^{(i)}\\}$.
Each $G^{(i)}_{m}$ contains the semantic atoms that correspond to class
$l^{(i)}_{m}$ from dataset $i$. According to the taxonomy construction process
(Section IV-A1), the extracted semantic atoms fully describe the semantics of
all classes from all selected datasets. As a consequence, an arbitrary dataset
class is represented by either a single or a combination of semantic atom(s).
Using this property, we can partition $\bm{\sigma}^{(i)}_{p}$ into groups
according to sets $G^{(i)}_{m}$ and accumulate their probabilities into a
reduced vector $\bm{s}_{p}^{(i)}\in{[0,1]}^{L^{(i)}}$ for each dataset $i$.
This process can be written concisely as:
$s^{(i)}_{p,m}(\bm{\sigma}_{p})=\sum_{\alpha\in
G^{(i)}_{m}}\sigma^{(i)}_{p,\alpha}~{},$ (12)
where for now, an integer number is assigned as “name” to every atom $\alpha$,
so that they can be used for indexing $\mathcal{A}=\\{1,2,\dots,A\\}$. Since
$\bm{\sigma}^{(i)}_{p}$ is a categorical distribution, then $\bm{s}^{(i)}_{p}$
is also a categorical distribution. Moreover, it contains classes that
correspond one-by-one to the ground truth $\bm{y}^{(i)}_{p}$. Thus, they can
be used in the standard cross-entropy loss formulation.
During batch-wise training, a batch can contain images from many datasets.
Without loss of generality, we will formulate the cross-entropy loss for a
single image $j$ from dataset $i$ in the batch. The label $\bm{y}^{(i)}_{j}$
(Eq. (3)) has shape $H\times W\times L^{(i)}$ (one-hot encoding), and the
output $\bm{\sigma}^{(i)}_{j}$ has shape $H\times W\times A$. By using a
single index $p\in P$ to enumerate spatial positions ($H,W$) and omitting from
the notation of $\bm{y}$ and $\bm{\sigma}$ the dataset index $i$ and image
index $j$, the cross-entropy loss can be expressed as:
$\text{Loss}_{j}\left(\bm{y},\bm{\sigma}\right)=-\dfrac{1}{\left|P\right|}\sum_{p\in\mathcal{P}}\sum_{m}y_{p,m}\log
s^{(i)}_{p,m}~{}.$ (13)
In the following, we derive the gradients of the loss wrt. the logits of the
network and prove that our method is a generalization of the standard
formulation. As this is independent of the dataset and the position indices,
we drop them for minimizing notation clutter. The logits
$\bm{\lambda}\in\mathbb{R}^{A}$ are the input of the softmax $\bm{\sigma}$,
where $\sigma_{i}(\bm{\lambda})=e^{\lambda_{i}}/\sum_{j}e^{\lambda_{j}}$ and
the converted outputs of the network can be expressed as
$\bm{s}\left(\bm{\sigma}\left(\bm{\lambda}\right)\right)$. Using the
backpropagation rule the gradient of the loss wrt. the logits is:
$\frac{\partial\text{Loss}}{\partial\bm{\lambda}}=\frac{\partial\text{Loss}}{\partial\bm{s}}\cdot\frac{\partial\bm{s}}{\partial\bm{\sigma}}\cdot\frac{\partial\bm{\sigma}}{\partial\bm{\lambda}}~{}.$
(14)
Since each pixel in the annotations has a single class (one-hot), _i.e_.
$y_{m}=1$ for class $m=m^{*}$ and $y_{m}=0,~{}m\neq m^{*}$, the loss of Eq.
(13) (omitting the summation over positions $p$) reduces to
$-\sum_{m}\llbracket m=m^{*}\rrbracket\log s_{m}=-\log s_{m^{*}}$, where
$\llbracket\cdot\rrbracket$ is the Iverson bracket. It is easy to show that
the partial derivatives of the factors in Eq. (14) are:
$\partial\text{Loss}/\partial s_{i}=-\llbracket i=i^{*}\rrbracket/\sigma_{i}$,
$\partial s_{i}/\partial\sigma_{j}=\llbracket j\in G_{i}\rrbracket$, and
$\partial\sigma_{i}/\partial\lambda_{j}=\sigma_{i}\left(\llbracket
i=j\rrbracket-\sigma_{j}\right)$. Substituting these into Eq. (14) it yields:
$\frac{\partial\text{Loss}}{\partial\lambda_{m}}=\sigma_{m}-\llbracket m\in
G_{m^{*}}\rrbracket~{},$ (15)
which is a mere generalization of the loss derivative in the original FCN
framework, which is
$\partial\text{Loss}/\partial\lambda_{m}=\sigma_{m}-\llbracket
m=m^{*}\rrbracket$. This property ensures comparable gradient flows between
FCN and our framework, and thus no architectural changes or loss modifications
are needed.
Figure 5: Creation of pseudo labels from weakly-annotated datasets before
training. For each pixel in the image a categorical probability vector is
created with elements corresponding to the set of annotated classes ($L=3$ in
this example). For each position of $\hat{\mathbf{y}}$, each element along the
third dimension is assigned a probability, _i.e_. the normalized number of
respective bounding boxes covering that pixel. During training, these pseudo
labels are further refined using input from the classifier itself, see Fig. 6.
### IV-B Combine datasets with different annotation types
This section describes how a single-backbone, single-classifier FCN is trained
on multiple weakly- or strongly- labeled datasets. For now, we assume that the
label spaces of all datasets are identical. This limitation is lifted in Sec.
IV-C, where combined training with different annotation types and conflicting
labels spaces is investigated.
As explained in Sec. III-B, the spatial localization of annotations in weakly-
labeled datasets, _e.g_. bounding boxes and image tags, is inadequate for
providing useful pixel-level supervision. However, if properly conditioned or
refined, these spatial locations have the potential to provide helpful cues
for increasing segmentation performance. A two-step approach is followed,
abiding to the FCN design principles without adding extra modules to the
network. First, as described in Sec. IV-B1, weak annotations from all datasets
are converted to pseudo per-pixel labels, so they can be seamlessly used
together with pixel labels from strongly-labeled datasets for pixel-wise
training. Second, during each training step, the pseudo labels are refined,
using only information from the network from this step, without requiring any
external knowledge. The process is analyzed in Sec. IV-B2.
#### IV-B1 Unifying weak and strong annotations
The objective of unifying heterogeneous annotations is to transform the weak
annotations into per-pixel pseudo labels, so they can be integrated in the
pixel-wise training loss. The pseudo labels are then refined during training,
to provide a best-effort approximation of the ideal fine labels (Sec. IV-B2).
We consider weak supervision from bounding boxes and image-level tags, as
listed in Tab. I. Bounding box annotations have orthogonal boundaries, which
rarely match the smooth object boundaries, _e.g_. poles, humans, bicycles,
while image tags have even coarser localization. We treat image tags as
bounding boxes that extend to the whole image. This forms a basis to handle
both annotation formats within a common formulation.
The core of the method involves representing the per-pixel label as a
categorical probability vector $\hat{\bm{y}}_{p}\in\left[0,1\right]^{L}$ over
the set of all classes $\mathcal{L}$ of the dataset it belongs to. This choice
enables including information from all bounding boxes, even if they heavily
overlap, and does not require hard choices to assign a single class to each
pixel, _e.g_. assigning randomly or by heuristics. The algorithm is described
in the following paragraph and visualized in Figure 5.
Given a weak label $\mathbf{y}$, we initiate a 3-D label canvas, with two
spatial dimensions, being equal to the label size, and a depth dimension with
size $L$ for the semantic classes. Each bounding box in $\mathcal{B}$ casts a
unity vote to all spatial locations (pixels) being covered by it, at a single
position in the depth dimension that corresponds to the semantic class of the
bounding box. After the voting is completed from all boxes, the label canvas
is normalized along the depth dimensions (semantic classes) to unity, by
dividing by the sum of votes for that position. Then, another 2-D slice is
concatenated along the same spatial dimensions, corresponding to the unlabeled
semantic class. Finally, for pixels that are not covered by any bounding box,
the unlabeled probability is set to unity. At this point, a valid categorical
probability vector for each image pixel is obtained, which can be directly
used as-is in the conventional per-pixel cross-entropy loss.
#### IV-B2 Supervising semantic atoms with unified annotations
In the previous section, it was sketched how weak annotations are transformed
into per-pixel pseudo-labels that are used in the cross-entropy loss
formulation after their spatial locality is refined. Here, the refinement of
these coarse labels is described. The refinement is performed in an online
fashion during training and generates more (spatially) localized labels for
supervision. It is achieved by applying two conditions to pseudo-labels that
omit supervision for uncertain or ambiguous pixels, _e.g_. pixels that may
reside outside of an object.
Figure 6: Refinement of pseudo labels during each training step using the
predictions of the network in that step.
We assume that a collection of $\mathbb{S}^{(S)}$ strongly-labeled and
$\mathbb{S}^{(W)}$ weakly-labeled datasets are given and all datasets have an
identical label space. First, the weak labels $\bm{y}^{(W)}$ of
$\mathbb{S}^{(W)}$ are converted into per-pixel pseudo-labels
$\hat{\mathbf{y}}$, using the procedure of Sec. IV-B1. Then, during a training
step, we rely on the network predictions, which is the best estimate it can be
attained, to improve the pseudo-labels. If $\bm{\sigma}_{p}\in[0,1]^{L}$ is
the softmax output for position $p$ for an image with weak labels, the
predictions can be expressed as
$\pi_{p}=\operatorname*{arg\,max}_{m}\sigma_{p,m}$. The refined pseudo-labels
$\tilde{\mathbf{y}}_{p}$ are obtained for pixel $p$ by keeping the pseudo-
label if the prediction agrees and the corresponding probability is higher
than a threshold $T$, as follows:
$\tilde{\mathbf{y}}_{p}=\begin{cases}\hat{\mathbf{y}}_{p},&\text{if}~{}\pi_{p}=\operatorname*{arg\,max}_{m}\hat{y}_{p,m}~{}\text{and}~{}\sigma_{p,\pi_{p}}\geq
T,\\\ \text{unlabeled},&\text{otherwise}\end{cases}$ (16)
The first condition of Eq. (16) is illustrated in Fig. 6. The second condition
refers to the magnitude of the probability of the predictions, which can be
view as a measure of confidence. Specifically, a heuristically chosen
threshold $T$ is used, which should be exceeded by the probability of the
highest predicted class, in order to be deemed reliable. This threshold
provides a good trade-off between utilizing enough weak labels, while
maintaining their confidence high. For the experiments, we have empirically
chosen $T=0.9$.
The final loss for the batch with images from both weakly-labeled $(W)$ and
strongly-labeled $(S)$ datasets is computed as:
$\text{Loss}=-\sum_{p\in\mathcal{P}}\sum_{j}z_{p,j}\log\sigma_{p,j}~{},$ (17)
where $\mathcal{P}=\mathcal{P}^{(S)}\cup\mathcal{P}^{(W)}$ is the set of all
pixels and $z_{p,j}$ is defined as:
$\bm{z}_{p}=\begin{cases}\frac{1}{\lvert\mathcal{P}^{(S)}\rvert}\bm{y}^{(S)}_{p},&p\in\mathcal{P}^{(S)}\\\
\frac{1}{\lvert\mathcal{P}^{(W)}\rvert}\tilde{\bm{y}}^{(W)}_{p},&p\in\mathcal{P}^{(W)}~{}.\end{cases}$
(18)
### IV-C Conflicting label spaces and annotation types
Sec. IV-A proposed a solution for label-space conflicts considering only
strongly-labeled datasets. Sec. IV-B proposed a solution for training networks
with multiple annotation types considering only datasets with identical label
spaces. The combination of the two approaches that is able to simultaneously
train networks with any datasets (case of last row of Tab. II) is described in
this section.
The extraction of semantic atoms (Sec. IV-A1) and the conversion of weak to
per-pixel annotations (Sec. IV-B1) can be directly applied to the selected
datasets. However, for supervising the semantic atoms, the formulas of Sec.
IV-A2, and IV-B2 cannot be directly applied for any semantic atom and a small
set of them requires a different handling. According to this distinction, the
semantic atoms are split into two sets $\mathcal{A}^{a}$ and $\mathcal{A}^{s}$
with classes that need special care. Each atom in $\mathcal{A}^{a}$ is either
a class with only strong labels, or with weak and strong labels from different
datasets. Each atom in $\mathcal{A}^{s}$ is strictly a class with weak labels
for which the condition of Eq. (6) holds. The localization cues of the atoms
in $\mathcal{A}^{s}$ are extremely sparse, due to their weak annotations and
the fact that they do not appear in strongly-labeled datasets. Consequently,
the refinement step (Eq. (16)) is ineffective for pixel-accurate segmentation.
As a solution, for these atoms (_e.g_. the traffic sign subclasses in the
taxonomy of Fig. 3), we use the parent (strongly-labeled) classes
$\mathcal{A}^{p}$ (_e.g_. traffic sign front) as cues for pixel-accurate
segmentation. Then, fine-grained semantics can be attained using
classification over $\mathcal{A}^{s}$.
The predictions of the classifier, as illustrated in Fig. 7, are over two sets
of classes: $\mathcal{A}^{ap}=\mathcal{A}^{a}\cup\mathcal{A}^{p}$ and
$\mathcal{A}^{s}$. For the subclasses in $\mathcal{A}^{s}$ the relationship
between them and the parent classes $\mathcal{A}^{p}$ is leveraged.
Specifically, the predictions of the corresponding parent class in
$\mathcal{A}^{p}$ are used to provide cues for the refinement process (for
example the predicted segmentation masks of the traffic sign front class are
used to refine the bounding boxes of the traffic sign subclasses of Fig. 3).
The classifier is trained using the losses from Eq. (13) and (17). During
inference, the subclasses of $\mathcal{A}^{s}$ simply replace their parent
classes from $\mathcal{A}^{p}$ in the final predictions.
## V Experimental evaluation
Figure 7: Classifier structure in case of conflicting label spaces and mixed
(strong and weak) supervision. The final predictions (right block) are the
predictions of classifier part $\mathcal{A}^{ap}$, where the pixels that are
assigned classes belonging to classifier part $\mathcal{A}^{p}$ are replaced
by subclass predictions of classifier part $\mathcal{A}^{s}$.
We conduct extensive experimentation with various dataset combinations to
validate our methodology for multi-dataset simultaneous training with mixed
supervision and conflicting label spaces. The results are assessed on three
directions: i) segmentation performance on seen datasets, _i.e_.
validation/testing splits of training datasets, ii) generalization on unseen
datasets, _i.e_. splits of datasets not used during training, and iii)
semantic knowledgeability: number of recognizable semantic concepts. Sec. V-A
describes the evaluation metrics and Sec. V-B discusses the technical details
of our experiments. The following three Sections V-C, V-D, V-E investigate the
three scenarios in multi-dataset training appearing in Table II. Finally, Sec.
V-F contains ablations for specific design choices in this work. A selection
of diverse datasets for street scene understanding with strong and weak
supervision are used. An overview of the employed datasets is shown in Table
III. For each dataset a collection of label spaces is defined.
### V-A Evaluation metrics
We use two metric families to quantify the performance of the models. The
first family consists of the standard Intersection over Union (IoU) metric
[16, 10] and averages of it (arithmetic – mIoU) to summarize performance and
generalizability over multiple classes and across different datasets. The
second family is based on a new metric, namely Knowledgeability that we define
in the following subsection.
Dataset name | | Labels | # imgs | Train./Eval. Lab. space
---|---|---|---|---
Training datasets | | | |
Cityscapes [10] | | pixel | 2,975 | C-28
Cityscapes Coarse [10] | | pixel, bbox | 19,997 | C-28
Cityscapes T. Signs [35] | | pixel | 2,975 | CT-62
Mapillary Vistas [11] | | pixel | 18,000 | V-66
Indian Driving D. [12] | | pixel | 20,000 | I-33
EuroCity-Persons [72] | | bbox | 40,000 | E-2
Map. Traffic Signs [73] | | bbox | 36,589 | T-51
Open Images [20] | | bbox, im. tag | 100,000 | O-51
Testing datasets | | | |
Cityscapes (val) [10] | | pixel | 500 | C-20
Cityscapes T. Signs [35] | | pixel | 500 | CT-34
Mapil. Vistas (val) [11] | | pixel | 2,000 | V-66
IDD (val) [12] | | pixel | 2,036 | I-33
Generalization (unseen) datasets | |
Wild Dash (val) [74] | | pixel | 70 | W-19
KITTI (train) [75] | | pixel | 200 | K-20
TABLE III: Overview of employed datasets for experimentation. The type of annotations, the number of images and the label spaces are shown. The networks are trained with the label spaces of the training datasets. The evaluation label spaces may be smaller than the training counterparts for the same dataset, due to missing classes in testing/generalization datasets or smaller official splits. Train datasets | Output label space | Accuracy [mIoU %] | Knowledgeability [$\mathcal{K}^{L(\text{dataset})}$ %]
---|---|---|---
Citys | IDD | Vistas | Zero-shot Generalization | Val-split Generalization | Unseen datasets | Seen datasets
WildDash | KITTI | Citys | IDD | Vistas | WildDash | KITTI | Citys | IDD | Vistas
✓ | | | C-20 | 27.8 | 48.0 | 63.0 | 29.8 | 22.5 | 39.2 | 43.1 | 50.3 | 30.4 | 26.0
| ✓ | | I-33 | 36.8 | 40.2 | 47.3 | 63.4 | 23.7 | 48.6 | 50.9 | 55.3 | 65.0 | 49.1
| | ✓ | V-66 | 42.4 | 49.5 | 67.6 | 41.0 | 40.9 | 53.2 | 57.5 | 60.3 | 53.0 | 63.5
✓ | ✓ | | HTSS-34 | 36.3 | 55.3 | 73.0 | 61.3 | 26.4 | 49.9 | 50.5 | 56.8 | 58.5 | 49.8
✓ | | ✓ | HTSS-66 | 44.4 | 51.6 | 61.2 | 43.7 | 43.1 | 60.0 | 62.4 | 62.3 | 61.0 | 62.9
| ✓ | ✓ | HTSS-68 | 44.0 | 53.8 | 69.2 | 58.0 | 42.6 | 59.5 | 63.0 | 60.1 | 62.4 | 63.0
✓ | ✓ | ✓ | HTSS-68 | 44.4 $\left\lceil+16.6\right\rceil$ | 56.5 $\left\lceil+16.3\right\rceil$ | 74.9 +11.9 | 57.9 -5.5 | 43.1 +2.2 | 64.2 $\left\lceil+25.0\right\rceil$ | 64.4 $\left\lceil+21.3\right\rceil$ | 66.7 +16.4 | 65.0 +0.0 | 68.3 +4.8
TABLE IV: HTSS on pixel-labeled datasets with conflicting label spaces.
Performance of combined training (bottom rows) compared to single-dataset
training (top rows). $L(\text{dataset})$ = 19, 20, 20, 33, 66 for the 5
datasets. $\left\lceil\text{+x}\right\rceil$: up to +x%.
Knowledgeability metric. This metric quantifies in a single value the semantic
richness of the output of a semantic segmentation system, by evaluating how
many semantic concepts (atoms) it can recognize with sufficient segmentation
accuracy. Using existing metrics, one way to achieve this is to report the
size of the system’s output label space and separately the IoU performance per
class. However, this approach has some pitfalls. First, merely reporting the
label space size is not a reliable metric for semantic richness of
predictions, since the IoU performance for some classes can be very low or
even zero. Second, IoU-based average aggregates, _e.g_. mIoU, do not reflect
the number of recognizable classes, because they assess a system purely at
segmentation level. Finally, these aggregates are intrinsically dependent on
the size of the evaluated label space: as the size increases, the difficulty
of assigning the correct class increases, eventually leading to mIoU reduction
(due to the smoothing properties of averaging). The new metric is designed to
explicitly consider the size of the label spaces of both the system output and
the evaluated dataset together with the segmentation performance for the
output classes.
The core of the metric is based on counting the number of classes that achieve
an IoU higher than a threshold $t$ wrt. the total number of classes $c$ that
are considered for computing the metric. To make the metric independent of the
need for proper selection of $t$, the counting is averaged over a set of
$N_{T}$ thresholds, which in this work are chosen to be equidistant, _i.e_.
$T=\\{0.0,~{}1/N_{T},~{}\dots,~{}1.0-1/N_{T}\\}$. Other values for T can be
chosen depending on the application and datasets specifics. Assuming an output
label space of a model that contains $L$ discreet semantic classes, the set of
all per-class IoUs $\mathcal{E}=\\{\text{IoU}_{i}\\}_{i=1}^{L}$ can be
constructed by evaluating the model output against the ground truth.
Subsequently, the set $\mathcal{E}$ is used to generate all the subsets
$\tilde{\mathcal{E}}_{t}=\\{\text{IoU}~{}|~{}\text{IoU}>t,~{}\text{IoU}\in\mathcal{E}\\}$
containing the IoUs above the threshold $t$ from $T$. To this end,
Knowledgeability is defined as the $c$-normalized number of classes averaged
over $T$:
$\mathcal{K}^{c}_{T}=\frac{1}{N_{T}}\sum_{t\in
T}\frac{\min(|\tilde{\mathcal{E}}_{t}|,c)}{c}~{}.$ (19)
This definition guarantees $\mathcal{K}^{c}_{T}$ to be between 0 and 1,
_i.e_.
$0\leq\mathcal{K}\leq\mathcal{K}_{\text{max}}=\min\left(L,c\right)/c\leq 1$,
which is achieved by creating the sets $\tilde{\mathcal{E}}_{t}$ using the
strictly greater condition ($\text{IoU}>t$) and by employing the $\min$
function. The bounds enable the use of the metric for comparison across
datasets with different number of classes and semantic segmentation systems
with different number of output classes. The new metric Knowledgeability
allows us to express the increase in the number of recognizable classes and at
the same time consider the performance on these classes, without the need to
choose a specific single threshold.
### V-B Implementation details
Convolutional network architecture. The convolutional network follows the PSP
architecture [76], since it provides a good trade-off between segmentation
performance, training time, and memory requirements. The backbone feature
extractor is a ResNet-50 [9] modified for segmentation by i) changing the
block strides and dilation rates [77, 76], ii) projecting the output
2048-dimension features to 256-dimension features to reduce memory
requirements, and iii) using a Pyramid Pooling Module [76], as described in
[35].
Hyper-parameter tuning and implementation details. Two factors that emerge in
multi-dataset training and that significantly affect the accuracy of CNNs for
segmentation are the batch size and the input image size. The global and per-
dataset batch size are connected with the robustness of the optimization
algorithm (SGD) and determine training balance across all classes. As the
number of the output classes increase, bigger batch sizes are required. The
second factor, _i.e_. the input image size of the feature extractor,
determines the scale and detail of the extracted features throughout the
network. Ideally one would like to use high batch sizes and image sizes given
the available (fixed) compute infrastructure. However, using more datasets for
multi-dataset training forces using larger batch sizes, to guarantee that all
dataset are represented in the batch. This in turn, requires reducing the
image size for satisfying of the available (fixed) compute infrastructure.
Each of our experiments (subsections) involve training with different number
of datasets and classes, thus we tune these two hyper-parameters separately
per experiment to obtain optimal performance given the constraints. This leads
to different baseline performance for each experiment and results can
therefore not be compared across different experiments. But, within a single
experiment, we guarantee that all methods achieve optimal results by finding
the empirically best trade-off between batch size and image size given the
memory constraints. The experiments are conducted on a machine with 4 Titan V
GPUs.
Training datasets | Output label space | Zero-shot generalization | Val-split generalization
---|---|---|---
pixel-labeled | bbox-labeled | WildDash (unseen) | Cityscapes (seen)
Citys | CitysC | ECP | CitysC | ECP-2 | CitysC-10 | All classes | ECP-2 | CitysC-10 | All classes
✓ | | | | C-20 | 12.9 | 16.7 | 23.0 | 65.0 | 69.6 | 61.3
✓ | ✓ | | | C-20 | 13.8 | 17.4 | 23.1 | 65.2 | 70.1 | 61.5
✓ | | ✓ | | HTSS-20 | 31.7 $+18.8$ | 20.7 $+4.0$ | 22.7 $-0.3$ | 66.7 $+1.7$ | 70.5 $+0.9$ | 61.3 $+0.0$
✓ | | | ✓ | HTSS-20 | 31.4 $\left\lceil+17.6\right\rceil$ | 20.8 $\left\lceil+3.4\right\rceil$ | 24.4 $\left\lceil+1.4\right\rceil$ | 65.9 $\left\lceil+0.9\right\rceil$ | 70.5 $\left\lceil+0.9\right\rceil$ | 63.4 $\left\lceil+2.1\right\rceil$
✓ | | ✓ | ✓ | HTSS-20 | 31.1 $\left\lceil+18.2\right\rceil$ | 26.2 $\left\lceil+9.5\right\rceil$ | 22.4 $\left\lceil-0.6\right\rceil$ | 64.0 $\left\lceil-1.0\right\rceil$ | 71.6 $\left\lceil+2.0\right\rceil$ | 61.7 $\left\lceil+0.4\right\rceil$
TABLE V: HTSS on pixel-labeled and bounding-box-labeled datasets with non-
conflicting label spaces. Accuracy [mIoU %] on seen and unseen datasets for
the specific class subsets (ECP-2, CitysC-10) that receive the extra weak
supervision from ECP and CitysC and all classes. The pixel-labeled CitysC (row
2) is used to set the oracle for the experiments involving the weakly-labeled
CitysC (rows 4, 5). C-20 and HTSS-20 label spaces coincide, the HTSS prefix is
used to emphasize mixed-supervision training using the HTSS framework.
$\left\lceil\text{+x}\right\rceil$: up to +x%.
### V-C Strong supervision, conflicting label spaces
In the first set of experiments (Tab. IV), we focus on combining pixel-labeled
datasets with conflicting label spaces (case of third row in Tab. II). This
scenario is commonly occurring, where different pixel-labeled datasets (for
semantic segmentation) are annotated at conflicting levels of semantic
granularity. In the experiments, the label spaces of three datasets
(Cityscapes, Vistas, IDD) are combined, as described in Sec. IV-A, resulting
in the taxonomy of Fig. 3 (without the MTS dataset). The direct solution of
training with the union of datasets and their label spaces is not applicable,
since the semantic conflicts among the label spaces introduce ambiguities in
the concatenated output label space. For example, the rider Cityscapes class
conflicts with the motorcyclist and bicyclist Vistas classes. These conflicts
are resolved by generating a universal taxonomy (Sec. IV-A).
We train on all combinations of three pixel-labeled datasets using the HTSS
methodology after solving the conflicts in their label spaces. We compare the
accuracy and knowledgeability (Sec. V-A) between HTSS networks against single-
dataset trainings. For these experiments, the input image size is $799\times
799$ and batch size formation is 1 image from Cityscapes, 2 from Vistas and 1
from IDD, for each GPU.
Accuracy results. As can be seen in Tab. IV, the HTSS-68 network trained on
all datasets outperforms 6 out of 7 other networks for seen (columns 3-5) or
unseen datasets (columns 1-2). The improvements are due to training a single
model with diverse images from multiple datasets (9 times more than Cityscapes
and 2 times more than Vistas), which is possible after solving conflicts in
the label spaces. An exception is the case of the IDD val split, where the
single-dataset training matches or outperforms HTSS networks. After careful
visual examination of the dataset, we have observed that the semantic
annotations of IDD have a high degree of overlapping concepts. For example,
the road class and the drivable fallback class have partially overlapping
definitions, _i.e_. they both contain semantic atoms like pothole or
crosswalk. This contradicts our hypothesis on assuming non-conflicting
semantic class definitions (refer to Eq. (5)) and possibly explains the
discrepancy in the results.
Knowledgeability results. The HTSS networks are able to segment, in a single
pass, an image over more semantic concepts with high attainable mIoU, as
demonstrated by Knowledgeability. Moreover, this is proportional to the size
of the output label space (_e.g_. columns 6, 7, 10).
### V-D Strong & weak supervision, non-conflicting label spaces
This section explores HTSS on a mix of strongly- and weakly-labeled datasets
that have non-conflicting label spaces. The hypothesis under investigation is
that a very large quantity of weak labels (exponentially more samples but with
poor localization) can be beneficial if used in combination with strong
labels, especially for under-represented classes.
Using the approach developed in Sec. IV-B we train HTSS networks on the pixel-
labeled Cityscapes and Cityscapes Coarse, the bounding-box-labeled ECP, and
the bounding-box-labeled Cityscapes Coarse, that we generated from the pixel
labels (see Tab. III). The label space used is the common 20 classes for
Cityscapes (C-20), which we call also HTSS-20, as it is a subset of the
complete taxonomy of Fig. 3. The results are provided in Tab. V. For these
experiments, the input image size is $699\times 699$ and batch size formation
is 1 image from Cityscapes, 2 from Cityscapes Coarse, and 2 from ECP, for each
GPU.
There are two interesting comparisons to discuss. First, training with
Cityscapes and Cityscapes Coarse irrespective of the supervision type (rows 2
and 4) should in principle give similar performance since the supervision
information is the same. However the HTSS network shows higher accuracy for
all class subsets either in the same or different image domains. Specifically,
the two underrepresented classes of Cityscapes (person and rider, _i.e_.
ECP-2) have the higher difference in accuracy by +18.8% for Wild Dash. This
indicates that the generated pixel pseudo-labels by HTSS (see Fig. 6) provide
more complete training cues than the original pixel labels of Cityscapes
Coarse. The second comparison (rows 1 and 3) highlights that bounding-box
labels from ECP (different image domain) increase the generalization accuracy
compared to
From the results of Tab. V we conclude that the HTSS framework can
successfully leverage weak supervision to increase segmentation accuracy on
selected classes (_e.g_. vulnerable road users, _i.e_. ECP-2) and at the same
time maintain or slightly improve the accuracy achieved by strong supervision
(Cityscapes).
A second experiment examines the segmentation accuracy of specific classes
when adding weak supervision from bounding boxes and image tags. The results
are provided in Tab. VI) and demonstrate that an increasing amount of weak
supervision improves the segmentation accuracy accordingly. Using weak
supervision from the Open Images dataset (rows 2 and 3) improves the average
mIoU accuracy up to +2.9%, while specific classes are substantially benefited
with an increase of up to +13.2% IoU. The inclusion of image-tag supervision,
improves or maintains IoUs for 6 out of 8 classes, however the improvement is
less significant compared to bounding-box supervision only. This shows that
weaker and less localized forms of supervision have smaller gains in
segmentation performance.
| | | Val-split generalization - Cityscapes (seen)
---|---|---|---
Train datasets | Bicycle | Bus | Car | Motorc. | Train | Truck | Vehicle mIoU | Person | Rider | Human mIoU
Citys | Open Im.
pixel | bbox | tag | | | | | | | | | |
✓ | | | 67.8 | 80.1 | 92.3 | 51.9 | 69.6 | 63.2 | 70.8 | 70.9 | 48.5 | 59.7
✓ | ✓ | | 68.7 | 82.1 | 92.9 | 50.2 | 69.8 | 71.9 | 72.6 | 72.5 | 51.2 | 61.9
✓ | ✓ | ✓ | 69.1 | 79.7 | 92.8 | 48.9 | 69.6 | 76.3 | 72.7 | 72.9 | 52.5 | 62.7
TABLE VI: HTSS on pixel-, bounding-box-, and image-tag- labeled images with non-conflicting label spaces. Accuracies (mIoU %) for the specific classes that receive extra supervision from the weakly-labeled Open Images dataset. Train datasets | | | | | |
---|---|---|---|---|---|---
pixel | bbox | | | | | |
Citys | CitysT | CitysT | MTS | Output label space | WildDash | Cityscapes T. Signs (seen)
W-19 | CitysT-14 | C-20 | CitysT-34
mIoU | mIoU | mIoU | mIoU | $\mathcal{K}^{34}$
✓ | | | | C-20 | 27.1 | n/a | 70.3 | n/a | 39.3
✓ | ✓ | | | CT-34 | 27.8 | 17.7 | 69.5 | 47.5 | 46.2
✓ | | ✓ | | HTSS-34 | 30.2 $\left\lceil+3.1\right\rceil$ | 17.0 | 69.8 $\left\lceil+0.3\right\rceil$ | 46.9 | 44.5 $\left\lceil+5.2\right\rceil$
✓ | | | ✓ | HTSS-70 | 28.9 | 11.6 | 70.7 | 45.6 | 44.3
TABLE VII: HTSS on pixel-labeled and bounding-box-labeled datasets with
conflicting label spaces. Performance on seen and unseen datasets for all
classes (W-19, CitysT-34) and for class subsets (C-20, CitysT-14). Cityscapes
Traffic Signs (CitysT) is originally a pixel-labeled dataset, which we convert
to bounding boxes [43] (bbox column). CitysT-34 = C-20 $\cup$ CitysT-14,
HTSS-70 = C-20 $\cup$ MTS-50.
### V-E Strong & weak supervision, conflicting label spaces
The most challenging scenario involves multi-dataset training with mixed
supervision and conflicting label spaces. To investigate this scenario, we
augment the label space of Cityscapes with traffic sign classes from the
bounding-box-labeled datasets Cityscapes Traffic Signs (14 t. signs) and MTS
(50 t. signs). In this case, the methodology developed in Sec. IV-C is
employed, the HTSS network is trained with input image size of $599\times
599$, and the batch size arrangement is 1 image from Cityscapes, 3 from
Cityscapes Traffic Signs, and 3 from Mapillary Traffic Signs, for each GPU.
Table VII shows the results. We evaluate all networks on the original
Cityscapes classes (C-20), the traffic sign classes subset (CitysT-14), all
CitysT-34 classes (CitysT-14 traffic signs + C-20), as well as, on the unseen
Wild Dash, whose classes are a subset of Cityscapes (W-19 $\subset$ C-20).
Comparing row 3 (HTSS-34) with the first two rows we observe that using the
HTSS methodology the network is able to learn from weak supervision and even
surpass the fully pixel-level supervised oracle (row 2) in some cases.
Moreover, the results of the last row indicate that using MTS, which has a
different image domain than Cityscapes, the network is able to segment
Cityscapes traffic signs to some extent (11.6%). Overall, these experiments
demonstrate the ability to: i) learn from mixed supervision, ii) learn from
the same or different image domains, and iii) maintain or slightly improve
generalization accuracy for the pixel-labeled classes (columns 1, 3).
((a)) Single-dataset training on Cityscapes.
((b)) HTSS on Cityscapes + Vistas + IDD.
Figure 8: Progress of features, while adding more datasets, as a 2-D t-SNE visualization, when using same t-SNE hyper-parameters. Clusters become less scattered (intra-class distance) and better separated (inter-class distance). Model | Conflicts resolution | Memory $\Delta$ Params | Inference $\Delta$ ms | Unseen dts. mmIoU | $\mathcal{K}^{66}$
---|---|---|---|---|---
one per dataset | post-proc. merging | $+5.4\cdot 10^{7}$ | $+127.1$ | 46.2 | 62.6
shared backbone, head per dataset | common classes | $+2.1\cdot 10^{5}$ | $+1.7$ | 34.2 | 45.6
post-proc. merging | $+2.1\cdot 10^{5}$ | $+1.8$ | 45.4 | 62.8
shared backbone, single head | common classes | reference | reference | 30.2 | 42.1
semantic atoms (ours) | $+5.1\cdot 10^{3}$ | $+0.0$ | 50.5 | 68.3
TABLE VIII: Common baselines methodologies for combining three pixel-labeled
datasets (Cityscapes, Vistas, IDD) with conflicting label spaces. All methods
use ResNet-50 backbones and softmax classifiers. The $\Delta$’s for the total
number of parameters (Params) and the single-image inference time (ms) are
w.r.t. the reference row 4, _i.e_. keeping only the common, non-conflicting
classes from all datasets.
### V-F Ablations and Insights
An analysis is provided for the conducted experiments. First, an ablation on
how the amount of weak supervision affects performance is presented in Tab. IX
for the experiment of Sec. V-E. An increasing number of images and bounding
boxes from the weakly-labeled dataset are added per step. We observe that as
weakly-labeled images are included, the segmentation performance increases
accordingly.
Second, we provide t-SNE plots in Fig. 8 for experiments from Tab. IV. The
plots capture the 2-D projections of the output of feature extractor before
and after adding multiple datasets. It can be seen that the representations
have better properties from the perspective of classificability/separability.
Finally, we provide comparisons of the HTSS methodology against various
baselines in Tab. VIII examining memory and time factors wrt. the attained
performance. The first three rows describe direct solutions using existing
trained networks and post processing for solving conflicts. The single-network
approach (fourth row) is the closest to HTSS, but resolves conflicts by
maintaining only the common classes. This leads to a significant loss in
Knowledgeability, as the number of recognizable classes reduce. Overall, the
HTSS approach uses a reduced number of parameters and performs fast inference,
since it uses a common backbone and a single classifier.
pixel | bbox-labeled | Cityscapes (C-20)
---|---|---
Citys | Open Images | mAcc | mIoU
✓ | - | 81.2 | 70.2
✓ | $1k$ images ($17.3k$ bboxes) | 80.7 | 69.8
✓ | $10k$ images ($140.4k$ bboxes) | 81.6 | 70.6
✓ | $100k$ images ($1185.8k$ bboxes) | 83.7 | 72.3
TABLE IX: Segmentation accuracy with different number of bounding boxes used
to generate pseudo labels from the weakly-labeled Open Images.
## VI Conclusion
We presented the HTSS framework for simultaneously training FCNs on multiple
heterogeneous datasets for semantic segmentation. We explored heterogeneous
training with various combinations of strongly (pixel) and weakly (bounding
boxes, image tags) labeled datasets. The experiments showed that HTSS
improved, in the majority of the combinations, three aspects of the predicted
results: i) the segmentation performance on test splits of seen (training)
datasets, generalization on unseen datasets, and awareness of semantic
concepts, expressed by the proposed Knowledgeability metric. Our HTSS
framework does not require any extra labeling for weakly-labeled datasets,
while the only manual step consists of defining a semantic taxonomy of the
label spaces of the employed datasets, solely in the case of conflicting label
spaces. HTSS allows supplementing the pixel-labeled training data with other
relevant datasets that otherwise would not be compatible. These properties
make the HTSS approach useful to many applications in which training data for
semantic segmentation are too scarce to achieve required performance.
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Panagiotis Meletis Panagiotis Meletis holds a PhD from the Eindhoven
University of Technology. He works on advancing deep learning techniques for
autonomous vehicles within the Mobile Perception Systems lab. His research
involves the design of novel algorithms towards holistic visual scene
understanding. Panagiotis received his MSc in Electrical and Computer
Engineering from the National Technical University of Athens in 2015. His
current research interests include deep learning techniques for machine
perception, automated visual reasoning, and explainable and sustainable
artificial intelligence.
---
Gijs Dubbelman PhD, is an associate professor with the Eindhoven University of
Technology and heading the Mobile Perception Research (MPS) lab, which aims to
improve the real-time perception capabilities of mobile sensor platforms
through research on artificial intelligence. Key research areas of the MPS lab
are 3-D computer vision, multi-modal pattern recognition, and deep learning.
Formerly, Gijs Dubbelman was a member of the Field Robotics Center of Carnegie
Mellon’s Robotics Institute, where he performed research on Visual-SLAM for
autonomous robots and vehicles.
---
|
# Factored couplings in multi-marginal optimal transport via difference of
convex programming
Quang Huy Tran
Univ. Bretagne-Sud, CNRS, IRISA
F-56000 Vannes
<EMAIL_ADDRESS>
&Hicham Janati
École Polytechnique, CMAP, UMR 7641
F-91120 Palaiseau
<EMAIL_ADDRESS>
Ievgen Redko
Univ Lyon, UJM-Saint-Etienne, CNRS, UMR 5516
F-42023 Saint-Etienne
<EMAIL_ADDRESS>
&Rémi Flamary
École Polytechnique, CMAP, UMR 7641
F-91120 Palaiseau
<EMAIL_ADDRESS>
&Nicolas Courty
Univ. Bretagne-Sud, CNRS, IRISA
F-56000 Vannes
<EMAIL_ADDRESS>
###### Abstract
Optimal transport (OT) theory underlies many emerging machine learning (ML)
methods nowadays solving a wide range of tasks such as generative modeling,
transfer learning and information retrieval. These latter works, however,
usually build upon a traditional OT setup with two distributions, while
leaving a more general multi-marginal OT formulation somewhat unexplored. In
this paper, we study the multi-marginal OT (MMOT) problem and unify several
popular OT methods under its umbrella by promoting structural information on
the coupling. We show that incorporating such structural information into MMOT
results in an instance of a different of convex (DC) programming problem
allowing us to solve it numerically. Despite high computational cost of the
latter procedure, the solutions provided by DC optimization are usually as
qualitative as those obtained using currently employed optimization schemes.
## 1 Introduction
Broadly speaking, the classic OT problem provides a principled approach for
transporting one probability distribution onto another following the principle
of the least effort. Such a problem, and the distance on the space of
probability distributions derived from it, arise in many areas of machine
learning (ML) including generative modeling, transfer learning and information
retrieval, where OT has been successfully applied. A natural extension of
classic OT, in which the admissible transport plan (a.k.a coupling) can have
more than two prescribed marginal distributions, is called the multi-marginal
optimal transport (MMOT) (Gangbo and Swiech, 1998). The latter has several
attractive properties: it enjoys the duality theory (Kellerer, 1984) and finds
connections with the Wasserstein barycenter problem (Agueh and Carlier, 2011)
used for data averaging and probabilistic graphical models (Haasler et al.,
2020). While being far less popular than the classic OT with two marginals,
MMOT is a very useful framework on its own with some notable recent
applications in generative adversarial networks (Cao et al., 2019), clustering
(Mi and Bento, 2020) and domain adaptation (Hui et al., 2018; He et al.,
2019), to name a few.
The recent success of OT in ML is often attributed to the entropic
regularization (Cuturi, 2013) where the authors imposed a constraint on the
coupling matrix forcing it to be closer to the independent coupling given by
the rank-one product of the marginals. Such a constraint leads to the
appearance of the strongly convex entropy term in the objective function and
allows the entropic OT problem to be solved efficiently using simple Sinkhorn-
Knopp matrix balancing algorithm. In addition to this, it was also noticed
that structural constraints allow to reduce the prohibitively high sample-
complexity of the classic OT problem (Genevay et al., 2019; Forrow et al.,
2019). While these and several other recent works (Lin et al., 2021; Scetbon
et al., 2021) addressed structural constraints in the classic OT problem with
two marginals, none of them considered a much more challenging case of doing
so in a multi-marginal setting. On the other hand, while the work of (Haasler
et al., 2020) considers the MMOT problem in which the cost tensor induced by a
graphical structure, it does not naturally promote the factorizability of
transportation plans.
#### Contributions
In this paper, we define and study a general MMOT problem with structural
penalization on the coupling matrix. We start by showing that a such
formulation includes several popular OT methods as special cases and allows to
gain deeper insights into them. We further consider a relaxed problem where
hard constraints are replaced by a regularization term and show that it leads
to an instance of the difference of convex programming problem. A numerical
study of the solutions obtained when solving the latter in cases of interest
highlights their competitive performance when compared to solutions provided
by the optimization strategies used previously.
## 2 Preliminary knowledge
#### Notations.
For each integer $n\geq 1$, we write $[n]:=\\{1,...,n\\}$. For each discrete
probability measure $\mu$ with finite support, its negative entropy is defined
as $H(\mu)=\langle\mu,\log\mu\rangle$, where the logarithm operator is
element-wise, with the convention that $0\log 0=0$. Here,
$\langle\cdot,\cdot\rangle$ denotes the Frobenius inner product. The Kullback-
Leibler divergence between two discrete probability measures $\mu$ and $\nu$
with finite supports is defined as
$\text{KL}(\mu|\nu)=\begin{cases}\langle\mu,\log\frac{\mu}{\nu}\rangle,\text{
if }\mu\text{ is absolutely continuous with respect to }\nu\\\ \infty,\text{
otherwise}.\end{cases}$
where the division operator in the logarithm is element-wise.
In what follows, we write $\mathcal{T}=(1,...,N)$, for some integer $N\geq 1$.
For any positive integers $a_{1},...,a_{N}$, we call
$P\in\mathbb{R}^{a_{1}\times...\times a_{N}}$ a $N$-D tensor. In particular, a
$1$-D tensor is a vector and $2$-D tensor is a matrix. A tensor is a
probability tensor if its entries are nonnegative and the sum of all entries
is $1$. Given $N$ probability vectors $\mu_{1},...,\mu_{N}$, we write
$\mu=(\mu_{n})_{n=1}^{N}$. We denote $\Sigma_{\mathcal{T}}$ the set of $N$-D
probability tensors and $U_{\mathcal{T}}\subset\Sigma_{\mathcal{T}}$ the set
of tensors whose $N$ marginal distributions are $\mu_{1},...,\mu_{N}$. Any
coupling in $U_{\mathcal{T}}$ is said to be admissible. We write
$\mu_{\mathcal{T}}:=\mu_{1}\otimes...\otimes\mu_{N}$ the tensor product (or
measure product) of $\mu_{1},...,\mu_{N}$.
#### Multi-marginal OT problem.
Given a collection of $N$ probability vectors
$\mu=(\mu_{n}\in\mathbb{R}^{a_{n}})_{n=1}^{N}$ and a $N$-D cost tensor
$C\in\mathbb{R}^{a_{1}\times...\times a_{N}}$, the MMOT problem reads
$\text{MMOT}(\mu)=\inf_{P\in U_{\mathcal{T}}}\langle C,P\rangle.$ (1)
In practice, such a formulation is intractable to optimize in a discrete
setting as it results in a linear program where the number of constraints
grows exponentially in $N$. A more tractable strategy for solving MMOT is to
consider the following entropic regularization problem
$\inf_{P\in U_{\mathcal{T}}}\langle
C,P\rangle+\varepsilon\text{KL}(P|\mu_{\mathcal{T}}).$ (2)
which can be solved using Sinkhorn’s algorithm Benamou et al. (2014). We refer
the interested reader to Supplementary materials for algorithmic details.
## 3 Factored Multi-marginal Optimal Transport
In this section, we first define a factored MMOT (F-MMOT) problem where we
seek to promote a structure on the optimal coupling given such as a
factorization into a tensor product. Interestingly, such a formulation can be
shown to include several other OT problems as special cases. Then, we
introduce a relaxed version called MMOT-DC where the factorization constraint
is smoothly promoted through a Kullback-Leibler penalty.
### 3.1 Motivation
Before a formal statement of our problem, we first give a couple of motivating
examples showing why and when structural constraints on the coupling matrix
can be beneficial. To this end, first note that a trivial example of the
usefulness of such constraints in OT is the famous entropic regularization.
Indeed, while most of the works define the latter by adding negative entropy
of the coupling to the classic OT objective function directly, the original
idea was to constraint the sought coupling to remain close (to some extent) to
a rank-one product of the two marginal distributions. The appearance of
negative entropy in the final objective function is then only a byproduct of
such constraint due to the decomposition of the KL divergence into a sum of
three terms with two of them being constant. Below we give two more examples
of real-world applications related to MMOT problem where a certain
decomposition imposed on the coupling tensor can be desirable.
#### Multi-source multi-target translation.
A popular task in computer vision is to match images across different domains
in order to perform the so-called image translation. Such tasks are often
tackled within the GAN framework where one source domain from which the
translation is performed, is matched with multiple target domains modeled
using generators. While MMOT was applied in this context by Cao et al. (2019)
when only one source was considered, its application in a multi-source setting
may benefit from structural constraints on the coupling tensor incorporating
the human prior on what target domains each source domain should be matched
to.
#### Multi-task reinforcement learning.
In this application, the goal is to learn individual policies for a set of
agents while taking into account the similarities between them and hoping that
the latter will improve the individual policies. A common approach is to
consider an objective function consisting of two terms where the first term is
concerned with learning individual policies, while the second forces a
consensus between them. Similar to the example considered above, MMOT problem
was used to promote the consensus across different agents’ policies in (Cohen
et al., 2021), even though such a consensus could have benefited from a prior
regarding the semantic relationships between the learned tasks.
### 3.2 Factored MMOT and its relaxation
We start by giving several definitions used in the following parts of the
paper.
###### Definition 3.1 (Tuple partition)
A sequence of tuples $(\mathcal{T}_{m})_{m=1}^{M}$, with $M\leq N$, is called
a partition of a $N$-tuple $\mathcal{T}$ if the tuples
$\mathcal{T}_{1},...,\mathcal{T}_{M}$ are nonempty and disjoint, and their
concatenation gives $\mathcal{T}$.
Here, we implicitly take into account the order of the tuple, which is not the
case for the partition of a set. If there exists a tuple in
$(\mathcal{T}_{m})_{m=1}^{M}$ which contains only one element, then we say
$(\mathcal{T}_{m})_{m=1}^{M}$ is degenerate.
###### Definition 3.2 (Marginal tensor)
Given a tensor $P\in\mathbb{R}^{a_{1}\times...\times a_{N}}$ and a partition
$(\mathcal{T}_{m})_{m=1}^{M}$ of $\mathcal{T}=(1,...,N)$, we call $P_{\\#m}$
the $\mathcal{T}_{m}$-marginal tensor, by summing $P$ over all dimensions not
in $\mathcal{T}_{m}$. We write $P_{\\#\mathcal{T}}=P_{\\#1}\otimes...\otimes
P_{\\#M}\in\mathbb{R}^{a_{1}\times...\times a_{N}}$ the tensor product of its
marginal tensors.
For example, for $M=N=2$ and $\mathcal{T}=(1,2)$, we have
$\mathcal{T}_{1}=(1)$ and $\mathcal{T}_{2}=(2)$. So, given a matrix
$P\in\mathbb{R}^{a_{1}\times a_{2}}$, its marginal tensors $P_{\\#1}$ and
$P_{\\#2}$ are vectors in $\mathbb{R}^{a_{1}}$ and $\mathbb{R}^{a_{2}}$,
respectively, defined by $(P_{\\#1})_{i}=\sum_{j}P_{ij}$ and
$(P_{\\#2})_{j}=\sum_{i}P_{ij}$ for $(i,j)\in[a_{1}]\times[a_{2}]$. The tensor
product $P_{\\#\mathcal{T}}\in\mathbb{R}^{a_{1}\times a_{2}}$ is defined by
$(P_{\\#\mathcal{T}})_{ij}=(P_{\\#1})_{i}(P_{\\#2})_{j}$.
Clearly, if $P$ is a probability tensor, then so are its marginal tensors.
Suppose $\mathcal{T}_{m}=(p,...,q)$ for $m=1,...,M$ and $1\leq p\leq q\leq N$.
We denote $\Sigma_{\mathcal{T}_{m}}$ the set of probability tensors in
$\mathbb{R}^{a_{p}\times...\times a_{q}}$ and $U_{\mathcal{T}_{m}}$ the set of
probability tensors in $\mathbb{R}^{a_{p}\times...\times a_{q}}$ whose
$(r)$-marginal vector is $\mu_{r}$, for every $r=p,...,q$. We define
$\mu_{\mathcal{T}_{m}}:=\mu_{p}\otimes...\otimes\mu_{q}$ by
$(\mu_{\mathcal{T}_{m}})_{i_{p}...i_{q}}:=\prod_{r}(\mu_{r})_{i_{r}}$, for
$i_{r}\in[a_{r}]$, with $r=p,...,q$. Clearly, we have
$\mu_{\mathcal{T}}=\mu_{\mathcal{T}_{1}}\otimes...\otimes\mu_{\mathcal{T}_{M}}$.
###### Definition 3.3 (Factored MMOT)
Given a partition $(\mathcal{T}_{m})_{m=1}^{M}$ of $\mathcal{T}$ and a
collection of histograms $\mu$, we consider the following OT problem
$\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}=\inf_{P\in\mathcal{F}_{M}}\langle
C,P\rangle,$ (F-MMOT)
where $\mathcal{F}_{M}\subset U_{\mathcal{T}}$ is the set of admissible
couplings which can be factorized as a tensor product of $M$ component
probability tensors in
$\Sigma_{\mathcal{T}_{1}},...,\Sigma_{\mathcal{T}_{M}}$.
Several remarks are in order here. First, one should note that the partition
considered above is in general not degenerate meaning that the decomposition
can involve tensors of an arbitrary order $<N$. Second, the decomposition in
this setting depicts the prior knowledge regarding the tuples of measures
which should be independent: the couplings for the measures from different
tuples will be degenerate and the optimal coupling tensor will be
reconstructed from couplings of each tuple separately. Third, suppose the
partition $(\mathcal{T}_{m})_{m=1}^{M}$ is not degenerate and $M=2$, i.e. the
tensor is factorized as product of two tensors, the problem F-MMOT is
equivalent to a variation of low nonnegative rank OT (see Appendix for a
proof).
As for the existence of the solution to this problem, we have that
$\mathcal{F}_{M}$ is compact because it is a close subset of the compact set
$U_{\mathcal{T}}$, which implies that F-MMOT always admits a solution.
Furthermore, observe that
$\begin{split}\mathcal{F}_{M}&=\\{P\in
U_{\mathcal{T}}:P=P_{1}\otimes...\otimes P_{M},\text{where
}P_{m}\in\Sigma_{\mathcal{T}_{m}},\forall m=1,...,M\\}\\\
&=\\{P\in\Sigma_{\mathcal{T}}:P=P_{1}\otimes...\otimes P_{M},\text{where
}P_{m}\in U_{\mathcal{T}_{m}},\forall m=1,...,M\\}.\end{split}$
Thus, the problem F-MMOT can be rewritten as
$\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}=\inf_{\begin{subarray}{c}P_{m}\in
U_{\mathcal{T}_{m}}\\\ \forall m=1,...,M\end{subarray}}\langle
C,P_{1}\otimes...\otimes P_{M}\rangle.$
So, if $\mathcal{T}_{1},...,\mathcal{T}_{M}$ are $2$-tuples and two marginal
distributions corresponding to each $U_{\mathcal{T}_{m}}$ are identical and
uniform, then by Birkhoff’s theorem (Birkhoff, 1946), F-MMOT admits an optimal
solution in which each component tensor $P_{m}$ is a permutation matrix.
#### COOT and GW as special cases.
When $N=4$ and $M=2$ with $\mathcal{T}_{1}=(a_{1},a_{2})$ and
$\mathcal{T}_{2}=(a_{3},a_{4})$, the problem F-MMOT becomes the CO-Optimal
transport (Redko et al., 2020), where the two component tensors are known as
sample and feature couplings. If furthermore, $a_{1}=a_{3},a_{2}=a_{4}$, and
$\mu_{1}=\mu_{3},\mu_{2}=\mu_{4}$, it becomes a lower bound of the discrete
Gromov-Wasserstein distance (Mémoli, 2011). This means that our formulation
can be seen as a generalization of several OT formulations.
Observe that if a probability tensor $P$ can be factorized as a tensor product
of probability tensors, i.e. $P=P_{1}\otimes...\otimes P_{M}$, then each
$P_{m}$ is also the $\mathcal{T}_{m}$-marginal tensor of $P$. This prompts us
to consider the following relaxation of factored MMOT, where the hard
constraint $\mathcal{F}_{M}$ is replaced by a regularization term.
###### Definition 3.4 (Relaxed Factored MMOT)
Given $\varepsilon\geq 0$, a partition $(\mathcal{T}_{m})_{m=1}^{M}$ of
$\mathcal{T}$ and a collection of measures $\mu$, we define the following
problem:
$\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}=\inf_{P\in
U_{\mathcal{T}}}\langle
C,P\rangle+\varepsilon\text{KL}(P|P_{\\#\mathcal{T}}).$ (MMOT-DC)
From the exposition above, one can guess that this relaxation is reminiscent
of the entropic regularization in MMOT and coincides with it when $M=N$. As
such it also recovers the classical entropic OT. One should note that the
choice of the KL divergence is not arbitrary and its advantage will become
clear when it comes to the algorithm. special case of MMOT-DC is when $M=N$,
we recover the entropic MMOT.
After having defined the two optimization problems, we now set on exploring
their theoretical properties.
### 3.3 Theoretical properties
Intuitively, the relaxed problem is expected to allow for solutions with a
lower value of the final objective function. We formally prove the validity of
this intuition below.
###### Proposition 3.1
(Preliminary properties)
1. 1.
$\forall\varepsilon\geq 0,\text{MMOT}(\mu)\leq\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}\leq\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$.
2. 2.
$\forall\varepsilon>0,\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}=0$ if and only
if $\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}=0$.
An interesting property of $\text{MMOT-DC}_{\varepsilon}$ is that it
interpolates between MMOT and F-MMOT. Informally, for very large
$\varepsilon$, the KL divergence term dominates, so the optimal transport
plans tend to be factorizable. On the other hand, for very small
$\varepsilon$, the KL divergence term becomes negligible and we approach MMOT.
The result below formalizes this intuition.
###### Proposition 3.2
(Interpolation between MMOT and F-MMOT) For any partition
$(\mathcal{T}_{m})_{m=1}^{M}$ of $\mathcal{T}$ and for $\varepsilon>0$, let
$P_{\varepsilon}$ be a minimiser of the problem $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$.
1. 1.
When $\varepsilon\to\infty$, one has $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}\to\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$.
In this case, any cluster point of the sequence of minimisers
$(P_{\varepsilon})_{\varepsilon}$ is a minimiser of
$\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$.
2. 2.
When $\varepsilon\to 0$, then $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}\to\text{MMOT}(\mu)$.
In this case, any cluster point of the sequence of minimisers
$(P_{\varepsilon})_{\varepsilon}$ is a minimiser of $\text{MMOT}(\mu)$.
#### GW distance revisited.
Somewhat surprisingly, the relaxation MMOT-DC also allows us to prove the
equality between GW distance and COOT in the discrete setting. Let
$\mathcal{X}$ be a finite subset (of size $m$) of a certain metric space.
Denote $C_{x}\in\mathbb{R}^{m\times m}$ its similarity matrix (e.g. distance
matrix). We define similarly the set $\mathcal{Y}$ of size $n$ and the
corresponding similarity matrix $C_{y}\in\mathbb{R}^{n\times n}$. We also
assign two discrete probability measures $\mu_{x}\in\mathbb{R}^{m}$ and
$\mu_{y}\in\mathbb{R}^{n}$ to $\mathcal{X}$ and $\mathcal{Y}$, respectively.
The GW distance is then defined as
$\text{GW}(C_{x},C_{y})=\inf_{P\in U(\mu_{x},\mu_{y})}\langle
L_{p}(C_{x},C_{y}),P\otimes P\rangle,$
and the COOT reads
$\text{COOT}(C_{x},C_{y})=\inf_{\begin{subarray}{c}P\in U(\mu_{x},\mu_{y})\\\
Q\in U(\mu_{x},\mu_{y})\end{subarray}}\langle L_{p}(C_{x},C_{y}),P\otimes
Q\rangle,$
where the $4$D tensor $L_{p}(X,Y)\in\mathbb{R}^{m\times n\times m\times n}$ is
defined by $\big{(}L_{p}(X,Y)\big{)}_{i,j,k,l}=|X_{i,k}-Y_{j,l}|^{p}$, for
some $p\geq 1$, and $U(\mu,\nu)$ is the set of couplings in
$\mathbb{R}^{m\times n}$ whose two marginal distributions are $\mu$ and $\nu$.
When $C_{x}$ and $C_{y}$ are two squared Euclidean distance matrices, and
$p=2$, it can be shown that the GW distance is equal to the COOT (Redko et
al., 2020). This is also true when $L_{p}(C_{x},C_{y})$ is a negative definite
kernel Séjourné et al. (2020). Here, we establish another case where this
equality still holds.
###### Corollary 3.3
If $L_{p}(C_{x},C_{y})$ is a (symmetric) Lipschitz function with respect to
both inputs, and induces a strictly positive definite kernel
$\exp\big{(}-\frac{L_{p}(C_{x},C_{y})}{\varepsilon}\big{)}$ on
$(\mathcal{X}\times\mathcal{Y})^{2}$, for every $\varepsilon>0$, then there
exists a solution $(P,Q)$ of COOT such that $P=Q$. As a consequence, we have
the equality between GW distance and COOT.
The proof relies on the connection between MMOT-DC and COOT shown in the
proposition 3.2, and that the two $\mathcal{T}_{1},\mathcal{T}_{2}$-marginal
matrices of the $4$-D solution of MMOT-DC are in fact identical, under the
assumption of the cost tensor. The proof of the second claim is deferred to
the Appendix.
## 4 Numerical solution
We now turn to computational aspects of the problem MMOT-DC. First, note that
the KL divergence term can be decomposed as
$\text{KL}(P|P_{\\#\mathcal{T}})=H(P)-\sum_{m=1}^{m}H(P_{\\#m}),$
where the function $H_{m}$ defined by $H_{m}(P):=H(P_{\\#m})$ is continuous
and convex with respect to $P$. Now, the problem MMOT-DC becomes
$\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}=\inf_{P\in
U_{\mathcal{T}}}\langle C,P\rangle+\varepsilon
H(P)-\varepsilon\sum_{m=1}^{M}H_{m}(P).$ (3)
This is nothing but a Difference of Convex (DC) programming problem (which
explains the name MMOT-DC), thanks to the convexity of the set
$U_{\mathcal{T}}$ and the entropy function $H$. Thus, it can be solved by the
DC algorithm (Tao and Souad, 1986; Pham and Thi, 1997) as follows: at the
iteration $t$,
1. 1.
Calculate $G^{(t)}\in\partial(\sum_{m=1}^{M}H_{m})(P^{(t)})$.
2. 2.
Solve $P^{(t+1)}\in\arg\min_{P\in U_{\mathcal{T}}}\langle C-\varepsilon
G^{(t)},P\rangle+\varepsilon H(P)$.
This algorithm is very easy to implement. Indeed, the second step is a
classical entropic-regularized MMOT problem and can be solved by the Sinkhorn
algorithm 3. In the first step, the gradient can be calculated explicitly. For
the sake of simplicity, we illustrate the calculation in a simple case, where
$M=2$ and $N=4$ with $\mathcal{T}_{1},\mathcal{T}_{2}$ are two $2$-tuples. The
function $H_{1}+H_{2}$ is continuous, so
$h^{(t)}=\nabla_{P}(H_{1}+H_{2})(P^{(t)})$. Given a $4$-D probability tensor
$P$, we have
$H_{1}(P)+H_{2}(P)=\sum_{i,j,k,l}P_{i,j,k,l}\log\big{(}\sum_{i,j}P_{i,j,k,l}\big{)}+P_{i,j,k,l}\log\big{(}\sum_{k,l}P_{i,j,k,l}\big{)}.$
So,
$\frac{\partial(H_{1}+H_{2})}{\partial
P_{i,j,k,l}}=\log\left(\sum_{i,j}P_{i,j,k,l}\right)+\frac{P_{i,j,k,l}}{\sum_{i,j}P_{i,j,k,l}}+\log\left(\sum_{k,l}P_{i,j,k,l}\right)+\frac{P_{i,j,k,l}}{\sum_{k,l}P_{i,j,k,l}}.$
The complete DC algorithm for the problem 3 can be found in the algorithm 1.
Algorithm 1 DC algorithm for the problem MMOT-DC
Input. Cost tensor $C$, partition $(\mathcal{T}_{m})_{m=1}^{M}$ of
$\mathcal{T}$, histograms $\mu_{1},...,\mu_{N}$, hyperparameter
$\varepsilon>0$, initialization $P^{(0)}$, tuple of initial dual vectors for
the Sinkhorn step $(f_{1}^{(0)},...,f_{N}^{(0)})$.
Output. Tensor $P\in U_{\mathcal{T}}$.
While not converge
1. 1.
Compute $G^{(t)}=\sum\limits_{m=1}^{M}\nabla_{P}H(P^{(t)}_{\\#m})$ the
gradient of teh concave term.
2. 2.
Solve
$P^{(t+1)}\in\arg\min_{P\in U_{\mathcal{T}}}\langle L(X,Y)-\varepsilon
G^{(t)},P\rangle+\varepsilon H(P),$
using the Sinkhorn algorithm 3, with the tuple of initial dual vectors
$(f_{1}^{(0)},...,f_{N}^{(0)})$.
We observed that initialization is crucial to the convergence of algorithm
which is not surprising for a non-convex problem. To accelerate the algorithm
for large $\varepsilon$, we propose to use the warm-start strategy, which is
similar to the one used in the entropic OT problem with very small
regularization parameter (Schmitzer, 2019). Its idea is simple: we consider an
increasing finite sequence $(\varepsilon_{n})_{n=0}^{N}$ approaching
$\varepsilon$ such that the solution $P_{\varepsilon_{0}}$ of the problem
$\text{MMOT-DC}_{\varepsilon_{0}}(X,Y)$ can be estimated quickly and
accurately using the initialization $P^{(0)}$. Then we solve each successive
problem $\text{MMOT-DC}_{\varepsilon_{n}}(X,Y)$ using the previous solution
$P_{\varepsilon_{n-1}}$ as initialization. Finally, the problem $\text{MMOT-
DC}_{\varepsilon}(X,Y)$ is solved using the solution $P_{\varepsilon_{N}}$ as
initialization.
Algorithm 2 DC algorithm with warm start for the problem MMOT-DC
Input. Cost tensor $C$, partition $(\mathcal{T}_{m})_{m=1}^{M}$ of
$\mathcal{T}$, histograms $\mu_{1},...,\mu_{N}$, hyperparameter
$\varepsilon>0$, initialization $P^{(0)}$, initial $\varepsilon_{0}>0$, step
size $s>1$, tuple of initial dual vectors $(f_{1}^{(0)},...,f_{N}^{(0)})$.
Output. Tensor $P\in U_{\mathcal{T}}$.
1. 1.
While $\varepsilon_{0}<\varepsilon$:
1. (a)
Using algorithm 1, solve the problem $\text{MMOT-DC}_{\varepsilon_{0}}(X,Y)$
with initialization $P^{(0)}$ and $(f_{1}^{(0)},...,f_{N}^{(0)})$ to find the
solution $P_{\varepsilon_{0}}$ and its associated tuple of dual vectors
$(f_{1}^{(\varepsilon_{0})},...,f_{N}^{(\varepsilon_{0})})$.
2. (b)
Set $P^{(0)}=P_{\varepsilon_{0}},f_{i}^{(0)}=f_{i}^{(\varepsilon_{0})}$, for
$i=1,...,N$.
3. (c)
Increase regularization: $\varepsilon_{0}:=s\varepsilon_{0}$.
2. 2.
Using algorithm 1, solve the problem $\text{MMOT-DC}_{\varepsilon}(X,Y)$ using
the initialization $P^{(0)}$ and $(f_{1}^{(0)},...,f_{N}^{(0)})$.
## 5 Experimental evaluation
In this section, we illustrate the use of MMOT-DC on simulated data. Rather
than performing experiments in full generality, we choose the setting where
$N=4$ and $M=2$ with $\mathcal{T}_{1}=(1,2)$ and $\mathcal{T}_{2}=(3,4)$, so
that we can compare MMOT-DC with other popular solvers of COOT and GW
distance. Given two matrices $X$ and $Y$, we always consider the $4$-D cost
tensor $C$, where $C_{i,j,k,l}=|X_{i,k}-Y_{j,l}|^{2}$. On the other hand, we
are not interested in the $4$-D minimiser of MMOT-DC, but only in its two
$\mathcal{T}_{1},\mathcal{T}_{2}$-marginal matrices.
#### Solving COOT on a toy example.
We generate a random matrix $X\in\mathbb{R}^{30\times 25}$, whose entries are
drawn independently from the uniform distribution on the interval $[0,1)$. We
equip the rows and columns of $X$ with two discrete uniform distributions on
$[30]$ and $[25]$. We fix two permutation matrices $P\in\mathbb{R}^{30\times
30}$ (called sample permutation) and $Q\in\mathbb{R}^{25\times 25}$ (called
feature permutation), then calculate $Y=PXQ$. We also equip the rows and
columns of $Y$ with two discrete uniform distributions on $[30]$ and $[25]$.
It is not difficult to see that $\text{COOT}(X,Y)=0$ because $(P,Q)$ is a
solution. As COOT is a special case of F-MMOT, we see that $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{2},\mu\big{)}=0$, for every
$\varepsilon>0$, by proposition 3.1. In this experiment, we will check if
marginalizing the minimizer of MMOT-DC allows us to recover the permutation
matrices $P$ and $Q$. As can be seen from the figure 1, MMOT-DC can recover
the permutation positions, for various values of $\varepsilon$. On the other
hand, it can not recover the true sparse permutation matrices because the
Sinkhorn algorithm applied to the MMOT problem implicitly results in a dense
tensor, thus having dense marginal matrices. For this reason, the loss only
remains very close to zero, but never exactly.
Figure 1: Couplings generated by COOT and MMOT-DC on the matrix recovering
task.
We also plot, with some abuse of notation, the histograms of the error between
the $(1,3),(1,4),(2,3),(2,4)$-marginal matrices of MMOT-DC and their
independent counterpart from F-MMOT. In this example, in theory, as the F-MMOT
optimal tensor $P$ can be factorized as $P=P_{12}\otimes P_{34}$, it is
immediate to see that $P_{12}=P_{14}=P_{23}=P_{24}\in\mathbb{R}^{30\times 25}$
are uniform matrices whose entries are $\frac{1}{750}$.
Figure 2: Histograms of difference between true independent marginal matrices
and their approximations. We see that the marginal matrices obtained by the
algorithm 1 approximate well the theoretical uniform matrices.
#### Quality of the MMOT-DC solutions.
Now, we consider the situation where the true matching between two matrices is
not known in advance and investigate the quality of the solutions returned by
MMOT-DC to solve the COOT and GW problems. This means that we will look at the
COOT loss $\langle C,P\otimes Q\rangle$, where the smaller the loss, the
better when using both exact COOT and GW solvers and our relaxation.
We generate two random matrices $X\in\mathbb{R}^{20\times 3}$ and
$Y\in\mathbb{R}^{30\times 2}$, whose entries are drawn independently from the
uniform distribution on the interval $[0,1)$. Then we calculate two
corresponding squared Euclidean distance matrices of size $20$ and $30$. Their
rows and columns are equipped with the discrete uniform distributions. In this
case, the COOT loss coincides with the GW distance (Redko et al., 2020).
We compare four solvers:
1. 1.
The Frank-Wolfe algorithm (Frank and Wolfe, 1956) to solve the GW (GW-FW).
2. 2.
The projected gradient algorithm to solve the entropic GW distance Peyré et
al. (2016) (EGW-PG). We choose the regularization parameter from
$\\{0.0008,0.0016,0.0032,0.0064,0.0128,0.0256\\}$ and pick the one which
corresponds to smallest COOT loss.
3. 3.
The Block Coordinate Descent algorithm to approximate COOT and its entropic
approximation Redko et al. (2020) (GW-BCD and EGW-BCD, respectively), where
two additional KL divergences corresponding to two couplings are introduced.
Both regularization parameters are tuned from
$\\{0,0.0005,0.001,0.005,0.01,0.05,0.1,0.5,1\\}$, where $0$ means that there
is no regularization term for the corresponding coupling and we pick the pair
whose COOT loss is the smallest.
4. 4.
The algorithm 1 to solve the MMOT-DC. We tune
$\varepsilon\in\\{1,1.4,1.8,2.2,2.6\\}$ and we pick the one which corresponds
to smallest COOT loss.
For GW-FW and EGW-PG, we use the implementation from the library PythonOT
(Flamary et al., 2021).
Given two random matrices, we record the COOT loss corresponding to the
solution generated by these three methods. We simulate this process $70$ times
and compare the overall performance of these methods. We can see in Table 1
the average value and standard deviation and the comparison for the values of
the loss between the different algorithms in Figure 3. The performance is
quite similar across methods with a slight advantage for EGW-PG. This is in
itself a very interesting result that has never been noted to the best of our
knowledge: the entropic version of GW can provide better solution than solving
the exact problem maybe because of the "convexification "of the problem due to
the entropic regularization. Our approach is also interestingly better than
the exact GW-FW which illustrates that the relaxation might help in finding
better solutions despite the non-convexity of the problem.
GW-FW | EGW-PG | GW-BCD | EGW-BCD | MMOT-DC
---|---|---|---|---
0.083 ($\pm$ 0.035) | 0.079 ($\pm$ 0.035) | 0.083 ($\pm$ 0.035) | 0.080 ($\pm$ 0.035) | 0.082 ($\pm$ 0.036)
Table 1: Average and standard deviation of COOT loss of the solvers. MMOT-DC
is competitive to other solvers, except for EGW-PG and EGW-BCD. Figure 3:
Scatter plots of MMOT-DC versus other solvers. In all four plots, the points
tend to concentrate around the line $y=x$, which indicates the comparable
performance of MMOT-DC. On the other hand, the top-right plot shows the clear
superiority of EGW-PG.
## 6 Discussion and conclusion
In this paper, we present a novel relaxation of the factorized MMOT problem
called MMOT-DC. More precisely, we replace the hard constraint on
factorization constraint by a smooth regularization term. The resulting
problem not only enjoys an interpolation property between MMOT and factorized
MMOT, but also is a DC problem, which can be solved easily by the DC
algorithm. We illustrate the use of MMOT-DC the via some simulated experiments
and show that it is competitive with the existing popular solvers of COOT and
GW distance. One limitation of the current DC algorithm is that, it is not
scalable because it requires storing a full-size tensor in the gradient step
computation. Thus, future work may focus on more efficiently designed
algorithms, in terms of both time and memory footprint. Moreover,
incorporating additional structure on the cost tensor may also be
computationally and practically beneficial. From a theoretical viewpoint, it
is also interesting to study the extension of MMOT-DC to the continuous
setting, which can potentially allow us to further understand the connection
between GW distance and COOT.
#### Acknowledgements.
The authors thank to Thibault Séjourné for the fruitful discussion on the
connection between GW distance and COOT. The authors thank to Titouan Vayer
for pointing out the error in the definition of the lower bound of GW
distance.
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## Appendix A Appendix
#### Derivation of the Sinkhorn algorithm in MMOT.
The corresponding entropic dual problem of the primal problem 2 reads
$\sup_{f_{n}\in\mathbb{R}^{a_{n}}}\sum_{n=1}^{N}\langle
f_{n},\mu_{n}\rangle-\varepsilon\sum_{i_{1},...,i_{N}}\exp\Big{(}\frac{\sum_{n}(f_{n})_{i_{n}}-C_{i_{1},...,i_{N}}}{\varepsilon}\Big{)}+\varepsilon.$
(4)
For each $n=1,...,N$ and $i_{n}\in[a_{n}]$, the first order optimality
condition reads
$0=(\mu_{n})_{i_{n}}-\exp\big{(}\frac{(f_{n})_{i_{n}}}{\varepsilon}\big{)}\sum_{i_{-n}}\exp\Big{(}\frac{\sum_{j\neq
n}(f_{j})_{i_{j}}-C_{i_{1},...,i_{N}}}{\varepsilon}\Big{)},$
where, with some abuse of notation, we write
$i_{-n}=(i_{1},...,i_{n-1},i_{n+1},...,i_{N})$. Or, equivalently
$(f_{n})_{i_{n}}=\varepsilon\log(\mu_{n})_{i_{n}}-\varepsilon\log\sum_{i_{-n}}\exp\Big{(}\frac{\sum_{j\neq
n}(f_{j})_{i_{j}}-C_{i_{1},...,i_{N}}}{\varepsilon}\Big{)},$
or even more compact form
$f_{n}=\varepsilon\log\mu_{n}-\varepsilon\log\sum_{i_{-n}}\exp\Big{(}\frac{\sum_{j\neq
n}(f_{j})_{i_{j}}-C_{\cdot,i_{-n}}}{\varepsilon}\Big{)}.$
Using the primal-dual relation, we obtain the minimiser of the primal problem
2 by
$P_{i_{1},...,i_{N}}=\exp\Big{(}\frac{\sum_{n}(f_{n})_{i_{n}}-C_{i_{1},...,i_{N}}}{\varepsilon}\Big{)},$
for $i_{n}\in[a_{n}]$, with $n=1,...,N$. Similar to the entropic OT, the
Sinkhorn algorithm 3 is also usually implemented in log-domain to avoid
numerical instability.
Algorithm 3 Sinkhorn algorithm for the entropic MMOT problem 2 from Benamou et
al. [2014]
Input. Histograms $\mu_{1},...,\mu_{N}$, hyperparameter $\varepsilon>0$, cost
tensor $C$ and tuple of initial dual vectors $(f^{(0)}_{1},...f^{(0)}_{N})$.
Output. Optimal transport plan $P$ and tuple of dual vectors
$(f_{1},...f_{N})$ (optional).
1. 1.
While not converge: for $n=1,...,N$,
$\begin{split}f^{(t+1)}_{n}&=\varepsilon\log\mu_{n}-\varepsilon\log\sum_{i_{-n}}\Big{[}\exp\Big{(}\frac{\sum_{j<n}(f^{(t+1)}_{j})_{i_{j}}+\sum_{j>n}(f^{(t)}_{j})_{i_{j}}-C_{\cdot,i_{-n}}}{\varepsilon}\Big{)}\Big{]}.\end{split}$
2. 2.
Return tensor $P$, where for $i_{n}\in[a_{n}]$, with $n=1,...,N$,
$P_{i_{1},...,i_{N}}=\exp\Big{(}\frac{\sum_{n}(f_{n})_{i_{n}}-C_{i_{1},...,i_{N}}}{\varepsilon}\Big{)}.$
(5)
#### F-MMOT of two components (i.e. $M=2$) as a variation of low nonnegative
rank OT.
For the sake of notational ease, we only consider the simplest case, but the
same argument holds in the general case. Consider $\mathcal{T}_{1}=(1,2)$ and
$\mathcal{T}_{2}=(3,4)$, then the factored MMOT problem reads
$\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{2},\mu\big{)}=\min_{P\in\mathcal{F}_{2}}\langle
C,P\rangle.$ (6)
First, we define three reshaping operations.
* •
vectorization: concatenates rows of a matrix into a vector.
$\text{vec}:\mathbb{R}^{m\times n}\to\mathbb{R}^{mn},$
where each element $A_{i,j}$ of the matrix $A\in\mathbb{R}^{m\times n}$ is
mapped to a unique element $b_{(i-1)n+j}$ of the vector $b\in\mathbb{R}^{mn}$,
with $A_{i,j}=b_{(i-1)n+j}$, for $i=1,...,m$ and $j=1,...,n$. Conversely, each
element $b_{k}$ is mapped to a unique element $A_{k//n,n-k\%n}$, for every
$k=1,...,mn$. Here, $k//n$ is the quotient of the division of $k$ by $n$ and
$k\%n$ is the remainder of this division, i.e. if $k=qn+r$, with $0\leq r<n$,
then $k//n=q$ and $k\%n=r$.
* •
Matrization: transforms a $4$D tensor to a $2$D tensor (matrix) by vectorizing
the first two and the last two dimensions of the tensor.
$\text{mat}:\mathbb{R}^{n_{1}\times n_{2}\times n_{3}\times
n_{4}}\to\mathbb{R}^{(n_{1}n_{2})\times(n_{3}n_{4})},$
where, similar to the vectorization, each element $P_{i,j,k,l}$ of the tensor
$P\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}\times n_{4}}$ is mapped to the
unique element $A_{(i-1)n_{2}+j,(k-1)n_{4}+l}$ of the matrix
$A\in\mathbb{R}^{(n_{1}n_{2})\times(n_{3}n_{4})}$, with
$P_{i,j,k,l}=A_{(i-1)n_{2}+j,(k-1)n_{4}+l}$.
* •
Concatenation: stacks vertically two equal-column matrices.
$\begin{split}\text{con}_{v}:&\mathbb{R}^{m\times d}\times\mathbb{R}^{n\times
d}\to\mathbb{R}^{(m+n)\times d}\\\
&\big{(}(u_{1},...,u_{m}),(v_{1},...,v_{n})\big{)}\to(u_{1},...,u_{m},v_{1},...,v_{n})^{T}.\end{split}$
Or, stacks horizontally two equal-row matrices
$\begin{split}\text{con}_{h}:&\mathbb{R}^{n\times p}\times\mathbb{R}^{n\times
q}\to\mathbb{R}^{n\times(p+q)}\\\
&\big{(}(u_{1},...,u_{p}),(v_{1},...,v_{q})\big{)}\to(u_{1},...,u_{p},v_{1},...,v_{q}).\end{split}$
###### Lemma A.1
For any $4$D tensor $\pi\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}\times
n_{4}}$, denote $P$ its matrisation. We have,
$\text{vec}\Big{(}\sum_{k,l}\pi_{\cdot,\cdot,k,l}\Big{)}=\sum_{n=1}^{n_{3}n_{4}}P_{\cdot,n}=P1_{n_{3}n_{4}},$
where $1_{n}$ is the vector of ones in $\mathbb{R}^{n}$.
#### Proof of lemma A.1.
For $(i,j)\in[n_{1}]\times[n_{2}]$, we have
$\text{vec}\Big{(}\sum_{k,l}\pi_{\cdot,\cdot,k,l}\Big{)}_{(i-1)n_{2}+j}=\sum_{k,l}\pi_{i,j,k,l}=\sum_{k,l}P_{(i-1)n_{2}+j,(k-1)n_{4}+l}=\sum_{n=1}^{n_{3}n_{4}}P_{(i-1)n_{2}+j,n}.$
Now, let $(e_{i})_{i=1}^{n_{1}n_{2}}$ be the standard basis vectors of
$\mathbb{R}^{(n_{1}n_{2})}$, i.e. $(e_{i})_{k}=1_{\\{i=k\\}}$. For each
$\pi\in U_{\mathcal{T}}$, denote $P$ its matrix form, then by lemma A.1, we
have, for $i\in[n_{1}]$,
$(\mu_{1})_{i}=\sum_{j}\sum_{k,l}\pi_{i,j,k,l}=\sum_{j=1}^{n_{2}}\sum_{n=1}^{n_{3}n_{4}}P_{(i-1)n_{2}+j,n},$
which can be written in matrix form as
$A_{1}^{T}P1_{n_{3}n_{4}}=\mu_{1}$
where the matrix
$A_{1}=\text{con}_{h}(v_{1},...,v_{n_{1}})\in\mathbb{R}^{(n_{1}n_{2})\times
n_{1}}$, with $v_{i}\in\mathbb{R}^{(n_{1}n_{2})}$, where
$v_{i}=\sum_{j=(i-1)n_{2}+1}^{in_{2}}e_{j}$, with $i\in[n_{1}]$. Similarly,
$A_{2}P1_{n_{3}n_{4}}=\mu_{2}$, where the matrix
$A_{2}=\text{con}_{h}(I_{n_{2}},...,I_{n_{2}})\in\mathbb{R}^{n_{2}\times(n_{1}n_{2})}$,
where $I_{n}\in\mathbb{R}^{n\times n}$ is the identity matrix. Both conditions
can be compactly written as
$A_{12}^{T}P1_{n_{3}n_{4}}=\mu_{12},$
where the matrix
$A_{12}=\text{con}_{h}(A_{1},A_{2}^{T})\in\mathbb{R}^{(n_{1}n_{2})\times(n_{1}+n_{2})}$
and $\mu_{12}=\text{con}_{v}(\mu_{1},\mu_{2})\in\mathbb{R}^{(n_{1}+n_{2})}$.
Note that $\mu_{12}$ is not a probability because its mass is $2$. The matrix
$A_{12}$ has exactly $2n_{1}n_{2}$ ones and the rest are zeros. Similarly, for
$A_{34}$ and $\mu_{34}$ defined in the same way as $A_{12}$ and $\mu_{12}$,
respectively, we establish the equality
$A_{34}^{T}P^{T}1_{n_{1}n_{2}}=\mu_{34}$. As a side remark, both matrices
$A_{12}^{T}$ and $A_{34}^{T}$ are totally unimodular, i.e. every square
submatrix has determinant $-1,0$, or $1$.
Figure 4: An example of the matrix $A_{12}$ when $n_{1}=2$ and $n_{2}=3$.
To handle the factorization constraint, first we recall the following concept.
###### Definition A.1
Given a nonnegative matrix $A$, we define its nonnegative rank by
$\text{rank}_{+}(A):=\min\big{\\{}r\geq 1:A=\sum_{i=1}^{r}M_{i},\text{ where
}\text{rank}(M_{i})=1,M_{i}\geq 0,\forall i\big{\\}}.$
By convention, zero matrix has zero (thus nonnegative) rank.
So, the constraint $P=P_{1}\otimes P_{2}$ is equivalent to
$\text{mat}(P)=\text{vec}(P_{1})\text{vec}(P_{2})^{T}$. By lemma 2.1 in [Cohen
and Rothblum, 1993], $\text{rank}_{+}(A)=1$ if and only if there exist two
nonnegative vectors $u,v$ such that $A=uv^{T}$. Thus, the factorization
constraint can be rewritten as $\text{rank}_{+}\big{(}\text{mat}(P)\big{)}=1$.
Denote $L=\text{mat}(C)$ and $M=n_{1}n_{2},N=n_{3}n_{4}$. The problem 6 can be
rewritten as
$\begin{split}\min_{Q\in\mathbb{R}^{M\times N}_{\geq 0}}&\langle L,Q\rangle\\\
\text{ such that }&A_{12}^{T}Q1_{N}=\mu_{12}\\\
&A_{34}^{T}Q^{T}1_{M}=\mu_{34}\\\ &\text{rank}_{+}(Q)=1.\end{split}$
#### Proof of proposition 3.1.
The inequality $\text{MMOT}(\mu)\leq\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$ follows from
the positivity of the KL divergence. On the other hand,
$\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}=\inf_{P\in\mathcal{F}_{M}}\langle
C,P\rangle+\varepsilon\text{KL}(P|P_{\\#}),$
because $\text{KL}(P|P_{\\#})=0$, for every $P\in\mathcal{F}_{M}$. As
$\mathcal{F}_{M}\subset U_{\mathcal{T}}$, we have $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}\leq\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$.
Now, if $\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}=0$, then
$\text{MMOT-DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}=0$.
Conversely, if $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}=0$, for
$\varepsilon>0$, then there exists $P^{*}\in U_{\mathcal{T}}$ such that
$\langle C,P^{*}\rangle=0$ and $P^{*}=P^{*}_{\\#}$. Thus $\langle
C,P^{*}_{\\#}\rangle=0$, which means
$\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}=0$.
#### Proof of proposition 3.2.
The function $\varepsilon\to\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$ is increasing
on $\mathbb{R}_{\geq 0}$ and bounded, thus admits a finite limit
$L\leq\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$, when
$\varepsilon\to\infty$, and a finite limit $l\geq\text{MMOT}(\mu)$, when
$\varepsilon\to 0$.
Let $P_{\varepsilon}$ be a solution of the problem $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$. As
$U_{\mathcal{T}}$ is compact, when either $\varepsilon\to 0$ or
$\varepsilon\to\infty$, one can extract a converging subsequence (after
reindexing) $(P_{\varepsilon_{k}})_{k}\to\widetilde{P}\in U_{\mathcal{T}}$,
when either $\varepsilon_{k}\to 0$ or $\varepsilon_{k}\to\infty$. Thus, the
convergence of the marginal distributions is also guaranteed, i.e
$(P_{\varepsilon_{k}})_{\\#m}\to\widetilde{P}_{\\#m}\in U_{\mathcal{T}_{m}}$,
for every $m=1,...,M$, which implies that
$P_{\varepsilon_{k}}-(P_{\varepsilon_{k}})_{\\#\mathcal{T}}\to\widetilde{P}-\widetilde{P}_{\\#\mathcal{T}}$.
When $\varepsilon\to 0$, let $P^{*}$ be a solution of the problem
$\text{MMOT}(\mu)$. Then,
$\langle C,P^{*}\rangle\leq\langle
C,P_{\varepsilon}\rangle+\varepsilon\text{KL}(P_{\varepsilon}|(P_{\varepsilon})_{\\#\mathcal{T}})\leq\langle
C,P^{*}\rangle+\varepsilon\text{KL}(P^{*}|P^{*}_{\\#\mathcal{T}}).$
By the sandwich theorem, when $\varepsilon\to 0$, we have $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}\to\langle
C,P^{*}\rangle=\text{MMOT}(\mu)$. Furthermore, as
$0\leq\langle C,P_{\varepsilon_{k}}\rangle-\langle
C,P^{*}\rangle\leq\varepsilon_{k}\text{KL}(P^{*}|P^{*}_{\\#\mathcal{T}}),$
when $\varepsilon_{k}\to 0$, it follows that $\langle
C,\widetilde{P}\rangle=\langle C,P^{*}\rangle$. So $\widetilde{P}$ is a
solution of the problem $\text{MMOT}(\mu)$. We conclude that any cluster point
of the sequence of minimisers of $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$ when
$\varepsilon\to 0$ is a minimiser of $\text{MMOT}(\mu)$. As a byproduct, since
$\text{KL}(P^{*}|P^{*}_{\\#\mathcal{T}})-\text{KL}(P_{\varepsilon_{k}}|(P_{\varepsilon_{k}})_{\\#\mathcal{T}})\geq\frac{\langle
C,P_{\varepsilon_{k}}\rangle-\langle C,P^{*}\rangle}{\varepsilon_{k}}\geq 0,$
we deduce that
$\text{KL}(\widetilde{P}|\widetilde{P}_{\\#\mathcal{T}})\leq\text{KL}(P^{*}|P^{*}_{\\#\mathcal{T}})$
(so the cluster point $\widetilde{P}$ has minimal mutual information).
On the other hand, when $\varepsilon\to\infty$, for
$\mu=\mu_{1}\otimes...\otimes\mu_{N}$, one has
$\langle C,\mu\rangle+\varepsilon\times 0\geq\langle
C,P_{\varepsilon}\rangle+\varepsilon\text{KL}(P_{\varepsilon}|(P_{\varepsilon})_{\\#\mathcal{T}})\geq\varepsilon\text{KL}(P_{\varepsilon}|(P_{\varepsilon})_{\\#\mathcal{T}}).$
Thus,
$0\leq\text{KL}(P_{\varepsilon}|(P_{\varepsilon})_{\\#\mathcal{T}})\leq\frac{1}{\varepsilon}\langle
C,\mu\rangle\to 0,\text{ when }\varepsilon\to\infty,$
which means $\text{KL}(P_{\varepsilon}|(P_{\varepsilon})_{\\#\mathcal{T}})\to
0$, when $\varepsilon\to\infty$. In particular, when
$\varepsilon_{k}\to\infty$, we have
$\text{KL}(P_{\varepsilon_{k}}|(P_{\varepsilon_{k}})_{\\#\mathcal{T}})\to 0$.
Thus, $\text{KL}(\widetilde{P}|\widetilde{P}_{\\#\mathcal{T}})=0$, which
implies $\widetilde{P}=\widetilde{P}_{\\#\mathcal{T}}$.
Now, as $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}\geq\langle
C,P_{\varepsilon}\rangle$, taking limit when $\varepsilon\to\infty$ gives us
$L\geq\langle C,\widetilde{P}\rangle=\langle
C,\widetilde{P}_{\\#\mathcal{T}}\rangle\geq\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$.
Thus $L=\langle
C,\widetilde{P}\rangle=\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$,
i.e. $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}\to\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$
when $\varepsilon\to\infty$. We also have that any cluster point of the
sequence of minimisers of $\text{MMOT-
DC}_{\varepsilon}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$ when
$\varepsilon\to\infty$ is a minimiser of
$\text{F-MMOT}\big{(}(\mathcal{T}_{m})_{m=1}^{M},\mu\big{)}$.
#### Proof of corollary 3.3.
In this proof, we write $C:=L_{p}(C_{x},C_{y})$, for notational ease. In the
setting of GW distance, we have $N=4,M=2$ and
$\mathcal{T}_{1}=(1,2),\mathcal{T}_{2}=(3,4)$. Given a solution
$P_{\varepsilon}$ of the problem MMOT-DC, let $Q_{i}\in
U\big{(}(P_{\varepsilon})_{\\#i},(P_{\varepsilon})_{\\#i}\big{)}\subset
U_{\mathcal{T}}$, for $i=1,2$. The optimality of $P_{\varepsilon}$ implies
that
$\langle
C,P_{\varepsilon}\rangle+\varepsilon\Big{[}H(P_{\varepsilon})-H\big{(}(P_{\varepsilon})_{\\#1}\big{)}-H\big{(}(P_{\varepsilon})_{\\#2}\big{)}\Big{]}\leq\langle
C,Q_{i}\rangle+\varepsilon\Big{[}H(Q_{i})-2H\big{(}(P_{\varepsilon})_{\\#i}\big{)}\Big{]}.$
Thus,
$2\big{(}\langle C,P_{\varepsilon}\rangle+\varepsilon
H(P_{\varepsilon})\big{)}\leq\sum_{i=1}^{2}\langle C,Q_{i}\rangle+\varepsilon
H(Q_{i}).$
As this is true for every $Q_{i}\in
U\big{(}(P_{\varepsilon})_{\\#i},(P_{\varepsilon})_{\\#i}\big{)}$, we have
$\begin{split}\frac{1}{2}\sum_{i=1}^{2}\text{OT}_{\varepsilon}\big{(}(P_{\varepsilon})_{\\#i},(P_{\varepsilon})_{\\#i}\big{)}&=\frac{1}{2}\sum_{i=1}^{2}\inf_{Q_{i}\in
U\big{(}(P_{\varepsilon})_{\\#i},(P_{\varepsilon})_{\\#i}\big{)}}\langle
C,Q_{i}\rangle+\varepsilon H(Q_{i})\\\ &\geq\langle
C,P_{\varepsilon}\rangle+\varepsilon H(P_{\varepsilon})\\\ &\geq\inf_{P\in
U\big{(}(P_{\varepsilon})_{\\#1},(P_{\varepsilon})_{\\#2}\big{)}}\langle
C,P\rangle+\varepsilon H(P)\\\
&=\text{OT}_{\varepsilon}\big{(}(P_{\varepsilon})_{\\#1},(P_{\varepsilon})_{\\#2}\big{)}.\end{split}$
The second inequality holds because $P_{\varepsilon}\in
U\big{(}(P_{\varepsilon})_{\\#1},(P_{\varepsilon})_{\\#2}\big{)}$. Thus,
$\text{OT}_{\varepsilon}\big{(}(P_{\varepsilon})_{\\#1},(P_{\varepsilon})_{\\#2}\big{)}-\frac{1}{2}\sum_{i=1}^{2}\text{OT}_{\varepsilon}\big{(}(P_{\varepsilon})_{\\#i},(P_{\varepsilon})_{\\#i}\big{)}\leq
0.$ (7)
The left-hand side of 7 is nothing but the Sinkhorn divergence
$\text{SD}_{\varepsilon}$ between $(P_{\varepsilon})_{\\#1}$ and
$(P_{\varepsilon})_{\\#2}$ [Ramdas et al., 2017]. As a strictly positive
definite kernel is necessarily universal in the finite setting (see for
example section 2.3 in [Borgwardt et al., 2006]), by theorem 1 in [Feydy et
al., 2019], the inequality 7 is in fact an equality and we must have
$(P_{\varepsilon})_{\\#1}=(P_{\varepsilon})_{\\#2}$.
Now, by proposition 3.2, when $\varepsilon\to\infty$, a cluster point
$P_{\\#1}\otimes P_{\\#2}$ of the sequence of minimisers
$(P_{\varepsilon})_{\varepsilon}$ induces a solution $(P^{(1)},P^{(2)})$ of
the COOT. The above result implies that $P_{\\#1}=P_{\\#2}$ and the equality
between GW and COOT then follows.
#### An empirical variation.
Intuitively, for sufficiently large $\varepsilon$, the minimisation of the KL
divergence is prioritised over the linear term in the objective function of
the MMOT-DC problem, which implies that the optimal tensor $P^{*}$ is "close"
to its corresponding tensor product $P^{*}_{\\#\mathcal{T}}$. So, instead of
calculating the gradient at $P$, one may calculate at $P_{\\#\mathcal{T}}$. In
this case, the gradient reads
$\begin{split}\sum_{m=1}^{M}\nabla_{P}H_{m}(P_{\\#\mathcal{T}})=\big{[}\log
P_{\\#1}+P_{\\#1}\big{]}\oplus...\oplus\big{[}\log
P_{\\#M}+P_{\\#M}\big{]},\end{split}$
where $\oplus$ represents the tensor sum operator between two arbitrary-size
tensors: $(P\oplus Q)_{i,j}:=P_{i}+Q_{j}$, where with some abuse of notation,
$i$ or $j$ can be understood as a tuple of indices. Thus, we avoid storing the
$N$-D gradient tensor (as in the algorithm 1) and only need to store $M$
smaller-size tensors. Not only saving the memory, this variation also seems to
be empirically competitive with the original algorithm 1, if not sometimes
better, in terms of COOT loss. The underlying reason might be related to the
approximate DCA scheme [Vo, 2015], where one replaces both steps in each DC
iteration by their approximation. We leave the formal theoretical
justification of this variation to the future work. We call this variation
MMOT-DC-v1 and use the same setup as in the experiment 5.
Figure 5: Scatter plots of MMOT-DC-v1 versus other solvers. In all four plots, the points tend to concentrate around the line $y=x$, which indicates the comparable performance of MMOT-DC-v1. On the other hand, the top-right plot shows the clear superiority of EGW-PG. MMOT-DC | MMOT-DC-v1
---|---
0.0822 ($\pm$ 0.0364) | 0.0820 ($\pm$ 0.0361)
Table 2: Average and standard deviation of COOT loss of MMOT-DC and MMOT-
DC-v1. The performance of the two algorithms is very similar.
|
# Directional-dependence of the event-by-event neutron-$\gamma$ multiplicity
correlations in 252Cf(sf)
Stefano Marin<EMAIL_ADDRESS>Department of Nuclear Engineering and
Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA Eoin
P. Sansevero<EMAIL_ADDRESS>Department of Nuclear Engineering and
Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA M.
Stephan Okar Department of Nuclear Engineering and Radiological Sciences,
University of Michigan, Ann Arbor, MI 48109, USA Isabel E. Hernandez
Department of Nuclear Engineering and Radiological Sciences, University of
Michigan, Ann Arbor, MI 48109, USA Ramona Vogt Nuclear and Chemical Sciences
Division, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
Physics and Astronomy Department, University of California, Davis, CA 95616,
USA Jørgen Randrup Nuclear Science Division, Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA Shaun D. Clarke Department of
Nuclear Engineering and Radiological Sciences, University of Michigan, Ann
Arbor, MI 48109, USA Vladimir A. Protopopescu Oak Ridge National Laboratory,
Oak Ridge, TN 37830, USA Sara A. Pozzi Department of Nuclear Engineering and
Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
###### Abstract
We differentiate the event-by-event n-$\gamma$ multiplicity data from
^252Cf(sf) with respect to the energies of the emitted particles as well as
their relative angles of emission. We determine that neutron emission enhances
$\gamma$-ray emission around $0.7$ and $1.2$ MeV, but the only directional
alignment was observed for $E_{\gamma}\leq 0.7$ MeV and tended to be parallel
and antiparallel to neutrons emitted in the same event. The emission of
$\gamma$ rays at other energies was determined to be nearly isotropic. The
presence of the emission and alignment enhancements is explained by positive
correlations between neutron emission and quadrupole $\gamma$-ray emission
along rotational bands in the de-exciting fragments. This observation
corroborates the hypothesis of positive correlations between the angular
momentum of a fragment and its intrinsic excitation energy. The results of
this work are especially relevant in view of the recent theoretical and
experimental interest in the generation of angular momentum in fission.
Specifically, we have determined an alignment of the fragments angular momenta
in a direction perpendicular to the direction of motion. We interpret the lack
of $n$-$\gamma$ angular correlations for fission fragments near closed shells
as a weakening of the alignment process for spherical nuclei. Lastly, we have
observed that statistical $\gamma$ rays are emitted isotropically, indicating
tha the average angular momentum removed by this radiation is small. These
results, and the analysis tools presented in this work, representing a
stepping stone for future analysis of $n$-$\gamma$ emission correlations and
their connection to angular momentum properties.
neutron-gamma multiplicity competition; fission fragment de-excitation
## I Introduction
Neutrons and $\gamma$ rays emitted from fission fragments reveal important
features of the nuclear fission process and the state of the fragments
immediately following fission. Among several open questions in fission, the
$n$-$\gamma$ angular correlations are particularly interesting because of
their intimate relation to the fission fragment angular momenta. The angular
momentum of a fragments plays a pivotal role in the emission of $\gamma$ rays
and the $n$-$\gamma$ angular distribution. The characterization of the
fragment angular momenta is one of the most important open questions in
fission physics.
The first experimental investigations of fission fragment angular momenta were
carried out in the 1960s [1] and 1970s [2] and some early theoretical work was
done in the 1980s on the character of the distribution of the fragment angular
momentum [3] and the underlying mechamism for its generation [4]. Due to the
advances in instrumentation, modeling, and computation since then, the topic
of fission fragment angular momenta has gained renewed interest in recent
years [5, 6, 7, 8, 9, 10, 11, 12]. Of particular interest are the event-by-
event correlations between fragment energy and angular momentum, the
directional alignment of the angular momentum with respect to the motion of
the fragment, and the correlations, both in magnitude and direction, between
the angular momenta of the two fragments.
The evaporation of neutrons from fragments is highly correlated with the
fragment intrinsic excitation energy [13, 14, 15], whereas $\gamma$-ray
emission correlates strongly with the fragment angular momentum [16, 17]. By
analysing the correlations between neutrons and $\gamma$ rays it is possible
to infer the underlying correlations between the fragment energy and angular
momentum. To reduce noise and systematic biases associated with the emission
of these particles, it is necessary to differentiate the emission based on the
kinematic properties of the emission, namely their kinetic energies and
directions.
In this work, we continue our analysis of the event-by-event neutron-$\gamma$
multiplicity correlations presented in Refs. [18, 19]. In Ref. [18], we
determined that neutron and $\gamma$-ray emissions are slightly negatively
correlated, as a result of energy and angular momentum conservation. The more
recent analysis of Ref. [19] analyzed how the correlations depend on the
energy of the emitted particles. We have observed predominantly negative
correlations with notable positive enhancements at specific $\gamma$-ray
energies: $E_{\gamma}\approx 0.7$ and $1.2$ MeV. With the aid of model
calculations, we concluded that the positive enhancements originated from
positive correlations between the angular momenta and energies of the
fragments in a fission event. In this work, we extend the previous
investigations by analysing the correlations differentiated with respect to
both energy and direction. The paper is structured as follows. In Section II,
we discuss the origins of the angular distribution of neutron and $\gamma$
radiation. In Section III, we present the new analysis of the experimental
data that takes into account both the energy and the angular dependence of the
emitted radiation. Section IV presents the analysis of the data collected
using the Chi-Nu liquid organic scintillator array at LANSCE, Los Alamos. The
experimental results and possible theoretical interpretations are also
discussed. Lastly, in Section V we discuss how the observed $n$-$\gamma$
emission alignments, and lack thereof, indicate that the fragment angular
momentum is polarized in a direction perpendicular to fragment direction of
motion, and the relationship between the fragment angular momentum and
excitation energy.
## II Sources of Angular Correlations
In the fragment center-of-mass frame (CoM), neutrons are emitted with mean
velocities comparable to the speed of the fragment in the lab frame. Thus, the
neutron kinematic boost effectively determines the angular distribution of
neutrons in the lab frame. The emission of neutrons in the CoM is often
approximated as isotropic and this approximation has been validated
experimentally [20, 13]. Other effects, related to the coupling of angular
momenta and the possibility of scission neutrons [21], would only result in
small corrections. Thus, we expect that neutrons will primarily follow the
direction of the fragment motion. Because the light and heavy fragments are
emitted emitted back-to-back in the CoM of the initial fissioning nucleus, the
angular distribution of neutrons appears as two distributions focused parallel
and anti-parallel to the motion of the fragments, i.e., the fission axis.
The kinematic focusing causes neutrons with greater kinetic energy in the lab
frame to be more tightly aligned with the fission axis and their distributions
more anisotropic in the lab frame. Larger lab-frame energies are also
associated with larger CoM energies, which biases the sample toward symmetric
fission (see Figs. 6 and 18 in Ref. [13]), i.e., fission events resulting in
two similar mass fragments.
The angular distribution of $\gamma$ rays is also affected by kinematic
boosting, an effect known as $\gamma$-ray aberration [22]. However, the
effects are significantly weaker given the relatively low velocity of
fragments. The effects of a weak aberration depend on the angular distribution
in the CoM, but can be approximated as a linear term in the cosine of the
angle of emission in the lab frame. The kinematic boosting of both neutrons
and $\gamma$ rays tend to make $n$-$\gamma$ angular distribution more parallel
when the particles are emitted by the same fragment, and more antiparallel
when the particles are emitted by different fragments. Because we observe
$n$-$\gamma$ correlations irrespective of the fragments emitting them, these
two effects tend to cancel each other. Thus, we do not expect the aberration
of $\gamma$ rays to have a dominant role in $n$-$\gamma$ angular correlations.
The coupling of the fragment angular momentum with that of emitted $\gamma$
rays gives rise to strong observable angular correlations [1]. We can observe
the angular correlations of $\gamma$ rays relative to the fission axis because
the angular momenta of the fragments is aligned perpendicular to the fission
axis [1, 23, 3, 24, 8].
The emission of $\gamma$ rays following fission is usually divided into two
stages: first, $\gamma$ rays are emitted in the continuum to dissipate the
intrinsic excitation energy left over after neutron evaporation; second,
$\gamma$ rays are emitted to dissipate the energy stored in the collective
degrees of freedom. We call the first type of $\gamma$ emission statistical,
because the transition strengths are determined from a statistical analysis of
the level densities. We call the second type of emission discrete, since the
transitions are determined by the available levels in the discrete region of
the level scheme. The angular distributions of these two categories of
$\gamma$ rays are very different.
Statistical $\gamma$-ray emission is assumed to be primarily electric dipole
radiation. The angular distributions of statistical $\gamma$ rays has been
described to be either isotropic [1], or aligned parallel to the fragment
angular momentum and thus perpendicular to the fission axis [25]. The
difference between these two alternatives lies in the angular momenta of the
initial and final states, $J_{i}$ and $J_{f}$ respectively. Transitions with
$J_{f}=J_{i}\pm 1$ contribute $\gamma$ rays emitted predominantly
perpendicular to the fission axis, whereas $J_{f}=J_{i}$ contribute $\gamma$
rays emitted predominantly parallel to it [26]. Depending on the proportion of
the two types of dipole transitions, the angular distributions of statistical
$\gamma$ rays can have different angular distributions.
Discrete emission along the yrast band is primarily electric quadrupole in
nature, although magnetic dipole contributions at the lowest energies have
also been observed. Discrete quadrupole emission along a rotational band tends
to be stretched, i.e., the angular momentum removed by the radiation is
maximized, $J_{f}=J_{i}-2$. Because of their stretched character, the angular
distribution of $\gamma$ rays from quadrupole band transitions are directed
approximately perpendicular to the angular momentum axis and are thus
predominantly parallel to the fission axis [26].
Based on the discussion presented above, we expect both neutron and
$\gamma$-ray emission to be correlated with the fission axis, with no directly
correlations between them. Intrinsic correlations between sequential emissions
are possible, even in the case of a nucleus that is not initially oriented.
These angular correlations arise because the fragment, as it de-excites from
energy level to energy level, is in a superposition of magnetic substates of
angular momentum. Angular momentum conservation dictates that the magnetic
quantum numbers of successive levels are entangled with one another,
introducing intrinsic correlations between them.
The intrinsic angular correlations between neutrons and $\gamma$ rays can be
quite strong. Thus, while we cannot currently exclude that these angular
correlations play an important role in the determination of $n$-$\gamma$,
there are several factors that reduce their strength. First, we expect these
intrinsic correlations only to affect the emission from a single fragment.
Second, the emission of other particles in the same decay sequence will
diminish the observed correlations. This is particularly important for the
correlations between neutrons and the discrete $\gamma$ rays, generally
emitted after $1-2$ statistical $\gamma$ rays have been emitted. Third,
intrinsic correlations can only be expected if neutrons are emitted with some
orbital angular momentum. It is usually assumed that neutrons below
approximately $1-2$ MeV are emitted predominantly as $s$-waves, thus
significantly reducing the effects of intrinsic angular correlations. A recent
investigation [5] showed that the optical model of the nucleus can predict
much larger values of the neutron orbital angular momentum, but more evidence
is needed. In light of these considerations, the intrinsic angular
correlations are not explicitly discussed in this paper, and will be
investigated in future work.
## III Experimental Analysis
In this analysis, we differentiate the covariance of the event-by-event
neutron and $\gamma$-ray multiplicities, $N_{n}$ and $N_{\gamma}$, with
respect to the angle between them, $\theta_{n\gamma}$, as well as their
energies, $E_{n}$ and $E_{\gamma}$. This analysis is an improvement of the
analysis presented in Ref. [19], which only differentiated correlations with
respect to energy. The differentiated normalized covariance,
$C_{E_{n}E_{\gamma}\theta_{n\gamma}}$, is
$C_{E_{n}E_{\gamma}\theta_{n\gamma}}=\frac{\partial^{3}\text{cov}(N_{n},N_{\gamma})}{\partial
E_{n}\partial
E_{\gamma}\partial\theta_{n\gamma}}\left[\frac{\partial}{\partial\theta_{n\gamma}}\left(\frac{\partial\langle
N_{n}\rangle}{\partial E_{n}}\frac{\partial\langle N_{\gamma}\rangle}{\partial
E_{\gamma}}\right)\right]^{-1}\ .$ (1)
The quantity $C_{E_{n}E_{\gamma}\theta_{n\gamma}}$ is bounded from below at
$-1$, but has no upper bound. The three differentiations we perform here serve
distinct purposes. Because the discrete level transitions tend to be lower in
energy than statistical emission, the differentiation with respect to
$E_{\gamma}$ helps to sharpen the separation between statistical and discrete
emission. The differentiation with respect to neutron energy sharpens the
neutron-$\gamma$ angular correlations by narrowing the angular distribution of
neutrons with respect to the fission axis. This differentiation also allows
the identification of correlations that exist due to sample biasing, and are
thus unrelated to the more interesting correlations between a fragment energy
and angular momentum. Lastly, the differentiation with respect to angle is
used to identify the angular momentum properties of $\gamma$ rays.
The data analyzed in this paper were collected with the Chi-Nu array at Los
Alamos National Laboratory. These data are the same as those analyzed in Refs.
[18, 19, 27, 28]. A detailed description of the experiment and the detector
can be found in those references. In this paper, we focus on the capabilities
of angular measurements with the Chi-Nu array. See Ref. [19] for a discussion
of the energy acceptance of the Chi-Nu detectors.
Because spontaneous fission lacks a preferred direction and the fission axis
is not experimentally measured, the directional distribution is measured
between pairs of emitted particles. Specifically, in the present experiment we
measure the neutron-$\gamma$ covariance between the measured neutron and
$\gamma$-ray multiplicities in two detectors whose geometric centers are
separated by an angle $\theta_{n\gamma}$. The normalization of the covariance,
the factor in square brackets in Eq. (1), is calculated from the product of
the mean measured multiplicities in each detector. Because of different gain
settings and distances to the source, the efficiencies of each detector varied
considerably. These variations in the individual detector behavior are
directly translated into variations in the product of efficiencies for each
detector pair. With $42$ active detectors during the experiment, a total of
$861$ detector pairs are possible. However, because we take the first detector
to detect a neutron and the second a $\gamma$-ray, each detector pair is
doubly degenerate. Thus, a total of $1722$ detector-pair combinations are
considered.
We show the angular efficiency of the system in the upper half of Fig. 1. Each
point in the polar plot represents a detector pair while the distance from the
origin represents the energy-averaged efficiency of the detector pair, the
product in the measured neutron-$\gamma$ multiplicities across all energies.
The red circle represents the average detector-pair efficiency across all
detector pairs. The average detector efficiency, expressed as the rate of
double counts per fission event in a specific detector pair $\langle
d_{n}d_{\gamma}\rangle$, is indicated on the figure. In the lower half of Fig.
1, we group detector pairs in angular bins. The size of the marker represents
the number of detector pairs included in that group, while the position of the
marker represents the average in angle and efficiency for all pairs in the
group. The legend indicates the number of detector pairs in each bin. The
standard deviations in both angle and efficiency are shown as error bars.
Figure 1: The angular efficiency of the Chi-Nu detection system. See text for
details.
Because the Chi-Nu array is hemispherical, angles between detector pairs are
neither isotropic nor symmetric with respect to $\pi/2$. The variations
observed in the efficiencies are significantly reduced if the mean of each
angular group is taken. In fact, we see from the lower half of Fig. 1 that the
mean pair efficiency of the angular groups falls close to the average over all
detector pairs. However, it should be noted that because of the hemispherical
geometry of Chi-Nu, we expect measurements to be biased toward neutrons and
$\gamma$ rays emitted at acute angles.
The angular resolution of the detection system is defined as the spread in the
angles measured by the experimental system when the particles are emitted at a
fixed angle. The resolution depends on the room return, where the radiation
interacts with materials around the detector system, or the detector system
itself, before being measured, and on the finite width of the detectors. Given
the large size of the liquid organic scintillators employed, the resolution
due to finite width of the detectors dominates, introducing an uncertainty in
the measured angles of $\approx 0.13$ rad. This result was confirmed by an
MCNPX-PoliMi [29] simulation of the detector array. Using the same simulation,
we have also determined that systematic biases are negligible: the mean angles
between measured particles will be the same as the emitted angles, while the
width of the measured angular distribution is broadened by angular resolution
effects. Cross talk effects are negligible because neutrons and $\gamma$ rays
are easily distinguished by the organic scintillators.
The unfolding of the neutron and $\gamma$-ray energies is performed using the
method described in Ref. [19]. This type of unfolding does not completely
recover the initial distribution, but addresses systematic biases in the
average spectra. Specifically, the unfolded distribution remains broadened
with a system-characteristic resolution. Because the angular response of Chi-
Nu does not introduce systematic biases and only introduces broadening, we do
not unfold the angle and take the angle between two particles to be the angle
between the centers of the detectors where the particles interacted. We
consider energy ranges of $1.0<E_{n}<7.4$ MeV and $0.24<E_{\gamma}<3$ MeV,
with bin widths of size $0.4$ and $0.16$ MeV for neutrons and $\gamma$ rays,
respectively.
## IV Results
For every fixed neutron and $\gamma$-ray energy $E_{n}$ and $E_{\gamma}$, we
obtain curves defining the magnitude of neutron-$\gamma$ correlations for
angles $\theta_{n\gamma}$ between the two emissions and extract the Legendre
coefficients from them. In the CoM frame, only even-order polynomials can
describe the angular distribution of $\gamma$ rays [26]. However, the
aberration of $\gamma$ rays can introduce odd-order, antisymmetric terms to
the angular distribution.
We have fit the experimental data using Legendre polynomial, and have
determined that the best fit was provided by retaining only the symmetric 0th
and 2nd order polynomials. Thus, the $n$-$\gamma$ correlations at fixed
energies are fit using
$C_{E_{n}E_{\gamma}\theta_{n\gamma}}=A_{0}(E_{n},E_{\gamma})+A_{2}(E_{n},E_{\gamma})P_{2}(\text{cos}\theta_{n\gamma})\
.$ (2)
Neutron-$\gamma$ correlation obtained from the data are shown in Fig. 2 for
several selected combinations of $E_{n}$ and $E_{\gamma}$. On the same plot,
we show the fit to the data retaining only the 0th and 2nd order polynomials.
The Legendre coefficients across all neutron and $\gamma$-ray energies are
shown in Fig. 3. Lastly, the statistical uncertainties of the coefficients,
determined from randomly resampling the data are also shown in Fig. 3.
Figure 2: Curves of $C_{E_{n},E_{\gamma}\theta_{n\gamma}}$ for selected
neutron and $\gamma$-ray energies. Legendre polynomial fits for
$0^{\text{th}}$ and $2^{\text{nd}}$ order are shown. Figure 3: Legendre
polynomials fit parameters to $C_{E_{n}E_{\gamma},\theta_{n\gamma}}$
The parameter $A_{0}$ has a simple physical interpretation: it represents the
magnitude of the $n$-$\gamma$ covariance averaged over all emission angles.
The result shown in Fig. 3 (a), as expected from our previous investigation
[19], shows structure developing in the regions $E_{\gamma}\approx 0.7$ and
$E_{\gamma}\approx 1.2$ MeV. Using model calculations, we showed that the
presence of the enhancement at $0.7$ MeV can be explained by positive
correlations between the fragment angular momentum and energy, increasing the
feeding of rotational band states with increasing neutron multiplicities, and
thus excitation energy. The enhancement at $1.2$ MeV is explained in part by
the same correlations and in part by a biasing of the fission samples towards
symmetric fission. For a more detailed analysis of the angle-independent
$n$-$\gamma$ correlations see Ref. [19].
The coefficient of the second Legendre polynomial, $A_{2}$, shown in Fig. 3
(c) shows the dependence of the correlations on the emission angle between
neutrons and $\gamma$ rays. Positive $A_{2}$ indicates $\gamma$ rays are
aligned predominantly along the direction of neutron emission, both parallel
and antiparallel, while negative $A_{2}$ indicates $\gamma$ rays are aligned
perpendicular to neutron emission.
The statistical uncertainties for both $A_{0}$ and $A_{2}$ are shown in Fig. 3
(b) and (d), respectively. The uncertainties are larger for the higher
energies, where fewer particles were measured. However, the uncertainties are
several times smaller than the magnitude of the Legendre coefficients in the
regions of enhancement that we discuss below.
We note enhanced positive structure at $E_{\gamma}\approx 0.7$ MeV in $A_{2}$.
This enhancement closely resembles the structure observed in $A_{0}$, but
extends to lower energies and does not extend to higher energies. Importantly,
we do not observe pronounced angular correlations at $E_{\gamma}\approx 1.2$
MeV, as we do in $A_{0}$. We observe a trend of enhanced correlations with
increasing neutron energies. These results are not surprising considering the
discussion presented in Sec. II. With increasing neutron lab-frame energies,
we bias towards more kinematically-boosted neutrons, thus the angle between
$\gamma$ rays and neutrons becomes more representative of the angle between
$\gamma$ rays and the fission axis. At high $E_{\gamma}$ the angular
correlations are much smaller and very close to $0$. Angular correlations are
also weak at the lowest $\gamma$-ray energies, but caution should be used in
interpreting this region as it borders the lower edge of the $E_{\gamma}$
acceptance and the unfolding might lead to artifacts. We do not observe
significant dependence on neutron energy in either the low or high
$E_{\gamma}$ region.
The alignment of $\gamma$ rays with the fission axis at $E_{\gamma}\approx
0.7$ MeV indicates that these $\gamma$ rays are predominantly from stretched
quadrupole transitions along rotational bands. As noted above, these
transitions generate $\gamma$ rays predominantly perpendicular to the angular
momentum, and thus parallel to the fission axis. This is not the first
experimental observation of these angular correlations, see for example Refs.
[30, 31, 1]. However, the results of this work combine these angular
correlations with the observed positive overall covariance, manifested as
$A_{0}$, between neutrons and $\gamma$ rays in this energy region, thus giving
further confirmation of the presence of positive correlations between the
fragment angular momentum and energy.
The positively-correlated region at $E_{\gamma}\approx 1.2$ MeV shows some
deviations from the expected behavior. In Ref. [19], we identified high-energy
transitions in spherical nuclei near the shell closure of ^132Sn as the main
contributor to the enhanced correlations. We expect these $\gamma$ rays to be
predominantly stretched, thus giving rise to positive angular correlations.
However, experimental observation shows that the angular correlations in this
region, while still positive, are significantly reduced with respect to the
enhancement at $E_{\gamma}\approx 0.7$ MeV. This reduction can occur if, in
spherical nuclei, the direction of the angular momentum is not strongly
oriented perpendicular to the fission axis, as is the case for more deformed
fragments.
In the higher-energy region of $E_{\gamma}\gtrapprox 1.8$ MeV the emission
should be dominated by statistical transitions. We observe that the $\gamma$
rays in this region are predominantly isotropic. This is in good agreement
with earlier observation by Val’skii et al. [30] and Hoffman [1], but in
disagreement with the simplified model of stretched $E1$ transitions mentioned
by Oberstedt et al. [31]. Therefore, the results of this analysis indicate
that the statistical transitions in the continuum are not dominantly
stretched: a significant component connects states of equal angular momentum.
In the low-energy region of $E_{\gamma}\approx 0.4$ MeV we again find $\gamma$
rays uncorrelated with the neutron direction, and hence the fission axis.
Val’skii [30] investigated the multipolar character of $\gamma$-ray
transitions and determined that, at the lowest energies and, significantly for
$E_{\gamma}<0.5$ MeV, $M1$ transitions become important. We can explain the
relative isotropy of these transitions at these energies, also observed by
Val’skii, by considering that these intraband transitions have a lower
probability of being stretched since, in some situations, two connected levels
in different bands will have the same angular momentum. Even for stretched
transitions, the angular momentum at low $E_{\gamma}$ will be reduced because
the contribution of $E2$ and $M1$ transitions are both important in this
energy region and their angular distributions carry opposite signs.
## V Concluding remarks
We have expanded our previous analysis of event-by-event $n$-$\gamma$ emission
correlations by considering the angle between the emitted particles in
addition to their individual energies. We have observed enhanced emission
correlations as well as alignments of the emitted particles for $\gamma$-ray
energies associated with rotational band transitions. We conclude that this
enhancement is related to positive correlations between the fragment energies
and angular momenta. Theoretical models can explain these correlations in
terms of excitations of rotational modes of the fragments during the fission
process [3, 8, 24]. With increasing excitation energy of the fissioning
system, these modes are excited more and give rise to an increase in the
fragment angular momenta.
An enhancement in the emission of isotropic discrete $\gamma$ rays was
observed at higher $\gamma$-ray energies. These emissions are predominantly
from stretched electric quadrupole transitions from heavy fragments with
masses close to the shell closure of ^132Sn. The results of our analysis
indicates that the angular momenta of these fragments are not strongly
polarized in the plane perpendicular to the fission axis. These results can be
explained by the nearly spherical shape of the fragments in this region, which
makes it harder to generate angular momentum and align it.
In addition to providing further evidence for positive correlations between
the energy and angular momentum of a fragment, the results also indicate that
statistical $\gamma$-ray emission is not dominantly stretched radiation. The
impact of transitions where the angular momentum of the initial and final
state is equivalent is strongly affected by the energy-dependent level
densities. It will be interesting to investigate the magnitude of the mixing
theoretically and compare model calculations with the experimental results
shown here. Understanding the magnitude of angular momentum carried by
statistical $\gamma$ rays is an essential step toward in the determination of
the fission fragment initial angular momenta.
Having experimentally determined the alignment of neutrons and $\gamma$ rays,
the next step will be to refine the current theoretical models and generate
predictions to compare to experiment. Along with several other important
observables, these correlations will provide significant help for the
refinement of the modeling of angular momentum in fission. The improved models
will, in turn, shed light on the fundamental dynamics of the fission process
[12] as well as provide predictions of $n$-$\gamma$ emission where
experimental data is lacking.
###### Acknowledgements.
S.M. thanks the Chi-Nu experimental group at LANSCE-LANL and M.J. Marcath for
sharing the experimental data used in this analysis. This work was in part
supported by the Office of Defense Nuclear Nonproliferation Research &
Development (DNN R&D), National Nuclear Security Administration, US Department
of Energy. This work was funded in-part by the Consortium for Monitoring,
Technology, and Verification under Department of Energy National Nuclear
Security Administration award number DE-NA0003920. The work of V.A.P. was
performed under the auspices of UT-Battelle, LLC under Contract No. DE-
AC05-00OR22725 with the U.S. Department of Energy. The work of R.V. was
performed under the auspices of the U.S. Department of Energy by Lawrence
Livermore National Laboratory under Contract DE-AC52-07NA27344. J. R.
acknowledges support from the Office of Nuclear Physics in the U.S. Department
of Energy under Contract DE-AC02-05CH11231.
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|
# Design a Win-Win Strategy That Is Fair to Both Service Providers and Tasks
When Rejection Is Not an Option
Yohai Trabelsi1 Pan Xu2 Sarit Kraus1
1Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
2New Jersey Institute of Technology, Newark, NJ, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Assigning tasks to service providers is a frequent procedure across various
applications. Often the tasks arrive dynamically while the service providers
remain static. Preventing task rejection caused by service provider overload
is of utmost significance. To ensure a positive experience in relevant
applications for both service providers and tasks, fairness must be
considered. To address the issue, we model the problem as an online matching
within a bipartite graph and tackle two minimax problems: one focuses on
minimizing the highest waiting time of a task, while the other aims to
minimize the highest workload of a service provider. We show that the second
problem can be expressed as a linear program and thus solved efficiently while
maintaining a reasonable approximation to the objective of the first problem.
We developed novel methods that utilize the two minimax problems. We conducted
extensive simulation experiments using real data and demonstrated that our
novel heuristics, based on the linear program, performed remarkably well.
## 1 Introduction
In resource allocation, numerous problems can be represented as online
matching in bipartite graphs. One side of the graph comprises service
providers (interchangeably called workers in this paper), while the other
consists of allocated task types. The graph’s edges indicate the
qualifications of service providers to perform tasks of specific types.
In online matching problems, a common scenario involves one dynamic side and
one static side. This dynamic-static setup finds application in various
contexts, such as matching riders(dynamic) to drivers(static) Dickerson et al.
(2021), connecting search queries(dynamic) to advertisers in sponsored
search(static) Delong et al. (2022), and facilitating the teleoperation of
autonomous vehicles (AVs) Ackerman Viden et al. (2023). The primary objective
in these problems is to optimize some criteria from the perspective of the
allocator.
Some other works are dedicated to optimizing allocation fairness. For example,
in the domain of ride-sourcing, a method to achieve allocation fairness was
proposed in Lesmana et al. (2019). Additionally, certain studies address cases
where fairness should be maintained for both online tasks and offline workers
Esmaeili et al. (2023).
Our work is motivated by the teleoperation of AVs that has garnered increasing
attention recently (e.g., Zhang (2020); Ackerman Viden et al. (2023); Tener
and Lanir (2022)). The primary role of teleoperation is to aid AVs by
intervening in challenging driving situations111As mentioned in Tener and
Lanir (2022), the AVs will need this intervention, at least in the near
future.. Ensuring a fair allocation of teleoperators to driving tasks is
crucial for enhancing the satisfaction of both teleoperators and AVs’ users.
Particularly, if certain intervention requests have significantly longer
waiting times or if some teleoperators are disproportionately busier than
others, such imbalances can lead to dissatisfaction among those affected. In
addition, as a person in the vehicle is awaiting the teleoperator’s
intervention, a rejection of a request is unacceptable. Another property of
this application is that the teleoperators (workers) are reusable, which means
they are ready to perform a new intervention request (task) once they finish a
previously allocated request.
We model the problem as online matching in a bipartite graph and propose
several approaches to optimize fairness for both the tasks (e.g., intervention
requests) and the workers (e.g., teleoperators) involved in the process. Our
notion of fairness is aligned with Rawls’ theory of justice Rawls (1999).
We introduce two minimax problems within the given context. The first concerns
fairness regarding tasks relative to waiting times, while the second focuses
on Rawlsian fairness for service providers based on their workload. In both
scenarios, task rejection is not permissible. We demonstrate that the second
problem can be efficiently formulated as a linear problem. Notably, the
solution to the second problem mirrors the first when task durations from each
worker conform to the same distribution. In cases where this isn’t true, we
show that the second problem’s solution approximates the first problem’s
solution, supported by a provable approximation ratio. Our study concludes
with extensive simulations that underscore the efficacy of these minimax
problems. Furthermore, we devise innovative heuristics that leverage the
minimax solutions. These heuristics enhance task fairness while preserving
favorable outcomes for worker fairness.
Our main contributions are: (1) We propose two models to promote fairness
among tasks and workers. (2) We present an LP-based algorithmic framework,
which can exactly solve fairness maximization among workers and approximately
among tasks, and we provide a tight approximation bound. (3) We empirically
implement and compare different methods, including several baselines, on
datasets involving the teleoperation of AVs.
### 1.1 Related Work
In this section, we describe previous works about fair allocation and
allocation with delays. Notably, to our knowledge, our work distinguishes
itself by being the first to consider fairness and allocation delays together.
#### Fair allocation
Some studies address fair allocation, focusing on only one side of the graph,
as seen in Ma et al. (2020). Although their fairness approach resembles ours,
it pertains solely to one side of the graph, which falls short of our
requirements. Other research, like Patro et al. (2020), deals with fairness in
recommendation systems. However, the fairness objectives in recommendation
systems significantly differ from those in task allocation contexts. Practical
solutions for enhancing fairness for both service providers and tasks are
explored in works such as Zhou et al. (2023). Regrettably, this branch of
research lacks theoretical performance bounds for their solutions. The
fairness principles in Esmaeili et al. (2023) closely align with ours. They
consider both workers (offline side) and tasks (online side), embracing
Rawlsian welfare Rawls (1958). Nonetheless, task rejection is permissible in
their scenario if workers are unavailable.
#### Allocation with delayed assignments
The original online matching problem was introduced in Karp et al. (1990),
where static nodes (workers) are instantly paired with dynamic nodes (tasks)
upon arrival. However, real scenarios often lack immediate worker availability
for tasks, prompting consideration for task execution delays over outright
rejection. Numerous works tackle resource allocation with potential task
delays. However, many of these approaches (e.g., Righter (1987); Li et al.
(2023)) prioritize utility maximization without factoring in task wait times
or worker workload. Some leverage reinforcement learning for such issues yet
often make batch decisions, leading to suboptimal outcomes. Moreover,
theoretical guarantees are frequently absent. An LP-based method for delayed
allocations is presented in Ackerman Viden et al. (2023), optimizing a complex
utility function that accounts for task waiting times but overlooks worker
workload.
Another pertinent domain involves queue admission control systems with
multiple classes. Here, diverse customer types (tasks) arrive dynamically, and
a decision-maker determines which task to accept, as demonstrated in Rigter et
al. (2022). However, several studies in this realm do not distinguish between
workers, while others permit task rejection. To our knowledge, the problem of
two-sided fair allocation when task rejection is not allowed has not yet been
addressed.
## 2 Preliminaries
$G$ | Input network graph $G=(I,J,E)$.
---|---
$I$ ($J$) | Set of worker (task) types.
$\mathcal{N}_{i}$ ($\mathcal{N}_{j}$) | Set of neighbors of $i$ ($j$).
$i\sim j$ ($j\sim i$) | Equivalent to $i\in\mathcal{N}_{j}$ ($j\in\mathcal{N}_{i}$).
$\lambda_{j}$ | Arrival rate of task type $j\in J$.
$\lambda_{i}$ | Arrival rate on worker $i\in I$.
$\mathrm{Exp}(\mu)$ | Exponential distribution of rate $\mu>0$.
$\mathrm{Exp}(\mu_{ij})$ | Service time taken by worker $i$ to service $j$.
$\rho_{i}\in[0,1]$ | Workload of worker $i\in I$.
$w_{j}$ | Expected (absolute) waiting time of $j$; see Eqn. (4).
$\bar{w}_{j}$ | Expected (relative) waiting time of $j$; see Eqn. (5).
$\kappa\geq 1$ | $\max_{i\in I}\big{(}\max_{j\sim i,j^{\prime}\sim i}\mu_{ij}/\mu_{i,j^{\prime}}\big{)}$.
Table 1: A glossary of notations throughout this paper.
Suppose we use a bipartite graph $G=(I,J,E)$ to model the worker-task network,
where $I$ denotes the set of offline workers (_e.g.,_ teleoperators), $J$ the
set of types of tasks, and an edge $e=(i,j)$ indicates the feasibility of
worker $i$ to serve the task (of type) $j$. Note that at certain points within
this paper, we abuse the notation by referring to $j$ as a task instead of a
task type. We also abuse the notation by referring to an edge $e=(i,j)$ as
$(ij)$. Tasks of type $j\in J$ arrive following an independent Poisson process
of rate $\lambda_{j}>0$. For each edge $e=(i,j)\in E$, we assume it takes
worker $i$ an exponentially distributed service time222This assumption is
justified in Devore (2008). Note that the theoretical analysis does not depend
on it. We could use any distribution if the mean and the variance of service
time are known. of rate $\mu_{ij}>0$ to complete a task of type $j$ (_i.e.,_
with mean of $1/\mu_{ij}$)333Note that the assumption does not necessarily
suggest the most likely outcome is for tasks to be finished in an extremely
short time. Consider a task type with an exponentially distributed service
time of rate $\mu$, denoted as $X=\mathrm{Exp}(\mu)$. We observe that for any
given threshold $a>0$, $\Pr[X\geq a]=e^{-\mu a}$, which can be close to one
when $\mu$ is small. . For each worker $i$ and task $j$, let
$\mathcal{N}_{i}\subseteq J$ and $\mathcal{N}_{j}\subseteq I$ denote the set
of neighbors of $i$ and $j$ in the graph $G$. The assigning rule is as
follows. Upon the arrival of a task of type $j$, we (as the central
coordinator) have to assign it to a feasible worker $i\in\mathcal{N}_{j}$
immediately: if $i$ is free (or available) at that time, then $i$ will serve
$j$ right away; otherwise, $j$ will join the virtual queue of $i$ and it will
stay there until being served by $i$.
### 2.1 Allocation Policy and Related Concepts
Consider an allocation policy $\pi(\mathbf{x})$ (possibly randomized),
characterized as a vector $\mathbf{x}=\\{x_{ij}|(ij)\in E\\}$, where
$x_{ij}\in[0,1]$ denotes the percentage of task (of type) $j$ assigned to and
served by worker $i$. In the following, we discuss a few important properties
and concepts related to $\pi(\mathbf{x})$. Let $\mathcal{Q}_{i}$ be the
virtual queue maintained by worker $i\in I$.
Arrival rate on $\mathcal{Q}_{i}$, denoted by $\lambda_{i}$. Observe that
$\mathbf{x}=(x_{ij})$ can be viewed alternatively as the probability that
$\pi$ assigns each arriving $j$ to $i$. Thus, we claim that $\mathcal{Q}_{i}$
admits a Poisson arrival process of rate
$\lambda_{i}:=\sum_{j\in\mathcal{N}_{i}}\lambda_{j}\cdot x_{ij}$. By the
property of the Poisson process (See section 2.3.2 at Gallager (2011)),
conditioning on the arrival of task (of type) $\bar{j}\in J$ on $i$, we claim
that $\Pr[\bar{j}=j]=x_{ij}\cdot\lambda_{j}/\lambda_{i}$ for each
$j\in\mathcal{N}_{i}$.
Service time on $\mathcal{Q}_{i}$, denoted by $\mathcal{S}_{i}$. The analysis
above shows that the task joining $\mathcal{Q}_{i}$ is of type
$j\in\mathcal{N}_{i}$ with probability equal to
$x_{ij}\cdot\lambda_{j}/\lambda_{i}$. Thus, the overall service time
$\mathcal{S}_{i}=\sum_{j\in\mathcal{N}_{i}}\chi_{ij}\cdot\mathrm{Exp}(\mu_{ij})$,
where $\chi_{ij}=1$ indicates that the task joining $i$ is of type $j$ with
$\mathsf{E}[\chi_{ij}]=x_{ij}\cdot\lambda_{j}/\lambda_{i}$, and
$\mathrm{Exp}(\mu_{ij})$ represents the exponentially distributed service time
of $i$ for $j$ of rate $\mu_{ij}$. Thus, $\mathcal{S}_{i}$ follows a
_hyperexponential distribution_ Gupta and Goyal (1964) with mean equal to
$\displaystyle
s_{i}:=\mathsf{E}[\mathcal{S}_{i}]=\sum_{j\in\mathcal{N}_{i}}(x_{ij}\lambda_{j})/(\lambda_{i}\mu_{ij}).$
(1)
Workload of worker $i$, denoted by $\rho_{i}$. By definition,
$\displaystyle\rho_{i}:=\lambda_{i}\cdot\mathsf{E}[\mathcal{S}_{i}]=\sum_{j\in\mathcal{N}_{i}}(x_{ij}\lambda_{j})/\mu_{ij},$
(2)
where $\rho_{i}$ can be re-interpreted as the probability that the worker $i$
is busy or the proportion of time the worker $i$ is busy averaged over a long
period. Note that $\rho_{i}<1$ is the key condition ensuring the virtual queue
$\mathcal{Q}_{i}$ can enter a stable state. This is also a condition we should
impose on every worker $i\in I$ when designing policy $\pi(\mathbf{x})$ since
otherwise, $i$ could always stay occupied in the long run (thus, not
acceptable to $i$) and every task $j$ assigned to $i$ could risk an infinitely
long waiting time (not acceptable to $j$).
Waiting time on worker $i$, denoted by $W_{i}$. By the analysis above, we see
that the queue $\mathcal{Q}_{i}$ on worker $i$ qualifies as an $M/G/1$ (using
the standard Kendall’s notation Kendall (1953)), which means it admits a
Poisson arrival process, a general service time distribution, and a single
worker. By the Pollaczek-Khinchin mean formula Asmussen (2003),
$\displaystyle
w_{i}:=\mathsf{E}[W_{i}]=\frac{\lambda_{i}\mathsf{E}[\mathcal{S}^{2}_{i}]}{2(1-\rho_{i})}=\frac{\sum_{j\in\mathcal{N}_{i}}x_{ij}\lambda_{j}/\mu^{2}_{ij}}{1-\sum_{j\in\mathcal{N}_{i}}x_{ij}\lambda_{j}/\mu_{ij}},$
(3)
where the numerator is equal to
$\displaystyle\lambda_{i}\cdot\mathsf{E}[\mathcal{S}_{i}^{2}]=\lambda_{i}\cdot\mathsf{E}\Big{[}\Big{(}\sum_{j\in\mathcal{N}_{i}}\chi_{ij}\cdot\mathrm{Exp}(\mu_{ij})\Big{)}^{2}\Big{]}$
$\displaystyle=\lambda_{i}\cdot\mathsf{E}\Big{[}\sum_{j\in\mathcal{N}_{i}}\chi_{ij}\cdot\mathrm{Exp}^{2}(\mu_{ij})\Big{]}=\lambda_{i}\cdot\sum_{j\in\mathcal{N}_{i}}\mathsf{E}\Big{[}\chi_{ij}\cdot\mathrm{Exp}^{2}(\mu_{ij})\Big{]}$
$\displaystyle=\lambda_{i}\cdot\sum_{j\in\mathcal{N}_{i}}(x_{ij}\lambda_{j}/\lambda_{i})\cdot(2/\mu^{2}_{ij})=2\sum_{j\in\mathcal{N}_{i}}(x_{ij}\lambda_{j}/\mu^{2}_{ij}).$
Absolute and relative waiting time of $j$, denoted by $W_{j}$ and
$\overline{W}_{j}$. Recall that under $\pi(\mathbf{x})$, a task $j$ will be
assigned to a feasible worker $i\in\mathcal{N}_{j}$ with probability $x_{ij}$.
Thus, the expected (absolute) waiting time of $j$ should be
$\displaystyle w_{j}:=\mathsf{E}[W_{j}]=\sum_{i\in\mathcal{N}_{j}}x_{ij}\cdot
w_{i},$ (4)
where $w_{i}$ is the expected waiting time on queue $\mathcal{Q}_{i}$, as
shown in (3). The _relative_ waiting time of $j$ on $\mathcal{Q}_{i}$ is
defined as the ratio of waiting time on $\mathcal{Q}_{i}$ to the service time
of $i$ for $j$, which has a mean of $1/\mu_{ij}$. Thus, the expected relative
waiting time of $j$ should be
$\displaystyle\bar{w}_{j}$
$\displaystyle:=\mathsf{E}[\overline{W}_{j}]=\sum_{i\in\mathcal{N}_{j}}x_{ij}\cdot
w_{i}/(1/\mu_{ij})$
$\displaystyle=\sum_{i\in\mathcal{N}_{j}}x_{ij}\cdot\mu_{ij}\cdot\frac{\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}^{2}}{1-\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}}.$ (5)
### 2.2 Two Fairness-Related Objectives
In this paper, we propose the following two fairness metrics and objectives
when optimizing a policy $\pi(\mathbf{x})$.
#### FAIR-T: Fairness promotion among tasks, denoted by ${\min\max_{j\in
J}\bar{w}_{j}}$
. We quantify the overall fairness among users achieved by policy
$\pi(\mathbf{x})$ as the maximum expected _relative_ waiting time among all
task types, _i.e.,_ $\max_{j\in J}\bar{w}_{j}$. A formula for calculating the
relative waiting time is shown in (5). Note that here we choose the relative
version instead of the absolute one (_i.e.,_ $\max_{j\in J}w_{j}$) following,
for example, the paper Maister and others (1984) that asserts that “the more
valuable the service, the longer the customer will wait.” A compelling example
is that:“Special checkout counters were originally provided because customers
with only a few items felt resentful at having to wait a long time for what
was seen as a simple transaction. Customers with a full cart of groceries were
much more inclined to tolerate lines.”
FAIR-S: Fairness promotion among workers, denoted by $\min\max_{i\in
I}\rho_{i}$. Recall that for each worker $i\in I$, the workload
$\rho_{i}\in(0,1)$, as defined in (2), captures the percentage of busy time on
worker $i$. Thus, the maximum workload, i.e., $\max_{i\in I}\rho_{i}$,
reflects the highest degree of being occupied among all workers under policy
$\pi(\mathbf{x})$. By opting for minimization of the maximum workload, denoted
by $\min\max_{i\in I}\rho_{i}$, we aim to minimize the occupation time of the
most occupied worker as substantially as feasible.
### 2.3 Two Optimization Programs
Consider an allocation policy $\pi(\mathbf{x})$ parameterized by
$\mathbf{x}=(x_{ij})$, where $x_{ij}$ with $(ij)\in E$ denotes the percentage
of task of type $j$ assigned to worker $i$. For ease of notation, we will use
$i\sim j$ (and $j\sim i$) to represent $i\in\mathcal{N}_{j}$ (and
$j\in\mathcal{N}_{i}$) throughout this paper. We formulate FAIR-T and FAIR-S
as minmax programs as follows.
$\displaystyle(\operatorname{\textbf{PT}})\min$ $\displaystyle\max_{j\in
J}\Big{(}\bar{w}_{j}=\sum_{i\sim
j}x_{ij}\cdot\mu_{ij}\cdot\frac{\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}^{2}}{1-\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}}\Big{)},$ (6) $\displaystyle
x_{j}:=\sum_{i\sim j}x_{ij}=1,~{}~{}\forall j\in J$ (7)
$\displaystyle\rho_{i}=\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}\leq 1,~{}~{}\forall i\in I$ (8)
$\displaystyle 0\leq x_{ij}\leq 1,~{}~{}\forall(ij)\in E.$ (9)
$\displaystyle(\operatorname{\textbf{PS}})~{}~{}\min$
$\displaystyle~{}~{}\max_{i}\rho_{i},$ (10) $\displaystyle x_{j}:=\sum_{i\sim
j}x_{ij}=1,$ $\displaystyle~{}~{}\forall j\in J$ (11)
$\displaystyle\rho_{i}=\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}\leq 1,$ $\displaystyle~{}~{}\forall i\in
I$ (12) $\displaystyle 0\leq x_{ij}\leq 1,$ $\displaystyle~{}~{}\forall(ij)\in
E.$ (13)
We refer to the above programs as $\operatorname{\textbf{PT}}$ and
$\operatorname{\textbf{PS}}$, respectively. Let $\mathbf{x}_{t}^{*}$ and
$\mathbf{x}^{*}_{s}$ be optimal solutions to $\operatorname{\textbf{PT}}$ and
$\operatorname{\textbf{PS}}$, respectively.
###### Lemma 1.
$\pi(\mathbf{x}_{t}^{*})$ and $\pi(\mathbf{x}_{s}^{*})$ are optimal policies
under FAIR-T and FAIR-S, respectively.
###### Proof.
We focus on showcasing the case of FAIR-T and the program
$\operatorname{\textbf{PT}}$. The proof for the other case is similar. Note
that the term shown in (6) captures the precise objective we aim to optimize.
To prove our claim, we need to demonstrate that all constraints in
$\operatorname{\textbf{PT}}$ hold true for any viable policy of
$\pi(\mathbf{x})$. Constraint (7) is reasonable because every policy must
assign each incoming task to a feasible worker without rejection, thereby
ensuring that the total percentages assigned for each type sum up to one.
Constraint (8) is valid as the workload of any worker (i.e., the percentage of
busy time) should not exceed one. Constraint (9) holds true since $x_{ij}$
represents the percentage of tasks of type $j$ assigned to worker $i$. ∎
Lemma 1 suggests that the optimal policies for FAIR-T and FAIR-S each can be
obtained by solving minmax programs represented by
$\operatorname{\textbf{PT}}$ and $\operatorname{\textbf{PS}}$ respectively.
Note that $\operatorname{\textbf{PS}}$ can be reformulated as a linear program
($\operatorname{LP}$) by introducing an auxiliary variable $\rho$ and
modifying the objective as $\min\rho$, along with additional constraints
$\rho\geq\rho_{i}$ for all $i\in I$. Consequently, we can efficiently solve
$\operatorname{\textbf{PS}}$ and obtain an optimal policy for FAIR-S. However,
for program $\operatorname{\textbf{PT}}$, the objective is non-linear and can
be neither convex nor concave even under very special settings, posing a
technical challenge for direct optimization; see detailed discussions in the
Appendix.
Nevertheless, under certain conditions, $\operatorname{\textbf{PT}}$ can be
effectively and accurately approximated by $\operatorname{\textbf{PS}}$, as
proven in Theorem 1.
###### Lemma 2.
The optimal values of $\operatorname{\textbf{PT}}$ and
$\operatorname{\textbf{PS}}$ each remain invariant if we treat any task type
$j\in J$ with an arrival rate of $\lambda_{j}$ as $k$ different online types,
each having the same set of neighbors as $j$, with an arrival rate of
$\lambda_{j}/k$ for any integer $k$.
The above lemma suggests that for fairness maximization among either workers
under metric FAIR-S or tasks under metric FAIR-T, we can assume without loss
of generality that all tasks take a uniform arrival rate by creating an
appropriate number of copies for each task type. In other words, the variation
among tasks’ arrival rates makes no difference to fairness promotion, compared
with the difference among service times. _In the remaining sections, we assume
without loss of generality that $\lambda_{j}=\lambda$ for all $j\in J$._
## 3 The Relation Between the Two Fairness Optimization Problems
Consider a general setting denoted by ${\boldsymbol{\mu}}:=(\mu_{ij})$, where
$\mu_{ij}$ with $(ij)\in E$ represents the parameter for the exponential
distribution of the service time taken by worker $i$ to serve task $j$. Let
$\eta_{t}({\boldsymbol{\mu}},\mathbf{x})$ denote the objective value of
$\operatorname{\textbf{PT}}({\boldsymbol{\mu}})$ with respect to the input
${\boldsymbol{\mu}}$ and a feasible solution $\mathbf{x}=(x_{ij})$. Similarly,
$\eta_{s}({\boldsymbol{\mu}},\mathbf{x})$ denotes the objective value of
$\operatorname{\textbf{PS}}({\boldsymbol{\mu}})$. When the context is clear,
we may omit either the first or second argument for $\eta_{t}$ and $\eta_{s}$.
For any given input ${\boldsymbol{\mu}}$, let
$\eta^{*}_{t}({\boldsymbol{\mu}})$ and $\eta_{s}^{*}({\boldsymbol{\mu}})$
denote the optimal values of $\operatorname{\textbf{PT}}({\boldsymbol{\mu}})$
and $\operatorname{\textbf{PS}}({\boldsymbol{\mu}})$ respectively.
###### Theorem 1.
Let $\mathbf{x}^{*}_{s}$ be an optimal solution to
$\operatorname{\textbf{PS}}({\boldsymbol{\mu}})$. We have
$\displaystyle\eta_{t}({\boldsymbol{\mu}},\mathbf{x}_{s}^{*})\leq\kappa^{3}\left(1+\Big{(}1-\frac{1}{\kappa}\Big{)}\cdot\frac{\eta^{*}_{s}({\boldsymbol{\mu}})}{1-\eta^{*}_{s}({\boldsymbol{\mu}})}\right)\cdot\eta_{t}^{*}({\boldsymbol{\mu}}),$
(14)
where $\kappa=\max_{i\in I}\big{(}\max_{j\sim i,j^{\prime}\sim
i}\mu_{ij}/\mu_{i,j^{\prime}}\big{)}\geq 1$, which captures the maximum
pairwise ratio among the expectations of all service time on each given
worker.
For a private case of Theorem 1\- where $\kappa=1$ we prove that
$\eta_{t}({\boldsymbol{\mu}},\mathbf{x}_{s}^{*})=\eta_{t}^{*}({\boldsymbol{\mu}})$
(Theorem 2).
These results serve as the bedrock of the whole proof for Theorem 1, which is
deferred to the Appendix for space reasons. Toward the proof of $\kappa=1$, we
first define the following minimax programs and show their equivalence to
$\operatorname{\textbf{PT}}$ and $\operatorname{\textbf{PS}}$, respectively,
for $\kappa=1$.
$\displaystyle(\overline{\operatorname{\textbf{PT}}})~{}~{}\min$
$\displaystyle~{}~{}\max_{j\in J}\Big{(}\bar{w}_{j}=\sum_{i\sim
j}\frac{x_{ij}}{1-\rho_{i}}-1\Big{)},$ (15) $\displaystyle\sum_{i\sim
j}x_{ij}=1,$ $\displaystyle~{}~{}\forall j\in J$ (16)
$\displaystyle\rho_{i}=(\lambda/\mu_{i})\sum_{j\sim i}x_{ij}\leq 1,$
$\displaystyle~{}~{}\forall i\in I$ (17) $\displaystyle 0\leq x_{ij}\leq 1,$
$\displaystyle~{}~{}(ij)\in E.$ (18)
$\displaystyle(\overline{\operatorname{\textbf{PS}}})~{}~{}\min$
$\displaystyle~{}~{}\max_{i\in I}\rho_{i},$ (19) $\displaystyle\sum_{i\sim
j}x_{ij}=1,$ $\displaystyle~{}~{}\forall j\in J$ (20)
$\displaystyle\rho_{i}=(\lambda/\mu_{i})\sum_{j\sim i}x_{ij}\leq 1,$
$\displaystyle~{}~{}\forall i\in I$ (21) $\displaystyle 0\leq x_{ij}\leq 1,$
$\displaystyle~{}~{}(ij)\in E.$ (22)
###### Lemma 3.
For $\kappa=1$, the programs $\operatorname{\textbf{PT}}$ and
$\overline{\operatorname{\textbf{PT}}}$ are equivalent and the programs
$\operatorname{\textbf{PS}}$ and $\overline{\operatorname{\textbf{PS}}}$ are
also equivalent.
###### Proof.
Note that $\kappa=1$ suggests that $\mu_{ij}$ takes some uniform value of
$\mu_{ij}=\mu_{i}$ for every $j\sim i$. Recall that $\lambda_{j}=\lambda$ for
all $j\in J$ due to Lemma 2. Under these assumptions, we see that the
expressions of $\rho_{i}$ and $\bar{w}_{j}$ in (2) and (5) can be simplified
as
$\displaystyle\rho_{i}=\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}=(\lambda/\mu_{i})\cdot\sum_{\ell\sim
i}x_{i\ell},$
$\displaystyle\bar{w}_{j}:=\mathsf{E}[\overline{W}_{j}]=\sum_{i\sim
j}x_{ij}\cdot w_{i}/(1/\mu_{ij})$ $\displaystyle=\sum_{i\sim
j}\frac{x_{ij}\cdot\mu_{i}\cdot\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i}^{2}}{1-\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i}}=\sum_{i\sim
j}\frac{x_{ij}\cdot\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i}}{1-\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i}}$ $\displaystyle=\sum_{i\sim
j}x_{ij}\Big{(}-1+\frac{1}{1-\rho_{i}}\Big{)}=-1+\sum_{i\sim
j}\frac{x_{ij}}{1-\rho_{i}},$
where the equality on the last line is due to $\sum_{i\sim j}x_{ij}=1$ for
every $j\in J$ (no rejection allowed). Substituting lines 17 and 21 with the
value of $\rho_{i}$ and line 15 with the value of $\bar{w}_{j}$ implies that
the programs $\operatorname{\textbf{PT}}$ and $\operatorname{\textbf{PS}}$ are
equivalent to $\overline{\operatorname{\textbf{PT}}}$ and
$\overline{\operatorname{\textbf{PS}}}$. ∎
Consider a given setting with ${\boldsymbol{\mu}}=(\mu_{ij})$ satisfying
$\mu_{ij}=\mu_{i}$ for all $j\sim i$ and $\lambda_{j}=\lambda$ for all $j\in
J$. For ease of notation, we use $\overline{\eta^{*}_{t}}$ and
$\overline{\eta^{*}_{s}}$ to denote optimal values of
$\overline{\operatorname{\textbf{PT}}}$ and
$\overline{\operatorname{\textbf{PS}}}$, respectively, with respect to the
given setting. By default, we assume both have feasible
solutions.444Infeasibility to either Program
$\overline{\operatorname{\textbf{PT}}}$ or
$\overline{\operatorname{\textbf{PS}}}$ suggests that no policy can lead to
meaningful fairness among tasks (finite max expected waiting time) or among
workers (a non-zero ratio of being free). We denote by
$\overline{\eta_{t}(\mathbf{x}_{s}^{*})}$ the value of
$\overline{\operatorname{\textbf{PT}}}$ on $\mathbf{x}_{s}^{*}$ in the given
setting.
It is tempting to prove that $\eta_{t}(\mathbf{x}_{s}^{*})=\eta_{t}^{*}$ for
$\kappa=1$ by showing that $\overline{\operatorname{\textbf{PT}}}$ and
$\overline{\operatorname{\textbf{PS}}}$ each possess an optimal solution such
that $\\{\rho_{i}\\}$ all take a uniform value, say $\rho$. Following this
“claim”, $\overline{\operatorname{\textbf{PT}}}$ is then reduced to $\min
1/(1-\rho)-1$ with $\rho=\rho_{i}$ for all $i\in I$, while
$\overline{\operatorname{\textbf{PS}}}$ is reduced to $\min\rho$ with
$\rho=\rho_{i}$ for all $i\in I$. This establishes Theorem 2 since $\min
1/(1-\rho)-1$ is equivalent to $\min\rho$. The example below disproves this
idea, unfortunately.
###### Example 1.
[_$\overline{\operatorname{\textbf{PT}}}$_ and
_$\overline{\operatorname{\textbf{PS}}}$_ each possess a unique optimal
solution with non-uniform values of $\\{\rho_{i}\\}$ and $\\{\bar{w}_{j}\\}$.]
Consider a graph $G=(I,J,E)$ such that $|I|=m=2$ and $|J|=n\gg 1$ (See Figure
1). The input setting is as follows. $\mu_{ij}=\mu$ for all $(ij)\in E$ and
$\lambda_{j}=\lambda$ for all $j\in J$. Let $\phi=\lambda/\mu$ with
$n\cdot\phi<1$. $i=2$ is connected to all $j\in J$, while $i=1$ is connected
only to $j=1$. We can verify that (1)
_$\overline{\operatorname{\textbf{PT}}}$_ and
_$\overline{\operatorname{\textbf{PS}}}$_ each have a _unique_ optimal
solution and the two are the same, which is $\mathbf{x}^{*}=(x_{ij})$ with
$x_{11}=1$, $x_{21}=0$, and $x_{2j}=1$ for all $1<j\leq n$; (2) for
_$\overline{\operatorname{\textbf{PS}}}$_ : $\rho_{1}(\mathbf{x}^{*})=\phi$,
and $\rho_{2}(\mathbf{x}^{*})=(n-1)\phi<1$; for
_$\overline{\operatorname{\textbf{PT}}}$_ :
$\bar{w}_{1}(\mathbf{x}^{*})=1/(1-\rho_{1}(\mathbf{x}^{*}))-1=1/(1-\phi)-1$
and $\bar{w}_{j}=1/(1-\rho_{2}(\mathbf{x}^{*}))-1=1/(1-(n-1)\phi)-1$ for
$1<j\leq n$.
We will now present two lemmas that establish together the correctness of
Theorem 2.
###### Lemma 4.
$\overline{\eta_{t}(\mathbf{x}_{s}^{*})}\leq 1/(1-\overline{\eta_{s}^{*}})-1$
###### Proof.
Since $\mathbf{x}^{*}=(x_{ij})$ is an optimal solution to
$\overline{\operatorname{\textbf{PS}}}$, $\overline{\eta_{s}^{*}}=\max_{i\in
I}\rho_{i}(\mathbf{x}^{*}):=\rho^{*}$. Observe that for each $j\in J$,
$\bar{w}_{j}(\mathbf{x}^{*})=\sum_{i\sim
j}\frac{x_{ij}}{1-\rho_{i}(\mathbf{x}^{*})}\leq\sum_{i\sim
j}\frac{x_{ij}}{1-\rho^{*}}=\frac{1}{1-\rho^{*}}-1,$
which suggests that
$\overline{\eta_{t}(\mathbf{x}^{*})}=\max_{j}\bar{w}_{j}(\mathbf{x}^{*})\leq
1/(1-\rho^{*})-1=1/(1-\overline{\eta^{*}_{s}})-1$. ∎
$i_{1}$$i_{2}$$j_{1}$$j_{2}$$j_{n}$
$\displaystyle\rho_{1}(\mathbf{x}^{*})=\phi,$
$\displaystyle\rho_{2}(\mathbf{x}^{*})=(n-1)\phi,$
$\displaystyle\bar{w}_{1}(\mathbf{x}^{*})=\frac{1}{1-\phi}-1,$
$\displaystyle\bar{w}_{j}(\mathbf{x}^{*})=\frac{1}{1-(n-1)\phi}-1.$
Figure 1: An example where $\overline{\operatorname{\textbf{PT}}}$ and
$\overline{\operatorname{\textbf{PS}}}$ each have a unique optimal solution
with non-uniform values of $\\{\rho_{i}\\}$ and $\\{\bar{w}_{j}\\}$, though
$\eta_{t}({\boldsymbol{\mu}},\mathbf{x}_{s}^{*})=\eta^{*}_{t}$ since the
programs share the same unique optimal solution.
###### Lemma 5.
$1/(1-\overline{\eta_{s}^{*}})-1\leq\overline{\eta_{t}^{*}}$.
The lemma’s proof is in the Appendix.
We’re now set to present results for $\kappa=1$.
###### Theorem 2.
Consider an input ${\boldsymbol{\mu}}=(\mu_{ij})$ with $\kappa=1$. Let
$\mathbf{x}_{s}^{*}$ be an optimal solution to
$\overline{\operatorname{\textbf{PS}}}$. We have that the value of
$\overline{\operatorname{\textbf{PT}}}$ on the solution of
$\mathbf{x}_{s}^{*}$ is equal to its optimal value, _i.e.,_
$\eta_{t}(\mathbf{x}_{s}^{*})=\eta_{t}^{*}$.
###### Proof.
The above two lemmas together imply that
$\overline{\eta_{t}(\mathbf{x}_{s}^{*})}\leq\overline{\eta_{t}^{*}}$.
$\mathbf{x}_{s}^{*}$ is feasible to $\overline{\operatorname{\textbf{PT}}}$
since $\overline{\operatorname{\textbf{PT}}}$ and
$\overline{\operatorname{\textbf{PS}}}$ share the same set of constraints, and
thus, $\overline{\eta_{t}(\mathbf{x}_{s}^{*})}\geq\overline{\eta_{t}^{*}}$,
which establishes Theorem 2. ∎
## 4 Experiments
### 4.1 Algorithms and Heuristics
This section presents an algorithm derived from solutions to one of the
minimax problems. We also describe a heuristic based on this algorithm, which
gives preference to assigning tasks to available workers, thereby enhancing
allocation through the effective workload of free workers. In addition, this
section introduces two real-time greedy heuristics, which function as baseline
methodologies.
#### Minimax problems based algorithm:
We first describe Algorithm 1. This algorithm has offline and online phases.
In the offline phase (line 2), a solution to one of the minimax problems is
computed. In the online phase (lines 4-7), when a task arrives, the task is
assigned to the queue of a worker according to the probabilities computed by
the program in the offline phase. This algorithm has two variants: One solves
$\operatorname{\textbf{PT}}$ in the offline phase while the other solves
$\operatorname{\textbf{PS}}$.
#### Minimax problems based heuristic:
A notable issue with Algorithm 1 is that tasks can wait for a busy worker
despite other available workers. This leads to suboptimal performance. To
address this, we create a heuristic based on Algorithm 1. Like Algorithm 1, in
Algorithm 2, task assignment probabilities are computed offline to mitigate
this problem. In the online phase, incoming tasks are assigned to free
workers. If multiple workers are free, their precomputed probabilities (from
the offline phase) are normalized to sum to 1. A worker is subsequently chosen
randomly, guided by these normalized probabilities. If there are no free
workers, the tasks are assigned according to their probabilities as in
Algorithm 1. Algorithm 2 describes this heuristic. As in Algorithm 1, there
are two variants of Algorithm 2: One solves $\operatorname{\textbf{PT}}$ in
the offline phase, while the other solves $\operatorname{\textbf{PS}}$.
This method targets reduced waiting times, especially during low-load periods.
However, this change might decrease worker workload or waiting times for other
tasks, as it deviates from calculated optimal probabilities. In practice, we
find that the trade-off for worker and task fairness is reasonable, given the
substantial benefits for all tasks’ fairness.
#### Computational complexity of Algorithms 1 and 2
Both algorithms 1 and 2 have offline and online phases. The offline phase is
identical for both algorithms and requires the solution of
$\operatorname{\textbf{PT}}$ or $\operatorname{\textbf{PS}}$. Following Cohen
et al. (2021), the runtime for solving the linear program-
$\operatorname{\textbf{PS}}$ can be as low as
$O^{*}(N^{2+1/6}\log(N/\delta))$, where $\delta$ is the relative accuracy and
$N=|E|$ is the number of edges in the graph $G$. We leave the complexity of
solving $\operatorname{\textbf{PT}}$ to future work. In any case, the
complexity of the offline phase dominates the complexity of the online phase.
1 Offline Phase:
2 Solve $\operatorname{\textbf{PT}}$ ($\operatorname{\textbf{PS}}$) and let
$\\{x_{ij}\\}$ be an optimal solution.
3 Online Phase:
4 for _each task of type $j$ that arrive on time $t$_ do
5 Let $\mathcal{Q}_{i}$ be a queue of worker $i$
6 Choose randomly a worker $i$ following the probabilities in $\\{x_{ij}\\}$
7
8 Update $\mathcal{Q}_{i}=\mathcal{Q}_{i}\cup\\{(j,t)\\}$.
9
Algorithm 1 An LP-based algorithm for FAIR-T and FAIR-S.
#### The two greedy heuristics
Similar to Ackerman Viden et al. (2023), we devised two greedy heuristics as
baselines for comparison. The first minimizes maximum task waiting times, and
the second minimizes maximum worker workload. In the first, incoming tasks are
assigned to workers with the shortest estimated waiting time, calculated by
summing average expected task durations for tasks in the queue. The elapsed
time for ongoing tasks is subtracted from their average duration to update
estimates. For the second, tasks are assigned to less utilized workers based
on current workload upon task arrival. This approach considers executed tasks,
using actual durations rather than expected durations. The first heuristic is
denoted as GTW (Greedy Task Waiting time) and the second as GWU (Greedy Worker
Workload).
### 4.2 Experimental Settings
We ran experiments on the teleoperation domain. As already mentioned in
Section 1, the teleoperation of AVs involves intervention tasks that are
assigned to the teleoperators who perform them. We adapted the dataset of
Ackerman Viden et al. (2023) for our two-sided fairness study. More details
about the experimental settings and additional experimental results have been
moved to the Appendix. Source code and data for running the experiments are
available at Trabelsi (2024).
Figure 2: Y axis is the maximum waiting time(in seconds) for a task(a,c and
e) and the max. worker workload(b,d and f). The X axis in (a,b) is the value
of $\kappa$(x axis) and the task load is of 120000 tasks per day. In (c,d),
the X axis is the task load and $\kappa$ is set to 1. In (e,f) the X axis is
for different balances of arrival distribution: first bar is for equal
distribution for each task type. In the second bar, the first type has
probability of 70% and the others have 10%. the other bars are defined
similarly for the second, third and forth task types (task load was 80000
tasks per day and $\kappa$ is set to 1). In the legend: SIM(PT) and SIM(PS)
denote Algorithm 1’s results for $\operatorname{\textbf{PT}}$ and
$\operatorname{\textbf{PS}}$ in simulation. SIM-F(PT) and SIM-F(PS) are
Algorithm 2’s results (in which we assign to a free worker first) for
$\operatorname{\textbf{PT}}$ and $\operatorname{\textbf{PS}}$ in simulation.
$GTW$ and $GWU$ are the results of the greedy heuristics targeting task
waiting time and worker workload in simulation. Error bars represent a
confidence interval of 0.95.
#### The tasks, their durations and their arrival rates
Our study built upon the four task types defined by Viden et al. (2023). Their
dataset provided valuable insights into the average duration times for each
teleoperator (worker) and task type in a simulation. We explored three
distinct approaches to define task duration in our experiments. All approaches
involved sampling durations from exponential distributions, but the difference
lay in the means of these distributions.
#### The teleoperators and the tasks they can perform
Using the dataset of Viden et al. (2023), we initially had 10 teleoperators
(workers) and 4 task types. We form a bipartite graph with 10 teleoperators on
one side and 4 task types on the other. The dataset provides average task
completion times for each teleoperator-task pair. An edge is established
between a teleoperator and a task type if their average time matches or
exceeds the task type’s median value. Following this process, a teleoperator
who consistently performed tasks slower than the median was identified and
subsequently excluded from the graph. More experiments on a synthetic dataset
in which the numbers of teleoperators and task types are varied can be found
in the Appendix.
1 Offline Phase:
2 Solve $\operatorname{\textbf{PT}}$ ($\operatorname{\textbf{PS}}$) and let
$\\{x_{ij}\\}$ be an optimal solution.
3 Online Phase:
4 for _each task of type $j$ that arrive on time $t$_ do
5 Let $\mathcal{Q}_{i}$ be a queue of worker $i$ and let $F$ be the subset of
free workers on time $t$
6 If $F\neq\emptyset$, randomly choose a free worker $i$ with probability
$\\{x_{ij}/\sum_{i^{\prime}\in F}{x_{i^{\prime}j}}\\}$
7 Otherwise choose randomly a worker $i$ following the probabilities in
$\\{x_{ij}\\}$
8 Update $\mathcal{Q}_{i}=\mathcal{Q}_{i}\cup\\{(j,t)\\}$.
Algorithm 2 A heuristic for FAIR-T (FAIR-S).
#### Experimental environment and more settings
Each experiment spanned a virtual 4-week period. Due to algorithmic
stochasticity, each experiment was repeated 10 times.
### 4.3 Results and Discussion
#### Effect of changing $\kappa$
In Figures 2(a,b), we illustrate the performance of various methods across
diverse $\kappa$ values. In Figure 2(a), we measure the maximum task waiting
time. We see that the gap between SIM($\operatorname{\textbf{PS}}$) and
SIM($\operatorname{\textbf{PT}}$), as well as the gap between
SIM-F($\operatorname{\textbf{PS}}$) and SIM-F($\operatorname{\textbf{PT}}$),
increase with $\kappa$. This aligns with the fact that with higher values of
$\kappa$, the approximation ratio of $\operatorname{\textbf{PS}}$’s solution
relative to $\operatorname{\textbf{PT}}$’s objective is greater. However, the
ratio between the different methods measured in practice is lower than the
worst-case theoretical ratio given by Theorem 1 (which is greater than
$\kappa^{3}$).
In Figure 2(b), we measure the maximum worker workload. The differences
between SIM($\operatorname{\textbf{PT}}$) vs SIM($\operatorname{\textbf{PS}}$)
are very small for $\kappa\leq 3$, but they become more significant for
$\kappa\in\\{4,5\\}$. Surprisingly, there is a different effect with
SIM-F($\operatorname{\textbf{PT}}$) and SIM-F($\operatorname{\textbf{PS}}$).
SIM-F($\operatorname{\textbf{PT}}$) performs slightly better than
SIM-F($\operatorname{\textbf{PS}}$). We conjecture that the initial selection
of free workers has a more detrimental effect in
SIM-F($\operatorname{\textbf{PS}}$), which integrates two distinctly different
methods, in contrast to the relatively similar approaches in
SIM-F($\operatorname{\textbf{PT}}$). We also see that for larger values of
$\kappa$, both SIM($\operatorname{\textbf{PT}}$) and
SIM($\operatorname{\textbf{PS}}$) perform worse than for lower values.
#### Effect of changing the task load
Figures 2(c,d) might help the teleoperation center’s owner decide whether the
current number of workers is sufficient. It is noticeable that in Figure 2(c)
there is a significant jump from 120000 to 140000 tasks per day. This means
that perhaps the owner should employ more workers in this case. Referring to
Figure 2(d) may lead us to similar conclusions. Employing more workers is
advisable if individual worker workload is excessively high.
#### Effect of changing the task balance
Figures 2(e,f) represent the performance of the different algorithms when
changing the task balance. The left bar represents an even distribution for
each task type (0.25). The second bar represents a higher probability for the
first type (0.7) and a lower probability for the other types (0.1). The other
bars are similarly defined for the other task types.
In Figures 2(e,f), higher arrival distribution of the first task type leads to
elevated waiting times and worker workload. Consequently, the teleoperation
center’s owner could enhance fairness by upskilling operators who are not
qualified for the task or hiring new ones proficient in it. Alternatively,
training could be provided to expedite task completion. The negligible error
bars in all figures show that the error approaches 0 if the experiments are
carried out over a sufficiently long period of time, as we have done.
#### Computed optimal values vs simulation values
In all experiments that we ran, the computed expected maximum waiting time
(OPT($\operatorname{\textbf{PT}}$)) and the computed expected maximum worker
workload (OPT($\operatorname{\textbf{PS}}$)) closely align with simulation-
derived values (SIM($\operatorname{\textbf{PT}}$) and
SIM($\operatorname{\textbf{PS}}$) respectively). Additionally, the alignment
of OPT($\operatorname{\textbf{PT}}$) and OPT($\operatorname{\textbf{PS}}$) at
$\kappa=1$ is consistent with Theorem 1.
#### Choosing the best algorithm
The heuristic GTW, which minimizes the maximum task waiting time, performs
well at maximum task waiting time and performs poorly at maximum worker
workload. Conversely, the greedy heuristic that minimizes the maximum worker
workload, GWU, performs well at the maximum worker workload and performs
poorly at maximum task waiting time. The methods that offer the best tradeoff
between two dimensions of fairness are SIM-F($\operatorname{\textbf{PT}}$) and
SIM-F($\operatorname{\textbf{PS}}$). However, since
$\operatorname{\textbf{PT}}$ is nonlinear, there is no tool that guarantees to
find an optimal solution for $\operatorname{\textbf{PT}}$, and therefore
$\operatorname{\textbf{PT}}$-dependent approaches such as
SIM-F($\operatorname{\textbf{PT}}$) might be unsolvable.
Therefore, if $\kappa=1$ or at least a small number close to $1$, we might
want to use SIM-F($\operatorname{\textbf{PS}}$). However,
SIM($\operatorname{\textbf{PS}}$) might be slightly better if worker workload
is more important than task waiting times (but still important). If $\kappa$
is large, it is advisable to consider using a tool that approximates a
solution for $\operatorname{\textbf{PT}}$ with
SIM-F($\operatorname{\textbf{PT}}$). The figures show that the available tools
work adequately in such cases, despite the lack of theoretical guarantees (at
least for small problems). Another option is to try both
SIM-F($\operatorname{\textbf{PT}}$) and SIM-F($\operatorname{\textbf{PS}}$)
and pick the one that gives the best results.
## 5 Conclusion
This paper addresses two-sided fairness problems represented as online
bipartite matching with accommodated delays. We introduce two minimax
problems: $\operatorname{\textbf{PT}}$ to minimize the maximum workload of
workers and $\operatorname{\textbf{PS}}$ to minimize the maximum waiting time
of tasks. We show that the second problem can be formulated as a linear
program and thus solved efficiently. Moreover, we showed that the policy using
a solution for $\operatorname{\textbf{PS}}$ approximates the solution for
$\operatorname{\textbf{PT}}$, and we then presented an upper bound on the
approximation ratio. Finally, we compared the performance of different
approaches (most of them used the solutions to the problems) and empirically
evaluated their performance.
Future research may explore different definitions of fairness. In addition, it
is promising to extend our approach to scenarios where workers are also
arriving dynamically. To demonstrate the need in such scenarios one might
consider the teleoperation application where teleoperators (workers) can join
or leave the crew. Considering different distributions for both task arrivals
and task durations can provide more depth and insights into the study.
Finally, it might be beneficial to consider some robust version, say,
minimization of the maximum possible absolute waiting time among users, which
is equivalent to the minimization of the maximum absolute waiting time among
all workers.
## Acknowledgements
This research has been partially supported by the Israel Science Foundation
under grant 1958/20 and the EU Project TAILOR under grant 952215. Work of Pan
Xu was partially supported by NSF CRII Award IIS-1948157.
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Technical Appendix
Paper title: Design a Win-Win Strategy That Is Fair to Both Service Providers
and Tasks When Rejection is Not an Option
Paper id: 185
## Appendix A Objective Function in Program $\operatorname{\textbf{PT}}$ Can
Be Neither Convex nor Concave (Even When $\kappa=1$)
We show that the optimization program $\operatorname{\textbf{PT}}$ can be
minimization of a non-convex function. Consider the example shown in Figure 3,
where $\lambda_{j}=\lambda$ for all $j\in J$, and $\mu_{ij}=\mu_{i},\forall
j\sim i,\forall i\in I$. Set $\phi_{i}=\lambda/\mu_{i}$, for each $i\in I$.
Let $x$ be the value on edge $(i=1,j=1)$, and thus, $1-x$ be that on edge
$(i=2,j=1)$. Similarly, let $y$ and $1-y$ be the values on edges $(i=3,j=2)$
and $(i=2,j=2)$. We can verify that
$\displaystyle\rho_{1}$ $\displaystyle=\phi_{1}\cdot
x,\rho_{2}=\phi_{2}(1-x+1-y),\rho_{3}=\phi_{3}\cdot y;$
$\displaystyle\bar{w}_{1}$ $\displaystyle=\frac{x}{1-\phi_{1}\cdot
x}+\frac{1-x}{1-\phi_{2}(1-x+1-y)}-1$ $\displaystyle\bar{w}_{2}$
$\displaystyle=\frac{y}{1-\phi_{3}\cdot y}+\frac{1-y}{1-\phi_{2}(1-x+1-y)}-1.$
Let ${\boldsymbol{\phi}}=(\phi_{1},\phi_{2},\phi_{3})$, and assume
$\phi_{1}\leq 1,\phi_{3}\leq 1,\phi_{2}\leq 1/2$. Under these assumptions, we
see the feasible region of $\operatorname{\textbf{PT}}$ can be reduced to
$\Omega=\\{(x,y):0\leq x,y\leq 1\\}$.
$\displaystyle f_{{\boldsymbol{\phi}}}(x,y)$
$\displaystyle=\max(\bar{w}_{1},\bar{w}_{2})=\max\left(\frac{x}{1-\phi_{1}\cdot
x}+\frac{1-x}{1-\phi_{2}(1-x+1-y)}-1,\frac{y}{1-\phi_{3}\cdot
y}+\frac{1-y}{1-\phi_{2}(1-x+1-y)}-1\right)$ $\displaystyle\Omega$
$\displaystyle=\\{(x,y):0\leq x,y\leq 1\\}.$
Furthermore, set $\phi_{1}=\phi_{3}=p\in[0,1]$ and $\phi_{2}=q\in[0,1/2]$.
Define
$\displaystyle g_{p,q}(x,y)=\frac{x}{1-p\cdot
x}+\frac{1-x}{1-q(2-x-y)}-1,~{}~{}h_{p,q}(x,y)=\frac{y}{1-p\cdot
y}+\frac{1-y}{1-q(2-x-y)}-1.$
Thus, the original program $\operatorname{\textbf{PT}}$ can be simplified as
$\displaystyle\min
f_{p,q}(x,y):=\max\Big{(}g_{p,q}(x,y),h_{p,q}(x,y)\Big{)}:~{}~{}0\leq x,y\leq
1.$ (23)
By symmetry, we can assume WLOG that $x\geq y$. Then we see
$\displaystyle f_{p,q}(x,y)$ $\displaystyle=h_{p,q}(x,y)=\frac{y}{1-p\cdot
y}+\frac{1-y}{1-q(2-x-y)}-1,$
$\displaystyle\mbox{~{}if~{}}2q+p^{2}xy\geq(p+q)\cdot(x+y),~{}0\leq y\leq
x\leq 1;$ $\displaystyle=g_{p,q}(x,y)=\frac{x}{1-p\cdot
x}+\frac{1-x}{1-q(2-x-y)}-1,$
$\displaystyle\mbox{~{}if~{}}2q+p^{2}xy\leq(p+q)\cdot(x+y),~{}0\leq y\leq
x\leq 1.$
We can verify that the region
$\Omega_{1}:=\\{(x,y):2q+p^{2}xy\geq(p+q)\cdot(x+y),~{}0\leq y\leq x\leq 1\\}$
is convex, while the function $f_{p,q}(x,y)=h_{p,q}(x,y)$ is _neither convex
nor concave_ over $\Omega_{1}$ under many settings of $(p,q)$, say, _e.g.,_
$p=0.7$ and $q=0.4$. This establishes $f_{p,q}(x,y)$ is neither convex or
concave over the original region $\Omega=\\{(x,y):0\leq x,y\leq 1\\}$.
$i_{1}$$i_{2}$$i_{3}$$j_{1}$$j_{2}$$x$$1-y$$1-x$$y$ Figure 3: A toy example
showing the objective function in Program $\operatorname{\textbf{PT}}$ can be
neither convex nor concave even when $\kappa=1$.
## Appendix B Proof of Lemma 2
Consider a given input setting characterized by
${\boldsymbol{\mu}}:=(\mu_{ij})$ and ${\boldsymbol{\lambda}}:=(\lambda_{j})$.
Recall that $\eta_{t}^{*}({\boldsymbol{\lambda}},{\boldsymbol{\mu}})$ and
$\eta_{s}^{*}({\boldsymbol{\lambda}},{\boldsymbol{\mu}})$ denote the optimal
values of $\operatorname{\textbf{PT}}$ and $\operatorname{\textbf{PS}}$ with
respect to ${\boldsymbol{\lambda}}$ and ${\boldsymbol{\mu}}$. We focus on the
case when $k=2$, and all the analysis can be straightforwardly generalized to
any generic integer $k$. Consider a modified version of
${\boldsymbol{\lambda}}$ in which the first request type $j=1$ is split into
two copies, $j=0$ and $j=1$, each having an arrival rate of $\lambda_{1}/2$.
Let $\bar{\boldsymbol{\lambda}}=(\bar{\lambda}_{j})$ be this modified version
with $\bar{\lambda}_{0}=\bar{\lambda}_{1}=\lambda_{1}/2$ and
$\bar{\lambda}_{j}=\lambda_{j}$ for all $j\geq 2$. Let $\bar{J}=J\cup\\{0\\}$
be the resulting set of online types.
###### Lemma 6.
$\eta_{t}^{*}({\boldsymbol{\lambda}},{\boldsymbol{\mu}})=\eta_{t}^{*}(\bar{\boldsymbol{\lambda}},{\boldsymbol{\mu}})$,
and
$\eta_{s}^{*}({\boldsymbol{\lambda}},{\boldsymbol{\mu}})=\eta_{s}^{*}(\bar{\boldsymbol{\lambda}},{\boldsymbol{\mu}})$,
###### Proof.
Let us focus on showing the first equality from Lemma 6. Suppose
$\mathbf{x}=(x_{ij})$ is an optimal solution to
$\operatorname{\textbf{PT}}({\boldsymbol{\lambda}},{\boldsymbol{\mu}})$.
Consider a modified solution $\mathbf{y}=(y_{ij})$ that satisfies: (1)
$y_{ij}=x_{ij}$ for all $j\geq 2$ and $i\sim j$, and (2)
$y_{i,0}=y_{i,1}=x_{i,1}$. Let $\rho_{i}(\mathbf{x})$ and
$\rho_{i}(\mathbf{y})$ be the values of $\rho_{i}$ with respect to
$\mathbf{x}$ under the setting of
$({\boldsymbol{\lambda}},{\boldsymbol{\mu}})$ and $\mathbf{y}$ under the
setting of $(\bar{\boldsymbol{\lambda}},{\boldsymbol{\mu}})$, respectively.
Similarly, let $\bar{w}_{j}(\mathbf{x})$ and $\bar{w}_{j}(\mathbf{y})$
represent the values of $\bar{w}_{j}$ with respect to $\mathbf{x}$ and
$\mathbf{y}$, respectively. We can verify that $\mathbf{y}$ is feasible for
$\operatorname{\textbf{PT}}(\bar{\boldsymbol{\lambda}},{\boldsymbol{\mu}})$.
Furthermore, $\rho_{i}(\mathbf{x})=\rho_{i}(\mathbf{y})$ for all $i\in I$, and
$\bar{w}_{j}(\mathbf{x})=\bar{w}_{j}(\mathbf{y})$ for all $j\geq 2$, and
$\bar{w}_{0}(\mathbf{y})=\bar{w}_{1}(\mathbf{y})=\bar{w}_{1}(\mathbf{x})$.
Thus, we claim that:
$\eta_{t}^{*}(\bar{\boldsymbol{\lambda}},{\boldsymbol{\mu}})\leq\max_{j\in\bar{J}}\bar{w}_{j}(\mathbf{y})=\max_{j\in
J}\bar{w}_{j}(\mathbf{x})=\eta_{t}^{*}({\boldsymbol{\lambda}},{\boldsymbol{\mu}}).$
Now we prove the other direction. Let $\mathbf{y}=(y_{ij})$ be an optimal
solution to
$\operatorname{\textbf{PT}}(\bar{\boldsymbol{\lambda}},{\boldsymbol{\mu}})$.
Consider a modified solution $\mathbf{x}=(x_{ij})$ that satisfies: (1)
$x_{ij}=y_{ij}$ for all $j\geq 2$ and $i\sim j$, and (2)
$x_{i,1}=(y_{i,0}+y_{i,1})/2$ for all $i\sim j$. We can verify that
$\mathbf{x}$ is feasible for
$\operatorname{\textbf{PT}}({\boldsymbol{\lambda}},{\boldsymbol{\mu}})$.
Furthermore, $\rho_{i}(\mathbf{x})=\rho_{i}(\mathbf{y})$ for all $i\in I$, and
$\bar{w}_{j}(\mathbf{x})=\bar{w}_{j}(\mathbf{y})$ for all $j\geq 2$, and
$\bar{w}_{1}(\mathbf{x})=(\bar{w}_{0}(\mathbf{y})+\bar{w}_{1}(\mathbf{y}))/2$.
Thus, we claim that:
$\eta_{t}^{*}({\boldsymbol{\lambda}},{\boldsymbol{\mu}})\leq\max_{j\in
J}\bar{w}_{j}(\mathbf{x})\leq\max_{j\in\bar{J}}\bar{w}_{j}(\mathbf{y})=\eta_{t}^{*}(\bar{\boldsymbol{\lambda}},{\boldsymbol{\mu}}).$
This shows that the first equality holds, and the second equality follows from
the same argument. ∎
## Appendix C Proof of Theorem 1
Now, we consider a general setting ${\boldsymbol{\mu}}=(\mu_{ij})$ with
$\kappa=\max_{i\in I}\max_{j\sim i,j^{\prime}\sim
i}\mu_{ij}/\mu_{i,j^{\prime}}\geq 1$. Consider such a virtual instance that
for each $i\in I$, all of $\mu_{ij}$ with $j\sim i$ are replaced with
${\mu}_{i}:=\max_{j\sim i}\mu_{ij}$. Let
$\bar{\boldsymbol{\mu}}=(\bar{\mu}_{ij})$ be the modified version of
${\boldsymbol{\mu}}$ such that $\bar{\mu}_{ij}=\mu_{i}$ for every $j\sim i$
and $i\in I$. By Claims 1 and 2, we see that
$\eta_{t}^{*}(\bar{\boldsymbol{\mu}})=-1+1/(1-\eta_{s}^{*}(\bar{\boldsymbol{\mu}}))$,
where $\eta_{t}^{*}(\bar{\boldsymbol{\mu}})$ and
$\eta_{s}^{*}(\bar{\boldsymbol{\mu}})$ denote the optimal values of
$\operatorname{\textbf{PT}}$ and $\operatorname{\textbf{PS}}$ under
$\bar{\boldsymbol{\mu}}$, respectively. Similarly,
$\eta_{t}^{*}({\boldsymbol{\mu}})$ and $\eta_{s}^{*}({\boldsymbol{\mu}})$
denote the optimal values of $\operatorname{\textbf{PT}}$ and
$\operatorname{\textbf{PS}}$ under ${\boldsymbol{\mu}}$, respectively.
###### Lemma 7.
(1)
$\eta_{t}^{*}({\boldsymbol{\mu}})\geq\eta_{t}^{*}(\bar{\boldsymbol{\mu}})/\kappa$;
(2)
$\eta_{s}^{*}(\bar{\boldsymbol{\mu}})\geq\eta_{s}^{*}({\boldsymbol{\mu}})/\kappa$.
###### Proof.
We prove the first inequality as follows. Consider any optimal solution
$\mathbf{x}=(x_{ij})$ to $\operatorname{\textbf{PT}}({\boldsymbol{\mu}})$.
Observe that for any $j\in J$,
$\displaystyle\bar{w}_{j}({\boldsymbol{\mu}},\mathbf{x})$
$\displaystyle=\sum_{i\sim j}x_{ij}\cdot\mu_{ij}\cdot\frac{\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}^{2}}{1-\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}}$
$\displaystyle\geq(1/\kappa)\cdot\sum_{i\sim j}x_{ij}\cdot\frac{\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}}{1-\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}}=(1/\kappa)\cdot\sum_{i\sim
j}x_{ij}\cdot\frac{\rho_{i}({\boldsymbol{\mu}},\mathbf{x})}{1-\rho({\boldsymbol{\mu}},\mathbf{x})}$
$\displaystyle=(1/\kappa)\cdot\sum_{i\sim
j}x_{ij}\cdot\Big{(}-1+\frac{1}{1-\rho_{i}({\boldsymbol{\mu}},\mathbf{x})}\Big{)}\geq(1/\kappa)\cdot\sum_{i\sim
j}x_{ij}\cdot\Big{(}-1+\frac{1}{1-\rho_{i}(\bar{\boldsymbol{\mu}},\mathbf{x})}\Big{)},$
where the inequality on the last line is due to
$\displaystyle\rho_{i}({\boldsymbol{\mu}},\mathbf{x})=\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}\geq\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i}=\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\bar{\mu}_{ij}=\rho_{i}(\bar{\boldsymbol{\mu}},\mathbf{x}).$
(24)
Therefore,
$\displaystyle\eta_{t}^{*}({\boldsymbol{\mu}})$ $\displaystyle=\max_{j\in
J}\bar{w}_{j}({\boldsymbol{\mu}},\mathbf{x})\geq(1/\kappa)\cdot\max_{j\in
J}\sum_{i\sim
j}x_{ij}\cdot\Big{(}-1+\frac{1}{1-\rho_{i}(\bar{\boldsymbol{\mu}},\mathbf{x})}\Big{)}\geq(1/\kappa)\cdot\eta_{t}^{*}(\bar{\boldsymbol{\mu}}),$
where the last inequality above is valid since $\mathbf{x}$ is optimal to
$\operatorname{\textbf{PT}}({\boldsymbol{\mu}})$, and thus, it is feasible to
$\operatorname{\textbf{PT}}(\bar{\boldsymbol{\mu}})$ since for each $j\in J$,
$\sum_{i\sim j}x_{ij}=1$ and for each $i\in I$,
$\rho_{i}(\bar{\boldsymbol{\mu}},\mathbf{x})\leq\rho_{i}({\boldsymbol{\mu}},\mathbf{x})\leq
1$ due to Inequality (24).
Now, we show the second one. Note that we assume by default that
$\operatorname{\textbf{PS}}({\boldsymbol{\mu}})$ is feasible with
$\eta^{*}_{b}({\boldsymbol{\mu}})\leq 1$. So is
$\operatorname{\textbf{PS}}(\bar{\boldsymbol{\mu}})$. We claim that the
optimal value of $\eta^{*}_{b}({\boldsymbol{\mu}})$ remains invariant after
removing the constraints of $\rho_{i}\leq 1$ for all $i\in I$. Let
$\operatorname{\textbf{PC}}$ be the program of $\operatorname{\textbf{PS}}$
after removing the constraints $\rho_{i}\leq 1$ for all $i\in I$, and suppose
$\eta_{c}({\boldsymbol{\mu}},\mathbf{x})$ is the corresponding value of
$\operatorname{\textbf{PC}}$ with respect to ${\boldsymbol{\mu}}$ and
$\mathbf{x}$. Observe that
$\eta_{s}^{*}({\boldsymbol{\mu}})=\eta_{c}^{*}({\boldsymbol{\mu}})$ and
$\eta_{s}^{*}(\bar{\boldsymbol{\mu}})=\eta_{c}^{*}(\bar{\boldsymbol{\mu}})$.
Consider an optimal solution $\mathbf{x}=(x_{ij})$ to
$\operatorname{\textbf{PS}}(\bar{\boldsymbol{\mu}})$. We can verify that it is
surely feasible to $\operatorname{\textbf{PC}}({\boldsymbol{\mu}})$. Thus,
$\displaystyle\eta^{*}_{b}(\bar{\boldsymbol{\mu}})$
$\displaystyle=\eta_{s}(\bar{\boldsymbol{\mu}},\mathbf{x})=\eta_{c}(\bar{\boldsymbol{\mu}},\mathbf{x})\geq\eta_{c}({\boldsymbol{\mu}},\mathbf{x})/\kappa\geq\eta_{c}^{*}({\boldsymbol{\mu}})/\kappa=\eta_{s}^{*}({\boldsymbol{\mu}})/\kappa,$
(25)
where (a) the first inequality on (25) follows from that
$\eta_{c}(\bar{\boldsymbol{\mu}},\mathbf{x})=\max_{i\in I}\sum_{\ell\sim
i}x_{i\ell}\cdot\lambda_{\ell}/\bar{\mu}_{i\ell}=\max_{i\in I}\sum_{\ell\sim
i}x_{i\ell}\cdot\lambda_{\ell}/\mu_{i}\geq\max_{i\in I}\sum_{\ell\sim
i}x_{i\ell}\cdot\lambda_{\ell}/(\kappa\cdot\mu_{i\ell})=\eta_{c}({\boldsymbol{\mu}},\mathbf{x})/\kappa;$
and (b) the second inequality on (25) is valid since $\mathbf{x}$ is feasible
to $\operatorname{\textbf{PC}}({\boldsymbol{\mu}})$ and
$\eta_{c}^{*}({\boldsymbol{\mu}})$ is the optimal value. ∎
###### Proof of Theorem 1.
Let $\mathbf{x}=(x_{ij})$ be an optimal solution to
$\operatorname{\textbf{PS}}({\boldsymbol{\mu}})$. Observe that $\mathbf{x}$ is
feasible to $\operatorname{\textbf{PT}}({\boldsymbol{\mu}})$ since the two
programs $\operatorname{\textbf{PT}}(\mu)$ and
$\operatorname{\textbf{PS}}(\mu)$ share the same set of constraints.
$\displaystyle\eta_{t}({\boldsymbol{\mu}},\mathbf{x})$
$\displaystyle=\max_{j\in J}\sum_{i\sim
j}x_{ij}\cdot\mu_{ij}\cdot\frac{\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}^{2}}{1-\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}}$ $\displaystyle\leq\max_{j\in
J}\kappa\cdot\sum_{i\sim j}x_{ij}\cdot\frac{\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}}{1-\sum_{\ell\sim
i}x_{i\ell}\lambda_{\ell}/\mu_{i\ell}}=\kappa\cdot\max_{j\in J}\sum_{i\sim
j}x_{ij}\cdot\Big{(}-1+\frac{1}{1-\rho_{i}({\boldsymbol{\mu}},\mathbf{x})}\Big{)}$
$\displaystyle\leq\kappa\cdot\Big{(}-1+\frac{1}{1-\max_{i\in
I}\rho_{i}({\boldsymbol{\mu}},\mathbf{x})}\Big{)}=\kappa\cdot\Big{(}-1+\frac{1}{1-\eta_{s}^{*}({\boldsymbol{\mu}})}\Big{)}$
$\displaystyle=\kappa\cdot{\frac{\eta_{s}^{*}({\boldsymbol{\mu}})}{1-\eta_{s}^{*}({\boldsymbol{\mu}})}}=\kappa\cdot{\frac{\eta_{s}^{*}(\bar{\boldsymbol{\mu}})}{1-\eta_{s}^{*}(\bar{\boldsymbol{\mu}})}}\cdot\Big{(}\frac{\eta_{s}^{*}({\boldsymbol{\mu}})}{1-\eta_{s}^{*}({\boldsymbol{\mu}})}\Big{)}/\Big{(}\frac{\eta_{s}^{*}(\bar{\boldsymbol{\mu}})}{1-\eta_{s}^{*}(\bar{\boldsymbol{\mu}})}\Big{)}$
$\displaystyle=\kappa\cdot\eta_{t}^{*}(\bar{\boldsymbol{\mu}})\cdot\frac{\eta_{s}^{*}({\boldsymbol{\mu}})}{\eta_{s}^{*}(\bar{\boldsymbol{\mu}})}\cdot\frac{1-\eta_{s}^{*}(\bar{\boldsymbol{\mu}})}{1-\eta_{s}^{*}({\boldsymbol{\mu}})}~{}~{}\big{(}\mbox{by
Claims~{}\ref{claim:main-3} and~{}\ref{claim:>=} on
$\bar{\boldsymbol{\mu}}$}\big{)}$
$\displaystyle\leq\kappa^{3}\cdot\eta_{t}^{*}({\boldsymbol{\mu}})\cdot\frac{1-\eta_{s}^{*}({\boldsymbol{\mu}})/\kappa}{1-\eta_{s}^{*}({\boldsymbol{\mu}})}=\kappa^{3}\cdot\eta_{t}^{*}({\boldsymbol{\mu}})\cdot\Big{(}1+\Big{(}1-\frac{1}{\kappa}\Big{)}\cdot\frac{\eta_{s}^{*}({\boldsymbol{\mu}})}{1-\eta_{s}^{*}({\boldsymbol{\mu}})}\Big{)}~{}~{}\big{(}\mbox{by
Lemma~{}\ref{lem:19-a}}\big{)}.$
∎
## Appendix D Proof of Lemma 5
We split the proof of Lemma 5 into the following two claims. For any
$S\subseteq J$ with $S\neq\emptyset$, let $\lambda(S)=\sum_{j\in
S}\lambda_{j}=\lambda\cdot|S|$, and $\mu(S)=\sum_{i\in\partial(S)}\mu_{i}$
with $\partial(S)=\\{i:\exists j\in S,i\sim j\\}$ being set of neighbors of
$S$.555Note that $\partial(S)$ includes all possible neighbors of nodes in
$S$, but nodes in $\partial(S)$ may have neighbors beyond $S$.
###### Claim 1.
$\eta_{s}^{*}=\max_{S\subseteq J,S\neq\emptyset}\lambda(S)/\mu(S)$.
###### Claim 2.
$\eta_{t}^{*}+1\geq 1/\big{(}1-\lambda(S)/\mu(S)\big{)}$ for any $S\subseteq
J$ with $S\neq\emptyset$.
The two claims above together establish Lemma 5. We present the proofs of the
two Claims.
###### Proof of Claim 1.
We first show $\eta_{s}^{*}\geq\lambda(S)/\mu(S)$ for any
$S\neq\emptyset,S\subseteq J$. Consider an optimal solution
$\mathbf{x}^{*}=(x_{ij})$ for $\overline{\operatorname{\textbf{PS}}}$. Let
$x_{i}=\sum_{j\sim i,j\in J}x_{ij}$.
$\displaystyle\eta_{s}^{*}=\max_{i\in
I}\rho_{i}(\mathbf{x}^{*})\geq\max_{i\in\partial(S)}\rho_{i}(\mathbf{x}^{*})=\max_{i\in\partial(S)}(\lambda
x_{i}/\mu_{i})$ $\displaystyle\geq\Big{(}\sum_{i\in\partial(S)}\lambda
x_{i}\Big{)}/\Big{(}\sum_{i\in\partial(S)}\mu_{i}\Big{)}\geq\lambda(S)/\mu(S),$
where the last inequality follows from
$\displaystyle\sum_{i\in\partial(S)}\lambda x_{i}$
$\displaystyle=\lambda\sum_{i\in\partial(S)}\sum_{j\sim i,j\in J}x_{ij}$
$\displaystyle\geq\lambda\sum_{j\in S}\sum_{i\in I,i\sim
j}x_{ij}=\lambda\cdot|S|=\lambda(S).$
Now we show $\eta_{s}^{*}\leq\max_{S\neq\emptyset,S\subseteq
J}\lambda(S)/\mu(S)$. For an optimal solution $\mathbf{x}^{*}=(x_{ij})$ of
$\overline{\operatorname{\textbf{PS}}}$, let $I(\mathbf{x}^{*})=\\{i\in
I:\rho_{i}(\mathbf{x}^{*})=\eta_{s}^{*}\\}$ be the set of node $i$ with
saturated load under $\mathbf{x}^{*}$. Let $\mathbf{y}^{*}=(y_{ij})$ be an
optimal solution of $\overline{\operatorname{\textbf{PS}}}$ such that
$I(\mathbf{y}^{*})$ has the smallest size. Let $J(\mathbf{y}^{*})=\\{j\in
J:\exists i\in I(\mathbf{y}^{*})\mbox{~{}with~{}}i\sim j,y_{ij}>0\\}$ be set
of non-zero neighbors of $I(\mathbf{y}^{*})$ with respect to $\mathbf{y}^{*}$.
We claim that $\partial(J(\mathbf{y}^{*}))=I(\mathbf{y}^{*})$. We show by
contradiction as follows. Suppose there is some $j\in J(\mathbf{y}^{*})$ such
that (1) there exists some $i\sim j,i\in I(\mathbf{y}^{*})$ with $y_{ij}>0$
and $\rho_{i}(\mathbf{y}^{*})=\eta_{s}^{*}$ and (2) there exists some
$\bar{i}\sim j$ with $\bar{i}\notin I(\mathbf{y}^{*})$, _i.e.,_
$\rho_{\bar{i}}(\mathbf{y}^{*})<\eta_{s}^{*}$. Consider the following
perturbation: $\bar{y}_{ij}\leftarrow y_{ij}-\epsilon$ and
$\bar{y}_{\bar{i},j}\leftarrow y_{\bar{i},j}+\epsilon$ for an appropriate
value of $\epsilon>0$, we could end up with another optimal solution that
either has a strictly smaller size of $I(\bar{\mathbf{y}}^{*})$ if
$|I(\mathbf{y}^{*})|>1$ or a strictly better optimal value if
$|I(\mathbf{y}^{*})|=1$, which contradicts our assumption. Observe that
$\eta_{s}^{*}=\rho_{i}(\mathbf{y}^{*})=(\lambda y_{i}/\mu_{i})$ for every
$i\in I(\mathbf{y}^{*})$ with $y_{i}=\sum_{j\sim i}y_{ij}$. Set
$S^{*}=J(\mathbf{y}^{*})$ with $\partial(S^{*})=I(\mathbf{y}^{*})$, we have
$\displaystyle\eta_{s}^{*}$ $\displaystyle=\Big{(}\sum_{i\in
I(\mathbf{y}^{*})}\lambda y_{i}\Big{)}/\Big{(}\sum_{i\in
I(\mathbf{y}^{*})}\mu_{i}\Big{)}=\Big{(}\sum_{i\in
I(\mathbf{y}^{*})}\lambda\sum_{j\sim i}y_{ij}\Big{)}/\mu(S^{*})$
$\displaystyle=\Big{(}\sum_{j\in J(\mathbf{y}^{*})}\lambda\sum_{i\sim
j}y_{ij}\Big{)}/\mu(S^{*})=\lambda(S^{*})/\mu(S^{*}).$
Therefore, we get $\eta_{s}^{*}\leq\max_{S\neq\emptyset,S\subseteq
J}\lambda(S)/\mu(S)$. ∎
###### Proof of Claim 2.
Consider any optimal solution $\mathbf{x}^{*}=(x_{ij})$ for
$\overline{\operatorname{\textbf{PT}}}$. Recall that
$\Lambda=\lambda(S)=\lambda\cdot|S|$ and
$\Phi=\mu(S)=\sum_{i\in\partial(S)}\mu_{i}$. We see
$\Lambda/\Phi=\lambda(S)/\mu(S)\leq\eta_{s}^{*}\leq 1$ due to Claim 1. Set
$x_{i}=\sum_{j\sim i,j\in J}x_{ij}$ and $x_{i}(S)=\sum_{j\sim i,j\in
S}x_{ij}$. Let $\rho_{i}(\mathbf{x}^{*})=(\lambda/\mu_{i})\sum_{j\sim i,j\in
J}x_{ij}=(\lambda/\mu_{i})\cdot x_{i}$ and
$\rho_{i,S}(\mathbf{x}^{*})=(\lambda/\mu_{i})\sum_{j\sim i,j\in
S}x_{ij}=(\lambda/\mu_{i})\cdot x_{i}(S)$. Thus,
$\rho_{i}(\mathbf{x}^{*})\geq\rho_{i,S}(\mathbf{x}^{*})$.
$\displaystyle\eta_{t}^{*}+1=\max_{j\in
J}\big{(}\bar{w}_{j}(\mathbf{x}_{a}^{*})+1\big{)}\geq\frac{1}{|S|}\sum_{j\in
S}\big{(}\bar{w}_{j}(\mathbf{x}_{a}^{*})+1\big{)}=\frac{1}{|S|}\sum_{j\in
S}\sum_{i\sim
j}\frac{x_{ij}}{1-\rho_{i}(\mathbf{x}^{*})}=\frac{1}{|S|}\sum_{i\in\partial(S)}\sum_{j\sim
i,j\in S}\frac{x_{ij}}{1-\rho_{i}(\mathbf{x}^{*})}$
$\displaystyle=\frac{1}{|S|}\sum_{i\in\partial(S)}\frac{x_{i}(S)}{1-\rho_{i}(\mathbf{x}^{*})}\geq\frac{1}{|S|}\sum_{i\in\partial(S)}\frac{x_{i}(S)}{1-\rho_{i,S}(\mathbf{x}^{*})}$
$\displaystyle=\frac{1}{\lambda\cdot|S|}\sum_{i\in\partial(S)}\mu_{i}\cdot\frac{(\lambda/\mu_{i})\cdot
x_{i}(S)}{1-\rho_{i,S}(\mathbf{x}^{*})}=\frac{1}{\Lambda}\sum_{i\in\partial(S)}\frac{\mu_{i}\cdot\rho_{i,S}(\mathbf{x}^{*})}{1-\rho_{i,S}(\mathbf{x}^{*})}=\frac{1}{\Lambda}\sum_{i\in\partial(S)}\mu_{i}\cdot\Big{(}-1+\frac{1}{1-\rho_{i,S}(\mathbf{x}^{*})}\Big{)}$
$\displaystyle=-\frac{\Phi}{\Lambda}+\frac{1}{\Lambda}\sum_{i\in\partial(S)}\frac{\mu_{i}}{1-\rho_{i,S}(\mathbf{x}^{*})}=-\frac{\Phi}{\Lambda}+\frac{1}{\Lambda}\sum_{i\in\partial(S)}\frac{\mu_{i}}{1-(\lambda/\mu_{i})\cdot
x_{i}(S)}=-\frac{\Phi}{\Lambda}+\frac{1}{\Lambda}\sum_{i\in\partial(S)}\frac{\mu_{i}^{2}}{\mu_{i}-\lambda\cdot
x_{i}(S)}.$
Set $z_{i}=\lambda\cdot x_{i}(S)$. Observe that (1)
$\displaystyle\sum_{i\in\partial(S)}z_{i}$
$\displaystyle=\sum_{i\in\partial(S)}\lambda\cdot
x_{i}(S)=\lambda\sum_{i\in\partial(S)}\sum_{j\sim i,j\in
S}x_{ij}=\lambda\cdot\sum_{j\in S}\sum_{i\sim j,i\in
I}x_{ij}=\lambda\cdot|S|=\Lambda,$
and (2) for any $i\in I$, we have $z_{i}\leq\mu_{i}$ since
$z_{i}/\mu_{i}=(\lambda/\mu_{i})\cdot
x_{i}(S)=\rho_{i,S}(\mathbf{x}^{*})\leq\rho_{i}(\mathbf{x}^{*})\leq 1$. For
any fixed values of $(\mu_{i})$, let
$g_{i}(z_{i}):=\mu_{i}^{2}/(\mu_{i}-z_{i})$ be a function of
$z_{i}\in[0,\mu_{i}]$. Consider a minimization program below,
$\displaystyle\Big{(}\min\sum_{i\in\partial(S)}g_{i}(z_{i}):0\leq
z_{i}\leq\mu_{i},\forall
i\in\partial(S);\sum_{i\in\partial(S)}z_{i}=\Lambda.\Big{)}$ (26)
By local perturbation, we claim that for any given $(\mu_{i})$, Program (26)
has a unique optimal solution $\mathbf{z}^{*}=(z_{i}^{*})$ such that
$\mu_{i}/(\mu_{i}-z^{*}_{i})$ takes a uniform value for every
$i\in\partial(S)$, and so does $z^{*}_{i}/\mu_{i}$. Let
$\beta=z^{*}_{i}/\mu_{i}$ for every $i\in\partial(S)$. We see that
$\beta=\sum_{i\in\partial(S)}z^{*}_{i}/\sum_{i\in\partial(S)}\mu_{i}=\Lambda/\Phi$.
Thus, we claim that
$\displaystyle\eta_{t}^{*}+1$
$\displaystyle\geq-\frac{\Phi}{\Lambda}+\frac{1}{\Lambda}\sum_{i\in\partial(S)}\frac{\mu_{i}^{2}}{\mu_{i}-\lambda\cdot
x_{i}(S)}=-\frac{\Phi}{\Lambda}+\frac{1}{\Lambda}\sum_{i\in\partial(S)}g_{i}(z_{i})$
$\displaystyle\geq-\frac{\Phi}{\Lambda}+\frac{1}{\Lambda}\sum_{i\in\partial(S)}g_{i}(z_{i}^{*})=-\frac{\Phi}{\Lambda}+\frac{1}{\Lambda}\sum_{i\in\partial(S)}\frac{\mu_{i}}{1-z_{i}^{*}/\mu_{i}}$
$\displaystyle=-\frac{\Phi}{\Lambda}+\frac{\Phi}{\Lambda}\cdot\frac{1}{1-\Lambda/\Phi}=\frac{\Phi}{\Lambda}\cdot\Big{(}\frac{1}{1-\Lambda/\Phi}-1\Big{)}=\frac{1}{1-\Lambda/\Phi}=\frac{1}{1-\lambda(S)/\mu(S)}.$
∎
## Appendix E Details for the Experiments Settings
### E.1 Approaches Used for Determining the Duration of the Tasks
We propose three approaches for determining the duration of the tasks. In the
first approach, we aim to study settings where $\kappa=1$. We calculated the
average duration time for each (human) teleoperator across all allowed tasks
and computed the overall average. This average was then used as the mean in
the exponential distribution. By setting $\kappa=1$ in this method, the
optimal solution of $\operatorname{\textbf{PS}}$ also became an optimal
solution of $\operatorname{\textbf{PT}}$. In the second approach, we aim at
varying $\kappa$ by assigning different mean values to the various tasks.
These mean values were chosen around the mean calculated in the previous
approach. This allowed us to assess the performance of our algorithms and
heuristics in scenarios where there was significant variance in the duration
times of the different tasks. Finally, the third approach considers the
average time taken by the teleoperators to perform the tasks in the simulation
for each specific teleoperator-task combination. This approach helps us to
make the settings as similar as possible to the real data.
The exact implementation of the second approach is as follows: The mean
duration for each teleoperator is the average task duration she is authorized
to undertake. The durations follow an exponential distribution with means
calculated as $(2\cdot\kappa\cdot x)/(1+k)$ and $2x-(2\cdot\kappa\cdot
x)/(1+k)$ for the first and last tasks respectively, where $x$ represents the
average task duration for the teleoperator. The average values range from 3.33
to 8 seconds, with a standard deviation 1.58. For the remaining tasks, the
means are uniformly selected within the range between the first and last task
means. In cases where a teleoperator is permitted to perform only one task,
the mean is set as $x$ for the exponential distribution. The experiments in
the main paper use the second approach, while both the second and third
approaches are presented in the appendix.
### E.2 Generating the Task Arrival Rates
To establish the task arrival rates, we followed a specific procedure. We used
the average number of task arrivals per day (100,000) used in Viden et al.
Ackerman Viden et al. [2023] as a basis and examined the neighborhood of this
number (e.g., 60000-140000). For each task type, we multiplied this average by
the weight assigned to that specific task type, resulting in the final
$\lambda$ parameter for the normal distribution of task types.
Having the task arrival rates, the actual arrival times are generated by the
following procedure: Using an exponential distribution with a mean of
$1/\lambda$ we calculate a set of arrival times. For each arrival time we
decide the type of the arriving task by using configurable probabilities. This
approach allowed us to model the arrival times of tasks and analyze the
system’s performance under various task-type balances.
### E.3 Experiments Environment and Some More Technical Details
In our experiments, we ran simulations using Python and Matlab. The simulation
ran for a (virtual) period of 4 weeks. The nonlinear minimax problem
$\operatorname{\textbf{PT}}$ was solved with the Matlab function fmincon and
the problem $\operatorname{\textbf{PS}}$ was cast as a linear program and
solved with the Matlab function linprog. Most of the experiments were carried
out on a Windows laptop. Other experiments were run on a Linux server with 98
cores to save time. It is important to emphasize that the
$\operatorname{\textbf{PS}}$ method is valid only when the available workers
can handle all the tasks. Therefore, we focus on problems for which
$\operatorname{\textbf{PS}}$ can provide a feasible solution.
## Appendix F Additional Figures for the Experiments Section
### F.1 Results for a Different Duration Distribution
Figures 4 and 5 are similar to figures 2(c,d,e,f) from the Experiments
section. The only difference is that the parameter $\mu_{i,j}$ is defined as
the real average time it takes the teleoperator $i$ to perform a task of type
$j$. Therefore, the value of $\kappa$ varies for the different workers.
Figure 4: Maximum Task Waiting Time (left) and Maximum worker workload (right)
under varying task arrival loads. The x-axis represents the expected number of
arrivals per day. In these graphs, the arrival distributions of all tasks are
equal.
Figure 5: Task waiting time (left) and worker workload (right) for different
arrival distributions: equal task arrival vs. one task type arriving seven
times more than the others (task load was 60000 requests per day).
### F.2 Results for Different Numbers of Task Types
In this section, we vary the number of task types and analyze the impact on
the maximum waiting time of tasks and the maximum workload of workers. In
Figure 7 we used the second approach(see section E.1) to define the task
duration parameter ($\mu_{i,j}$) for tasks 1-4, while in Figure 6 we used the
third approach. In both Figures, to define the parameters for a new task not
present in the dataset (5 and 6), we selected a mean value uniformly at random
from the range 4 and 11 (the means of the original tasks) and used the
standard deviation of the actual duration values to define a normal
distribution. This normal distribution was used to determine the duration for
each worker. To obtain a reasonable duration, we set a minimum duration of at
least 1 second for all workers.
The graph of the input network was created according to these durations. An
edge exists between a worker and a task type only if the average time required
by the worker to complete a task of that type is at most equal to the median
duration for that task type. In the final step, the duration parameters for
the new tasks were determined based on these durations, using the second and
third approaches for Figures 7 and 6 respectively.
In Figures 6 and 7 we see that having only 2 task types leads to a worse
performance both in the maximum waiting time of the tasks and in the maximum
worker workload measures. For 3-6 task types, we find that the performance in
Figure 6 decreases for both measures as the number of task types increases (at
least for some methods), while the performance in Figure 7 is very similar in
this range.
We conclude that although the number of task types is irrelevant when
distributed uniformly among the workers, it is quite important if they are
distributed differently (as in the real-data). This fact should be taken into
account when modeling task types in an application.
Figure 6: Task waiting time (left) and worker workload (right) for different
numbers of task types(task load was 90000 requests per day; $\mu_{i,j}$ varies
according to the real and synthetic data and $\kappa$ is concluded
accordingly).
Figure 7: Task waiting time (left) and worker workload (right) for different
numbers of task types(task load was 120000 requests per day and $\kappa=1$).
### F.3 Results for Different Numbers of Workers
In this section, we vary the number of workers and analyze the effects on the
maximum waiting time of the tasks and the maximum workload of the workers. In
Figure 8 we used the second approach(section E.1) to define the task duration
parameter ($\mu_{i,j}$) for workers 1-9, while in Figure 9 we used the third
approach. In both Figures, we have chosen a (uniformly) random duration from
the range of existing durations to define the average durations for a new
worker (10 to 14). The graph of the input network was defined using these
durations, where an edge between a worker and a task type exists only if the
average duration of the worker performing a task of that type is at most equal
to the median duration for that task type. Finally, the duration parameters of
the new tasks were determined based on these durations according to the second
and third approaches.
We find that, as expected, the performance of the system improves as the
number of workers increases. Therefore, when deciding how many workers to
acquire, we should look for a reasonable balance between system performance
and the cost of additional workers.
Figure 8: Task waiting time (left) and worker workload (right) for different
numbers of workers(task load was 90000 requests per day and $\kappa$ is
determined by the values of $\mu_{i,j}$).
Figure 9: Task waiting time (left) and worker workload (right) for different
numbers of workers types(task load was 90000 requests per day and $\kappa=1$
).
|
# Comparison theorems for multi-dimensional BSDEs with jumps and applications
to constrained stochastic linear-quadratic control
Ying Hu , Xiaomin Shi and Zuo Quan Xu Y. Hu: Univ Rennes, CNRS, IRMAR-UMR
6625, F-35000 Rennes, France<EMAIL_ADDRESS>X. Shi: School of
Statistics and Mathematics, Shandong University of Finance and Economics,
Jinan 250100, China<EMAIL_ADDRESS>Z. Xu: Department of Applied
Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong.
<EMAIL_ADDRESS>
###### Abstract.
In this paper, we, for the first time, establish two comparison theorems for
multi-dimensional backward stochastic differential equations with jumps. Our
approach is novel and completely different from the existing results for one-
dimensional case. Using these and other delicate tools, we then construct
solutions to coupled two-dimensional stochastic Riccati equation with jumps in
both standard and singular cases. In the end, these results are applied to
solve a cone-constrained stochastic linear-quadratic and a mean-variance
portfolio selection problem with jumps. Different from no jump problems, the
optimal (relative) state processes may change their signs, which is of course
due to the presence of jumps.
### Keywords:
Backward stochastic differential equations with jumps, multi-dimensional
comparison theorem, stochastic Riccati equation with jumps, cone-constrained
linear-quadratic control, mean-variance problem.
### Mathematics Subject Classification (2020):
60H30. 60J76. 93E20. 91G10.
## 1\. Introduction
The study of backward stochastic differential equations (BSDEs, for short) can
be dated back to Bismut [3], who studied the linear case, as an adjoint
equation in the Pontryagin stochastic maximum principle. The general Lipschitz
continuous case was later resolved in the seminal paper of Pardoux and Peng
[28]. Since then, BSDEs have attracted strong interest of many researchers and
found widely applications in partial differential equations, stochastic
control, stochastic differential game and mathematical finance; see, e.g., [6,
8, 9, 10, 12, 16, 30]. In particular, the solvability of quadratic BSDEs in
one-dimensional case was firstly obtained in Kobylanski [20], and then
generalized to multi-dimensional case by [11, 15, 25, 38].
BSDEs that are driven by a Brownian motion and a Poisson random measure, which
are named as BSDE with jumps (BSDEJ) in this paper, were firstly tackled by
Tang and Li [37], then followed notably by Barles, Buckdahn and Pardoux [2],
Royer [33], Quenes and Sulem [32] in the Lipschitz case. Quadratic BSDEs with
jumps and their applications in utility maximization problems have also been
investigated; see, e.g., Antonelli and Mancini [1], Kazi-Tani, Possamaï and
Zhou [17], Laeven and Stadje [21], Morlais [26, 27] among many others. Please
refer to Papapantoleon, Possamaï, Saplaouras [29] for a synopsis of these
topics.
BSDEs arising from stochastic linear quadratic (LQ) control problems, called
the stochastic Riccati equations (SREs), form an important class of BSDEs. In
these BSDEs, the first unknown variable appears on the denominator and the
second unknown variable grows quadratically in the generator. These features
distinguish them from those well-studied BSDEs with Lipschitz or quadratic
growth generators, so that they have to be studied by new methods.
Bismut [4] firstly found that a linear state feedback form optimal control for
a stochastic LQ control problem is available, provided that its associated SRE
admits a solution in some suitable space. Unfortunately he could not show the
existence of such a solution in general. Nowadays numerous progresses have
been made in solving SREs. Kohlmann and Tang [19] resolved the existence and
uniqueness issues for one-dimensional SREs, then Tang [35, 36] resolved the
matrix-valued case using the stochastic maximum principle and dynamic
programming method respectively. Sun, Xiong and Yong [34] studied the
indefinite case. Inspired by Tang’s [36] dynamic programming method, Zhang,
Dong and Meng [39] established the existence and uniqueness of solutions to
SREs with jumps (SREJ). Li, Wu and Yu [22] studied the indefinite case using a
so-called relax compensator.
Motivated by the mean-variance (MV) portfolio selection problem with no-
shorting constraints, Hu and Zhou [16] studied cone-constrained stochastic LQ
problem and found that the optimal control takes a piecewise ($0$ is the
unique segment point) linear state feedback form. The associated SRE is a two-
dimensional, but decoupled, BSDE. Hence it can be treated separately as two
one-dimensional BSDEs. The solvability was established with the aid of
quadratic BSDE theory and truncation techniques. The decoupling phenomenon
lies in the fact that the optimal state process will not change its sign
(namely not cross $0$), i.e. it will stay positive (resp. negative) if the
initial state is positive (resp. negative). Dong [7] generalized the model in
[16] to incorporate a jump by the enlargement of filtration framework. The
corresponding SRE is a coupled two-dimensional BSDEJ, whose solvability is
obtained by solving two recursive systems of BSDEs driven only by Brownian
motions. This decomposition approach works only in the filtration enlargement
theory; see also Kharroubi, Lim and Ngoupeyou [18], Hu, Shi and Xu [14] for
the unconstrained or regime switching case. Czichowsky and Schweizer [5]
extended the cone-constrained MV model to a general semi-martingale framework,
but they can not solve the two-dimensional SREJ. They claimed that “finding a
solution by general BSDE techniques seems a formidable challenge” in [5,
Remark 4.8].
This paper is intended as an attempt to cope with the formidable challenge
indicated in [5]. Our main contribution is to resolve the solvability of a
two-dimensional coupled SREJ in the Wiener-Poisson world via pure BSDE
techniques. Although one can consider the more general semi-martingale
framework, we will focus on the Wiener-Poisson world as SREJ in this case
takes more concrete structures for presentation and illustration. We establish
the solvability for both standard and singular cases, containing the SREJ
emerging in the cone-constrained MV problem as an special example. Since the
existing approximation procedures in Kohlmann and Tang [19] and our previous
work [13] can not be applied to the present problem, we provide a new
approximation procedure to achieve the goal.
A crucial and novel tool used in the approximation approach is our new
comparison theorems for BSDEJs. We establish two comparison theorems which
seem to be the first ones for multi-dimensional case. The first one requires a
locally Lipschitz condition for one generator (see Remark 2.3) and works for
bounded state processes, whereas the second one requires the globally
Lipschitz condition for both generators and works for square integrable state
processes.
Most of existing comparison theorems for BSDEJs require the condition
$\gamma>-1$ (see Remark 2.2) or even stronger $\gamma>-1+\varepsilon$ in order
to utilize the Girsanov theorem; see, e.g., Barles, Buckdahn and Pardoux [2]
and Royer [33]. To the best of our knowledge, Quenez and Sulem’s [32]
comparison theorem is the only one that relaxes the condition to
$\gamma\geq-1$. Without resort to the Girsanov theorem, they used the
conditional expectation representation of one-dimensional linear BSDEJs to
establish their comparison theorem. Nevertheless all of these existing
comparison theorems for BSDEJs can only deal with one-dimensional case. In our
approximation procedure, however, the SREJ is a fully coupled two-dimensional
BSDEJ, therefore comparison theorems for multi-dimensional BSDEJs are strongly
appealing. It is worth pointing that the conditional expectation
representation method used in [32] cannot be applied to multi-dimensional
BSDEJs. In this paper, we propose a completely different approach to establish
our comparison theorems for multi-dimensional BSDEJs for the first time. We
achieve the goal by directly analyzing $((\delta Y_{t})^{+})^{2}$ with the aid
of the Meyer-Itô formula and utilizing a tricky elementary inequality (Lemma
2.1) that works for $\gamma\geq-1$. Note one cannot expect to extend the
results to the case $\gamma<-1$ since counter-examples do exist in this case;
see [2, Remark 2.7].
With the help of the new comparison theorems for multi-dimensional BSDEJs, we
can construct solutions to the two-dimensional coupled SREJ in both standard
and singular cases. We then apply the result to solve a cone-constrained
stochastic LQ problem with jumps and obtain the efficient portfolio for a MV
problem with jumps. It is worth pointing out that even without the cone-
constraint, MV problems with jumps have not been investigated thoroughly. Lim
[23] studied such a problem, but he assumed all the coefficients are
predictable with respect to the Brownian filtration, rendered the
corresponding SRE exactly the same as that in the model without jumps. On the
other hand, Zhang, Dong and Meng [39] examined stochastic LQ problems with
jumps, but they assumed the control weight in the running cost is uniformly
positive so that their result cannot solve the corresponding MV problem where
the control weight is 0. We will not only solve the MV problem with jumps, but
also incorporate convex cone-constraint, especially covering the famous no-
shorting constraints. By adding cone-constraint, the associated SREJ becomes a
fully coupled two-dimensional BSDEJ, thus causing notably nontrivial
difficulty in its solvability.
The rest part of this paper is organized as follows. Section 2 is devoted to
proving two comparison theorems for multi-dimensional BSDEJs. In Section 3, we
study a cone-constrained stochastic LQ control problem with jumps and prove
the existence and uniqueness of solution to the associated SREJ. In Section 4,
we solve a cone-constrained MV problem. Appendix A provides a heuristical
derivation of the SREJ. A lengthy and complementary proof of Theorem 3.1 is
relegated to Appendix B.
## 2\. Comparison theorems for multi-dimensional BSDEJs
Let $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ be a fixed complete filtered
probability space. The filtration $\mathbb{F}=\\{\mathcal{F}_{t},t\geq 0\\}$
is generated by two independent random sources augmented by all
$\mathbb{P}$-null sets: one is a standard $n$-dimensional Brownian motion
$W_{t}=(W_{1,t},\ldots,W_{n,t})^{\top}$, and the other one is a Poisson random
measure $N(\operatorname{d}\\!t,\operatorname{d}\\!e)$ defined on
${\mathbb{R}}_{+}\times\mathcal{E}$ induced by a stationary Poisson point
process with a stationary compensator (intensity measure) given by
$\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$ satisfying
$\nu(\mathcal{E})<\infty$, where
$\mathcal{\mathcal{E}}\subseteq{\mathbb{R}}^{\ell}\setminus\\{0\\}$ is a
nonempty Borel subset of the $\ell$-dimensional Euclidean space
${\mathbb{R}}^{\ell}$. We use an increasing sequence
$\\{T_{n}\\}_{n\in\mathbb{N}}$ to denote the jump times of underlying Poisson
point process. The compensated Poisson random measure is denoted by
$\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e)$. For the ease of
notations, we only consider one-dimensional Poisson random measure, although
the results of this paper can be generalized to the multi-dimensional case
without essential difficulties. Throughout the paper, let $T$ denote a fixed
positive constant, $\mathcal{P}$ denote the $\mathbb{F}$-predictable
$\sigma$-field on $\Omega\times[0,T]$, and $\mathcal{B}(\mathcal{E})$ denote
the Borel $\sigma$-algebra of $\mathcal{E}$.
We denote by ${\mathbb{R}}^{\ell}$ the set of $\ell$-dimensional column
vectors, by ${\mathbb{R}}^{\ell}_{+}$ the set of vectors in
${\mathbb{R}}^{\ell}$ whose components are nonnegative, by
${\mathbb{R}}^{\ell\times n}$ the set of $\ell\times n$ real matrices, and by
$\mathbb{S}^{n}$ the set of symmetric $n\times n$ real matrices. Therefore,
${\mathbb{R}}^{\ell}\equiv{\mathbb{R}}^{\ell\times 1}$. For any vector $Y$, we
denote $Y_{i}$ as its $i$-th component. For any matrix $M=(m_{ij})$, we denote
its transpose by $M^{\top}$, and its norm by $|M|=\sqrt{\sum_{ij}m_{ij}^{2}}$.
If $M\in\mathbb{S}^{n}$ is positive definite (resp. positive semidefinite), we
write $M>$ (resp. $\geq$) $0.$ We write $A>$ (resp. $\geq$) $B$ if
$A,B\in\mathbb{S}^{n}$ and $A-B>$ (resp. $\geq$) $0.$ We use the standard
notations $x^{+}=\max\\{x,0\\}$ and $x^{-}=\max\\{-x,0\\}$ for
$x\in{\mathbb{R}}$ and define a set $\mathcal{M}=\\{1,2,...,\ell\\}$. We will
use the elementary inequality $|a^{\top}b|\leq c|a|^{2}+\frac{|b|^{2}}{2c}$
for any $a,b\in{\mathbb{R}}^{n},c>0$ frequently throughout the paper without
claim.
We use the following spaces throughout the paper:
$\displaystyle L^{2}_{\mathcal{F}_{T}}(\Omega;{\mathbb{R}})$
$\displaystyle=\Big{\\{}\xi:\Omega\to{\mathbb{R}}\;\Big{|}\;\mbox{$\xi$ is
$\mathcal{F}_{T}$-measurable, and ${\mathbb{E}}|\xi|^{2}<\infty$}\Big{\\}},$
$\displaystyle L^{\infty}_{\mathcal{F}_{T}}(\Omega;{\mathbb{R}})$
$\displaystyle=\Big{\\{}\xi:\Omega\to{\mathbb{R}}\;\Big{|}\;\mbox{$\xi$ is
$\mathcal{F}_{T}$-measurable, and essentially bounded}\Big{\\}},$
$\displaystyle L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}})$
$\displaystyle=\Big{\\{}\phi:\Omega\times[0,T]\to{\mathbb{R}}\;\Big{|}\;\mbox{$\phi$
is $\mathcal{P}$-measurable and
${\mathbb{E}}\int_{0}^{T}|\phi_{t}|^{2}\operatorname{d}\\!t<\infty$}\Big{\\}},$
$\displaystyle L^{\infty}_{\mathbb{F}}(0,T;{\mathbb{R}})$
$\displaystyle=\Big{\\{}\phi:\Omega\times[0,T]\to{\mathbb{R}}\;\Big{|}\;\mbox{$\phi$
is $\mathcal{P}$-measurable and essentially bounded}\Big{\\}},$ $\displaystyle
L^{2,\nu}({\mathbb{R}})$
$\displaystyle=\Big{\\{}\phi:\mathcal{E}\to{\mathbb{R}}\;\Big{|}\;\mbox{$\phi$
is $\mathcal{B}(\mathcal{E})$-measurable and
$||\phi||^{2}_{\nu}:=\int_{\mathcal{E}}|\phi(e)|^{2}\nu(\operatorname{d}\\!e)<\infty$}\Big{\\}},$
$\displaystyle L^{\infty,\nu}({\mathbb{R}})$
$\displaystyle=\Big{\\{}\phi:\mathcal{E}\to{\mathbb{R}}\;\Big{|}\;\mbox{$\phi$
is $\mathcal{B}(\mathcal{E})$-measurable and essentially bounded w.r.t.
$\operatorname{d}\\!\nu$}\Big{\\}},$ $\displaystyle
L^{2,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}})$
$\displaystyle=\Big{\\{}\phi:\Omega\times[0,T]\times\mathcal{E}\to{\mathbb{R}}\;\Big{|}\;\mbox{$\phi$
is $\mathcal{P}\otimes\mathcal{B}(\mathcal{E})$-measurable }$
$\displaystyle\qquad\quad\ \mbox{and
${\mathbb{E}}\int_{0}^{T}\int_{\mathcal{E}}|\phi_{t}(e)|^{2}\nu(\operatorname{d}\\!e)\operatorname{d}\\!t<\infty$}\Big{\\}},$
$\displaystyle L^{\infty,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}})$
$\displaystyle=\Big{\\{}\phi:\Omega\times[0,T]\times\mathcal{E}\to{\mathbb{R}}\;\Big{|}\;\phi\mbox{
is $\mathcal{P}\otimes\mathcal{B}(\mathcal{E})$-measurable and}$
$\displaystyle\qquad\quad\mbox{ essentially bounded w.r.t.
$\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\otimes\operatorname{d}\\!\nu$}\Big{\\}},$
$\displaystyle S^{2}_{\mathbb{F}}(0,T;{\mathbb{R}})$
$\displaystyle=\Big{\\{}\phi:\Omega\times[0,T]\to{\mathbb{R}}\;\Big{|}\;(\phi_{t})_{0\leq
t\leq T}\mbox{ is c\\`{a}d-l\\`{a}g, $\mathbb{F}$-adapted}$
$\displaystyle\qquad\quad\mbox{ and ${\mathbb{E}}\sup_{0\leq t\leq
T}|\phi_{t}|^{2}<\infty$}\Big{\\}},$ $\displaystyle
S^{\infty}_{\mathbb{F}}(0,T;{\mathbb{R}})$
$\displaystyle=\Big{\\{}\phi:\Omega\times[0,T]\to{\mathbb{R}}\;\Big{|}\;(\phi_{t})_{0\leq
t\leq T}\mbox{ is c\\`{a}d-l\\`{a}g, $\mathbb{F}$-adapted}$
$\displaystyle\qquad\quad\mbox{ and essentially bounded}\Big{\\}}.$
These definitions are generalized in the obvious way to the cases that
${\mathbb{R}}$ is replaced by ${\mathbb{R}}^{n}$, ${\mathbb{R}}^{n\times\ell}$
or $\mathbb{S}^{n}$. Arguments $s$, $t$ and $\omega$, or statements “almost
surely” (a.s.) and “almost everywhere” (a.e.), may be suppressed for
simplicity in many circumstances when no confusion occurs. We shall use $c$ to
represent a generic positive constant which can be different from line to
line. All the equations and inequalities in subsequent analysis shall be
understood in the sense that $\operatorname{d}\\!\mathbb{P}$-a.s. or
$\operatorname{d}\\!\nu$-a.e. or
$\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.}$ or
$\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\otimes\operatorname{d}\\!\nu$-a.e.
etc.
In this paper, any $\ell$-dimensional backward stochastic differential
equation with jumps (BSDEJ) (on $[0,T]$) is characterized by a pair $(\xi,f)$,
in which $\xi:\Omega\to{\mathbb{R}}^{\ell}$ is called the terminal value which
is an $\mathcal{F}_{T}$-measurable random vector, and
$f:\Omega\times[0,T]\times{\mathbb{R}}^{\ell}\times{\mathbb{R}}^{n\times\ell}\times
L^{2,\nu}({\mathbb{R}}^{\ell})\to{\mathbb{R}}^{\ell}$ is called the generator
which is a
$\mathcal{P}\otimes\mathcal{B}({\mathbb{R}}^{\ell})\otimes\mathcal{B}({\mathbb{R}}^{n\times\ell})\otimes\mathcal{B}(L^{2,\nu}({\mathbb{R}}^{\ell}))$-measurable
process. We call the BSDEJ $\ell$-dimensional as its state process is
${\mathbb{R}}^{\ell}$-valued. We often rewrite in its component form for the
ease of presentations.
### 2.1. Comparison theorem for bounded processes
We first prove a comparison theorem where the state processes are essentially
bounded.
###### Theorem 2.1.
Suppose, for every $i\in\mathcal{M}$,
$\displaystyle(Y_{i},Z_{i},\Phi_{i}),(\overline{Y}_{i},\overline{Z}_{i},\overline{\Phi}_{i})\in
S^{\infty}_{\mathbb{F}}(0,T;{\mathbb{R}})\times
L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}}^{n})\times
L^{2,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}}),$
and they satisfy BSDEJs
$\displaystyle Y_{i,t}$
$\displaystyle=\xi_{i}+\int_{t}^{T}f_{i}(s,Y_{s-},Z_{i,s},\Phi_{s})\operatorname{d}\\!s$
(2.1)
$\displaystyle\qquad-\int_{t}^{T}Z_{i,s}^{\top}\operatorname{d}\\!W_{s}-\int_{t}^{T}\int_{\mathcal{E}}\Phi_{i,s}(e)\widetilde{N}(\operatorname{d}\\!s,\operatorname{d}\\!e)~{}\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.},$
and
$\displaystyle\overline{Y}_{i,t}$
$\displaystyle=\overline{\xi}_{i}+\int_{t}^{T}\overline{f}_{i}(s,\overline{Y}_{s-},\overline{Z}_{i,s},\overline{\Phi}_{s})\operatorname{d}\\!s$
(2.2)
$\displaystyle\qquad-\int_{t}^{T}\overline{Z}_{i,s}^{\top}\operatorname{d}\\!W_{s}-\int_{t}^{T}\int_{\mathcal{E}}\overline{\Phi}_{i,s}(e)\widetilde{N}(\operatorname{d}\\!s,\operatorname{d}\\!e)~{}\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.}.$
Also suppose that, for all $i\in\mathcal{M}$ and $s\in[0,T]$,
1. (1)
$\xi_{i}\leq\overline{\xi}_{i}$;
2. (2)
there exists a constant $c>0$ such that
$\displaystyle\quad\;f_{i}(s,Y_{s-},Z_{i,s},\Phi_{1,s},\cdots,\Phi_{i,s},\cdots,\Phi_{\ell,s})$
$\displaystyle\qquad\quad-
f_{i}(s,Y_{s-},Z_{i,s},\Phi_{1,s},\cdots,\overline{\Phi}_{i,s},\cdots,\Phi_{\ell,s})$
$\displaystyle\leq
c\int_{\mathcal{E}}(\Phi_{i,s}(e)-\overline{\Phi}_{i,s}(e))^{+}\nu(\operatorname{d}\\!e)+\int_{\mathcal{E}}|\Phi_{i,s}(e)-\overline{\Phi}_{i,s}(e)|\nu(\operatorname{d}\\!e);$
3. (3)
there exists a constant $c>0$ such that
$\displaystyle\quad\;f_{i}(s,Y_{s-},Z_{i,s},\Phi_{1,s},\cdots,\overline{\Phi}_{i,s},\cdots,\Phi_{\ell,s})-f_{i}(s,\overline{Y}_{s-},\overline{Z}_{i,s},\overline{\Phi}_{s})$
$\displaystyle\leq c\Big{(}|Y_{i,s-}-\overline{Y}_{i,s-}|+\sum_{j\neq
i}(Y_{j,s-}-\overline{Y}_{j,s-})^{+}+|Z_{i,s}-\overline{Z}_{i,s}|$
$\displaystyle\qquad\quad+\sum_{j\neq
i}\int_{\mathcal{E}}(Y_{j,s-}+\Phi_{j,s}(e)-\overline{Y}_{j,s-}-\overline{\Phi}_{j,s}(e))^{+}\nu(\operatorname{d}\\!e)\Big{)};~{}\text{and}$
4. (4)
$f_{i}(s,\overline{Y}_{s-},\overline{Z}_{i,s},\overline{\Phi}_{s})\leq\overline{f}_{i}(s,\overline{Y}_{s-},\overline{Z}_{i,s},\overline{\Phi}_{s}).$
Then $Y_{i}\leq\overline{Y}_{i}$ for all $i\in\mathcal{M}$.
To prove this theorem, we need the following critical elementary result.
###### Lemma 2.1.
For all $(x,y)\in{\mathbb{R}}\times{\mathbb{R}}$ and $c\geq-1$, we have
$\displaystyle[(x+y)^{+}]^{2}-(x^{+})^{2}-2(1+c)x^{+}y\geq-(c^{2}\vee
1)(x^{+})^{2}.$
Proof: There are three cases:
* •
If $x\leq 0$, then
$\displaystyle[(x+y)^{+}]^{2}-(x^{+})^{2}-2(1+c)x^{+}y=[(x+y)^{+}]^{2}\geq
0=-(c^{2}\vee 1)(x^{+})^{2}.$
* •
If $y\leq 0$, then, since $c\geq-1$,
$\displaystyle[(x+y)^{+}]^{2}-(x^{+})^{2}-2(1+c)x^{+}y\geq[(x+y)^{+}]^{2}-(x^{+})^{2}\geq-(x^{+})^{2}\geq-(c^{2}\vee
1)(x^{+})^{2}.$
* •
If $x\geq 0$ and $y\geq 0$, then
$\displaystyle[(x+y)^{+}]^{2}-(x^{+})^{2}-2(1+c)x^{+}y=y^{2}-2cxy$
$\displaystyle=(y-cx)^{2}-c^{2}x^{2}$ $\displaystyle\geq-(c^{2}\vee
1)x^{2}=-(c^{2}\vee 1)(x^{+})^{2}.\qquad$
The proof is complete. $\Box$
Proof of Theorem 2.1.
For $t\in[0,T]$ and $i\in\mathcal{M}$, set
$\delta Y_{i,t}=Y_{i,t}-\overline{Y}_{i,t},\ \delta
Z_{i,t}=Z_{i,t}-\overline{Z}_{i,t},\
\delta\Phi_{i,t}=\Phi_{i,t}-\overline{\Phi}_{i,t}.$
Applying the Meyer-Itô formula [31, Chapter IV, Theorem 70] to $(\delta
Y_{i,t})^{+}$, we get,
$\displaystyle\operatorname{d}\\!\;(\delta Y_{i,t})^{+}$
$\displaystyle=-\mathbf{1}_{\\{\delta
Y_{i,t-}>0\\}}[f_{i}(t,Y_{t-},Z_{i,t},\Phi_{t})-\overline{f}_{i}(t,\overline{Y}_{t-},\overline{Z}_{i,t},\overline{\Phi}_{t})]\operatorname{d}\\!t$
$\displaystyle\quad\;+\int_{\mathcal{E}}[(\delta
Y_{i,t-}+\delta\Phi_{i,t}(e))^{+}-(\delta
Y_{i,t-}^{i})^{+}-\mathbf{1}_{\\{\delta
Y_{i,t-}>0\\}}\delta\Phi_{i,t}(e)]\nu(\operatorname{d}\\!e)\operatorname{d}\\!t+\frac{1}{2}\operatorname{d}\\!L_{i,t}$
$\displaystyle\quad\;+\mathbf{1}_{\\{\delta Y_{i,t-}>0\\}}\delta
Z_{i,t}^{\top}\operatorname{d}\\!W_{t}+\int_{\mathcal{E}}[(\delta
Y_{i,t-}+\delta\Phi_{i,t}(e))^{+}-(\delta
Y_{i,t-})^{+}]\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),$
where $L_{i,t}$ is the local time of $\delta Y_{i,t}$ at $0$. Since $\delta
Y_{i,t-}\operatorname{d}\\!L_{i,t}=0$, applying Itô’s formula to $((\delta
Y_{i,t})^{+})^{2}$ yields
$\displaystyle\operatorname{d}\\!\;((\delta Y_{i,t})^{+})^{2}$
$\displaystyle=-2(\delta
Y_{i,t-})^{+}[f_{i}(t,Y_{t-},Z_{i,t},\Phi_{t})-\overline{f}_{i}(t,\overline{Y}_{t-},\overline{Z}_{i,t},\overline{\Phi}_{t})]\operatorname{d}\\!t+\mathbf{1}_{\\{\delta
Y_{i,t-}>0\\}}|\delta Z_{i,t}|^{2}\operatorname{d}\\!t$
$\displaystyle\quad\;+\int_{\mathcal{E}}[((\delta
Y_{i,t-}+\delta\Phi_{i,t}(e))^{+})^{2}-((\delta Y_{i,t-})^{+})^{2}-2(\delta
Y_{i,t-})^{+}\delta\Phi_{i,t}(e)]\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
(2.3) $\displaystyle\quad\;+2(\delta Y_{i,t-})^{+}\delta
Z_{i,t}^{\top}\operatorname{d}\\!W_{t}+\int_{\mathcal{E}}[((\delta
Y_{i,t-}+\delta\Phi_{i,t}(e))^{+})^{2}-((\delta
Y_{i,t-})^{+})^{2}]\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e).$
Using the condition 4 and inserting two zero-sum terms, we get
$\displaystyle\quad\;f_{i}(t,Y_{t-},Z_{i,t},\Phi_{t})-\overline{f}_{i}(t,\overline{Y}_{t-},\overline{Z}_{i,t},\overline{\Phi}_{t})$
$\displaystyle\leq
f_{i}(t,Y_{t-},Z_{i,t},\Phi_{t})-f_{i}(t,\overline{Y}_{t-},\overline{Z}_{i,t},\overline{\Phi}_{t})$
$\displaystyle=[f_{i}(t,Y_{t-},Z_{i,t},\Phi_{t})-f_{i}(t,Y_{t-},Z_{i,t},\Phi_{1,t},\cdots,\overline{\Phi}_{i,t},\cdots,\Phi_{\ell,t})]$
$\displaystyle\quad\;+[f_{i}(t,Y_{t-},Z_{i,t},\Phi_{1,t},\cdots,\overline{\Phi}_{i,t},\cdots,\Phi_{\ell,t})-f_{i}(t,\overline{Y}_{t-},\overline{Z}_{i,t},\overline{\Phi}_{t})].$
By the conditions 2, the first difference on the right hand side (RHS) in
above is upper bounded by
$\int_{\mathcal{E}}\gamma_{i,t}(e)\delta\Phi_{i,t}(e)\nu(\operatorname{d}\\!e),$
where
$\displaystyle\gamma_{i,t}(e)=\begin{cases}c&\quad\text{ if
}\delta\Phi_{i,t}(e)\geq 0;\\\ -1&\quad\text{ if
}\delta\Phi_{i,t}(e)<0.\end{cases}$
By the conditions 3, the second difference on the RHS is upper bounded
respectively by
$c\Big{(}|\delta Y_{i,t-}|+\sum_{j\neq i}(\delta Y_{j,t-})^{+}+|\delta
Z_{i,t}|+\sum_{j\neq i}\int_{\mathcal{E}}(\delta
Y_{j,t-}+\delta\Phi_{j,t}(e))^{+}\nu(\operatorname{d}\\!e)\Big{)}.$
Using these estimates and $\nu(\mathcal{E})<\infty$, we deduce that
$\displaystyle 2(\delta
Y_{i,t-})^{+}[f_{i}(t,Y_{t-},Z_{i,t},\Phi_{t})-\overline{f}_{i}(t,\overline{Y}_{t-},\overline{Z}_{i,t},\overline{\Phi}_{t})]$
$\displaystyle\leq c\sum_{i=1}^{\ell}((\delta
Y_{i,t-})^{+})^{2}+\mathbf{1}_{\\{\delta Y_{i,t-}>0\\}}|\delta
Z_{i,t}|^{2}+\mathbf{1}_{\\{\delta Y_{i,t-}>0\\}}\sum_{j\neq
i}\int_{\mathcal{E}}((\delta
Y_{j,t-}+\delta\Phi_{j,t}(e))^{+})^{2}\nu(\operatorname{d}\\!e)$ (2.4)
$\displaystyle\quad\;+2(\delta
Y_{i,t-})^{+}\int_{\mathcal{E}}\gamma_{i,t}(e)\delta\Phi_{i,t}(e)\nu(\operatorname{d}\\!e).$
Integrating from $t$ to $T$ in (2.1), taking conditional expectation and using
(2.1), we obtain
$\displaystyle\quad((\delta Y_{i,t})^{+})^{2}$
$\displaystyle\leq\mathbb{E}_{t}\int_{t}^{T}\Big{(}c\sum_{i=1}^{\ell}((\delta
Y_{i,s-})^{+})^{2}+\mathbf{1}_{\\{\delta Y_{i,s-}>0\\}}\sum_{j\neq
i}\int_{\mathcal{E}}((\delta
Y_{j,s-}+\delta\Phi_{j,s}(e))^{+})^{2}\nu(\operatorname{d}\\!e)\Big{)}\operatorname{d}\\!s$
$\displaystyle-\mathbb{E}_{t}\int_{t}^{T}\int_{\mathcal{E}}\Big{[}((\delta
Y_{i,s-}+\delta\Phi_{i,s}(e))^{+})^{2}-((\delta
Y_{i,s-})^{+})^{2}-2(1+\gamma_{i,s}(e))(\delta
Y_{i,s-})^{+}\delta\Phi_{i,s}(e)\Big{]}\nu(\operatorname{d}\\!e)\operatorname{d}\\!s.$
Because $\gamma_{i}\in L^{\infty,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}})$ and
$\gamma_{i}\geq-1$, it follows from Lemma 2.1 that
(2.5) $-[((\delta Y_{i,s-}+\delta\Phi_{i,s}(e))^{+})^{2}-((\delta
Y_{i,s-})^{+})^{2}-2(1+\gamma_{i,s}(e))(\delta
Y_{i,s-})^{+}\delta\Phi_{i,s}(e)]\\\ \qquad\leq(\gamma_{i,s}(e)^{2}\vee
1)((\delta Y_{i,s-})^{+})^{2}\leq c((\delta Y_{i,s-})^{+})^{2}.$
Combining the above estimates and using $\nu(\mathcal{E})<\infty$, we obtain
(2.6) $\displaystyle((\delta Y_{i,t})^{+})^{2}$ $\displaystyle\leq
c\mathbb{E}_{t}\int_{t}^{T}\sum_{i=1}^{\ell}((\delta
Y_{i,s-})^{+})^{2}\operatorname{d}\\!s+\sum_{j\neq
i}{\mathbb{E}}_{t}\int_{t}^{T}\int_{\mathcal{E}}((\delta
Y_{j,s-}+\delta\Phi_{j,s}(e))^{+})^{2}\nu(\operatorname{d}\\!e)\operatorname{d}\\!s,$
where the constant $c$ is independent of $t$, $T$ and $i$.
Note that
$\displaystyle{\mathbb{E}}_{t}\int_{t}^{T}\int_{\mathcal{E}}((\delta
Y_{j,s-}+\delta\Phi_{j,s})^{+})^{2}\nu(\operatorname{d}\\!e)\operatorname{d}\\!s$
$\displaystyle={\mathbb{E}}_{t}\int_{t}^{T}\int_{\mathcal{E}}((\delta
Y_{j,s-}+\delta\Phi_{j,s}(e))^{+})^{2}N(\operatorname{d}\\!s,\operatorname{d}\\!e)$
$\displaystyle={\mathbb{E}}_{t}\Big{[}\sum_{n\in\mathbb{N},\ t<T_{n}\leq
T}((\delta Y_{j,T_{n}-}+\delta\Phi_{j,T_{n}}(\Delta
U_{T_{n}}))^{+})^{2}\Big{]}$ (2.7)
$\displaystyle={\mathbb{E}}_{t}\Big{[}\sum_{n\in\mathbb{N},\ t<T_{n}\leq
T}((\delta Y_{j,T_{n}})^{+})^{2}\Big{]},$
where
$U_{t}:=\int_{0}^{t}\int_{\mathcal{E}}zN(\operatorname{d}\\!s,\operatorname{d}\\!e)$,
$\Delta U_{T_{n}}:=U_{T_{n}}-U_{T_{n}-}$, recalling that
$\\{T_{n}\\}_{n\in\mathbb{N}}$ denotes of jump times of underlying Poisson
point process. Substituting (2.1) into (2.6) yields,
$\displaystyle((\delta Y_{i,t})^{+})^{2}$ $\displaystyle\leq
c\mathbb{E}_{t}\int_{t}^{T}\sum_{i=1}^{\ell}((\delta
Y_{i,s-})^{+})^{2}\operatorname{d}\\!s+\sum_{j\neq
i}{\mathbb{E}}_{t}\Big{[}\sum_{n\in\mathbb{N},\ t<T_{n}\leq T}((\delta
Y_{j,T_{n}})^{+})^{2}\Big{]}.$
Since the jumps of $\delta Y$ are accountable, we can replace $\delta
Y_{i,s-}$ by $\delta Y_{i,s}$ in the above integral to get
(2.8) $\displaystyle((\delta Y_{i,t})^{+})^{2}$ $\displaystyle\leq
c\mathbb{E}_{t}\int_{t}^{T}\sum_{i=1}^{\ell}((\delta
Y_{i,s})^{+})^{2}\operatorname{d}\\!s+\sum_{j\neq
i}{\mathbb{E}}_{t}\Big{[}\sum_{n\in\mathbb{N},\ t<T_{n}\leq T}((\delta
Y_{j,T_{n}})^{+})^{2}\Big{]}.$
For any constant $h\in(0,T]$, set
$M(h):=\operatorname*{ess\;sup}\limits_{(t,i)\in[T-h,T]\times\mathcal{M}}((\delta
Y_{i,t})^{+})^{2},$
which is finite since $\delta Y$ is bounded. For any $t\in[T-h,T]$, we obtain
from (2.8) that
$\displaystyle((\delta Y_{i,t})^{+})^{2}$ $\displaystyle\leq
c\int_{t}^{T}\sum_{i=1}^{\ell}M(h)\operatorname{d}\\!s+\sum_{j\neq
i}{\mathbb{E}}_{t}\Big{[}\sum_{n\in\mathbb{N},\ t<T_{n}\leq T}M(h)\Big{]}$
$\displaystyle=c\ell M(h)(T-t)+M(h)\sum_{j\neq
i}{\mathbb{E}}_{t}\int_{t}^{T}\int_{\mathcal{E}}1N(\operatorname{d}\\!s,\operatorname{d}\\!e)$
$\displaystyle=c\ell M(h)(T-t)+M(h)(\ell-1)\nu(\mathcal{E})(T-t)$
$\displaystyle\leq(c\ell+(\ell-1)\nu(\mathcal{E}))M(h)h.$
Taking essential supreme over $(t,i)\in[T-h,T]\times\mathcal{M}$ on both sides
leads to
(2.9) $\displaystyle M(h)$
$\displaystyle\leq(c\ell+(\ell-1)\nu(\mathcal{E}))M(h)h.$
Setting $h=\min\\{1/(c\ell+(\ell-1)\nu(\mathcal{E})+1),T\\}$ from now on. It
then follows from above that $M(h)=0$, thus $\delta Y_{i,t}\leq 0$ for all
$t\in[T-h,T]$. Similarly, using $\delta Y_{i,T-h}\leq 0$ and repeating the
above argument on $[0\vee(T-2h),T-h]$, one can get $\delta Y_{i,t}\leq 0$ for
all $t\in[0\vee(T-2h),T-h]$. Repeating this procedure, the desired comparison
result follows. $\Box$
###### Remark 2.1.
If the inequalities in the conditions 1 and 4 are reversed, then so is the
conclusion.
###### Remark 2.2.
It is not hard to see that the condition 2 is equivalent to that there exists
a process $\gamma_{i}\in L^{\infty,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}})$ with
$\gamma_{i}\geq-1$ such that
$\displaystyle\quad\;f_{i}(s,Y_{s-},Z_{i,s},\Phi_{1,s},\cdots,\Phi_{i,s},\cdots,\Phi_{\ell,s})$
$\displaystyle\qquad\quad-
f_{i}(s,Y_{s-},Z_{i,s},\Phi_{1,s},\cdots,\overline{\Phi}_{i,s},\cdots,\Phi_{\ell,s})$
$\displaystyle\leq\int_{\mathcal{E}}\gamma_{i,s}(e)(\Phi_{i,s}(e)-\overline{\Phi}_{i,s}(e))\nu(\operatorname{d}\\!e).$
Most of existing comparison theorems for BSDEJs require the condition
$\gamma>-1$ or even stronger $\gamma>-1+\varepsilon$ in order to utilize the
Girsanov theorem; see, e.g., Barles, Buckdahn and Pardoux [2] and Royer [33].
Our requirement, namely $\gamma\geq-1$, is the same as Quenez and Sulem’s
[32]. But all these existing comparison theorems work for one-dimensional
BSDEJs only.
###### Remark 2.3.
The condition 3 holds if, for every $K>0$, there exists a constant $c>0$
(depending on $K$) such that
$f_{i}(s,y,z,\phi)-f_{i}(s,\overline{y},\overline{z},\overline{\phi})\\\
\qquad\leq c\Big{(}|y_{i}-\overline{y}_{i}|+\sum_{j\neq
i}(y_{j}-\overline{y}_{j})^{+}+|z-\overline{z}|+\sum_{j\neq
i}\int_{\mathcal{E}}(y_{j}-\overline{y}_{j}+\phi_{j}(e)-\overline{\phi}_{j}(e))^{+}\nu(\operatorname{d}\\!e)\Big{)}$
holds for all $(y,z,\phi)$ and
$(\overline{y},\overline{z},\overline{\phi})\in{\mathbb{R}}^{\ell}\times{\mathbb{R}}^{n}\times
L^{2,\nu}({\mathbb{R}}^{\ell})$ satisfying $\phi_{i}\equiv\overline{\phi}_{i}$
and $|y|+|\overline{y}|\leq K$. Since $|y|+|\overline{y}|\leq K$, it is a
locally Lipschitz condition w.r.t $y$. The condition implies $f_{i}$ is
increasing w.r.t $y_{j}$ and $\phi_{j}$ for every $j\neq i$. Also, the term
$\sum_{j\neq i}(y_{j}-\overline{y}_{j})^{+}$ can be removed if there does
exist jump, i.e. $\nu(\mathcal{E})>0$.
###### Remark 2.4.
We call a generator $f$ is Lipschitz in $(y,z,\phi)$ with Lipschitz constant
$c$ if
$\displaystyle|f(\omega,t,y,z,\phi)-f(\omega,t,\overline{y},\overline{z},\overline{\phi})|\leq
c(|y-\overline{y}|+|z-\overline{z}|+||\phi-\overline{\phi}||_{\nu})~{}\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.}$
holds for all $(y,z,\phi)$,
$(\overline{y},\overline{z},\overline{\phi})\in{\mathbb{R}}^{\ell}\times{\mathbb{R}}^{n\times\ell}\times
L^{2,\nu}({\mathbb{R}}^{\ell})$. Then the condition 3 holds if
1. (1)
$f_{i}(s,y,z,\phi)$ is Lipschitz in $(y,z,\phi)$;
2. (2)
$f_{i}(s,y,z,\phi)$ is increasing w.r.t $y_{j}$ for every $j\neq i$; and
3. (3)
there exists a constant $c>0$ such that
$\displaystyle
f_{i}(s,Y_{s-},Z_{i,s},\Phi_{1,s},...,\Phi_{i-1,s},\overline{\Phi}_{i,s},\Phi_{i+1,s},...,\Phi_{\ell,s})$
$\displaystyle\qquad-
f_{i}(s,\overline{Y}_{1,s-},...,\overline{Y}_{i-1,s-},Y_{i,s-},\overline{Y}_{i+1,s-},,...,\overline{Y}_{\ell,s-},Z_{i,s-},\overline{\Phi}_{s-})$
$\displaystyle\leq c\sum_{j\neq
i}\int_{\mathcal{E}}(Y_{j,s-}+\Phi_{j,s}(e)-\overline{Y}_{j,s-}-\overline{\Phi}_{j,s}(e))^{+}\nu(\operatorname{d}\\!e).$
###### Remark 2.5.
In (2.5) the condition $\gamma\in
L^{\infty,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}}^{\ell})$ can be replaced by the
following weaker one: there exist constants $0<h,\varepsilon<1$ such that
$\operatorname*{ess\;sup}_{t\in[0,T]}\mathbb{E}_{t}\int_{t}^{T\wedge(t+h)}\int_{\mathcal{E}}|\gamma_{s}(e)|^{2}\nu(\operatorname{d}\\!e)\operatorname{d}\\!s\leq
1-\varepsilon.$
This condition is satisfied, for instance, when
$\int_{\mathcal{E}}|\gamma_{\cdot}(e)|^{2}\nu(\operatorname{d}\\!e)\in
L^{\infty}_{\mathbb{F}}(0,T;{\mathbb{R}})$. Indeed, the above condition
implies, for $t\in[T-h,T]$,
$\displaystyle\mathbb{E}_{t}\int_{t}^{T}((\delta
Y_{i,s-})^{+})^{2}\int_{\mathcal{E}}(\gamma_{i,s}(e)^{2}\vee
1)\nu(\operatorname{d}\\!e)\operatorname{d}\\!s$ $\displaystyle\leq
M(h)~{}\mathbb{E}_{t}\int_{t}^{T}\int_{\mathcal{E}}(\gamma_{i,s}(e)^{2}+1)\nu(\operatorname{d}\\!e)\operatorname{d}\\!s$
$\displaystyle\leq
M(h)(1-\varepsilon+h\nu(\mathcal{E}))\leq(1-\varepsilon/2)M(h),$
by choosing $h$ small enough. This together with (2.1) and (2.1) will lead to
an estimate similar to (2.9) in the above proof.
### 2.2. Comparison theorem for square integrable processes
Theorem 2.1 requires the state processes to be bounded, which may be too
restrictive for applications. The following result relaxes this assumption to
square integrable processes, but we have to in addition assume that both $f$
and $\overline{f}$ are globally Lipschitz.
###### Theorem 2.2.
We shall use the same notations as in Theorem 2.1. Suppose, for all
$i\in\mathcal{M}$,
$\displaystyle(Y_{i},Z_{i},\Phi_{i}),(\overline{Y}_{i},\overline{Z}_{i},\overline{\Phi}_{i})\in
S^{2}_{\mathbb{F}}(0,T;{\mathbb{R}})\times
L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}}^{n})\times
L^{2,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}}),$
and they satisfy the BSDEJs (2.1) and (2.1). Also suppose that, for every
$i\in\mathcal{M}$,
1. (1)
the conditions 1, 2, 3 and 4 hold;
2. (2)
$f_{i}(\cdot,0,0,0)$ and $\overline{f}_{i}(\cdot,0,0,0)\in
L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}})$;
3. (3)
both $f_{i}$ and $\overline{f}_{i}$ are Lipschitz in $(y,z,\phi)$.
Then $Y_{i}\leq\overline{Y}_{i}$ for all $i\in\mathcal{M}$.
Proof: For each $m\geq 1$ and $i\in\mathcal{M}$, we denote
$\displaystyle\xi^{m}_{i}=\xi_{i}\mathbf{1}_{|\xi|+|\overline{\xi}|\leq
m},~{}~{}f^{m}_{i}(t,y,z,\phi)=f_{i}(t,y,z,\phi)\mathbf{1}_{|f(t,0,0,0)|+|\overline{f}(t,0,0,0)|\leq
m},$
$\displaystyle\overline{\xi}^{m}_{i}=\overline{\xi}_{i}\mathbf{1}_{|\xi|+|\overline{\xi}|\leq
m},~{}~{}\overline{f}^{m}_{i}(t,y,z,\phi)=\overline{f}_{i}(t,y,z,\phi)\mathbf{1}_{|f(t,0,0,0)|+|\overline{f}(t,0,0,0)|\leq
m}.$
Note that $\xi^{m}_{i}$, $\overline{\xi}^{m}_{i}$, $f^{m}_{i}(\cdot,0,0,0)$
and $\overline{f}^{m}_{i}(\cdot,0,0,0)$ are bounded by $m$ and the generators
$f^{m}=(f^{m}_{1},...,f^{m}_{\ell})$ and
$\overline{f}^{m}=(\overline{f}^{m}_{1},...,\overline{f}^{m}_{\ell})$ are both
Lipschitz in $(y,z,\phi)$ with the same Lipschitz constant as $f$ and
$\overline{f}$. It then follows from [37, Theorem 2.4] or [2, Theorem 2.1,
Proposition 2.2] that the following BSDEJs:
$\displaystyle Y_{i,t}^{m}$
$\displaystyle=\xi^{m}_{i}+\int_{t}^{T}f^{m}_{i}(s,Y_{s-}^{m},Z_{i,s}^{m},\Phi_{s}^{m})\operatorname{d}\\!s$
$\displaystyle\qquad\qquad\qquad-\int_{t}^{T}(Z_{i,s}^{m})^{\top}\operatorname{d}\\!W_{s}-\int_{t}^{T}\int_{\mathcal{E}}\Phi_{i,s}^{m}(e)\widetilde{N}(\operatorname{d}\\!s,\operatorname{d}\\!e)~{}\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.},\
i\in\mathcal{M},$
and
$\displaystyle\overline{Y}_{i,t}^{m}$
$\displaystyle=\overline{\xi}^{m}_{i}+\int_{t}^{T}\overline{f}^{m}_{i}(s,\overline{Y}_{s-}^{m},\overline{Z}_{i,s}^{m},\overline{\Phi}_{s}^{m})\operatorname{d}\\!s$
$\displaystyle\qquad\qquad\qquad-\int_{t}^{T}(\overline{Z}_{i,s}^{m})^{\top}\operatorname{d}\\!W_{s}-\int_{t}^{T}\int_{\mathcal{E}}\overline{\Phi}_{i,s}^{m}(e)\widetilde{N}(\operatorname{d}\\!s,\operatorname{d}\\!e)~{}\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.},\
i\in\mathcal{M},$
admit unique solutions $(Y^{m},Z^{m},\Phi^{m})$ and
$(\overline{Y}^{m},\overline{Z}^{m},\overline{\Phi}^{m})$ respectively, such
that
$\displaystyle(Y^{m}_{i},Z^{m}_{i},\Phi^{m}_{i}),~{}(\overline{Y}^{m}_{i},\overline{Z}^{m}_{i},\overline{\Phi}^{m}_{i})\in
S^{2}_{\mathbb{F}}(0,T;{\mathbb{R}})\times
L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}}^{n})\times
L_{\mathcal{P}}^{2,\nu}(0,T;{\mathbb{R}})\ \mbox{ for all}\ i\in\mathcal{M}.$
We temporally suppose that
(2.10) $\displaystyle Y^{m}_{i},~{}~{}\overline{Y}^{m}_{i}\in
S^{\infty}_{\mathbb{F}}(0,T;{\mathbb{R}})\ \mbox{ for all}\ i\in\mathcal{M}.$
Then applying Theorem 2.1 leads to
(2.11) $\displaystyle Y_{i}^{m}\leq\overline{Y}_{i}^{m}\ \mbox{for all}\
i\in\mathcal{M}.$
From [2, Proposition 2.2], we know there is constant $c>0$ independent of $m$
such that
$\displaystyle{\mathbb{E}}\Big{[}\sup_{0\leq t\leq
T}|Y_{t}-Y_{t}^{m}|^{2}\Big{]}\leq
c{\mathbb{E}}\Big{[}|\xi-\xi^{m}|^{2}+\int_{0}^{T}|f(t,Y_{t},Z_{t},\Phi_{t})-f^{m}(t,Y_{t},Z_{t},\Phi_{t})|^{2}\operatorname{d}\\!t\Big{]},$
$\displaystyle{\mathbb{E}}\Big{[}\sup_{0\leq t\leq
T}|\overline{Y}_{t}-\overline{Y}_{t}^{m}|^{2}\Big{]}\leq
c{\mathbb{E}}\Big{[}|\overline{\xi}-\overline{\xi}^{m}|^{2}+\int_{0}^{T}|\overline{f}(t,\overline{Y}_{t},\overline{Z}_{t},\overline{\Phi}_{t})-\overline{f}^{m}(t,\overline{Y}_{t},\overline{Z}_{t},\overline{\Phi}_{t})|^{2}\operatorname{d}\\!t\Big{]}.$
These estimates together with the definitions of
$\xi^{m},\overline{\xi}^{m},f^{m},\overline{f}^{m}$ and the dominated
convergence theorem lead to
$\displaystyle\lim_{m\to\infty}{\mathbb{E}}\Big{[}\sup_{0\leq t\leq
T}|Y_{t}-Y_{t}^{m}|^{2}+\sup_{0\leq t\leq
T}|\overline{Y}_{t}-\overline{Y}_{t}^{m}|^{2}\Big{]}=0.$
Applying the elementary inequalities $(x^{+})^{2}\leq
2(y^{+})^{2}+2(x-y)^{2}$, $(x+y)^{2}\leq 2x^{2}+2y^{2}$ for $x$,
$y\in{\mathbb{R}}$ and (2.11), we have
$\displaystyle\quad{\mathbb{E}}\Big{[}\sup_{0\leq t\leq
T}\sum_{i=1}^{\ell}[(Y_{i,t}-\overline{Y}_{i,t})^{+}]^{2}\Big{]}$
$\displaystyle\leq{\mathbb{E}}\Big{[}2\sup_{0\leq t\leq
T}\sum_{i=1}^{\ell}[(Y_{i,t}^{m}-\overline{Y}_{i,t}^{m})^{+}]^{2}+2\sup_{0\leq
t\leq
T}\sum_{i=1}^{\ell}(Y_{i,t}-Y_{i,t}^{m}+\overline{Y}^{m}_{i,t}-\overline{Y}_{i,t})^{2}\Big{]}$
$\displaystyle={\mathbb{E}}\Big{[}2\sup_{0\leq t\leq
T}\sum_{i=1}^{\ell}(Y_{i,t}-Y_{i,t}^{m}+\overline{Y}^{m}_{i,t}-\overline{Y}_{i,t})^{2}\Big{]}$
$\displaystyle\leq{\mathbb{E}}\Big{[}4\sup_{0\leq t\leq
T}\sum_{i=1}^{\ell}(Y_{i,t}-Y_{i,t}^{m})^{2}+4\sup_{0\leq t\leq
T}\sum_{i=1}^{\ell}(\overline{Y}^{m}_{i,t}-\overline{Y}_{i,t})^{2}\Big{]}.$
Sending $m\to\infty$ in the above, we get the desired result
$Y_{i}\leq\overline{Y}_{i}$ for all $i\in\mathcal{M}$.
It remains to establish (2.10). To this end, let $\beta>0$ be a large constant
to be chosen later. Applying Itô’s formula to $e^{\beta t}(Y^{m}_{i,t})^{2}$,
for each $i\in\mathcal{M}$, yields
$\displaystyle\quad\;e^{\beta
t}(Y^{m}_{i,t})^{2}+{\mathbb{E}}_{t}\int_{t}^{T}e^{\beta
s}\Big{(}\beta(Y^{m}_{i,s})^{2}+|Z^{m}_{i,s}|^{2}+||\Phi^{m}_{i,s}||^{2}_{\nu}\Big{)}\operatorname{d}\\!s$
$\displaystyle={\mathbb{E}}_{t}[e^{\beta
T}(\xi^{m}_{i})^{2}]+{\mathbb{E}}_{t}\int_{t}^{T}2e^{\beta
s}Y^{m}_{i,s-}f^{m}_{i}(s,Y_{s-}^{m},Z_{i,s}^{m},\Phi_{s}^{m})\operatorname{d}\\!s$
$\displaystyle\leq m^{2}e^{\beta T}+{\mathbb{E}}_{t}\int_{t}^{T}2e^{\beta
s}|Y^{m}_{s-}||f^{m}_{i}(s,Y_{s-}^{m},Z_{i,s}^{m},\Phi_{s}^{m})-f^{m}_{i}(s,0,0,0)|\operatorname{d}\\!s$
$\displaystyle\qquad\qquad\;+{\mathbb{E}}_{t}\int_{t}^{T}2e^{\beta
s}|Y^{m}_{s-}||f^{m}_{i}(s,0,0,0)|\operatorname{d}\\!s$ $\displaystyle\leq
m^{2}e^{\beta T}+{\mathbb{E}}_{t}\int_{t}^{T}2e^{\beta
s}|Y^{m}_{s-}|c\Big{(}|Y_{s-}^{m}|+|Z_{i,s}^{m}|+||\Phi_{s}^{m}||_{\nu}\Big{)}\operatorname{d}\\!s$
$\displaystyle\qquad\qquad\;+{\mathbb{E}}_{t}\int_{t}^{T}e^{\beta
s}|Y^{m}_{s-}|^{2}+{\mathbb{E}}_{t}\int_{t}^{T}e^{\beta
s}|f^{m}_{i}(s,0,0,0)|^{2}\operatorname{d}\\!s$ $\displaystyle\leq
m^{2}(1+T)e^{\beta T}+{\mathbb{E}}_{t}\int_{t}^{T}e^{\beta
s}\Big{(}c|Y_{s-}^{m}|^{2}+|Z_{i,s}^{m}|^{2}+\frac{1}{\ell}||\Phi_{s}^{m}||_{\nu}^{2}\Big{)}\operatorname{d}\\!s,$
where the last constant $c$ does not depend on $t$, $\beta$ and $i$. Canceling
the common terms involving $|Z_{i,s}^{m}|^{2}$, we get
$\displaystyle\quad e^{\beta
t}(Y^{m}_{i,t})^{2}+{\mathbb{E}}_{t}\int_{t}^{T}e^{\beta
s}\Big{(}\beta(Y^{m}_{i,s})^{2}+||\Phi_{i,s}^{m}||_{\nu}^{2}\Big{)}\operatorname{d}\\!s$
$\displaystyle\leq m^{2}(1+T)e^{\beta T}+{\mathbb{E}}_{t}\int_{t}^{T}e^{\beta
s}\Big{(}c|Y_{s-}^{m}|^{2}+\frac{1}{\ell}||\Phi_{s}^{m}||_{\nu}^{2}\Big{)}\operatorname{d}\\!s.$
Summing $i$ from $1$ to $\ell$ gives
$\displaystyle\quad e^{\beta
t}|Y^{m}_{t}|^{2}+{\mathbb{E}}_{t}\int_{t}^{T}e^{\beta
s}\Big{(}\beta|Y^{m}_{t}|^{2}+||\Phi^{m}_{t}||^{2}_{\nu}\Big{)}\operatorname{d}\\!s$
$\displaystyle\leq\ell m^{2}(1+T)e^{\beta
T}+{\mathbb{E}}_{t}\int_{t}^{T}e^{\beta
s}\Big{(}c\ell|Y_{s-}^{m}|^{2}+||\Phi_{s}^{m}||_{\nu}\Big{)}\operatorname{d}\\!s$
$\displaystyle=\ell m^{2}(1+T)e^{\beta T}+{\mathbb{E}}_{t}\int_{t}^{T}e^{\beta
s}\Big{(}c\ell|Y_{s}^{m}|^{2}+||\Phi_{s}^{m}||_{\nu}\Big{)}\operatorname{d}\\!s,$
where the last equation is due to the fact that the jumps of $Y$ are
accountable. By setting $\beta=c\ell$ and canceling the common integrals in
the above estimate, we obtain $Y^{m}\in
S^{\infty}_{\mathbb{F}}(0,T;{\mathbb{R}}^{\ell})$. The assertion for
$(\overline{Y}^{m}_{i},\overline{Z}^{m}_{i},\overline{\Phi}^{m}_{i})$ in
(2.10) can be similarly proved. This completes the proof. $\Box$
## 3\. A stochastic LQ control problem with jumps and the related two-
dimensional BSDEJ
### 3.1. Cone-constrained stochastic LQ control with jumps
Consider the following ${\mathbb{R}}$-valued linear stochastic differential
equation (SDE):
(3.1)
$\displaystyle\begin{cases}\operatorname{d}\\!X_{t}=\left[A_{t}X_{t-}+B_{t}^{\top}u_{t}\right]\operatorname{d}\\!t+\left[C_{t}X_{t-}+D_{t}u_{t}\right]^{\top}\operatorname{d}\\!W_{t}\\\
\qquad\qquad\qquad+\int_{\mathcal{E}}\left[E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t}\right]\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\
t\in[0,T],\\\ X_{0}=x,\end{cases}$
where $A,\ B,\ C,\ D$ are all $\mathcal{P}$-measurable processes, and
$E(\cdot),\ F(\cdot)$ are
$\mathcal{P}\otimes\mathcal{B}(\mathcal{E})$-measurable stochastic processes
of suitable size, $x\in{\mathbb{R}}$ is known.
Let $\Pi$ be a given closed cone in ${\mathbb{R}}^{m}$, so if $u\in\Pi$, then
$\lambda u\in\Pi$ for all $\lambda\geq 0$. It is used to represent the
constraint set for controls. The class of admissible controls is defined as
the set
$\displaystyle\mathcal{U}:=\Big{\\{}u\in
L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}}^{m})\;\Big{|}\;u_{t}\in\Pi,~{}\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.}\Big{\\}}.$
If $u\in\mathcal{U}$, then (3.1) admits a unique solution $X$, and we call
$(X,u)$ an admissible pair.
The cone-constrained stochastic LQ problem is stated as follows:
(3.2) $\displaystyle\begin{cases}\mathrm{Minimize}&\ J(x,u)\\\ \mbox{subject
to}&\ (X,u)\mbox{ is admissible for}\ \eqref{state},\end{cases}$
where the cost functional is given as the following quadratic form
(3.3) $\displaystyle J(x,u):=$
$\displaystyle\mathbb{E}\Big{[}\int_{0}^{T}\Big{(}Q_{t}X_{t}^{2}+u_{t}^{\top}R_{t}u_{t}+2X_{t}S_{t}^{\top}u_{t}\Big{)}\operatorname{d}\\!t+GX_{T}^{2}\Big{]}.$
The associated value function is defined as
$\displaystyle V(x):=\inf_{u\in\mathcal{U}}J(x,u).$
Problem (3.2) is said to be solvable (at $x$), if there exists a control
$u^{*}\in\mathcal{U}$ such that
$\displaystyle-\infty<J(x,u^{*})\leq J(x,u),\quad\forall\;u\in\mathcal{U},$
in which case, $u^{*}$ is called an optimal control for problem (3.2), and the
optimal value is
$\displaystyle V(x)=J(x,u^{*}).$
Our aim is to solve problem (3.2).
We put the following assumptions on the coefficients in this section.
###### Assumption 3.1 (Bounded coefficients).
It holds that
$\displaystyle\begin{cases}A\in L^{\infty}_{\mathbb{F}}(0,T;{\mathbb{R}}),\
B\in L_{\mathbb{F}}^{\infty}(0,T;{\mathbb{R}}^{m}),\ C\in
L_{\mathbb{F}}^{\infty}(0,T;{\mathbb{R}}^{n}),\\\ D\in
L_{\mathbb{F}}^{\infty}(0,T;{\mathbb{R}}^{n\times m}),\ E\in
L^{\infty,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}}),\ F\in
L_{\mathcal{P}}^{\infty,\nu}(0,T;{\mathbb{R}}^{m}),\\\ Q\in
L_{\mathbb{F}}^{\infty}(0,T;{\mathbb{R}}_{+}),\ R\in
L_{\mathbb{F}}^{\infty}(0,T;\mathbb{S}^{m}),\ S\in
L_{\mathbb{F}}^{\infty}(0,T;{\mathbb{R}}^{m}),\ G\in
L_{\mathcal{F}_{T}}^{\infty}(\Omega;{\mathbb{R}}_{+}).\end{cases}$
###### Assumption 3.2 (Standard case).
It holds that $\left(\begin{smallmatrix}R&S\\\
S^{\top}&Q\end{smallmatrix}\right)\geq 0$, and there exists a constant
$\delta>0$ such that $R\geq\delta\mathbf{1}_{m}$, where $\mathbf{1}_{m}$
denotes the $m$-dimensional identity matrix.
###### Assumption 3.3 (Singular case).
It holds that $\left(\begin{smallmatrix}R&S\\\
S^{\top}&Q\end{smallmatrix}\right)\geq 0$ and there exists a constant
$\delta>0$ such that $G\geq\delta$ and
$D^{\top}D+\int_{\mathcal{E}}F(e)F(e)^{\top}\nu(\operatorname{d}\\!e)\geq\delta\mathbf{1}_{m}$.
### 3.2. Coupled SRE with jumps
Nowadays, it is well known that solutions to stochastic LQ problems depend
heavily on the solvability of the related SREs. We now introduce the
associated SRE for our problem (3.2).111We will give a heuristic derivation in
Appendix A for the readers’ convenience. See also Dong [7] for a special SRE
with single jump stems from the theory of filtration enlargement.
For
$(\omega,t,v,P_{i},\Lambda,\Gamma_{i})\in\Omega\times[0,T]\times\Pi\times{\mathbb{R}}\times{\mathbb{R}}^{m}\times
L^{\infty,\nu}({\mathbb{R}})$, $i=1,2$ define the following mappings:
$\displaystyle H_{1}(\omega,t,v,P_{1},P_{2},\Lambda,\Gamma_{1},\Gamma_{2})$
$\displaystyle:=v^{\top}(R+P_{1}D^{\top}D)v+2(P_{1}(B+D^{\top}C)+D^{\top}\Lambda+S)^{\top}v$
$\displaystyle\qquad+\int_{\mathcal{E}}\Big{[}(P_{1}+\Gamma_{1})\Big{(}((1+E+F^{\top}v)^{+})^{2}-1\Big{)}-2P_{1}(E+F^{\top}v)$
$\displaystyle\qquad\qquad\quad+(P_{2}+\Gamma_{2})((1+E+F^{\top}v)^{-})^{2}\Big{]}\nu(\operatorname{d}\\!e),$
$\displaystyle H_{2}(\omega,t,v,P_{1},P_{2},\Lambda,\Gamma_{1},\Gamma_{2})$
$\displaystyle:=v^{\top}(R+P_{2}D^{\top}D)v-2(P_{2}(B+D^{\top}C)+D^{\top}\Lambda+S)^{\top}v$
$\displaystyle\qquad+\int_{\mathcal{E}}\Big{[}(P_{2}+\Gamma_{2})\Big{(}((-1-E+F^{\top}v)^{-})^{2}-1\Big{)}+2P_{2}(-E+F^{\top}v)$
$\displaystyle\qquad\qquad\quad+(P_{1}+\Gamma_{1})\Gamma((-1-E+F^{\top}v)^{+})^{2}\Big{]}\nu(\operatorname{d}\\!e),$
and set
(3.4) $\displaystyle
H_{1}^{*}(\omega,t,P_{1},P_{2},\Lambda,\Gamma_{1},\Gamma_{2}):=\inf_{v\in\Pi}H_{1}(\omega,t,v,P_{1},P_{2},\Lambda,\Gamma_{1},\Gamma_{2}),$
(3.5) $\displaystyle
H_{2}^{*}(\omega,t,P_{1},P_{2},\Lambda,\Gamma_{1},\Gamma_{2}):=\inf_{v\in\Pi}H_{2}(\omega,t,v,P_{1},P_{2},\Lambda,\Gamma_{1},\Gamma_{2}).$
The associated SRE for our problem (3.2) is given as follows:
(3.6)
$\displaystyle\begin{cases}\operatorname{d}\\!P_{1,t}=-\Big{[}(2A+C^{\top}C)P_{1,t-}+2C^{\top}\Lambda_{1,t}+Q+H_{1}^{*}(t,P_{1,t-},P_{2,t-},\Lambda_{1,t},\Gamma_{1,t},\Gamma_{2,t})\Big{]}\operatorname{d}\\!t\\\
\hfill+\Lambda_{1,t}^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\Gamma_{1,t}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\qquad\quad\\\
\operatorname{d}\\!P_{2,t}=-\Big{[}(2A+C^{\top}C)P_{2,t-}+2C^{\top}\Lambda_{2,t}+Q+H_{2}^{*}(t,P_{1,t-},P_{2,t-},\Lambda_{2,t},\Gamma_{1,t},\Gamma_{2,t})\Big{]}\operatorname{d}\\!t\\\
\hfill+\Lambda_{2,t}^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\Gamma_{2,t}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\qquad\quad\\\
P_{1,T}=G,\ P_{2,T}=G,\\\ P_{1,t}\geq 0,\ P_{1,t-}+\Gamma_{1,t}\geq 0,\
P_{2,t}\geq 0,\ P_{2,t-}+\Gamma_{2,t}\geq 0.\end{cases}$
This is a new two-dimensional coupled nonlinear BSDEJ.
###### Remark 3.1.
Hu and Zhou [16] studied a cone-constrained LQ problem without jumps; the
associated SREs [16, Eq. (3.5) and (3.6)] are decoupled, so that one can solve
$P_{1}$ and $P_{2}$ separately. As is well-known $P_{1}$ and $P_{2}$
correspond to the optimal value with positive and negative initial state. When
there is no jump in the model, the optimal state process does not change sign,
so that only one of $P_{1}$ and $P_{2}$ is involved. Therefore, they are
decoupled.
Things become notably different in models with jumps. Because of jumps, the
sign of the optimal state process may switch between positive and negative
values, so $P_{1}$ and $P_{2}$ are coupled together and one cannot treat them
separately. So our SRE (3.6) is actually a system of coupled BSDEJs whose
solvability is far from trivial compared to the decoupled BSDEJs in [16, Eq.
(3.5) and (3.6)].
If all the coefficients in Assumption 3.1 are predictable with respect to the
Brownian filtration, then $\Gamma_{1}=\Gamma_{2}=0$ and the SRE becomes a two-
dimensional coupled BSDE without jumps. Even without jumps, the BSDE is still
new and cannot be covered by existing results on multi-dimensional BSDEs; see,
e.g., Fan, Hu and Tang [11], Hu and Tang [15].
###### Remark 3.2.
If $\Pi$ is symmetric, namely, $-v\in\Pi$ whenever $v\in\Pi$, then
$H_{1}^{*}=H_{2}^{*}$ and (3.6) will degenerate to one equation since
$(P_{1},\Lambda_{1},\Gamma_{1})=(P_{2},\Lambda_{2},\Gamma_{2})$. In
particular, if there is no control constraint, that is,
$\Pi={\mathbb{R}}^{m}$, then both $H_{1}^{*}$ and $H_{2}^{*}$ are equal to
$\displaystyle(P+\Gamma)E^{2}+2\Gamma E$
$\displaystyle+\Big{(}P(B+D^{\top}C)+D^{\top}\Lambda+S+\int_{\mathcal{E}}((P+\Gamma)E+\Gamma)F\nu(\operatorname{d}\\!e)\Big{)}^{\top}$
$\displaystyle\qquad\times\Big{(}R+PD^{\top}D+\int_{\mathcal{E}}(P+\Gamma)FF^{\top}\nu(\operatorname{d}\\!e)\Big{)}^{-1}$
$\displaystyle\qquad\times\Big{(}P(B+D^{\top}C)+D^{\top}\Lambda+S+\int_{\mathcal{E}}((P+\Gamma)E+\Gamma)F\nu(\operatorname{d}\\!e)\Big{)}.$
Under $\Pi={\mathbb{R}}^{m}$, Zhang, Dong and Meng [39] addressed the
solvability of the matrix-valued SREJ under the assumption
$R\geq\delta\mathbf{1}_{m}$ and $S\equiv 0$. By contrast, we will solve the
BSDEJ (3.6) in both standard and singular cases for general cone $\Pi$.
###### Definition 3.1.
A stochastic process
$(P_{1},\Lambda_{1},\Gamma_{1},P_{2},\Lambda_{2},\Gamma_{2})$ is called a
solution to the BSDEJ (3.6) if it satisfies (3.6), and
$(P_{i},\Lambda_{i},\Gamma_{i})\in
S^{\infty}_{\mathbb{F}}(0,T;{\mathbb{R}})\times
L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}}^{m})\times
L^{\infty,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}})$, $i=1,2$. The solution is
called nonnegative if $P_{i}\geq 0$, and called uniformly positive if
$P_{i}\geq c$ for some deterministic constant $c>0$, $i=1,2$.
### 3.3. Existence of solution to the BSDEJ (3.6)
Dong [7] constructed a solution to a SRE with single jump using two recursive
systems of BSDEs driven only by Brownian motions. His decomposition approach
is tailor made in the filtration enlargement framework, hence fails in the
Poisson random measure model which accommodates probably accountable jumps.
Czichowsky and Schweizer [5] characterized the optimal value process of a
cone-constrained mean-variance problem in terms of a coupled system of BSDEs
[5, Eq.(4.18)] in a semimartingale model. They claimed in [5, Remark 4.8] that
_“Due to the coupling term coming from $\mathfrak{h}$, the BSDE system (4.18)
is very complicated. It has a nonlinear non-Lipschitz generator plus a
generator with jumps, so that finding a solution by general BSDE techniques
seems a formidable challenge”_. We now respond to this formidable challenge in
the Wiener-Poisson world by providing a proof of the existence of solution to
(3.6) by pure BSDE techniques.
###### Theorem 3.1 (Existence in Standard case).
Suppose Assumptions 3.1, 3.2 hold, then the BSDEJ (3.6) admits a nonnegative
solution $(P_{1},\Lambda_{1},\Gamma_{1},P_{2},\Lambda_{2},\Gamma_{2})$.
Proof: For $k=1,2,...$, define maps
(3.7) $\displaystyle
H_{1}^{k}(\omega,t,P_{1},P_{2},\Lambda_{1},\Gamma_{1},\Gamma_{2}):=\inf_{v\in\Pi,|v|\leq
k}H_{1}(\omega,t,v,P_{1},P_{2},\Lambda_{1},\Gamma_{1},\Gamma_{2}),$ (3.8)
$\displaystyle
H_{2}^{k}(\omega,t,P_{1},P_{2},\Lambda_{2},\Gamma_{1},\Gamma_{2}):=\inf_{v\in\Pi,|v|\leq
k}H_{2}(\omega,t,v,P_{1},P_{2},\Lambda_{2},\Gamma_{1},\Gamma_{2}).$
Then they are uniformly Lipschitz in
$(P_{1},\Lambda_{1},\Gamma_{1},P_{2},\Lambda_{2},\Gamma_{2})$ and decreasingly
approach to
$H_{1}^{*}(\omega,t,P_{1},\Lambda_{1},\Gamma_{1},P_{2},\Gamma_{2})$ and
$H_{2}^{*}(\omega,t,P_{2},\Lambda_{2},\Gamma_{2},P_{1},\Gamma_{1})$
respectively as $k$ goes to infinity.
For each $k$, the following BSDE
(3.9)
$\displaystyle\begin{cases}\operatorname{d}\\!P_{1,t}^{k}=-\Big{[}(2A+C^{\top}C)P_{1,t-}^{k}+2C^{\top}\Lambda_{1,t}^{k}+Q+H_{1}^{k}(t,P_{1,t-}^{k},P_{2,t-}^{k},\Lambda_{1,t}^{k},\Gamma_{1,t}^{k},\Gamma_{2,t}^{k})\Big{]}\operatorname{d}\\!t\\\
\hfill+(\Lambda_{1,t}^{k})^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\Gamma_{1,t}^{k}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\qquad\quad\\\
\operatorname{d}\\!P_{2,t}^{k}=-\Big{[}(2A+C^{\top}C)P_{2,t-}^{k}+2C^{\top}\Lambda_{2,t}^{k}+Q+H_{2}^{k}(t,P_{1,t-}^{k},P_{2,t-}^{k},\Lambda_{2,t}^{k},\Gamma_{1,t}^{k},\Gamma_{2,t}^{k})\Big{]}\operatorname{d}\\!t\\\
\hfill+(\Lambda_{2,t}^{k})^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\Gamma_{2,t}^{k}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\qquad\quad\\\
P_{1,T}^{k}=G,\ P_{2,T}^{k}=G,\end{cases}$
is a two-dimensional BSDEJ with a Lipschitz generator, so by [37, Lemma 2.4],
it admits a unique solution
$(P_{1}^{k},\Lambda_{1}^{k},\Gamma_{1}^{k},P_{2}^{k},\Lambda_{2}^{k},\Gamma_{2}^{k})$
such that
$(P_{i}^{k},\Lambda_{i}^{k},\Gamma_{i}^{k})\in
S^{2}_{\mathbb{F}}(0,T;{\mathbb{R}})\times
L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}}^{n})\times
L^{2,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}}),\ i=1,2.$
From the definition of $H_{1}^{k}$, we have
$\displaystyle\qquad
H_{1}^{k}(t,P_{1},P_{2},\Lambda_{1},\Gamma_{1},\Gamma_{2})-H_{1}^{k}(t,P_{1},P^{\prime}_{2},\Lambda_{1},\Gamma_{1},\Gamma^{\prime}_{2})$
$\displaystyle\leq\sup_{v\in\Pi,|v|\leq
k}\int_{\mathcal{E}}((1+E+F^{\top}v)^{-})^{2}(P_{2}+\Gamma_{2}(e)-P^{\prime}_{2}-\Gamma^{\prime}_{2}(e))\nu(\operatorname{d}\\!e)$
$\displaystyle\leq
c_{k}\int_{\mathcal{E}}(P_{2}+\Gamma_{2}(e)-P^{\prime}_{2}-\Gamma^{\prime}_{2}(e))^{+}\nu(\operatorname{d}\\!e),$
and
$\displaystyle\qquad
H_{1}^{k}(t,P_{1},P_{2},\Lambda_{1},\Gamma_{1},\Gamma_{2})-H_{1}^{k}(t,P_{1},P_{2},\Lambda_{1},\Gamma^{\prime}_{1},\Gamma_{2})$
$\displaystyle\leq\sup_{v\in\Pi,|v|\leq
k}\int_{\mathcal{E}}(\Gamma_{1}(e)-\Gamma^{\prime}_{1}(e))\Big{(}((1+E+F^{\top}v)^{+})^{2}-1\Big{)}\nu(\operatorname{d}\\!e)$
$\displaystyle\leq\sup_{v\in\Pi,|v|\leq
k}\int_{\mathcal{E}}(\Gamma_{1}(e)-\Gamma^{\prime}_{1}(e))((1+E+F^{\top}v)^{+})^{2}\nu(\operatorname{d}\\!e)+\int_{\mathcal{E}}|\Gamma_{1}(e)-\Gamma^{\prime}_{1}(e)|\nu(\operatorname{d}\\!e)$
$\displaystyle\leq
c_{k}\int_{\mathcal{E}}(\Gamma_{1}(e)-\Gamma^{\prime}_{1}(e))^{+}\nu(\operatorname{d}\\!e)+\int_{\mathcal{E}}|\Gamma_{1}(e)-\Gamma^{\prime}_{1}(e)|\nu(\operatorname{d}\\!e),$
where $c_{k}<\infty$ is defined as
$\displaystyle c_{k}$ $\displaystyle=\operatorname*{ess\;sup}_{v\in\Pi,|v|\leq
k}|1+E+F^{\top}v|^{2},~{}\nu(\operatorname{d}\\!e)\textrm{-a.e..}$
Similar estimates for $H_{2}^{k}$ can be established. Hence according to
Theorem 2.2, $P^{k}_{i}$ is decreasing in $k$, for $i=1,2$.
Next, we show that the sequence $\\{P_{i}^{k}\\}_{k=1,2,...}$ is nonnegative
and uniformly bounded from above, for $i=1,2$.
From Assumption 3.1, there exists a constant $c>0$ such that
$\displaystyle
2A+C^{\top}C+\int_{\mathcal{E}}E(e)^{2}\nu(\operatorname{d}\\!e)\leq c,\ Q\leq
c,\ G\leq c.$
It is easy to check that
$(\overline{P}_{1,t},\overline{\Lambda}_{1,t},\overline{\Gamma}_{1,t})=(\overline{P}_{2,t},\overline{\Lambda}_{2,t},\overline{\Gamma}_{2,t})=((c+1)e^{c(T-t)}-1,0,0)$
satisfies the following two-dimensional BSDEJ
(3.10)
$\displaystyle\begin{cases}\operatorname{d}\\!\overline{P}_{1}=-\Big{[}c\overline{P}_{1}+C^{\top}\Lambda_{1}+c+\int_{\mathcal{E}}\Big{(}\overline{\Gamma}_{1}\big{(}((1+E)^{+})^{2})-1\big{)}+\overline{\Gamma}_{2}((1+E)^{-})^{2}\Big{)}\nu(\operatorname{d}\\!e)\Big{]}\operatorname{d}\\!t\\\
\hfill+\overline{\Lambda}_{1}^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\overline{\Gamma}_{1}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\qquad\quad\\\
\operatorname{d}\\!\overline{P}_{2}=-\Big{[}c\overline{P}_{2}+C^{\top}\Lambda_{2}+c+\int_{\mathcal{E}}\Big{(}\overline{\Gamma}_{2}\big{(}((1+E)^{+})^{2})-1\big{)}+\overline{\Gamma}_{2}((1+E)^{-})^{2}\Big{)}\nu(\operatorname{d}\\!e)\Big{]}\operatorname{d}\\!t\\\
\hfill+\overline{\Lambda}_{2}^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\overline{\Gamma}_{2}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\qquad\quad\\\
\overline{P}_{1,T}=c,~{}\overline{P}_{2,T}=c.\end{cases}$
By the definition of $H_{1}^{k}$, we have
$\displaystyle
H_{1}^{k}(t,\overline{P}_{1},\overline{P}_{2},\overline{\Lambda}_{1},\overline{\Gamma}_{1},\overline{\Gamma}_{2})$
$\displaystyle\leq
H_{1}(t,0,\overline{P}_{1},\overline{P}_{2},\overline{\Lambda}_{1},\overline{\Gamma}_{1},\overline{\Gamma}_{2})$
$\displaystyle=\int_{\mathcal{E}}\Big{(}\overline{P}_{1}E^{2}+\overline{\Gamma}_{1}\big{(}((1+E)^{+})^{2})-1\big{)}+\overline{\Gamma}_{1}((1+E)^{-})^{2}\Big{)}\nu(\operatorname{d}\\!e),$
so
$\displaystyle\quad(2A+C^{\top}C)\overline{P}_{1}+2C^{\top}\overline{\Lambda}_{1}+Q+H_{1}^{k}(t,\overline{P}_{1},\overline{P}_{2},\overline{\Lambda}_{1},\overline{\Gamma}_{1},\overline{\Gamma}_{2})$
$\displaystyle\leq
c\overline{P}_{1}+C^{\top}\overline{\Lambda}_{1}+c+\int_{\mathcal{E}}\Big{(}\overline{\Gamma}_{1}\big{(}((1+E)^{+})^{2})-1\big{)}+\overline{\Gamma}_{1}((1+E)^{-})^{2}\Big{)}\nu(\operatorname{d}\\!e).$
Similarly, we have
$\displaystyle\quad(2A+C^{\top}C)\overline{P}_{2}+2C^{\top}\overline{\Lambda}_{2}+Q+H_{2}^{k}(t,\overline{P}_{1},\overline{P}_{2},\overline{\Lambda}_{2},\overline{\Gamma}_{1},\overline{\Gamma}_{2})$
$\displaystyle\leq
c\overline{P}_{2}+C^{\top}\Lambda_{2}+c+\int_{\mathcal{E}}\Big{(}\overline{\Gamma}_{2}\big{(}((1+E)^{+})^{2})-1\big{)}+\overline{\Gamma}_{2}((1+E)^{-})^{2}\Big{)}\nu(\operatorname{d}\\!e).$
Keeping the above two inequalities in mind, applying Theorem 2.2 to BSDEJs
(3.9) and (3.10), we have for $i=1,2$, $k=1,2,...$
(3.11) $\displaystyle P_{i,t}^{k}\leq\overline{P}_{i,t}\leq M,$
where $M:=(c+1)e^{cT}-1$.
On the other hand, notice that
$(\underline{P}_{1,t},\underline{\Lambda}_{1,t},\underline{\Gamma}_{1,t})=(\underline{P}_{2,t},\underline{\Lambda}_{2,t},\underline{\Gamma}_{2,t}):=(0,0,0)$
satisfies
$\displaystyle\begin{cases}\operatorname{d}\\!\underline{P}=\underline{\Lambda}^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\underline{\Gamma}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\\\
\underline{P}_{T}=0,\end{cases}$
and
$\displaystyle\quad(2A+C^{\top}C)\underline{P}_{1}+2C^{\top}\underline{\Lambda}_{1}+Q+H_{1}^{k}(t,\underline{P}_{1},\underline{P}_{2},\underline{\Lambda}_{1},\underline{\Gamma}_{1},\underline{\Gamma}_{2})$
$\displaystyle\geq
Q+\inf_{v\in{\mathbb{R}}^{m}}(v^{\top}Rv+2S^{\top}v)=Q-S^{\top}R^{-1}S\geq 0,$
thanks to Assumption 3.2. Hence, by Theorem 2.2 again,
(3.12) $\displaystyle P^{k}_{i,t}\geq\underline{P}_{t}=0,\ i=1,2,~{}k=1,2,...$
Notice, for $i=1,2$,
$\displaystyle{\mathbb{E}}\int_{0}^{T}\int_{\mathcal{E}}\mathbf{1}_{\\{P_{i,t-}^{k}+\Gamma_{i,t}^{k}(e)<0\\}}\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
$\displaystyle={\mathbb{E}}\int_{0}^{T}\int_{\mathcal{E}}\mathbf{1}_{\\{P_{i,t-}^{k}+\Gamma_{i,t}^{k}(e)<0\\}}N(\operatorname{d}\\!t,\operatorname{d}\\!e)$
$\displaystyle={\mathbb{E}}\Big{[}\sum_{n\in\mathbb{N},T_{n}\leq
T}\mathbf{1}_{\\{P_{i,T_{n}-}^{k}+\Gamma_{i,T_{n}}^{k}(-\Delta
U_{T_{n}})<0\\}}\Big{]}$
$\displaystyle={\mathbb{E}}\Big{[}\sum_{n\in\mathbb{N},T_{n}\leq
T}\mathbf{1}_{\\{P_{i,T_{n}}^{k}<0\\}}\Big{]}=0,$
where
$U_{t}=\int_{0}^{t}\int_{\mathcal{E}}eN(\operatorname{d}\\!t,\operatorname{d}\\!e)$
and $\Delta U_{T_{n}}=U_{T_{n}}-U_{T_{n}-}$, hence,
(3.13) $\displaystyle P_{i,t-}^{k}+\Gamma_{i,t}^{k}\geq 0.$
Similarly, we can establish
(3.14) $\displaystyle P_{i,t-}^{k}+\Gamma_{i,t}^{k}\leq M.$
Now we obtain
$-M\leq-P_{i,t-}^{k}\leq\Gamma_{i,t}^{k}\leq M-P_{i,t-}^{k}\leq M.$
Hence, $\Gamma_{i}^{k}$, $k=1,2,\cdots,$ are uniformly bounded by $M$, and
thus belong to $L^{\infty,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}})$.
Since $P_{i}^{k}$ is decreasing w.r.t. $k$, we can define
$P_{i,t}:=\lim_{k\to\infty}P_{i,t}^{k},\ i=1,2$. Combining (3.11) and (3.12),
it follows
$0\leq P_{i,t}\leq M,\ i=1,2,~{}t\in[0,T].$
Applying Itô’s formula to $(P_{1,t}^{k})^{2}$, we deduce that
$\displaystyle\begin{cases}\operatorname{d}\\!\;(P_{1,t}^{k})^{2}=\Big{\\{}-2P_{1}^{k}\Big{[}(2A+C^{\top}C)P_{1,t-}^{k}+2C^{\top}\Lambda_{1,t}^{k}+Q+H_{1}^{k}(t,P_{1,t-}^{k},P_{2,t-}^{k},\Lambda_{1,t}^{k},\Gamma_{1,t}^{k},\Gamma_{2,t}^{k})\Big{]}\\\
\qquad\qquad\qquad+|\Lambda_{1}^{k}|^{2}+\int_{\mathcal{E}}\Gamma_{1}^{k}(e)^{2}\nu(\operatorname{d}\\!e)\Big{\\}}\operatorname{d}\\!t\\\
\hfill+2P_{1}^{k}(\Lambda_{1}^{k})^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}[(P_{1,t-}^{k}+\Gamma_{1,t}^{k}(e))^{2}-(P_{1,t-}^{k})^{2}]\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\qquad\quad\\\
(P_{1,T}^{k})^{2}=G^{2}.\end{cases}$
Since $0\leq P_{i}^{k},P_{i}^{k}+\Gamma_{i}^{k}\leq M,i=1,2$, and
$H_{1}^{k}\leq\int_{\mathcal{E}}\Big{[}(P_{1}+\Gamma_{1})\Big{(}((1+E)^{+})^{2}-1\Big{)}-2P_{1}E+(P_{2}+\Gamma_{2})((1+E)^{-})^{2}\Big{]}\nu(\operatorname{d}\\!e)\leq
c,$
by taking expectation on both sides in above and integrating over $[0,T]$, we
have
(3.15)
$\displaystyle\quad(P_{1,0}^{k})^{2}+\frac{1}{2}{\mathbb{E}}\int_{0}^{T}|\Lambda_{1}^{k}|^{2}\operatorname{d}\\!s+{\mathbb{E}}\int_{0}^{T}\int_{\mathcal{E}}\Gamma_{1}^{k}(e)^{2}\nu(\operatorname{d}\\!e)\operatorname{d}\\!s\leq
c,$
where $c>0$ is a constant independent of $k$. Therefore, the sequence
$(\Lambda_{1}^{k},\Gamma_{1}^{k})$, $k=1,2,\cdots,$ is bounded in
$L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}}^{n})\times
L^{2,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}})$, thus we can extract a subsequence
(which is still denoted by $(\Lambda_{1}^{k},\Gamma_{1}^{k})$) converging in
the weak sense to some $(\Lambda_{1},\Gamma_{1})\in
L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}}^{n})\times
L^{2,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}})$. Similar considerations applying to
$(P_{2,t}^{k})^{2}$ yield some $(\Lambda_{2},\Gamma_{2})\in
L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}}^{n})\times
L^{2,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}})$ which is the weak limit of
$(\Lambda_{2}^{k},\Gamma_{2}^{k})$.
Following Kobylanski’s argument [20, Proposition 2.4] (see also Antonelli and
Mancini [1, Theorem 1], Kohlmann and Tang [19, Theorem 2.1]), we establish in
Appendix B the following strong convergence:
###### Lemma 3.1.
It holds that
(3.16)
$\displaystyle\lim_{k\to\infty}{\mathbb{E}}\int_{0}^{T}|\Lambda_{i}^{k}-\Lambda_{i}|^{2}\operatorname{d}\\!t=0,\
\lim_{k\to\infty}{\mathbb{E}}\int_{0}^{T}\int_{\mathcal{E}}|\Gamma_{i}^{k}-\Gamma_{i}|^{2}\nu(\operatorname{d}\\!e)\operatorname{d}\\!t=0,\
i=1,2.$
Furthermore, $(P_{1},\Lambda_{1},\Gamma_{1},P_{2},\Lambda_{2},\Gamma_{2})$ is
a nonnegative solution to the BSDEJ (3.6).
This completes the proof. $\Box$
###### Theorem 3.2 (Existence in Singular case).
Suppose Assumptions 3.1, 3.3 hold, then the BSDEJ (3.6) admits a uniformly
positive solution
$(P_{1},\Lambda_{1},\Gamma_{1},P_{2},\Lambda_{2},\Gamma_{2})$.
Proof: Similar to the proof of Theorem 3.1, one can show the existence of a
nonnegative solution
$(P_{1},\Lambda_{1},\Gamma_{1},P_{2},\Lambda_{2},\Gamma_{2})$ to the BSDEJ
(3.6), so we omit the details. We only give a sketch on how to find a
uniformly positive lower bound for such a solution.
Under Assumptions 3.1, 3.3, there exists constant $c_{2}>0$, such that
$2A+C^{\top}C+\int_{\mathcal{E}}E^{2}\nu(\operatorname{d}\\!e)-\delta^{-1}\Big{|}B+D^{\top}C\pm\int_{\mathcal{E}}EF\nu(\operatorname{d}\\!e)\Big{|}^{2}\geq-
c_{2},$
where $\delta$ is the constant in Assumption 3.3. Notice
$(\underline{P}_{1,t},\underline{\Lambda}_{1,t},\underline{\Gamma}_{1,t})=(\underline{P}_{2,t},\underline{\Lambda}_{2,t},\underline{\Gamma}_{2,t}):=(\delta
e^{-c_{2}(T-t)},0,0)$ solves the following BSDEJ
(3.17)
$\displaystyle\begin{cases}\operatorname{d}\\!\underline{P}=-(-c_{2}\underline{P}+C^{\top}\underline{\Lambda})\operatorname{d}\\!t+\underline{\Lambda}^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\underline{\Gamma}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\\\
\underline{P}_{T}=\delta.\end{cases}$
We have the following estimates,
$\displaystyle\qquad
H_{1}^{k}(t,\underline{P}_{1},\underline{P}_{2},\underline{\Lambda}_{1},\underline{\Gamma}_{1},\underline{\Gamma}_{2})$
$\displaystyle\geq\inf_{v\in{\mathbb{R}}^{m}}H_{1}(t,v,\underline{P}_{1},\underline{P}_{2},\underline{\Lambda}_{1},\underline{\Gamma}_{1},\underline{\Gamma}_{2})$
$\displaystyle=\inf_{v\in{\mathbb{R}}^{m}}H_{1}(t,v,\underline{P}_{1},\underline{P}_{2},0,0,0)$
$\displaystyle\geq\inf_{v\in{\mathbb{R}}^{m}}\Big{[}v^{\top}Rv+2S^{\top}v\Big{]}+\underline{P}_{1}\int_{\mathcal{E}}E^{2}\nu(\operatorname{d}\\!e)$
$\displaystyle\qquad+\underline{P}_{1}\inf_{v\in{\mathbb{R}}^{m}}\Big{[}v^{\top}(D^{\top}D+\int_{\mathcal{E}}F(e)F(e)^{\top}\nu(\operatorname{d}\\!e))v+2\Big{(}B+D^{\top}C+\int_{\mathcal{E}}EF\nu(\operatorname{d}\\!e)\Big{)}^{\top}v\Big{]}$
$\displaystyle\geq-Q+\underline{P}_{1}\Big{[}\int_{\mathcal{E}}E^{2}\nu(\operatorname{d}\\!e)-\delta^{-1}\Big{|}B+D^{\top}C+\int_{\mathcal{E}}EF\nu(\operatorname{d}\\!e)\Big{|}^{2}\Big{]},$
where we used $\underline{\Lambda}_{1}=0$,
$\underline{\Gamma}_{1}=\underline{\Gamma}_{2}=0$ in the equality,
$\underline{P}_{1}=\underline{P}_{2}>0$ in the second inequality, and
$\left(\begin{smallmatrix}R&S\\\ S^{\top}&Q\end{smallmatrix}\right)\geq
0,\quad
D^{\top}D+\int_{\mathcal{E}}F(e)F(e)^{\top}\nu(\operatorname{d}\\!e)\geq\delta\mathbf{1}_{m}$
in the last inequality. Similar result also holds for
$H_{2}^{k}(t,\underline{P}_{1},\underline{P}_{2},\underline{\Lambda}_{2},\underline{\Gamma}_{1},\underline{\Gamma}_{2})$.
Applying Theorem 2.2 to the BSDEJs (3.9) and (3.17), we get, for $i=1,2$,
(3.18) $\displaystyle P_{i,t}^{k}\geq\underline{P}_{i,t}=\delta
e^{-c_{2}(T-t)}\geq\delta e^{-c_{2}T},\ t\in[0,T],$
which leads to the desired lower bound. $\Box$
### 3.4. Solution to the LQ problem (3.2)
In this subsection we will present an explicit solution to the LQ problem
(3.2) in terms of solutions to the BSDEJ (3.6).
For $P_{i}>0$, $\Lambda_{i}\in{\mathbb{R}}^{n}$, $\Gamma_{i}\in L^{2,\nu},\
i=1,2$, define
$\displaystyle\hat{v}_{1}(\omega,t,P_{1},P_{2},\Lambda_{1},\Gamma_{1},\Gamma_{2})=\operatorname*{argmin}_{v\in\Pi}H_{1}(\omega,t,v,P_{1},P_{2},\Lambda_{1},\Gamma_{1},\Gamma_{2}),$
(3.19)
$\displaystyle\hat{v}_{2}(\omega,t,P_{1},P_{2},\Lambda_{2},\Gamma_{1},\Gamma_{2})=\operatorname*{argmin}_{v\in\Pi}H_{1}(\omega,t,v,P_{1},P_{2},\Lambda_{2},\Gamma_{1},\Gamma_{2}).$
###### Theorem 3.3.
Let $(P_{i},\Lambda_{i},\Gamma_{i})\in
S^{\infty}_{\mathbb{F}}(0,T;{\mathbb{R}})\times
L^{2}_{\mathbb{F}}(0,T;{\mathbb{R}}^{m})\times
L^{\infty,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}}),\ i=1,2$, be a nonnegative (in
Standard case), or uniformly positive (in Singular case) solution to the BSDEJ
(3.6). Then the state feedback control given by
$\displaystyle
u^{*}(t,X)=\hat{v}_{1}(\omega,t,P_{1,t-},P_{2,t-},\Lambda_{1,t},\Gamma_{1,t},\Gamma_{2,t})X_{t-}^{+}+\hat{v}_{2}(\omega,t,P_{1,t-},P_{2,t-},\Lambda_{1,t},\Gamma_{1,t},\Gamma_{2,t})X_{t-}^{-},$
is optimal for the LQ problem (3.2). Moreover, the optimal value is
$\displaystyle V(x)=P_{1,0}(x^{+})^{2}+P_{2,0}(x^{-})^{2}.$
The proof of Theorem 3.3 is standard, and thus omitted here; please see [16,
Theorem 5.2] or [39, Theorem 5.2] for the standard verification argument.
As a byproduct of Theorem 3.3, we have the following uniqueness result.
###### Theorem 3.4.
Suppose Assumptions 3.1 and 3.2 (resp. Assumptions 3.1 and 3.3) hold, then the
BSDEJ (3.6) admits at most one nonnegative (resp. uniformly positive)
solution.
It seems a challenging task to establish this result by pure BSDE techniques.
## 4\. Application to mean-variance portfolio selection problem
Consider a financial market consisting of a risk-free asset (the money market
instrument or bond) whose price is $S_{0}$ and $m$ risky securities (the
stocks) whose prices are $S_{1},\ldots,S_{m}$. And assume $m\leq n$, i.e. the
number of risky securities is no more than the dimension of the Brownian
motion. The asset prices $S_{k}$, $k=0,1,\ldots,m,$ are driven by stochastic
differential equations (SDEs):
$\displaystyle\begin{cases}\operatorname{d}\\!S_{0,t}=r_{t}S_{0,t}\operatorname{d}\\!t,\\\
S_{0,0}=s_{0},\end{cases}$
and
$\displaystyle\begin{cases}\operatorname{d}\\!S_{k,t}=S_{k,t}\Big{(}(\mu_{k,t}+r_{t})\operatorname{d}\\!t+\sum\limits_{j=1}^{n}\sigma_{kj,t}\operatorname{d}\\!W_{j,t}+\int_{\mathcal{E}}F_{k,t}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e)\Big{)},\\\
S_{k,0}=s_{k},\end{cases}$
where, for every $k=1,\ldots,m$, $r$ is the interest rate process, $\mu_{k}$,
$\sigma_{k}:=(\sigma_{k1},\ldots,\sigma_{kn})$ and $F_{k}$ are the mean excess
return rate process and volatility rate process of the $k$-th risky security.
Define the vectors $\mu=(\mu_{1},\ldots,\mu_{m})^{\top}$,
$F=(F_{1},\ldots,F_{m})^{\top}$ and matrix
$\displaystyle\sigma=\left(\begin{array}[]{c}\sigma_{1}\\\ \vdots\\\
\sigma_{m}\\\ \end{array}\right)\equiv(\sigma_{kj})_{m\times n},\ \text{for}\
\text{each}\ i\in\mathcal{M}.$
We shall assume, in this section,
###### Assumption 4.1.
The interest rate $r$ is a bounded deterministic measurable function of $t$,
$\mu\in L_{\mathbb{F}}^{\infty}(0,T;{\mathbb{R}}^{m}),\ \sigma\in
L_{\mathbb{F}}^{\infty}(0,T;{\mathbb{R}}^{m\times n}),\ F\in
L_{\mathcal{P}}^{\nu,\infty}(0,T;{\mathbb{R}}^{m}),$
and there exists a constant $\delta>0$ such that
$\sigma\sigma^{\top}+\int_{\mathcal{E}}F(e)F(e)^{\top}\nu(\operatorname{d}\\!e)\geq\delta\mathbf{1}_{m}$
for all $t\in[0,T]$.
A small investor, whose actions cannot affect the asset prices, will decide at
every time $t\in[0,T]$ the amount $\pi_{j,t}$ of his wealth to invest in the
$j$-th risky asset, $j=1,\ldots,m$. The vector process
$\pi:=(\pi_{1},\ldots,\pi_{m})^{\top}$ is called a portfolio of the investor.
Then the investor’s self-financing wealth process $X$ corresponding to a
portfolio $\pi$ is the unique strong solution of the SDE:
(4.1)
$\displaystyle\begin{cases}\operatorname{d}\\!X_{t}=[r_{t}X_{t-}+\pi_{t}^{\top}\mu_{t}]\operatorname{d}\\!t+\pi_{t}^{\top}\sigma_{t}\operatorname{d}\\!W_{t}+\int_{\mathcal{E}}\pi_{t}^{\top}F_{t}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\\\
X_{0}=x.\end{cases}$
The admissible portfolio set is defined as
$\displaystyle\mathcal{U}=\Big{\\{}\pi\in
L^{2}_{\mathbb{F}}(0,T;\mathbb{R}^{m})\;\Big{|}\;\pi_{t}\in\Pi~{}\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.}\Big{\\}},$
where $\Pi\in{\mathbb{R}}^{m}$ is a given closed convex cone. For instance,
$\Pi={\mathbb{R}}^{m}$ means there is no trading constraint; while
$\Pi={\mathbb{R}}_{+}^{m}$ means shorting is not allowed in the market. For
any $\pi\in\mathcal{U}$, the SDE (4.1) has a unique strong solution. Different
from the previous sections, in this section we request the constraint set
$\Pi$ to be convex in order to apply the dual approach below.
For a given expectation level $z\geq
xe^{\int_{0}^{T}r_{s}\operatorname{d}\\!s}$, the investor’s mean-variance
problem is to
$\displaystyle\mathrm{Minimize}$
$\displaystyle\quad\mathrm{Var}(X_{T})\equiv{\mathbb{E}}[X_{T}^{2}-z^{2}],$
(4.2) $\displaystyle\mathrm{s.t.}$
$\displaystyle\quad\begin{cases}{\mathbb{E}}[X_{T}]=z,\\\
\pi\in\mathcal{U}.\end{cases}$
###### Remark 4.1.
Lim [23] studied a mean-variance problem with jumps without portfolio
constraints, i.e. $\Pi={\mathbb{R}}^{m}$. In his model, all the coefficients
in (4.1) are assumed to be predictable with respect to the Brownian motion
filtration, so no jump term has entered into his SRE, which is exactly the
same one as in the model without jumps.
We shall say that the mean-variance problem (4.2) is feasible for a given
level $z\geq xe^{\int_{0}^{T}r_{s}\operatorname{d}\\!s}$ if there is a
portfolio $\pi\in\mathcal{U}$ which satisfies the target constraint
${\mathbb{E}}[X_{T}]=z$. An optimal portfolio to (4.2) is called an efficient
portfolio corresponding to $z$ and the corresponding
$(\sqrt{\mathrm{Var}(X_{T})},z)$ is called an efficient point. The set of all
efficient points, with $z\geq xe^{\int_{0}^{T}r_{s}\operatorname{d}\\!s}$, is
called the efficient frontier.
Define the dual cone of $\Pi$ as
$\widehat{\Pi}:=\Big{\\{}y\in{\mathbb{R}}^{m}\;\Big{|}\;x^{\top}y\leq 0\mbox{
for all $x\in\Pi$}\Big{\\}}.$
The following result gives an equivalent condition for the feasibility of
(4.2). The proof is exactly the same as [13, Theorem 5.3], so we omit it.
###### Theorem 4.1 (Feasibility).
Under assumption 4.1, the mean-variance problem (4.2) is feasible for any
$z\geq xe^{\int_{0}^{T}r_{t}\operatorname{d}\\!t}$ if and only if
(4.3)
$\int_{0}^{T}\mathbb{P}(\mu_{t}\notin\widehat{\Pi})\operatorname{d}\\!t>0.$
For the rest of this section, we will always assume (4.3) holds.
The way to solve (4.2) is rather clear nowadays. To deal with the constraint
${\mathbb{E}}[X_{T}]=z$, we introduce a Lagrange multiplier
$-2\lambda\in{\mathbb{R}}$ and obtain the following _relaxed_ optimization
problem:
(4.4) $\displaystyle\inf$ $\displaystyle\quad
J(x,\pi,\lambda;z)={\mathbb{E}}[(X_{T}-\lambda)^{2}]-(\lambda-z)^{2},$
$\displaystyle\mathrm{s.t.}$ $\displaystyle\quad\pi\in\mathcal{U}.$
Denote its optimal value as
$V(x,\lambda;z)=\inf_{\pi\in\mathcal{U}}J(x,\pi,\lambda;z).$
According to the Lagrange duality theorem (see Luenberger [24])
(4.5)
$\displaystyle\inf_{\pi\in\mathcal{U},{\mathbb{E}}[X_{T}]=z}\mathrm{Var}(X_{T})=\sup_{\lambda\in{\mathbb{R}}}V(x,\lambda;z).$
So we can solve the problem (4.4) by a two-step procedure: Firstly determine
$V(x,\lambda;z)$ for every $\lambda$, and then try to find a $\lambda^{*}$ to
maximize $\lambda\mapsto V(x,\lambda;z)$.
The relaxed problem (4.4) is a special stochastic LQ problem (3.2) studied in
Section 3, where
(4.6) $\displaystyle A=r,\ B=\mu,\ C=0,\ D=\sigma^{\top},\ E=0,\ Q=0,\ R=0,\
S=0,\ G=1.$
The associated BSDEJ (3.6) becomes
(4.7)
$\displaystyle\begin{cases}\operatorname{d}\\!P_{1,t}=-\Big{[}2rP_{1,t-}+H_{1}^{*}(t,P_{1,t-},P_{2,t-},\Lambda_{1,t},\Gamma_{1,t},\Gamma_{2,t})\Big{]}\operatorname{d}\\!t+\Lambda_{1,t}^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\Gamma_{1,t}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\\\
\operatorname{d}\\!P_{2,t}=-\Big{[}2rP_{2,t-}+H_{2}^{*}(t,P_{1,t-},P_{2,t-},\Lambda_{2,t},\Gamma_{1,t},\Gamma_{2,t})\Big{]}\operatorname{d}\\!t+\Lambda_{2,t}^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\Gamma_{2,t}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\\\
P_{1,T}=1,\ P_{2,T}=1,\end{cases}$
where $H_{1}^{*},\ H_{2}^{*}$, $\hat{v}_{1}$, $\hat{v}_{2}$ are defined as in
(3.4), (3.5) and (3.4) with coefficients given in (4.6):
$\displaystyle
H_{1}(\omega,t,v,P_{1},P_{2},\Lambda_{1},\Gamma_{1},\Gamma_{2})$
$\displaystyle=P_{1}v^{\top}\sigma\sigma^{\top}v+2(P_{1}\mu+\sigma\Lambda_{1})^{\top}v$
$\displaystyle\qquad+\int_{\mathcal{E}}\Big{[}(P_{1}+\Gamma_{1})\Big{(}((1+F^{\top}v)^{+})^{2}-1\Big{)}-2P_{1}F^{\top}v$
$\displaystyle\qquad\qquad+(P_{2}+\Gamma_{2})((1+F^{\top}v)^{-})^{2}\Big{]}\nu(\operatorname{d}\\!e),$
$\displaystyle
H_{2}(\omega,t,v,P_{1},P_{2},\Lambda_{2},\Gamma_{1},\Gamma_{2})$
$\displaystyle=P_{2}v^{\top}\sigma\sigma^{\top}v-2(P_{2}\mu+\sigma\Lambda_{2})^{\top}v$
$\displaystyle\qquad+\int_{\mathcal{E}}\Big{[}(P_{2}+\Gamma_{2})\Big{(}((-1+F^{\top}v)^{-})^{2}-1\Big{)}+2P_{2}F^{\top}v$
$\displaystyle\qquad\qquad+(P_{1}+\Gamma_{1})((-1+F^{\top}v)^{+})^{2}\Big{]}\nu(\operatorname{d}\\!e).$
Clearly, Theorems 3.2 and 3.4 can be applied to the BSDEJ (4.7) to ensure that
it admits a unique uniformly positive solution
$(P_{1},\Lambda_{1},\Gamma_{1},P_{2},\Lambda_{2},\Gamma_{2})$. Accordingly,
Theorem 3.3 leads to the following solution to the relaxed problem (4.4).
###### Theorem 4.2.
Let $(P_{1},\Lambda_{1},\Gamma_{1},P_{2},\Lambda_{2},\Gamma_{2})$ be the
unique uniformly positive solution to (4.7). Then the state feedback control
given by
$\displaystyle\pi^{*}(t,X)$
$\displaystyle=\hat{v}_{1}(\omega,t,P_{1},P_{2},\Lambda_{1},\Gamma_{1},\Gamma_{2})\Big{(}X_{t-}-\lambda
e^{-\int_{t}^{T}r_{s}\operatorname{d}\\!s}\Big{)}^{+}$ (4.8)
$\displaystyle\qquad+\hat{v}_{2}(\omega,t,P_{1},P_{2},\Lambda_{2},\Gamma_{1},\Gamma_{2})\Big{(}X_{t-}-\lambda
e^{-\int_{t}^{T}r_{s}\operatorname{d}\\!s}\Big{)}^{-},$
is optimal for the LQ problem (3.2). Moreover, the optimal value is
$\displaystyle V(x,\lambda;z)=P_{1,0}\Big{[}\Big{(}x-\lambda
e^{-\int_{0}^{T}r_{s}\operatorname{d}\\!s}\Big{)}^{+}\Big{]}^{2}+P_{2,0}\Big{[}\Big{(}x-\lambda
e^{-\int_{0}^{T}r_{s}\operatorname{d}\\!s}\Big{)}^{-}\Big{]}^{2}-(\lambda-z)^{2}.$
This resolves the first step problem. To solve the second step problem, i.e.,
to maximize $\lambda\mapsto V(x,\lambda;z)$, the following result is critical.
###### Lemma 4.1.
Assume Assumption 4.1 and conditioin (4.3) hold. Then
(4.9) $\displaystyle P_{1,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}-1\leq
0,\ P_{2,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}-1<0.$
Proof: Applying Itô’s formula to
$P_{2,t}e^{-2\int_{t}^{T}r_{s}\operatorname{d}\\!s}$ on $[0,T]$, we have
(4.10) $\displaystyle
1-P_{2,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}=-{\mathbb{E}}\int_{0}^{T}H_{2}^{*}(t,P_{2,t-},\Lambda_{2,t},\Gamma_{2,t},P_{1,t-},\Gamma_{1,t})\operatorname{d}\\!t.$
Since
$H_{2}^{*}(t,P_{2,t-},\Lambda_{2,t},\Gamma_{2,t},P_{1,t-},\Gamma_{1,t})\leq 0$
by its very definition, it follows
$P_{2,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}-1\leq 0$. Similarly, we
can prove that $P_{1,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}-1\leq 0$.
It remains to prove the strict inequality
$P_{2,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}-1<0$. Suppose, on the
contrary, $P_{2,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}-1=0$. It then
follows from (4.10) that
$H_{2}^{*}(t,P_{1,t-},P_{2,t-},\Lambda_{2,t},\Gamma_{1,t},\Gamma_{2,t})=0~{}\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.}$.
Thus we deduce, from the uniqueness (Theorem 3.4) of solution to the BSDE
(4.7), that $P_{2,t}=e^{2\int_{t}^{T}r_{s}\operatorname{d}\\!s}$,
$\Lambda_{2,t}=0$ and $\Gamma_{2,t}=0$.
On the other hand,
(4.11)
$\displaystyle((1-F^{\top}v)^{+})^{2}+2F^{\top}v-1\leq(1-F^{\top}v)^{2}+2F^{\top}v-1=|F^{\top}v|^{2}.$
Since $F\in L_{\mathcal{P}}^{\nu,\infty}(0,T;{\mathbb{R}}^{m})$, there exists
$c_{1}>0$ such that $|F(e)|\leq c_{1}$ for almost all $e\in\mathcal{E}$. Hence
(4.12) $\displaystyle 1-F^{\top}v\geq 0,\ \mbox{if}\ v\in\Pi\ \mbox{and }\
|v|\leq c_{1}^{-1}.$
Combining (4.11) and (4.12), we have
$\displaystyle H_{2}^{*}(t,P_{1},P_{2},0,\Gamma_{1},0)$
$\displaystyle=\inf_{v\in\Pi}\Big{[}P_{2}v^{\top}\sigma\sigma^{\top}v-2P_{2}\mu^{\top}v+\int_{\mathcal{E}}\Big{[}P_{2}\Big{(}((1-F^{\top}v)^{+})^{2}+2F^{\top}v-1\Big{)}$
$\displaystyle\qquad\qquad+(P_{1}+\Gamma_{1})((1-F^{\top}v)^{-})^{2}\Big{]}\nu(\operatorname{d}\\!e)\Big{]}$
$\displaystyle\leq\inf_{v\in\Pi,|v|\leq
c_{1}^{-1}}\Big{[}P_{2}v^{\top}\Big{(}\sigma\sigma^{\top}+\int_{\mathcal{E}}FF^{\top}\nu(\operatorname{d}\\!e)\Big{)}v-2P_{2}\mu^{\top}v\Big{]}$
$\displaystyle\leq P_{2}\inf_{v\in\Pi,|v|\leq
c_{1}^{-1}}\Big{(}c|v|^{2}-2\mu^{\top}v\Big{)}.$
Note (4.3) implies that there exists $\mathcal{O}\subseteq\Omega\times[0,T]$
such that
$\mu\notin\widehat{\Pi}~{}\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.}$
on $\mathcal{O}$. Hence, there exists $v_{0}\in\Pi$ such that
$\mu^{\top}v_{0}>0$ on $\mathcal{O}$. By choosing $v_{1}=\varepsilon v_{0}$
with $\varepsilon>0$ being sufficiently small so that $|v_{1}|\leq
c_{1}^{-1}$, we get
$\displaystyle H_{2}^{*}(t,P_{1},P_{2},0,\Gamma_{1},0)$ $\displaystyle\leq
P_{2}\varepsilon\Big{(}c\varepsilon|v_{0}|^{2}-2\mu^{\top}v_{0}\Big{)}~{}\textrm{on
$\mathcal{O}$.}$
The RHS is negative for sufficiently small $\varepsilon>0$ on $\mathcal{O}$,
leading to a contraction. Therefore
$P_{2,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}-1<0$. $\Box$
To find a $\lambda^{*}$ to maximize $\lambda\mapsto V(x,\lambda;z)$, we do
some tedious calculation (using (4.9)) and obtain
$\max_{\lambda}V(x,\lambda;z)=V(x,\lambda^{*};z)=\frac{P_{2,0}}{1-P_{2,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}}\Big{(}x-ze^{-\int_{0}^{T}r_{s}\operatorname{d}\\!s}\Big{)}^{2},$
where
$\lambda^{*}=\frac{z-xP_{2,0}e^{-\int_{0}^{T}r_{s}\operatorname{d}\\!s}}{1-P_{2,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}}.$
The above analysis boils down to the following solution to the mean-variance
problem (4.2).
###### Theorem 4.3.
Let $(P_{1},\Lambda_{1},\Gamma_{1},P_{2},\Lambda_{2},\Gamma_{2})$ be the
unique uniformly positive solution to (4.7). Then the state feedback portfolio
given by
$\displaystyle\pi^{*}(t,X)$
$\displaystyle=\hat{v}_{1}(\omega,t,P_{1},P_{2},\Lambda_{1},\Gamma_{1},\Gamma_{2})\Big{(}X_{t-}-\lambda^{*}e^{-\int_{t}^{T}r_{s}\operatorname{d}\\!s}\Big{)}^{+}$
(4.13)
$\displaystyle\qquad+\hat{v}_{2}(\omega,t,P_{1},P_{2},\Lambda_{1},\Gamma_{2},\Gamma_{2})\Big{(}X_{t-}-\lambda^{*}e^{-\int_{t}^{T}r_{s}\operatorname{d}\\!s}\Big{)}^{-},$
is optimal to the mean-variance problem (4.2). Moreover, the efficient
frontier is determined by
$\displaystyle\mathrm{Var}(X_{T})=\frac{P_{2,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}}{1-P_{2,0}e^{-2\int_{0}^{T}r_{s}\operatorname{d}\\!s}}\Big{(}{\mathbb{E}}[X_{T}]-xe^{\int_{0}^{T}r_{s}\operatorname{d}\\!s}\Big{)}^{2},$
where ${\mathbb{E}}[X_{T}]\geq xe^{\int_{0}^{T}r_{s}\operatorname{d}\\!s}$.
###### Remark 4.2.
In the constrained mean-variance model without jumps studied in Hu and Zhou
[16], the efficient portfolio only takes the second term on the RHS of (4.3),
i.e. the optimal wealth $X_{t}$ will never exceed
$\lambda^{*}e^{-\int_{t}^{T}r_{s}\operatorname{d}\\!s}$ on $[0,T]$, and it
only depends on $(P_{2},\Lambda_{2})$ as $\hat{v}_{2}$ does. But in our cone-
constrained MV problem with jumps, the optimal wealth $X_{t}$ will probably
cross the bliss point $\lambda^{*}e^{-\int_{t}^{T}r_{s}\operatorname{d}\\!s}$.
## Appendix A Heuristic derivation of the BSDEJ (3.6)
By the Meyer-Itô formula [31, Theorem 70], we have
$\displaystyle\operatorname{d}\\!X_{t}^{+}$
$\displaystyle=\mathbf{1}_{\\{X_{t-}>0\\}}\Big{[}\big{(}A_{t}X_{t-}+B_{t}^{\top}u_{t}-\int_{\mathcal{E}}(E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})\nu(\operatorname{d}\\!e)\big{)}\operatorname{d}\\!t+(C_{t}X_{t-}+D_{t}u_{t})^{\top}\operatorname{d}\\!W_{t}\Big{]}$
$\displaystyle\qquad+\int_{\mathcal{E}}\Big{[}(X_{t-}+E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})^{+}-X_{t-}^{+}\Big{]}N(\operatorname{d}\\!t,\operatorname{d}\\!e)+\frac{1}{2}\operatorname{d}\\!L_{t},$
and
$\displaystyle\operatorname{d}\\!X_{t}^{-}$
$\displaystyle=-\mathbf{1}_{\\{X_{t-}\leq
0\\}}\Big{[}\big{(}A_{t}X_{t-}+B_{t}^{\top}u_{t}-\int_{\mathcal{E}}(E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})\nu(\operatorname{d}\\!e)\big{)}\operatorname{d}\\!t+(C_{t}X_{t-}+D_{t}u_{t})^{\top}\operatorname{d}\\!W_{t}\Big{]}$
$\displaystyle\qquad+\int_{\mathcal{E}}\Big{[}(X_{t-}+E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})^{-}-X_{t-}^{-}\Big{]}N(\operatorname{d}\\!t,\operatorname{d}\\!e)+\frac{1}{2}\operatorname{d}\\!L_{t},$
where $L$ is the local time of $X$ at $0$. Since
$X_{t}^{\pm}\operatorname{d}\\!L_{t}=0$, applying the Itô formula yields
$\displaystyle\operatorname{d}\\!\;(X_{t}^{+})^{2}$
$\displaystyle=\mathbf{1}_{\\{X_{t-}>0\\}}\Big{[}2X_{t-}^{+}\big{(}A_{t}X_{t-}+B_{t}^{\top}u_{t}-\int_{\mathcal{E}}(E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})\nu(\operatorname{d}\\!e)\big{)}+|C_{t}X_{t-}+D_{t}u_{t}|^{2}\Big{]}\operatorname{d}\\!t$
$\displaystyle\qquad+\mathbf{1}_{\\{X_{t-}>0\\}}2X_{t-}^{+}(C_{t}X_{t-}+D_{t}u_{t})^{\top}\operatorname{d}\\!W_{t}$
$\displaystyle\qquad+\int_{\mathcal{E}}\Big{[}((X_{t-}+E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})^{+})^{2}-(X_{t-}^{+})^{2}\Big{]}N(\operatorname{d}\\!t,\operatorname{d}\\!e),$
and
$\displaystyle\operatorname{d}\\!\;(X_{t}^{-})^{2}$
$\displaystyle=\mathbf{1}_{\\{X_{t-}\leq
0\\}}\Big{[}-2X_{t-}^{-}\big{(}A_{t}X_{t-}+B_{t}^{\top}u_{t}-\int_{\mathcal{E}}(E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})\nu(\operatorname{d}\\!e)\big{)}+|C_{t}X_{t-}+D_{t}u_{t}|^{2}\Big{]}\operatorname{d}\\!t$
$\displaystyle\qquad-\mathbf{1}_{\\{X_{t-}\leq
0\\}}2X_{t-}^{-}(C_{t}X_{t-}+D_{t}u_{t})^{\top}\operatorname{d}\\!W_{t}$
$\displaystyle\qquad+\int_{\mathcal{E}}\Big{[}((X_{t-}+E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})^{-})^{2}-(X_{t-}^{-})^{2}\Big{]}N(\operatorname{d}\\!t,\operatorname{d}\\!e).$
Assume that $P_{1}$ and $P_{2}$ are semimartingales of the following form:
$\displaystyle\begin{cases}\operatorname{d}\\!P_{1}=-f_{1}\operatorname{d}\\!t+\Lambda_{1}^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\Gamma_{1}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\\\
P_{1,T}=G,\end{cases}$
and
$\displaystyle\begin{cases}\operatorname{d}\\!P_{2}=-f_{2}\operatorname{d}\\!t+\Lambda_{2}^{\top}\operatorname{d}\\!W+\int_{\mathcal{E}}\Gamma_{2}(e)\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),\\\
P_{2,T}=G.\end{cases}$
Applying Itô’s formula to $P_{1,t}(X_{t}^{+})^{2}$,
$\displaystyle\operatorname{d}\\!P_{1,t}(X_{t}^{+})^{2}$
$\displaystyle=P_{1,t-}\mathbf{1}_{\\{X_{t-}>0\\}}\Big{[}2X_{t-}^{+}(A_{t}X_{t-}+B_{t}^{\top}u_{t})+|C_{t}X_{t-}+D_{t}u_{t}|^{2}\Big{]}\operatorname{d}\\!t$
$\displaystyle\qquad+2X_{t-}^{+}(C_{t}X_{t-}+D_{t}u_{t})^{\top}\Lambda_{1,t}\operatorname{d}\\!t$
$\displaystyle\qquad-2P_{1,t-}X_{t-}^{+}\mathbf{1}_{\\{X_{t-}>0\\}}\int_{\mathcal{E}}(E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
$\displaystyle\qquad+\int_{\mathcal{E}}(P_{1,t-}+\Gamma_{1,t}(e))\Big{(}((X_{t-}+E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})^{+})^{2}-(X_{t-}^{+})^{2}\Big{)}\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
$\displaystyle\qquad-(X_{t-}^{+})^{2}f_{1}\operatorname{d}\\!t+\Big{[}(X_{t-}^{+})2\Lambda_{1,t}+2P_{1,t-}X_{t-}^{+}(C_{t}X_{t-}+D_{t}u_{t})\Big{]}^{\top}\operatorname{d}\\!W$
$\displaystyle\qquad+\int_{\mathcal{E}}\Big{[}(P_{1,t-}+\Gamma_{1}(e))[((X_{t-}+E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})^{+})^{2}-(X_{t-}^{+})^{2}]$
$\displaystyle\qquad\qquad\qquad+(X_{t-}^{+})^{2}\Gamma_{1}(e)\Big{]}\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e),$
and applying Itô’s formula to $P_{2,t}(X_{t}^{-})^{2}$,
$\displaystyle\operatorname{d}\\!P_{2,t}(X_{t}^{-})^{2}$
$\displaystyle=P_{2,t-}\mathbf{1}_{\\{X_{t-}\leq
0\\}}\Big{[}-2X_{t-}^{-}(A_{t}X_{t-}+B_{t}^{\top}u_{t})+|C_{t}X_{t-}+D_{t}u_{t}|^{2}\Big{]}\operatorname{d}\\!t$
$\displaystyle\qquad-2X_{t-}^{-}(C_{t}X_{t-}+D_{t}u_{t})^{\top}\Lambda_{2,t}\operatorname{d}\\!t$
$\displaystyle\qquad+2P_{2,t-}X_{t-}^{-}\mathbf{1}_{\\{X_{t-}\leq
0\\}}\int_{\mathcal{E}}(E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
$\displaystyle\qquad+\int_{\mathcal{E}}(P_{2,t-}+\Gamma_{2,t}(e))\Big{(}((X_{t-}+E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})^{-})^{2}-(X_{t-}^{-})^{2}\Big{)}\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
$\displaystyle\qquad-(X_{t-}^{-})^{2}f_{2}\operatorname{d}\\!t+\Big{[}(X_{t-}^{-})^{2}\Lambda_{2,t}-2P_{2,t-}X_{t-}^{-}(C_{t}X_{t-}+D_{t}u_{t})\Big{]}^{\top}\operatorname{d}\\!W$
$\displaystyle\qquad+\int_{\mathcal{E}}\Big{[}(P_{2,t-}+\Gamma_{2}(e))[((X_{t-}+E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})^{-})^{2}-(X_{t-}^{-})^{2}]$
$\displaystyle\qquad\qquad\qquad+(X_{t-}^{-})^{2}\Gamma_{2}(e)\Big{]}\widetilde{N}(\operatorname{d}\\!t,\operatorname{d}\\!e).$
Then
$\displaystyle J(x;u)$
$\displaystyle=\mathbb{E}\Big{[}\int_{0}^{T}\Big{(}Q_{t}X_{t}^{2}+u_{t}^{\top}R_{t}u_{t}+2X_{t}S_{t}^{\top}u_{t}\Big{)}\operatorname{d}\\!t+GX_{T}^{2}\Big{]}$
$\displaystyle=P_{1,0}(x^{+})^{2}+P_{2,0}(x^{-})^{2}+\mathbb{E}\int_{0}^{T}\Big{(}Q_{t}X_{t}^{2}+u_{t}^{\top}R_{t}u_{t}+2X_{t}S_{t}^{\top}u_{t}\Big{)}\operatorname{d}\\!t$
$\displaystyle\qquad+P_{1,t-}\mathbf{1}_{\\{X_{t-}>0\\}}\Big{[}2X_{t-}^{+}(A_{t}X_{t-}+B_{t}^{\top}u_{t})+|C_{t}X_{t-}+D_{t}u_{t}|^{2}\Big{]}\operatorname{d}\\!t$
$\displaystyle\qquad+2X_{t-}^{+}(C_{t}X_{t-}+D_{t}u_{t})^{\top}\Lambda_{1,t}\operatorname{d}\\!t$
$\displaystyle\qquad-2P_{1,t-}X_{t-}^{+}\mathbf{1}_{\\{X_{t-}>0\\}}\int_{\mathcal{E}}(E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
$\displaystyle\qquad+\int_{\mathcal{E}}(P_{1,t-}+\Gamma_{1,t}(e))\Big{(}((X_{t-}+E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})^{+})^{2}-(X_{t-}^{+})^{2}\Big{)}\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
$\displaystyle\qquad-(X_{t-}^{+})^{2}f_{1}\operatorname{d}\\!t-(X_{t-}^{-})^{2}f_{2}\operatorname{d}\\!t$
$\displaystyle\qquad+P_{2,t-}\mathbf{1}_{\\{X_{t-}\leq
0\\}}\Big{[}-2X_{t-}^{-}(A_{t}X_{t-}+B_{t}^{\top}u_{t})+|C_{t}X_{t-}+D_{t}u_{t}|^{2}\Big{]}\operatorname{d}\\!t$
$\displaystyle\qquad-2X_{t-}^{-}(C_{t}X_{t-}+D_{t}u_{t})^{\top}\Lambda_{2,t}\operatorname{d}\\!t$
$\displaystyle\qquad+2P_{2,t-}X_{t-}^{-}\mathbf{1}_{\\{X_{t-}\leq
0\\}}\int_{\mathcal{E}}(E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
$\displaystyle\qquad+\int_{\mathcal{E}}(P_{2,t-}+\Gamma_{2,t}(e))\Big{(}((X_{t-}+E_{t}(e)X_{t-}+F_{t}(e)^{\top}u_{t})^{-})^{2}-(X_{t-}^{-})^{2}\Big{)}\nu(\operatorname{d}\\!e)\operatorname{d}\\!t.$
Denote $\phi(X_{t-},u_{t})$ be the integrand on the RHS of the above equation.
* •
If $X_{t-}>0$, then
$\displaystyle\phi(X_{t-},u_{t})$
$\displaystyle=\Big{[}Q+v^{\top}(R+P_{1,t-}D^{\top}D)v+2S^{\top}v+P_{1,t-}(2A+C^{\top}C)+2C^{\top}\Lambda_{1,t}$
$\displaystyle\qquad+2(P_{1,t-}(B+D^{\top}C)+D^{\top}\Lambda_{1})^{\top}v$
$\displaystyle\qquad-2P_{1,t-}\int_{\mathcal{E}}(E+F^{\top}v)\nu(\operatorname{d}\\!e)+\int_{\mathcal{E}}(P_{1,t-}+\Gamma_{1,t-})\Big{(}((1+E+F^{\top}v)^{+})^{2}-1\Big{)}\nu(\operatorname{d}\\!e)$
$\displaystyle\qquad-
f_{1}+\int_{\mathcal{E}}(P_{2,t-}+\Gamma_{2,t-})((1+E+F^{\top}v)^{-})^{2}\nu(\operatorname{d}\\!e)\Big{]}X_{t-}^{2},$
where $v_{t}=\frac{u_{t}}{|X_{t-}|}$.
* •
If $X_{t-}<0$, then
$\displaystyle\phi(X_{t-},u_{t})$
$\displaystyle=\Big{[}Q+v^{\top}(R+P_{2,t-}D^{\top}D)v-2S^{\top}v+P_{2,t-}(2A+C^{\top}C)+2C^{\top}\Lambda_{2,t}$
$\displaystyle\qquad-2(P_{1,t-}(B+D^{\top}C)+D^{\top}\Lambda_{2})^{\top}v$
$\displaystyle\qquad-2P_{2,t-}\int_{\mathcal{E}}(E-F^{\top}v)\nu(\operatorname{d}\\!e)+\int_{\mathcal{E}}(P_{2,t-}+\Gamma_{2,t-})\Big{(}((-1-E+F^{\top}v)^{-})^{2}-1\Big{)}\nu(\operatorname{d}\\!e)$
$\displaystyle\qquad-
f_{2}+\int_{\mathcal{E}}(P_{1,t-}+\Gamma_{1,t-})((-1-E+F^{\top}v)^{+})^{2}\nu(\operatorname{d}\\!e)\Big{]}X_{t-}^{2},$
where $v_{t}=\frac{u_{t}}{|X_{t-}|}$.
* •
If $X_{t-}=0$, then $\phi(0,0)=0$ and
$\displaystyle\phi(X_{t-},u_{t})$
$\displaystyle=u_{t}^{\top}Ru_{t}+P_{2,t-}|Du_{t}|^{2}+\int_{\mathcal{E}}(P_{1,t-}+\Gamma_{1,t})((F^{\top}u_{t})^{+})^{2}\nu(\operatorname{d}\\!e)$
$\displaystyle\qquad+\int_{\mathcal{E}}(P_{2,t}+\Gamma_{2,t-})((F^{\top}u_{t})^{-})^{2}\nu(\operatorname{d}\\!e)\geq
0.$
In order to ensure $\min_{u\in\Pi}\phi(X_{t},u)=0$, it is evident to take
$f_{1}$ and $f_{2}$ in the form of (3.4) and (3.5).
## Appendix B Proof of Lemma 3.1
For any positive integers $k<l$, set
$P_{i,t}^{k,l}:=P_{i,t}^{k}-P_{i,t}^{l}\geq 0,\
\Lambda_{i,t}^{k,l}:=\Lambda_{i,t}^{k}-\Lambda_{i,t}^{l},\
\Gamma_{i,t}^{k,l}:=\Gamma_{i,t}^{k}-\Gamma_{i,t}^{l},\ i=1,2.$
Let $\kappa>0$ be a constant to be specified later, and write
$\Psi(x)=\frac{1}{\kappa}\big{(}e^{\kappa x}-\kappa x-1\big{)}.$
Notice $\Psi^{\prime}(x)\geq 0$ for $x\geq 0$. Applying Itô’s formula to
$\Psi(P_{1,t}^{k,l})$, we get
$\displaystyle\quad\Psi(P_{1,0}^{k,l})+\frac{1}{2}{\mathbb{E}}\int_{0}^{T}\Psi^{\prime\prime}(P_{1,t}^{k,l})|\Lambda_{1}^{k,l}|^{2}\operatorname{d}\\!t$
$\displaystyle\qquad\qquad+{\mathbb{E}}\int_{0}^{T}\int_{\mathcal{E}}\Big{[}\Psi(P_{1,t-}^{k,l}+\Gamma_{1,t}^{k,l})-\Psi(P_{1,t-}^{k,l})-\Psi^{\prime}(P_{1,t-}^{k,l})\Gamma_{1}^{k,l}\Big{]}\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
$\displaystyle=\Psi(0)+{\mathbb{E}}\int_{0}^{T}\Psi^{\prime}(P_{1,t}^{k,l})\Big{[}(2A+C^{\top}C)P^{k,l}_{1,t-}+2C^{\top}\Lambda^{k,l}_{1,t}$
$\displaystyle\qquad+H_{1}^{k}(t,P_{1,t-}^{k},P_{2,t-}^{k},\Lambda_{1,t}^{k},\Gamma_{1,t}^{k},\Gamma_{2,t}^{k})-H_{1}^{l}(t,P_{1,t-}^{l},P_{2,t-}^{l},\Lambda_{1,t}^{l},\Gamma_{1,t}^{l},\Gamma_{2,t}^{l})\Big{]}\operatorname{d}\\!t.$
Using the following fact:
$\displaystyle\Psi(0)=0,\ P_{1,t}^{k,l}\geq 0,\
\Psi^{\prime}(P_{1,t}^{k,l})=e^{\kappa P_{1,t}^{k,l}}-1\geq 0,\ H_{1}^{l}\geq
H_{1}^{*},$ $\displaystyle
H_{1}^{k}\leq\int_{\mathcal{E}}\Big{[}(P_{1}+\Gamma_{1})\Big{(}((1+E)^{+})^{2}-1\Big{)}-2P_{1}E+(P_{2}+\Gamma_{2})((1+E)^{-})^{2}\Big{]}\nu(\operatorname{d}\\!e)\leq
c,$
where $c$ is independent of $k$ and $l$, we obtain
$\displaystyle\quad\Psi(P_{1,0}^{k,l})+\frac{1}{2}{\mathbb{E}}\int_{0}^{T}\Psi^{\prime\prime}(P_{1,t}^{k,l})|\Lambda_{1}^{k,l}|^{2}\operatorname{d}\\!t$
$\displaystyle\qquad\qquad+{\mathbb{E}}\int_{0}^{T}\int_{\mathcal{E}}\Big{[}\Psi(P_{1,t-}^{k,l}+\Gamma_{1,t}^{k,l})-\Psi(P_{1,t-}^{k,l})-\Psi^{\prime}(P_{1,t-}^{k,l})\Gamma_{1}^{k,l}\Big{]}\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
$\displaystyle\leq{\mathbb{E}}\int_{0}^{T}\Psi^{\prime}(P_{1,t}^{k,l})\Big{[}c+2C^{\top}\Lambda_{1,t}^{k,l}-H_{1}^{*}(t,P_{1,t-}^{l},P_{2,t-}^{l},\Lambda_{1,t}^{l},\Gamma_{1,t}^{l},\Gamma_{2,t}^{l})\Big{]}\operatorname{d}\\!t.$
Keeping in mind $P^{l}_{i}$, $\Gamma^{l}_{i}$, $i=1,2$, are uniformly bounded,
we have the following estimates:
$\displaystyle-
H_{1}^{*}(t,P_{1,t-}^{l},P_{2,t-}^{l},\Lambda_{1,t}^{l},\Gamma_{1,t}^{l},\Gamma_{2,t}^{l})$
$\displaystyle\leq-\inf_{v\in{\mathbb{R}}^{m}}H_{1}(t,0,P_{1,t-}^{l},P_{2,t-}^{l},\Lambda_{1,t}^{l},\Gamma_{1,t}^{l},\Gamma_{2,t}^{l})$
$\displaystyle\leq c+c|\Lambda_{1}^{l}|^{2}$ $\displaystyle\leq
c+3c(|\Lambda_{1}^{k,l}|^{2}+|\Lambda_{1}^{k}-\Lambda_{1}|^{2}+|\Lambda_{1}|^{2}),$
where $c>0$ is a constant independent of $l$ and $k$. The above estimates lead
to
$\displaystyle\quad\Psi(P_{1,0}^{k,l})+{\mathbb{E}}\int_{0}^{T}\Big{(}\frac{1}{2}\Psi^{\prime\prime}(P_{1,t}^{k,l})-3c\Psi^{\prime}(P_{1,t}^{k,l})\Big{)}|\Lambda_{1}^{k,l}|^{2}\operatorname{d}\\!t$
$\displaystyle\qquad\qquad+{\mathbb{E}}\int_{0}^{T}\int_{\mathcal{E}}\Big{[}\Psi(P_{1,t-}^{k,l}+\Gamma_{1,t}^{k,l})-\Psi(P_{1,t-}^{k,l})-\Psi^{\prime}(P_{1,t-}^{k,l})\Gamma_{1}^{k,l}\Big{]}\nu(\operatorname{d}\\!e)\operatorname{d}\\!t$
(B.1)
$\displaystyle\leq{\mathbb{E}}\int_{0}^{T}\Psi^{\prime}(P_{1,t}^{k,l})\Big{[}2c+2C^{\top}\Lambda_{1,t}^{k,l}+3c|\Lambda_{1}^{k}-\Lambda_{1}|^{2}+3c|\Lambda_{1}|^{2})\Big{]}\operatorname{d}\\!t.$
Take $\kappa=12c$. Then
$\frac{1}{2}\Psi^{\prime\prime}(x)-3c\Psi^{\prime}(x)=3c(e^{\kappa
x}+1)=3c\Psi^{\prime}(x)+6c\geq 6c$ for $x\geq 0$. So by the dominated
convergence theorem, the sequence
$\sqrt{\frac{1}{2}\Psi^{\prime\prime}(P_{1,t}^{k,l})-3c\Psi^{\prime}(P_{1,t}^{k,l})}$
converges strongly to
$\sqrt{3c\Psi^{\prime}(P_{1,t}^{k}-P_{1,t})+6c},$
as $l\to\infty$, and they are uniformly bounded. Therefore,
$\sqrt{3c\Psi^{\prime}(P_{1,t}^{k,l})+6c}\;\Lambda_{1}^{k,l}$
converges weakly to
$\sqrt{3c\Psi^{\prime}(P_{1,t}^{k}-P_{1,t})+6c}\;(\Lambda_{1}^{k}-\Lambda_{1}).$
By the mean value theorem and the uniformly boundedness of $P_{1,t-}^{k,l}$
and $\Gamma_{1,t-}^{k,l}$, we obtain
(B.2)
$\displaystyle\Psi(P_{1,t-}^{k,l}+\Gamma_{1,t}^{k,l})-\Psi(P_{1,t-}^{k,l})-\Psi^{\prime}(P_{1,t-}^{k,l})\Gamma_{1}^{k,l}$
$\displaystyle=\frac{1}{\kappa}e^{\kappa
P_{1,t-}^{k,l}}\Big{[}e^{\kappa\Gamma_{1,t}^{k,l}}-\kappa\Gamma_{1,t}^{k,l}-1\Big{]}\geq\varepsilon|\Gamma_{1,t}^{k,l}|^{2},$
for some constant $\varepsilon>0$ independent of $l$ and $k$. We then get from
(B.1) that
$\displaystyle\quad{\mathbb{E}}\int_{0}^{T}\big{(}3c\Psi^{\prime}(P_{1}^{k}-P_{1})+6c\big{)}|\Lambda_{1}^{k}-\Lambda_{1}|^{2}\operatorname{d}\\!t$
$\displaystyle\leq\varliminf_{l\to\infty}{\mathbb{E}}\int_{0}^{T}\big{(}3c\Psi^{\prime}(P_{1}^{k,l})+6c\big{)}|\Lambda_{1}^{k,l}|^{2}\operatorname{d}\\!t$
$\displaystyle\leq{\mathbb{E}}\int_{0}^{T}\Psi^{\prime}(P_{1}^{k}-P_{1})\Big{[}2c+2C^{\top}(\Lambda_{1}^{k}-\Lambda_{1})+3c|\Lambda_{1}^{k}-\Lambda_{1}|^{2}+3c|\Lambda_{1}|^{2})\Big{]}\operatorname{d}\\!s.$
Canceling the common terms, it yields
$\displaystyle{\mathbb{E}}\int_{0}^{T}6c|\Lambda_{1}^{k}-\Lambda_{1}|^{2}\operatorname{d}\\!s\leq{\mathbb{E}}\int_{0}^{T}\Psi^{\prime}(P_{1}^{k}-P_{1})\Big{[}2c+2C^{\top}(\Lambda_{1}^{k}-\Lambda_{1})+3c|\Lambda_{1}|^{2})\Big{]}\operatorname{d}\\!s.$
By passing to the limit $k\to\infty$, applying dominated convergence theorem
and noticing $\Psi^{\prime}(0)=0$, we have
$\displaystyle\lim_{k\to\infty}{\mathbb{E}}\int_{0}^{T}|\Lambda_{1}^{k}-\Lambda_{1}|^{2}\operatorname{d}\\!t=0.$
Using (B.2), we can similarly get
$\displaystyle\lim_{k\to\infty}{\mathbb{E}}\int_{0}^{T}\int_{\mathcal{E}}|\Gamma_{1}^{k}-\Gamma_{1}|^{2}\nu(\operatorname{d}\\!e)\operatorname{d}\\!t=0.$
Because $\Gamma_{1}^{k}$, $k=1,2,\cdots,$ are uniformly bounded in
$L^{\infty,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}})$, we conclude that
$\Gamma_{1}\in L^{\infty,\nu}_{\mathcal{P}}(0,T;{\mathbb{R}})$.
Along appropriate subsequence (which is still denoted by
$(\Lambda_{1}^{k},\Gamma_{1}^{k})$) we may obtain
$\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.}$
convergence of
$\displaystyle\int_{t}^{T}(\Lambda_{1}^{k})^{\top}\operatorname{d}\\!W+\int_{t}^{T}\int_{\mathcal{E}}\Gamma_{1}^{k}(e)\widetilde{N}(\operatorname{d}\\!s,\operatorname{d}\\!e)\to\int_{t}^{T}\Lambda_{1}^{\top}\operatorname{d}\\!W+\int_{t}^{T}\int_{\mathcal{E}}\Gamma_{1}(e)\widetilde{N}(\operatorname{d}\\!s,dy),$
and
$\displaystyle\lim_{k\to\infty}\int_{t}^{T}\Big{[}(2A+C^{\top}C)P_{1,t-}^{k}+2C^{\top}\Lambda_{1,t}^{k}+Q\Big{]}\operatorname{d}\\!s=\int_{t}^{T}\Big{[}(2A+C^{\top}C)P_{1,t-}+2C^{\top}\Lambda_{1,t}+Q\Big{]}\operatorname{d}\\!s.$
We now turn to prove
(B.3)
$\displaystyle\lim_{k\to\infty}\int_{t}^{T}H_{1}^{k}(s,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})\operatorname{d}\\!s=\int_{t}^{T}H_{1}^{*}(s,P_{1,s-},P_{2,s-},\Lambda_{1,s},\Gamma_{1,s},\Gamma_{2,s})\operatorname{d}\\!s.$
We have
$\displaystyle\quad|H_{1}^{k}(s,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})-H_{1}^{*}(s,P_{1,s-},P_{2,s-},\Lambda_{1,s},\Gamma_{1,s},\Gamma_{2,s})|$
$\displaystyle\leq|H_{1}^{k}(s,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})-H_{1}^{*}(s,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})|$
$\displaystyle\qquad+|H_{1}^{*}(s,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})-H_{1}^{*}(s,P_{1,s-},P_{2,s-},\Lambda_{1,s},\Gamma_{1,s},\Gamma_{2,s})|.$
Recall that
$\Lambda_{1,s}^{k}\to\Lambda_{1,s}~{}\operatorname{d}\\!\mathbb{P}\otimes\operatorname{d}\\!t\textrm{-a.e.}$,
so there exists $k_{1}(\omega,s)$ such that $|\Lambda_{1,s}^{k}|\leq
1+|\Lambda_{1,s}|$ for $k\geq k_{1}$. Notice that
$H_{1}(s,0,P_{1,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},P_{2,s-}^{k}+\Gamma_{2,s}^{k})$
is upper bounded by some $c>0$, and
$\displaystyle
H_{1}(s,v,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})$
$\displaystyle\geq\delta|v|^{2}-2c_{1}|v|(1+|\Lambda_{1}^{k}|)-c_{1}$
$\displaystyle\geq\delta|v|^{2}-2c_{1}|v|(2+|\Lambda_{1}|)-c_{1}\geq c,$
if $|v|>c_{2}(1+|\Lambda_{1,s}|)$ with $c_{2}>0$ being sufficiently large.
Hence, for $k\geq\max\\{c_{2}(1+|\Lambda_{1,s}|),k_{1}\\}$, we have
$\displaystyle
H_{1}^{*}(s,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})$
$\displaystyle=\inf_{\begin{subarray}{c}v\in\Pi\\\ |v|\leq
c_{2}(1+|\Lambda_{1}|)\end{subarray}}H_{1}(s,v,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})$
$\displaystyle\geq\inf_{\begin{subarray}{c}v\in\Pi\\\ |v|\leq
k\end{subarray}}H_{1}(s,v,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})$
$\displaystyle=H_{1}^{k}(s,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k}).$
We also have the reverse inequality $H_{1}^{*}\leq H_{1}^{k}$ by definition.
Therefore, when $k\geq\max\\{c_{2}(1+|\Lambda_{1,s}|),k_{1}\\}$, $H_{1}^{k}$
and $H_{1}^{*}$ coincide at
$(s,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})$.
Notice that
$\displaystyle\quad|H_{1}^{*}(s,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})-H_{1}^{*}(s,P_{1,s-},P_{2,s-},\Lambda_{1,s},\Gamma_{1,s},\Gamma_{2,s}))|$
$\displaystyle\leq\sup_{\begin{subarray}{c}v\in\Pi\\\ |v|\leq
c_{2}(1+|\Lambda_{1}|)\end{subarray}}|H_{1}(s,v,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})-H_{1}(s,v,P_{1,s-},P_{2,s-},\Lambda_{1,s},\Gamma_{1,s},\Gamma_{2,s}))|,$
one can easily see, from the definition of $H_{1}$, that as long as
$|P_{1}^{k}-P_{1}|+|\Lambda_{1}^{k}-\Lambda_{1}|+\int_{\mathcal{E}}|\Gamma_{1}^{k}-\Gamma_{1}|\nu(\operatorname{d}\\!e)+|P_{2}^{k}-P_{2}|+\int_{\mathcal{E}}|\Gamma_{2}^{k}-\Gamma_{2}|\nu(\operatorname{d}\\!e)\to
0,$
we have
$\lim_{k\to\infty}|H_{1}^{*}(s,P_{1,s-}^{k},P_{2,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},\Gamma_{2,s}^{k})-H_{1}^{*}(s,P_{1,s-},P_{2,s-},\Lambda_{1,s},\Gamma_{1,s},\Gamma_{2,s}))|=0.$
Since
$\displaystyle|H_{1}^{k}(s,P_{1,s-}^{k},\Lambda_{1,s}^{k},\Gamma_{1,s}^{k},P_{2,s-}^{k},\Gamma_{2,s}^{k})|\leq
c(1+|\Lambda_{1,s}^{k}|^{2}),$
the dominated convergence theorem leads to (B.3).
Now it is standard to show that
$\lim_{k\to\infty}{\mathbb{E}}\Big{[}\sup_{t\in[0,T]}|P^{k}_{1,t}-P_{1,t}|\Big{]}=0;$
please refer to Antonelli and Mancini [1, Theorem 1] for details.
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|
Yerim<EMAIL_ADDRESS>
Nur Suriza<EMAIL_ADDRESS>
Sang-Chul<EMAIL_ADDRESS>
Department of Electrical
and Computer Engineering
Inha University
Incheon, South Korea Local Feature Extraction from Salient Regions
# Local Feature Extraction from Salient Regions by Feature Map Transformation
###### Abstract
Local feature matching is essential for many applications, such as
localization and 3D reconstruction. However, it is challenging to match
feature points accurately in various camera viewpoints and illumination
conditions. In this paper, we propose a framework that robustly extracts and
describes salient local features regardless of changing light and viewpoints.
The framework suppresses illumination variations and encourages structural
information to ignore the noise from light and to focus on edges. We classify
the elements in the feature covariance matrix, an implicit feature map
information, into two components. Our model extracts feature points from
salient regions leading to reduced incorrect matches. In our experiments, the
proposed method achieved higher accuracy than the state-of-the-art methods in
the public dataset, such as HPatches, Aachen Day-Night, and ETH, which
especially show highly variant viewpoints and illumination.
## 1 Introduction
Extracting and describing local features for matching is essential, especially
in computer vision tasks that involve image matching, searching, tracking, and
3D reconstruction [Heinly et al.(2015)Heinly, Schonberger, Dunn, and Frahm,
Svärm et al.(2016)Svärm, Enqvist, Kahl, and Oskarsson, Noh et al.(2017)Noh,
Araujo, Sim, Weyand, and Han]. Feature matching focuses on three main phases
when given two similar images are to be matched: feature detection, feature
description, and feature matching [Lowe(2004), Cheng et al.(2014)Cheng, Leng,
Wu, Cui, and Lu]. The primary goal of feature matching is to optimize matching
accuracy while minimizing the memory footprint of earlier applications. The
extracted features should be sparse, highly repeatable, and precise. Each
image’s salient features, such as its corners, are initially recognized as
interest points during the detection phase. Then, local descriptors are
extracted based on the neighborhood regions of these interest points and used
in the matching algorithms.
Classical approaches [Lowe(2004), Bay et al.(2006)Bay, Tuytelaars, and Van
Gool] concentrate on the detect-then-describe method, where they first detect
the points by analyzing the gradient of the image before describing the points
with directional information. Furthermore, these approaches [Lowe(2004), Bay
et al.(2006)Bay, Tuytelaars, and Van Gool] have shifted research trends by the
emergence of deep learning methods [Tian et al.(2017)Tian, Fan, and Wu,
Zagoruyko and Komodakis(2015), DeTone et al.(2018)DeTone, Malisiewicz, and
Rabinovich]. Since deep convolutional neural networks (DCNNs) can
automatically learn features, mimic traditional detector behaviors, and
process complex images, CNN methods have achieved remarkable performance than
before [Luo et al.(2020)Luo, Zhou, Bai, Chen, Zhang, Yao, Li, Fang, and Quan,
Tian et al.(2020)Tian, Balntas, Ng, Barroso-Laguna, Demiris, and Mikolajczyk,
Tyszkiewicz et al.(2020)Tyszkiewicz, Fua, and Trulls]. These data-driven
methods concentrate on sparse points by leveraging descriptors’ information
for corresponding points. At the same time, several networks [DeTone et
al.(2018)DeTone, Malisiewicz, and Rabinovich, Revaud et al.(2019)Revaud, De
Souza, Humenberger, and Weinzaepfel] attempted to achieve better performance
by influencing detection and description simultaneously, with improved
repeatability and sparsity of detected points. Detector-free local feature
matcher [Sun et al.(2021)Sun, Shen, Wang, Bao, and Zhou, Zhou et
al.(2021)Zhou, Sattler, and Leal-Taixe] and a decoupled pipeline for a
detection and description module were also studied [Li et al.(2022)Li, Wang,
Liu, Ran, Xu, and Guo].
Despite these achievements, there is insufficient consideration for light and
structure information in an image. Examining this information to robustly
locate and define matched points, regardless of camera viewpoint or
illumination variance, is critical. Nighttime images are challenging due to
the uncertainties of light and structure [Sun et al.(2021)Sun, Shen, Wang,
Bao, and Zhou]. When viewpoints change significantly, it is also difficult to
match correctly [Balntas et al.(2017)Balntas, Lenc, Vedaldi, and Mikolajczyk].
Although some studies investigated this viewpoint and light information to
apply in the local feature domain, they used only hand-crafted ways such as
detecting corners or simply rotating features [Pautrat et al.(2020)Pautrat,
Larsson, Oswald, and Pollefeys, Liu et al.(2019)Liu, Shen, Lin, Peng, Bao, and
Zhou, Melekhov et al.(2020)Melekhov, Brostow, Kannala, and Turmukhambetov].
In this work, we propose a new strategy that uses both style and structure
information to address the issue of mismatches in image variance.
Specifically, we apply the concept of Instance Selective Whitening (ISW) loss,
introduced by RobustNet [Choi et al.(2021)Choi, Jung, Yun, Kim, Kim, and
Choo], where the features are transformed with implicit information about the
style component to have robustness under variations in light. Since there are
limitations in this idea and considers only the style factor, we revised ISW
to consider the structure factor and apply it to the local feature field.
Furthermore, we focus on salient points to reduce matching time.
Contributions. In this paper, we propose a framework that addresses the
problems from a different light and structural information. We first extract
features using Feature Map Generation (FMG) module. Then, Feature Map
Transformation (FMT) module divides extracted features into two components:
style and structure matrix. Each component independently learns the
information gathered by the learned feature. Consequently, regardless of any
changes in any component, the feature map will still select salient and
matchable points. We introduce a loss function to maximize the influence of
the structure information and minimize the style information in the feature
map. The main contributions are summarized as follows:
* •
We overcome the limitation of feature matching in image variance by
distinguishing between structure- and style-dependent features and
transforming the feature maps.
* •
We propose Feature Map Transformation (FMT) module, exploiting an existing
style transfer concept that concentrates only on style components, to
transform the feature map while training to make it focus on salient features.
* •
Extensive experiments on different benchmark datasets demonstrate that the
proposed method can achieve high accuracy in matching tasks in a short time
and with fewer parameters.
## 2 Related Work
Local feature learning. The joint learning of feature detectors and
descriptors requires a unified network to construct feature maps and allows
the two tasks to share the majority of computations for improved performance.
DELF [Noh et al.(2017)Noh, Araujo, Sim, Weyand, and Han] proposed an image
retrieval technique that learns local features as a by-product of a
classification loss combined with an attention mechanism to improve
performance on large-scale images. It outperforms images under changing light
conditions but has limitations in terms of structural variation. SuperPoint
[DeTone et al.(2018)DeTone, Malisiewicz, and Rabinovich] suggested a method
for learning from the manual annotation of significant points on simple images
such as corners and edges. However, because of their low repeatability and
descriptor accuracy, it has many outliers, so the matched points tend to be
mistakenly judged. R2D2 [Revaud et al.(2019)Revaud, De Souza, Humenberger, and
Weinzaepfel] has overcome this issue by learning the descriptor reliability in
parallel with the detection and description phases and only selecting both
repeatable and reliable keypoints with respect to the descriptor.
To find only the matchable points, D2-Net [Dusmanu et al.(2019)Dusmanu, Rocco,
Pajdla, Pollefeys, Sivic, Torii, and Sattler] proposed a describe-and-detect
method for joint detection and description that uses a single CNN with shared
weights. The detection is based on the entire channel’s local maxima and the
shape map’s spatial dimensions. DISK [Tyszkiewicz et al.(2020)Tyszkiewicz,
Fua, and Trulls] applied reinforcement learning on an end-to-end network
inspired by D2-Net [Dusmanu et al.(2019)Dusmanu, Rocco, Pajdla, Pollefeys,
Sivic, Torii, and Sattler] that relied on policy gradients. Furthermore,
ASLFeat [Luo et al.(2020)Luo, Zhou, Bai, Chen, Zhang, Yao, Li, Fang, and Quan]
demonstrated significant improvement using a score map that used local shape
estimation to select matching points.
Feature covariance. Previous research [Gatys et al.(2015)Gatys, Ecker, and
Bethge, Gatys et al.(2016)Gatys, Ecker, and Bethge] proposed that image style
information is considered via feature correlations such as a gram or a
covariance matrix. Since then, feature correlation has been applied to several
different research areas, including style transfer [Li et al.(2017)Li, Fang,
Yang, Wang, Lu, and Yang], image-to-image translation [Cho et al.(2019)Cho,
Choi, Park, Shin, and Choo], domain adaptation [Roy et al.(2019)Roy, Siarohin,
Sangineto, Bulo, Sebe, and Ricci, Sun and Saenko(2016)], and network
architecture [Luo(2017), Pan et al.(2019)Pan, Zhan, Shi, Tang, and Luo, Huang
et al.(2018)Huang, Yang, Lang, and Deng]. Whitening transformation (WT) [Li et
al.(2017)Li, Fang, Yang, Wang, Lu, and Yang, Cho et al.(2019)Cho, Choi, Park,
Shin, and Choo, Pan et al.(2019)Pan, Zhan, Shi, Tang, and Luo], which
eliminates feature correlation and assigns unit variance to each feature, aids
in the removal of style information from the feature representations.
Since region-specific styles and region-invariant content are simultaneously
written to the covariance vector of the feature maps, whitening all the
correlation components reduce feature identification and distort the
boundaries of objects [Li et al.(2017)Li, Fang, Yang, Wang, Lu, and Yang, Li
et al.(2018)Li, Liu, Li, Yang, and Kautz]. RobustNet [Choi et al.(2021)Choi,
Jung, Yun, Kim, Kim, and Choo] proposed the ISW loss, which extracted only the
style information to solve the problem. We want to focus on style and
structure information, so we modify the ISW loss to satisfy our objective.
## 3 Method
### 3.1 Feature Map Generation Module
Feature Map Generation (FMG) module first extracts the features of an image
pair, $I_{1}$ and $I_{2}$ independently, which outputs two branches:
descriptors and point extraction feature maps. The point extraction branch
consists of two feature maps. One produces another with a 1$\times$1
convolution layer; the former is a reliability map $\mathbf{S}$ and the latter
in repeatability map $\mathbf{R}$. The covariance matrix derived from the
descriptor map $\mathbf{X}$ is used to transform the feature map to focus on
saliency with style and structure information. Then FMG module uses feature
maps $\mathbf{X}$, $\mathbf{S}$, and $\mathbf{R}$ to calculate the loss
functions for repeatability and reliability. The FMG module’s network
architecture differs from R2D2 [Revaud et al.(2019)Revaud, De Souza,
Humenberger, and Weinzaepfel] but uses the same loss functions in this part.
So only the architecture part will be described. The proposed method pipeline
is shown in Figure 1.
Figure 1: Proposed framework. Network consisting of a feature map generation
module and a transformation module is shown. The reliability map $\mathbf{S}$
and repeatability map $\mathbf{R}$ learn the regions of points of interest,
while the covariance matrix derived from the descriptor map $\mathbf{X}$ is
used to transform the feature map to focus on saliency with style and
structure information.
As introduced in MobileNet [Howard et al.(2017)Howard, Zhu, Chen,
Kalenichenko, Wang, Weyand, Andreetto, and Adam], depthwise separable
convolution (DSC) is a factorization convolution method that significantly
reduces computation and model size with new representations. Motivated by
this, we used the DSC to focus on the salient area. Inspired by [Revaud et
al.(2019)Revaud, De Souza, Humenberger, and Weinzaepfel, Luo et al.(2020)Luo,
Zhou, Bai, Chen, Zhang, Yao, Li, Fang, and Quan], we adopt a modified L2Net
[Tian et al.(2017)Tian, Fan, and Wu]—where the last layer is replaced by a set
of three consecutive layers—in our backbone network to extract feature
information from an image pair. Since the input images are image pairs, two
backbone networks are needed. We use weight sharing in the DSC layer when
adding it behind the backbone network to reduce the model weight. Furthermore,
the relationship between descriptors and points of interest can be maintained
by sharing the weights. The backbone network then generates three feature
maps, $\mathbf{X}$ by $\ell$2 normalization, $\mathbf{S}$ by the element-wise
square, and $\mathbf{R}$ obtained from $\mathbf{S}$ with a 1$\times$1
convolution layer. In contrast to [Revaud et al.(2019)Revaud, De Souza,
Humenberger, and Weinzaepfel], $\mathbf{S}$ and $\mathbf{R}$ come from the
same branch since they depend on one another. We assumed that $\mathbf{S}$ is
a feature map that learns a point that reduces the matching distance, which
might affect in high repeatability rate in $\mathbf{R}$. Therefore, the model
weight gets lighter and develops more robust features through information
sharing.
FMG module calculates the reliability loss, $\mathcal{L}_{reli}$, to get
discriminative feature points. Let $\mathbf{X}_{ij}$ be the local descriptor
in each pixel $(i,j)$ of the image $I_{1}$; we then predict the individual
reliability scores $\mathbf{S}_{ij}$ from $\mathbf{X}_{ij}$ and
$\mathbf{X}^{{}^{\prime}}_{uv}$. Here, we specify the exact coordinate $(u,v)$
that corresponds to $(i,j)$, knowing the ground truth correspondence mapping
$T$, where $T\in\mathbb{R}^{H\times W\times 2}$ is the ground truth
correspondence between image $I_{1}$ and $I_{2}$. $\mathbf{X}_{ij}$ is
compared with $\mathbf{X}^{{}^{\prime}}_{uv}$, where
$\mathbf{X}^{{}^{\prime}}_{uv}$ is extracted from $I_{2}$. Then, average
precision is used to calculate $\mathcal{L}_{reli}$, optimized with a
differentiable approximation [He et al.(2018)He, Lu, and Sclaroff, Revaud et
al.(2019)Revaud, De Souza, Humenberger, and Weinzaepfel], using
$\mathbf{S}_{ij}$.
In addition, FMG module calculates the repeatability loss,
$\mathcal{L}_{repeat}$, for extracting repeatable feature points as in [Revaud
et al.(2019)Revaud, De Souza, Humenberger, and Weinzaepfel]. It uses peakiness
prediction and similarity between feature pairs from input pair images. For
similarity, Let $\mathbf{R}$ and $\mathbf{R^{{}^{\prime}}}$ be the
repeatability maps corresponding to $I_{1}$ and $I_{2}$. We set
$\mathbf{R}^{{}^{\prime}}_{T}$ to be a map in which $\mathbf{R^{{}^{\prime}}}$
is transformed by the ground truth homography relationship between image pairs
$I_{1}$ and $I_{2}$. Because the prime objective is to predict keypoints with
high repeatability, we train the network so that the positions of the local
maxima in $\mathbf{R}$ are covariant to the actual picture transformations,
such as viewpoint structures or light shifts. Assuming that all the local
maxima of $\mathbf{R}$ coincide with the local maxima of
$\mathbf{R}^{{}^{\prime}}_{T}$, we define the loss function. The basic concept
is to maximize the cosine similarity between $\mathbf{R}$ and
$\mathbf{R}^{{}^{\prime}}_{T}$ such that the two heatmaps are identical and
their maxima identically coincide. The loss may remain at a particular
constant that may terminate the learning process, so we prevent this using the
peakiness prediction. The final repeatability loss is calculated by
considering both similarity and the peakiness of the input image pair.
### 3.2 Feature Map Transformation Module
One problem with feature matching is structural perspective and lighting
(style) differences. The style corresponds to noise, such as weather and
light. Concentrating on the structure, which relates to the image’s point of
view or the edges, can result in better results. Feature Map Transformation
(FMT) module suppresses the style information by performing task adaptation
and supplementing the existing WT loss with an extensive transformation. We
improve the previous loss by including structural information to overcome the
limitations of using only existing styles.
Style/Structure Covariance Matrix. Previous studies [Li et al.(2019)Li, Zhang,
Yang, Liu, Song, and Hospedales, Choi et al.(2021)Choi, Jung, Yun, Kim, Kim,
and Choo] claimed that applying WT to each instance of style transfer could
successfully erase style information. WT is a linear transformation that
equalizes the variance term in each channel to one and reduces the covariances
between channels to zero. The intermediate feature map is
$\textbf{X}\in\mathbb{R}^{C\times HW}$, where $C$ is the number of channels,
and $H$ and $W$ are the height and width of the feature map, respectively. The
covariances between pairs of channels can be defined as follows:
$\Sigma=\frac{1}{HW}\left(\textbf{X}-\mathbf{\mu}\cdot\textbf{O}^{\top}\right)\left(\textbf{X}-\mathbf{\mu}\cdot\textbf{O}^{\top}\right)^{\top}\in\mathbb{R}^{C\times
C}$ (1)
where $\textbf{O}\in\mathbb{R}^{HW}$ is a column vector of ones, and the
$\mathbf{\mu}$ and $\Sigma$ are the mean vector and covariance matrix,
respectively. When the loss function is designed so that the elements of the
covariance matrix $\Sigma$ decrease, the feature extraction is less affected
by the style element in extracting features from the input image. This is
because the feature map generation lacks the information of style elements.
Based on this concept, we adopt a method of transforming feature maps using
style elements. Furthermore, we use information about the structure, the
leftover elements in the gram matrix beside style elements, to design the loss
function in the direction of expansion rather than suppression.
Figure 2: Transformed feature map. Comparison of feature maps from each
network indicates the tendency in which we change the feature map; eliminates
style information refers to noise, while structure complexity enlarges.
Transformation Loss. In order to transform and manipulate feature maps using
the aforementioned style and structure information, we introduce the FMT
module which use feature map X to calculate $\mathcal{L}_{cov}$. This is done
since the descriptor has the most information among the three output feature
maps from the FMG module. FMT module then separates style and structure
characteristics from the feature representation’s higher-order statistics by
selectively modifying each attribute differently. Replacing the old feature
map X with a standardized feature map $\mathbf{X}_{s}$ simplifies the
optimization process of both the diagonal and off-diagonal elements of the
covariance matrix simultaneously [Ulyanov et al.(2016)Ulyanov, Vedaldi, and
Lempitsky, Choi et al.(2021)Choi, Jung, Yun, Kim, Kim, and Choo]. $\Sigma_{s}$
is a matrix made from the standardized feature map $\mathbf{X}_{s}$, and we
define $\Sigma_{s}$ as follows:
$\Sigma_{s}=\frac{1}{HW}\mathbf{X}_{s}\cdot\mathbf{X}_{s}^{\top}\in\mathbb{R}^{C\times
C}$ (2)
After obtaining the covariance matrices $\Sigma_{s}(I_{1})$ and
$\Sigma_{s}(I_{2})$, we calculate the difference between the two matrices to
produce matrix $\Sigma_{C}$, as defined in Eq. 3. Matrix $\Sigma_{C}$
indicates the sensitivity of the corresponding covariance to the photometric
transformation [Choi et al.(2021)Choi, Jung, Yun, Kim, Kim, and Choo].
Elements with a high variance value retain the style information, whereas
elements with a low variance value retain the structure information. We use
absolute value notation with vertical bars.
$\Sigma_{C}=\left|\Sigma_{s}(I_{1})-\Sigma_{s}(I_{2})\right|$ (3)
We cluster the style and structural components in equal amounts, using the
mean value to determine the threshold for separating the two components. If a
matrix $\Sigma_{C}$ element is greater than the threshold, that element is
classified as a style factor, while the rest of the elements are classified as
structural factors. This definition is established because the prominent
factors of the matrix $\Sigma_{C}$ are assumed to imply the style factor [Choi
et al.(2021)Choi, Jung, Yun, Kim, Kim, and Choo]. In this case, the style
factor refers to changes in light or color, and the structural factor refers
to complexities with many objects, edges, or viewpoints. The proposed loss
function, $\mathcal{L}_{cov}$ is formulated as follows:
$\mathcal{L}_{cov}=\begin{cases}\mathbb{E}\left[\left\|\Sigma_{C}\odot\mathbf{M}_{sty}\right\|_{1}\right]&\text{
if }\Sigma_{C}>\mathbf{\mu}\\\ \
1-\mathbb{E}\left[\left\|\Sigma_{C}\odot\mathbf{M}_{str}\right\|_{1}\right]&\text{
otherwise}\end{cases}$ (4)
where $\mathbb{E}$ and $\mathbf{M}\in\mathbb{R}^{C\times C}$ are the
arithmetic mean and a mask matrix. $\mathbf{M}_{sty}$ and $\mathbf{M}_{str}$
are masks that select style and structure values, and $\odot$ is element-wise
multiplication. Finally, the total loss function can be represented by Eq. 5,
with each weight $\lambda_{i}$ are empirically tuned to the optimal ratio of
1:1:2. We define $i\in\left\\{1,2,3\right\\}$ because the number of loss terms
is three. We strengthen the argument that adding the transformation loss is
superior in selecting only salient features when comparing the feature map of
R2D2 [Revaud et al.(2019)Revaud, De Souza, Humenberger, and Weinzaepfel] with
ours in Figure 2.
$\mathcal{L}_{total}=\lambda_{1}\cdot\mathcal{L}_{reli}+\lambda_{2}\cdot\mathcal{L}_{repeat}+\lambda_{3}\cdot\mathcal{L}_{cov}$
(5)
This aggregated loss function is used to select salient points to minimize the
prediction of the less informative regions, such as the sky or ground. FMT is
inducing the feature map transformation so that the proposed transformation
loss function could extract robust features if the location where the image is
taken is in the same place, regardless of the change in structure and style.
## 4 Experiment
### 4.1 Implementation details
Training. We apply Adam to optimize the network for 25 epochs with a fixed
learning rate of 0.0001, a weight decay of 0.0005, and a batch size of 8 pairs
of cropped images of 192 by 192 pixels, as in R2D2 [Revaud et al.(2019)Revaud,
De Souza, Humenberger, and Weinzaepfel]. Our experiment used the training
dataset and ground-truth correspondences used in R2D2. Since our model uses
the modified version of R2D2, we trained the network from scratch.
Nonetheless, we fixed the patch size N used in the repeatability loss to 16 in
all training parts to improve the performance of the transformation loss.
Testing. We used the sum of different scales of images to diversify the
resolution of the feature maps at test time. The descriptors were interpolated
at the modified locations. This multi-scale feature extraction enables the
extraction of more tentative keypoints and provides improved localization.
Experiment Settings. This study used an NVIDIA GeForce RTX 3090 GPU and CUDA
toolkit, version 11.2, with Python 3 and PyTorch 1.8 in the training
environment.
Figure 3: Quantitative Results on HPatches. Comparison in terms of MMA on the
HPatches dataset.
MMA@3 Overall Illumi. Viewp. Hes.Aff. [Perd’och et al.(2009)Perd’och, Chum,
and Matas] 56.24 51.35 60.79 DELF(new) [Noh et al.(2017)Noh, Araujo, Sim,
Weyand, and Han] 49.43 89.73 12.02 SuperPoint [DeTone et al.(2018)DeTone,
Malisiewicz, and Rabinovich] 64.45 69.38 59.88 LF-Net [Ono et al.(2018)Ono,
Trulls, Fua, and Yi] 53.01 57.31 49.02 D2-Net [Dusmanu et al.(2019)Dusmanu,
Rocco, Pajdla, Pollefeys, Sivic, Torii, and Sattler] 39.76 44.99 34.91 R2D2
[Revaud et al.(2019)Revaud, De Souza, Humenberger, and Weinzaepfel] 70.06
75.56 64.96 ASLFeat [Luo et al.(2020)Luo, Zhou, Bai, Chen, Zhang, Yao, Li,
Fang, and Quan] 72.28 75.47 68.28 DISK [Tyszkiewicz et al.(2020)Tyszkiewicz,
Fua, and Trulls] 75.34 79.43 71.53 Ours 78.41 83.22 73.94
Table 1: MMA@3 on HPatches. Comparison at 3px threshold.
### 4.2 Feature Matching
Quantitative Evaluation. We evaluated the performance of selecting meaningful
points by calculating mean matching accuracy (MMA). If the distance between
the converted point and the reference point exists within the threshold based
on the ground truth homography metric, the converted point is classified as a
correct conversion point. Figure 3 illustrates the comparisons on the
H-Patches dataset [Balntas et al.(2017)Balntas, Lenc, Vedaldi, and
Mikolajczyk], with MMA measured at various error thresholds. Figure 3 was
drawn from the cache data provided by the D2-Net [Dusmanu et al.(2019)Dusmanu,
Rocco, Pajdla, Pollefeys, Sivic, Torii, and Sattler] repository. The
comparison was performed with DELF [Noh et al.(2017)Noh, Araujo, Sim, Weyand,
and Han], SuperPoint [DeTone et al.(2018)DeTone, Malisiewicz, and Rabinovich],
LF-Net [Ono et al.(2018)Ono, Trulls, Fua, and Yi] mono and multi-scale D2-Net
[Dusmanu et al.(2019)Dusmanu, Rocco, Pajdla, Pollefeys, Sivic, Torii, and
Sattler], R2D2 [Revaud et al.(2019)Revaud, De Souza, Humenberger, and
Weinzaepfel], ASLFeat [Luo et al.(2020)Luo, Zhou, Bai, Chen, Zhang, Yao, Li,
Fang, and Quan], DISK [Tyszkiewicz et al.(2020)Tyszkiewicz, Fua, and Trulls],
and a hand-crafted Hessian affine detector with a RootSIFT descriptor
[Perd’och et al.(2009)Perd’och, Chum, and Matas]. Our network denoted as
"Ours" outperformed almost all state-of-the-art networks. Regarding
illumination, DELF [Noh et al.(2017)Noh, Araujo, Sim, Weyand, and Han]
outperformed our method since it identifies key points in a low-resolution
feature map with a fixed grid. However, ours still exhibited the highest
performance in terms of the five-pixel threshold because of the fixed grid of
keypoints without spatial variation in this subgroup. The overall scores for
the illumination dataset and viewpoints and their respective individual scores
are presented in Table 1, with the MMA threshold set to 3.
Qualitative Evaluation. Figure 4 shows the comparison between our baseline
network R2D2 [Revaud et al.(2019)Revaud, De Souza, Humenberger, and
Weinzaepfel] and the current state-of-the-art method DISK [Tyszkiewicz et
al.(2020)Tyszkiewicz, Fua, and Trulls] with a severe change in illumination
and viewpoint in the first and second row, respectively. The experiment was
done with the nearest neighborhood matching with 3 points of error threshold.
When we look at the yellow box, we see that we succeeded in focusing on
structural information. The structurally less focused network failed to match
in this part. In the red box, it can be seen that the error rate is lowered by
focusing less on the background or natural objects corresponding to noise. The
matching time is also shorter, shown in Figure 5.
### 4.3 Visual Localization
In a local reconstruction job [Sattler et al.(2017)Sattler, Torii, Sivic,
Pollefeys, Taira, Okutomi, and Pajdla], we evaluate our technique on the
Aachen Day-Night dataset v1.1 [Sattler et al.(2018)Sattler, Maddern, Toft,
Torii, Hammarstrand, Stenborg, Safari, Okutomi, Pollefeys, Sivic, et al.] as
in D2-Net [Dusmanu et al.(2019)Dusmanu, Rocco, Pajdla, Pollefeys, Sivic,
Torii, and Sattler]. In this section, we present the results of an optical
localization task. Given the daytime photographs with known camera positions,
our goal is to identify the nighttime image of the same area. The known
location of the daytime photos in each set is used to triangulate the 3D
structure of the scene after extensive feature matching. Finally, these 3D
models are used to locate the query photographs taken at night. We followed
the guidelines for Visual Localization Benchmark, for which we used our
matches as input for a pre-defined visual localization pipeline based on
COLMAP [Schonberger and Frahm(2016), Schönberger et al.(2016)Schönberger,
Zheng, Frahm, and Pollefeys]. We also adopted hierarchical localization
[Sarlin et al.(2019)Sarlin, Cadena, Siegwart, and Dymczyk] in every network
for higher performance. This pipeline was then used to build an SfM model with
the registered test photos. We used NetVlad [Arandjelovic et
al.(2016)Arandjelovic, Gronat, Torii, Pajdla, and Sivic] for the global
feature and the nearest neighborhood for matching. The percentages of properly
localized photos under three error levels are reported in Table 2. Our results
demonstrate our method’s strong generalization capabilities because of its
high localization performance compared with SuperPoint [DeTone et
al.(2018)DeTone, Malisiewicz, and Rabinovich], D2-Net [Dusmanu et
al.(2019)Dusmanu, Rocco, Pajdla, Pollefeys, Sivic, Torii, and Sattler], R2D2
[Revaud et al.(2019)Revaud, De Souza, Humenberger, and Weinzaepfel] and DISK
[Tyszkiewicz et al.(2020)Tyszkiewicz, Fua, and Trulls]. Our network performed
fairly well when compared with matcher methods [Sun et al.(2021)Sun, Shen,
Wang, Bao, and Zhou, Zhou et al.(2021)Zhou, Sattler, and Leal-Taixe].
Figure 4: Qualitative Results on HPatches. There are illustrated pairs of
illumination and viewpoint variations. Green dots represent points that are
correctly matched, whereas red dots represent points that are incorrectly
matched. The boxed area provides significantly better outcomes than previous
methods by focusing on the salient region.
Day Night 0.5m, $2^{\circ}$ 1m, $5^{\circ}$ 5m, $10^{\circ}$ 0.5m, $2^{\circ}$
1m, $5^{\circ}$ 5m, $10^{\circ}$ SuperPoint [DeTone et al.(2018)DeTone,
Malisiewicz, and Rabinovich] 85.3 91.9 94.5 58.6 74.3 85.9 D2-Net [Dusmanu et
al.(2019)Dusmanu, Rocco, Pajdla, Pollefeys, Sivic, Torii, and Sattler] 81.6
89.3 96.2 62.8 80.6 92.7 R2D2 [Revaud et al.(2019)Revaud, De Souza,
Humenberger, and Weinzaepfel] 89.9 95.4 98.4 69.6 85.9 96.3 DISK [Tyszkiewicz
et al.(2020)Tyszkiewicz, Fua, and Trulls] - - - 72.3 86.4 97.9 Ours 90.4 96.1
98.9 72.3 89.0 96.9 LoFTR∗ [Sun et al.(2021)Sun, Shen, Wang, Bao, and Zhou] -
- - 72.8 88.5 99.0 Patch2Pix∗ [Zhou et al.(2021)Zhou, Sattler, and Leal-Taixe]
86.4 93.0 97.5 72.3 88.5 97.9
Table 2: Aachen Evaluation. Comparison on Aachen for the visual localization
task. * are marked for matcher methods.
Figure 5: Matching Time. Comparison of matching time.
### 4.4 3D Reconstruction
For 3D reconstruction evaluation, we used ETH-Microsoft Dataset [Schonberger
et al.(2017)Schonberger, Hardmeier, Sattler, and Pollefeys] and for the
evaluation protocols, we ran SfM algorithm by COLMAP [Schonberger and
Frahm(2016), Schönberger et al.(2016)Schönberger, Zheng, Frahm, and
Pollefeys]. In Table 3, we compare our method with the results presented in
[Luo et al.(2020)Luo, Zhou, Bai, Chen, Zhang, Yao, Li, Fang, and Quan], but
only the jointly learned models. Compared with other methods, since the number
of matched points was less than 100K, we do not provide the comparison with
R2D2 [Revaud et al.(2019)Revaud, De Souza, Humenberger, and Weinzaepfel]. For
sparse reconstruction, we report the number of registered images ($\\#$ Reg),
the number of sparse points ($\\#$ Sparse), tracked length (Track), and
reprojection error (Reproj). Table 3 data shows that our network performs
favorably against previous methods on the 3D reconstruction task. Furthermore,
the number of registered images obtained the best value, and the number of
sparse points also had the best or second-order value. It can be interpreted
that the selection of salient points was made well.
Madrid Metropolis 1344 images Gendarmenmarkt 1463 images Tower of London 1576
images $\\#$ Reg $\\#$ Sparse Track Reproj $\\#$ Reg $\\#$ Sparse Track Reproj
$\\#$ Reg $\\#$ Sparse Track Reproj SuperPoint [DeTone et al.(2018)DeTone,
Malisiewicz, and Rabinovich] 438 29K 9.03 1.02px 967 93K 7.22 1.03px 681 52K
8.67 0.96px D2-Net [Dusmanu et al.(2019)Dusmanu, Rocco, Pajdla, Pollefeys,
Sivic, Torii, and Sattler] 495 144K 6.39 1.35px 965 310K 5.55 1.28px 708 287K
5.20 1.34px ASLFeat [Luo et al.(2020)Luo, Zhou, Bai, Chen, Zhang, Yao, Li,
Fang, and Quan] 649 129K 9.56 0.95px 1061 320K 8.98 1.05px 846 252K 13.16
0.95px Ours 766 142K 8.13 1.19px 1316 516K 6.81 1.19px 1186 315K 8.63 1.21px
Table 3: ETH Evaluation. 3D reconstruction held with ETH-Microsoft Dataset.
HPatches Aachen (Day) Aachen (Night) Overall Illum Viewp 0.5m, 2∘ 1m, 5∘ 5m,
10∘ 0.5m, 2∘ 1m, 5∘ 5m, 10∘ w/o $\mathcal{L}_{sty}\&\mathcal{L}_{str}$ 70.06
75.56 64.96 89.9 95.4 98.4 69.6 85.9 96.3 w/o $\mathcal{L}_{sty}$ 72.08 78.04
66.55 89.8 96.1 98.7 73.3 88.5 95.3 w/o $\mathcal{L}_{str}$ 76.53 81.43 71.99
89.7 95.8 98.5 69.6 84.8 96.3 w/o DSC 76.89 81.95 72.19 89.9 95.3 98.2 69.1
85.9 93.7 w/ All 78.41 83.22 73.94 90.4 96.1 98.9 72.3 89.0 96.9
Table 4: Ablation Studies. Ablation experiment on MMA@3 and visual
localization to see how each component affect transformation loss. We show
that the best results occur when style and structure loss are included with
depth-wise convolution.
### 4.5 Ablation Studies
We validated the significance of the components that comprise our suggested
transformation loss function by performing two ablation studies. We studied
the presence of style and structural component losses to determine how each
characteristic contributes to learning. $\mathcal{L}_{sty}$ and
$\mathcal{L}_{str}$ denote loss function that use same subscript in the masks
of Eq. 4. The result in Table 4 reveals that the matching accuracy is not only
influenced by the style factor but also by the structure factor. This
experiment confirms the efficiency of ISW when applied to our method and its
ability to provide additional information about the structure. Furthermore,
ablation studies were conducted with and without DSC layer. The performance
improved using DSC layer, with a model weight reduction of 2 MB.
## 5 Conclusion
We proposed a robust network using self-transformation loss, which transforms
a feature map that contributes to the repeatability of local features. We
separated structure and style characteristics by clustering the covariance
vector and influenced the feature of each characteristic. The feature maps
were unified to make two feature maps closer by reducing the light component
and sharpening the structure component. Consequently, this step increases the
repeatability of the points, reduces outliers, and assigns robustly matched
descriptors. A comparison with similar research reveals that our method more
effectively extracts robust matching points in various scenes. Nevertheless,
the current work has limitations. Points tend to be retrieved by clustering
them in a particular region. This analysis reveals that adaptively selecting
from sparse points is a promising avenue for future research.
## 6 Acknowledgment
This work was supported in part by the National Research Foundation of Korea
(NRF) under Grant NRF-2021R1A2C2010893 and in part by Institute of Information
& communications Technology Planning & Evaluation (IITP) grant funded by the
Korea government(MSIT) (No.RS-2022-00155915, Artificial Intelligence
Convergence Innovation Human Resources Development (Inha University).
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# The linear span of uniform matrix product states
Claudia De Lazzari Dipartimento di Matematica, Università di Trento, Via
Sommarive 14, 38123 Povo (TN), Italy<EMAIL_ADDRESS>, Harshit J
Motwani Department of Mathematics: Algebra and Geometry, Ghent University,
9000 Gent, Belgium<EMAIL_ADDRESS>and Tim Seynnaeve
Mathematical Institute, University of Bern, Alpeneggstrasse 22, 3012 Bern,
Switzerland<EMAIL_ADDRESS>
###### Abstract.
The variety of uniform matrix product states arises both in algebraic geometry
as a natural generalization of the Veronese variety, and in quantum many-body
physics as a model for a translation-invariant system of sites placed on a
ring. Using methods from linear algebra, representation theory, and invariant
theory of matrices, we study the linear span of this variety.
###### Key words and phrases:
Matrix product states, invariant theory of matrices
###### 2020 Mathematics Subject Classification:
15A69, 20G05, 81P45
## Introduction
Tensor networks are a powerful tool for the description of tensors living in a
high dimensional space. Since their original conception in quantum many-body
physics [AKLT88, FNW92, ÖR95], they have found a wide range of applications in
different fields, such as numerical tensor calculus [Ose11, Hac12, CLO+16,
BSU16], geometric complexity theory [LQY12], graphical models [RS19], and
machine learning [CLO+16, CCX+18]. Methods from differential and complex
geometry were introduced in the study of these varieties of tensors in
[HMOV14] and some important developments were achieved using methods from
algebraic geometry and representation theory, see for instance [BBM15, LQY12,
CM14, CMS19, CLVW20, CGFW21].
In this article, we focus on a particular type of tensor networks: uniform
matrix product states. From a quantum mechanics perspective, they model
translation-invariant physical systems of sites placed on a ring. The geometry
of uniform matrix product states has been extensively studied [PGVWC07,
HMOV14, CM14, CMS19], but our understanding of them is still far from
complete, and several fundamental mathematical problems remain open.
Our geometric point of view is the following: we fix a tensor space
$(\mathbb{C}^{n})^{\otimes d}$, and consider the set of all tensors in this
space that admit a translation-invariant matrix product state representation,
with a given bond dimension $m$. After taking the closure of this set, we
obtain an algebraic variety, which we denote by $\operatorname{uMPS}(m,n,d)$.
The ultimate goal would be to obtain a complete description of this variety,
i.e. to find all defining equations. Phrased in this generality, this question
is likely to be intractable. Indeed, even just determining which _linear_
equations, if any, vanish on our variety is poorly understood. This is
precisely the goal of this paper:
###### Problem 0.1.
Describe the linear span $\langle\operatorname{uMPS}(m,n,d)\rangle$ of the
variety $\operatorname{uMPS}(m,n,d)$. More precisely:
* •
What is the dimension of $\langle\operatorname{uMPS}(m,n,d)\rangle$? (1.4)
* •
For which parameters $m,n,d\in\mathbb{N}$ does
$\langle\operatorname{uMPS}(m,n,d)\rangle$ fill the ambient space? (1.6)
* •
How does $\langle\operatorname{uMPS}(m,n,d)\rangle$ decompose as a
$GL_{n}$-representation? (1.9)
The variety $\operatorname{uMPS}(m,n,d)$ does not only arise from tensor
networks, it is also a very natural geometric construction in its own right.
Indeed, as we will soon see, $\operatorname{uMPS}(m,n,d)$ is the closed image
of the polynomial map that takes as input an $n$-tuple of $m\times
m$-matrices, and outputs the traces of all $d$-fold products of the given
matrices. In this way, it is a natural generalization of the classical
Veronese variety. Our main Problem 0.1 is therefore equivalent to the
following:
###### Problem 0.2.
Let $A_{0},\ldots,A_{n-1}$ be $m\times m$ matrices with generic entries. Which
linear relations hold between the polynomials
$\operatorname{Tr}(A_{i_{1}}\cdots A_{i_{d}}),$
where $(i_{1},\ldots,i_{d})\in[n]^{d}$?
The ring generated by all polynomials $\operatorname{Tr}(A_{i_{1}}\cdots
A_{i_{d}})$, where the generic matrices are fixed but we allow $d$ to vary, is
known as the _trace algebra_. In his classical work [Pro76], Procesi described
how to obtain all relations between the generators of this ring in principle.
Note however, that the question we are asking is slightly different: we are
only interested in relations between traces of matrix products of a _fixed
length_ $d$.
This article is divided into three sections. In Section 1, we define the
variety of uniform matrix product states, and describe its natural symmetries.
In Section 2, we undertake a computational study of the space
$\langle\operatorname{uMPS}(m,n,d)\rangle$ in the smallest nontrivial case
$m=n=2$. We describe an algorithm that can compute this space, viewed as a
$GL_{2}$-representation. Based on this result, we obtain a conjectured formula
for the character (and in particular: the dimension) of
$\langle\operatorname{uMPS}(2,2,d)\rangle$. Section 3 contains our main
theoretical results. In Section 3.1, we introduce a powerful method to find
linear equations that vanish on $\langle\operatorname{uMPS}(m,n,d)\rangle$,
based on the Cayley-Hamilton theorem. As a corollary, we show that for
$d\geq(m+1)(m+2)$, the linear span $\langle\operatorname{uMPS}(m,n,d)\rangle$
does not fill its natural ambient space
$\operatorname{Cyc}^{d}(\mathbb{C}^{n})$, the space of cyclically symmetric
tensors; significantly improving the state of the art. In Section 3.2 we study
the special case $m=n=2$, based on the computations in Section 3.1. Using the
trace parametrization we show an upper bound on the dimension of
$\langle\operatorname{uMPS}(2,2,d)\rangle$ which is close to optimal, and we
take some first steps towards proving our conjectured character formula, again
using our Cayley-Hamilton technique.
## Acknowledgements
The third author would like to thank Alessandra Bernardi, Jarosław Buczyński,
Joseph Landsberg, and Frank Verstraete for many helpful discussions. The
second author is supported by FWO grants (G023721N, G0F5921N) and UGent BOF
grants (BOF21/DOC/182, STA/201909/038).
## 1\. Preliminaries
We once and for all fix three parameters $m,n,d\in\mathbb{N}$. We will
consider tensors in the space $(\mathbb{C}^{n})^{\otimes d}$. The standard
basis of $\mathbb{C}^{n}$ will be written as $\\{e_{0},\ldots,e_{n-1}\\}$. We
abbreviate the set $\\{0,\ldots,n-1\\}$ to $[n]$.
###### Definition 1.1.
The uniform Matrix Product State parametrization is given by the map
(1) $\displaystyle\begin{split}\varphi:(\mathbb{C}^{m\times
m})^{n}&\to(\mathbb{C}^{n})^{\otimes d}\\\
(A_{0},\dots,A_{n-1})&\mapsto\sum_{0\leq i_{1},\dots,i_{d}\leq
n-1}\operatorname{Tr}(A_{i_{1}}\cdots A_{i_{d}})\
e_{i_{1}}\otimes\cdots\otimes e_{i_{d}}.\end{split}$
We denote the image of this map by $\operatorname{uMPS}^{\circ}(m,n,d)$. The
closure of $\operatorname{uMPS}^{\circ}(m,n,d)$ in the Euclidean or
equivalently Zariski topology over the complex numbers, is the algebraic
variety of uniform matrix product states, denoted by
${\operatorname{uMPS}(m,n,d)}$.
###### Remark 1.2.
If we think of $(\mathbb{C}^{m\times m})^{n}$ as the space of $m\times m\times
n$ tensors, then the $\operatorname{uMPS}$ parametrization takes a tensor in
this space and contracts it $d$ times with itself in a circle; see Figure 1.
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ Figure 1. Graphical
representation of map (1) defining $\operatorname{uMPS}(m,n,d)$. The weights
on the edges represent the dimensions of the vector space along which the
tensor contraction is taking place. There are total $d$ tensors involved with
dimensions $m\times m\times n$. The parameter $m$ is called bond dimension and
$n$ is called physical dimension.
###### Remark 1.3.
An alternative name for $\operatorname{uMPS}$ is _translation invariant matrix
product states with periodic boundary conditions_ [PGVWC07, CM14]. The name
“uniform matrix product states” is sometimes reserved for the thermodynamic
limit, where the number $d$ of sites approaches infinity. Our terminology is
consistent with [HMOV14, CMS19].
As mentioned in the introduction, the main question we will try to answer in
this article is the following:
###### Question 1.4.
Determine the linear span of $\operatorname{uMPS}(m,n,d)$; i.e. the smallest
vector subspace of $(\mathbb{C}^{n})^{\otimes d}$ containing
$\operatorname{uMPS}(m,n,d)$. In particular: what is the dimension of this
space?
### 1.1. Cyclic and symmetric invariance
The space $(\mathbb{C}^{n})^{\otimes d}$ comes equipped with an action of the
symmetric group $\mathfrak{S}_{d}$: for $\sigma\in\mathfrak{S}_{d}$ and
$\omega=v_{1}\otimes\cdots\otimes v_{d}\in(\mathbb{C}^{n})^{\otimes d}$ we
have
$\sigma\cdot(v_{1}\otimes\cdots\otimes
v_{d})=v_{\sigma^{-1}(1)}\otimes\cdots\otimes v_{\sigma^{-1}(d)}.$
The symmetric group $\mathfrak{S}_{d}$ naturally contains the cyclic group
$C_{d}$ and the dihedral group $D_{2d}$ as subgroups. To be precise: let
$r,s\in\mathfrak{S}_{d}$ be the cyclic permutation and reflection defined
respectively by
$r(i)=i+1(\operatorname{mod}d)\quad\text{and}\quad s(i)=d+1-i,$
then $C_{d}\subseteq\mathfrak{S}_{d}$ is the cyclic subgroup generated by $r$,
and $D_{2d}$ is the subgroup generated by $r$ and $s$. The _cyclically
symmetric tensors_ and _dihedrally symmetric tensors_ are then the elements of
$(\mathbb{C}^{n})^{\otimes d}$ that are invariant under the action of these
subgroups:
$\displaystyle\operatorname{Cyc}^{d}(\mathbb{C}^{n}):=\\{\omega\in(\mathbb{C}^{n})^{\otimes
d}\mid\sigma\cdot\omega=\omega\quad\forall\sigma\in C_{d}\\},$
$\displaystyle\operatorname{Dih}^{d}(\mathbb{C}^{n}):=\\{\omega\in(\mathbb{C}^{n})^{\otimes
d}\mid\sigma\cdot\omega=\omega\quad\forall\sigma\in D_{2d}\\}.$
Note that both are linear subspaces of $(\mathbb{C}^{n})^{\otimes d}$, and
that
$\operatorname{Dih}^{d}(\mathbb{C}^{n})\subseteq\operatorname{Cyc}^{d}(\mathbb{C}^{n})$,
where the inclusion is strict unless $d\leq 2$ or $n=2$ and $d\leq 5$.
###### Observation 1.5.
The set $\operatorname{uMPS}(m,n,d)$ is a subset of the space of cyclically
invariant tensors
$\operatorname{Cyc}^{d}(\mathbb{C}^{n})\subset(\mathbb{C}^{n})^{\otimes d}$
because of the trace invariance under cyclic permutations of the matrices:
given $M_{1},\dots,M_{d}\in\mathbb{C}^{m\times m}$ then
$\operatorname{Tr}(M_{1}\cdots M_{d})=\operatorname{Tr}(M_{\sigma(1)}\cdots
M_{\sigma(d)}),$
for $\sigma\in C_{d}$.
In other words, we can think of $\operatorname{uMPS}(m,n,d)$ as a subvariety
of the ambient space $\operatorname{Cyc}^{d}(\mathbb{C}^{n})$. As noted in
[CMS19, Corollary 3.18], if we fix $n$ and $d$ and let $m$ grow, the space
$\operatorname{uMPS}(m,n,d)$ will eventually fill this entire ambient space.
###### Question 1.6.
For fixed $n$ and $d$, what is the smallest $m$ such that
$\langle\operatorname{uMPS}(m,n,d)\rangle=\operatorname{Cyc}^{d}(\mathbb{C}^{m})$?
It is known that for $m=d$, equality holds [CMS19, Proposition 3.11]. On the
other hand, it follows from a dimension count ([CMS19, Theorem 3.14], see also
[NV18]) that if $d\gg m$, the inclusion
$\langle\operatorname{uMPS}(m,n,d)\rangle\subset\operatorname{Cyc}^{d}(\mathbb{C}^{n})$
is strict. In Section 3.1 we will prove that already for $d=O(m^{2})$, we have
a strict inclusion.
###### Observation 1.7.
In the case $m=n=2$, we have the stronger inclusion
$\operatorname{uMPS}(2,2,d)\subseteq\operatorname{Dih}^{d}(\mathbb{C}^{n}).$
This is a consequence of the identity $\operatorname{Tr}(A_{i_{1}}\cdots
A_{i_{d}})=\operatorname{Tr}(A_{i_{d}}\cdots A_{i_{1}})$, which holds for any
pair of $2\times 2$ matrices $A_{0}$, $A_{1}$ and sequence
$i_{1},\ldots,i_{d}$ with $i_{j}\in\\{0,1\\}$. See [Gre14, Theorem 1.1]
###### Example 1.8.
We consider $\operatorname{uMPS}(2,2,6)$. For every
$A_{0},A_{1}\in\mathbb{C}^{2\times 2}$, we have the trace relation
$\operatorname{Tr}(A_{0}^{2}A_{1}^{2}A_{0}A_{1})=\operatorname{Tr}(A_{1}A_{0}A_{1}^{2}A_{0}^{2})$,
that does not come from invariance of trace under cyclic permutations of the
matrices.
### 1.2. $GL_{n}$-invariance
The general linear group $GL_{n}$ naturally acts on the space
$(\mathbb{C}^{n})^{\otimes d}$: given $g\in GL_{n}$ and
$\omega=v_{1}\otimes\cdots\otimes v_{d}\in(\mathbb{C}^{n})^{\otimes d}$, we
have
(2) $g\cdot(v_{1}\otimes\cdots\otimes v_{N})=(g\cdot
v_{1})\otimes\cdots\otimes(g\cdot v_{N}).$
Clearly, $\operatorname{Cyc}^{d}(\mathbb{C}^{n})$ and
$\operatorname{Dih}^{d}(\mathbb{C}^{n})$ are invariant under this action. The
following computation shows that $\operatorname{uMPS}(m,n,d)$ is invariant
under this action as well:
(3) $\displaystyle g\cdot\varphi(A_{1},\dots,A_{n})$
$\displaystyle=\sum_{j_{1},\dots,j_{d}=1}^{n}\operatorname{Tr}(A_{j_{1}}\cdots
A_{j_{d}})\
(\sum_{i_{1}=1}^{n}{g_{i_{1},j_{1}}e_{i_{1}}})\otimes\cdots\otimes(\sum_{i_{d}=1}^{n}{g_{i_{d},j_{d}}e_{i_{d}}})$
$\displaystyle=\sum_{j_{1},\dots,j_{d}=1}^{n}\sum_{i_{1},\dots,i_{d}=1}^{n}g_{i_{1},j_{1}}\cdots
g_{i_{d},j_{d}}\operatorname{Tr}(A_{j_{1}}\cdots A_{j_{d}})\
e_{i_{1}}\otimes\cdots\otimes e_{i_{d}}$
$\displaystyle=\sum_{j_{1},\dots,j_{d}=1}^{n}\sum_{i_{1},\dots,i_{d}=1}^{n}\operatorname{Tr}(g_{i_{1},j_{1}}A_{j_{1}}\cdots
g_{i_{d},j_{d}}A_{j_{d}})\ e_{i_{1}}\otimes\cdots\otimes e_{i_{d}}$
$\displaystyle=\sum_{i_{1},\dots,i_{d}=1}^{n}\operatorname{Tr}\Big{[}\Big{(}\sum_{j_{1}=1}^{n}g_{i_{1},j_{1}}A_{j_{1}}\Big{)}\cdots\Big{(}\sum_{j_{d}=1}^{n}g_{i_{d},j_{d}}A_{j_{d}}\Big{)}\Big{]}e_{i_{1}}\otimes\cdots\otimes
e_{i_{d}}$
$\displaystyle=\varphi\Big{(}\sum_{j=1}^{n}g_{1,j}A_{j},\dots,\sum_{j=1}^{n}g_{n,j}A_{j}\Big{)}.$
This means that the space $\langle\operatorname{uMPS}(m,n,d)\rangle$ we are
interested in is naturally a representation of $GL_{n}$. In the remainder of
this subsection, we briefly recall what we will use about representation
theory of $GL_{n}$, and fix notation.
We once and for all fix a torus $T\subset GL_{n}$, consisting of all diagonal
matrices; and identify $T$ with $(\mathbb{C}^{*})^{n}$. For
$\lambda=(\lambda_{0},\ldots,\lambda_{n-1})\in\mathbb{Z}^{n}$ and
$t=\operatorname{diag}(t_{0},\ldots,t_{n-1})\in T$, we write
$t^{\lambda}=t_{0}^{\lambda_{0}}\cdots
t_{n-1}^{\lambda_{n-1}}\in\mathbb{C}^{*}$. Let $V$ be any representation of
$GL_{n}$. For any $\lambda\in\mathbb{Z}^{n}$, the _weight space_ $V_{\lambda}$
is defined as
$V_{\lambda}=\\{v\in V\mid t\cdot v=t^{\lambda}v\quad\forall t\in T\\}.$
It is a well-known fact from representation theory that
$V=\bigoplus_{\lambda\in\mathbb{Z}^{n}}{V_{\lambda}}$
as vector spaces; and that knowing the dimensions of the weight spaces
uniquely determines the representation $V$ up to isomorphism. The polynomial
$\chi_{V}=\sum_{\lambda\in\mathbb{Z}^{n}}{(\dim V_{\lambda})t^{\lambda}}.$
is known as the _character_ of $V$. If we view our representation as a
morphism $\rho:GL_{n}\to GL(V)$, the character $\chi_{V}$ is equal to
$\operatorname{Tr}(\rho(\operatorname{diag}(t_{0},\ldots,t_{n-1})))$.
So we can refine our main Question 1.4 to:
###### Question 1.9.
Let $V=\langle\operatorname{uMPS}(m,n,d)\rangle$, viewed as a
$GL_{n}$-representation. For every weight $\lambda\in\mathbb{Z}^{n}$,
determine the dimension of the weight space $V_{\lambda}$.
### 1.3. Words, necklaces and bracelets
As a warm-up, let us consider the classical representation
$V=(\mathbb{C}^{n})^{\otimes d}$. Its character is equal to
$(t_{1}+\cdots+t_{n})^{d}$, and by expanding we find that $\dim{V_{\lambda}}$
is equal to the multinomial coefficient
$\binom{d}{\lambda_{1},\ldots,\lambda_{n}}$ if $\sum_{\lambda_{i}}=d$, and
zero otherwise. We can also see this in terms of coordinates, and this will be
useful later:
###### Definition 1.10.
A _word_ of length $d$ on the alphabet $[n]$ is just an ordered tuple
$I=(i_{1},\ldots,i_{d})$, with $i_{j}\in[n]$. The _weight_ of a word $I$ is a
tuple $w(I)=(w_{0},\ldots,w_{n-1})\in\mathbb{Z}^{n}$, where $w_{i}$ is the
number of entries in $I$ that are equal to $i$.
For every word $I=(i_{1},\ldots,i_{d})$, we can define a vector
$e_{I}:=e_{i_{1}}\otimes\cdots\otimes e_{i_{d}}$. The space
$(\mathbb{C}^{n})^{\otimes d}$ has a basis given by the $e_{I}$, where $I$
runs over all words of length $d$ on the alphabet $[n]$. In addition, every
$e_{I}$ is a weight vector of weight $w(I)$. So the dimension of the weight
space $V_{\lambda}$ is the number of words of weight $\lambda$, which is
indeed the multinomial coefficient
$\binom{d}{\lambda_{1},\ldots,\lambda_{n}}$.
We move on to the subrepresentations $\operatorname{Cyc}^{d}(\mathbb{C}^{n})$
and $\operatorname{Dih}^{d}(\mathbb{C}^{n})$, the natural ambient spaces for
uniform matrix product states.
###### Definition 1.11.
A _necklace_ (of length $d$ on the alphabet $[n]$) is an equivalence class of
words, where two words are equivalent if they agree up to the action $C_{d}$.
A _bracelet_ (of length $d$ on the alphabet $[n]$) is an equivalence class of
words, where two words are equivalent if they agree up to the action $D_{2d}$.
For a fixed necklace or bracelet, all words in the equivalence class clearly
have the same weight; this is the _weight_ of the necklace or bracelet. We
denote by $N(n,d)$, resp. $B(n,d)$, the set of necklaces, resp. bracelets, of
length $d$ on $[n]$ and by $N_{\lambda}(n,d)\subset N(n,d)$, resp.
$B_{\lambda}(n,d)\subset B(n,d)$, the subset of elements of weight
$\lambda\in\mathbb{Z}^{n}$.
To every necklace $N\in N(n,d)$, we associate a basis vector
$e_{N}:=\frac{1}{d}\sum_{\sigma\in C_{d}}{\sigma\cdot e_{I}}$, where $I$ is
any representative of $N$. Then $\operatorname{Cyc}^{d}(\mathbb{C}^{n})$ has a
basis given by $\\{e_{N}:\ N\in N(n,d)\\}$. Moreover, $e_{N}$ is a weight
vector of weight $w(N)$, hence we find that the dimension of the weight space
of weight $\lambda$ is given by the number $|N_{\lambda}(n,d)|$ of necklaces
of weight $\lambda$.
###### Remark 1.12.
The number $|N(n,d)|$ of necklaces of length $d$ on $[n]$ can be counted using
Polya’s enumeration theorem, see for instance [Sta13, Theorem 7.10]:
(4)
$\dim\operatorname{Cyc}^{d}(\mathbb{C}^{n})=|N(n,d)|=\frac{1}{d}\sum_{l|d}\varphi(l)n^{\frac{d}{l}},$
where $\varphi$ is Euler’s totient function. There is also a formula for
$|N_{\lambda}(n,d)|=\dim\operatorname{Cyc}^{d}(\mathbb{C}^{n})_{\lambda}$: it
is equal to the coefficient of $x_{0}^{\lambda_{0}}\cdots
x_{n-1}^{\lambda_{n-1}}$ in the polynomial
$\frac{1}{d}\sum_{\ell\mid
d}{(x_{0}^{d/\ell}+\cdots+x_{n-1}^{d/\ell})^{\ell}}\varphi(\frac{d}{\ell}).$
To every bracelet $b\in B(n,d)$, we associate a basis vector
$e_{b}:=\frac{1}{2d}\sum_{\sigma\in D_{2d}}{\sigma\cdot e_{I}}$, where $I$ is
a representative of $b$. Then $\operatorname{Dih}^{d}(\mathbb{C}^{n})$ has a
basis given by $\\{e_{b}:\ b\in B(n,d)\\}$, and the dimension of the weight
space of weight $\lambda$ is given by the number $|B_{\lambda}(n,d)|$ of
bracelets of weight $\lambda$.
###### Remark 1.13.
The number of bracelets of length $d$ on $[n]$ is given by
(5)
$\dim\operatorname{Dih}^{d}(\mathbb{C}^{n})=|B(n,d)|=\begin{cases}\frac{1}{2}|N(n,d)|+\frac{1}{4}(n+1)n^{d/2}&\text{for
}d\text{ even,}\\\ \frac{1}{2}|N(n,d)|+\frac{1}{4}n^{(d+1)/2}&\text{for
}d\text{ odd.}\end{cases}$
We only state the formula for $|B_{\lambda}(n,d)|$ in the case of binary
bracelets (i.e. $n=2$), as that is the only case that is relevant to us:
$|B_{(w,d-w)}(2,d)|=\begin{cases*}\left(\frac{1}{2d}\sum_{l|\gcd(d,w)}{\varphi(l)\binom{\frac{d}{l}}{\frac{w}{l}}}\right)+\frac{1}{2}\binom{\frac{d}{2}-1}{\frac{w-1}{2}}&\text{for
$w$ odd,}\\\
\left(\frac{1}{2d}\sum_{l|\gcd(d,w)}{\varphi(l)\binom{\frac{d}{l}}{\frac{w}{l}}}\right)+\frac{1}{2}\binom{\frac{d}{2}}{\frac{w}{2}}&\text{for
$w$ even.}\end{cases*}$
## 2\. Computations
In this section, we describe how to computationally answer 1.9 for fixed
parameters. We focus on the smallest interesting case $m=n=2$. In this case we
are dealing with representations of $GL_{2}$, so the weights are in
$\mathbb{Z}^{2}$. Moreover, the only occurring weights in
$(\mathbb{C}^{2})^{\otimes d}$ are $t_{0}^{w}t_{1}^{d-w}$ for $w=0,\ldots,d$.
For subrepresentations of $(\mathbb{C}^{2})^{\otimes d}$, we will abbreviate
the weight spaces $V_{(w,d-w)}$ to $V_{w}$. Our goal is to determine the
dimension of the weight spaces $\langle\operatorname{uMPS}(2,2,d)\rangle_{w}$.
All of our dimension counts in this section and the next use the following
easy observation.
###### Observation 2.1.
For $p_{1},\ldots,p_{N}\in\mathbb{C}[y_{1},\ldots,y_{s}]$ polynomials, and $X$
the image of the polynomial map
(6) $\displaystyle\begin{split}\mathbb{C}^{s}&\to\mathbb{C}^{N}\\\
(y_{1},\ldots,y_{s})&\mapsto(p_{1}(y_{1},\ldots,y_{s}),\ldots,p_{N}(y_{1},\ldots,y_{s})),\end{split}$
a linear equation $\sum{\alpha_{i}x_{i}}$ vanishes on $X$ if and only if the
identity $\sum{\alpha_{i}p_{i}}=0$ holds in the polynomial ring
$\mathbb{C}[y_{1},\ldots,y_{s}]$. In particular, the dimension of the linear
span of $X$ is equal to the dimension of the subspace of
$\mathbb{C}[y_{1},\ldots,y_{s}]$ spanned by the $p_{i}$’s.
### 2.1. The trace parametrization
If we directly use Definition 1.1, we see that $\operatorname{uMPS}(2,2,d)$ is
the closed image of a polynomial map
$\mathbb{C}^{8}\to\operatorname{Dih}^{d}(\mathbb{C}^{2})$. However, in this
specific case there is an alternative parametrization by $\mathbb{C}^{5}$
instead of $\mathbb{C}^{8}$: the _trace parametrization_ , which appears to be
computationally more efficient in practice. It is based on the connection
between uniform matrix product states and invariant theory of matrices.
###### Definition 2.2.
Let
$R=\mathbb{C}[(a_{ij}^{k})_{1\leq i,j\leq m;1\leq k\leq n}]$
be the polynomial ring in $m^{2}n$ variables, and for $k=0,\ldots,n-1$, let
$A_{k}:=(a_{ij}^{k})_{1\leq i,j\leq m}$ be generic $m\times m$ matrices. The
_trace algebra_ $\mathcal{C}_{m,n}$ is the subalgebra of $R$ generated by the
polynomials $\operatorname{Tr}(A_{i_{1}}\cdots A_{i_{s}})$, where
$(i_{1},\ldots,i_{s})$ runs over all words (or equivalently: necklaces) in
$[n]$.
###### Remark 2.3.
The trace algebra is precisely the subring of $R$ consisting of all elements
that are invariant under simultaneous conjugation:
$f\in\mathcal{C}_{m,n}\iff
f(P^{-1}A_{0}P,\ldots,P^{-1}A_{n-1}P)=f(A_{0},\ldots,A_{n-1})\quad\forall P\in
GL_{m}.$
This is known as the _first fundamental theorem_ in the invariant theory of
matrices [Sib68, Pro76].
###### Proposition 2.4 ([Sib68, Corollary 2]).
The trace algebra $\mathcal{C}_{2,2}$ is generated by the following five
polynomials:
$\operatorname{Tr}(A_{0}),\operatorname{Tr}(A_{1}),\operatorname{Tr}(A_{0}^{2}),\operatorname{Tr}(A_{0}A_{1}),\operatorname{Tr}(A_{1}^{2}),$
and moreover, there are no polynomial relations between these generators.
###### Corollary 2.5.
For every bracelet $b=(b_{1},\ldots,b_{k})$, there is a unique polynomial
$P_{b}(T_{0},T_{1},T_{00},T_{01},T_{11})\in\mathbb{C}[T_{0},T_{1},T_{00},T_{01},T_{11}]$
such that for every pair $(A_{0},A_{1})$ of $2\times 2$ matrices, the
following equality holds:
(7) $\operatorname{Tr}(A_{b_{1}}\cdots
A_{b_{k}})=P_{b}(\operatorname{Tr}(A_{0}),\operatorname{Tr}(A_{1}),\operatorname{Tr}(A_{0}^{2}),\operatorname{Tr}(A_{0}A_{1}),\operatorname{Tr}(A_{1}^{2})).$
###### Remark 2.6.
If we give the ring $\mathbb{C}[T_{0},T_{1},T_{00},T_{01},T_{11}]$ a grading
by putting $\deg(T_{0})=\deg(T_{1})=1$ and
$\deg(T_{00})=\deg(T_{01})=\deg(T_{11})=2$, then the polynomial $P_{b}$ is
homogeneous of degree $\operatorname{length}(b)$.
The above means that $\operatorname{uMPS}(2,2,d)$ is the image of the
polynomial map
(8)
$\displaystyle\begin{split}\psi:\mathbb{C}^{5}\to&\operatorname{Dih}^{d}(\mathbb{C}^{2})\\\
(T_{0},T_{1},T_{00},T_{01},T_{11})\mapsto&\sum_{b}{P_{b}(T_{0},T_{1},T_{00},T_{01},T_{11})e_{b}},\end{split}$
where $b$ runs over all bracelets of length $d$. This is the trace
parametrization.
In order to compute the polynomials $P_{b}$, for bracelets of length $3$, one
verifies that
$\displaystyle P_{000}$
$\displaystyle=-\frac{1}{2}T_{0}^{3}+\frac{3}{2}T_{0}T_{00},$ $\displaystyle
P_{100}$
$\displaystyle=-\frac{1}{2}T_{0}^{2}T_{1}+\frac{1}{2}T_{1}T_{00}+T_{0}T_{01},$
$\displaystyle P_{110}$
$\displaystyle=-\frac{1}{2}T_{0}T_{1}^{2}+\frac{1}{2}T_{0}T_{11}+T_{1}T_{01},$
$\displaystyle P_{111}$
$\displaystyle=-\frac{1}{2}T_{1}^{3}+\frac{3}{2}T_{1}T_{11}.$
For bracelets of length $\geq 4$, we can inductively use the following
identity [Sib68], which holds for every quadruple $(A,B,C,D)$ of $2\times
2$-matrices:
$\displaystyle 2\operatorname{Tr}(ABCD)=$
$\displaystyle\operatorname{Tr}(A)\left(\operatorname{Tr}(BCD)-\operatorname{Tr}(B)\operatorname{Tr}(CD)\right)+\operatorname{Tr}(B)\left(\operatorname{Tr}(CDA)-\operatorname{Tr}(C)\operatorname{Tr}(DA)\right)$
$\displaystyle+\operatorname{Tr}(C)\left(\operatorname{Tr}(DAB)-\operatorname{Tr}(D)\operatorname{Tr}(AB)\right)+\operatorname{Tr}(D)\left(\operatorname{Tr}(ABC)-\operatorname{Tr}(A)\operatorname{Tr}(BC)\right)$
(9)
$\displaystyle-\operatorname{Tr}(AC)\operatorname{Tr}(BD)+\operatorname{Tr}(AB)\operatorname{Tr}(CD)+\operatorname{Tr}(AD)\operatorname{Tr}(BC)$
$\displaystyle+\operatorname{Tr}(A)\operatorname{Tr}(B)\operatorname{Tr}(C)\operatorname{Tr}(D).$
### 2.2. Computing the character
The weight space $\langle\operatorname{uMPS}(2,2,d)\rangle_{w}$ is the image
of the map
$\displaystyle\mathbb{C}^{5}\to$
$\displaystyle\operatorname{Dih}^{d}(\mathbb{C}^{2})_{w}$
$\displaystyle(T_{0},T_{1},T_{00},T_{01},T_{11})\mapsto$
$\displaystyle\sum_{b}{P_{b}(T_{0},T_{1},T_{00},T_{01},T_{11})e_{b}},$
where $b$ ranges over all bracelets of weight $w$. By 2.1, we need to compute
the dimension of the linear subspace of
$\mathbb{C}[T_{0},T_{1},T_{00},T_{01},T_{11}]$ spanned by the $P_{b}$’s. This
can be computed by putting the coefficients of the $P_{b}$’s in a matrix and
computing its rank.
Data: $d$
Result: Character of the representation
$\langle\operatorname{uMPS}(2,2,d)\rangle$
$T_{0}\leftarrow\operatorname{Tr}(A_{0})$;
$T_{1}\leftarrow\operatorname{Tr}(A_{1})$;
$T_{00}\leftarrow\operatorname{Tr}(A_{0}^{2})$;
$T_{01}\leftarrow\operatorname{Tr}(A_{0}A_{1})$;
$T_{11}\leftarrow\operatorname{Tr}(A_{1}^{2})$;
for _$\ell=3$ to d_ do
$bracelets$ = bracelets of length $\ell$;
for _$b$ in $bracelets$_ do
P[b] $\leftarrow$ $\operatorname{Tr}(A_{b_{1}}\cdots A_{b_{\ell}})$, expressed
in the $T_{i}$;
end for
end for
for _$w=0$ to $d$_ do
List the monomials $\mathbf{y}^{\alpha_{1}},\ldots,\mathbf{y}^{\alpha_{t}}$
appearing in the $P[b]$, where $b$ ranges over all bracelets of weight $w$;
Write $P[b]=\sum_{j}\beta_{b,j}\mathbf{y}^{\alpha_{j}}$;
Compute the rank of the matrix $(\beta_{b,j})_{b,j}$;
end for
Algorithm 1 Linear span of $\langle\operatorname{uMPS}(2,2,d)\rangle$
Putting everything together, we obtain Algorithm 1, which for a given $d$
computes the character (in particular the dimension) of
$\langle\operatorname{uMPS}(2,2,d)\rangle$. We implemented this algorithm in
SageMath [DLMS22].
The results are summarized in Footnote 1, where we write
$D_{w}:=\dim\langle\operatorname{uMPS}(2,2,d)\rangle_{w}$. For $d<8$, the
space $\langle\operatorname{uMPS}(2,2,d)\rangle$ is equal to the ambient space
$\operatorname{Dih}^{2}(\mathbb{C}^{n})$.
Table 1. Character of $\langle\operatorname{uMPS}(2,2,d)\rangle$. Since
$D_{w}=D_{d-w}$111More generally: for every $GL_{2}$-representation $V$ we
have that $V_{(b_{0},b_{1})}=V_{(b_{1},b_{0})}$., we only list $D_{w}$ for
$w\leq\lceil\frac{d}{2}\rceil$.
$\begin{array}[]{l|lllllllllll|l|l}\hline\cr\hline\cr
d&D_{0}&D_{1}&D_{2}&D_{3}&D_{4}&D_{5}&D_{6}&D_{7}&D_{8}&D_{9}&D_{10}&\dim\langle\operatorname{uMPS}(2,2,d)\rangle&\dim\operatorname{Dih}^{2}(\mathbb{C}^{n})\\\
\hline\cr 8&1&1&4&5&7&&&&&&&29&30\\\ 9&1&1&4&6&8&&&&&&&40&46\\\
10&1&1&5&7&11&11&&&&&&61&78\\\ 11&1&1&5&8&12&14&&&&&&82&126\\\
12&1&1&6&9&15&17&20&&&&&118&224\\\ 13&1&1&6&10&16&20&23&&&&&154&380\\\
14&1&1&7&11&19&23&29&29&&&&211&687\\\ 15&1&1&7&12&20&26&32&35&&&&268&1224\\\
16&1&1&8&13&23&29&38&41&45&&&353&2250\\\
17&1&1&8&14&24&32&41&47&51&&&438&4112\\\
18&1&1&9&15&27&35&47&53&61&61&&559&7685\\\
19&1&1&9&16&28&38&50&59&67&71&&680&14310\\\
20&1&1&10&17&31&41&56&65&77&81&86&846&27012\\\ \hline\cr\hline\cr\end{array}$
Based on these computations, we make the following conjecture:
###### Conjecture 2.7.
$\dim\langle\operatorname{uMPS}(2,2,d)\rangle_{w}=\begin{cases}1+\frac{d(v-1)v}{2}-\frac{2(v-1)v(2v-1)}{3}+v\lfloor\frac{d}{2}\rfloor-2v^{2}+v&\text{
for $w=2v$,}\\\ 1+\frac{dv(v+1)}{2}-\frac{2v(v+1)(2v+1)}{3}&\text{for
$w=2v+1$.}\end{cases}$
This would in particular imply
###### Conjecture 2.8 (Corollary of Conjecture 2.7)).
$\dim\langle\operatorname{uMPS}(2,2,d)\rangle=\begin{cases}\frac{1}{192}(d^{4}-4d^{2}+192d+192)&\text{
for $d$ even,}\\\ \frac{1}{192}(d^{4}-10d^{2}+192d+201)&\text{ for $d$
odd.}\end{cases}$
### 2.3. Higher degree equations
Equations of tensor varieties such as secant varieties of the Segre variety or
Veronese variety are well known to be hard to compute. For uniform matrix
product states, Critch and Morton gave a complete description of the ideal of
$\operatorname{uMPS}(2,2,4)$ and $\operatorname{uMPS}(2,2,5)$ and several
linear equation of $\operatorname{uMPS}(2,2,d)$ are given for $d$ until $12$,
c.f. [CM14]. The generators of the ideal of $\operatorname{uMPS}(2,2,d)$ for
$d=4,5,6$ are given in [CMS19].
Our algorithm from the previous section computes the linear span of
$\operatorname{uMPS}(2,2,d)$, which is equivalent to computing the degree $1$
part of its defining ideal. With some small modifications, one can obtain an
algorithm to compute the degree $k$ part of the defining ideal, viewed as a
$GL_{2}$-representation. In this paper we will only give a brief sketch of
this algorithm. For a more detailed account, including the use of raising
operators to speed up the algorithm, we refer the reader to [DL22].
We first consider a slightly more general setting: let $X$ be any variety that
is given as the closed image of a homogeneous polynomial map
$p:\mathbb{C}^{s}\to\mathbb{C}^{N}$ as in (6), and let $I$ be its
(homogeneous) defining ideal. Suppose that $p$ is compatible with a given
$GL_{n}$-action on $\mathbb{C}^{s}$ and $\mathbb{C}^{N}$. Then for every
$k\in\mathbb{N}$, the degree $k$ part $I_{k}$ is naturally a
$GL_{n}$-representation. Fix a weight $w$ and consider the map
$p^{k}_{w}:\mathbb{C}^{s}\xrightarrow{p}\mathbb{C}^{N}\xrightarrow{\nu_{k}}S^{k}\mathbb{C}^{N}\xrightarrow{}(S^{k}\mathbb{C}^{N})_{w},$
where $\nu_{k}$ is the $k$’th Veronese embedding. Then the weight space
$I_{k,w}\subseteq(S^{k}\mathbb{C}^{N})_{w}^{*}$ is equal to
$\langle\mathrm{Im}\;(p^{k}_{w})\rangle^{\perp}\cong\left({S^{k}\mathbb{C}^{N}}/{\langle\mathrm{Im}\;(p^{k}_{w})\rangle}\right)^{*}.$
The character of this representation is given by the difference of the
characters of $\langle\mathrm{Im}\;(p^{k}_{w})\rangle$ and
$S^{k}\mathbb{C}^{N}$. The latter character can be computed from the character
of $\mathbb{C}^{N}$, and hence we are left to compute the character of
$\langle\mathrm{Im}\;(p^{k}_{w})\rangle$.
In the case $X=\operatorname{uMPS}(2,2,d)$, we can compute the character of
$\langle\mathrm{Im}\;(p^{k}_{w})\rangle$ using a minor modification of
Algorithm 1, where in the second loop we replace the collection of polynomials
$\\{P[b]\mid\operatorname{weight}(b)=w\\}$ with their $k$-fold products
$\left\\{\prod_{j=1}^{k}P[b_{j}]\mid\sum_{j}\operatorname{weight}(b_{j})=w\right\\}.$
Using this algorithm, we computed $I(\operatorname{uMPS}(2,2,d))_{2}$ for
$d\leq 10$ and $I(\operatorname{uMPS}(2,2,d))_{3}$ for $d\leq 9$. Our code is
available at [DLMS22], and our results are summarized in Tables 2 and 3.
Table 2. Character of $I(\operatorname{uMPS}(2,2,d))^{*}_{2}$. Since
$D_{w}=D_{2d-w}$22footnotemark: 2, we only list $D_{w}$ for $w\leq d$.
$\begin{array}[]{l|llllllll}\hline\cr\hline\cr
d&D_{3}&D_{4}&D_{5}&D_{6}&D_{7}&D_{8}&D_{9}&D_{10}\\\ \hline\cr
6&0&1&1&2&&&&\\\ 7&0&1&3&6&7&&&\\\ 8&0&5&10&25&32&42&&\\\
9&1&7&21&48&79&110&119&\\\ 10&1&14&38&100&176&290&360&408\\\
\hline\cr\hline\cr\end{array}$ Table 3. Character of
$I(\operatorname{uMPS}(2,2,d))^{*}_{3}$. Since $D_{w}=D_{3d-w}$33footnotemark:
3, we only list $D_{w}$ for $w\leq\lceil\frac{3d}{2}\rceil$.
$\begin{array}[]{l|lllllllllll}\hline\cr\hline\cr
d&D_{3}&D_{4}&D_{5}&D_{6}&D_{7}&D_{8}&D_{9}&D_{10}&D_{11}&D_{12}&D_{13}\\\
\hline\cr 6&0&1&2&8&11&17&17&&&&\\\ 7&0&1&4&15&29&49&67&77&&&\\\
8&0&5&14&51&101&198&292&414&478&532&\\\
9&1&7&26&83&191&388&671&1039&1431&1784&1983\\\ \hline\cr\hline\cr\end{array}$
## 3\. Results
### 3.1. Linear relations via Cayley-Hamilton
We recall the classical Cayley-Hamilton theorem. We use this result in order
to find linear equations for ${\operatorname{uMPS}(m,n,d+k)}$, $k\geq m$ based
on linear equations for ${\operatorname{uMPS}(m,n,N)}$, $N=d,d+1,\dots,d+m-1$.
###### Theorem 3.1 (Cayley-Hamilton).
Let $A\in\mathbb{C}^{m\times m}$ be a $m\times m$ complex matrix and
$p_{A}(\lambda)=\det(\lambda\mathrm{Id}_{m}-A)$ its characteristic polynomial.
Then $p_{A}(A)=0$.
In fact, the only thing we will use is the following statement, which (since
$\deg p_{A}=m$) immediately follows from Theorem 3.1.
###### Corollary 3.2.
Let $A\in\mathbb{C}^{m\times m}$. Then $A^{k}$, for $k\geq m$ can be written
as a linear combination of its previous powers.
###### Lemma 3.3.
Let $c=(c_{1},\ldots,c_{s})\in\mathbb{C}^{s}$ be a vector of coefficients and
$\\{i_{\ell}^{j}\\}_{1\leq\ell\leq d,1\leq j\leq s}$ be indices; with
$i_{\ell}^{j}\in[n]$. Assume that for every $n$-tuple $(A_{0},\ldots,A_{n-1})$
of $m\times m$ matrices and every $k<m$ the following identity holds:
(10) $\sum_{j=1}^{s}c_{j}\operatorname{Tr}(A_{i_{1}^{j}}\cdots
A_{i_{d}^{j}}A_{0}^{k})=0.$
Then the same identity holds for arbitrary $k\in\mathbb{N}$.
###### Proof.
We use induction on $k\geq m$. By Corollary 3.2, $A_{0}^{k}$ can be written as
a linear combination of its previous powers $A_{0}^{l}$, for $l=0,\dots,m-1$.
Therefore, we have
$\displaystyle\sum_{j=1}^{s}c_{j}\operatorname{Tr}(A_{i_{1}^{j}}\cdots
A_{i_{d}^{j}}A_{0}^{k})$
$\displaystyle=\sum_{j=1}^{s}c_{j}\operatorname{Tr}\Big{[}A_{i_{1}^{j}}\cdots
A_{i_{d}^{j}}\Big{(}\sum_{l=0}^{m-1}\gamma_{l}A_{0}^{l}\Big{)}\Big{]}$
$\displaystyle=\sum_{j=1}^{s}c_{j}\sum_{l=0}^{m-1}\gamma_{l}\big{(}\operatorname{Tr}(A_{i_{1}^{j}}\cdots
A_{i_{d}^{j}}A_{0}^{l})\big{)}$
$\displaystyle=\sum_{l=0}^{m-1}\gamma_{l}\Big{(}\sum_{j=1}^{s}c_{j}\big{(}\operatorname{Tr}(A_{i_{1}^{j}}\cdots
A_{i_{d}^{j}}A_{0}^{l})\big{)}\Big{)}=0.$
∎
The usefulness of Lemma 3.3 stems from the fact that one can find expressions
of the form (10) which are trivial for small $k$, in the sense that they
follow from the cyclic invariance of the trace, but nontrivial for large $k$.
We illustrate this in the example below.
###### Example 3.4.
We show that for any $2\times 2$ matrices $A_{0},A_{1},A_{2},A_{3}$ and any
$k\geq 0$, the following identity holds
$\displaystyle\operatorname{Tr}(A_{1}A_{2}A_{0}A_{3}A_{0}^{k})+\operatorname{Tr}(A_{2}A_{3}A_{0}A_{1}A_{0}^{k})+\operatorname{Tr}(A_{3}A_{1}A_{0}A_{2}A_{0}^{k})$
$\displaystyle=$
$\displaystyle\operatorname{Tr}(A_{1}A_{0}A_{2}A_{3}A_{0}^{k})+\operatorname{Tr}(A_{2}A_{0}A_{3}A_{1}A_{0}^{k})+\operatorname{Tr}(A_{3}A_{0}A_{1}A_{2}A_{0}^{k}).$
By the above argument, it suffices to show the identity for $k=0$ and $k=1$.
But these both follow from cyclic invariance of the trace:
$\displaystyle{\hbox
to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{1}A_{2}A_{0}A_{3})$}\hss}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rule[-2.99998pt]{60.11678pt}{0.5pt}}+{\hbox
to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{2}A_{3}A_{0}A_{1})$}\hss}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rule[-2.99998pt]{60.11678pt}{0.5pt}}+{\hbox
to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{3}A_{1}A_{0}A_{2})$}\hss}\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle=$ $\displaystyle{\hbox
to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{1}A_{0}A_{2}A_{3})$}\hss}\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color<EMAIL_ADDRESS>to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{2}A_{0}A_{3}A_{1})$}\hss}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rule[-2.99998pt]{60.11678pt}{0.5pt}}+{\hbox
to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{3}A_{0}A_{1}A_{2})$}\hss}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rule[-2.99998pt]{60.11678pt}{0.5pt}}$
$\displaystyle{\hbox
to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{1}A_{2}A_{0}A_{3}A_{0})$}\hss}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rule[-2.99998pt]{70.4168pt}{0.5pt}}+{\hbox
to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{2}A_{3}A_{0}A_{1}A_{0})$}\hss}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rule[-2.99998pt]{70.4168pt}{0.5pt}}+{\hbox
to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{3}A_{1}A_{0}A_{2}A_{0})$}\hss}\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle=$ $\displaystyle{\hbox
to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{1}A_{0}A_{2}A_{3}A_{0})$}\hss}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rule[-2.99998pt]{70.4168pt}{0.5pt}}+{\hbox
to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{2}A_{0}A_{3}A_{1}A_{0})$}\hss}\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color<EMAIL_ADDRESS>to0.0pt{\leavevmode\hbox{\set@color$\operatorname{Tr}(A_{3}A_{0}A_{1}A_{2}A_{0})$}\hss}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\rule[-2.99998pt]{70.4168pt}{0.5pt}}.$
This immediately gives us a nontrivial linear relation on
$\operatorname{uMPS}(2,4,d)$, for $d\geq 6$. But we can also find linear
relations on $\operatorname{uMPS}(2,2,d)$. For instance, if we put $k=2$,
$A_{2}=A_{1}^{2}$ and $A_{3}=A_{0}A_{1}$, we find
$\displaystyle\operatorname{Tr}(A_{1}^{3}A_{0}^{2}A_{1}A_{0}^{2})+\operatorname{Tr}(A_{1}^{2}A_{0}A_{1}A_{0}A_{1}A_{0}^{2})+\operatorname{Tr}(A_{1}^{2}A_{0}^{3}A_{1}^{2}A_{0})$
$\displaystyle=$
$\displaystyle\operatorname{Tr}(A_{1}^{2}A_{0}A_{1}A_{0}^{2}A_{1}A_{0})+\operatorname{Tr}(A_{1}^{2}A_{0}^{2}A_{1}^{2}A_{0}^{2})+\operatorname{Tr}(A_{1}^{3}A_{0}^{3}A_{1}A_{0}),$
which is the unique linear relation on $\operatorname{uMPS}(2,2,8)$ that
doesn’t follow from dihedral symmetry.
###### Theorem 3.5.
Let $A_{0},\ldots,A_{m},B$ be $m\times m$ matrices. Then for every
$\ell\in\mathbb{N}$ it holds that
(11) $\sum_{\sigma\in\mathfrak{S}_{m},\tau\in
C_{m+1}}{\operatorname{sgn}(\sigma)\operatorname{sgn}(\tau)\operatorname{Tr}(A_{\tau(0)}B^{\sigma(0)}A_{\tau(1)}B^{\sigma(1)}\cdots
A_{\tau(m-1)}B^{\sigma(m-1)}A_{\tau(m)}B^{\ell})}=0.$
Here $\mathfrak{S}_{m}$ is the symmetric group acting on
$\\{0,1,\ldots,m-1\\}$, and $C_{m+1}$ is the cyclic group acting on
$\\{0,1,\ldots,m\\}$.
###### Proof.
We will first show the statement for $\ell\in\\{0,1,\ldots,m-1\\}$. So let us
fix such an $\ell$. We will write
$T(\sigma,\tau):=\operatorname{Tr}(A_{\tau(0)}B^{\sigma(0)}A_{\tau(1)}B^{\sigma(1)}\cdots
A_{\tau(m-1)}B^{\sigma(m-1)}A_{\tau(m)}B^{\ell}).$
Let us write $c_{a}$ for the permutation that cyclically permutes the first
$a$ elements. Precisely
$c_{a}(i)=\begin{cases}i+1&\text{for }i<a-1\\\ 0&\text{for }i=a-1\\\
i&\text{for }i>a-1.\end{cases}$
Step 1. For $\sigma\in\mathfrak{S}_{m}$ and $\tau\in C_{m+1}$, we define
$\displaystyle\widetilde{\sigma}$ $\displaystyle:=\sigma\circ
c_{\sigma^{-1}(\ell)+1}^{-1}\circ c_{m}^{\sigma^{-1}(\ell)+1}$
$\displaystyle\widetilde{\tau}$ $\displaystyle:=\tau\circ
c_{m+1}^{\sigma^{-1}(\ell)+1}$
we have $T(\sigma,\tau)=T(\widetilde{\sigma},\widetilde{\tau}).$
Indeed, if we write $k=\sigma^{-1}(\ell)$, then
$\displaystyle T(\sigma,\tau)$
$\displaystyle=\operatorname{Tr}(A_{\tau(0)}B^{\sigma(0)}\cdots
A_{\tau(k)}B^{\sigma(k)}A_{\tau(k+1)}B^{\sigma(k+1)}\cdots
A_{\tau(m-1)}B^{\sigma(m-1)}A_{\tau(m)}B^{\sigma(k)})$
$\displaystyle=\operatorname{Tr}(A_{\tau(k+1)}B^{\sigma(k+1)}\cdots
A_{\tau(m)}B^{\sigma(k)}A_{\tau(0)}B^{\sigma(0)}\cdots
A_{\tau(k-1)}B^{\sigma(k-1)}A_{\tau(k)}B^{\sigma(k)})$
$\displaystyle=T(\widetilde{\sigma},\widetilde{\tau}).$
Where for the last step, note that
* •
$k+1=c_{m+1}^{k+1}(0)$, …, $m=c_{m+1}^{k+1}(m-k-1)$, $0=c_{m+1}^{k+1}(m-k)$,
…, $k=c_{m+1}^{k+1}(m)$.
* •
$k+1=c_{k+1}^{-1}(c_{m}^{k+1}(0))$, …, $m-1=c_{k+1}^{-1}(c_{m}^{k+1}(m-k-2))$,
$k=c_{k+1}^{-1}(c_{m}^{k+1}(m-k-1))$, $0=c_{k+1}^{-1}(c_{m}^{k+1}(m-k))$, …,
$k-1=c_{k+1}^{-1}(c_{m}^{k+1}(m-1))$.
Step 2. Note that the assignment
$\displaystyle\mathfrak{S}_{m}\times C_{m+1}$
$\displaystyle\to\mathfrak{S}_{m}\times C_{m+1}$ $\displaystyle(\sigma,\tau)$
$\displaystyle\mapsto(\widetilde{\sigma},\widetilde{\tau})$
is an involution. Indeed, we have
$\displaystyle\widetilde{\sigma}^{-1}(\ell)=c_{m}^{-\sigma^{-1}(\ell)-1}(c_{\sigma^{-1}(\ell)+1}(\sigma^{-1}(\ell)))=c_{m}^{-\sigma^{-1}(\ell)-1}(0)=m-\sigma^{-1}(\ell)-1.$
So, again writing $k=\sigma^{-1}(\ell)$
$\displaystyle\widetilde{\widetilde{\sigma}}$
$\displaystyle=\widetilde{\sigma}\circ
c_{\widetilde{\sigma}^{-1}(\ell)+1}^{-1}\circ
c_{m}^{\widetilde{\sigma}^{-1}(\ell)+1}$ $\displaystyle=\sigma\circ
c_{k+1}^{-1}\circ c_{m}^{k+1}\circ c_{m-k}^{-1}\circ c_{m}^{m-k}$
$\displaystyle=\sigma.$
To see the last equality:
* •
For $i<k$:
$c_{k+1}^{-1}(c_{m}^{k+1}(c_{m-k}^{-1}(c_{m}^{m-k}(i))))=c_{k+1}^{-1}(c_{m}^{k+1}(c_{m-k}^{-1}(m-k+i)))=\\\
c_{k+1}^{-1}(c_{m}^{k+1}(m-k+i))=c_{k+1}^{-1}(i+1)=i$.
* •
For $i=k$:
$c_{k+1}^{-1}(c_{m}^{k+1}(c_{m-k}^{-1}(c_{m}^{m-k}(k))))=c_{k+1}^{-1}(c_{m}^{k+1}(c_{m-k}^{-1}(0)))=\\\
c_{k+1}^{-1}(c_{m}^{k+1}(m-k-1))=c_{k+1}^{-1}(0)=k$.
* •
For $i>k$:
$c_{k+1}^{-1}(c_{m}^{k+1}(c_{m-k}^{-1}(c_{m}^{m-k}(i))))=c_{k+1}^{-1}(c_{m}^{k+1}(c_{m-k}^{-1}(i-k)))=\\\
c_{k+1}^{-1}(c_{m}^{k+1}(i-k-1))=c_{k+1}^{-1}(i)=i$.
And furthermore $\widetilde{\widetilde{\tau}}=\tau\circ c_{m+1}^{k+1}\circ
c_{m+1}^{m-k}=\tau$.
Step 3. Note that
$\displaystyle\operatorname{sgn}(\widetilde{\sigma})\operatorname{sgn}(\widetilde{\tau})$
$\displaystyle=(-1)^{k+(k+1)(m-1)+(k+1)m}\operatorname{sgn}(\sigma)\operatorname{sgn}(\tau)$
$\displaystyle=-\operatorname{sgn}(\sigma)\operatorname{sgn}(\tau).$
From Step 1 we have that
$T(\sigma,\tau)=T(\widetilde{\sigma},\widetilde{\tau})$. There will be
cancellations of terms in (11) as
$\operatorname{sgn}(\widetilde{\sigma})\operatorname{sgn}(\widetilde{\tau})=-\operatorname{sgn}(\sigma)\operatorname{sgn}(\tau)$
from Step 3. Finally using Step 2 we ensure that all terms will cancel out
therefore establishing the given identity for $0\leq l\leq m-1$. Now using
Lemma 3.3, we conclude that the given identity (11) holds for all
$l\in\mathbb{N}$. ∎
###### Corollary 3.6.
If $n\geq 3$ and $d\geq\frac{(m+1)(m+2)}{2}$, then
${\operatorname{uMPS}(m,n,d)}$ is contained in a proper linear subspace of the
space of cyclically invariant tensors.
###### Proof.
Let $\ell\geq m$ and let $\mathfrak{S}_{m+1}$ denote the symmetric group
acting on $\\{0,1,\ldots,m-1,\ell\\}$. Then we can rewrite (11) as follows:
(12)
$\sum_{\sigma\in\mathfrak{S}_{m+1}}{\operatorname{sgn}(\sigma)\operatorname{Tr}(A_{0}B^{\sigma(0)}A_{1}B^{\sigma(1)}\cdots
A_{m-1}B^{\sigma(m-1)}A_{m}B^{\sigma(\ell)})}=0.$
Let $X_{0},X_{1},X_{2}$ be $m\times m$ matrices, and in (12) substitute
$A_{0}=X_{0}$, $B=X_{1}$, and $A_{i}=X_{2}$ for $i=1,\ldots,m$. Note that even
after that substitution, the ternary bracelets corresponding to the $(m+1)!$
terms in (12) are all distinct. Hence no two terms will cancel, and we get a
nontrivial linear relation on $\operatorname{uMPS}(m,3,d)$, where
$d=1+2+\cdots+(m-1)+\ell+(m+1)\geq 1+2+\cdots+(m+1)=\binom{m+2}{2}$. ∎
###### Remark 3.7.
With a bit more care, one can also get nontrivial relations on
$\operatorname{uMPS}(m,2,d)$ in this way. For instance if we take $\ell=m$ and
in (12) we substitute $A_{0}=X_{0}X_{1}^{m+1}X_{0}$, $B=X_{1}$, and
$A_{i}=X_{0}$ for $i=1,\ldots,m$, one verifies that again no terms cancel, and
hence we found a nontrivial linear relation on $\operatorname{uMPS}(m,2,d)$,
where $d=\binom{m+3}{2}$.
### 3.2. Linear equations for $\operatorname{uMPS}(2,2,d)$
From the trace parametrization, we can give an upper bound on
$\dim\langle\operatorname{uMPS}(2,2,d)\rangle$.
###### Theorem 3.8.
For every $d\in\mathbb{N}$, we have the inequality
$\dim\langle\operatorname{uMPS}(2,2,d)\rangle\leq\begin{cases}\frac{1}{192}(d+6)(d+4)^{2}(d+2)&\text{for
$d$ even},\\\ \frac{1}{192}(d+7)(d+5)(d+3)(d+1)&\text{for $d$
odd}.\end{cases}$
###### Proof.
It follows from (8) and Remark 2.6 that
$\dim\langle\operatorname{uMPS}(2,2,d)\rangle$ can be at most the number of
degree $d$ monomials in $\mathbb{C}[T_{0},T_{1},T_{00},T_{01},T_{11}]$.
Counting these monomials gives the above formula. ∎
Note that asymptotically for $d\to\infty$, the above bound agrees with our
conjectured formula in 2.8.
As in the previous section, we abbreviate “weight
$\lambda=(w,d-w)\in\mathbb{Z}^{2}$” to “weight $w$”. In the rest of this
section, we provide a proof of Conjecture 2.7 in the cases $w=0,1,2,3$.
Consider the parametrization of $\operatorname{uMPS}(2,2,d)$ in coordinates
$\displaystyle\varphi:(\mathbb{C}^{2\times 2})^{2}$
$\displaystyle\to\operatorname{Dih}^{d}(\mathbb{C}^{2})$
$\displaystyle(A_{0},A_{1})$
$\displaystyle\mapsto(\operatorname{Tr}(A_{0}^{d}),\operatorname{Tr}(A_{0}^{d-1}A_{1}),\dots,\operatorname{Tr}(A_{1}^{d})).$
It is in particular a polynomial map in the unknown entries of the matrices
$A_{0},A_{1}\in\mathbb{C}^{2\times 2}$, we denote by
$A_{0}=\begin{pmatrix}a_{1}&a_{2}\\\ a_{3}&a_{4}\end{pmatrix},\quad
A_{1}=\begin{pmatrix}b_{1}&b_{2}\\\ b_{3}&b_{4}\end{pmatrix}.$
We will write
$T_{i_{1}\dots i_{d}}:=\operatorname{Tr}(A_{i_{1}}\dots
A_{i_{d}})\in\mathbb{C}[a_{1},\dots,b_{4}]_{d}\quad\text{and}\quad
W_{w}:=\langle T_{b}:b\in B_{w}(2,d)\rangle$
By 2.1, we have
$\dim\langle\operatorname{uMPS}(2,2,d)\rangle_{w}=\dim W_{w}.$
The cases $w=0$ and $w=1$ are easy:
###### Proposition 3.9.
The space $W_{0}$ is a $1$-dimensional vector space generated by the
polynomial $T_{0,\dots,0}=\operatorname{Tr}(A_{0}^{d})$. The space $W_{1}$ is
a $1$-dimensional vector space generated by the polynomial $T_{10\dots
0}=\operatorname{Tr}(A_{1}A_{0}^{d-1})$.
###### Proof.
If $w=0$ then $b=({0\dots 0})\in B_{0}(2,d)$ is the only binary bracelet of
weight zero, and if $w=1$ then $b=({10\dots 0})\in B_{1}(2,d)$ is the only
binary bracelet of weight $1$. ∎
We now turn to the case $w=2$. Then 2.7 states that $\dim
W_{w}=\lfloor\frac{d}{2}\rfloor$. But $\lfloor\frac{d}{2}\rfloor$ is exactly
the number $B_{2}(2,d)$ of generators $T_{b}$ of $W_{w}$; hence we need to
show that they are linearly independent.
###### Proposition 3.10.
The polynomials $\\{T_{b}:b\in B_{2}(2,d)\\}$ are linearly independent.
###### Proof.
Note that
$W_{2}=\langle T_{10^{i}10^{d-2-i}}\mid
i=0,\dots\lfloor\frac{d}{2}\rfloor-1\rangle.$
If we make the following substitutions:
$A_{1}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\quad
A_{0}=\begin{pmatrix}1&0\\\ 0&x\end{pmatrix}$
our generators $T_{b}$ become
(13)
$T_{10^{i}10^{d-2-i}}=\operatorname{Tr}(A_{1}A_{0}^{i}A_{1}A_{0}^{d-2-i})=x^{i}+x^{d-2-i}\quad
i=0,\dots\lfloor\frac{d}{2}\rfloor-1.$
Since for the given choice of $A_{0},A_{1}$ the polynomials (13) are
$\lfloor\frac{d}{2}\rfloor$ linearly independent polynomials, the same holds
for a generic choice of matrices. ∎
Finally, we prove the case $w=3$. In this case our conjectured formula states
that $\dim W_{w}=d-3$. Consider the following subset of $B_{3}(2,d)$:
$\widetilde{B}_{3}:=\\{b\in B_{3}(2,d):b\text{ contains }11\text{ {or}
}101\\}\subset B_{3}(2,d).$
###### Lemma 3.11.
The cardinality of $\widetilde{B}_{3}$ equals $d-3$.
###### Proof.
The cardinality of $\widetilde{B}_{3}$ is the sum of the number of binary
bracelets of weight $3$ containing $11$ and the number of binary bracelets of
weight $3$ containing $101$ but not $11$, that are $\lceil\frac{d-2}{2}\rceil$
and $\lceil\frac{d-5}{2}\rceil$ respectively. Therefore the cardinality of
$\widetilde{B}_{3}$ is
$\lceil\frac{d-2}{2}\rceil+\lceil\frac{d-5}{2}\rceil=d-3.$
∎
In order to prove the case $w=3$ we need to show that
$\\{T_{b}:b\in\widetilde{B}_{3}\\}$ is a basis of $W_{3}$. We first show
linear independence:
###### Lemma 3.12.
The polynomials $\\{T_{b}:b\in\widetilde{B}_{3}\\}$ are linearly independent.
###### Proof.
We will show that the polynomials are linearly independent even after the
following substitution:
$A_{1}=\begin{pmatrix}0&1\\\ 1&1\end{pmatrix},\quad
A_{0}=\begin{pmatrix}1&0\\\ 0&x\end{pmatrix}.$
Then $W_{3}$ is spanned by the following polynomials:
$\displaystyle f_{b}$
$\displaystyle:=T_{110^{b}10^{d-b-3}}=x^{b}+x^{d-b-3}+2x^{d-3}\quad
b\in\\{0,\dots,\lfloor\frac{d-3}{2}\rfloor\\},$ $\displaystyle g_{b}$
$\displaystyle:=T_{1010^{b}10^{d-b-4}}=x^{b+1}+x^{d-b-3}+x^{d-4}+x^{d-3}\quad
b\in\\{1,\dots,\lfloor\frac{d-4}{2}\rfloor\\}.$
We now simply have to put the coefficients of these polynomials in a matrix
and show it has full rank. For $d$ even the matrix of coefficients is given by
$S=\begin{pmatrix}1&&&&\dots&&&&&3\\\ &1&&&&&&&1&2\\\ &&1&&&&&1&&2\\\
\vdots&&&\ddots&&&\iddots&&&\vdots\\\ 0&&&&1&1&&&&2\\\ 0&0&1&0&&\dots&&&2&1\\\
&&0&1&&&&1&1&1\\\ \vdots&&&&\ddots&&\iddots&&&\vdots\\\
0&&&\dots&&2&\dots&0&1&1\end{pmatrix}$
and for $d$ odd, given by
$S=\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&&&&&\dots&&&&0&3\\\
&1&&&&&&&0&1&2\\\ &&1&&&&&&1&0&2\\\
\vdots&&&\ddots&&&&\iddots&&\vdots&\vdots\\\ &&&&1&&1&&&0&2\\\
&&&&&2&&&&0&2\\\ 0&0&1&0&&&\dots&&0&2&1\\\ &&0&1&0&&&&1&1&1\\\
&&&&\ddots&&&\iddots&&\vdots&\vdots\\\ 0&0&&&&1&1&&0&1&1\end{pmatrix}.$
By elementary row operations, we can reduce the left upper part to a diagonal
matrix of order $\lfloor\frac{d-1}{2}\rfloor$. The left lower part is filled
with zeros. The (rectangular) right lower block of dimension
$\lfloor\frac{d-4}{2}\rfloor\times\lfloor\frac{d-2}{2}\rfloor$ can be put in
the following upper triangular forms, for $d$ even and odd respectively
$\begin{pmatrix}0&&\dots&-1&2&-1\\\ \vdots&&-1&1&1&-1\\\
&\iddots&\iddots&0&1&-1\\\ -1&1&&\vdots&\vdots&\vdots\\\
2&0&\dots&0&1&1\end{pmatrix}\to\begin{pmatrix}2&0&\dots&&&1&1\\\
0&2&&&&3&-1\\\ &&\ddots&&&5&-3\\\ \vdots&&&&&\vdots&\vdots\\\
0&\dots&&0&2&*&*\\\ 0&\dots&&&0&*&*\end{pmatrix}\quad\text{for }d\text{
even,}$ $\begin{pmatrix}0&&\dots&-1&2&-1\\\ \vdots&&-1&1&1&-1\\\
&\iddots&\iddots&0&1&-1\\\ &&&\vdots&\vdots&\vdots\\\ -1&1&&&&-1\\\
1&0&\dots&0&1&0\end{pmatrix}\to\begin{pmatrix}1&0&\dots&&0&1&0\\\
0&1&0&\dots&0&2&-1\\\ \vdots&0&\ddots&&&3&-2\\\ &&&&&\vdots&\vdots\\\
&&&0&1&*&*\\\ &&&&0&*&*\end{pmatrix}\quad\text{for }d=2\text{ odd.}$
Both the obtained blocks have rank $\lfloor\frac{d-4}{2}\rfloor$. We have that
the rank of $S$ is
$\lfloor\frac{d-1}{2}\rfloor+\lfloor\frac{d-4}{2}\rfloor=d-3$, and this
concludes the proof. ∎
We finish our proof by showing that $\\{T_{b}:b\in\widetilde{B}_{3}\\}$ spans
$W_{3}$:
###### Lemma 3.13.
Every polynomial
$T_{10^{a}10^{b}10^{c}}=\operatorname{Tr}(A_{1}A_{0}^{a}A_{1}A_{0}^{b}A_{1}A_{0}^{c})$,
with $1<a\leq b\leq c$, $a+b+c=d-3$ is an element of the linear span $\langle
T_{b}:b\in\widetilde{B}_{3}\rangle$.
###### Proof.
Notice that the elements of $B_{3}(2,n)\setminus\widetilde{B}_{3}$ can be
written without loss of generality in the form
$10^{a}10^{b}10^{c},\quad\text{with }1<a\leq b\leq c.$
We use induction on $a$. If $a=0$ and $a=1$ then
$(10^{a}10^{b}10^{c})\in\widetilde{B}_{3}$ and we are done. If we substitute
$A_{1}\to A_{1}A_{0}^{a-1}$, $A_{2}\to A_{1}A_{0}^{b}$ and $A_{3}\to A_{1}$ in
the equation given by Theorem 3.5 We get
$\displaystyle\operatorname{Tr}(A_{1}A_{0}^{a-1}A_{1}A_{0}^{b+1}A_{1}A_{0}^{c})+\operatorname{Tr}(A_{1}A_{0}^{b}A_{1}A_{0}A_{1}A_{0}^{a+c-1})+\operatorname{Tr}(A_{1}^{2}A_{0}^{a}A_{1}A_{0}^{b+c})=$
$\displaystyle\operatorname{Tr}(A_{1}A_{0}^{a}A_{1}A_{0}^{b}A_{1}A_{0}^{c})+\operatorname{Tr}(A_{1}A_{0}^{b+1}A_{1}^{2}A_{0}^{a+c-1})+\operatorname{Tr}(A_{1}A_{0}A_{1}A_{0}^{a-1}A_{1}A_{0}^{b+c}).$
Reordering the summands we obtain
$\displaystyle
T_{10^{a}10^{b}10^{c}}=(T_{10^{b}1010^{a+c-1}}+T_{110^{a}10^{b+c}}-T_{10^{b+1}110^{a+c-1}}-T_{1010^{a-1}10^{b+c}})+T_{10^{a-1}10^{b+1}10^{c}}.$
All terms in the parenthesis have as subscript an elements of
$\widetilde{B}_{3}$, and the last term is in
$\langle\\{T_{b}:b\in\widetilde{B}_{3}\\}\rangle$ by the induction hypothesis.
This concludes the proof. ∎
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|
# The implications of high BH spins on the origin of BH-BH mergers
A. Olejak11affiliationmark: , K. Belczynski11affiliationmark: 1 Nicolaus
Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18,
00-716 Warsaw, Poland, (aleksandra.olejak<EMAIL_ADDRESS>
###### Abstract
The LIGO/Virgo collaboration has reported $50$ black hole—black hole (BH-BH)
mergers and $8$ candidates recovered from digging deeper into the detectors
noise. The majority of these mergers have low effective spins pointing toward
low BH spins and efficient angular momentum transport (AM) in massive stars as
proposed by several models (e.g., the Tayler-Spruit dynamo). However, out of
these $58$ mergers, $7$ are consistent with having high effective spin
parameter ($\chi_{\rm eff}>0.3$). Additionally, $2$ events seem to have high
effective spins sourced from the spin of the primary (more massive) BH. These
particular observations could be used to discriminate between the isolated
binary and dynamical formation channels. It might seem that high BH spins
point to a dynamical origin if AM in stars is efficient and forms low-spinning
BHs. In such a case dynamical formation is required to produce second and
third generations of BH-BH mergers with typically high-spinning BHs. Here we
show, however, that isolated binary BH-BH formation naturally reproduces such
highly spinning BHs. Our models start with efficient AM in massive stars that
is needed to reproduce the majority of BH-BH mergers with low effective spins.
Later, some of the binaries are subject to a tidal spin-up allowing the
formation of a moderate fraction ($\sim 10\%$) of BH-BH mergers with high
effective spins ($\chi_{\rm eff}\gtrsim 0.4-0.5$). In addition,
isolated–binary evolution can produce a small fraction of BH-BH mergers with
almost maximally spinning primary BHs. Therefore, the formation scenario of
these atypical BH-BH mergers remains to be found.
###### Subject headings:
stars: black holes, compact objects, massive stars
## 1\. Introduction
The LIGO/Virgo collaboration has announced detection of gravitational waves
from $\sim 50$ double black hole (BH-BH) mergers (Abbott et al., 2019a, b;
Fishbach & Holz, 2020; Abbott et al., 2021a). Additional $8$ BH-BH merger
candidates have been recently reported (Abbott et al., 2021b). The majority of
all these events have low effective spins parameters: $\chi_{\rm
eff}=\frac{m_{1}a_{\rm 1}\cos\theta_{1}+m_{2}a_{\rm
2}\cos\theta_{2}}{m_{1}+m_{2}}\approx 0,$ where $m_{i}$ are BH masses, $a_{\rm
i}=cJ_{i}/Gm_{i}^{2}$ are dimensionless BH spin magnitudes ($J_{i}$ being the
BH angular momentum (AM), $c$ the speed of light, $G$ the gravitational
constant), and $\theta_{i}$ are angles between the individual BH spins and the
system orbital AM.
However, among the dectections there are also several BH-BH mergers which are
characterized by higher (non-zero) positive effective spins. In Table 1 we
list the parameters of the five BH-BH mergers with highest effective spins
reported by Abbott et al. (2021a) with additional two high effective spin
systems reported by Abbott et al. (2021b).
Table 1BH-BH mergers with high effective spins
No. | Namea | $\chi_{\rm eff}$ | $m_{1}$ | $m_{2}$ | $a_{1}$
---|---|---|---|---|---
1 | GW190517 | $0.52^{+0.19}_{-0.19}$ | $37.4^{+11.7}_{-7.6}$ | $25.3^{+7.0}_{-7.3}$ | –
2 | GW170729 | $0.37^{+0.21}_{-0.25}$ | $50.2^{+16.2}_{-10.2}$ | $34.0^{+9.1}_{-10.1}$ | –
3 | GW190620 | $0.33^{+0.22}_{-0.25}$ | $57.1^{+16.0}_{-12.7}$ | $35.5^{+12.2}_{-12.3}$ | –
4 | GW190519 | $0.31^{+0.20}_{-0.22}$ | $66.0^{+10.7}_{-12.0}$ | $40.5^{+11.0}_{-11.1}$ | –
5 | GW190706 | $0.28^{+0.26}_{-0.29}$ | $67.0^{+14.6}_{-13.3}$ | $38.2^{+14.6}_{-13.3}$ | –
6 | GW190403 | $0.70^{+0.15}_{-0.27}$ | $88.0^{+28.2}_{-32.9}$ | $22.1^{+23.8}_{-9.0}$ | $0.92^{+0.07}_{-0.22}$
7 | GW190805 | $0.35^{+0.30}_{-0.36}$ | $48.2^{+17.5}_{-12.5}$ | $32.0^{+13.4}_{-11.4}$ | $0.74^{+0.22}_{-0.60}$
a: Names are abbreviated. We include candidate detections as full
astrophysical events. Parameters of first $5$ events are from original
LIGO/Virgo analysis (Abbott et al., 2021a), while the remaining $2$ are from
deeper search into the detectors noise (Abbott et al., 2021b).
The formation of close BH-BH systems is an open issue with several formation
channels proposed and discussed in the context of the LIGO/Virgo mergers. The
major formation scenarios include the isolated binary evolution (Bond & Carr,
1984; Tutukov & Yungelson, 1993; Lipunov et al., 1997; Voss & Tauris, 2003;
Belczynski et al., 2010b; Dominik et al., 2012; Kinugawa et al., 2014; Hartwig
et al., 2016; de Mink & Mandel, 2016; Mandel & de Mink, 2016; Marchant et al.,
2016; Spera et al., 2016; Belczynski et al., 2016a; Eldridge & Stanway, 2016;
Woosley, 2016; van den Heuvel et al., 2017; Stevenson et al., 2017; Kruckow et
al., 2018; Hainich et al., 2018; Marchant et al., 2018; Spera et al., 2019;
Neijssel et al., 2019; du Buisson et al., 2020; Bavera et al., 2020, 2021; Qin
et al., 2021), the dense stellar system dynamical channel (Portegies Zwart &
McMillan, 2000; Miller & Hamilton, 2002b, a; Portegies Zwart et al., 2004;
Gültekin et al., 2004, 2006; O’Leary et al., 2007; Sadowski et al., 2008;
Downing et al., 2010; Antonini & Perets, 2012a; Benacquista & Downing, 2013;
Mennekens & Vanbeveren, 2014; Bae et al., 2014; Chatterjee et al., 2016;
Mapelli, 2016; Hurley et al., 2016; Rodriguez et al., 2016; VanLandingham et
al., 2016; Askar et al., 2017; Arca-Sedda & Capuzzo-Dolcetta, 2017; Samsing,
2017; Morawski et al., 2018; Banerjee, 2018; Di Carlo et al., 2019; Zevin et
al., 2019; Rodriguez et al., 2018; Perna et al., 2019; Kremer et al., 2020),
isolated multiple (triple, quadruple) systems (Antonini et al., 2017; Silsbee
& Tremaine, 2017; Arca-Sedda et al., 2018; Liu & Lai, 2018; Fragione & Kocsis,
2019), mergers of binaries in galactic nuclei (Antonini & Perets, 2012b;
Hamers et al., 2018; Hoang et al., 2018; Fragione et al., 2019) and primordial
BH formation (Sasaki et al., 2016; Green, 2017; Clesse & García-Bellido, 2017;
Carr & Silk, 2018; De Luca et al., 2020).
BH spins and their orientations can play an important role in distinguishing
between various BH-BH formation models. If the BH spins are not small, then
their orientation may possibly distinguish between a binary evolution origin
(predominantly aligned spins) and dynamical formation channels (more or less
isotropic distribution of spin orientations). If the BHs formed out of stars
have small spins (Spruit, 2002; Zaldarriaga et al., 2017; Hotokezaka & Piran,
2017; Fuller et al., 2019; Qin et al., 2019; Olejak et al., 2020; Bavera et
al., 2020; Belczynski et al., 2020) then BH-BH mergers with high effective
spins may challenge their isolated evolution origin. In dense stellar
clusters, BHs may merge several times easily producing BHs with high spins and
making a dynamical channel a prime site for such events (Gerosa & Berti, 2017;
Fishbach et al., 2017). However, the assumption about the BH natal spin (and
the AM transport efficiency) also plays a role in the effective spin
distribution for the dynamical channel (Banerjee, 2021).
In this study we show that the current understanding of stellar/binary
astrophysics (Belczynski et al., 2021) and the degeneracy between the
different formation channels do not allow for such a simple test of the origin
of the LIGO/Virgo BH-BH mergers. To demonstrate this we show that although the
isolated binary evolution channel produces mostly BH-BH mergers with low
effective spins, a small but significant fraction of mergers is expected to
have moderate or even high effective spins. Despite the assumption that stars
slow down their rotation due to efficient AM transport, we find that tidal
interactions are capable of spinning up some stars allowing formation of
rapidly spinning BHs (Detmers et al., 2008; Kushnir et al., 2017; Qin et al.,
2018).
## 2\. Method
We use the population synthesis code StarTrack (Belczynski et al., 2002, 2008)
with a model of star formation rates and metallicity distribution based on
Madau & Dickinson (2014) described in Belczynski et al. (2020). We employ the
delayed core-collapse supernova (SN) engine for neutron star/BH mass
calculation (Fryer et al., 2012), with weak mass loss from pulsation pair
instability supernovae (Belczynski et al., 2016b). BH natal kicks are
calculated from a Maxwellian distribution with
$\sigma=265{\rm~{}km}{\rm~{}s}^{-1}$ and decreased by fallback during core-
collapse; this makes a significant fraction of BHs form without a natal kick
(Mirabel & Rodrigues, 2003). We assume our standard wind losses for massive
O/B stars (Vink et al., 2001) and LBV winds (specific prescriptions for these
winds are listed in Sec. 2.2 of Belczynski et al., 2010a). BH natal spins are
calculated under the assumption that AM in massive stars is transported by the
Tayler-Spruit magnetic dynamo (Spruit, 2002) as adopted in the MESA stellar
evolutionary code (Paxton et al., 2015). Such BH natal spins take values in
the range $a\in 0.05-0.15$ (see Belczynski et al., 2020). Note that the
modified classic Tayler-Spruit dynamo with a different non-linear saturation
mechanism of the Tayler instability (Fuller et al., 2019; Fuller & Ma, 2019)
causes larger magnetic field amplitudes, more efficient AM transport and even
lower final natal spins $(a\sim 0.01)$. BH spin may be increased if the
immediate BH progenitors (Wolf-Rayet: WR) stars in close binaries are subject
to tidal spin-up. In our calculations for BH-WR, WR-BH and WR-WR binary
systems with orbital periods in the range $P_{\rm orb}=0.1-1.3$ d the BH natal
spin magnitude is fit from WR star spun-up MESA models (see eq.15 of
Belczynski et al. (2020)), while for systems with $P_{\rm orb}<0.1$d the BH
spin is taken to be equal to $a=1$. BH spins may also be increased by
accretion in binary systems. We treat accretion onto a compact object during
Roche lobe overflow (RLOF) and from stellar winds using the analytic
approximations presented in King et al. (2001) and Mondal et al. (2020). In
the adopted approach the accumulation of matter on a BH is very inefficient so
accretion does not noticeably affect the final BH spin. However note that,
e.g. van Son et al. (2020) or Bavera et al. (2021) tested different super
Eddington accretion prescriptions finding that some BHs may be significantly
spun-up by accretion.
For common the envelope (CE) evolution we assume a $100\%$ ($\alpha_{\rm
CE}=1$) orbital energy transfer for CE ejection and we adopt $5\%$ Bondi
accretion rate onto the BHs during CE (Ricker & Taam, 2008; MacLeod & Ramirez-
Ruiz, 2015; MacLeod et al., 2017). During the stable RLOF (whether it is a
thermal- or nuclear-timescale mass transfer: TTMT/NTMT) we adopt the following
input physics. If an accretor is a compact object (neutron star or BH) we
allow for super-Eddington accretion with excess transferred mass lost with an
AM specific to the accretor (Mondal et al., 2020). In all other cases, we
allow a fraction of the transferred mass of $f_{\rm a}=0.5$ to be lost from
the binary with a specific AM of the binary orbit $j_{\rm loss}=1.0$
(expressed in units of $2\pi A^{2}/P_{\rm orb}$, $A$ being an orbital
separation; see eq. 33 of Belczynski et al. (2008)).
RLOF stability is an important issue in the context of BH-BH system formation
in the framework of the isolated binary evolution (Neijssel et al., 2019;
Olejak et al., 2021; Gallegos-Garcia et al., 2021; Belczynski et al., 2021).
In the standard StarTrack evolution we impose rather liberal limits for CE
(dynamical-timescale RLOF) to develop (see Belczynski et al. (2008): binaries
with a donor star more massive than $2-3$ times the mass of the accretor are
subject to CE. In this model (for simplicity tagged here as CE model) the vast
majority of BH-BH mergers form through CE evolution, although we find some
cases ($\lesssim 1\%$) of BH-BH merger formation without any CE event. In the
alternative model (non-CE model, detailed description in Olejak et al., 2021)
we allow CE to be suppressed for some systems even with mass ratio as high as
$6-8$ (Pavlovskii et al., 2017). In this model the majority of the BH-BH
mergers form without any CE event (the orbital decrease is obtained through
angular momentum loss during stable RLOF), although some ($<10\%$) BH-BH
mergers form with the assistance of CE.
For each model we calculate the evolution of $64$ million massive, Population
I/II binary systems. We use the star formation history and chemical evolution
of the Universe to obtain the BH-BH merger properties within an approximate
reach of LIGO/Virgo (redshift $z<1$). We use the same method as described in
Belczynski et al. (2020).
## 3\. Results
Figure 1 shows a typical example of binary system evolution without a CE phase
leading to the formation a BH-BH merger with a tidally spun-up primary BH
(restricted RLOF stability criteria; Olejak et al., 2021). The rather unequal-
mass massive stellar system ($112{~{}M}_{\odot}$ and $68{~{}M}_{\odot}$) with
a metallicity of $Z=0.002$ goes through two RLOF events. The RLOF I is
initiated by the more massive star; first by an NTMT when the donor is still
on the main-sequence and then through a TTMT when the donor evolves off main-
sequence. After the RLOF I, the system mass ratio is reversed: the initially
more massive star lost over $80\%$ of its mass while the companion gained
$\sim 40{~{}M}_{\odot}$. Next, the initially more massive star ends its
evolution directly collapsing to the less–massive (secondary) BH with a mass
of $m_{2}=15{~{}M}_{\odot}$ and spin $a_{2}=0.14$. When the companion star
expands, it initiates a second stable RLOF. At the onset of RLOF II the system
has highly unequal masses: the donor is almost $6.5$ times more massive than
the BH. The thermal timescale for a donor with mass $M_{\rm don}\approx
97{~{}M}_{\odot}$, radius $R_{\rm don}\approx 300{~{}R}_{\odot}$ and
luminosity $L_{\rm don}\approx 3\times 10^{6}L_{\odot}$ (parameters at the
RLOF II onset 111Such parameter values are inline with other predictions for
massive stars e.g. using Geneva stellar evolution code (Yusof et al., 2013). )
calculated with the formula by Kalogera & Webbink (1996), is $\tau_{\rm
th}\approx 330$ $\rm{yr}.$ It corresponds to a very high mass transfer rate
$\dot{M}=M_{\rm don}/\tau_{\rm th}\approx 0.3{\rm~{}M}_{\odot}{\rm~{}yr}^{-1}$
which does not allow the BH to accrete much mass (despite the fact that we
allow for super-Eddington accretion). Half of the donors mass is lost from the
binary with the specific AM of the BH (as the matter was transferred to the
vicinity of the BH accretor). This has a huge effect on the orbital separation
which decreases from $A=467{~{}R}_{\odot}$ to only $A=7.1{~{}R}_{\odot}$.
After RLOF II the binary consists of a BH and a WR star that are close enough
to allow for the tidal spin-up of the WR star. Finally, the WR star directly
collapses to the more massive (primary) BH with a mass
$m_{1}=36{~{}M}_{\odot}$ and spin $a_{1}=0.68$. The BH-BH system merges after
$\sim 67$ Myr.
Figure 2 shows a typical CE evolution scenario (standard StarTrack RLOF
stability criteria) leading to the formation a BH-BH merger with both BHs
spun-up by tidal interactions. At the beginning, the binary system of two
$\sim 36{~{}M}_{\odot}$ stars with $Z=0.0025$ is on a wide ($A\approx
1340{~{}R}_{\odot}$) and eccentric orbit ($e=0.1$). When the initially more
massive star expands the system goes through a stable RLOF, after which the
donor looses its H-rich envelope and the orbit circularizes. Soon after RLOF
I, the system goes through another (unstable) RLOF initiated by the initially
less massive companion star. The ensuing CE evolution leads to significant
orbital contraction from $A=3100{~{}R}_{\odot}$ to $A=4.5{~{}R}_{\odot}$ and
leaves two WR stars subject to strong tidal interactions. Both stars end their
evolution at a similar time forming via supernovae explosions two $\sim
9{~{}M}_{\odot}$ BHs. At the formation, both BHs get significant natal kicks
that makes the system orbit larger $A\approx 19{~{}R}_{\odot}$ and eccentric
$e=0.44$, leading to a merger time of $\sim 6.7$ Gyr.
In Table 2 we present the statistical spin properties of BH-BH systems merging
at redshifts $z<1$ for the two tested RLOF stability criteria models. In the
rows $1-6$ we list the percentage of the BH-BH mergers with effective–spin
parameter values $\chi_{\rm eff}>0.0,\ 0.1,\ 0.2,\ 0.3,\ 0.4,\ 0.5$. In the
rows $7-9$ we list the percentages of BH-BH mergers with a highly spinning
primary BH $a_{1}>0.5,\ 0.7,\ 0.9$ while the rows $10-12$ give the percentages
of mergers with a highly spinning secondary BH $a_{2}>0.5,\ 0.7,\ 0.9$. The
full distribution of the primary–spin, the secondary–spin and the
effective–spin parameter for both the CE and non-CE evolution, is plotted in
Figure 3 in APPENDIX A.
Figure 1.— Typical example of non-CE evolutionary scenario leading to the
formation of BH-BH merger with tidally spun-up primary: $a_{1}=0.68$ and
$\chi_{\rm eff}=0.52$. Binary system goes through two phases of RLOF with
episodes of nuclear and thermal timescale mass transfer. RLOF I ends with the
system mass ratio reversal. After RLOF II the system orbital separation
significantly decreases and WR star is a subject to tidal spin-up by a BH.
Soon thereafter the close BH-BH system is formed with a short merger time of
$\sim 67$ Myr (see Sec. 3). Figure 2.— Typical example of evolutionary
scenario with CE phase leading to the formation of BH-BH merger with
$a_{1}=0.79$, $a_{2}=0.79$ and $\chi_{\rm eff}=0.77$. First, the binary system
goes through stable RLOF phase with episodes of nuclear and thermal timescale
mass transfer initiated by the initially more massive star. Then initially
less massive star expands and initiates CE, after which the orbital separation
is significantly decreased. After CE, binary hosts two compact WR stars that
are subject to tidal spin-up. Both stars explode as supernovae and form BHs on
eccentric orbit with merger time of $\sim 6.7$ Gyr (see Sec. 3). Table
2Predictions for BH-BH mergers from binary evolution
No. | conditiona | CE model | non-CE model
---|---|---|---
1 | $\chi_{\rm eff}>0.0$ | 97% | 93%
2 | $\chi_{\rm eff}>0.1$ | 95% | 85%
3 | $\chi_{\rm eff}>0.2$ | 70% | 60%
4 | $\chi_{\rm eff}>0.3$ | 36% | 39%
5 | $\chi_{\rm eff}>0.4$ | 10% | 21%
6 | $\chi_{\rm eff}>0.5$ | 2% | 7%
7 | $a_{1}>0.5$ | 3% | 34%
8 | $a_{1}>0.7$ | 2% | 15%
9 | $a_{1}>0.9$ | 1% | 1%
10 | $a_{2}>0.5$ | 52% | 11%
11 | $a_{2}>0.7$ | 33% | 7%
12 | $a_{2}>0.9$ | 12% | 2%
a: We list fractions of BH-BH mergers (redshift $z<1$) produced in our two
population synthesis models satisfying a given condition.
## 4\. Discussion and Conclusions
The rapidly increasing number of detected BH-BH mergers does allow for some
general population statements (Roulet et al., 2021; Galaudage et al., 2021;
Abbott et al., 2021b). It appears that (i) majority ($\sim 70-90\%$) of BH-BH
mergers have low effective spins consistent with $\chi_{\rm eff}\approx 0$ and
that (ii) small fraction ($\sim 10-30\%$) of mergers have positive non-zero
spins that can be as high as $\chi_{\rm eff}\gtrsim 0.5$. Additionally, the
population is consistent with (iii) no systems having negative effective spins
and (iv) a not isotropic distribution of effective spins (which could indicate
dynamical origin). Finally, (v) there is at least one case of a primary BH
(more massive) in a BH-BH merger with very high spin ($a_{1}>0.7$ at 90%
credibility). These properties are noted to be broadly consistent with BH-BH
mergers being formed in an isolated binary evolution.
In our study we have tested whether we can reproduce the above spin
characteristics with our binary evolution models that employ efficient AM
transport in massive stars and that impose tidal spin-up of compact massive
Wolf-Rayet stars in close binaries. The two presented models employ our
standard input physics but allow for the formation of BH-BH mergers assisted
either by a CE or by a stable RLOF. We find that the observed population and
its spin characteristics (i–v) is consistent with our
isolated–binary–evolution predictions (see Tab. 2). In particular, we find
that the majority of BH-BH mergers have small positive effective spins: $\sim
70\%$ mergers have $0<\chi_{\rm eff}<0.3$ (efficient AM transport), while a
small fraction have significant spins: $36-39\%$ mergers have $\chi_{\rm
eff}>0.3$ and $2-7\%$ mergers have $\chi_{\rm eff}>0.5$ (tidal spin-up). The
fraction of systems with negative effective spins is small ($3-7\%$) as most
BHs do not receive strong natal kicks in our simulations. Individual BH spins
can reach high values. A large fraction ($11-52\%$) of secondary BHs may have
significant spin values ($a_{2}>0.5$) as it is the less massive stars that are
most often subject to tidal spin-up. Nevertheless, primary BHs may also form
with high spins ($3-34\%$ with $a_{1}>0.5$) if both stars have similar masses
and both are subject to tidal spin-up (see Fig. 2) or due to mass ratio
reversal caused by the RLOF (see Fig. 1). We also note the formation of a
small fraction of almost maximally spinning BHs: $2-12\%$ for $a_{2}>0.9$
(secondary BH) and $1\%$ for $a_{1}>0.9$ (primary BH). These results on
effective spins and individual BH spins are consistent with the current
LIGO/Virgo population of BH-BH mergers. Note that Qin et al. (2021) came to
different conclusions, finding the high-spinning detections challenging for
the Tayler-Spruit dynamo, especially for the unequal mass event with a high
spinning primary (GW190403). Our non-CE model reproduces this type of mergers
due to the mass ratio reversal (see Fig. 1). In this channel, at the onset of
the second stable RLOF, the donor may be even 5-6 times more massive than the
accretor, ending as an unequal mass $(q\leq 0.4)$ BH-BH merger. Qin et al.
(2021) have not considered the case of a stable RLOF in such unequal mass
systems.
The above fractions correspond to just two different modes of spinning-up
during the classical isolated binary BH-BH formation. Had we varied several
other factors that influence BH spins and their orientations in BH-BH mergers,
the ranges of these fractions would have broadened. Some obvious physical
processes that can affect BH spins and their orientations include: initial
star spin alignment (or lack thereof) with the binary AM, the alignment of
stellar spins (or lack thereof) during RLOF phases, the treatment of
accretion, the initial mass ratio distribution that can alter the ratio of
systems going through stable and unstable (CE) RLOF, and the natal kicks that
can misalign spin orientations. Above all, the three major uncertainties
include the initial stellar rotation of stars forming BHs, the efficiency of
AM transport and the strength of tides in close binary systems. All of the
above are only weakly constrained. Note that this is a proof-of-principle
study that is limited only to BH spins in BH-BH mergers. In particular, we did
not try to match BH masses and BH-BH merger rates for the highly spinning
LIGO/Virgo BHs. In this study we have only shown that it is possible to
produce highly spinning BHs by tidal interactions of stars in close binaries
in evolution that includes and does not include CE. Our two examples of
evolution (Fig. 1 and 2) have much smaller masses than the LIGO/Virgo mergers
with highly spinning BHs (Tab. 1). Note, however, that we have not used here
the input physics that allows for the formation of BHs with mass over
$50{~{}M}_{\odot}$. Such model is already incorporated and tested within our
population synthesis code (Belczynski, 2020). An attempt to match all observed
parameters simultaneously is projected to happen in the future when LIGO/Virgo
will deliver a larger sample of highly spinning BHs.
Given the results presented in this study, alas limited only to BH spins, we
conclude that (i) the isolated binary evolution channel reproduces well the BH
spins of the LIGO/Virgo mergers (ii) if, in fact, the binary channel is
producing the majority of the LIGO/Virgo BH-BH mergers, then this indicates
that the AM transport is efficient in massive stars and the tidal interactions
in close binaries are strong.
We thank the anonymous reviewer, Jean-Pierre Lasota, Ilya Mandel and Sambaran
Banerjee for their useful comments on the manuscript. KB and AO acknowledge
support from the Polish National Science Center (NCN) grant Maestro
(2018/30/A/ST9/00050).
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## APPENDIX A
Figure 3.— Distribution of primary BH spin ($a_{1}$) – top panel; secondary
BH spin ($a_{2}$) – middle panel; effective spin parameter ($\chi_{\rm eff}$)
– bottom panel; of BH-BH mergers at redshifts $z<1.0$. The results are for two
tested models: the non-CE model plotted with red line and the CE model plotted
with blue line. The figure is a supplement to the statistical spin predictions
shown in Table 2 and described in Section 3.
|
# Arbitrage equilibria in active matter systems
Venkat Venkatasubramanian<EMAIL_ADDRESS>Complex Resilient Intelligent
Systems Laboratory, Department of Chemical Engineering, Columbia University,
New York, NY 10027 Abhishek Sivaram Department of Chemical and Biochemical
Engineering, Technical University of Denmark, 2800 Kongens Lyngby, Denmark N.
Sanjeevrajan Department of Materials Engineering, Indian Institute of
Technology-Madras, Chennai, India 600036 Arun Sankar School of Electrical,
Computer, and Energy Engineering, Arizona State University, Tempe, AZ
## Abstract
The motility-induced phase separation (MIPS) phenomenon in active matter has
been of great interest for the past decade or so. A central conceptual puzzle
is that this behavior, which is generally characterized as a nonequilibrium
phenomenon, can yet be explained using simple equilibrium models of
thermodynamics. Here, we address this problem using a new theory, statistical
teleodynamics, which is a conceptual synthesis of game theory and statistical
mechanics. In this framework, active agents compete in their pursuit of
maximum effective utility, and this self-organizing dynamics results in an
arbitrage equilibrium in which all agents have the same effective utility. We
show that MIPS is an example of arbitrage equilibrium and that it is
mathematically equivalent to other phase-separation phenomena in entirely
different domains, such as sociology and economics. As examples, we present
the behavior of Janus particles in a potential trap and the effect of
chemotaxis on MIPS.
## I Introduction
Active matter describes systems composed of a large number of self-actualizing
dynamical agents that consume and dissipate energy and exhibit interesting
macroscopic behaviors [1, 2, 3, 4, 5, 6, 7]. Biological examples of such self-
organizing systems include bacteria, ants, birds, mussels, etc. Nonliving
active matter examples include self-propelled Janus particles, layers of
vibrated granular rods, and so on. A central conceptual puzzle in our evolving
understanding of active matter is why and when a collection of active agents
that looks like an out-of-equilibrium system on the microscopic scale behaves
macroscopically like a simple equilibrium system of passive matter [5, 7, 8,
6, 9, 10].
Recently, a new framework called statistical teleodynamics has been developed
to predict the macroscopic emergent behavior of active matter systems that
comprise biological, ecological, and socioeconomic agents [11, 12, 13, 14, 15,
16]. Statistical teleodynamics is a natural generalization of statistical
thermodynamics for goal-driven agents in active matter. It is a conceptual
synthesis of the central concepts and techniques of population games theory
with those of statistical mechanics towards a unified theory of emergent
equilibrium phenomena and pattern formation in active matter. The name comes
from the Greek word telos, which means goal. Just as the dynamical behavior of
molecules is driven by thermal agitation (hence thermodynamics), the dynamics
of purposeful agents are driven by the pursuit of their goals and hence
teleodynamics.
The fundamental quantity in statistical teleodynamics is the effective utility
of an agent, which measures the net benefit of an agent after subtracting all
the costs the agent incurred in acquiring it. All agents in a population
compete to increase their effective utilities. Population game theory proves
that this competitive pursuit of maximum utility, under certain conditions,
leads to an equilibrium, called the arbitrage equilibrium, where the effective
utilities of all agents are equal [17].
There is an important philosophical difference between statistical
teleodynamics and statistical thermodynamics. Statistical teleodynamics
acknowledges the importance of recognizing the individual active agent and its
behavioral properties explicitly in developing a bottom-up analytical
framework of emergent phenomena. It also accounts overtly for the role of an
agent’s purpose (e.g., survival and growth) and the naturally attendant
concept of the pursuit of maximum utility. The role of purpose is not
acknowledged in statistical mechanics, as there is no way to express that
concept in that framework. However, it is the central concept in game theory
and hence fits in naturally in statistical teleodynamics.
Since our theory is a bottom-up emergentist framework, it emphasizes the agent
perspective in contrast to statistical mechanics, which stresses the system
view. For example, whenever equilibrium is addressed in statistical mechanics,
it is usually formulated in terms of minimizing free energy, which is a top-
down system perspective. However, in statistical teleodynamics, while the
system view is also present via the maximization of the game-theoretic
potential (as we discuss in the following sections), the equality of effective
utilities for all agents as the equilibrium criterion from the agent
perspective is conspicuously recognized and exploited.
In our previous work, using statistical teleodynamics, we have shown that
emergent self-organized behaviors of active agents in different disciplines,
such as bacterial chemotaxis [13], ant crater formation [13], flocking of
birds [14], mussel bed patterning [15], social segregation [16], and income
distributions [12], can be understood as the result of arbitrage equilibria in
their respective contexts. The self-actualizing agents in these studies are
examples of living agents found in biology, ecology, sociology, and economics.
Therefore, they are self-driven by survival purpose and the pursuit of maximum
utility.
Although the statistical teleodynamics framework was developed to model and
predict the emergent behavior of a large population of living goal-driven
agents, we believe that it could also be helpful in understanding the emergent
behavior of nonliving self-actualizing agents, such as Janus particles.
Although nonliving agents are not purposeful, their self-actualizing behavior
appears seemingly purposeful, as if they are pursuing some goal in their
persistent dynamics.
In this paper, by applying the statistical teleodynamics framework, we show
that the nonliving active matter systems are in arbitrage equilibria. Under
certain mathematical conditions that we discuss in the rest of the paper, this
arbitrage equilibrium is equivalent to a statistical or thermodynamic
equilibrium, thereby resolving the above-mentioned conceptual puzzle.
The remainder of the paper is organized as follows. First, we introduce the
statistical teleodynamics framework. Then, we show how the self-organizing
dynamics of ant-crater formation is similar to the self-actualizing behavior
of Janus particles in a potential trap [18], and the emergent distribution for
both weak and strong traps is indeed an equilibrium distribution, a Weibull
distribution. We then present a model game-theoretic system that predicts the
emergent macroscopic behavior of a population of agents that spontaneously
segregate under certain conditions at arbitrage equilibrium. We show how this
arbitrage equilibrium is equivalent to motility-induced phase separation
(MIPS) [5, 7, 6, 19]. We further extend this discussion to develop a utility-
based model of chemotaxis-driven MIPS [20]. In all these cases, we show that
the final configurations are indeed in arbitrage equilibrium, which is
equivalent to thermodynamic or statistical equilibrium.
## II Statistical Teleodynamics, Potential Games, and Arbitrage Equilibrium
As noted, statistical teleodynamics is a synthesis of the central concepts and
techniques of population games theory with those of statistical mechanics. The
theory of population games is concerned with predicting the final outcome(s)
of a large population of goal-driven agents. Given a large collection of
strategically interacting rational agents, where each agent is trying to
decide and execute the best possible course of actions that maximizes the
agent’s payoff or utility in light of similar strategies executed by the other
agents, can we predict which strategies would be executed and what outcomes
are likely [21, 17]? In particular, one would like to know whether such a game
would lead to an equilibrium situation.
For some population games, one can identify a single scalar-valued global
function, called a potential ($\phi(\boldsymbol{x})$), that captures the
necessary information about the utilities (where $\boldsymbol{x}$ is the state
vector of the system). The gradient of the potential is the payoff or utility.
Such games are called potential games [22, 17, 21, 23]. A potential game
reaches strategic equilibrium, called Nash equilibrium, when the potential
$\phi(\boldsymbol{x})$ is maximized. Furthermore, this equilibrium is unique
if $\phi(\boldsymbol{x})$ is strictly concave (i.e.,
$\partial^{2}\phi/\partial^{2}x<0$) [17].
In potential games, the utility $h_{i}$ of an agent in state $i$ is the
gradient of potential $\phi(\boldsymbol{x})$, i.e.,
${h}_{i}(\boldsymbol{x})\equiv{\partial\phi(\boldsymbol{x})}/{\partial x_{i}}$
(1)
where $x_{i}=N_{i}/N$ and $\boldsymbol{x}$ is the population vector. $N_{i}$
is the number of agents in state $i$, and $N$ is the total number of agents.
Therefore, we have
$\displaystyle\phi(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\int{h}_{i}(\boldsymbol{x}){d}x_{i}$ (2)
where $n$ is the total number of states.
To determine the maximum potential, one can use the method of Lagrange
multipliers with $L$ as Lagrangian and $\lambda$ as the Lagrange multiplier
for the constraint $\sum_{i=1}^{n}x_{i}=1$:
$L=\phi+\lambda(1-\sum_{i=1}^{n}x_{i})$ (3)
All agents enjoy the same utility in equilibrium, i.e., $h_{i}=h^{*}$. It is
an arbitrage equilibrium [24] in which agents are no longer incentivized to
switch states, as all states provide the same utility $h^{*}$. In other words,
equilibrium is reached when the opportunity for arbitrage, i.e., the ability
to increase one’s utility simply by switching to another option or state at no
cost, disappears. Thus, the maximization of $\phi$ and $h_{i}=h^{*}$ are
equivalent when the equilibrium is unique (i.e., $\phi(\boldsymbol{x})$ is
strictly concave [17]), and both specify the same outcome, namely, an
arbitrage equilibrium. The former stipulates it from the top-down system
perspective, whereas the latter is the bottom-up agent perspective. Thus, this
formulation exhibits the duality property.
Just as mechanical equilibrium is reached when the forces balance each other
equally, thermal equilibrium is reached when the temperatures are equal, and
phase equilibrium is achieved when the chemical potentials are equal, our
theory demonstrates that a system of active agents will reach an arbitrage
equilibrium when their effective utilities are equal. Whenever the effective
utility is of a particular mathematical form (as explained in Sections VI and
VII), this game-theoretic Nash equilibrium is equivalent to the statistical
Boltzmann equilibrium. In fact, our theory reveals the critical insight that
both living and non-living agents are driven by arbitrage opportunities
towards equilibrium, except that their arbitrage currencies are different. For
nonliving matter, the currency is the chemical potential, whereas for living
matter, the effective utility.
Although the chemical potential (and free energy) is appropriate for
describing nonliving physicochemical systems, its usage for living agents,
such as bacteria, ants, birds, and so on, seems a bit awkward. We believe that
effective utility (and the game-theoretic potential) is a more natural choice
for active agents driven by survival and growth goals found in biology,
ecology, sociology, and economics. Thus, the utility-oriented perspective
helps us to extend the concepts and techniques of statistical thermodynamics
more naturally to teleological agents by smoothly connecting with game theory.
This is what has been accomplished by statistical teleodynamics.
We wish to emphasize that we are not claiming that the lower forms of living
agents such as bacteria and ants pursue the survival goal and strategies
rationally. Our view is that the biological survival instincts of such agents
cause particular dynamical behaviors that evolved over millions of years to
help them improve their survival chances. Therefore, they act in a goal-driven
manner _instinctively_ , which can be modeled using our framework of the
pursuit of maximum utility or survival fitness.
Our goal is to identify the fundamental principles and mechanisms of self-
organization of goal-driven agents. Towards that, we develop simple models
that offer an appropriate coarse-grained description. The spirit of our
modeling is similar to that of the van der Waals or the Ising model in
statistical thermodynamics.
## III Ant crater model and Janus particles
As an example of an active matter system, the dynamical behavior of self-
actualized Janus particles in a potential trap has attracted considerable
attention [18, 25]. Takatori et al. [18] discuss the behavior of Janus
particles in two regimes: (i) weak trap ($\alpha<1$) and (ii) strong trap
($\alpha>1$). They observe that the particle behavior in the weak regime can
be considered an equilibrium outcome, whereas, in the strong regime, they
suggest a nonequilibrium behavior.
We approach this system from the perspective of statistical teleodynamics. For
us, this system resembles the behavior of a large population of ants building
an ant colony underground. The dynamics of this activity involves the
transport of sand grains by ants from an underground nest to the surface. The
resulting grain pile, called the ant crater, is of a particular shape known as
the Weibull distribution [13].
Since grain transport involves an effort that increases with the distance the
ant travels, the ants would prefer to drop the grains sooner rather than later
to minimize the effort. However, if they drop them too close to the nest, the
sand grains pile could collapse back into the nest, which would mean more work
for them later on. Therefore, ants innately balance the need to transport
grains as far away as possible while trying to minimize the effort (i.e., the
disutility) expended in doing so.
As we showed recently [13], this process can be modeled considering the
effective utility of ants and their self-organizing competitive dynamics. In
our model, the utility of an ant is determined by three factors. The first
factor is the utility or benefit that it gains from having a home, the nest,
given by $b>0$.
The second factor describes the disutility (i.e., the cost) it incurs by
transporting the grains away from the nest. We assume that the ants move
outward radially from the nest with some average velocity $v$. We model the
rate at which the ants drop off the grains as $sr^{a-1}$, where $r$ is the
distance it travels from the nest to the drop-off point ($s>0$ and $a>1$ are
constant parameters). The disutility of the effort $W$ an ant expends then
depends on how much grain it carries and for how long. This results in
$\displaystyle
W=\int_{0}^{t}sr^{a-1}dt=\int_{0}^{r}sr^{a-1}\frac{dr}{v}=\frac{sr^{a}}{va}=\frac{\omega
r^{a}}{a}$
where $\omega=s/v$. In chemical engineering, $W$ is known as the Damköhler
number which quantifies the ratio of the time scales of flow to that of the
reaction. Higher values of Damköhler number, in this case, suggest a higher
propensity of an ant to drop off the grains.
The third factor accounts for the disutility of competition among ants. As
ants ($N_{i}$) try to crowd at the same location ($r_{i}$) to drop off their
grains, this term forces them to spread out to minimize the cost of the
competition. As Venkatasubramanian et al. [13] discuss, this term is modeled
as $-\ln N_{i}$.
Combining all three, the effective utility $h_{i}$ that an ant gains by
dropping a grain at a distance $r_{i}$ is given by
$h_{i}(r_{i},N_{i})=b-\frac{\omega r_{i}^{a}}{a}-\ln N_{i}$ (4)
The potential for this system then becomes
$\displaystyle\phi(\mathbf{x})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\int h_{i}(\mathbf{x})dx_{i}$ (5) $\displaystyle=$
$\displaystyle b-\frac{\omega}{a}\langle
r^{a}\rangle+\frac{1}{N}\ln\frac{N!}{\prod_{i=1}^{N}(Nx_{i})!}$ (6)
where $\langle r^{a}\rangle$ is the expectation of the quantity $r^{a}$, based
on the locations of the ants ($N$ is the total number of ants). As the reader
might recognize, the last term in Eq. 6 is entropy. Therefore, by maximizing
potential $\phi$, one is equivalently maximizing entropy subject to the
constraints in the first two terms. This deep connection between statistical
mechanics (through entropy) and game theory (through potential) has been
discussed in great detail here [12, 24]. We discuss this connection at some
length in Section V.
As Venkatasubramanian et al. [13] show, there is a unique arbitrage
equilibrium outcome for this collective behavior, where all ants have the same
utility, i.e., $h_{i}=h^{*}$. Therefore, we have
$h^{*}=b-\frac{\omega r_{i}^{a}}{a}-\ln N_{i}^{*}$ (7)
which can be rearranged to show
$\displaystyle x_{i}^{*}=\frac{N_{i}^{*}}{N}=\frac{\exp\left(-\dfrac{\omega
r_{i}^{a}}{a}\right)}{\sum_{j}\exp\left(-\dfrac{\omega r_{j}^{a}}{a}\right)}$
(8)
where $N_{i}^{*}$ is the value in equilibrium. In the continuum limit, the
states are continuous, where the state is defined as the radial location $r$.
This results in the simplification $x_{i}^{*}=N_{i}^{*}/N=\rho^{*}(r)2\pi
rdr/N$, where $\rho^{*}(r)$ is the number density of ants at location $r$ at
equilibrium. With this result in Eq (8), it can be shown that the number
density of ants follows the distribution,
$\rho^{*}(r)=\frac{A}{r}\exp\left({-\frac{\omega r^{a}}{a}}\right)$ (9)
where $A$ is a constant that satisfies the boundary condition of a constant
flux of ants from the center of the nest.
Note that this emergent distribution is that of the number of ants. Given this
distribution of $\rho^{*}(r)$, the grain distribution can be calculated using
a cumulative distribution given by
$\displaystyle F(r)=\frac{\int_{0}^{r}(sr^{a-1}\rho^{*})2\pi
r~{}dr}{\int_{0}^{{\infty}}(sr^{a-1}\rho^{*})2\pi r~{}dr}$
This gives,
$\displaystyle F(r)=1-\exp{\left(-\frac{\omega r^{a}}{a}\right)}$
with the grain distribution $f(r)$, given by
$\displaystyle f(r)=\frac{dF}{dr}=\omega r^{a-1}\exp\left(-\frac{\omega
r^{a}}{a}\right)$ (10)
which is the Weibull distribution of ant craters that is observed empirically.
It is important to emphasize that this distribution results from an
equilibrium, namely, the arbitrage equilibrium. This is not a nonequilibrium
or far-from-equilibrium outcome.
### III.1 Janus particles in a potential trap
We now consider the Janus particles from this perspective. We will show that
the Janus particle dynamics can be considered as an arbitrage equilibrium
outcome in both the weak and strong traps.
Our analysis is based on the work of Takatori et al. [18]. They report on the
behavior of self-propelled Janus particles that are in an acoustic trap whose
strength can be tuned. The trap force is modeled by
$F^{trap}(r)=-kr\exp(-2(r/w)^{2})$ (11)
where $k$ is the spring constant and $w$ is the width of the trap. They show
that the probability distribution $P(r)$ due to the active Brownian motion of
the swimmer is given by
$P(\bar{r})(U_{0}\tau_{R})^{2}=(\alpha/\pi)\exp(-\alpha\bar{r}^{2})$ (12)
where $\bar{r}=r/(U_{0}\tau_{R})$ and $\alpha:=k\tau_{R}/\zeta$ is the
nondimensional trap.
In our approach, we consider the “struggle” of Janus particles swimming
against the trap force to be similar to the effort expended by ants
transporting sand grains. Therefore, we propose that the effective utility
$H_{i}$ for a Janus particle be
$H_{i}(r_{i},N_{i})=-\frac{\omega r_{i}^{a}}{a}-\ln N_{i}$ (13)
where the first term is the disutility of the work done by the Janus particle
against the trap force, and the second term is the disutility of the
competition among the particles, as before. We do not need the benefit term
$b$ here, as it is not relevant for Janus particles. Following the same
analysis for the dynamics of the ants, we conclude that the Janus particles
will also reach the same arbitrage equilibrium outcome of $H_{i}=H^{*}$ with
the probability distribution given by the Weibull distribution
$\displaystyle P(r)=\omega r^{a-1}\exp\left(-\frac{\omega r^{a}}{a}\right)$
(14)
Now, we expect that the exponent $a$ would depend on the trap strength
$\alpha$. As the trap strength increases, the work done by the particles
against the trap force increases super linearly with distance $r$. We
postulate that this nonlinear dependence can be modeled as
$\displaystyle a=1+\alpha+0.5\alpha^{2}$ (15)
We obtained the best fit for Eq. 14 for the weak ($\alpha<1$) and strong
($\alpha>1$) trap force data reported by Takatori et al. [18] (Figs. 2a and 2b
in their paper). The best-fit plots are shown in Figs. 1-4. As we can see, the
Weibull distribution fits both regimes (weak and strong) quite well (the
$R^{2}$ values are reported in the figures). We determined the exponent $a$
from the fitted distributions and used Eq. 15 to estimate the $\hat{\alpha}$
values. We see that the estimated values ($\hat{\alpha}$ shown in the figures)
are in good agreement with the actual $\alpha$ values reported by Takatori et
al. [18].
Thus, the data appear to support our prediction that the final distributions
of the Janus particles in both the weak- and strong-trap regimes are the
results of arbitrage equilibria. They are not out-of-equilibrium or
nonequilibrium systems, but arbitrage equilibrium systems. More experimental
and simulation studies are needed at other values of $\alpha$ to confirm this
more conclusively. In addition, it would be helpful to derive the first term
on the right-hand side of Eq. 13, and Eq. 15, from the first principles.
Figure 1: Weibull distribution fit for the weak trap: Experimental data Figure
2: Weibull distribution fit for the weak trap: Simulation data Figure 3:
Weibull distribution fit for the strong trap: Experimental data Figure 4:
Weibull distribution fit for the strong trap: Simulation data
## IV Schelling-agents model and MIPS
We now consider another game-theoretic model system to address motility-
induced phase separation (MIPS) from a statistical teleodynamics perspective
[13]. Here again, we demonstrate the emergence of arbitrage equilibrium as a
result of the self-organizing dynamics of a large population of active agents
competing for benefits. There are numerous examples of such behavior in the
living world. In ecology, for example, we have the formation of mussel beds in
the sea [26, 27, 28, 15]; in sociology, the social segregation of different
groups [29, 30]; and in economics, the emergence of income inequality in
societies [12].
We develop our model by considering a large lattice of local neighborhoods or
blocks, each with $M$ sites that agents can occupy. There are $n$ such blocks,
$nM$ sites, and a total of $N$ agents, with an average agent density of
$\rho_{0}=N/(nM)$. The state of an agent is defined by specifying the block
$i$ in which it is located, and the state of the system is defined by
specifying the number of agents, $N_{i}$, in block $i$, for all blocks
($i\in\\{1,\dots,n\\}$ ). Let block $i$ also have $V_{i}$ vacant sites, so
$V_{i}=M-N_{i}$.
Next, as we did in the case of the ant crater-Janus particles, we define the
effective utility, $h_{i}$, for agents in block $i$, which agents try to
maximize by moving to better locations (i.e., other blocks), if possible. The
effective utility is, again, the net sum of the benefits minus the costs, but
the benefits and costs here are different from those of the ant-crater
situation, as one might expect.
Here, an agent prefers to have more members in its neighborhood, as this
aggregation provides certain benefits. In ecology, for example, for mussels,
it improves the chances of survival of mussels against predators and sea wave
stress [28, 27]. In sociology, it increases social benefits such as meeting
life partners, finding better jobs, etc. [30]. Therefore, this affinity
benefit term, which represents cooperation between agents, is proportional to
the number of agents in its neighborhood. We model this as $\alpha N_{i}$,
where $\alpha>0$ is a parameter.
However, this affinity benefit comes with a cost. As more and more agents
aggregate, they all compete for the same limited resources in the
neighborhood. This congestion cost is widely modeled by the quadratic
disutility term $-\beta N_{i}^{2}$ ($\beta>0$) [12, 31].
In addition, agents also balance two competing search strategies -
exploitation and exploration. Exploitation takes advantage of the
opportunities in the immediate, local neighborhood of the agents. On the other
hand, exploration examines possibilities outside. This is a widely used
strategy in biology, ecology, and sociology. For example, a genetic mutation
can be thought of as exploitation, which is searching locally in the design
space, while crossover is exploration, which is searching more globally.
Regarding exploration, the agents derive a benefit from having a large number
of vacant sites to potentially move to in the future should such a need arise.
This is the instinct to explore other opportunities, as new vacant sites are
potentially new sources of food and other benefits. We call this resource the
option benefit term as agents have the option to move elsewhere if needed.
Again, following Venkatasubramanian [12, 24, 13], we model this as
$\gamma\ln(M-N_{i})$, $\gamma>0$. The logarithmic function captures the
diminishing utility of this option, a commonly used feature in economics and
game theory. As before, this benefit is also associated with a cost due to
competition among agents for these vacant sites. As in the ant-crater case, we
model this competition cost as $-\delta\ln N_{i}$, $\delta>0$ [11, 12, 24].
Combining all these, we have the following effective utility function $h_{i}$
for the agents in block $i$ as,
$\displaystyle h_{i}(N_{i})=\alpha N_{i}-\beta
N_{i}^{2}+\gamma\ln(M-N_{i})-\delta\ln N_{i}$ (16)
Intuitively, the first two terms in the equation model the benefit-cost trade-
off in the exploitation behavior while the last two model a similar trade-off
in exploration.
Rewriting this in terms of the density ($\rho_{i}$) of agents in block $i$,
$\rho_{i}=N_{i}/M$, and absorbing the constant $M$ into $\alpha$ and $\beta$,
we have
$\displaystyle
h_{i}(\boldsymbol{\rho})=\alpha\rho_{i}-\beta\rho_{i}^{2}+\gamma\ln(1-\rho_{i})-\delta\ln\rho_{i}$
(17)
Note that in certain cases, agents may not occupy all $M$ sites and only a
fraction of sites can be occupied due to restrictions such as steric factors.
This corresponds to a maximum occupancy density ($\rho_{\text{max}}$). In such
a situation the utility of vacant sites will be
$\ln(1-\rho_{i}/\rho_{\text{max}})$, resulting in the formulation,
$\displaystyle
h_{i}(\boldsymbol{\rho})=\alpha\rho_{i}-\beta\rho_{i}^{2}+\gamma\ln(1-\rho_{i}/\rho_{\mathrm{max}})-\delta\ln\rho_{i}$
(18)
We can set $\delta=1$ without any loss of generality. In addition, we set
$\gamma=1$ and $\rho_{\mathrm{max}}=1$ to gain analytical simplicity, but
these can be relaxed later if necessary. Therefore, we now have
$\displaystyle
h_{i}(\boldsymbol{\rho})=\alpha\rho_{i}-\beta\rho_{i}^{2}+\ln(1-\rho_{i})-\ln\rho_{i}$
(19)
For simplicity, we define $u(\rho_{i})=\alpha\rho_{i}-\beta\rho_{i}^{2}$.
Therefore, the potential $\phi(\boldsymbol{\rho})$ in Eq. 2 becomes
$\displaystyle\phi(\boldsymbol{\rho})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\int
h_{i}(\boldsymbol{x})dx_{i}=\frac{M}{N}\sum_{i=1}^{n}\int
h_{i}(\boldsymbol{\rho})d\rho_{i}$ $\displaystyle=$
$\displaystyle\frac{M}{N}\sum_{i=1}^{n}\int_{0}^{\rho_{i}}\left[u(\rho)+\ln(1-\rho)-\ln\rho\right]d\rho$
One can generalize the discrete formulation to a continuous one by replacing
$\rho_{i}$ by $\rho(r)$, where the density is a continuous function of radius
$r$ of the neighborhood as demonstrated by Sivaram and Venkatasubramanian [14]
in the self-organized flocking behavior of birds.
Now, according to the theory of potential games [17], an arbitrage equilibrium
is reached when the potential is maximized. We can determine the equilibrium
utility, $h^{*}$, by the Lagrangian multiplier approach mentioned above (Eq.
3), but there exists a simpler alternative that is more convenient for our
purposes here. To analyze the equilibrium behavior, we can take the simpler
agent-based perspective and exploit the fact that at equilibrium all agents
have the same effective utility, i.e., $h_{i}=h^{*}$, for all $i$. In other
words,
$\displaystyle\alpha\rho^{*}-\beta\rho^{*2}+\ln(1-\rho^{*})-\ln\rho^{*}=h^{*}$
(21)
We explore numerically the behavior of $h$ as a function of $\rho$ (Eq. 19),
as shown in Fig. 5 ($\beta=0$, different $\alpha$) and Fig. 6 ($\alpha=6$,
different $\beta$). As we can see, these two plots are qualitatively similar.
Below a threshold value of $\alpha$ and $\beta$, the utility function is
monotonic and has a unique density (blue curve) for a given utility value.
Above the threshold, the utility is non-monotonic (green curve) and can have
multiple density values for the same utility. This behavior is known as the
van der Waals loop in thermodynamics. In mathematical terms, this cubic-like
function has three real and positive zeros. The red dotted line shows this.
The orange curve represents the threshold behavior.
Figure 5: Effective Utility vs Density: $h$ vs $\rho$ for different $\alpha$.
The black points are the spinodal points
($\rho_{s1}=0.211,h_{s1}=2.585;\rho_{s2}=0.789,h_{s2}=3.415$). The red points
are the binodal points
($\rho_{b1}=0.071,h_{b1}=3.00;\rho_{b2}=0.929,h_{b2}=3.00$). Figure 6:
Effective Utility vs Density: $h$ vs $\rho$ for different $\beta$ The red
points are binodal points. The black points are the spinodal points
($\rho_{s1}$ = 0.237, $h_{s1}$ = 2.535; $\rho_{s2}$ = 0.685, $h_{s2}$ =
2.864). The red points are the binodal points ($\rho_{b1}$ = 0.117, $h_{b1}$ =
2.708; $\rho_{b2}$ = 0.829, $h_{b2}$ = 2.708).
Note that whether all agents remain in a single phase of uniform density
dispersed throughout the region or separate into various groups is determined
by the slope $\partial h/\partial\rho\big{|}_{\rho^{*}}$, which is the second
derivative of $\phi$, $\partial^{2}\phi/\partial^{2}\rho\big{|}_{\rho^{*}}$,
and the zeros of $h(\rho)$. This behavior is mathematically equivalent to
spinodal decomposition in thermodynamics, widely studied, for example, in the
phase separation of alloys and polymer blends [32, 33].
In thermodynamics, the phase between the spinodal points (discussed below in
more detail) is unstable because it corresponds to increasing the free energy
of the system, and hence the single phase splits into two phases of different
densities to lower the free energy. For the same reason, the phases between
the spinodal and binodal points are metastable, and the phases at the binodal
points are stable.
A similar behavior happens here in statistical teleodynamics as well. Here,
the goal is to maximize the potential $\phi$ in (LABEL:eq:schelling-
potential). Thus, in Fig 5 and Fig. 6, we observe that for $\alpha=0$ (blue
curve) and $\alpha=4$ (orange curve), $\partial h/\partial\rho\leq 0$ (i.e.
negative slope; recall that $\partial h/\partial\rho$ =
$\partial^{2}\phi/\partial^{2}\rho$). In such a parameter regime, phase
separation does not occur. However, for higher values of $\alpha$, regions
with $\partial h/\partial\rho>0$ (i.e., positive slope), phase separation
develops.
We better understand this from Fig. 7. The upper part of this figure shows the
potential ($\phi$) vs the density ($\rho$) curve (in green) for $\alpha=6$,
$\beta=0$. The plotted equation is
$\phi=\alpha\frac{\rho^{2}}{2}-\beta\frac{\rho^{3}}{3}-\rho\ln{\rho}-(1-\rho)\ln{(1-\rho)}-2.6\rho$
(22)
The linear term $2.6\rho$ is subtracted from the actual potential function as
a way of rescaling to highlight the double hump nature of the $\phi-\rho$
curve. This subtraction is done purely for illustrative purposes only, as this
double hump otherwise is not so visible in the scale of the figure. In all our
calculations and simulations, this subtraction is not needed and hence is not
done.
The spinodal points are shown as black dots, where $\partial
h/\partial\rho\big{|}_{\rho^{*}}$ =
$\partial^{2}\phi/\partial^{2}\rho\big{|}_{\rho^{*}}=0$. The corresponding
spinodal points are also shown in Fig. 5 as black dots on the green curve
($\alpha=6$, $\beta=0$). Fig. 7 also shows the binodal points (in red,
connected by the common tangent line), where $\partial
h/\partial\rho\big{|}_{\rho^{*}}$ =
$\partial^{2}\phi/\partial^{2}\rho\big{|}_{\rho^{*}}<0$. The corresponding
binodal points are seen in Fig. 5 as red points connected by the red dotted
line. As we see, the two binodal points enjoy the same effective utility
(3.00), which is the arbitrage equilibrium.
The bottom part of Fig. 7 shows the loci of binodal points (red curve) and of
spinodal points (black curve) for different values of $\alpha$ ($\beta=0$). As
$\alpha$ changes, the binodal and spinodal points change, and for $\alpha>4$
($\beta=0$) they disappear. Within the spinodal region, shown in dark gray,
the miscibility gap, a single phase of uniform density is unstable and would
split into two phases of different densities. The reason is that the potential
$\phi$ of a large agent group here is less than the sum of the two potentials
of the low-density group and the high-density group at the binodal points. We
see this geometrically from the common tangent line connecting the binodal
points to be above the single-phase green curve between the spinodal points.
Agents in such regions will be self-driven towards the high-density binodal
point to increase their utility. So $\phi$ increases, and the system splits
into two groups of different densities.
Thus, for the green curve in Fig. 5, a self-organized, utility-driven, stable
phase separation occurs spontaneously at the binodal points (red dotted line)
at the arbitrage equilibrium. While the miscibility gap is unstable, the
region immediately outside of it, between the black and red curves, is
metastable. Beyond the red curve, one has a stable single phase of uniform
density - no phase separation here.
Figure 7: Game potential ($\phi$) curve and the spinodal and binodal points.
For $\alpha=6,\beta=0$, the spinodal densities are 0.211 and 0.789; the
binodal densities are 0.071 and 0.929.
In summary, for high values of $\alpha$ (e.g., green curve in Fig. 5),
combined with average densities in the miscibility gap, we observe the
spontaneous emergence of two phases, high and low density groups of agents, at
arbitrage equilibrium, driven by the self-actuated pursuit of maximum utility
by the agents.
In Fig. 8, we show the region (shaded in yellow) within which the phase
separation occurs at the arbitrage equilibrium for the values of the average
density $\rho_{0}$, $\alpha$, and $\beta$ within the region. In Fig. 9 and
Fig. 10, we show the 2-D slices of the yellow region of spontaneous phase
separation. For a given value of $\alpha$, $\beta$, and $\rho_{0}$, they show
the loci of the two densities (i.e., low- and high-density groups) of the
corresponding equilibrium states of the agents.
Intuitively, in the high-density phase, agents derive so much more benefit
from the affinity term (due to the high $\alpha$) that it more than
compensates for the disutilities due to congestion and competition, thus
yielding a high effective utility. Similarly, in the low-density phase, the
benefits of reduced congestion and lower competition combined with increased
option benefit more than compensate for the loss of utility from the affinity
term. Thus, every agent enjoys the same effective utility $h^{*}$ in one phase
or the other at the arbitrage equilibrium. This causes equilibrium because, as
noted, there is no more arbitrage incentive left for agents to switch
neighborhoods.
As noted, this analysis is mathematically equivalent to spinodal decomposition
in statistical thermodynamics, with an important difference. In statistical
thermodynamics, agents try to minimize their chemical potentials and the free
energy of the system. Here, in statistical teleodynamics, agents try to
maximize their utilities ($h_{i}$) and the game-theoretic potential ($\phi$).
In thermodynamics, chemical potentials are equal in phase equilibrium. In
teleodynamics, the effective utilities are equal in arbitrage equilibrium.
The parallel is striking, but it is not surprising because, as
Venkatasubramanian has shown [12, 13], statistical teleodynamics is a
generalization of statistical thermodynamics for goal-driven agents.
Therefore, given this mathematical equivalence, one should expect to observe
“macroscopic” phenomena generally associated with thermodynamics (such as
phase separation and equilibrium) in entirely different contexts (such as MIPS
or social segregation in socioeconomic systems). The “microscopic” mechanisms
of the self-organizing dynamics might differ in different contexts. Thus, the
driving force for the movements of nonliving agents could be temperature,
pressure, or chemical potential gradients, whereas the driving force for
living agents is effective utility. As noted, since our theory is “mesoscopic”
in character, it is agnostic to the “microscopic” details.
Figure 8: Phase separation region at the arbitrage equilibrium Figure 9:
Constant $\alpha$ (vertical) 2-D slices of the yellow region Figure 10:
Constant $\beta$ (horizontal) 2-D slices of the yellow region
### IV.1 Agent-based simulation results
We tested our model using an agent-based simulation on a $300\times 300$
lattice (90,000 cells total), for $\alpha=6$, $\beta=0$, and for different $N$
($N$ = 22,500, 45,000, 55,000). For details of the simulations, the reader is
referred to the Methods section.
Figure 11: Equilibrium patterns for different average densities, $\rho_{0}$,
at the end of 10,000 iterations, for $\alpha=6,$ $\beta=0$. (A), (B), and (C)
represent, respectively, the sparsely distributed dots pattern (I) obtained
for 22,500 agents ($\rho_{0}=0.25$), the corresponding density distribution at
the end of 10,000 iterations, and the average utility evolution over the
iterations. Similar results are shown for the labyrinthine pattern (II) in
(D), (E), and (F) for 45,000 agents ($\rho_{0}=0.5$), and for the “gapped”
pattern (III) in (G), (H), and (I) for 55,000 agents ($\rho_{0}=0.61$).
In our simulations, we observe (Fig. 11) the three basic types of patterns, or
“macroscopic” states, namely, (I) sparsely distributed dots, (II) labyrinthine
or worm-like structures, and (III) “gapped” patterns that are seen
empirically, for example, in mussel beds [27] for different mussel densities.
As one might expect, the size of the interaction neighborhood (see the Methods
section) plays a role in determining the specific details, i.e., the
“microscopic” features, of these “macroscopic” states. That is, for example,
the detailed “microscopic” features of the labyrinthine or worm-like
structures might look different for different neighborhood sizes, but their
“macroscopic” state would remain labyrinthine. Thus, the basic “macroscopic”
states are found to be robust. The “macroscopic” state transitions appear like
phase transitions, moving from category I to category II to III as the density
increases.
We believe that these “macroscopic” and “microscopic” characteristics reflect
the structure of the phase-space landscape of $\phi$. Thus, as the agents move
around in the physical space, the system wanders around in the phase-space
landscape, settling into one state or the other. Since a “macroscopic” state
could be achieved via many different “microscopic” states, i.e.,
_multiplicity_ , one gets different “microscopic” outcomes in different
simulation runs, while the “macroscopic” outcome remains the same and robust.
The “macroscopic” states are the attractors seen in many nonlinear dynamical
systems.
The corresponding density histograms and the evolution of average utility as a
function of iteration are also shown in Fig. 11. For this configuration (i.e.,
$\alpha=6$, $\beta=0$), the spinodal densities (the black points in Fig. 5 and
Fig. 7) are 0.211 and 0.789, and the binodal densities (the red points in Fig.
5 and Fig. 7) are 0.071 and 0.929.
A word of caution as we discuss the results. As noted, our simple coarse-
grained model is equivalent in spirit to the van der Waals or the Ising model
in statistical mechanics. Therefore, we do not expect our analytical model and
associated simulations to capture all the nuances and complexities of the
real-world patterns in active matter systems.
As predicted by our theory, the density histograms show (Fig. 11, B, E, H)
spinodal decomposition for the three agent populations ($N$ = 22,500, 45,000,
55,000). That is, there are two phases, one with low-density groups and the
other with high-density groups. These two densities are around the binodal
densities predicted by the theory. Although the theory predicts two sharp
binodal density values ($\rho_{1}$ = 0.071 and $\rho_{2}$ = 0.929), it would
be hard to see such precise results in the simulation for one main reason.
Theoretical predictions are based on concepts from statistical mechanics,
which only work well for an extremely large number of agents (such as the
Avogadro number of molecules, $\sim 10^{23}$). This is when the statistical
estimates and outcomes are extremely precise, as in the case of, for example,
alloys in materials science. In our simulation, we have only 22,500 - 55,500
agents. So, the statistics is not that precise. Therefore, one should expect
to see a distribution of values instead of singular peaks. That is what we
observe in our simulations.
We also observe that the distributions around the low binodal density are
narrower, whereas they are broader at the high binodal density. The reason is
the following. As we see in Fig. 5, an individual agent reaches its maximum
utility at the upper spinodal point at the spinodal density of 0.789, while
the entire agent bed reaches its maximum potential $\phi$ (and hence the
arbitrage equilibrium) at the binodal density of 0.929 as seen from Fig. 7.
Thus, in high-density clusters, individual agents constantly compete to reach
the upper spinodal point (density = 0.789) of higher individual utility, while
the competition of the other agents to reach the same state drives the agent
bed away from the spinodal point to the binodal point (density = 0.929).
Therefore, agents mainly bounce around between these two points, the spinodal
density of 0.789 and the binodal density of 0.929, with a weighted average
density of about 0.85 (see Table 1) right in the middle.
We also observe in Fig. 11 (C, F, I) that the average utility improves as the
simulation proceeds, as the agents maneuver around to increase their effective
utilities, and then finally settles and fluctuates around the arbitrage
equilibrium value.
The key statistics are summarized in Table 1. The spinodal and binodal
densities are the same for the three different cases of $N$, because
$\alpha=6,\beta=0$ for all cases (see Fig. 5, green curve). We also find that
the average utility of Phase-1 is almost the same as that of the corresponding
Phase-2, as predicted by the theory. Thus, we see that a vast majority
(86-90%) of the agents are in their arbitrage equilibrium states, either in
Phase-1 or Phase-2.
Table 1: Summary of key metrics in phase separation
Category N Spinodal densities Binodal densities Phase-1 Phase-2 $\rho_{s1}$
$\rho_{s2}$ $\rho_{b1}$ $\rho_{b2}$ Share of agents($\%$) Avg. density Avg.
utility Share of agents($\%$) Avg. density Avg. utility Sparsely distributed
dots 22,500 0.211 0.789 0.071 0.929 33.28 0.052 3.217 52.63 0.855 3.330
Labyrinthine 45,000 0.211 0.789 0.071 0.929 8.71 0.051 3.234 79.85 0.850 3.338
Gapped 55,000 0.211 0.789 0.071 0.929 4.32 0.050 3.250 85.42 0.848 3.341
### IV.2 Stability of the Arbitrage Equilibrium
We can determine the stability of this equilibrium by performing a Lyapunov
stability analysis [11, 12]. A Lyapunov function $V$ is a continuously
differentiable function that takes positive values everywhere except at the
equilibrium point (i.e., $V$ is positive definite), and decreases (or is not
increasing) along every trajectory traversed by the dynamical system
($\dot{V}$ is negative definite or negative semi-definite). A dynamical system
is locally stable at equilibrium if $\dot{V}$ is negative semi-definite and is
asymptotically stable if $\dot{V}$ is negative definite.
Following Venkatasubramanian [12], we identify our Lyapunov function
$V(\boldsymbol{\rho})$
$\displaystyle
V(\boldsymbol{\rho})=\phi^{*}(\boldsymbol{\rho})-\phi(\boldsymbol{\rho})$ (23)
where $\phi^{*}$ is the potential at the arbitrage equilibrium (AE) (recall
that $\phi^{*}$ is at its maximum at AE) and $\phi(\boldsymbol{\rho})$ is the
potential at any other state. Note that $V(\boldsymbol{\rho})$ has the
desirable properties we seek: (i) $V(\boldsymbol{\rho^{*}})$ = 0 at AE and
$V(\boldsymbol{\rho})$ $>$ 0 elsewhere, i.e., $V(\boldsymbol{\rho})$ is
positive definite; (ii) Since $\phi(\boldsymbol{\rho})$ increases as it
approaches the maximum, $V(\boldsymbol{\rho})$ decreases with time, so it is
easy to see that $\dot{V}$ is negative definite. Therefore, the arbitrage
equilibrium is not only stable but also asymptotically stable.
Figure 12: Stability analysis. (A) Equilibrium configuration of agents at the
end of 10,000 iterations (B) disturbed configuration at 10,001${}^{\text{th}}$
iteration (C) equilibrium configuration at the end of 15,000 iteration (D)
evolution of average utility over iterations. The sharp decrease in the
average utility at 10,001${}^{\text{th}}$ is due to the disturbance introduced
at 10,001${}^{\text{th}}$ iteration.
Our simulation results confirm this theoretical prediction (see Fig. 12). We
show the stability results for the configuration of Fig. 11-A, as an example.
After the agents population reached equilibrium (10,000 iterations), we
disturbed the equilibrium state by randomly changing the positions of the
agents. As a result, the average utility of the population goes down as seen
from the sharp drop at the 10,001th iteration in Fig. 12-D. Fig. 12-B shows
the disturbed state of the agent groups. The simulation is then continued from
this new disturbed far-from-equilibrium state. As we see from Fig. 12-C, the
agent population recovers quickly to reach the original category-I
“macroscopic“ state even though some of the “microscopic” features are
different this time. The reader might have noticed that two small groups in
Fig. 12-A have merged to become one larger group in two different locations in
Fig. 12-C. This phenomenon is called Ostwald Ripening in materials science. We
also notice that the average utility is back to its old level.
This analysis shows that the arbitrage equilibrium region is not only stable,
but asymptotically stable. That is, the “macroscopic” structures are resilient
and self-healing. Given the speed of recovery, it could possibly be
exponentially stable, but we have not proved this analytically here. It is
interesting to observe that this result is similar to that of the dynamics of
the income game [11, 12] and the dynamics of flocking birds [14].
## V Connection with Statistical Thermodynamics
To appreciate how statistical teleodynamics is similar to and different from
statistical thermodynamics, let us consider a familiar example, the
thermodynamic system of gas molecules in a container, from the perspective of
statistical teleodynamics. We call this the Thermodynamic Game [11, 12]. Now,
real gas molecules are, of course, purpose-free, and hence do not pursue
maximum utility. However, in this game-theoretic formulation, we show that our
imaginary molecules-like agents, when they pursue a particular form of
“utility” as given in Eq. 24, behave like gas molecules. So, approaching this
from the perspective of a potential game, we introduce the following
“utility,” $h_{i}$, for our molecular agents in state $i$:
${h}_{i}(E_{i},N_{i})=-\beta E_{i}-\ln N_{i}$ (24)
where $E_{i}$ is the energy of an agent in state $i$, $N_{i}$ is the number of
agents in state $i$, $\beta=1/k_{B}T$, $k_{B}$ is the Boltzmann constant and
$T$ is temperature. The first term models the tendency of molecules to prefer
low-energy states (since our “molecular” agent tries to maximize its utility,
the negative sign leads to smaller values of $E_{i}$). The $-\ln N_{i}$ term
models the disutility of competition we have used in the sections above. This
term models the “restless” nature of molecules and their propensity to spread
out. This is so because the $-\ln N_{i}$ term incentivizes the agent to leave
its current location of higher $N_{i}$ to a location of lower $N_{i}$ all the
time.
By integrating this effective utility, we obtain the potential
$\phi(\mathbf{x})$ as
$\phi(\mathbf{x})=-\frac{\beta}{N}E+\frac{1}{N}\ln\frac{N!}{\prod_{i=1}^{n}(Nx_{i})!}$
(25)
where $E=N\sum_{i=1}^{n}x_{i}E_{i}$ is the total energy that is conserved, $N$
is the total number of molecules, $n$ is the total number of states, and
$x_{i}=N_{i}/N$.
This game reaches a unique Nash equilibrium when $\phi(\mathbf{x})$ is
maximized [17]. To determine the equilibrium distribution, we maximize the
Lagrangian given by Eq. 3
$L=\phi+\lambda(1-\sum_{i=1}^{n}x_{i})$
and obtain the well-known Boltzmann exponential distribution of energy at
equilibrium
$x_{i}^{*}=\frac{\exp(-\beta E_{i})}{\sum_{j=1}^{n}\exp(-\beta E_{j})}$ (26)
Thus, the arbitrage equilibrium of this game is the same as the statistical
thermodynamic equilibrium, as expected. The critical insight here is, as
Venkatasubramanian et al. first showed [11], that the second term in Eq. 25 is
the same as entropy (except for the Boltzmann constant). Thus, maximizing
potential in population game theory is equivalent to maximizing entropy in
statistical mechanics subject to the constraints given by the first term in
Eq. 25, i.e., the constraint on total energy $E$.
This is a deep and beautiful connection between statistical mechanics and
potential game theory. This fundamental connection allows us to generalize
statistical thermodynamics to statistical teleodynamics and lays the
foundation towards a universal theory of emergent equilibrium behavior of both
purpose-free and purpose-driven agents.
Pursuing this line of enquiry some more, we recognize another important
connection. From Eq. 25, we have
$\phi=-\frac{1}{Nk_{B}T}(E-TS)=-\frac{\beta}{N}A$ (27)
where $A=E-TS$ is the Helmholtz free energy. Indeed, in statistical
thermodynamics, $A$ is called a thermodynamic potential. Again, we see the
correspondence between the game-theoretic potential and the thermodynamic
potential. Furthermore, we see the correspondence between utility and the
chemical potential, with an important difference. Active agents try to
increase their utilities, whereas thermodynamic agents try to decrease their
chemical potential. In the Thermodynamic Game, an arbitrage equilibrium is
reached when all agents have the same effective utility, which is equivalent
to the chemical potential being equal in thermodynamics. In fact, our theory
reveals the critical insight that both living and non-living agents are driven
by arbitrage opportunities towards equilibrium, except that their arbitrage
currencies are different. For thermodynamic agents, the currency is the
chemical potential, whereas for living agents, the utility. Thus, we see that
statistical teleodynamics is a natural generalization of statistical
thermodynamics for goal-driven agents.
The perspective of statistical teleodynamics reveals an underappreciated
insight in statistical mechanics, which is the recognition that when $N_{i}$
(or $x_{i}$) follows the exponential distribution in Eq. 26, $h_{i}=h^{*}$ for
all molecules. In other words, it is an invariant. As noted, this is the
defining criterion for arbitrage equilibrium. Thus, the exponential
distribution is a curve of constant effective utility (or equivalently
constant chemical potential), for all values of $E_{i}$. That is, it is an
isoutility curve for $h_{i}$ defined by Eq. 24. This turns out to be a
particularly valuable insight in the context of the emergence of a fair income
distribution in the dynamics of the free market [12].
This recognition reveals another valuable insight that is also not readily
appreciated in statistical mechanics. We realize that we do not necessarily
have to maximize the potential $\phi(x)$ to derive the equilibrium
distribution (when the equilibrium is unique, i.e.,
$\partial^{2}\phi/\partial^{2}x<0$). We can adopt the agent perspective and
recognize that equilibrium is reached when all agents enjoy the same utility
$h_{i}=h^{*}$. Therefore, we have
${h}_{i}={h^{*}}=-\beta E_{i}-\ln N_{i}^{*},~{}~{}i\in\\{1,\dots,n\\}$ (28)
where $N_{i}^{*}$ is given by the equilibrium distribution. From this, it is
easy to derive the Boltzmann energy distribution (Eq. 26) by rearranging and
solving for $N_{i}^{*}$. In fact, this is the trick we used to derive Eq. 8
from Eq. 7.
Thus, we see that without necessarily invoking the system perspective of
maximizing the potential (which is equivalent to maximizing entropy with
constraints), we can derive the Boltzmann distribution easily. This important
property is not seen that clearly in statistical mechanics. In statistical
mechanics, we typically maximize entropy or minimize Gibbs free energy to
arrive at equilibrium results. That is, the emphasis is on the system
perspective; the individual agent’s view is not given importance. This is one
of the important philosophical differences between statistical thermodynamics
and statistical teleodynamics. While the former is generally a top-down,
system-oriented perspective, the latter is decidedly a bottom-up, agent-
oriented perspective.
Both Eq. 27 and Eq. 24 reveal another interesting feature of the statistical
teleodynamics framework with respect to thermodynamic laws [34, 35, 36]. Since
maximizing potential $\phi$ is the same as maximizing entropy $S$ subject to
the constraint on $E$ (Eq. 27), we see the two laws of thermodynamics embedded
in this equation. The second term is the root of the second law of maximizing
entropy and the first term is the source of the first law of energy
conservation. Similarly, in Eq. 24, the two laws are embedded in the same
manner. Eq. 27 is the system-based view of the two laws, while Eq. 24 is the
agent-based view.
Therefore, we realize that the non-entropic terms in utility and potential
serve the role of constraints on entropy maximization. This interpretation
provides a deeper understanding of the effective utility functions (and their
potential versions) of living agents, such as ants. Consider the effective
utility of an ant given by Eq. 4. The $-\ln N_{i}$ term, as before,
corresponds to the second law of entropy maximization, and the other two terms
play the role of the first law of enforcing constraints. However, unlike the
thermodynamics first law of energy conservation, the teleodynamics “first law”
does not enforce conservation, but rather constraints. Thus, we see that the
teleodynamics “first law” is the general case, and the thermodynamic first law
is a special case where the constraint on the total energy is also the
conservation of it. The “first law” of teleodynamics allows for more
complicated and “non-thermodynamic” constraints to be imposed on entropy
maximization. We also see that the “second law” of teleodynamics is similar to
the second law of thermodynamics, but with the broader concept of arbitrage
equilibrium. The ”zeroth law” of teleodynamics is that all agents continually
try to increase their effective utilities. This is essentially the Darwinian
survival-of-the-fittest principle in biology and the Smithian constant pursuit
of self-interest principle in economics.
## VI Connection with MIPS
It is instructive to compare the phase separation phenomena described by our
analysis of non-thermodynamic systems, such as mussel beds [15] or social
groups [16], with MIPS. MIPS is generally described as a non-equilibrium or
out-of-equilibrium behavior of active matter systems [5, 7, 6, 19, 3, 4].
Based on our analysis above, we argue that MIPS is indeed an equilibrium
phenomenon, but a different kind of equilibrium, an arbitrage equilibrium. As
we show above, under certain conditions, this arbitrage equilibrium is
equivalent to a statistical or thermodynamic equilibrium. In this section, we
explore this perspective at some length.
We start with the work reported by Takatori and Brady [6]. In particular, four
of their equations are relevant to our discussion here. They are for the
active pressure ($\mathrm{\Pi^{act}}$), Helmholtz free energy
($\mathrm{F^{act}}$), Gibbs free energy ($\mathrm{G^{act}}$), and the chemical
potential ($\mathrm{\mu^{act}}$) of the self-propelled particles,
respectively, as given below (we have changed their volume fraction symbol
$\phi$ to $\theta$ so that it is not confused with our $\phi$ which is the
game-theoretic potential).
$\displaystyle\Pi^{act}$ $\displaystyle=$ $\displaystyle
nk_{s}T_{s}[1-\theta-\theta^{2}+3\theta\mathrm{Pe_{R}}(1-\theta/\theta_{0})^{-1}]$
(29)
$\displaystyle\frac{F^{act}}{Nk_{s}T_{s}}=\ln\theta-\frac{\theta(\theta+2)}{2}-3\mathrm{Pe_{R}}\theta_{0}\ln(1-\theta/\theta_{0})$
$\displaystyle+F^{0}(k_{s}T_{s},\mathrm{Pe_{R}})$ (30)
$\displaystyle\frac{G^{act}}{Nk_{s}T_{s}}$ $\displaystyle=$
$\displaystyle\frac{F^{act}}{Nk_{s}T_{s}}+\frac{\Pi^{act}}{nk_{s}T_{s}}$ (31)
$\displaystyle\mu^{act}(k_{s}T_{s},\theta,\mathrm{Pe_{R}})=\mu^{0}(k_{s}T_{s},\mathrm{Pe_{R}})+k_{s}T_{s}\ln\theta$
$\displaystyle+k_{s}T_{s}\ln\Gamma(\theta,\mathrm{Pe_{R}})$ (32)
where $\theta$ is the volume fraction, $\theta_{0}$ is the volume fraction at
close packing, $N$ is the number of active swimmers, $k_{s}T_{s}=\zeta
U_{0}^{2}\tau_{R}/6$, $U_{0}$ is the intrinsic swim speed, $\tau_{R}$ is the
reorientation time, $\zeta$ is the hydrodynamic drag factor, $\mathrm{Pe_{R}}$
is the Peclet number, $\mathrm{\Pi^{act}}$ is the active pressure,
$\mathrm{F^{act}}$ is the non-equilibrium Helmholtz free energy,
$\mathrm{G^{act}}$ is the nonequilibrium Gibbs free energy, $\mathrm{F^{0}}$
is the reference state Helmholtz free energy, $\mu^{act}$ is the
nonequilibrium chemical potential, $\mu^{0}$ is the reference state of the
chemical potential, $\Gamma$ is a nonlinear expression (for more details on
these quantities, see [6]).
In Eq. 32, the second term on the right-hand side represents the entropic,
ideal-gas contribution to the chemical potential. The third term is the
nonideal term that is the analog of enthalpic attraction between active
swimmers, and is represented by $\Gamma(\theta,Pe_{R})$ that resembles the
fugacity coefficient in classical thermodynamics.
We observe that our Eq. 22 is equivalent to Eq. 31. The game-theoretic
potential is the negative of the Gibbs free energy, since our active agents
maximize utility rather than minimizing the chemical potential. Our Fig. 7
showing the spinodal and binodal regions is equivalent to Figure 3 in their
paper (except for sign reversal due to game potential $\phi=-G^{act}$). Note
that we are not saying that they are equal; they are equivalent in the sense
that both produce a function ($\phi$ or $G^{act}$) with two maxima (for
$\phi$) or equivalently two minima (for $G^{act}$) so that the common tangent
line can be drawn to determine the binodal points. Thus, they are two
equivalent models of phase separation in active matter systems.
Now, from Eq. 18 ($\delta=1$), we have
$\displaystyle
h_{i}(\boldsymbol{\rho})=\alpha\rho_{i}-\beta\rho_{i}^{2}+\gamma\ln(1-\rho_{i}/\rho_{\mathrm{max}})-\ln\rho_{i}$
(33)
The nonequilibrium chemical potential $\mu^{act}$ (Eq. 32) is equivalent to
our effective utility $h_{i}$ in Eq. 33. We see that our entropic term
$-\ln\rho_{i}$ corresponds to their $k_{s}T_{s}\ln\theta$ (the signs are
opposite because the utility is negative of the chemical potential), and the
rest are reflected in $\ln\Gamma(\theta,\mathrm{Pe_{R}})$, where
$\Gamma(\theta,\mathrm{Pe_{R}})$ is given by [6]
$\displaystyle\Gamma(\theta,\mathrm{Pe_{R}})=(1-\theta/\theta_{0})^{-3\theta_{0}\mathrm{Pe_{R}}}\exp[\theta^{3}-\theta^{2}/2$
$\displaystyle+3\mathrm{Pe_{R}}\theta_{0}(1-\theta_{0})/(1-\theta/\theta_{0})$
$\displaystyle-3\theta(1-\theta_{0}\mathrm{Pe_{R}})]$ (34)
and therefore
$\displaystyle\ln\Gamma(\theta,\mathrm{Pe_{R}})=-3\mathrm{Pe_{R}}\theta_{0}\ln(1-\theta/\theta_{0})+(\theta^{3}-\theta^{2}/2)$
$\displaystyle+3\mathrm{Pe_{R}}\theta_{0}(1-\theta_{0})/(1-\theta/\theta_{0})$
$\displaystyle-3\theta(1-\theta_{0}\mathrm{Pe_{R}})$
Their $\ln\Gamma(\theta,\mathrm{Pe_{R}})$ is much more complicated than our
equivalent terms, as it captures all the detailed phenomenology of the active
particle dynamics, whereas our model, again, is agnostic of such details. We
show in Fig. 13 the fit of Eq. 33 to the data from Eq. 32, for $\theta_{0}=1$
($\rho_{\mathrm{max}}=1$) and $Pe_{\mathrm{R}}=0.05$, as an example. We see
that despite ignoring the complicated details in Eq. 32, the simpler Eq. 33
fits the true $\mu^{act}$ quite well.
The purpose of this exercise is not to accurately predict Eq. 32 using Eq. 33.
Obviously, Eq. 32 will always do better than Eq. 33 as it incorporates more of
the “microscopic” details of the dynamics that Eq. 33 ignores. The objective
is to show that the simpler “mesoscopic” generic model in Eq. 33, which can be
used as a template for MIPS in different domains including non-physicochemical
phenomena such as social segregation [16], has sufficient predictive and
explanatory power to do just as well as the more detailed customized model in
Eq. 32.
Figure 13: Effective utility vs Density with $\theta_{0}=1$ and
$\mathrm{Pe_{R}}=0.05$
Therefore, we argue that what Takatori and Brady describe as “nonequilibrium
chemical potential” and “free energy of nonequilibrium active matter” in their
paper are actually equilibrium quantities, namely, arbitrage equilibrium
quantities. As our analysis above (Figures 7-10), and their own Figures 1 and
3 [6], show these active agents are in phase equilibrium, since their
effective utilities are equal.
This phase separation is not driven by thermodynamic quantities such as the
chemical potential and Gibbs free energy, but rather by other factors that are
relevant to the “microscopic” mechanisms in the domain of the agents. These
mechanisms could differ in different domains. They are different, for example,
for bacteria in biology [13], ants that ferry sand grains, mussels in the sea
[27], birds in a flock [14], and humans in social and economic settings [12,
16]. But they all belong to the same universality class, which is governed by
the same mathematical structures and conditions described in Eq. 1-4, and Eq.
19 of statistical teleodynamics. This mathematical structure determines the
properties of the arbitrage equilibrium. When the competition cost in the
effective utilities of agents has the form $-\ln N_{i}$ (or $-\ln\rho)$, this
arbitrage equilibrium corresponds to the thermodynamic or statistical
equilibrium, as we discussed in Section V.
We see a similar correspondence for active Brownian particles whose velocities
are density dependent, as discussed by Cates and Tailleur [7]. They write the
chemical potential $\mu$ of active particles without Brownian diffusion as
$\mu=\mu_{\mathrm{id}}+\mu_{\mathrm{ex}}=\ln\rho(\mathbf{r})+\ln
v(\rho(\mathbf{r}))$ (36)
where the propulsion speed $v(\rho)$ monotonically decreases with increasing
density. In our framework, this would be equivalent to the effective utility
$\displaystyle h(\rho)=-\ln v(\rho(\mathbf{r}))-\ln\rho(\mathbf{r})$ (37)
As Cates and Tailleur [7] show, this dynamics will lead to phase separation if
there exists a concave region in the free energy corresponding to Eq. 36. In
our case, this would correspond to the existence of a convex region in the
potential $\phi$ as seen in Eq. 22 and Fig. 7.
However, a more complete version of this model should include the utility
provided by the empty spaces in which the particles can potentially move. This
becomes particularly important for high particle density $\rho$ when the empty
space becomes valuable for motile particles. This is what we call an option
benefit in Eq. 19. Therefore, Eq. 37 now becomes
$\displaystyle h(\rho)=-\ln
v(\rho(\mathbf{r}))+\gamma\ln(1-\rho(\mathbf{r}))-\ln\rho(\mathbf{r})$ (38)
Furthermore, if the propulsion speed decays exponentially with respect to
density, as Cates and Tailleur suggest [7], then we have
$v(\rho)=\exp(-\alpha_{v}\rho)$, which leads to
$\displaystyle
h(\rho)=\alpha_{v}\rho+\gamma\ln(1-\rho(\mathbf{r}))-\ln\rho(\mathbf{r})$ (39)
For $\gamma=1$, $\alpha_{v}>4$ results in a non-monotonic cubic profile (i.e.,
the van der Waals loop), as seen in Fig. 5, and hence phase separation for the
appropriate densities.
### VI.1 Connection with Chemotaxis and MIPS
Zhao et al. [20] discuss MIPS in the context of chemotaxis, where there is a
directed motion of the active particle along the chemical gradient. While they
predict and explain the effect of chemotaxis on MIPS from a dynamic
perspective, here we do the same from the perspective of statistical
teleodynamics.
Venkatasubramanian et al. [13] proposed the effective utility of a particle in
a chemoattractant environment as
$\displaystyle h_{i}=\alpha c_{i}-\ln n_{i}$ (40)
where $c_{i}$ is the concentration of the chemoattractant at a given state
$i$, and $n_{i}$ is the number of particles at the state. The first term
corresponds to an affinity for states (in this case, regions) with higher
concentrations of the chemoattractant, and the second term corresponds to the
entropic component as before. They also derive the game-theoretic potential,
$\phi(\mathbf{x})$ as
$\displaystyle\phi(\mathbf{x})=\alpha\frac{C}{N}+\frac{1}{N}\ln\frac{N!}{\prod_{i=1}^{n}(Nx_{i})!}$
(41)
where $N$ is the number of active particles, $x_{i}=n_{i}/N$, and $C$ is the
total amount of chemoattractant. This formulation is equivalent to that of
O’Byrne and Tailleur [10].
In the continuum limit, $c_{i}$ can be replaced by $c(\mathbf{r})$, the
concentration distribution of the chemoattractant in the particle environment
at the location $r$. Similarly, $N_{i}$ is replaced by $\rho(\mathbf{r})$, the
density of the active particle at a given location. Now, the utility becomes
$\displaystyle h(c,\rho)=\alpha c(\mathbf{r})-\ln\rho(\mathbf{r})+\ln N$ (42)
and similarly, the game theoretic potential
$\displaystyle\phi=\frac{\int\alpha
c(\mathbf{r})\rho(\mathbf{r})d\mathbf{r}}{N}-\frac{1}{N}\int\rho(\mathbf{r})\ln\rho(\mathbf{r})d\mathbf{r}$
(43)
O’Byrne and Tailleur [10], who proposed a coarse-grained diffusive model for
active matter dynamics driven by a chemorepellant concentration field $c$,
describe the resultant coarse-grain dynamics using an effective free energy
functional $\mathcal{F}$ and deterministic flux $J_{D}$.
$\displaystyle
J_{D}=-D\rho\nabla\left\\{\left[v_{1}+v_{0}\frac{\alpha_{1}+(d-1)\Gamma_{1}}{\alpha_{0}+(d-1)\Gamma_{0}}\right]\frac{c}{dD}+\log\rho\right\\}$
(44)
where $J_{D}=\nabla\left(\delta\mathcal{F}/\delta\rho\right)$, the gradient of
the functional derivative of the free energy ($\rho$ is the particle density).
$\displaystyle\delta\mathcal{F}/\delta\rho=-D\left\\{\left[v_{1}+v_{0}\frac{\alpha_{1}+(d-1)\Gamma_{1}}{\alpha_{0}+(d-1)\Gamma_{0}}\right]\frac{c}{dD}+\log\rho\right\\}$
(45)
We identify that this functional derivative,
$\left(\delta\mathcal{F}/\delta\rho\right)$, is the chemical potential of the
active particle, which is the negative of the utility in Eq. 42. Once again,
as before, we see the equivalence of these two frameworks. Observe that our
Eq. 42 maps to Eq. 45 upto a proportionality constant with
$\alpha=-\left[v_{1}+v_{0}\dfrac{\alpha_{1}+(d-1)\Gamma_{1}}{\alpha_{0}+(d-1)\Gamma_{0}}\right]\dfrac{1}{dD}$.
The simulations performed in the study (with the parameter values:
$v_{0}=1,v_{1}=0.2,\alpha_{0}=50,\alpha_{1}=\Gamma_{1}=0$) correspond to
$\alpha<0$ in our formulation, the chemorepellant case.
If the initial amount of the chemoattractant is fixed, as the particles
consume the chemoattractant, its local concentration decreases with increasing
local density $\rho(r)$ of the particles. This decrease can be modeled as
$c(\mathbf{r})=-k^{\prime}\rho[\mathbf{r}]$, giving density-dependent utility
as,
$\displaystyle h(\rho)=-\alpha^{\prime}\rho-\ln\rho+\text{constant}$ (46)
where $\alpha^{\prime}=\alpha k^{\prime}$. Now, consider the density-dependent
velocity model of the active Brownian particles. Incorporating that dynamics
(Eq. 39) into Eq. 46, we have
$\displaystyle h(\rho)$ $\displaystyle=$
$\displaystyle(\alpha_{v}-\alpha^{\prime})\rho(\mathbf{r})+\gamma\ln(1-\rho(\mathbf{r}))-\ln\rho(\mathbf{r})$
(47)
Rewriting Eq. 47 in terms of the chemical potential ($\mu^{act}$), we have
$\displaystyle\mu^{act}(\rho)$ $\displaystyle=$
$\displaystyle-(\alpha_{v}-\alpha^{\prime})\rho(\mathbf{r})-\gamma\ln(1-\rho(\mathbf{r}))+\ln\rho(\mathbf{r})$
This system can show phase separation depending on the $\alpha_{v}$ and
$\alpha^{\prime}$ values. We show, for example, three cases ($\gamma=1)$: (i)
no chemotaxis: $\alpha_{v}=9$ and $\alpha^{\prime}=0$, (ii) weak chemotaxis:
$\alpha_{v}=9$ and $\alpha^{\prime}=2$, and (iii) strong chemotaxis:
$\alpha_{v}=9$ and $\alpha^{\prime}=7$ in Fig. 14. We see that the non-
monotonocity of $h$ changes depending on the values of $\alpha_{v}$ and
$\alpha^{\prime}$, thus determining whether a phase separation occurs or not.
Specifically, the nonmonotonic cubic profile (i.e., the van der Waals loop)
occurs when $\alpha_{v}-\alpha^{\prime}>4$. We see that a strong presence of
chemotaxis can prevent phase separation (blue curve).
Figure 14: Effect of chemotaxis on utility based on Eq. 47 with $\gamma=1$
## VII Conclusions
In developing a theory of emergent behavior of self-propelled agents that
dynamically change states, one faces three questions from the very beginning:
(i) Why do the agents change states? (ii) How do they change states? (iii)
What happens to the system, i.e., the population of such agents, eventually?
The answers to the first two questions depend on the “microscopic” fundamental
mechanisms of the particular situation. For example, gas molecules change
states driven by thermal agitation, ants by pheromone signals, flocking birds
by visual and auditory cues, humans by socioeconomic considerations, and so
on. Thus, the “microscopic” details of why and how agents move around depend
on the appropriate fundamental mechanisms of physics, chemistry, biology,
ecology, sociology, or economics, as the case may be. Statistical
teleodynamics is agnostic of such “microscopic” mechanisms. It has a
“mesoscopic” view of the agents, as it answers the question of what happens
eventually.
Towards this goal, we have developed simple models that offer an appropriate
coarse-grained description that is not restricted by system-specific details
and nuances, but without losing key conceptual insights and relevance to
empirical macroscopic phenomena. The spirit of our modeling is similar to that
of van der Waals in developing his equation of state.
In our theory, the central concept is the effective utility $h_{i}$ of an
agent. All agents constantly compete to increase their utilities in an
environment with resource constraints. The resources may be energy, space,
food, money, etc. This competition, under certain conditions, eventually leads
to an arbitrage equilibrium in which all agents have the same effective
utility.
Given the “mesoscopic” coarse-graining in our models, the effective utility
$h_{i}$ is agnostic of the “microscopic” details of its components. Consider,
for example, the emergent behaviors of ants and Janus particles. Their
“microscopic” mechanisms of motion, i.e., the “whys” and “hows,” are very
different for the two cases. However, these different “microscopic” mechanisms
result in the same “mesoscopic” model of their effective utilities (Eq. 4 and
Eq. 13), thereby predicting the same “macroscopic” emergent behavior, namely,
the Weibull distribution.
We find a similar situation with motility-induced phase separation phenomena.
As our mathematical analysis shows in Section V, the necessary and sufficient
conditions for phase separation are, respectively: (i) the effective utility
function at arbitrage equilibrium ($h^{*}$) must be at least a cubic or cubic-
equivalent of the order parameter (e.g., $\rho$ in Eq. 21) with three distinct
real and positive zeros, and (ii) the initial value of the order parameter is
within the spinodal region of the miscibility gap (e.g., Figs. 7-8).
Again, these “mesoscopic” requirements can be met by using many different
“microscopic” mechanisms. Janus particles in the work of Takatori et al. [6]
do it one way, but mussels in the sea use different mechanisms for their
pattern formation [37]. In sociology and economics, agents use entirely
different “microscopic” socioeconomic processes to dynamically switch states
that lead to social and economic segregation [12, 29, 16].
Comparing our model in Eq. 21 and Eq. 22 with that of Takatori et al. [6] in
Eq. 29-32, we see that the latter is more complicated, as it reflects the
“microscopic” details of this dynamics. In our simpler model, such details
have been avoided. However, their phase diagram (Fig. 1 in [6]) and the Gibbs
free energy vs. volume fraction plot (Fig. 3 in [6]) have the same qualitative
features as our model shown in Fig. 7-8. These qualitative features are the
existence of two minima in the Gibbs free energy (corresponding to two maxima
in our game-theoretic potential $\phi$) and the miscibility gap in the phase
diagram.
Although these motility-induced phase separation phenomena are generally
characterized as nonequilibrium or out-of-equilibrium phenomena [4, 7, 6], our
analysis recognizes them as the result of the arbitrage equilibrium. As we
discussed in Section V, this is, in principle, the same as thermodynamic
equilibrium, with the only difference being the arbitrage currency. In
thermodynamics, the currency is the chemical potential, and in teleodynamics,
it is the effective utility. Otherwise, the mathematical structure in both
situations is the same. This resolves the puzzle noted by many [9, 10, 5, 7,
8, 6] that some active matter systems that look like out-of-equilibrium
systems at the microscopic scale behave macroscopically like simple
equilibrium systems of passive matter.
Statistical teleodynamics is applicable to the entire range of the agency
spectrum going from purpose-free thermodynamic agents (e.g., molecules) to
purpose-driven rational agents (e.g., humans) with the different kinds of
biological and ecological agents (e.g., bacteria, ants, birds, mussels, etc.)
situated somewhere in between in this spectrum of self-actualizing
capabilities. This is summarized in Table 2, which lists utility function
templates in different domains. In this paper, we have already discussed
examples of the thermodynamic game, ant-crater formation, and social
segregation. In economics, the emergence of the income distribution can be
modeled by [11, 12]
$h_{i}=\alpha\ln S_{i}-\beta\left(\ln S_{i}\right)^{2}-\ln N_{i}.$ (49)
where the first term is the benefit of income ($S_{i}$), the second is the
cost of work expended to earn this income, and the last is the cost of
competition.
In ecology, the flocking behavior of bird-like agents is modeled by [14]
$h_{i}=\alpha N_{i}-\beta N_{i}^{2}+\gamma N_{i}l_{i}-\ln N_{i}$ (50)
where the first term is the security benefit of having many birds in the
neighborhood, the second is the congestion cost of these neighbors, the third
is the alignment ($l_{i}$) benefit of flying in the same direction as the
neighbors, and the last is again the cost of competition.
We wish to stress that all of these systems reach arbitrage equilibria. Some
of the equilibria have well-known distributions as outcomes, such as
exponential (energy), lognormal (income), or Weibull (ant craters, Janus
particles). But some others have “messy” distributions, as in the case of
social segregation and birds flocking. The specific outcome depends on the
non-entropic terms in the effective utility function, i.e., the “first law” of
teleodynamics terms that enforce the constraints on entropy maximization.
Table 2: Utility functions in different domains
Domain | System | Utility function ($h_{i}$)
---|---|---
Physics | Energy distribution | $-\beta E_{i}-\ln N_{i}$
Physics | Janus particles | $-\dfrac{\omega r_{i}^{a}}{a}-\ln N_{i}$
Biology | Bacterial chemotaxis | $\alpha c_{i}-\ln N_{i}$
Ecology | Ant craters | $b-\dfrac{\omega r_{i}^{a}}{a}-\ln N_{i}$
Ecology | Birds flocking | $\alpha N_{i}-\beta N_{i}^{2}+\gamma N_{i}l_{i}-\ln N_{i}$
Sociology | Social segregation | $\eta N_{i}-\xi N_{i}^{2}+\ln(H-N_{i})-\ln N_{i}$
Economics | Income distribution | $\alpha\ln S_{i}-\beta\left(\ln S_{i}\right)^{2}-\ln N_{i}$
By comparing the mathematical structure of the various effective utility
functions, we observe a certain universality across the different domains.
They are all based on benefit-cost trade-offs, but the actual nature of the
benefits and costs depend on the details of the specific domain, as one would
expect. The main requirement to belong to this universality class is that the
disutility due to competition can be modeled (or at least reasonably
approximated) as $-\ln N_{i}$ (discrete case) or $-\ln\rho$ (continuous case).
This is a critical requirement, as Kanbur and Venkatasubramanian explain [24].
This agent-based property directly leads to the system-wide property of
entropy, thereby connecting the agents and the system in a cohesive
mathematical framework. This term also facilitates the integration of
potential game theory with statistical mechanics, paving the way for a
universal theory of emergent equilibrium phenomena in active and passive
matter.
One might argue that our theory is not that different from what has already
been proposed to explain MIPS using chemical the potential and free-energy-
based approaches reported in the literature [7, 6]. We agree that the
conventional thermodynamic perspective is suitable for physicochemical
systems, although even here, one has to invoke the new concept of
“nonequilibrium chemical potential” [7, 25]. Thus, conventional concepts based
on thermodynamics are already proving inadequate for handling even
physicochemical systems. We believe that it would be even more problematic for
higher-order living agents, such as those listed in Table 2. For example, how
would one relate the salary $S_{i}$ of an economic agent to its
“nonequilibrium chemical potential”? A utility-based perspective is much more
intuitive and, hence, a natural framework. The other important contribution,
we believe, is the conceptual progress we have made in recognizing that many
of these so-called nonequilibrium systems are actually systems in equilibrium,
an arbitrage equilibrium.
Our analysis suggests that the pursuit of maximum utility or survival fitness,
combined with competitive dynamics under constraints, could be a universal
self-organizing mechanism for active matter. In biology, in general, the
search for improving one’s fitness occurs in the design space of genetic
features. Here, mutation and crossover operations facilitate movements in the
feature space such that an agent improves itself genetically via Darwinian
evolution to increase its utility, i.e., survival fitness. In economics, on
the other hand, agents search in the products and/or services space so that
they can offer better products/services to improve their economic survival
fitness in a competitive marketplace.
Thus, this mechanism is reminiscent of Adam Smith’s invisible hand. In all
these different domains, each agent pursues its own self-interest to increase
its own $h_{i}$, but a stable collective order emerges spontaneously as a
result of the competitive dynamics among all agents under constraints.
Therefore, we suggest that it is a pair of invisible hands: one is the pursuit
of maximum utility and the other the competitive dynamics under constraints.
We need both principles for the arbitrage equilibrium to emerge spontaneously.
By formulating equilibria in active matter more broadly, unrestricted by the
narrow confines of thermodynamics, statistical teleodynamics and arbitrage
equilibria open up conceptual possibilities that coherently accommodate active
matter in the entire range of nonliving to living agents more naturally.
However, what we have presented is a van der Waals-like version of statistical
teleodynamics. Much more needs to be done to further develop this framework to
address challenging emergent phenomena in physics, chemistry, biology,
ecology, sociology, and economics.
## Methods
The agent-based simulation was performed using Python. We distributed agents
on a 2-D $300\times 300$ grid with 90,000 cells. Three simulation studies are
reported in this paper – with 22,500 agents, 45,000 agents, and 55,000 agents.
For each case, initially, the agents were randomly distributed on the grid,
with each agent occupying one cell.
The dynamical evolution of the system is determined by two neighborhoods
around an agent $i$. One is the local neighborhood of _interaction_ , which is
an area with 49 cells that surround the agent $i$ (including the cell $i$ is
occupying). The other is the _exploration_ neighborhood (which is larger than
the interaction neighborhood and contains it) within which an agent $i$ can
explore and move to another cell to improve its utility $h_{i}$. The
exploration neighborhood has 1680 cells. The neighborhood sizes are parameters
that can be varied to balance the need to allow for complex patterns to emerge
at arbitrage equilibrium and the need to accomplish this in a reasonable
amount of computational time. We found that our combination (49 and 1680)
accomplishes this well.
The density of agents in any cell is defined as the ratio of the number of
agents in the interaction neighborhood to the total number of cells in the
neighborhood. At each iteration, every agent is given the opportunity to move
to a vacant cell in the exploration neighborhood where it would have higher
utility than its current cell. If the agent does not find a vacant cell, it
chooses to stay at its current location. After an agent moves, its utility,
and its neighbors’ density and utility are updated. The simulations were
carried out for 10,000 iterations, at which time the system typically reached
the arbitrage equilibrium.
## Acknowledgements
The first author thanks John Brady, Kyle Bishop, and Chris Durning for helpful
discussions. This work was supported in part by a grant to the Center for
Managing Systemic Risk from Columbia University.
## Author Contributions
VV: Conceptualization, Theory, Methodology, Analysis, Investigation,
Supervision, Funding Acquisition, and Writing; A. Sivaram: Methodology,
Analysis, Investigation, and Writing; NS: Software development and analysis;
A. Sankar: Software development and analysis.
The authors have no conflicts of interest to declare.
## Data Availablity and Reproducibility Statement
Data for Fig. 1-4 are from Takatori et al. [18]. Figures 5-6 are plotted using
Eq. 19. Figures 7-10 are plotted using Eq. 22. Figures 11-12 are from agent-
based simulations, the code of which will be made available upon request to
the corresponding author. We are currently updating the code for better ease
of use. Figure 13 is plotted using the equation. 32-33, and Figure 14 using
Eq. 48.
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|
# Unknown Domain Inconsistency Minimization
for Domain Generalization
Seungjae Shin 1, HeeSun Bae11footnotemark: 1 1, Byeonghu Na1, Yoon-Yeong Kim2
& Il-Chul Moon1,3
1Department of Industrial and Systems Engineering, KAIST
2Department of Statistics, University of Seoul, 3summary.ai
<EMAIL_ADDRESS><EMAIL_ADDRESS>Equal
contribution
###### Abstract
The objective of domain generalization (DG) is to enhance the transferability
of the model learned from a source domain to unobserved domains. To prevent
overfitting to a specific domain, Sharpness-Aware Minimization (SAM) reduces
source domain’s loss sharpness. Although SAM variants have delivered
significant improvements in DG, we highlight that there’s still potential for
improvement in generalizing to unknown domains through the exploration on data
space. This paper introduces an objective rooted in both parameter and data
perturbed regions for domain generalization, coined Unknown Domain
Inconsistency Minimization (UDIM). UDIM reduces the loss landscape
inconsistency between source domain and unknown domains. As unknown domains
are inaccessible, these domains are empirically crafted by perturbing
instances from the source domain dataset. In particular, by aligning the loss
landscape acquired in the source domain to the loss landscape of perturbed
domains, we expect to achieve generalization grounded on these flat minima for
the unknown domains. Theoretically, we validate that merging SAM optimization
with the UDIM objective establishes an upper bound for the true objective of
the DG task. In an empirical aspect, UDIM consistently outperforms SAM
variants across multiple DG benchmark datasets. Notably, UDIM shows
statistically significant improvements in scenarios with more restrictive
domain information, underscoring UDIM’s generalization capability in unseen
domains. Our code is available at https://github.com/SJShin-AI/UDIM.
## 1 Introduction
Domain Generalization (DG) (Zhou et al., 2022; Wang et al., 2022) focuses on
domain shift that arises when training and testing occur across distinct
domains, i.e. a domain of real pictures in training, and a separate domain of
cartoon images in testing. The objective of DG is to train a model on a given
source domain dataset, and generalizes well on other unobserved domains. To
address the domain discrepancy between the source domain and other domains,
various methods have been proposed: 1) alignment-based methods (Li et al.,
2021a; Wald et al., 2021); 2) augmentation-based methods (Qiao et al., 2020a;
Zhou et al., 2021); and 3) regularization-based methods (Arjovsky et al.,
2019; Krueger et al., 2021; Rame et al., 2022). While these methodologies have
demonstrated promising results, they often underperform in settings where
given domain information is particularly limited (Wang et al., 2021b; Qiao et
al., 2020b). Also, most methods lack theoretical guarantees on the
minimization of the target risk at the distribution level.
In contrast to the aforementioned methods, Sharpness-aware optimizers, which
flatten the loss landscape over a perturbed parameter region, demonstrate
promising performances in DG tasks (Zhang et al., 2023b; Wang et al., 2023).
By optimizing the perturbed local parameter regions, these approaches relieve
the model overfitting to a specific domain, thereby enhancing the adaptability
of the model across various domains. Also, this concept has a solid
theoretical foundation based on the parameter space analysis with PAC-Bayes
theories (McAllester, 1999; Dziugaite & Roy, 2017).
While the perturbation methods based on the parameter space have shown
promising improvements in DG tasks, this paper theoretically claims that
perturbation rooted in the data space is essential for robust generalization
to unobserved domains. Accordingly, this paper introduces an objective that
leverages both parameter and data perturbed regions for domain generalization.
In implementation, our objective minimizes the loss landscape discrepancy
between a source domain and unknown domains, where unknown domains are
emulated by perturbing instances from the source domain datasets. Recognizing
the loss landscape discrepancy as an Inconsistency score across different
domains, we name our objective as Unknown Domain Inconsistency Minimization
(UDIM).
Introduction of UDIM on the DG framework has two contributions. First, we
theoretically prove that the integration of sharpness-aware optimization and
UDIM objective becomes the upper bound of population risk for all feasible
domains, without introducing unoptimizable terms. Second, we reformulate the
UDIM objective into a practically implementable term. This is accomplished
from deriving the worst-case perturbations for both parameter space and data
space, each in a closed-form expression. Our experiments on various DG
benchmark datasets illustrate that UDIM consistently improves the
generalization ability of parameter-region based methods. Moreover, we found
that these improvements become more significant as domain information becomes
more limited.
## 2 Preliminary
### 2.1 Problem Definition of Domain Generalization
This paper investigates the task of domain generalization in the context of
multi-class classification (Arjovsky et al., 2019; Sagawa et al., 2019; Nam et
al., 2021). We define an input as $x\in\mathbb{R}^{d}$ and its associated
class label as $y\in\\{1,..,C\\}$. Let $\mathscr{D}_{e}$ represent a
distribution of $e$-th domain. $\mathcal{E}$ is a set of indices for all
domains, and $\mathscr{D}:=\\{\mathscr{D}_{e}\\}_{e\in\mathcal{E}}$ denotes
the set of distributions for all domains, where every domain shares the same
class set. For instance, let’s hypothesize that video streams from autonomous
cars are being collected, and the data collection at days and nights will
constitute two distinct domains. A sampled dataset from the $e$-th domain is
denoted by $D_{e}=\left\\{(x_{i},y_{i})\right\\}_{i=1}^{n_{e}}$ where
$(x_{i},y_{i})\sim\mathscr{D}_{e}$ and $n_{e}$ is the number of data instances
of $e$-th domain.
Throughout this paper, let $\theta\in\Theta$ represents a parameter of trained
model $f_{\theta}$, where $\Theta$ is a set of model parameters. Using
$D_{e}$, we define a loss function as
$\mathcal{L}_{D_{e}}(\theta)=\frac{1}{n_{e}}\sum_{(x_{i},y_{i})\in
D_{e}}\ell(f_{\theta}(x_{i}),y_{i})$, where we sometimes denote
$\ell(f_{\theta}(x_{i}),y_{i})$ as $\ell(x_{i},\theta)$. The population risk
for domain $e$ is given by
$\mathcal{L}_{\mathscr{D}_{e}}(\theta)=\mathbb{E}_{(x,y)\sim\mathscr{D}_{e}}[\ell(f_{\theta}(x),y)]$.
Then, the population risk over all domains is defined as
$\mathcal{L}_{\mathscr{D}}(\theta)=\sum_{e\in\mathcal{E}}p(e)\mathcal{L}_{\mathscr{D}_{e}}(\theta)$,
where $p(e)$ represents the occurrence probability of domain $e$.
In essence, the primary goal of training a model, $f_{\theta}$, is to minimize
the population risk, $\mathcal{L}_{\mathscr{D}}(\theta)$. In practical
scenarios, we only have access to datasets derived from a subset of all
domains. We call these accessible domains and datasets as source domains and
source domain datasets, respectively; denoted as
$\mathscr{D}_{S}=\\{\mathscr{D}_{s}\\}_{s\in\mathcal{S}}$ and
$D_{S}=\\{D_{s}\\}_{s\in\mathcal{S}}$ where $\mathcal{S}$ is the set of
indexes for source domains. As $\mathscr{D}_{S}\neq\mathscr{D}$, $D_{S}$
deviates from the distribution $\mathscr{D}$ under the sampling bias of
$\mathscr{D}_{S}$. As a consequence, a model parameter
${\theta}^{*}_{S}=\text{argmin}_{\theta}\mathcal{L}_{D_{S}}(\theta)$, which is
trained exclusively on $D_{S}$, might not be optimal for
$\mathcal{L}_{\mathscr{D}}(\theta)$. Accordingly, domain generalization
emerges as a pivotal task to optimize
$\theta^{*}=\text{argmin}_{\theta}\mathcal{L}_{\mathscr{D}}(\theta)$ by only
utilizing the source domain dataset, $D_{S}$.
### 2.2 Variants of Sharpness-Aware Minimization for Domain Generalization
Recently, a new research area has emerged by considering optimization over the
parameter space (Foret et al., 2020; Kwon et al., 2021). Several studies have
focused on the problem of $\theta$ overfitted to a training dataset (Wang et
al., 2023; Zhang et al., 2023b; a). These studies confirmed that optimization
on the perturbed parameter region improves the generalization performance of
the model. To construct a model that can adapt to unknown domains, it is
imperative that an optimized parameter point is not overfitted to the source
domain datasets. Accordingly, there were some studies to utilize the parameter
perturbation in order to avoid such overfitting, as elaborated below (Table
1).
Among variations in Table 1, Sharpness-Aware Minimization (SAM) (Foret et al.,
2020) is the most basic form of the parameter perturbation, which regularizes
the local region of $\theta$ to be the flat minima on the loss curvature as
$\min_{\theta}\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+\|\theta\|_{2}$.
Here, $\epsilon$ is the perturbation vector to $\theta$; and $\rho$ is the
maximum size of the perturbation vector. Subsequently, methodologies for
regularizing stronger sharpness were introduced (Zhang et al., 2023b; Wang et
al., 2023; Zhang et al., 2023a). These approaches exhibited incremental
improvements in domain generalization tasks. Check Appendix A for more
explanations.
Table 1: Objectives of SAM variants for DG
Method | Objective
---|---
SAM (Foret et al., 2020) | $\max\limits_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)$
GAM (Zhang et al., 2023b) | $\mathcal{L}_{D_{s}}(\theta)+\rho\max\limits_{\|\epsilon\|_{2}\leq\rho}\|\nabla\mathcal{L}_{D_{s}}(\theta+\epsilon)\|$
SAGM (Wang et al., 2023) | $\mathcal{L}_{D_{s}}(\theta)+\mathcal{L}_{D_{s}}(\theta+\rho\nabla\mathcal{L}_{D_{s}}(\theta)\big{/}\|\nabla\mathcal{L}_{D_{s}}(\theta)\|-\alpha\nabla\mathcal{L}_{D_{s}}(\theta))$
FAM (Zhang et al., 2023a) | $\mathcal{L}_{D_{s}}(\theta)+\max\limits_{\|\epsilon\|_{2}\leq\rho}\left(\mathcal{L}_{D_{s}}(\theta+\epsilon)-\mathcal{L}_{D_{s}}(\theta)\right)+\rho\max\limits_{\|\epsilon\|_{2}\leq\rho}\|\nabla\mathcal{L}_{D_{s}}(\theta+\epsilon)\|$
We are motivated by this characteristic to extend the parameter perturbation
towards data perturbation, which could be effective for the exploration of
unknown domains. None of SAM variants proposed data-based perturbation for
unknown domain discovery. Fundamentally, SAM variants predominantly focus on
identifying the flat minima for the given source domain dataset, $D_{S}$. In
Section 3.1, we highlight that finding flat minima in the source domain cannot
theoretically ensure generalization to unknown target domains. Consequently,
we demonstrate that generalization should be obtained from unobserved domains,
rather than solely the source domain. Since SAM imposes the perturbation
radius of $\rho$ on only $\theta$, we hypothesize $D_{s}$ could be perturbed
by additional mechanism to generalize $D_{s}$ toward $\mathscr{D}$.
While SAM minimizes the loss over $\rho$-ball region of $\theta$, its actual
implementation minimizes the maximally perturbed loss w.r.t.
$\theta+\epsilon^{*}$; where the maximal perturbation, $\epsilon^{*}$, is
approximated in a closed-form solution via Taylor expansion as follows:
$\displaystyle\operatorname*{max}_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)\approx\mathcal{L}_{D_{s}}(\theta+\epsilon^{*})\text{
where
}\epsilon^{*}\approx\rho\cdot\text{sign}(\nabla_{\theta}\mathcal{L}_{D_{s}}(\theta))\frac{|\nabla_{\theta}\mathcal{L}_{D_{s}}(\theta)|}{\|\nabla_{\theta}\mathcal{L}_{D_{s}}(\theta)\|_{2}^{2}}.$
(1)
The existence of this closed-form solution of $\epsilon^{*}$ simplifies the
learning procedure of SAM by avoiding min-max game and its subsequent
problems, such as oscillation of parameters (Chu et al., 2019). This closed-
form solution can also be applied to the perturbation on the data space.
## 3 Method
### 3.1 Motivation : Beyond the Source domain-based Flatness
Based on the context of domain generalization, Theorem 3.1 derives the
relationship between the SAM loss on the source domain dataset, denoted as
$\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)$, and the
generalization loss on an arbitrary unknown domain,
$\mathcal{L}_{\mathscr{D}_{e}}(\theta)$, for a model parameter
$\theta\in\Theta$ as follows:
###### Theorem 3.1.
(Rangwani et al., 2022) For $\theta\in\Theta$ and arbitrary domain
$\mathscr{D}_{e}\in\mathscr{D}$, with probability at least $1-\delta$ over
realized dataset $D_{s}$ from $\mathscr{D}_{s}$ with $|D_{s}|=n$, the
following holds under some technical conditions on
$\mathcal{L}_{\mathscr{D}_{e}}(\theta)$, where
$h_{0}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a strictly increasing
function.
$\displaystyle\mathcal{L}_{\mathscr{D}_{e}}(\theta)$
$\displaystyle\leq\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+\mathcal{D}^{\text{f}}({\mathscr{D}_{s}}||{\mathscr{D}_{e}})+h_{0}(\frac{\|\theta\|_{2}^{2}}{\rho^{2}})$
(2)
Theorem 3.1 provides an upper bound for
$\mathcal{L}_{\mathscr{D}_{e}}(\theta)$. The second term of this upper bound,
represented as $\mathcal{D}^{\text{f}}({\mathscr{D}_{s}}||{\mathscr{D}_{e}})$,
corresponds to the distribution discrepancy between $\mathscr{D}_{s}$ and
$\mathscr{D}_{e}$. Notably, $\mathcal{D}^{\text{f}}$ denotes the discrepancy
based on $f$-divergence. When $e\neq s$, $D_{e}$ is inaccessible in the
context of domain generalization. Accordingly, the SAM optimization on $D_{s}$
leaves an unoptimizable term in its upper bound, posing challenges for the
domain generalization.
In a setting that only $D_{s}$ and model parameter $\theta$ are accessible,
generating unseen domain data becomes infeasible. Nontheless, by perturbing
$D_{s}$ towards the direction that is most sensitive given $\theta$, we can
emulate the worst-case scenario for an unobserved domain (Sagawa et al.,
2019). While parameters trained via the SAM optimizer may exhibit flat regions
based on the source domain dataset, $D_{s}$, there is no theoretic study on
the flat minima under unobserved domains. By identifying the worst-case
scenario that maximizes the loss landscape difference between domains, our
methodology seeks the generalization across the unknown domains.
### 3.2 UDIM : Unknown Domain Inconsistency Minimization
This section proposes an objective based on both parameter and data perturbed
regions for domain generalization, coined Unknown Domain Inconsistency
Minimization (UDIM). UDIM minimizes the loss landscape discrepancy between the
source domain and unknown domains, where unknown domains are empirically
realized by perturbing instances from the source domain dataset, $D_{s}$.
Let
$\Theta_{\theta,\rho}=\\{\theta^{{}^{\prime}}|\|\theta^{{}^{\prime}}-\theta\|_{2}\leq\rho\\}$,
which is $\rho$-ball perturbed region of a specific parameter point $\theta$.
When training a parameter $\theta$ on an arbitrary domain dataset $D_{e}$ with
the SAM optimizer, some regions within $\Theta_{\theta,\rho}$ are expected to
be optimized as flat regions for source domain. Following the notation of
Parascandolo et al. (2020b), define
$N^{\gamma,\rho}_{e,\theta}:=\\{\theta^{{}^{\prime}}\in\Theta_{\theta,\rho}\text{
}|\text{
}\big{|}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{e}}(\theta)\big{|}\leq\gamma\\}$,
which is the region in $\Theta_{\theta,\rho}$ where the loss value of $\theta$
deviates by no more than $\gamma$. Given small enough values of
$\mathcal{L}_{D_{e}}(\theta)$ and $\gamma$, $N^{\gamma,\rho}_{e,\theta}$ could
be recognized as a flat minima for $e$-th domain.
We aim to utilize the flat minima of $\theta$ obtained through training on
$D_{s}$ using the SAM optimizer, where we represent $s$-th domain as our
source domain. The goal is to regularize the loss and its corresponding
landscape of unknown domains, so that the flat minima of the unknown domain
aligns with that of the source domain. By optimizing the domain of worst-case
deviation in the loss landscape from $D_{s}$, we facilitate regularization
across a range of intermediary domains. Eq. 3 formalizes our motivation, which
is cross-domain inconsistency score:
$\displaystyle\mathcal{I}^{\gamma}_{s}(\theta)=\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}|\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}})|$
(3)
In the above equation, the inner maximization seeks the worst-case parameter
point $\theta^{{}^{\prime}}$ that amplifies the domain-wise loss disparity,
while the outer maximization identifies $e$-th domain that maximizes
$\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}|\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}})|$.
Parascandolo et al. (2020a) utilizes an equation similar to Eq. 3, and this
equation is also employed to indicate $\theta$ with a loss surface that is
invariant to the environment changes. Our methodology differs from
Parascandolo et al. (2020a), which simply uses inconsistency as a metric, in
that we employ it directly as the objective we aim to optimize. Let’s assume
that $\theta$ exhibits sufficiently low loss along with flat loss landscape of
$D_{s}$. Further, if we can identify $\theta$ with a low value of
$\mathcal{I}^{\gamma}_{s}(\theta)$, then this $\theta$ would also demonstrate
consistently good generalization for unknown domains. This motivation leads to
the UDIM’s objective, which specifically targets the reduction of the cross-
domain inconsistency score across unobserved domains.
(a) Changes in the loss landscape across domains based on the perturbed
parameter space of $\theta$: initial state (left), post-SAM optimization
(center), and subsequent to UDIM application (right).
(b) Changes in domain-wise inconsistency sharpness based on data space of
$D_{s}$ before (left) and after (right) applying UDIM.
Figure 1: Illustration of our model, UDIM, based on parameter space (a) and
data space (b). (a) We define flatness within a perturbed region by minimizing
the inconsistency loss relative to the unknown domains, around the flat region
derived from the source domain. (b) Furthermore, by reducing the domain-wise
inconsistency within the input perturbed regions, where $\rho_{x}$ denotes
perturbation length, our method can also be interpreted as an data space
perspective of SAM.
##### Objective Formulation of UDIM
While we define the cross-domain inconsistency as
$\mathcal{I}^{\gamma}_{s}(\theta)$, the final formulation of UDIM is the
optimization of $\theta$ regarding both source and unknown domain losses from
the flat-minima perspective. Eq. 4 is the parameter optimization of our
proposal:
$\displaystyle\min_{\theta}\Big{(}\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+\lambda_{1}\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}|\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}})|+\lambda_{2}\|\theta\|_{2}\Big{)}$
(4)
The objective of UDIM consists of three components. The first term is a SAM
loss on $D_{s}$, guiding $\theta$ toward a flat minima in the context of
$D_{s}$. Concurrently, as $\theta$ provides flattened local landscape, the
second term weighted with $\lambda_{1}$ is a region-based loss disparity
between the worst-case domain $e$ and the source domain. Subsequently, the
algorithm seeks to diminish the region-based loss disparity between the worst-
case domain and the source domain. Figure 1 (a) illustrates the update of the
loss landscape in the parameter-space for each domain, based on the
optimization of Eq. 4. As SAM optimization on $D_{s}$ progresses,
$N^{\gamma,\rho}_{s,\theta}$ is expected to be broadened. However, SAM
optimization does not imply the minimization of
$\mathcal{I}^{\gamma}_{s}(\theta)$ (center). Given that the optimization of
$\mathcal{I}^{\gamma}_{s}(\theta)$ is conducted on
$N^{\gamma,\rho}_{s,\theta}$, we expect the formation of flat minima spanning
all domains in the optimal state (rightmost).
Figure 2: Inconsistency score of each method on PACS training dataset (X-axis:
training iteration). Y-axis is depicted in a log-scale.
The optimization of Eq. 4 can also be interpreted as minimizing another form
of sharpness in the data space. Let’s suppose all other domains lie within a
finite perturbation of the source domain’s dataset. In the context of the data
space over the $D_{s}$, the optimization can be seen as identifying the worst-
case perturbed dataset, $D_{e}$, by evaluating $\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}|\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}})|$.
If we view the perturbation of the $D_{s}$, not as discrete choices but as a
continuum within the data-space; our optimization can be viewed as minimizing
the sharpness of domain-wise inconsistency in the data space. Figure 1 (b)
illustrates this interpretation. After such optimization, the resulting
parameter $\theta$ can offer the consistent generalization over domains
located within the perturbed data space.
At this juncture, a crucial consideration is whether the SAM optimization on
$D_{s}$ focuses on minimizing the second term,
$\mathcal{I}^{\gamma}_{s}(\theta)$, or not. We illustrate the limited
capability of SAM variants in minimizing the second term, by an analytical
illustration of Figure 1 (a); and by an empirical demonstration of Figure 2.
Therefore, UDIM covers the optimization area that SAM does not operate.
##### Theoretical Analysis of UDIM
Given the definition of
$\Theta_{\theta,\rho}=\\{\theta^{{}^{\prime}}|\|\theta^{{}^{\prime}}-\theta\|_{2}\leq\rho\\}$,
we introduce
$\Theta_{\theta,\rho{{}^{\prime}}}=\operatorname*{arg\,max}_{\Theta_{\theta,\hat{\rho}}\subseteq
N^{\gamma,\rho}_{s,\theta}}\hat{\rho}$, which is the largest
$\rho{{}^{\prime}}$-ball region around $\theta$ in
$N^{\gamma,\rho}_{s,\theta}$.111For the sake of simplicity, we do not inject
the notation of $s$ or $\gamma$ in $\Theta_{\theta,\rho{{}^{\prime}}}$.
Theorem 3.2 introduces the generalization bound of Eq. 4, which is the
objective of UDIM. Theorem 3.2 states that Eq. 4 can become the upper bound of
$\mathcal{L}_{\mathscr{D}}(\theta)$, which is the population risk over all
domains.
###### Theorem 3.2.
For $\theta\in\Theta$ and arbitrary domain $e\in\mathcal{E}$, with probability
at least $1-\delta$ over realized dataset $D_{e}$ from $\mathscr{D}_{e}$, the
following holds under technical conditions on
$\mathcal{L}_{\mathscr{D}_{e}}(\theta)$ and $\mathcal{L}_{D_{e}}(\theta)$,
where $h:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a strictly increasing
function. (Proof in Appendix B.1.)
$\displaystyle\mathcal{L}_{\mathscr{D}}(\theta)$
$\displaystyle\leq\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+(1-\frac{1}{|\mathcal{E}|})\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}|\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}})|+h(\frac{\|\theta\|_{2}^{2}}{\rho^{2}})$
(5)
In such a case, Theorem 3.2 retains the same form of weight decay term as
presented in Theorem 3.1. Unlike Theorem 3.1, which contains terms that are
inherently unoptimizable, our objective is capable of minimizing every term
encompassed in the upper bound of Theorem 3.2 because we do not have an
inaccessible term,
$\mathcal{D}^{\text{f}}({\mathscr{D}_{s}}||{\mathscr{D}_{e}})$.
### 3.3 Implementation of UDIM
This section reformulates the second term of Eq. 4 into an implementable form.
We first formalize the perturbation of $D_{s}$ to emulate the worst-case
domain for $\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}|\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}})|$.
##### Inconsistency-Aware Domain Perturbation on $D_{s}$
For clarification, we explain the perturbation process based on an arbitrary
input instance $x$ from $D_{s}$. Given that the magnitude of the perturbation
vector for input $x$ is constrained to $\rho_{x}$, the perturbed input
$\tilde{x}$ can be expressed as follows:
$\displaystyle\tilde{x}=x+\underset{\epsilon_{x}:\|\epsilon_{x}\|_{2}\leq\rho_{x}}{\text{argmax}}\max\limits_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\Big{(}\ell(x+\epsilon_{x},\theta^{{}^{\prime}})-\ell(x,\theta^{{}^{\prime}})\Big{)}\approx
x+\underset{\epsilon_{x}:\|\epsilon_{x}\|_{2}\leq\rho_{x}}{\text{argmax}}\max\limits_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\ell(x+\epsilon_{x},\theta^{{}^{\prime}})$ (6)
$\displaystyle\underset{\text{1st
Taylor}}{\approx}x+\underset{\epsilon_{x}:\|\epsilon_{x}\|_{2}\leq\rho_{x}}{\text{argmax}}\Big{(}\ell(x+\epsilon_{x},\theta)+\rho^{\prime}\|\nabla_{\theta}\ell(x+\epsilon_{x},\theta)\|_{2}\Big{)}$
(7)
Assuming $N^{\gamma,\rho}_{s,\theta}$ as flat minima of $\theta$,
$\ell(x,\theta^{{}^{\prime}})$ is almost invariant for
$\theta^{{}^{\prime}}\in N^{\gamma,\rho}_{s,\theta}$. We assume that the
invariant value of $\ell(x,\theta^{{}^{\prime}})$ does not significantly
influence the direction of $x$’s perturbation because it becomes almost
constant in $N^{\gamma,\rho}_{s,\theta}$. Therefore, we cancel out this term
in Eq. 6.
Additionally, as we cannot specify the regional shape of
$N^{\gamma,\rho}_{s,\theta}$, it is infeasible to search maximal point
$\theta^{{}^{\prime}}\in N^{\gamma,\rho}_{s,\theta}$. As a consequence, we
utilize $\Theta_{\theta,\rho{{}^{\prime}}}$, which is the largest
$\rho{{}^{\prime}}$-ball region within $N^{\gamma,\rho}_{s,\theta}$, to
approximately search the maximal point $\theta^{{}^{\prime}}$. It should be
noted that $\Theta_{\theta,\rho{{}^{\prime}}}\subseteq
N^{\gamma,\rho}_{s,\theta}\subseteq\Theta_{\theta,\rho}$, where we assume that
$\rho{{}^{\prime}}$ gradually approaches $\rho$ during the SAM optimization.
Through the first-order Taylor expansion222We empirically found out that
extending it into second-order does not affect the resulting performance. for
the maximum point within $\Theta_{\rho{{}^{\prime}}}$, we can design the
aforementioned perturbation loss as Eq. 7. Consequently, the perturbation is
carried out in a direction that maximizes the loss and
$\rho{{}^{\prime}}$-weighted gradient norm of the original input $x$.
##### Inconsistency Minimization on $\theta$
After the perturbation on $x\in D_{s}$, we get an inconsistency-aware
perturbed dataset, $\tilde{D}_{s}$; which approximates the worst-case of
unobserved domains. Accordingly, we can formulate the optimization of
$\mathcal{I}^{\gamma}_{s}(\theta)$ based on $\theta$ as
$\min_{\theta}\max\limits_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\Big{(}\mathcal{L}_{\tilde{D}_{s}}(\theta^{{}^{\prime}})-\mathcal{L}_{{D}_{s}}(\theta^{{}^{\prime}})\Big{)}$.
For the above min-max optimization, we approximate the search for the maximum
parameter $\theta^{{}^{\prime}}$ in a closed-form using second order taylor-
expansion, similar to the approach of SAM in Eq. 1.
$\displaystyle\max\limits_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\Big{(}\mathcal{L}_{\tilde{D}_{s}}(\theta^{{}^{\prime}})-\mathcal{L}_{{D}_{s}}(\theta^{{}^{\prime}})\Big{)}{\approx}\mathcal{L}_{\tilde{D}_{s}}(\theta)-\mathcal{L}_{{D}_{s}}(\theta)+\rho{{}^{\prime}}\|\nabla_{\theta}\mathcal{L}_{\tilde{D}_{s}}(\theta)\|_{2}+\max\limits_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\frac{1}{2}\theta^{{}^{\prime}\top}\mathbf{H}_{\tilde{D}_{s}}\theta^{{}^{\prime}}$
(8)
$\displaystyle=\Big{(}\mathcal{L}_{\tilde{D}_{s}}(\theta)-\mathcal{L}_{{D}_{s}}(\theta)\Big{)}+\rho{{}^{\prime}}\|\nabla_{\theta}\mathcal{L}_{\tilde{D}_{s}}(\theta)\|_{2}+\gamma\max\limits_{i}\lambda^{\tilde{D}_{s}}_{i}/\lambda^{{D}_{s}}_{i}$
(9)
Full derivation and approximation procedure of Eq. 8 and Eq. 9 are in Appendix
B.2. $\mathbf{H}_{\tilde{D}_{s}}$ in Eq. 8 denotes a Hessian matrix of the
perturbed dataset, $\tilde{D}_{s}$. Also, $\lambda^{\tilde{D}_{s}}_{i}$ in Eq.
9 denotes $i$-th eigenvalue of $\mathbf{H}_{\tilde{D}_{s}}$. Finally, Eq. 9
becomes the objective with three components: 1) the loss difference between
$\tilde{D}_{s}$ and $D_{s}$, 2) the gradient norm of $\tilde{D}_{s}$ and 3)
the maximum eigenvalue ratio between $\tilde{D}_{s}$ and $D_{s}$. Note that
$\lambda^{\tilde{D}_{s}}_{i}/\lambda^{{D}_{s}}_{i}$ is minimized when
$\mathbf{H}_{\tilde{D}_{s}}$ becomes equivalent to $\mathbf{H}_{D_{s}}$.
While Eq. 9 is differentiable with respect to $\theta$ and can thus be
utilized as a tractable objective, computing the Hessian matrix for an over-
parameterized $\theta$ is computationally demanding. In line with Rame et al.
(2022), we replace the objective with the Hessian matrix (Eq. 9) with an
optimization based on gradient variance. Accordingly, Eq. 10 represents the
gradient variance-based objective as an optimization with respect to $\theta$
as follows: (See detailed derivations in Appendix B.3)
$\displaystyle\min_{\theta}\rho{{}^{\prime}}\|\nabla_{\theta}\mathcal{L}_{\tilde{D}_{s}}(\theta)\|_{2}+\|\text{Var}(\mathbf{G}_{\tilde{D}_{s}})-\text{Var}(\mathbf{G}_{{D}_{s}})\|_{2}$
(10)
Here, ${\mathbf{g}}_{i}$ is a per-sample gradient for $i$-th sample; and
$\mathbf{G}_{{D}}=\\{{\mathbf{g}}_{i}\\}^{|D|}_{i=1}$ is a set of per-sample
gradient for $x_{i}\in D$. Accordingly, the variance of $\mathbf{G}_{{D}}$,
which we denote as $\text{Var}(\mathbf{G}_{{D}})$, is calculated as
$\text{Var}(\mathbf{G}_{{D}})=\frac{1}{|D|-1}\sum^{|D|}_{i=1}\big{(}\mathbf{g}_{i}-\bar{\mathbf{g}}\big{)}^{2}$.
Matching the gradient variances between two distinct datasets encapsulates a
specific form of loss matching between them (Rame et al., 2022). This allows
us to unify loss matching and Hessian matching under a single optimization
using $\text{Var}(\mathbf{G}_{{D}})$.
Summary Our methodology applies the perturbation technique to both input $x$
and parameter $\theta$. Particularly, the perturbation on inputs and
parameters is necessary to minimize $\mathcal{I}^{\gamma}_{s}(\theta)$ under
unknown domains, which is the unique contribution of this work. Ablation study
in Section 4.3 supports that UDIM, which is the combination of the Eq. 7 and
Eq. 10, yields the best performance compared to various perturbations and
optimizations. Algorithm of UDIM is in Appendix C.
A single iteration of UDIM optimization with SAM loss can be described as the
following procedure:
1\. Construction of $\tilde{D_{s}}$ via Inconsistency-Aware Domain
Perturbation on $D_{s}$
$\displaystyle\tilde{D_{s}}=\\{(\tilde{x}_{i},y_{i})\,|\,(x_{i},y_{i})\in
D_{s}\\}{\text{ where
}}\tilde{x}_{i}=x_{i}+\rho_{x}\frac{\nabla_{x_{i}}\big{(}\ell(x,\theta_{t})+\rho^{\prime}\|\nabla_{\theta_{t}}\ell(x,\theta_{t})\|_{2}\big{)}}{\big{\|}\nabla_{x_{i}}\big{(}\ell(x,\theta_{t})+\rho^{\prime}\|\nabla_{\theta_{t}}\ell(x,\theta_{t})\|_{2}\big{)}\big{\|}_{2}}$
(11)
2\. SAM loss and Inconsistency Minimization on the current parameter
$\theta_{t}$
$\displaystyle\theta_{t+1}=\theta_{t}-\eta\nabla_{\theta_{t}}\Big{(}\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta_{t}+\epsilon)+\rho{{}^{\prime}}\|\nabla_{\theta_{t}}\mathcal{L}_{\tilde{D}_{s}}(\theta_{t})\|_{2}+\|\text{Var}(\mathbf{G}_{\tilde{D}_{s}})-\text{Var}(\mathbf{G}_{{D}_{s}})\|_{2}+\lambda_{2}\|\theta_{t}\|_{2}\Big{)}$
(12)
Table 2: Test accuracy for CIFAR-10-C. Each level states the severity of
corruption. Bold is the best case of each column or improved performances with
the respective sharpness-based optimizers.
Method | level1 | level2 | level3 | level4 | level5 | Avg
---|---|---|---|---|---|---
LOO DG Based | ERM | 75.9$\pm$ 0.5 | 72.9$\pm$ 0.4 | 70.0$\pm$ 0.4 | 65.9$\pm$ 0.4 | 59.9$\pm$ 0.5 | 68.9
IRM (Arjovsky et al., 2019) | 37.6$\pm$ 2.7 | 36.0$\pm$ 2.8 | 34.6$\pm$ 2.6 | 32.8$\pm$ 2.1 | 30.8$\pm$ 1.9 | 34.3
GroupDRO (Sagawa et al., 2019) | 76.0$\pm$ 0.1 | 72.9$\pm$ 0.1 | 69.8$\pm$ 0.2 | 65.5$\pm$ 0.3 | 59.5$\pm$ 0.5 | 68.7
OrgMixup (Zhang et al., 2018) | 77.1$\pm$ 0.0 | 74.2$\pm$ 0.1 | 71.4$\pm$ 0.1 | 67.4$\pm$ 0.2 | 61.2$\pm$ 0.1 | 70.3
Mixup (Yan et al., 2020) | 76.3$\pm$ 0.3 | 73.2$\pm$ 0.2 | 70.2$\pm$ 0.2 | 66.1$\pm$ 0.1 | 60.1$\pm$ 0.1 | 69.2
CutMix (Yun et al., 2019) | 77.9$\pm$ 0.0 | 74.2$\pm$ 0.1 | 70.8$\pm$ 0.2 | 66.3$\pm$ 0.3 | 60.0$\pm$ 0.4 | 69.8
MTL (Blanchard et al., 2021) | 75.6$\pm$ 0.5 | 72.7$\pm$ 0.4 | 69.9$\pm$ 0.2 | 65.9$\pm$ 0.0 | 60.2$\pm$ 0.3 | 68.9
MMD (Li et al., 2018b) | 76.4$\pm$ 0.1 | 73.2$\pm$ 0.2 | 70.0$\pm$ 0.3 | 65.7$\pm$ 0.3 | 59.6$\pm$ 0.5 | 69.0
CORAL Sun & Saenko (2016) | 76.0$\pm$ 0.4 | 72.9$\pm$ 0.2 | 69.9$\pm$ 0.0 | 65.8$\pm$ 0.1 | 59.6$\pm$ 0.1 | 68.8
SagNet (Nam et al., 2021) | 76.6$\pm$ 0.2 | 73.6$\pm$ 0.3 | 70.5$\pm$ 0.4 | 66.4$\pm$ 0.4 | 60.1$\pm$ 0.4 | 69.5
ARM (Zhang et al., 2021) | 75.7$\pm$ 0.1 | 72.9$\pm$ 0.1 | 69.9$\pm$ 0.2 | 65.9$\pm$ 0.2 | 59.8$\pm$ 0.3 | 68.8
DANN (Ganin et al., 2016) | 75.4$\pm$0.4 | 72.6$\pm$0.3 | 69.7$\pm$0.2 | 65.6$\pm$0.0 | 59.6$\pm$0.2 | 68.6
CDANN (Li et al., 2018c) | 75.3$\pm$0.2 | 72.3$\pm$0.2 | 69.4$\pm$0.2 | 65.3$\pm$0.1 | 59.4$\pm$0.2 | 68.3
VREx Krueger et al. (2021) | 76.0$\pm$ 0.2 | 73.0$\pm$ 0.2 | 70.0$\pm$ 0.2 | 66.0$\pm$ 0.1 | 60.0$\pm$ 0.2 | 69.0
RSC (Huang et al., 2020) | 76.1$\pm$ 0.4 | 73.2$\pm$ 0.5 | 70.1$\pm$ 0.5 | 66.2$\pm$ 0.5 | 60.1$\pm$ 0.5 | 69.1
Fishr (Rame et al., 2022) | 76.3$\pm$ 0.3 | 73.4$\pm$ 0.3 | 70.4$\pm$ 0.5 | 66.3$\pm$ 0.8 | 60.1$\pm$ 1.1 | 69.3
SDG Based | M-ADA (Qiao et al., 2020a) | 77.2$\pm$0.2 | 74.2$\pm$0.1 | 71.2$\pm$0.0 | 67.1$\pm$0.1 | 61.1$\pm$0.1 | 70.2
LTD (Wang et al., 2021a) | 75.3$\pm$0.2 | 73.0$\pm$0.0 | 70.6$\pm$0.0 | 67.2$\pm$0.2 | 61.7$\pm$0.2 | 69.6
SAM Based | SAM (Foret et al., 2020) | 79.0$\pm$0.3 | 76.0$\pm$0.3 | 72.9$\pm$0.3 | 68.7$\pm$0.2 | 62.5$\pm$0.3 | 71.8
UDIM w/ SAM | 80.3$\pm$0.0 | 77.7$\pm$0.1 | 75.1$\pm$0.0 | 71.5$\pm$0.1 | 66.2$\pm$0.1 | 74.2
SAGM (Wang et al., 2023) | 79.0$\pm$0.1 | 76.2$\pm$0.0 | 73.2$\pm$0.2 | 69.0$\pm$0.3 | 62.7$\pm$0.4 | 72.0
UDIM w/ SAGM | 80.1$\pm$0.1 | 77.5$\pm$0.1 | 74.8$\pm$0.1 | 71.2$\pm$0.2 | 65.9$\pm$0.2 | 73.9
GAM (Zhang et al., 2023b) | 79.5$\pm$0.1 | 76.8$\pm$0.1 | 74.0$\pm$0.2 | 69.9$\pm$0.1 | 64.1$\pm$0.2 | 72.8
UDIM w/ GAM | 81.4$\pm$0.1 | 78.9$\pm$0.0 | 76.3$\pm$0.0 | 72.8$\pm$0.1 | 67.4$\pm$0.1 | 75.3
## 4 Experiment
### 4.1 Implementation
Datasets and Implementation Details We validate the efficacy of our method,
UDIM, via experiments across multiple datasets for domain generalization.
First, we conducted evaluation on CIFAR-10-C (Hendrycks & Dietterich, 2019), a
synthetic dataset that emulates various domains by applying several synthetic
corruptions to CIFAR-10 (Krizhevsky et al., 2009). Furthermore, we extend our
evaluation to real-world datasets with multiple domains, namely PACS (Li et
al., 2017), OfficeHome (Venkateswara et al., 2017), and DomainNet (Peng et
al., 2019). Since UDIM can operate regardless of the number of source domains,
we evaluate UDIM under both scenarios on real-world datasets: 1) when multiple
domains are presented in the source (Leave-One-Out Domain Generalization,
LOODG); and 2) when a single domain serves as the source (Single Source Domain
Generalization, SDG). Unless specified, we report the mean and standard
deviation of accuracies from three replications. Appendix D.1 provides
information on the dataset and our implementations.
Implementation of UDIM To leverage a parameter $\theta$ exhibitng flat minima
on $D_{s}$ during the minimization of $\mathcal{I}^{\gamma}_{s}(\theta)$, we
perform a warm-up training on $\theta$ with the SAM loss on $D_{s}$. While
keeping the total number of iterations consistent with other methods, we
allocate initial iterations for warm-up based on the SAM loss. As a result, we
expect to optimize Eq. 4 with a sufficiently wide region of
$N^{\gamma,\rho}_{s,\theta}$ for sufficiently low $\gamma$.
The gradient variance, $\text{Var}(\mathbf{G}_{{D}_{s}})$ in Eq. 10,
necessitates the costly computation of per-sample gradients with respect to
$\theta$. We utilize BackPACK (Dangel et al., 2020), which provides the faster
computation of per-sample gradients. Also, we compute the gradient variance
only for the classifier parameters, which is an efficient practice to improve
the performance with low computational costs (Shin et al., 2023). Appendix D.1
specifies additional hyperparameter settings of UDIM.
Baselines for Comparison Since our approach, UDIM, is tested under both LOODG
and SDG scenarios, we employed methods tailored for each scenario as
baselines. These include strategies for robust optimization (Arjovsky et al.,
2019; Sagawa et al., 2019) and augmentations for novel domain discovery (Zhang
et al., 2018; Yun et al., 2019; Nam et al., 2021). Appendix D.2 enumerates
baselines for comparisons. For methods that leverage relationships between
source domains, a straightforward application is not feasible in SDG
scenarios. In such cases, we treated each batch as if it came from a different
domain to measure experimental performance. ERM means the base model trained
with standard classification loss. We also utilize the sharpness-based
approaches as the baselines. Note that the first term of Eq. 4 (SAM loss on
$D_{s}$) can be substituted with objectives for similar flatness outcomes,
such as SAGM (Wang et al., 2023) and GAM (Zhang et al., 2023b).
Table 3: Test accuracy for PACS. Each column represents test domain for LOODG,
and train domain for SDG. ∗ denotes that the performances are from its
original paper. Bold indicates the best case of each column or improved
performances when combined with the sharpness-based optimizers.
| Leave-One-Out Source Domain Generalization | Single Source Domain Generalization
---|---|---
Method | Art | Cartoon | Photo | Sketch | Avg | Art | Cartoon | Photo | Sketch | Avg
Fishr∗ (Best among LOODG methods) | 88.4∗$\pm$0.2 | 78.7∗$\pm$0.7 | 97.0∗$\pm$0.1 | 77.8∗$\pm$2.0 | 85.5∗ | 75.9$\pm$1.7 | 81.1$\pm$0.7 | 46.9$\pm$0.7 | 57.2$\pm$4.4 | 65.3
RIDG (Chen et al., 2023b) | 86.3$\pm$1.1 | 81.0$\pm$1.0 | 97.4$\pm$0.7 | 77.5$\pm$2.5 | 85.5 | 76.2$\pm$1.4 | 80.0$\pm$1.8 | 48.5$\pm$2.8 | 54.8$\pm$2.4 | 64.9
ITTA (Chen et al., 2023a) | 87.9$\pm$1.4 | 78.6$\pm$2.7 | 96.2$\pm$0.2 | 80.7$\pm$2.2 | 85.8 | 78.4$\pm$1.5 | 79.8$\pm$1.3 | 56.5$\pm$3.7 | 60.7$\pm$0.9 | 68.8
M-ADA | 85.5$\pm$0.7 | 80.7$\pm$1.5 | 97.2$\pm$0.5 | 78.4$\pm$1.4 | 85.4 | 78.0$\pm$1.1 | 79.5$\pm$1.2 | 47.1$\pm$0.4 | 55.7$\pm$0.5 | 65.1
LTD | 85.7$\pm$1.9 | 79.9$\pm$0.9 | 96.9$\pm$0.5 | 83.3$\pm$0.5 | 86.4 | 76.8$\pm$0.7 | 82.5$\pm$0.4 | 56.2$\pm$2.5 | 53.6$\pm$1.4 | 67.3
SAM | 86.8$\pm$0.6 | 79.6$\pm$1.4 | 96.8$\pm$0.1 | 80.2$\pm$0.7 | 85.9 | 77.7$\pm$1.1 | 80.5$\pm$0.6 | 46.7$\pm$1.1 | 54.2$\pm$1.5 | 64.8
UDIM w/ SAM | 88.5$\pm$0.1 | 86.1$\pm$0.1 | 97.3$\pm$0.1 | 82.7$\pm$0.1 | 88.7 | 81.5$\pm$0.1 | 85.3$\pm$0.4 | 67.4$\pm$0.8 | 64.6$\pm$1.7 | 74.7
SAGM | 85.3$\pm$2.5 | 80.9$\pm$1.1 | 97.1$\pm$0.4 | 77.8$\pm$0.5 | 85.3 | 78.9$\pm$1.2 | 79.8$\pm$1.0 | 44.7$\pm$1.8 | 55.6$\pm$1.1 | 64.8
UDIM w/ SAGM | 88.9$\pm$0.2 | 86.2$\pm$0.3 | 97.4$\pm$0.4 | 79.5$\pm$0.8 | 88.0 | 81.6$\pm$0.3 | 84.8$\pm$1.2 | 68.1$\pm$0.8 | 63.3$\pm$0.9 | 74.5
GAM | 85.5$\pm$0.6 | 81.1$\pm$1.0 | 96.4$\pm$0.2 | 81.0$\pm$1.7 | 86.0 | 79.1$\pm$1.3 | 79.7$\pm$0.9 | 46.3$\pm$0.6 | 56.6$\pm$1.1 | 65.4
UDIM w/ GAM | 87.1$\pm$0.9 | 86.3$\pm$0.4 | 97.2$\pm$0.1 | 81.8$\pm$1.1 | 88.1 | 82.4$\pm$0.9 | 84.2$\pm$0.4 | 68.8$\pm$0.8 | 64.0$\pm$0.7 | 74.9
(a) Sensitivity analyses of UDIM on $\rho^{\prime}$, $\lambda_{1}$, and
$\rho_{x}$.
(b) Acc over iterations
Figure 3: (a) Sensitivity analyses of UDIM. (b) test accuracy plot of UDIM and
sharpness-based approaches based on training iterations. Shaded regions
represent standard deviation.
### 4.2 Classification Accuracies on Various Benchmark Datasets
To assess the effectiveness of each method under unknown domains, we present
the accuracy results on unknown target domains. Table 2 shows the results on
CIFAR-10-C. The sharpness-based methods exhibit excellent performances
compared to the other lines of methods. This underscores the importance of
training that avoids overfitting to a specific domain. By adding UDIM
(specifically, the inconsistency term of Eq. 4) to these SAM-based approaches,
we consistently observed improved performances compared to the same approaches
without UDIM.
Table 3 shows the results on the PACS dataset. Similar to the results in Table
2, UDIM consistently outperforms the existing baselines across each scenario.
This improvement is particularly amplified in the single source scenario.
Unlike the Leave-One-Out scenario, the single source scenario is more
challenging as it requires generalizing to unknown domains using information
from a single domain. These results emphasize the robust generalization
capability of UDIM under unknown domains.
This section aims to examine the efficacy of UDIM by analyzing the sensitivity
of each hyper-parameter used in UDIM’s implementation. Additionally, We
perform an ablation study by composing each part of the UDIM’s objective in
Eq. 12. Unless specified, each experiment is carried out in the Single source
domain generalization scenario using the PACS dataset.
### 4.3 Sensitivity Analyses and Ablation Study
Figure 4: Ablation study of UDIM
Sensitivity Analyses Figure 3 (a) shows the sensitivity of
$\rho^{{}^{\prime}}$, $\lambda_{1}$ and $\rho_{x}$, which are main hyper-
parameters of UDIM, under the feasible range of value lists. As a default
setting of UDIM, we set $\rho^{{}^{\prime}}$=0.05, $\lambda_{1}$=1 and
$\rho_{x}$=1. In implementation, we multiply $\rho_{x}$ by the unnormalized
gradient of $x$. Therefore, the values presented in the ablation study have a
larger scale. Also, we compare the performances between SAM and UDIM w/ SAM to
compare the effectiveness of UDIM objective. Each figure demonstrates that
UDIM’s performance remains robust and favorable, invariant to the changes in
each hyper-parameter. Figure 3 (b) presents the test accuracies over training
iterations while varying the sharpness-based approaches used alongside UDIM.
Regardless of which method is used in conjunction, additional performance
improvements over the iterations are observed compared to the original SAM
variants.
Ablation Studies Figure 4 presents the ablation results of UDIM, which were
carried out by replacing a subpart of the UDIM’s objective with alternative
candidates and subsequently assessing the performances. We conduct various
ablations: ’SourceOpt’ represents the optimization method for $D_{s}$,
’Perturb’ indicates the perturbation method utilized to emulate unknown
domains, and ’PerturbOpt’ indicates the optimization for the perturbed dataset
$\tilde{D}_{s}$. Appendix D.3 enumerates each ablation candidate and its
implementation. UDIM, depicted by the red bar, consistently outperforms in all
ablations, suggesting the effectiveness of our derived objective formulation.
(a) SAM
(b) SAGM
(c) GAM
(d) UDIM
Figure 5: Sharpness plots for models trained using various methods: the upper
plot shows sharpness on the perturbed parameter space, while the lower plot
displays sharpness on the perturbed data space. The colormap of each row is
normalized into the same scale for fair comparison.
### 4.4 Sharpness Analyses
Figure 1 claims that UDIM would reduce the suggested sharpness in both
parameter space and data space. To support this claim with experiments, Figure
5 enumerates the sharpness plots for models trained with sharpness-based
methods and those trained with UDIM. These figures are obtained by training
models in the single source domain setting of the PACS dataset, and each plot
is drawn utilizing target domain datasets, which are not utilized for
training.
The top row of Figure 5 represents the measurement of sharpness in the
parameter space by introducing a random perturbation to the parameter
$\theta$. Additionally, to examine the sharpness in the perturbed data space
of unobserved domains, the bottom side of Figure 5 illustrates sharpness based
on the input-perturbed region of the target domain datasets. Current SAM
variants struggle to maintain sufficient flatness within both the perturbed
parameter space and data space. On the other hand, UDIM effectively preserves
flatness in the perturbed parameter space and the data space of unknown
domains. Within the region, the model trained using UDIM also exhibits a lower
loss value compared to other methods. Through preserving flatness in each
space, we confirm that the optimization with UDIM, both in parameter and data
space, has practical effectiveness.
## 5 Conclusion
We introduce UDIM, a novel approach to minimize the discrepancy in the loss
landscape between the source domain and unobserved domains. Combined with SAM
variants, UDIM consistently improves generalization performance on unobserved
domains. This performance is achieved by perturbing both domain and parameter
spaces, where UDIM optimization is conducted based on the iterative update
between the dataset and the parameter. Experimental results demonstrate
accuracy gains, up to $9.9\%$ in some settings, by adopting UDIM in current
sharpness-based approaches.
#### Acknowledgments
This research was supported by AI Technology Development for Commonsense
Extraction, Reasoning, and Inference from Heterogeneous Data(IITP) funded by
the Ministry of Science and ICT(2022-0-00077).
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## Appendix A Explanation of Sharpness Variants for Domain Generalization
Gradient norm-Aware Minimization (GAM) Zhang et al. (2023b) introduces first-
order flatness, which minimizes a maximal gradient norm within a perturbation
radius, to regularize a stronger flatness than SAM. Accordingly, GAM seeks
minima with uniformly flat curvature across all directions.
Sharpness-Aware Gradient Matching (SAGM) Wang et al. (2023) minimizes an
original loss, the corresponding perturbed loss, and the gap between them.
This optimization aims to identify a minima that is both flat and possesses a
sufficiently low loss value. Interpreting the given formula, this optimization
inherently regularizes the gradient alignment between the original loss and
the perturbed loss.
Flatness-Aware Minimization (FAM) Zhang et al. (2023a) concurrently optimizes
both zeroth-order and first-order flatness to identify flatter minima. To
compute various sharpness metric on the different order, it incurs a higher
computational cost.
## Appendix B Proofs and Discussions
### B.1 Proof for Theorem 3.2
First, we provide some theorem, definition, and assumptions needed to prove
the Theorem 3.2.
###### Theorem B.1.
(Foret et al., 2020) For any $\rho>0$ which satisfies
$\mathcal{L}_{\mathscr{D}_{e}}(\theta)\leq\mathbb{E}_{\epsilon\sim
p(\epsilon)}\mathcal{L}_{\mathscr{D}_{e}}(\theta+\epsilon)$, with probability
at least $1-\delta$ over realized dataset $D_{e}$ from $\mathscr{D}_{e}$ with
$|D_{e}|=n$, the following holds under some technical conditions on
$\mathcal{L}_{\mathscr{D}_{e}}(\theta)$:
$\mathcal{L}_{\mathscr{D}_{e}}(\theta)\leq\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{e}}(\theta+\epsilon)+h_{e}(\frac{\|\theta\|_{2}^{2}}{\rho^{2}}),$
where $h_{e}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a strictly increasing
function.
###### Definition B.2.
Let
$\Theta_{\theta,\rho}=\\{\theta^{{}^{\prime}}|\|\theta^{{}^{\prime}}-\theta\|_{2}\leq\rho\\}$
and
$N^{\gamma,\rho}_{e,\theta}=\\{\theta^{{}^{\prime}}\in\Theta_{\theta,\rho}|~{}|\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{e}}(\theta)|\leq\gamma\\}$.
###### Assumption B.3.
$\mathcal{L}_{\mathscr{D}}(\theta)\leq\mathbb{E}_{\epsilon\sim
p(\epsilon)}\mathcal{L}_{\mathscr{D}}(\theta+\epsilon)$ where
$p(\epsilon)\sim\mathcal{N}(0,\sigma^{2}I)$ for some $\sigma>0$.
###### Assumption B.4.
$\max_{e\in\mathcal{E}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})\geq\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}})$
for all $\theta^{{}^{\prime}}\in N^{\gamma,\rho}_{s,\theta}$.
In practice, Assumption B.4 is acceptable. Contrary to the dataset $D_{e}$
from an unobserved domain $e$, $D_{s}$ is a source domain dataset provided to
us and hence, available for training. Moreover, the region of
$N^{\gamma,\rho}_{s,\theta}$ would be flat with respect to local minima,
$\theta^{*}$, from the perspective of the source domain dataset. Therefore,
the perturbed loss on the source domain dataset,
$L_{D_{s}}(\theta^{{}^{\prime}})$, would be likely to have a sufficiently low
value.
###### Theorem B.5.
For $\theta\in\Theta$ and arbitrary domain $e\in\mathcal{E}$, with probability
at least $1-\delta$ over realized dataset $D_{e}$ from $\mathscr{D}_{e}$ with
$|D_{e}|=n$, the following holds under some technical conditions on
$\mathcal{L}_{\mathscr{D}_{e}}(\theta)$.
$\displaystyle\mathcal{L}_{\mathscr{D}}(\theta)$
$\displaystyle\leq\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+(1-\frac{1}{|\mathcal{E}|})\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}|\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}})|+h(\frac{\|\theta\|_{2}^{2}}{\rho^{2}})$
(13)
where $h:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a strictly increasing
function.
###### Proof.
For the derivation of Theorem 3.2, we assume the case of single source domain
generalization, where $s$ represents a single domain in $\mathcal{E}$. It
should be noted that the number of available source domain does not affect the
validity of this proof because we can consider multiple source domains as a
single source domain by $D_{s}=\cup_{i\in\mathcal{S}}D_{i}$. Based on the
definition,
$\mathcal{L}_{\mathscr{D}}(\theta)=\frac{1}{|\mathcal{E}|}\sum_{e\in\mathcal{E}}\mathcal{L}_{\mathscr{D}_{e}}(\theta)=\frac{1}{|\mathcal{E}|}\Big{(}\mathcal{L}_{\mathscr{D}_{s}}(\theta)+\sum_{e\in\mathcal{E},e\neq
s}\mathcal{L}_{\mathscr{D}_{e}}(\theta)\Big{)}$. From Theorem B.1, we can
derive the generalization bound of a source domain $s$ as follows:
$\displaystyle\mathcal{L}_{\mathscr{D}_{s}}(\theta)\leq\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+h_{s}(\frac{\|\theta\|_{2}^{2}}{\rho^{2}})$
(14)
where $h_{s}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a strictly increasing
function. To define the upper bound of
$\mathcal{L}_{\mathscr{D}}(\theta^{*})$, we need to find the upper bound of
the remaining term, $\sum_{e\in\mathcal{E},e\neq
s}\mathcal{L}_{\mathscr{D}_{e}}(\theta)$. Here, we introduce a parameter set
$\Theta_{\rho{{}^{\prime}}}:=\operatorname*{arg\,max}_{\Theta_{\hat{\rho}}\subseteq
N^{\gamma,\rho}_{s,\theta}}\hat{\rho}$, which is the largest
$\rho{{}^{\prime}}$-ball region around $\theta$ in
$N^{\gamma,\rho}_{s,\theta}$. Then, we can construct inequality as follows:
$\displaystyle\max_{\|\epsilon\|_{2}\leq\rho{{}^{\prime}}}\mathcal{L}_{D_{e}}(\theta+\epsilon)\leq\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})\leq\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{e}}(\theta+\epsilon)$
(15)
This is because $\Theta_{\rho^{\prime}}\subset
N^{\gamma,\rho}_{s,\theta}\subset\Theta_{\theta,\rho}$. Similar to Foret et
al. (2020); Kim et al. (2022), we make use of the following result from
Laurent & Massart (2000):
$\displaystyle z\sim\mathcal{N}(0,\sigma^{2}I)\Rightarrow\|z\|^{2}_{2}\leq
k\sigma^{2}\Bigg{(}1+\sqrt{\frac{\log{n}}{k}}\Bigg{)}^{2}\,\,\text{with
probability at least}\,\,1-\frac{1}{\sqrt{n}}$ (16)
We set $\rho=\sigma(\sqrt{k}+\sqrt{\log n})$. Then, it enables us to connect
expected perturbed loss and maximum loss as follows:
$\displaystyle\mathbb{E}_{\epsilon\sim\mathcal{N}(0,\sigma^{2}{I})}\Big{[}\mathcal{L}_{D_{e}}(\theta+\epsilon)\Big{]}\leq(1-\frac{1}{\sqrt{n}})\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{e}}(\theta+\epsilon)+\frac{1}{\sqrt{n}}l_{e,max}$
(17)
Here, $l_{e,max}$ is the maximum loss bound when $\|z\|^{2}_{2}\geq\rho^{2}$.
Also, we introduce $\sigma^{\prime}$ where
$\rho^{\prime}=\sigma^{\prime}(\sqrt{k}+\sqrt{\log n})$. Then, similar to Eq.
17, we also derive the equation for $\rho^{\prime}$ as follows:
$\displaystyle\mathbb{E}_{\epsilon\sim\mathcal{N}(0,(\sigma^{{}^{\prime}})^{2}{I})}\Big{[}\mathcal{L}_{D_{e}}(\theta+\epsilon)\Big{]}$
$\displaystyle\leq(1-\frac{1}{\sqrt{n}})\max_{\|\epsilon\|_{2}\leq\rho^{\prime}}\mathcal{L}_{D_{e}}(\theta+\epsilon)+\frac{1}{\sqrt{n}}l^{\prime}_{e,max}$
(18) $\displaystyle\leq(1-\frac{1}{\sqrt{n}})\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})+\frac{1}{\sqrt{n}}l^{\prime}_{e,max}$
(19)
where $l^{\prime}_{e,max}$ is the maximum loss bound when
$\|z\|^{2}_{2}\geq(\rho^{\prime})^{2}$. Then, we have the below derivation by
summing up all $e\neq s$ and using the fact that $l_{e,max}\leq
l^{\prime}_{e,max}$:
$\displaystyle\frac{1}{(|\mathcal{E}|-1)}\sum_{e\in\mathcal{E},e\neq
s}\mathbb{E}_{\epsilon\sim\mathcal{N}(0,{\sigma^{{}^{\prime}}}^{2}{I})}\Big{[}\mathcal{L}_{D_{e}}(\theta+\epsilon)\Big{]}$
(20)
$\displaystyle\leq(1-\frac{1}{\sqrt{n}})\frac{1}{(|\mathcal{E}|-1)}\sum_{e\in\mathcal{E},e\neq
s}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})+\frac{1}{\sqrt{n}}\frac{1}{(|\mathcal{E}|-1)}\sum_{e\in\mathcal{E},e\neq
s}l^{\prime}_{e,max}$ (21)
$\displaystyle\leq(1-\frac{1}{\sqrt{n}})\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})+\frac{1}{\sqrt{n}}\max_{e\in\mathcal{E}}l^{\prime}_{e,max}$
(22)
Using Assumption B.3 and PAC-Bayesian generalization bound (McAllester, 1999;
Dziugaite & Roy, 2017; Foret et al., 2020), we find the upper bound of the sum
of target domain losses.
$\displaystyle\frac{1}{(|\mathcal{E}|-1)}\sum_{e\in\mathcal{E},e\neq
s}\mathcal{L}_{\mathscr{D}_{e}}(\theta)\leq\frac{1}{(|\mathcal{E}|-1)}\sum_{e\in\mathcal{E},e\neq
s}\mathbb{E}_{\epsilon\sim\mathcal{N}(0,(\sigma^{\prime})^{2}I)}[\mathcal{L}_{\mathscr{D}_{e}}(\theta+\epsilon)]$
(23) $\displaystyle\leq\frac{1}{(|\mathcal{E}|-1)}\sum_{e\in\mathcal{E},e\neq
s}\\{\mathbb{E}_{\epsilon\sim\mathcal{N}(0,(\sigma^{\prime})^{2}I)}[\mathcal{L}_{{D}_{e}}(\theta+\epsilon)]+h_{e}(\frac{\|\theta\|_{2}^{2}}{\rho^{2}})\\}$
(24)
$\displaystyle\leq(1-\frac{1}{\sqrt{n}})\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})+\frac{1}{\sqrt{n}}\max_{e\in\mathcal{E}}l^{\prime}_{e,max}+\frac{1}{(|\mathcal{E}|-1)}\sum_{e\in\mathcal{E},e\neq
s}h_{e}(\frac{\|\theta\|_{2}^{2}}{\rho^{2}})$ (25) $\displaystyle\Rightarrow$
$\displaystyle\sum_{e\in\mathcal{E},e\neq
s}\mathcal{L}_{\mathscr{D}_{e}}(\theta)\leq(|\mathcal{E}|-1)(1-\frac{1}{\sqrt{n}})\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})+\tilde{h}(\frac{\|\theta\|_{2}^{2}}{\rho^{2}})$
(26)
where $h_{e},\tilde{h}$ are strictly increasing functions. We use the fact
that the sum of strictly increasing functions is also a strictly increasing
function. By integrating results of Eq. 14 and 26,
$\displaystyle\frac{1}{|\mathcal{E}|}\Big{(}\mathcal{L}_{\mathscr{D}_{s}}(\theta)+\sum_{e\in\mathcal{E},e\neq
s}\mathcal{L}_{\mathscr{D}_{e}}(\theta)\Big{)}\leq$
$\displaystyle\frac{1}{|\mathcal{E}|}\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)$
$\displaystyle+(1-\frac{1}{|\mathcal{E}|})(1-\frac{1}{\sqrt{n}})\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})+h(\frac{\|\theta\|_{2}^{2}}{\rho^{2}})$
(27)
where $h:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a strictly increasing
function. The first and second terms of RHS in the above inequality are upper
bounded by the maximum perturbed loss for source domain and unknown domain
inconsistency loss.
$\displaystyle\frac{1}{|\mathcal{E}|}\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+(1-\frac{1}{|\mathcal{E}|})(1-\frac{1}{\sqrt{n}})\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})$ (28)
$\displaystyle\leq\frac{1}{|\mathcal{E}|}\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+(1-\frac{1}{|\mathcal{E}|})\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})$ (29)
$\displaystyle=\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+(1-\frac{1}{|\mathcal{E}|})\Big{(}\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)\Big{)}$
(30)
$\displaystyle\leq\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+(1-\frac{1}{|\mathcal{E}|})\Big{(}\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}})\Big{)}$
(31)
$\displaystyle\leq\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+(1-\frac{1}{|\mathcal{E}|})\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}(\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}}))$
(32)
$\displaystyle=\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+(1-\frac{1}{|\mathcal{E}|})\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}|\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}})|$
(33)
The last equality comes from the Assumption B.4. To sum up, we can derive the
upper bound of the population loss for whole domain using the maximum
perturbed loss for source domain and unknown domain inconsistency loss with
weight decay term.
$\displaystyle\mathcal{L}_{\mathscr{D}}(\theta)\leq\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)+(1-\frac{1}{|\mathcal{E}|})\max_{e\in\mathcal{E}}\max_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}|\mathcal{L}_{D_{e}}(\theta^{{}^{\prime}})-\mathcal{L}_{D_{s}}(\theta^{{}^{\prime}})|+h(\frac{\|\theta\|_{2}^{2}}{\rho^{2}})$
(34)
∎
### B.2 Detailed Explanation on Approximation
Here, we show the full derivation of Eq. 8 and 9 as follows.
$\displaystyle\max\limits_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\Big{(}\mathcal{L}_{\tilde{D}_{s}}(\theta^{{}^{\prime}})-\mathcal{L}_{{D}_{s}}(\theta^{{}^{\prime}})\Big{)}\approx\max\limits_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\mathcal{L}_{\tilde{D}_{s}}(\theta^{{}^{\prime}})-\mathcal{L}_{{D}_{s}}(\theta)+\gamma^{{}^{\prime}}$
(35) $\displaystyle\underset{\text{ 2nd
Taylor}}{\approx}\mathcal{L}_{\tilde{D}_{s}}(\theta)-\mathcal{L}_{{D}_{s}}(\theta)+\rho{{}^{\prime}}\|\nabla_{\theta}\mathcal{L}_{\tilde{D}_{s}}(\theta)\|_{2}+\max\limits_{\theta^{{}^{\prime}}\in
N^{\gamma,\rho}_{s,\theta}}\frac{1}{2}\theta^{{}^{\prime}\top}\mathbf{H}_{\tilde{D}_{s}}\theta^{{}^{\prime}}$
(36)
$\displaystyle=\Big{(}\mathcal{L}_{\tilde{D}_{s}}(\theta)-\mathcal{L}_{{D}_{s}}(\theta)\Big{)}+\rho{{}^{\prime}}\|\nabla_{\theta}\mathcal{L}_{\tilde{D}_{s}}(\theta)\|_{2}+\gamma\max\limits_{i}\lambda^{\tilde{D}_{s}}_{i}/\lambda^{{D}_{s}}_{i}$
(37)
### B.3 Discussion on Hessian matrix and Gradient Variance
##### Hessian Matching
This section first discusses how the Hessian matrix matching between two
different datasets could be substituted by the gradient variance matching for
the respective datasets. It should be noted that we follow the motivation and
derivation of Rame et al. (2022), and this section just re-formalize the
derivations based on our notations. ${\mathbf{g}}_{i}$ is a per-sample
gradient for $i$-th sample; and
$\mathbf{G}_{{D}}=\\{{\mathbf{g}}_{i}\\}^{|D|}_{i=1}$ is a set of per-sample
gradient for $x_{i}\in D$. Accordingly, the variance of $\mathbf{G}_{{D}}$,
which we denote as $\text{Var}(\mathbf{G}_{{D}})$, is calculated as
$\text{Var}(\mathbf{G}_{{D}})=\frac{1}{|D|-1}\sum^{|D|}_{i=1}\big{(}\mathbf{g}_{i}-\bar{\mathbf{g}}\big{)}^{2}$.
We first revisit the Eq. 37, which is our intermediate objective as follows:
$\displaystyle\Big{(}\mathcal{L}_{\tilde{D}_{s}}(\theta)-\mathcal{L}_{{D}_{s}}(\theta)\Big{)}+\rho{{}^{\prime}}\|\nabla_{\theta}\mathcal{L}_{\tilde{D}_{s}}(\theta)\|_{2}+\gamma\max\limits_{i}\lambda^{\tilde{D}_{s}}_{i}/\lambda^{{D}_{s}}_{i}$
(38)
The last term of Eq. 38,
$\max\limits_{i}\lambda^{\tilde{D}_{s}}_{i}/\lambda^{{D}_{s}}_{i}$, refers to
the maximum eigenvalue ratio of the Hessian matrices between two different
datasets, $\tilde{D}_{s}$ and ${D}_{s}$. This ratio is minimized when Hessian
matrices of $\tilde{D}_{s}$ and ${D}_{s}$ becomes equivalent. Then,
$\max\limits_{i}\lambda^{\tilde{D}_{s}}_{i}/\lambda^{{D}_{s}}_{i}$ is
approximated to the hessian matrix matching between $\tilde{D}_{s}$ and
${D}_{s}$ as $\|\mathbf{H}_{\tilde{D}_{s}}-\mathbf{H}_{{D}_{s}}\|_{2}$.
Computing the full Hessian matrix in over-parameterized networks is
computationally challenging. Therefore, we express the formula using the
diagonalized Hessian matrix, denoted as $\hat{\mathbf{H}}_{{D}_{s}}$, which
results in
$\|\hat{\mathbf{H}}_{\tilde{D}_{s}}-\hat{\mathbf{H}}_{{D}_{s}}\|_{2}$.
Let the Fisher Information Matrix (Rame et al., 2022) as
${\bm{F}}=\sum_{i=1}^{n}\mathbb{E}_{\hat{y}\sim
P_{\theta}(\cdot|x_{i})}\left[\nabla_{\theta}\log
p_{\theta}(\hat{y}|x_{i})\nabla_{\theta}\log
p_{\theta}(\hat{y}|x_{i})^{\top}\right]$, where $p_{\theta}(\cdot|x_{i})$ is
the density of $f_{\theta}$ on a input instance $x$. Fisher Information Matrix
(FIM) approximates the Hessian $\mathbf{H}$ with theoretically probably
bounded errors under mild assumptions Schraudolph (2002). Then, diagonalized
Hessian matrix matching between $\tilde{D}_{s}$ and ${D}_{s}$,
$\|\hat{\mathbf{H}}_{\tilde{D}_{s}}-\hat{\mathbf{H}}_{{D}_{s}}\|_{2}$ could be
replaced by $\|\hat{{\bm{F}}}_{\tilde{D}_{s}}-\hat{{\bm{F}}}_{{D}_{s}}\|_{2}$,
where $\hat{{\bm{F}}}$ also denotes the diagonalized version of ${\bm{F}}$.
Empirically, $\hat{{\bm{F}}}$ is equivalent to the gradient variance of the
trained model, $f_{\theta}$. This finally confirms the validation of our
objective, gradient variance difference between $\tilde{D}_{s}$ and ${D}_{s}$
as
$\|\text{Var}(\mathbf{G}_{\tilde{D}_{s}})-\text{Var}(\mathbf{G}_{{D}_{s}})\|_{2}$.
Table 2 on the main paper of Rame et al. (2022) empirically supports that the
similarity between Hessian diagonals and gradient variances is over 99.99$\%$.
##### Loss Matching
Matching the gradient variances for all parameters of our model, $f_{\theta}$,
incurs significant computational overhead. In this study, we restrict the
gradient variance matching to a subset of entire parameters, specificall y
selecting the classifier parameters. In this section, we demonstrate that by
matching gradient variances of two different datasets based on the classifier
parameters, it inherently achieve the loss matching across those datasets.
For simplicity in notation, we refer an arbitrary domain index as $e$. Let
$x_{e}^{i}$ represent the $i$-th sample and $y_{e}^{i}$ be its corresponding
target class label. We denote $z_{e}^{i}\in\mathbb{R}^{d}$ as the features for
this $i$-th sample from domain $e$. The associated classifier layer $W$ is
characterized by weights $\\{w_{k}\\}_{k=1}^{d}$ and bias $b$. First, we
assume the mean squared error as our loss function for our analysis. For the
$i$-th sample, the gradient of the loss with respect to $b$ is given by
$\nabla_{b}\ell(f_{\theta}(x_{e}^{i}),y_{e}^{i})=(f_{\theta}(x_{e}^{i})-y_{e}^{i})$.
Hence, the gradient variance based on the parameter $b$ for domain $e$ is
given by:
$\text{Var}(\mathbf{G}_{D_{e}}^{b})=\frac{1}{n_{e}}\sum_{i=1}^{n_{e}}(f_{\theta}(x_{e}^{i})-y_{e}^{i})^{2}$,
which directly aligns with the mean squared error (MSE) between the
predictions and the target labels in domain $e$. Considering our objective,
$\|\text{Var}(\mathbf{G}_{\tilde{D}_{s}})-\text{Var}(\mathbf{G}_{{D}_{s}})\|_{2}$,
the gradient variance matching on $b$ could be recognized as mean squared
error loss matching, which is
$\|\frac{1}{|\tilde{D}_{s}|}\sum_{(x^{i},y^{i})\in\tilde{D}_{s}}(f_{\theta}(x^{i})-y^{i})^{2}-\frac{1}{|{D}_{s}|}\sum_{(x^{j},y^{j})\in{D}_{s}}(f_{\theta}(x^{j})-y^{j})^{2}\|_{2}$.
##### Analysis on the remaining term
We also investigate the gradient variance matching based on
$\\{w_{k}\\}_{k=1}^{d}\in W$, which are remaining part of the classifier
parameter $W$. The gradients with respect to the $w_{k}$ is derived as
$\nabla_{w_{k}}\ell(y_{e}^{i},\hat{y}_{e}^{i})=(\hat{y}_{e}^{i}-y_{e}^{i})z_{e}^{i,k}$.
Thus, the uncentered gradient variance in $w_{k}$ for domain $e$ is:
$\text{Var}(\textbf{G}_{D_{e}}^{w_{k}})=\frac{1}{n_{e}}\sum_{i=1}^{n_{e}}\left((\hat{y}_{e}^{i}-y_{e}^{i})z_{e}^{i,k}\right)^{2}$.
Different from the case of $b$, $\text{Var}(\textbf{G}_{D_{e}}^{w_{k}})$ adds
a weight $z_{e}^{i,k}$ on the mean squared error. As $z_{e}^{i,k}$ act as a
weight, gradient variance matching on $w_{k}$ still learns toward matching the
MSE loss between two different datasets.
## Appendix C Algorithm of UDIM
Here, we present the algorithm of UDIM as follows.
Input: Source dataset $D_{s}$; perturbation threshold for model parameter
$\theta$ and data, $\rho^{\prime}$ and $\rho_{x}$; learning rate $\eta$;
warmup ratio for source domain flatness, $p$; number of total training
iterations, $N$; Other hyperpameters.
Output: Trained model $f_{\theta}(x)$
for _$t=1,...,N/p$_ do
Warmup $f_{\theta}(x)$ with SAM optimization as Eq. 1
end for
for _$t=N/p,...,N$_ do
Define $D_{B}=\\{(x_{i},y_{i})\\}_{i=1}^{|B|}$ i.i.d. sampled from $D_{s}$
Make $\tilde{D}_{B}=\\{(\tilde{x}_{i},y_{i})\\}_{i=1}^{|B|}$, with
$\tilde{x}_{i}$ for all $i$ as
$\tilde{x}_{i}=x_{i}+\rho_{x}\frac{\nabla_{x_{i}}\big{(}\ell(x,\theta_{t})+\rho^{\prime}\|\nabla_{\theta_{t}}\ell(x,\theta_{t})\|_{2}\big{)}}{\big{\|}\nabla_{x_{i}}\big{(}\ell(x,\theta_{t})+\rho^{\prime}\|\nabla_{\theta_{t}}\ell(x,\theta_{t})\|_{2}\big{)}\big{\|}_{2}}$
Update $f_{\theta}(x)$ by
$\theta_{t+1}\leftarrow\theta_{t}-\eta\nabla_{\theta_{t}}\Big{(}\max\limits_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{B}}(\theta_{t}+\epsilon)+\rho{{}^{\prime}}\|\nabla_{\theta_{t}}\mathcal{L}_{\tilde{D}_{B}}(\theta_{t})\|_{2}+\|\text{Var}(\mathbf{G}_{\tilde{D}_{B}})-\text{Var}(\mathbf{G}_{D_{B}})\|_{2}+\lambda_{2}\|\theta_{t}\|_{2}\Big{)}$
end for
Algorithm 1 Training algorithm of UDIM w/ SAM
We represent the algorithm of UDIM with SAM as a default setting. It should be
noted that our method can be orthogonally utilized with other sharpness-based
optimization methods.
## Appendix D Experiment
### D.1 Implementation details
##### Dataset Explanation
* •
PACS (Li et al., 2017) comprises of four domains, which are photos, arts,
cartoons and sketches. This dataset contains 9,991 images. It consists of 7
classes.
* •
OfficeHome (Venkateswara et al., 2017) includes four domains, which are art,
clipart, product and real. This dataset contains 15,588 images. It consists of
65 classes.
* •
DomainNet (Peng et al., 2019) consists of six domains, which are clipart,
infograph, painting, quickdraw, real and sketch. This dataset contains 586,575
images. It consists of 345 classes.
* •
CIFAR-10-C (Hendrycks & Dietterich, 2019) has been utilized to evaluate the
robustness of a trained classifier. Images are corrupted from CIFAR10
(Krizhevsky et al., 2009) test dataset under 5 levels of severity. Corruption
types include, brightness, contrast, defocus-blur, elastic-transform, fog,
frost, gaussian-blur, gaussian-noise, glass-blur, impulse-noise, jpeg-
compression, motion-blur, pixelate, saturate, shot-noise, snow, spatter,
speckle-noise, zoom-blur total of 19 types.
##### Network Architecture and Optimization
We use ResNet-18 (He et al., 2016) for CIFAR-10-C and ResNet-50 (He et al.,
2016) for other datasets pretrained on ImageNet (Deng et al., 2009) and use
Adam (Kingma & Ba, 2014) optimizer basically. learning rate is set as $3\times
10^{-5}$ following Wang et al. (2023). For calculating the gradient related
materials, e.g. Hessian, we utilize BackPACK (Dangel et al., 2020) package.
##### Experimental settings
For PACS and OfficeHome, we trained for total of 5,000 iterations. For
DomainNet, we trained for 15,000 iterations. For CIFAR-10, since it usually
trains for 100 epochs, we translate it to iterations, which becomes total of
$781\times 100=78,100$ iterations. Unless specified, we use batch size as 32
for PACS, OfficeHome, and DomainNet and 64 for CIFAR-10-C. For other
hyperparameters, we follow the experiment settings of Wang et al. (2023)
unless specified. Although our method mainly focuses on the domain
generalization, our concept could be also effectively utilized for domain
adaptation Csurka (2017) and open-set domain adaptation Jang et al. (2022).
##### Hyperparameter setting of UDIM
The main hyperparameters of UDIM is $\rho,\rho^{{}^{\prime}}$, $\lambda_{1}$
and $\rho_{x}$. Throughout all experiments, we set $\rho$=0.05 without any
hyperparameter tuning. For $\rho^{{}^{\prime}}$, we used values [0.01, 0.025,
0.05], and in most experiments, the value 0.05 consistently showed good
performance. It should be noted that a warm-up using the SAM loss is required
before the full UDIM optimization to ensure that $\rho^{{}^{\prime}}$=0.05 can
be utilized validly. For both $\lambda_{1}$ and $\rho_{x}$, we used values in
the range [1,10] and reported the best performances observed among these
results. Our methodology applies perturbations to each instance of the
original source domain dataset, effectively doubling the number of unique
instances in a single batch compared to experiments for the baselinse. As a
result, we utilized half the batch size of other baselines.
##### Evaluation Detail
For reporting the model performance, model selection criterion is important.
We get the test performance whose accuracy for source validation dataset is
best. For PACS and OfficeHome, we evaluated every 100 iterations and for
DomainNet, we evaluated every 1,000 iterations for Leave-One-Out Source Domain
Generalization and every 5,000 iterations for Single Source Domain
Generalization.
### D.2 Baseline description
In this paragraph, we explain baselines that we used for comparison.
Specifically, we compare our method with (1) methods whose objectives are
mainly related to Leave-One Out Source Domain Generalization, (2) methods
which are mainly modeled for Single Source Domain Generalization, and (3)
sharpness-aware minimization related methods, as we reported in tables
repeatedly.
IRM (Arjovsky et al., 2019) tries to learn a data representation such that the
optimal classifier matches for all training distributions. Specifically, it
minimizes the empirical risk and the regularization term, the multiplication
of samples’ gradients, to motivate the invariance of the predictor.
GroupDRO (Sagawa et al., 2019) minimizes the loss by giving different weight
to each domain. Weight term for each domain is proportional to the domain’s
current loss.
OrgMixup (Zhang et al., 2018) represents the naive mixup technique which is
generally utilized in machine learning community to boost generalization.
Mixup (Yan et al., 2020) is a mixup among domains.
Cutmix (Yun et al., 2019) is another skill which is widely used in machine
learning community to boost generalization. Specifically, it mixes up parts of
inputs randomly by pixel-wise.
Mixstyle (Zhou et al., 2021) mix up the statistics (specifically, mean and
standard deviation) of the feature. The mixed feature statistics are applied
to the style-normalized input. We did not consider the domain label.
MTL (Blanchard et al., 2021) considers the exponential moving average (EMA) of
features.
MLDG (Li et al., 2018a) is a meta learning based method for domain
generalization. Specifically, it simulates the domain shift between train and
test during training procedure by synthesizing virtual testing domains within
each mini-batch. Then it optimizes meta loss using the synthesized dataset.
MMD (Li et al., 2018b) minimizes the discrepancy of feature distributions in a
every domain pair-wise manner, while minimizing the empirical risk for source
domains.
CORAL (Sun & Saenko, 2016) is similar to MMD. However, while MMD employs the
gaussian kernel to measure the feature discrepancy, CORAL aligns the second-
order statistics between different distributions with a nonlinear
transformation. This alignment is achieved by matching the correlations of
layer activations in deep neural networks.
SagNet (Nam et al., 2021) disentangles style features from class categories to
prevent bias. Specifically, it makes two networks, content network and style
network, and trains both networks to be invariant to other counterpart by
giving randomized features (updating the content network with randomized
styled features and vice versa).
ARM (Zhang et al., 2021) represents adaptive risk minimization. Specifically,
it makes an adaptive risk representing context.
DANN represents Domain Adversarial Neural Networks, and it iteratively trains
a discriminator which discriminates domain and a featurizer to learn a feature
which becomes invariant to domain information.
CDANN is class conditional version of DANN.
VREx (Krueger et al., 2021) controls the discrepancy between domains by
minimizing the variance of loss between domains.
RSC (Huang et al., 2020) challenges the dominant features of training domain
(by masking some specific percentage of dominant gradient), so it can focus on
label-related domain invariant features.
Fishr (Rame et al., 2022) approximates the hessian as the variance of gradient
matrix, and they align the gradient variance of each domain.
M-ADA (Qiao et al., 2020a) perturbs input data to simulate the unseen domain
data, yet with adequate regularization not to make the data be too far from
the original one. The adversarial perturbation direction is affected by the
wasserstein autoencoder. Note that this method is specifically designed for
Single source domain generalization.
LTD (Wang et al., 2021a) perturbs source domain data with augmentation
network, maximize the mutual information between the original feature and the
perturbed feature so that the perturbed feature is not too far from the
original feature (with contrastive loss), and maximize likelihood of the
original feature. Note that this method is also specifically designed for
Single source domain generalization.
SAM (Foret et al., 2020) is an optimization technique to consider the
sharpness of loss surface. It first perturbs parameter to its worst direction,
gets gradient and update the calculated gradient at the original parameter
point.
SAGM (Wang et al., 2023) minimizes an original loss, the corresponding
perturbed loss, and the gap between them. This optimization aims to identify a
minima that is both flat and possesses a sufficiently low loss value.
Interpreting the given formula, this optimization inherently regularizes the
gradient alignment between the original loss and the perturbed loss.
GAM (Zhang et al., 2023b) introduces first-order flatness, which minimizes a
maximal gradient norm within a perturbation radius, to regularize a stronger
flatness than SAM. Accordingly, GAM seeks minima with uniformly flat curvature
across all directions.
RIDG (Chen et al., 2023b) presents a new approach in deep neural networks
focusing on decision-making in the classifier layer, diverging from the
traditional emphasis on features. It introduces a ’rationale matrix’, derived
from the relationship between features and weights, to guide decisions for
each input. A novel regularization term is proposed to align each sample’s
rationale with the class’s mean, enhancing stability across samples and
domains.
ITTA (Chen et al., 2023a) proposes an Improved Test-Time Adaptation (ITTA)
method for domain generalization. ITTA uses a learnable consistency loss for
the TTT task to better align with the main prediction task and introduces
adaptive parameters in the model, recommending updates solely during the test
phase. This approach aims to address the issues of auxiliary task selection
and parameter updating in test-time training.
### D.3 ablation
Figure 4 of the main paper presents the ablation results of UDIM, which were
carried out by replacing a subpart of the UDIM’s objective with alternative
candidates and subsequently assessing the performances. This section
enumerates enumerates each ablation candidate and its implementation. We
conduct various ablations: ’SourceOpt’ represents the optimization method for
$D_{s}$, ’Perturb’ indicates the perturbation method utilized to emulate
unknown domains, and ’PerturbOpt’ indicates the optimization for the perturbed
dataset $\tilde{D}_{s}$. It should be noted that each ablation means that only
the specific part is substituted, while keeping the other parts of UDIM
unchanged.
##### SourceOpt
The optimization for the source domain dataset in UDIM is originally based on
the SAM loss, which is
$\max_{\|\epsilon\|_{2}\leq\rho}\mathcal{L}_{D_{s}}(\theta+\epsilon)$. To
verify the significance of flatness modeling, we replaced the original
optimization with simple ERM, which we refer to as the ERM ablation.
##### Perturb
The perturbation process of UDIM is conducted based on Eq. 11. We substituted
the perturbation method in Eq. 11 with traditional adversarial attack
techniques, which are the cases of FGSM (Goodfellow et al., 2014) and PGD
(Madry et al., 2017), to compare their performance outcomes.
##### PerturbOpt
Lastly, to observe the ablation for inconsistency minimization between the
perturbed domain dataset $\tilde{D}_{s}$ and $D_{s}$, we replaced the
optimization in Eq. 10 with both ERM and SAM-based optimizations. Each case in
the PerturbOpt ablation is denoted as ERM or SAM.
### D.4 Additional Results
In this section, we report more results that we did not report in the main
paper due to the space issue.
Table 4 shows the model performance of total baselines (At the table 3 of the
main paper, there are only the model performance of some baselines). As we can
see in the table, our method, UDIM, consistently improves SAM-based
optimization variants and show best performances for each column. We mark -
for training failure case (when the model performance is near 1$\%$).
Table 4: Test accuracy for PACS. For Leave-One-Out Source Domain
Generalization, each column represents test domain, and train domain for
Single Source Domain Generalization. ∗ denotes performances are from its
original paper considering LOODG. For SDG scenario, we generated the
experiment results for all baselines. Bold indicates the best case of each
column or improved performances when combined with the respective sharpness-
based optimizers.
| Leave-One-Out Source Domain Generalization | Single Source Domain Generalization
---|---|---
Method | Art | Cartoon | Photo | Sketch | Avg | Art | Cartoon | Photo | Sketch | Avg
ERM | 86.9$\pm$2.3 | 79.5$\pm$1.5 | 96.6$\pm$0.5 | 78.2$\pm$4.1 | 85.3 | 79.9$\pm$0.9 | 79.9$\pm$0.8 | 48.1$\pm$5.8 | 59.6$\pm$1.1 | 66.9
IRM∗ | 85.0∗$\pm$1.6 | 77.6∗$\pm$0.9 | 96.7∗$\pm$0.3 | 78.5∗$\pm$2.6 | 84.4∗ | 73.3$\pm$1.5 | 77.8$\pm$2.3 | 46.9$\pm$0.8 | 49.7$\pm$3.0 | 61.9
GroupDRO | 84.8$\pm$2.2 | 79.4$\pm$1.2 | 97.3$\pm$0.3 | 75.8$\pm$1.0 | 84.3 | 79.0$\pm$0.5 | 79.0$\pm$0.6 | 42.0$\pm$2.9 | 60.8$\pm$3.9 | 65.2
OrgMixup | 87.7$\pm$0.3 | 77.4$\pm$1.3 | 97.6$\pm$0.3 | 76.3$\pm$0.9 | 84.8 | 74.5$\pm$1.1 | 79.8$\pm$0.4 | 46.8$\pm$2.1 | 55.5$\pm$1.9 | 64.1
Mixup | 86.9$\pm$1.3 | 78.2$\pm$0.7 | 97.8$\pm$0.4 | 73.7$\pm$2.9 | 84.2 | 77.4$\pm$1.4 | 80.0$\pm$1.2 | 47.3$\pm$1.6 | 58.2$\pm$1.2 | 65.7
CutMix | 80.5$\pm$0.7 | 75.7$\pm$1.4 | 97.0$\pm$0.5 | 74.8$\pm$1.7 | 82.0 | 71.1$\pm$0.5 | 76.4$\pm$3.1 | 37.7$\pm$0.3 | 50.4$\pm$4.0 | 58.9
Mixstyle | 84.4$\pm$2.3 | 80.4$\pm$0.6 | 95.6$\pm$0.1 | 80.5$\pm$1.1 | 85.2 | 78.1$\pm$2.8 | 78.8$\pm$1.1 | 56.1$\pm$3.9 | 54.7$\pm$2.9 | 66.9
MTL | 85.4$\pm$2.2 | 78.8$\pm$2.2 | 96.5$\pm$0.2 | 74.4$\pm$2.0 | 83.8 | 76.7$\pm$1.2 | 78.7$\pm$1.7 | 44.7$\pm$2.0 | 59.5$\pm$1.4 | 64.9
MLDG | 87.7$\pm$0.6 | 77.5$\pm$0.7 | 96.6$\pm$0.6 | 75.3$\pm$1.9 | 84.3 | - | - | - | - | -
MMD∗ | 84.5∗$\pm$0.6 | 79.7∗$\pm$0.7 | 97.5∗$\pm$0.4 | 78.1∗$\pm$1.3 | 85.0∗ | 75.4$\pm$1.1 | 80.1$\pm$0.5 | 45.2$\pm$1.2 | 58.2$\pm$0.6 | 64.7
CORAL∗ | 87.7∗$\pm$0.6 | 79.2∗$\pm$1.1 | 97.6∗$\pm$0.0 | 79.4∗$\pm$0.7 | 86.0∗ | 76.3$\pm$0.8 | 79.2$\pm$2.0 | 45.9$\pm$1.7 | 57.0$\pm$1.4 | 64.6
SagNet | 87.1$\pm$1.1 | 78.0$\pm$1.9 | 96.8$\pm$0.2 | 78.4$\pm$1.4 | 85.1 | 77.4$\pm$0.0 | 78.9$\pm$1.8 | 47.6$\pm$2.4 | 56.4$\pm$4.0 | 65.1
ARM | 86.4$\pm$0.1 | 78.8$\pm$0.6 | 96.1$\pm$0.1 | 75.1$\pm$3.3 | 84.1 | 76.2$\pm$0.5 | 75.5$\pm$4.0 | 45.2$\pm$5.7 | 61.9$\pm$2.0 | 64.7
DANN∗ | 85.9∗$\pm$0.5 | 79.9∗$\pm$1.4 | 97.6∗$\pm$0.2 | 75.2∗$\pm$2.8 | 84.6∗ | 79.0$\pm$1.4 | 76.5$\pm$2.0 | 48.7$\pm$2.1 | 57.9$\pm$4.7 | 65.5
CDANN∗ | 84.0∗$\pm$0.9 | 78.5∗$\pm$1.5 | 97.0∗$\pm$0.4 | 71.8∗$\pm$3.9 | 82.8∗ | 78.5$\pm$1.5 | 78.7$\pm$2.0 | 48.3$\pm$3.1 | 56.9$\pm$2.2 | 65.6
VREx | 87.2$\pm$0.5 | 77.8$\pm$0.8 | 96.8$\pm$0.3 | 75.2$\pm$3.4 | 84.3 | 75.3$\pm$2.1 | 80.2$\pm$0.4 | 44.9$\pm$2.8 | 56.8$\pm$2.6 | 64.3
RSC | 81.0$\pm$0.7 | 77.6$\pm$1.0 | 95.3$\pm$0.8 | 75.0$\pm$1.4 | 82.2 | 68.9$\pm$2.3 | 70.6$\pm$3.6 | 41.1$\pm$3.1 | 45.9$\pm$3.1 | 56.6
Fishr∗ | 88.4∗$\pm$0.2 | 78.7∗$\pm$0.7 | 97.0∗$\pm$0.1 | 77.8∗$\pm$2.0 | 85.5∗ | 75.9$\pm$1.7 | 81.1$\pm$0.7 | 46.9$\pm$0.7 | 57.2$\pm$4.4 | 65.3
RIDG | 86.3$\pm$1.1 | 81.0$\pm$1.0 | 97.4$\pm$0.7 | 77.5$\pm$2.5 | 85.5 | 76.2$\pm$1.4 | 80.0$\pm$1.8 | 48.5$\pm$2.8 | 54.8$\pm$2.4 | 64.9
ITTA | 87.9$\pm$1.4 | 78.6$\pm$2.7 | 96.2$\pm$0.2 | 80.7$\pm$2.2 | 85.8 | 78.4$\pm$1.5 | 79.8$\pm$1.3 | 56.5$\pm$3.7 | 60.7$\pm$0.9 | 68.8
M-ADA | 85.5$\pm$0.7 | 80.7$\pm$1.5 | 97.2$\pm$0.5 | 78.4$\pm$1.4 | 85.4 | 78.0$\pm$1.1 | 79.5$\pm$1.2 | 47.1$\pm$0.4 | 55.7$\pm$0.5 | 65.1
LTD | 85.7$\pm$1.9 | 79.9$\pm$0.9 | 96.9$\pm$0.5 | 83.3$\pm$0.5 | 86.4 | 76.8$\pm$0.7 | 82.5$\pm$0.4 | 56.2$\pm$2.5 | 53.6$\pm$1.4 | 67.3
SAM | 86.8$\pm$0.6 | 79.6$\pm$1.4 | 96.8$\pm$0.1 | 80.2$\pm$0.7 | 85.9 | 77.7$\pm$1.1 | 80.5$\pm$0.6 | 46.7$\pm$1.1 | 54.2$\pm$1.5 | 64.8
UDIM w/ SAM | 88.5$\pm$0.1 | 86.1$\pm$0.1 | 97.3$\pm$0.1 | 82.7$\pm$0.1 | 88.7 | 81.5$\pm$0.1 | 85.3$\pm$0.4 | 67.4$\pm$0.8 | 64.6$\pm$1.7 | 74.7
SAGM | 85.3$\pm$2.5 | 80.9$\pm$1.1 | 97.1$\pm$0.4 | 77.8$\pm$0.5 | 85.3 | 78.9$\pm$1.2 | 79.8$\pm$1.0 | 44.7$\pm$1.8 | 55.6$\pm$1.1 | 64.8
UDIM w/ SAGM | 88.9$\pm$0.2 | 86.2$\pm$0.3 | 97.4$\pm$0.4 | 79.5$\pm$0.8 | 88.0 | 81.6$\pm$0.3 | 84.8$\pm$1.2 | 68.1$\pm$0.8 | 63.3$\pm$0.9 | 74.5
GAM | 85.5$\pm$0.6 | 81.1$\pm$1.0 | 96.4$\pm$0.2 | 81.0$\pm$1.7 | 86.0 | 79.1$\pm$1.3 | 79.7$\pm$0.9 | 46.3$\pm$0.6 | 56.6$\pm$1.1 | 65.4
UDIM w/ GAM | 87.1$\pm$0.9 | 86.3$\pm$0.4 | 97.2$\pm$0.1 | 81.8$\pm$1.1 | 88.1 | 82.4$\pm$0.9 | 84.2$\pm$0.4 | 68.8$\pm$0.8 | 64.0$\pm$0.7 | 74.9
Table 5: Results on OfficeHome dataset. ∗ represents we got the results from
Wang et al. (2023) and ′ from Rame et al. (2022) considering Leave-One-Out
Source Domain Generalization. For Single Source Domain Generalization case, we
report the model performances generated under our experiment setting.
| Leave-One-Out Source Domain Generalization | Single Source Domain Generalization
---|---|---
Method | Art | Clipart | Product | Real World | Avg | Art | Clipart | Product | Real World | Avg
ERM | 61.4$\pm$1.0 | 53.5$\pm$0.2 | 75.9$\pm$0.2 | 77.1$\pm$0.2 | 67.0 | 55.6$\pm$0.6 | 52.8$\pm$1.6 | 50.3$\pm$1.1 | 59.4$\pm$0.3 | 54.5
IRM∗ | 61.8∗$\pm$1.0 | 52.3∗$\pm$1.0 | 75.2∗$\pm$0.8 | 77.2∗$\pm$1.1 | 66.6∗ | 54.9$\pm$0.3 | 53.2$\pm$0.6 | 48.6$\pm$0.7 | 59.2$\pm$0.1 | 54.0
GroupDRO | 61.3$\pm$2.0 | 53.3$\pm$0.4 | 75.4$\pm$0.3 | 76.0$\pm$1.0 | 66.5 | 55.1$\pm$0.2 | 52.0$\pm$0.5 | 50.3$\pm$1.1 | 59.3$\pm$0.3 | 54.2
OrgMixup | 63.7$\pm$1.1 | 55.4$\pm$0.1 | 77.1$\pm$0.2 | 78.9$\pm$0.6 | 68.8 | 56.0$\pm$1.1 | 54.4$\pm$1.3 | 50.4$\pm$0.4 | 61.0$\pm$0.7 | 55.5
Mixup | 64.1$\pm$0.6 | 54.9$\pm$0.7 | 76.6$\pm$0.4 | 78.7$\pm$0.5 | 68.6 | 55.5$\pm$0.6 | 54.1$\pm$1.1 | 49.4$\pm$1.7 | 59.4$\pm$0.6 | 54.6
CutMix | 63.2$\pm$0.3 | 52.1$\pm$1.7 | 77.2$\pm$0.8 | 78.1$\pm$0.6 | 67.7 | 53.5$\pm$0.8 | 52.2$\pm$1.2 | 47.7$\pm$1.7 | 60.2$\pm$0.4 | 53.4
Mixstyle∗ | 51.1∗$\pm$0.3 | 53.2∗$\pm$0.4 | 68.2∗$\pm$0.7 | 69.2∗$\pm$0.6 | 60.4 | 44.3$\pm$0.5 | 29.8$\pm$1.2 | 33.6$\pm$0.5 | 48.5$\pm$0.9 | 39.0
MTL | 60.1$\pm$0.5 | 52.0$\pm$0.3 | 75.7$\pm$0.3 | 77.2$\pm$0.4 | 66.3 | 55.3$\pm$0.3 | 53.3$\pm$0.4 | 49.0$\pm$0.3 | 60.4$\pm$0.1 | 54.5
MLDG∗ | 63.7∗$\pm$0.3 | 54.5∗$\pm$0.6 | 75.9∗$\pm$0.4 | 78.6∗$\pm$0.1 | 68.2∗ | - | - | - | - | -
MMD∗ | 63.0∗$\pm$0.1 | 53.7∗$\pm$0.9 | 76.1∗$\pm$0.3 | 78.1∗$\pm$0.5 | 67.7∗ | 55.1$\pm$0.2 | 52.0$\pm$0.5 | 50.3$\pm$1.1 | 59.3$\pm$0.3 | 54.2
CORAL | 64.1$\pm$0.5 | 54.5$\pm$1.7 | 76.2$\pm$0.4 | 77.8$\pm$0.5 | 68.2 | 55.6$\pm$0.6 | 52.8$\pm$1.6 | 50.3$\pm$1.1 | 59.4$\pm$0.3 | 54.5
SagNet | 62.3$\pm$1.1 | 51.7$\pm$0.3 | 75.4$\pm$1.0 | 78.1$\pm$0.2 | 66.9 | 56.9$\pm$1.2 | 53.4$\pm$2.1 | 50.8$\pm$0.3 | 61.2$\pm$0.8 | 55.6
ARM | 59.9$\pm$0.6 | 51.8$\pm$0.5 | 73.3$\pm$0.5 | 75.7$\pm$0.8 | 65.2 | 55.0$\pm$0.1 | 51.6$\pm$1.1 | 47.3$\pm$0.8 | 59.3$\pm$0.7 | 53.3
DANN∗ | 59.9∗$\pm$1.3 | 53.0∗$\pm$0.3 | 73.6∗$\pm$0.7 | 76.9∗$\pm$0.5 | 65.9∗ | 55.2$\pm$0.8 | 49.3$\pm$1.5 | 48.4$\pm$1.5 | 58.4$\pm$0.2 | 52.8
CDANN∗ | 61.5∗$\pm$1.4 | 50.4∗$\pm$2.4 | 74.4∗$\pm$0.9 | 76.6∗$\pm$0.8 | 65.7∗ | 55.2$\pm$0.7 | 49.9$\pm$1.4 | 47.6$\pm$1.3 | 58.6$\pm$0.5 | 52.8
VREx | 61.4$\pm$1.0 | 52.2$\pm$0.2 | 76.1$\pm$0.6 | 77.6$\pm$0.9 | 66.8 | 55.5$\pm$0.6 | 52.6$\pm$0.2 | 49.1$\pm$1.0 | 59.3$\pm$0.6 | 54.1
RSC | - | - | - | - | - | - | - | - | - | -
Fishr${}^{{}^{\prime}}$ | 62.4$\pm$0.5 | 54.4$\pm$0.4 | 76.2$\pm$0.5 | 78.3$\pm$0.1 | 67.8 | 55.1$\pm$0.4 | 51.2$\pm$0.1 | 49.2$\pm$1.0 | 59.9$\pm$1.4 | 53.9
RIDG | 63.6$\pm$0.7 | 55.0$\pm$0.9 | 76.0$\pm$0.8 | 77.5$\pm$0.7 | 68.0 | 56.8$\pm$0.5 | 55.4$\pm$0.7 | 50.5$\pm$0.3 | 60.9$\pm$0.1 | 55.9
ITTA | 61.8$\pm$0.9 | 57.0$\pm$1.0 | 74.3$\pm$0.3 | 77.3$\pm$0.3 | 67.6 | 56.0$\pm$0.4 | 51.5$\pm$0.8 | 50.5$\pm$0.6 | 61.6$\pm$0.4 | 54.9
SAM | 62.2$\pm$0.7 | 55.9$\pm$0.1 | 77.0$\pm$0.9 | 78.8$\pm$0.6 | 68.5 | 56.9$\pm$0.4 | 53.8$\pm$1.1 | 50.9$\pm$0.7 | 61.5$\pm$0.8 | 55.8
UDIM (w/ SAM) | 63.5$\pm$1.3 | 58.6$\pm$0.4 | 76.9$\pm$0.6 | 79.1$\pm$0.3 | 69.5 | 58.1$\pm$0.6 | 55.0$\pm$0.9 | 53.8$\pm$0.1 | 64.3$\pm$0.2 | 57.8
SAGM | 63.1$\pm$2.1 | 56.2$\pm$0.4 | 77.3$\pm$0.2 | 78.4$\pm$0.4 | 68.8 | 57.7$\pm$0.3 | 54.8$\pm$1.0 | 51.5$\pm$1.2 | 61.4$\pm$0.1 | 56.3
UDIM (w/ SAGM) | 64.4$\pm$0.3 | 57.3$\pm$0.5 | 77.1$\pm$0.4 | 79.1$\pm$0.3 | 69.5 | 58.5$\pm$0.4 | 55.7$\pm$0.6 | 54.5$\pm$0.1 | 64.5$\pm$0.4 | 58.3
GAM | 64.0$\pm$0.5 | 58.6$\pm$1.2 | 77.5$\pm$0.1 | 79.3$\pm$0.2 | 69.8 | 59.4$\pm$0.6 | 56.1$\pm$0.9 | 53.3$\pm$0.6 | 63.4$\pm$0.2 | 58.1
UDIM (w/ GAM) | 64.2$\pm$0.3 | 57.4$\pm$0.9 | 77.5$\pm$0.1 | 79.3$\pm$0.3 | 69.6 | 58.7$\pm$0.3 | 55.7$\pm$0.0 | 53.6$\pm$0.4 | 64.4$\pm$0.0 | 58.1
Table 5 and 7 shows the model performances for OfficeHome (Venkateswara et
al., 2017) and DomainNet (Peng et al., 2019) dataset, respectively. Similar to
the above tables, UDIM shows performance improvement over sharpness-aware
baselines, and good performances for each column consistently.
Table 6: Results on DomainNet datasets - Leave-One-Out (Multi) Source Domain
Generalization
Method | clipart | infograph | painting | quickdraw | real | sketch | Avg
---|---|---|---|---|---|---|---
ERM | 62.8$\pm$0.4 | 20.2$\pm$0.3 | 50.3$\pm$0.3 | 13.7$\pm$0.5 | 63.7$\pm$0.2 | 52.1$\pm$0.5 | 43.8
IRM | 48.5$\pm$2.8 | 15.0$\pm$1.5 | 38.3$\pm$4.3 | 10.9$\pm$0.5 | 48.2$\pm$5.2 | 42.3$\pm$3.1 | 33.9
GroupDRO | 47.2$\pm$0.5 | 17.5$\pm$0.4 | 33.8$\pm$0.5 | 9.3$\pm$0.3 | 51.6$\pm$0.4 | 40.1$\pm$0.6 | 33.3
MTL | 57.9$\pm$0.5 | 18.5$\pm$0.4 | 46.0$\pm$0.1 | 12.5$\pm$0.1 | 59.5$\pm$0.3 | 49.2$\pm$0.1 | 40.6
MLDG | 59.1$\pm$0.2 | 19.1$\pm$0.3 | 45.8$\pm$0.7 | 13.4$\pm$0.3 | 59.6$\pm$0.2 | 50.2$\pm$0.4 | 41.2
MMD | 32.1$\pm$13.3 | 11.0$\pm$4.6 | 26.8$\pm$11.3 | 8.7$\pm$2.1 | 32.7$\pm$13.8 | 28.9$\pm$11.9 | 23.4
CORAL | 59.2$\pm$0.1 | 19.7$\pm$0.2 | 46.6$\pm$0.3 | 13.4$\pm$0.4 | 59.8$\pm$0.2 | 50.1$\pm$0.6 | 41.5
SagNet | 57.7$\pm$0.3 | 19.0$\pm$0.2 | 45.3$\pm$0.3 | 12.7$\pm$0.5 | 58.1$\pm$0.5 | 48.8$\pm$0.2 | 40.3
ARM | 49.7$\pm$0.3 | 16.3$\pm$0.5 | 40.9$\pm$1.1 | 9.4$\pm$0.1 | 53.4$\pm$0.4 | 43.5$\pm$0.4 | 35.5
DANN∗ | 53.8∗$\pm 0.7$ | 17.8∗$\pm 0.3$ | 43.5∗$\pm 0.3$ | 11.9∗$\pm 0.5$ | 56.4∗$\pm 0.3$ | 46.7∗$\pm 0.5$ | 38.4∗
CDANN∗ | 53.4∗$\pm 0.4$ | 18.3∗$\pm 0.7$ | 44.8∗$\pm 0.3$ | 12.9∗$\pm 0.2$ | 57.5∗$\pm 0.4$ | 46.7∗$\pm 0.2$ | 38.9∗
VREx | 47.3$\pm$3.5 | 16.0$\pm$1.5 | 35.8$\pm$4.6 | 10.9$\pm$0.3 | 49.6$\pm$4.9 | 42.0$\pm$3.0 | 33.6
RSC | 55.0$\pm$1.2 | 18.3$\pm$0.5 | 44.4$\pm$0.6 | 12.2$\pm$0.2 | 55.7$\pm$0.7 | 47.8$\pm$0.9 | 38.9
Fishr∗ | 58.2∗$\pm$0.5 | 20.2∗$\pm$0.2 | 47.7∗$\pm$0.3 | 12.7∗$\pm$0.2 | 60.3∗$\pm$0.2 | 50.8∗$\pm$0.1 | 41.7∗
SAM | 64.5$\pm$0.3 | 20.7$\pm$0.2 | 50.2$\pm$0.1 | 15.1$\pm$0.3 | 62.6$\pm$0.2 | 52.7$\pm$0.3 | 44.3
UDIM (w/ SAM) | 63.5$\pm$0.1 | 21.01$\pm$0.1 | 50.63$\pm$0.1 | 14.76$\pm$0.1 | 62.5$\pm$0.1 | 53.39$\pm$0.1 | 44.3
Table 7: Results on DomainNet datasets - Single Source Domain Generalization
Method | clipart | infograph | painting | quickdraw | real | sketch | Avg
---|---|---|---|---|---|---|---
ERM | 27.5$\pm$0.5 | 26.6$\pm$0.1 | 28.5$\pm$0.3 | 7.1$\pm$0.1 | 28.9$\pm$0.4 | 29.4$\pm$0.3 | 24.7
IRM | - | - | - | - | - | - | -
GroupDRO | 27.6$\pm$0.4 | 26.6$\pm$0.8 | 28.9$\pm$0.3 | 7.3$\pm$0.3 | 28.7$\pm$0.2 | 29.4$\pm$0.7 | 24.8
OrgMixup | 28.3$\pm$0.1 | 27.3$\pm$0.1 | 29.4$\pm$0.3 | 7.7$\pm$0.4 | 30.2$\pm$0.2 | 30.0$\pm$0.7 | 25.5
Mixup | 27.5$\pm$0.3 | 26.9$\pm$0.8 | 28.6$\pm$0.4 | 7.0$\pm$0.1 | 28.7$\pm$0.2 | 29.6$\pm$0.6 | 24.7
CutMix | 28.0$\pm$0.2 | 26.5$\pm$0.2 | 28.4$\pm$0.6 | 6.4$\pm$0.1 | 29.0$\pm$0.3 | 29.7$\pm$0.8 | 24.6
Mixstyle | 18.3$\pm$0.8 | 14.5$\pm$0.3 | 19.6$\pm$0.2 | 5.0$\pm$0.1 | 20.5$\pm$1.3 | 21.3$\pm$0.8 | 16.5
MTL | 27.3$\pm$0.2 | 26.6$\pm$0.3 | 28.7$\pm$0.1 | 7.8$\pm$0.1 | 28.2$\pm$0.2 | 28.7$\pm$0.6 | 24.5
MLDG | - | - | - | - | - | - | -
MMD | 27.6$\pm$0.4 | 26.6$\pm$0.7 | 28.5$\pm$0.3 | 7.4$\pm$0.3 | 28.9$\pm$0.3 | 29.5$\pm$0.9 | 24.8
CORAL | 18.1$\pm$12.8 | 17.4$\pm$12.3 | 18.9$\pm$13.4 | 4.8$\pm$3.4 | 19.5$\pm$13.8 | 20.2$\pm$14.3 | 16.5
SagNet | 27.6$\pm$0.3 | 25.6$\pm$0.5 | 28.5$\pm$0.7 | 7.3$\pm$0.2 | 28.8$\pm$0.5 | 28.8$\pm$0.8 | 24.4
ARM | 17.0$\pm$12.0 | 16.8$\pm$11.9 | 17.7$\pm$12.5 | 4.1$\pm$2.9 | 17.5$\pm$12.4 | 18.8$\pm$13.3 | 15.3
DANN | 26.8$\pm$0.8 | 25.2$\pm$0.9 | 28.0$\pm$0.4 | 6.8$\pm$0.6 | 27.6$\pm$0.2 | 28.0$\pm$0.2 | 23.8
CDANN | 26.8$\pm$0.7 | 25.8$\pm$0.2 | 27.6$\pm$0.2 | 6.8$\pm$0.6 | 27.6$\pm$0.2 | 28.0$\pm$0.2 | 23.8
VREx | 18.6$\pm$13.1 | 17.3$\pm$12.3 | 19.3$\pm$13.6 | 4.7$\pm$3.3 | 19.3$\pm$13.7 | 19.9$\pm$14.1 | 16.5
RSC | - | - | - | - | - | - | -
Fishr | 30.0$\pm$0.3 | 26.6$\pm$0.7 | 28.9$\pm$0.3 | 7.5$\pm$0.7 | 28.4$\pm$0.7 | 28.9$\pm$0.3 | 24.7
SAM | 28.4$\pm$0.2 | 26.9$\pm$0.1 | 29.1$\pm$0.4 | 6.9$\pm$0.5 | 30.0$\pm$0.2 | 29.8$\pm$0.7 | 25.2
UDIM (w/ SAM) | 30.0$\pm$0.1 | 23.8$\pm$0.4 | 31.0$\pm$0.1 | 12.6$\pm$0.1 | 30.7$\pm$0.2 | 34.0$\pm$0.3 | 27.0
### D.5 Additional Analyses
In this section, we additionally provide analysis on 1) Comparison with Shui
et al. (2022), which utilzes both 1) data-based perturbation and 2) parameter-
based regularization on their own framework. Afterwards, we provide visual
inspections on the perturbed domain instances, which are constructed by Eq 11.
#### D.5.1 Analytical Comparison with Shui et al. (2022)
UDIM and Shui et al. (2022) share a similar direction, as both methodologies
involve 1) generating virtual samples through distinct data perturbation
methods, and 2) implementing their own types of regularization on these
samples.
Shui et al. (2022) introduced novel regularization techniques for the
embedding function based on theoretical analysis. Their approach involves
establishing an upper bound on the balanced error rate in the test
environment, which is obtained from the combination of the balanced error
rates in the source environments, the feature-conditional invariance, and the
smoothness of the embedding function. In particular, they focused on
minimizing the smoothness term by reducing the Frobenius norm of the Jacobian
matrix with respect to the embedding function. To implement this
regularization term over unobserved regions, they use virtual samples
generated by a linear combination of samples from each source.
On the other hand, UDIM also introduces an upper bound on the generalization
loss in an unknown domain. This bound includes the SAM loss on the source
domain and a region-based loss disparity between the worst-case domain and the
source domain. This disparity is implemented by a gradient variance-based
objective outlined in Eq. 10. The objective involves a perturbed dataset
constructed by perturbing samples, where the direction of perturbation is
determined by the inconsistencies described in Eq. 7.
#### D.5.2 Analytic comparison with domain augmentation and adversarial
attack
The proposed method, UDIM, involves applying arbitrary perturbation to a given
instance to create a new instance, which is similar to domain augmentation and
adversarial attacks. However, UDIM has the following differences and
advantages compared to them.
##### Comparison with domain augmentation
First, we taxonomize domain augmentation based on its dependency on learning
signals such as parameters or loss, dividing it into not-learnable domain
augmentation Zhang et al. (2018); Zhou et al. (2021); Li et al. (2021b) and
learnable domain augmentation Zhang et al. (2018); Zhou et al. (2021); Li et
al. (2021b). Please note that we briefly cite the representative methodologies
among wide range of augmentation techniques.
While non-learnable domain augmentations are effective in generating new
styles and may generalize well to specific types of domains, it does not
guarantee generalization across wide range of unseen domains, as discussed in
the Theorem 3.2. In contrast, UDIM’s data perturbation method is designed to
generate perturbations towards the most vulnerable or worst domain from a
parameter space perspective, enabling a reduction of the generalization bound
in Eq 5, even in scenarios involving numerous unobserved domains.
Additionally, it’s important to note that these domain augmentation techniques
could be applied orthogonally to the UDIM framework, for instance, by
implementing domain augmentation prior to UDIM’s data perturbation.
Learnable augmentations, similar to UDIM, determine its augmentation direction
based on the current parameter response. However, these methodologies do not
link their augmentation with a theoretical analysis to assure minimization of
the target objective, which is left-hand side of Eq 5 of our manuscript.
UDIM’s data perturbation impacts the generalization bound from a parameter
perspective as it takes into account a parameter loss curvature information,
rather than just a single parameter point, when determining perturbations.
##### Comparison with adversarial attack
Adversarial attacks also introduce perturbations in the direction most
vulnerable to the current parameters, but methodologies like FGSM Goodfellow
et al. (2014) and PGD Madry et al. (2017) do not consider the local parameter
curvature in their perturbation process. By integrating perturbations on
instances with attention to parameter loss curvature; and parameter
perturbation, we facilitate the modeling of inconsistency in unknown domains,
as described in Eq 3. Having said that, Kim et al. (2023) also utilize worst-
case instance selection on the active learning framework by utilizing the
parameter perturbation. In the literature of coreset selection Feldman (2020),
Shin et al. (2023) also utilizes the perturbed parameter region to attain
samples which effectively represent whole dataset.
From a mathematical perspective, UDIM’s data perturbation involves receiving
not only gradients related to the simple cross-entropy loss but also
additional gradients concerning the norm of gradient, as elaborated in Eq 7.
##### Combination of domain augmentation with SAM optimizer
We accordingly report the experimental results of models, which combines
various domain augmentation techniques with SAM optimization in Table 8. We
reported the average test accuracy for each domain in each setting. Applying
SAM optimization to data instances of augmented domains led to mixed results:
some methodologies improved, others didn’t, but all still under-performed
compared to UDIM. We hypothesize that the observed performance decline in
certain augmentations combined with the SAM optimizer might stem from an
unstable learning process. This instability may arise from attempting to
minimize sharpness in the perturbed domain prematurely, before ensuring
flatness in the source domain.
Method | Leave-One-Out Source Domain Generalization | Single Source Domain Generalization
---|---|---
SAM | 85.9% | 64.8%
SAM w/ Mixup | 84.51% | 62.28%
SAM w/ Mixstyle | 86.56% | 68.59%
SAM w/ Simple Augment Li et al. (2021b) | 86.26% | 64.97%
SAM w/ advstyle Zhong et al. (2022) | 85.28% | 61.02%
UDIM | 88.7% | 74.7%
Table 8: Performance comparison of models using combinations of domain
augmentation variants and SAM optimizer, alongside UDIM model, on the PACS
dataset.
#### D.5.3 Visual Inspection on the Perturbed Domain
In this section, we aim to illustrate how perturbed instances are created
depending on the size of $\rho$, which represents the magnitude of data
perturbation. Each plot utilizes the PACS dataset and a model trained under
the Single Source Domain Generalization setting.
Figure 6 displays instances perturbed through the original UDIM methodology.
As the perturbation size increases, we can observe a gradual distortion of the
image’s semantics, highlighting the importance of selecting an appropriate
perturbation size.
Figure 7 shows the scenario where the data perturbation of the original UDIM
method is applied to the input channel, instead of input pixels. This approach
of perturbing the channel over pixels has the advantage of maintaining the
basic shape and line information of the image, ensuring the preservation of
essential visual features.
Figure 6: Pixel-perturbed domain instances by UDIM model trained on PACS
dataset by varying the perturbation size Figure 7: Channel-perturbed domain
instances by UDIM model trained on PACS dataset by varying the perturbation
size
|
# Towards End-to-End Synthetic Speech Detection
Guang Hua, _Member, IEEE_ , Andrew Beng Jin Teoh, , and Haijian Zhang This
work was supported by the 2020–2021 International Scholar Exchange Fellowship
(ISEF) Program at the Chey Institute for Advanced Studies, South Korea.
_(Corresponding Author: Andrew Beng Jin Teoh)_ G. Hua and H. Zhang are with
the School of Electronic Information, Wuhan University, Wuhan 430072, China
(e-mail<EMAIL_ADDRESS>[email protected]).A. B. J. Teoh is with the
School of Electrical and Electronic Engineering, College of Engineering,
Yonsei University, Seoul 120749, South Korea (e-mail: [email protected]).
###### Abstract
The constant Q transform (CQT) has been shown to be one of the most effective
speech signal pre-transforms to facilitate synthetic speech detection,
followed by either hand-crafted (subband) constant Q cepstral coefficient
(CQCC) feature extraction and a back-end binary classifier, or a deep neural
network (DNN) directly for further feature extraction and classification.
Despite the rich literature on such a pipeline, we show in this paper that the
pre-transform and hand-crafted features could simply be replaced by end-to-end
DNNs. Specifically, we experimentally verify that by only using standard
components, a light-weight neural network could outperform the state-of-the-
art methods for the ASVspoof2019 challenge. The proposed model is termed Time-
domain Synthetic Speech Detection Net (TSSDNet), having ResNet- or Inception-
style structures. We further demonstrate that the proposed models also have
attractive generalization capability. Trained on ASVspoof2019, they could
achieve promising detection performance when tested on disjoint ASVspoof2015,
significantly better than the existing cross-dataset results. This paper
reveals the great potential of end-to-end DNNs for synthetic speech detection,
without hand-crafted features.
###### Index Terms:
Synthetic speech detection, speech forensics, ASVspoof2019, ASVspoof2015,
cross-dataset testing, end-to-end.
## I Introduction
The success of deep learning technology has shifted the paradigm of speech
synthesis from the classic hidden Markov model based framework [1] to neural
speech synthesis. Equipped with powerful deep neural network (DNN)
architectures e.g., [2], and fueled by massive training data, today’s text-to-
speech (TTS) systems could synthesize high quality speech that is hard to be
distinguished from human voices. Despite the multitude of benefits, these
advances have also improved the quality of voice spoofing attacks, including
voice conversion [3], impersonation [4], cloning [5], etc., posing new
challenges to synthetic speech detection.
For nearly a decade, the combination of a front-end feature extractor and a
back-end binary classifier is the _de facto_ framework for synthetic speech
detection. Within this framework, an overwhelming majority of the existing
works have focused on the development of hand-crafted front-end features,
including fundamental frequency, power spectrum, octave spectrum, linear
frequency cepstral coefficient (LFCC), mel-frequency cepstral coefficient
(MFCC), cepstral mean and variance (CMVN), cochlear filter cepstral
coefficient (CFCC), filter bank based cepstral coefficient, linear prediction
cepstral coefficient (LPCC), modified group delay (MGD), relative phase shift
(RPS), constant Q cepstral coefficient (CQCC), and many of their variations
and combinations [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Usually, one or a few
of these features are used to train a Gaussian mixture model (GMM) or a
support vector machine (SVM) for classification. Taking the advantage of DNNs
in classification tasks, multilayer perceptron (MLP) and convolutional neural
network (CNN) based classifiers have been used to replace the conventional
back-end classifiers [16, 17, 18, 15, 19, 20, 21, 22]. On the other side, DNN
structures have also been used at the front-end to facilitate feature
extraction [23, 22, 24, 25], followed by conventional classifiers. DNNs can
also work across the front- and back-end, with pre-transformed features as
input [26, 27, 28, 29, 30].
Figure 1: Relationship between the existing front-end$\rightarrow$back-end
pipeline and the proposed end-to-end framework for synthetic speech detection.
Among hand-crafted features, CQCC has been found to be the best choice, which
is also the baseline feature in the ASVspoof2019 challenge [31]. Recently,
Yang _et al._ developed a set of subband CQCC features for better detection
performance [19]. Subsequently, Das _et al._[21] further fused $8$ hand-
crafted features, followed by an MLP classifier. For deep learning based
approach, Lavrentyeva _et al._ [29] proposed the use of FFT, LFCC, and CMVN,
followed by a CNN for classification, while using CQT as model input, Li _et
al._ [30] incorporated the so called Res2Net structure and squeeze-and-
excitation (SE) block. With score level fusion, Lavrentyeva _et al._ [29] and
Li _et al._ [30] have achieved the state-of-the-art performance on
ASVspoof2019 dataset.
Based on the above overview, the existing mainstream workflow for synthetic
speech detection is summarized in the brown blocks of Fig. 1. It can be seen
that a time-frequency transform (e.g., CQT) of the speech waveform before
hand-crafted feature extraction (e.g., CQCC), or before feeding the data into
a DNN, has become an implicit standard routine in the existing works. However,
since DNNs are best known for their excellent capability of feature
extraction, there naturally arises a question of whether it is necessary to
apply these pre-transforms. In fact, these transforms usually discard some
information about the observed speech signal. For example, the CQT feature,
more precisely the log power spectrum of the CQT [30], does not have the phase
information of the signal. To further generate the CQCC, even more information
will be discarded [19]. From hand-crafted feature engineering point of view, a
good feature captures discriminative information between classes and is also
compact in size, but the same principle may not apply to the DNN regime.
In this paper, we show that the pre-transforms, as well as the hand-crafted
features, are in fact not a must for DNN based synthetic speech detection.
Despite the rich hand-crafted features, we experimentally verify that via the
use of standard DNN structures, an end-to-end light-weight neural network with
mere speech waveform could achieve even better results. Our proposal is
motivated by recent works analyzing raw-waveform based DNNs [32] and the
attempt of applying end-to-end DNNs to related speech processing tasks, e.g.,
speech separation [33]. The proposed model is thus termed as Time-domain
Synthetic Speech Detection Net (TSSDNet). We note that the first work on end-
to-end synthetic speech detection was probably carried out by Muckenhirn _et
al._ [34], in which a basic feedforward sequential CNN was used. It was tested
on older datasets, not achieving the state-of-the-art results. In our design
of the TSSDNet, two types of advanced CNN structures are considered, including
ResNet-style skip connection with $1\times 1$ kernels [35] and Inception-style
parallel convolutions [36], respectively. We demonstrate that via proper
training, the proposed networks outperform the state-of-the-art hand-crafted
feature based detectors as well as DNN based ones on the challenging
ASVspoof2019 dataset [31]. To analyze practical merits of the proposed
methods, we further perform a cross-dataset evaluation between ASVspoof2019
and ASVspoof2015[37]111The ASVspoof2017 dataset is not considered in this
paper because it only has replay attack. Although replay attack is seen to be
considered together with synthesis attack, the underlying mechanism is very
different. The physical access portion of ASVspoof2019 is also excluded for
the same reason., demonstrating their promising generalization capability.
(a) ResNet style, Res-TSSDNet.
(b) Inception style, Inc-TSSDNet.
Figure 2: Structures of the proposed models, where all the conv layers apply
“SAME” padding, and in all local max pooling layers, stride$=$kernel size.
$M$: number of stacked ResNet- and Inception-Style modules. $C_{\text{R}}$:
number of channels in Res-TSSDNet. $C_{\text{I}}$: number of channels in Inc-
TSSDNet.
## II The Proposed Models
In many deep learning tasks such as object recognition or semantic
understanding, it has been found that generally the deeper the network, the
better the performance [35, 36, 38]. However, in synthetic speech detection,
the critical feature is the artifact left behind data forgery, which may not
contain any semantic information. Since deeper features are more towards
higher level semantic information, which may not be suitable to represent the
subtle forgery artifacts, we hypothesize that the network for synthetic speech
detection should be relatively shallower. Grabbing the essence of the popular
ResNet [35] and Inception network [36], the proposed end-to-end TSSDNets are
designed as follows.
### II-A Model Structure
The proposed Res-TSSDNet and Inc-TSSDNet are depicted in Fig. 2 (a) and (b),
respectively, which share the same first layer, $3$ final fully-connected
linear layers, and global max pooling before the linear layers. The ResNet-
style and Inception-style blocks are repeated for $M$ times, respectively, and
batch normalization (BN) is applied in both networks. $C_{\text{R}}$ and
$C_{\text{I}}$ denote the number of channels in the corresponding modules,
which may vary across layers. Noticeably, to increase the receptive field and
control model complexity, dilated convolutions [39] with dilation $d$ are
incorporated in the Inc-TSSDNet, which is different from the original
Inception network [36]. All the convolution layers apply “SAME” padding with
$\text{stride}=1$, while for the pooling layers the stride equals to the
corresponding kernel size.
### II-B Training Strategy
#### II-B1 Data Preparation
Normally, the training data contain raw speech recordings with varied
durations. To align the training data, we adopt the treatment in [30]. In
[30], the training examples are truncated or repeated until the duration is
$6.4$ seconds to generate the CQT feature, while here we keep every example
with $6$ seconds, with the default $16$ kHz sample rate. These $6$-second
examples are then directly fed into the networks for end-to-end training.
Since all convolution layers have “SAME” padding, the length of feature vector
($9.6\times 10^{4}$ at input) is reduced solely by the pooling layers. Note
that hand-crafted feature based method, e.g., CQCC [19], is insensitive to the
length of recording since all time slices contribute to classifier training.
Batch size is set to $32$. .
#### II-B2 Weighted Cross-Entropy Loss
Considering the fact that in general data-driven media content forgery
detection tasks, the number of genuine examples is usually much less than the
number of fake ones, we apply weighted cross-entropy (WCE) loss during the
training phase to cope with data imbalance. Let $\\{x_{i},y_{i}\\}$ compose
the labeled training set, where $\forall i$, label $y_{i}\in\\{0,1\\}$, then
the WCE loss is given by
${\mathop{\rm
WCE}\nolimits}\left({{\bf{z}},{y_{i}}}\right)=-{w_{{y_{i}}}}\log\left({{z_{{y_{i}}}}}\right),$
(1)
where $\mathbf{z}=[z_{0},z_{1}]$ contains the softmax probabilities of the $2$
classes, and $w_{y_{i}}$ is the inverse ratio of label $y_{i}$ in the training
set. For all the training processes, we use the Adam [40] optimizer and
default settings. Exponential learning rate decay with a multiplicative factor
of $0.95$ is adopted. The model yielding the lowest equal error rate (EER) on
development set within $100$ epochs is selected for evaluation.
#### II-B3 Mixup Regularization
For practical forensic merits, the trained model is expected to generalize to
unseen attacks, and the ASVspoof datasets have been specially designed for
this purpose. In this paper, we consider the mixup regularization [41] as a
booster to further improve the generalization capability. Specifically, it
uses a set of mixed examples and labels, instead of the original set, to train
the network, i.e.,
$\tilde{x}_{i}=\lambda x_{i}+(1-\lambda)x_{j},\quad\tilde{y}_{i}=\lambda
y_{i}+(1-\lambda)y_{j},$ (2)
where $\\{x_{i},y_{i}\\}$ and $\\{x_{j},y_{j}\\}$ are two randomly selected
training pairs, $\lambda\sim\mathop{\rm Beta}(\alpha,\alpha)$, and
$\alpha\in(0,\infty)$ is a hyperparameter. The implementation of the mixup
regularization could be carried out via the following equivalent loss
function,
${{\mathop{\rm
CE}}_{{\rm{mixup}}}}\left({\tilde{\bf{z}},{y_{i}},{{y}_{j}}}\right)=\lambda{\mathop{\rm
CE}}(\tilde{\bf{z}},{y_{i}})+(1-\lambda){\mathop{\rm
CE}}(\tilde{\bf{z}},{{y}_{j}}),$ (3)
where $\tilde{\bf{z}}$ contains the softmax probabilities from mixed examples,
and ${\mathop{\rm CE}}(\cdot,\cdot)$ is the standard cross-entropy (CE) loss,
equivalent to setting $w_{0}=w_{1}$ in (1).
## III Results
We first present the main results obtained by the proposed networks in
comparison with the benchmark and the state-of-the-art solutions on the latest
ASVspoof2019 dataset. We then perform ablation study, followed by cross-
dataset evaluation on the ASVspoof2015 dataset. All the results are generated
using a single GeForce GTX 1080 or 1080Ti GPU. PyTorch implementations of the
proposed TSSDNets are available at: _https://github.com/ghuawhu/end-to-end-
synthetic-speech-detection_.
TABLE I: EER (%) of the proposed and state-of-the-art methods on ASVspoof2019 LA dev and eval sets, $M=4$, $C_{\text{L}}=\\{64,32\\}$, $C_{\text{R}}=\\{32,64,128,128\\}$, $C_{\text{I}}=\\{8,16,32,32\\}$. Method | #Param | Dev | Eval
---|---|---|---
Baseline LFCC+GMM [42] | - | $0.43$ | $9.57$
Baseline CQCC+GMM [42] | - | $2.71$ | $8.09$
Subband CQCC+MLP [19] | - | - | $8.04$
$8$ Features+MLP[21] | - | $0.00$ | $4.13$
Spec+VGG+SincNet [28] | $>4.32$M | $0.00$ | $8.01$
Spec+CQCC+ResNet+SE [27] | $5.80$M | $0.00$ | $6.70$
FFT+CNN [29] | $10.2$M | $0.04$ | $4.53$
$3$ Features+CNN [29] | $30.6$M | $0.00$ | $1.86$
CQT+Res2Net+SE [30] | $0.92$M | $0.43$ | $2.50$
$3$ Features+Res2Net+SE [30] | $2.76$M | $0.00$ | $1.89$
CQT+2D-Res-TSSDNet | $0.97$M | $0.59$ | $5.89$
End-to-End Res-TSSDNet | $0.35$M | $0.74$ | $\mathbf{1.64}$
End-to-End Inc-TSSDNet | $\mathbf{0.09}$M | $1.09$ | $4.04$
### III-A Main Results
The comparison of the results in terms of EER obtained on the logical access
(LA) development and evaluation sets of the ASVspoof2019 challenge is
presented in Table I, where the 2D-Res-TSSDNet is the 2D version of the Res-
TSSDNet, having the same architecture except that all the convolution and
pooling ($2\times 2$ pooling) layers use 2D kernels instead.
TABLE II: Ablation study of Res-TSSDNet and Inc-TSSDNet, using ASVspoof2019 LA
eval EER (%), $C_{\text{L}}=\\{64,32\\}$.
| $M$ | $C_{\text{R}}$ | $1\times 1$ | #Param | Eval
---|---|---|---|---|---
Res-TSSDNet | $3$ | $\\{32,64,128\\}$ | Yes | $0.18$M | $11.37$
$4$ | $\\{32,64,128,128\\}$ | No | $0.32$M | $2.69$
$4$ | $\\{32,64,128,128\\}$ | Yes | $0.35$M | $\mathbf{1.64}$
$5$ | $\\{32,64,128,128,128\\}$ | No | $0.47$M | $5.14$
$5$ | $\\{32,64,128,128,128\\}$ | Yes | $0.51$M | $4.58$
| $M$ | $C_{\text{I}}$ | Dilation $d$ | #Param | Eval
---|---|---|---|---|---
Inc-TSSDNet | $3$ | $\\{8,16,32\\}$ | $\\{2^{0},\ldots,2^{3}\\}$ | $0.04$M | $10.39$
$4$ | $\\{8,16,32,32\\}$ | $\\{2^{0},\ldots,2^{3}\\}$ | $0.09$M | $4.04$
$5$ | $\\{8,16,32,64,64\\}$ | $\\{2^{0},\ldots,2^{3}\\}$ | $0.35$M | $5.31$
$4$ | $\\{8,16,32,32\\}$ | $\\{2^{0},\ldots,2^{7}\\}$ | $0.34$M | $\mathbf{3.75}$
$5$ | $\\{8,16,32,64,64\\}$ | $\\{2^{0},\ldots,2^{7}\\}$ | $1.34$M | $4.20$
We make the following remarks from the main results. i) The works of [19] and
[21] represent the best results of sophisticated hand-crafted feature
engineering plus an MLP as the back-end classifier. ii) The majority of recent
works belong to the type of pre-transform (or light feature engineering) plus
DNNs to further perform feature extraction and classification. iii) All the
works incorporating DNNs rely on the feature and model fusion for performance
improvement, and in [29] and [30], the fused results have achieved EERs below
$2\%$. iv) The 2D-Res-TSSDNet result is obtained with experimental settings
identical to [30] without fusion, and it can be seen that when working with 2D
pre-transform input, the use of advanced DNN components, i.e., Res2Net and SE,
becomes very necessary. v) Most importantly, the proposed Res-TSSDNet is a
single end-to-end network (no fusion, no feature engineering), containing less
than a half of trainable weights than the one in [30] and only about one-tenth
than in [29], but it achieves the overall lowest evaluation EER by a clear
margin. vi) Lastly, the Inc-TSSDNet is extremely light, having only $0.09$M
parameters, but it could still achieve an EER lower than those from [27, 28,
29] heavy models.
### III-B Ablation Study
We first perform ablation study by varying the depth or width of the networks,
and the results are summarized in Table II. For the Res-TSSDNet, the column
“$1\times 1$” indicates whether the “skip connection” in Fig. 2 (a) is used.
It can be seen that going either shallower or deeper will result in the raise
of EER, while with the use of ResNet skip connection, the network could
achieve $1.05\%$ EER reduction over the one without using it. Similarly for
the Inc-TSSDNet, the sweet spot also lies in the moderate depth or width.
We further perform intra-model sensitivity analysis using the two proposed
end-to-end models in Table I. Fixing all hyperparameters, the two models are
trained from scratch using ASVspoof2019 training set for over $30$ times, and
the dev and eval EERs are summarized in Fig. 3. It can be seen that the EERs
of both models are bounded within certain ranges (except one outlier eval EER
$>4\%$ for the Res-TSSDNet). The Inc-TSSDNet yields tighter dev EERs over the
Res-TSSDNet, but eval EERs of the former are clearly higher. We can see from
Fig. 3 and Table II that the intra-model differences may be as significant as
the differences from model configurations. Relatively lighter models are hence
recommended for their better trade-offs between accuracy and efficiency.
Figure 3: Intra-model performance on ASVspoof2019.
In addition, we have also discovered that i) changing all the activations from
ReLU to leaky or parametric ReLU does not lead to a clear performance
difference; ii) The first layer with a $1\times 7$ convolution kernel, adopted
from ResNet setting, is slightly better than using a $1\times 3$ kernel; iii)
Global max pooling is found to be more effective than global average pooling
before the linear layers for both networks, but for the 2D-Res-TSSDNet, we
stick to global average pooling; iv) The EERs of using standard CE are
slightly higher than those using the WCE; v) Duration of training example also
matters. Experimental results using $5$-second truncation yielded a slight
performance degradation, but when $2$-second truncation is applied, the EER on
evaluation set increased drastically.
### III-C Cross-Dataset Testing
We now perform the cross-dataset experiments. Since the ASVspoof2015 training
set contains relatively old speech synthesis methods, we focus on using
networks trained on the training set of more advanced ASVspoof2019 to test on
the dev and eval sets of ASVspoof2015. The intra- and inter-dataset EERs are
presented in Table III. It can be seen that the GMMs learned from LFCC and
CQCC features in ASVspoof2019 are generally inconsistent with the data in
ASVspoof2015. For the best Res-TSSDNet on ASVspoof2019, it could not
generalize to ASVspoof2015 either, whose EERs indicate almost
indistinguishable softmax probability distributions for real and fake classes.
However, by incorporating mixup regularization and increasing the level of
mixup level $\alpha$, we observe that the Res-TSSDNet can significantly reduce
the cross dataset EERs to less than $2\%$, while slightly sacrificing the
performance on the original dataset. Further, all the Inc-TSSDNets have very
attractive generalization capability even for the lightest model. The $M=5$,
$8$-branch version yields the best cross-dataset performance with $1.96\%$
eval EER. This is a significant score compared to the existing cross-dataset
results as reported in [43, 44, 45]. Noticeably in [45], also trained on
ASVspoof2019 training set and tested on ASVspoof2015, the use of the CQT based
features could only achieve EERs greater than $20\%$ (see Table 2 in [45]).
For completeness, the detection error trade-off (DET) curves on ASVspoof2015
evaluation set using a few methods in Table III are provided in Fig. 4.
TABLE III: EER (%) of networks trained on ASVspoof2019 training set, tested on ASVspoof2015 dev and eval sets. Method | 2019 | 2015
---|---|---
Eval | Dev | Eval
Baseline LFCC+GMM [42] | $9.57$ | $19.82$ | $15.91$
Baseline CQCC+GMM [42] | $8.09$ | $47.72$ | $39.90$
Res-TSSDNet | $1.64$ | $39.42$ | $42.52$
Mixup, $\alpha=0.1$, Res-TSSDNet | $2.07$ | $5.48$ | $5.46$
Mixup, $\alpha=0.5$, Res-TSSDNet | $2.29$ | $3.50$ | $5.75$
Mixup, $\alpha=1.0$, Res-TSSDNet | $2.16$ | $\mathbf{0.71}$ | $\mathbf{1.95}$
$M=3$, $4$-branch, Inc-TSSDNet | $10.39$ | ${5.31}$ | ${5.24}$
$M=4$, $4$-branch, Inc-TSSDNet | $4.04$ | ${2.78}$ | ${3.29}$
$M=4$, $8$-branch, Inc-TSSDNet | $3.75$ | ${1.84}$ | $2.16$
$M=5$, $8$-branch, Inc-TSSDNet | $4.20$ | $\mathbf{1.31}$ | $\mathbf{1.96}$
Figure 4: DET curves of cross-dataset testing on ASVspoof2015 eval set.
## IV Conclusion
We have shown that a light-weight end-to-end neural network, significantly
different from the exiting front- and back-end pipeline, could achieve to date
the best synthetic speech detection results. It reduces the ASVspoof2019 eval
EER by a clear margin compared to much heavier networks fed by pre-transform
inputs, sophisticated hand-crafted features plus MLP classifiers, or the
fusion of many systems of such kinds. We have further shown via cross-dataset
testing that the proposed networks could also generalize to unseen dataset. In
the ongoing ASVspoof2021 challenge, a new speech deepfake (DF) detection task
is introduced specially for synthetic deepfake speech detection, and end-to-
end methods are being given more attention, e.g., the RawNet2 [46] is used as
a baseline.
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# Stability analysis and control of decision-making of miners in blockchain
Kosuke Toda Graduate School of Engineering Science, Osaka University,
Machikaneyama 1-3, Toyonaka-shi, Osaka, 560–8531, Japan. Naomi Kuze Graduate
School of Engineering Science, Osaka University, Machikaneyama 1-3, Toyonaka-
shi, Osaka, 560–8531, Japan. Toshimitsu Ushio Graduate School of Engineering
Science, Osaka University, Machikaneyama 1-3, Toyonaka-shi, Osaka, 560–8531,
Japan.
###### Abstract
To maintain blockchain-based services with ensuring its security, it is an
important issue how to decide a mining reward so that the number of miners
participating in the mining increases. We propose a dynamical model of
decision-making for miners using an evolutionary game approach and analyze the
stability of equilibrium points of the proposed model. The proposed model is
described by the 1st-order differential equation. So, it is simple but its
theoretical analysis gives an insight into the characteristics of the
decision-making. Through the analysis of the equilibrium points, we show the
transcritical bifurcations and hysteresis phenomena of the equilibrium points.
We also design a controller that determines the mining reward based on the
number of participating miners to stabilize the state where all miners
participate in the mining. Numerical simulation shows that there is a trade-
off in the choice of the design parameters.
Keywords— B lockchain, Proof-of-work, Decision-making, Evolutionary Game,
Bifurcation, Hysteresis, Feedback control.
## 1 Introduction
Blockchain is a distributed ledger technology for recording transactions that
underlies various fields such as digital currency like Bitcoin [1], data
sharing [2], and computer security [3]. Blockchain-based services use
cryptography to record transactions as a chain of blocks. A block consists of
a block header and transaction data. The block header contains a cryptographic
hash of its previous block, which makes blockchain-based services resistant to
tampering. In these services, participants called miners create blocks in a
distributed manner, and the longest chain of blocks is considered to be
legitimate. When a miner succeeds in creating a block, he/she gets a reward
called a mining reward.
Blockchain-based services approve transactions through a consensus algorithm.
As a consensus algorithm, proof-of-work (PoW) is typically used. In this
algorithm, the mining difficulty is set using a scalar value called a nonce in
the block header. To create a block, miners must find a nonce such that the
cryptographic hash value for the previous block satisfies specific conditions.
The process of creating blocks is called a mining. In general, a cryptographic
hash value for a block is unique according to the nonce contained in the
block. Moreover, a nonce that satisfies the specific conditions cannot be
calculated directly. As a result, an exhaustive search imposes a large
computational cost on miners, which contributes to the resistance to
tampering. Because transaction approvals depend on miner calculations (such
calculations are very costly and require a lot of energy [4, 5]), the
participation of many miners is needed to maintain blockchain-based services
and ensure blockchain system security [6, 7]. Therefore, it is important to
analyze the decision-making problem of whether miners participate in the
mining according to the energy consumption and mining rewards.
Game theory is used to analyze interactions among rational decision-makers.
Many studies have adopted game theory to analyze blockchain-related issues
with PoW [8], such as decision-making problems in the mining considering the
energy consumption [9, 10]. Evolutionary game theory has been used as a
powerful mathematical tool for analyzing dynamical models of evolutionary
selection [11]. Dynamical characteristics of the selection process are modeled
by replicator dynamics. Control methods for the replicator dynamics have been
studied in [12, 13, 14]. Evolutionary game models and replicator dynamics are
also used in analyzing blockchain-related issues such as mining pool selection
problems [15, 16], and attack scenarios [17].
We previously focused on a decision-making problem of whether miners
participate in the mining according to the energy consumption and the mining
rewards, and modeled it as a non-cooperative game. Through theoretical and
numerical analysis, we showed the property of Nash equilibria [18]. However,
in this study, we assumed that, once miners choose a strategy (i.e.,
participation in the mining or not), they do not change their strategies.
Practically, the miners may decide to participate in the mining dynamically
based on their current earned mining rewards. It is important to analyze such
a dynamical decision-making process.
In this paper, we propose a dynamical model of the decision-making problem for
miners, by applying an evolutionary game approach. We analyze the stability of
its equilibrium points and show the existence of transcritical bifurcations
and hysteresis phenomena with the coexistence of two asymptotically stable
equilibrium points: one corresponds to the state where all miners participate
in the mining and the other to the state where the number of participating
miners is minimum. The former equilibrium point is preferable to maintain
blockchain-based services. We propose a controller that determines the mining
reward based on the number of current participating miners so as to stabilize
the equilibrium point if at least one miner participates in the initial time.
The remainder of this paper is organized as follows. In Section 2, we propose
an evolutionary game-based dynamical model of the decision-making process. In
Section 3, we analyze the stability of its equilibrium point. In Section 4, we
design a state feedback controller to let all miners participate in the
mining.
## 2 Dynamical model of decision-making
We assume that miners in a blockchain network are partitioned into two sets
$\mathcal{M}$ and $\mathcal{N}$, where miners in $\mathcal{M}$ always
participate in the mining and those in $\mathcal{N}$ have two strategies,
participating in the mining (strategy $s_{k}=1$) and not participating in the
mining (strategy $s_{k}=0$), where $k\in\mathcal{N}$. Note that
$\mathcal{M}\cap\mathcal{N}=\emptyset$. Denoted by $m$ and $n$ are the
cardinalities of $\mathcal{M}$ and $\mathcal{N}$, respectively (we assume
$m\geq 1$ and $n\geq 1$). We define $x_{0}$ and $x_{1}$ as the ratios of
miners in $\mathcal{N}$ that choose strategies $0$ and $1$, respectively. Note
that
$\displaystyle x_{0}+x_{1}=1.$ (2.1)
Miners need to find a nonce such that the first $h$ bits of the hash of the
block are all $0$. Then, $D=2^{h}$ is the difficulty parameter and $1/D$ is
the probability that a miner creates a block with one hash calculation [19].
When miner $k\in\mathcal{M}\cup\mathcal{N}$ participates in the mining, he/she
needs a cost $c$ per unit operating time. The average number $w_{k}=f_{k}(c)$
of hash queries calculated per unit operating time by miner $k$ depends on the
cost $c$, and we assume that $f_{k}(c)$ is the same for all miners. In this
paper, for simplicity, we assume $f_{k}(c)=vc\,(v>0)$ for any
$k\in\mathcal{M}\cup\mathcal{N}$.
The mining of blocks can be described as a Poisson process [20, 21]. That is,
the block creation time is exponentially distributed [22]. The rate
$\lambda_{k}$ of the Poisson process of miner $k\in\mathcal{M}\cup\mathcal{N}$
is given by $\lambda_{k}=w_{k}/D$ [21]111 Note that a combination of
independent Poisson processes is still a Poisson process. Thus, the rate of
the Poisson process of all miners is written as
$\sum_{i\in\mathcal{M}\cup\mathcal{N}}\lambda_{i}$. . If miner $k$ chooses
$s_{k}=1$, then the rate of the Poisson process is
$\lambda_{k}=s_{k}f_{k}(c)/D=s_{k}c/d$ (we define $d=D/v$, in this paper). Let
$R$ be the mining reward. Based on the previous work [18], the expected reward
$R_{k}$ and the expected cost $CS_{k}$ for the mining of miner $k$ are
calculated as follows.
$\displaystyle
R_{k}=\frac{\lambda_{k}}{\sum_{i\in\mathcal{M}\cup\mathcal{N}}\lambda_{i}}R=\frac{Rs_{k}}{m+nx_{1}},$
(2.2) $\displaystyle
CS_{k}=\frac{c\lambda_{k}}{(\sum_{i\in\mathcal{M}\cup\mathcal{N}}\lambda_{i})^{2}}=\frac{ds_{k}}{(m+nx_{1})^{2}}.$
(2.3)
We define the utility function $u_{i}(x_{0},x_{1})$ of miners that choose the
strategy $i\in\\{0,\ 1\\}$ as
$\displaystyle u_{i}(x_{0},x_{1})=\begin{cases}0&\mbox{if}\;\;i=0,\\\
\frac{1}{m+nx_{1}}\left(R-\frac{d}{m+nx_{1}}\right)&\mbox{if}\;\;i=1,\end{cases}$
(2.4)
which means that the utility of a miner who participates in the mining is the
difference between the expected reward $R_{k}$ and the expected cost $CS_{k}$.
Based on the principle of the evolutionary game [11], the dynamics of the
ratio of miners that choose the strategy $i$ is given by
$\displaystyle\frac{\dot{x}_{i}}{x_{i}}=u_{i}(x_{0},x_{1})-\bar{u}(x_{0},x_{1})\;(i=0,1),$
(2.5)
where $\bar{u}(x_{0},x_{1})=\sum_{i=0}^{1}x_{i}u_{i}(x_{0},x_{1})$ is the
average utility of all miners. According to (2.4), (2.5) is rewritten as
$\displaystyle\dot{x}_{1}=-\dot{x}_{0}=\frac{x_{1}(1-x_{1})}{m+nx_{1}}\left(R-\frac{d}{m+nx_{1}}\right)\eqqcolon\varphi_{R}(x_{1}).$
(2.6)
Thus, the dynamics of the decision-making of miners is described by the above
1st-order differential equation and the reward $R$ plays an important role in
the decision-making of the miners for the participation in the mining. In the
following, we investigate stability and stabilization of equilibrium points of
(2.6). For that purpose, the concept of a basin of attraction [23] is
important. Let $\xi(t;x_{1}^{\mathrm{init}})$ be the solution of (2.6) that
starts from an initial state $x_{1}^{\mathrm{init}}$ at time $t=0$. For a
given asymptotically stable equilibrium point $x_{1}^{\prime}$ of (2.6), the
basin of attraction is defined as the set of all initial states
$x_{1}^{\mathrm{init}}$ such that $\xi(t;x_{1}^{\mathrm{init}})$ is defined
for all $t\geq 0$ and
$\lim_{t\to\infty}\xi(t;x_{1}^{\mathrm{init}})=x_{1}^{\prime}$.
## 3 Stability analysis
In this section, we investigate the stability of the equilibrium point
$x_{1}=0,\ 1,\ x_{1}^{*}$ of (2.6), where
$\displaystyle x_{1}^{*}=\frac{1}{n}\left(\frac{d}{R}-m\right).$ (3.1)
When $1/(m+n)<R/d<1/m$, the equilibrium point $x_{1}^{*}$ satisfies
$0<x_{1}^{*}<1$. This equilibrium point is the state where the utility
$u_{1}(1-x_{1}^{*},x_{1}^{*})$ is equal to $0$, that is, the utility for the
strategy $0$ is equal to that for the strategy $1$.
We investigate the local stability of the three equilibrium points. The
derivative of $\varphi_{R}(x_{1})$ with respect to $x_{1}$ is
$\displaystyle\frac{\partial\varphi_{R}(x_{1})}{\partial
x_{1}}=\left(-\frac{x_{1}}{m+nx_{1}}+\frac{1-x_{1}}{m+nx_{1}}-\frac{nx_{1}(1-x_{1})}{(m+nx_{1})^{2}}\right)$
$\displaystyle\hskip
56.9055pt\times\left(R-\frac{d}{m+nx_{1}}\right)+\frac{dnx_{1}(1-x_{1})}{(m+nx_{1})^{3}}.$
(3.2)
Thus, we obtain
$\displaystyle\left.\frac{\partial\varphi_{R}(x_{1})}{\partial
x_{1}}\right|_{x_{1}=0}=\frac{1}{m}\left(R-\frac{d}{m}\right),$ (3.3)
$\displaystyle\left.\frac{\partial\varphi_{R}(x_{1})}{\partial
x_{1}}\right|_{x_{1}=1}=-\frac{1}{m+n}\left(R-\frac{d}{m+n}\right),$ (3.4)
$\displaystyle\left.\frac{\partial\varphi_{R}(x_{1})}{\partial
x_{1}}\right|_{x_{1}=x_{1}^{*}}=\frac{R^{3}}{nd^{2}}\left(\frac{d}{R}-m\right)\left((m+n)-\frac{d}{R}\right).$
(3.5)
Table 1: The relation between $R/d$ and the stability of equilibrium points. Condition for $R/d$ | $x_{1}=0$ | $x_{1}=1$ | $x_{1}=x_{1}^{*}$
---|---|---|---
$R/d<1/(m+n)$ | S | U | S
$1/(m+n)<R/d<1/m$ | S | S | U
$R/d>1/m$ | U | S | S
Thus, we have their stability conditions as shown in Table 1, where S (resp.
U) represents an asymptotically stable (resp. unstable) point.
Figure 1: The $m-R/d$ parameter plane where $n$ is fixed to $n=2$.
Fig. 1 shows the $m-R/d$ parameter plane with $n=2$ where the meaning of each
region is as follows. In the region $(A)$, both $x_{1}=1$ and $x_{1}^{*}<0$
are asymptotically stable equilibrium points, and the basin of attraction of
$x_{1}=1$ is $(0,\ \infty)$, that is, every solution of (2.6) starting in
$(0,1]$ converges to $1$. In the region $(B)$, both $x_{1}=0$ and $x_{1}=1$
are asymptotically stable equilibrium points, and basins of attraction of
$x_{1}=0$ and $x_{1}=1$ are $(-\infty,\ x_{1}^{*})$ and $(x_{1}^{*},\
\infty)$, respectively, that is, every solution of (2.6) starting in $[0,\
x_{1}^{*})$ converges to $0$, and every solution of (2.6) starting in
$(x_{1}^{*},\ 1]$ converges to $1$. In the region $(C)$, both $x_{1}=0$ and
$x_{1}^{*}>0$ are asymptotically stable equilibrium points, and the basin of
attraction of $x_{1}=0$ is $(-\infty,\ 1)$, that is, every solution of (2.6)
starting in $[0,\ 1)$ converges to $0$.
Figure 2: The relation between $R/d$ and the stability of equilibrium points
when $m=n=2$.
Shown in Fig. 2 is a bifurcation diagram with respect to the bifurcation
parameter $R/d$, where $m=n=2$. The solid (resp. dashed) line represents an
asymptotically stable (resp. unstable) equilibrium point. Two curves of
equilibrium points pass through $(x_{1},R)=(1,d/(m+n))$ (resp.
$(x_{1},R)=(0,d/m)$), one given by $x_{1}=x_{1}^{*}$, the other by $x_{1}=1$
(resp. $x_{1}=0$). Both curves exist on both sides of $R=d/(m+n)$ (resp.
$R=d/m$). The stability along each curve exchanges on passing through
$R=d/(m+n)$ (resp. $R=d/m$). Thus, the exchange of stability (known as a
transcritical bifurcation) [24] is observed when
$(x_{1},R)=(0,d/m),(1,d/(m+n))$. We show in A that the vector field (2.6)
satisfies the condition of the transcritical bifurcation shown in [24].
Since the values of $x_{i}\,(i=0,1)$ satisfy $0\leq x_{i}\leq 1$, we observe
jump phenomena owing to these transcritical bifurcations. Moreover, $R>d/m$
needs to be satisfied so that all miners in $\mathcal{N}$ participate in the
mining. It is noted that, once the miners participate in the mining, they
continue to participate in the mining until the reward $R$ becomes $d/(m+n)$.
Thus, a hysteresis phenomenon of the equilibrium points is observed.
Figure 3: Trajectories of (2.6) when $m=n=2$, $d=100$, and $R=40$, from
$x_{1}^{\mathrm{init}}=0.1$ (blue) and $x_{1}^{\mathrm{init}}=0.9$ (red).
Fig. 3 shows trajectories of (2.6) from an initial state
$x_{1}^{\mathrm{init}}=0.1$ (blue) and $x_{1}^{\mathrm{init}}=0.9$ (red). When
$d/(m+n)<R<d/m$, both $x_{1}=0$ and $x_{1}=1$ are asymptotically stable points
whose basins of attraction are $(-\infty,\ x_{1}^{*})$ and $(x_{1}^{*},\
\infty)$, respectively. Thus, the number of miners who participate in the
mining converges to $0$ if the initial ratio is less than $x_{1}^{*}$ because
their utility is negative and they prefer non-participation.
## 4 Stabilization
The result in Section 3 implies that no miner in $\mathcal{N}$ participates in
the mining in the steady state when the mining reward $R^{*}$ satisfies
$R^{*}<d/(m+n)$. When the mining reward $R^{*}$ satisfies $d/(m+n)<R^{*}<d/m$,
miners’ behaviors depend on the initial state $x_{1}^{\mathrm{init}}$, i.e.,
no miner in $\mathcal{N}$ participates in the mining in the steady state when
$x_{1}^{\mathrm{init}}<x_{1}^{*}$. We propose a state feedback controller to
adjust the reward based on the ratio $x_{1}$ so that all miners in
$\mathcal{N}$ participate in the mining, i.e., let every trajectory of $x_{1}$
with its initial state in $(0,\ 1]$ converge to $1$.
### 4.1 Case where $R^{*}<d/(m+n)$
First, we show that $x_{1}=1$ cannot be stabilized when $R^{*}<d/(m+n)$. We
consider the following state feedback controller $R_{1}(x_{1})$ that adjusts
the reward based on the ratio $x_{1}$.
$\displaystyle R=R_{1}(x_{1}),\;R_{1}(1)=R^{*}<\frac{d}{m+n}.$ (4.1)
The controlled trajectory of $x_{1}$ by (4.1) is described by
$\displaystyle\dot{x}_{1}=\frac{x_{1}(1-x_{1})}{m+nx_{1}}\left(R_{1}(x_{1})-\frac{d}{m+n}\right)\eqqcolon\psi_{R}(x_{1}).$
(4.2)
The derivative of $\psi_{R}(x_{1})$ with respect to $x_{1}$ is
$\displaystyle\frac{\partial\psi_{R}(x_{1})}{\partial
x_{1}}=\left(-\frac{x_{1}}{m+nx_{1}}+\frac{1-x_{1}}{m+nx_{1}}-\frac{nx_{1}(1-x_{1})}{(m+nx_{1})^{2}}\right)$
$\displaystyle\hskip
42.67912pt\times\left(R_{1}(x_{1})-\frac{d}{m+nx_{1}}\right)$
$\displaystyle\hskip
48.36967pt+\frac{x_{1}(1-x_{1})}{m+nx_{1}}\left(\frac{\partial
R_{1}(x_{1})}{\partial x_{1}}+\frac{dn}{(m+nx_{1})^{2}}\right).$ (4.3)
We obtain
$\displaystyle\left.\frac{\partial\psi_{R}(x_{1})}{\partial
x_{1}}\right|_{x_{1}=1}=-\frac{1}{m+n}\left(R_{1}(1)-\frac{d}{m+n}\right)>0.$
(4.4)
Therefore, the unstable equilibrium point $x_{1}=1$ cannot be stabilized even
if the feedback controller is used.
### 4.2 Case where $d/(m+n)<R^{*}<d/m$
Next, we show that $x_{1}=1$ can be an asymptotically stable equilibrium point
whose basin of attraction is $(0,\ 1]$ with a state feedback controller. We
introduce the following state feedback controller $R_{2}(x_{1})$ to adjust the
reward based on the ratio $x_{1}$,
$\displaystyle R=R_{2}(x_{1})=R^{*}+\Delta R(x_{1}),\;\Delta R(1)=0,$ (4.5)
and let every trajectory of $x_{1}$ with its initial state in $(0,\ 1]$
converge to $1$.
#### 4.2.1 The condition of the feedback gain
We give $\bar{x}_{1}$ satisfying $x_{1}^{*}<\bar{x}_{1}\leq 1$ and
$\varepsilon>0$. For a given reward $R^{*}\in(d/(m+n),d/m)$, let $\Delta
R(x_{1})$ be
$\displaystyle\Delta
R(x_{1})=\begin{cases}K(\bar{x}_{1}-x_{1})&\mbox{if}\;\;x_{1}<x_{1}^{*}+\varepsilon,\\\
0&\mbox{otherwise},\end{cases}$ (4.6)
where $K>0$ is a feedback gain. We obtain a condition for the gain $K$ and
$\varepsilon$ such that every trajectory of $x_{1}$ with its initial state in
$(0,\ 1]$ converges to $1$ as in Proposition 1.
###### Proposition 1.
Assume $d/(m+n)<R^{*}<d/m$. Let $\zeta_{R}(x_{1})$ be
$\displaystyle\zeta_{R}(x_{1})$ $\displaystyle\coloneqq-
Knx_{1}^{2}+(Kn\bar{x}_{1}-Km+R^{*}n)x_{1}$ $\displaystyle\hskip
71.13188pt+(R^{*}m+Km\bar{x}_{1}-d),$ (4.7)
and let $\alpha,\beta\;(\alpha<\beta)$ be real solutions of the quadratic
equation $\zeta_{R}(x_{1})=0$. Then, every trajectory of $x_{1}$ with its
initial state in $(0,\ 1]$ converges to $1$ if the gain $K$ and $\varepsilon$
satisfy
$\displaystyle K>\frac{d-R^{*}m}{m\bar{x}_{1}}\ (>0),$ (4.8) $\displaystyle
0<\varepsilon\begin{cases}<\beta-x_{1}^{*}&{\rm if}\;\;\beta<1,\\\ \leq
1-x_{1}^{*}&{\rm if}\;\;\beta\geq 1.\end{cases}$ (4.9)
###### Proof.
With the controller (4.5) and (4.6), the dynamics of $x_{1}$
($x_{1}<x_{1}^{*}+\varepsilon$) is given by
$\displaystyle\dot{x}_{1}=\eta_{R}(x_{1}),$ (4.10)
$\displaystyle\eta_{R}(x_{1})\coloneqq\frac{x_{1}(1-x_{1})}{m+nx_{1}}\left(R^{*}+K(\bar{x}_{1}-x_{1})-\frac{d}{m+nx_{1}}\right).$
(4.11)
According to (4.7), $\eta_{R}(x_{1})$ can be rewritten as
$\displaystyle\eta_{R}(x_{1})=\frac{x_{1}(1-x_{1})\zeta_{R}(x_{1})}{(m+nx_{1})^{2}}.$
(4.12)
First, we prove that the quadratic equation $\zeta_{R}(x_{1})=0$ has two
distinct real solutions under (4.8). We have
$\displaystyle\zeta_{R}(x_{1}^{*})=K(\bar{x}_{1}-x_{1}^{*})(m+nx_{1})>0,$
(4.13)
which implies with (4.8) that the quadratic equation $\zeta_{R}(x_{1})=0$ has
two distinct real solutions $\alpha,\beta$ satisfying
$\alpha<x_{1}^{*}<\beta$.
Next, we prove $\alpha<0$ under (4.8). We obtain
$\displaystyle\zeta_{R}(0)$ $\displaystyle=m\bar{x}_{1}K-(d-R^{*}m)$
$\displaystyle>m\bar{x}_{1}\frac{d-R^{*}m}{m\bar{x}_{1}}-(d-R^{*}m)=0,$ (4.14)
from (4.7) and (4.8). Since $\zeta_{R}(x_{1})$ is a convex upward quadratic
function, the smaller solution $\alpha$ of $\zeta_{R}(x_{1})=0$ satisfies
$\alpha<0$.
Finally, we prove that the system controlled by (4.5) satisfies
$\dot{x}_{1}>0$ for any $x_{1}\in(0,\ 1)$ under (4.8) and (4.9). When
$\beta<1$, $\eta_{R}(x_{1})>0$ for any $x_{1}\in(0,\ \beta)$ from (4.12). It
is obvious that $\dot{x}_{1}>0$ for any $x_{1}\in(0,x_{1}^{*}+\varepsilon)$
from (4.9). For any $x_{1}\in[x_{1}^{*}+\varepsilon,1)$, $\dot{x}_{1}>0$
because $K=0$. Thus,
$\displaystyle\dot{x}_{1}>0\;\mbox{for}\;\mbox{any}\;x_{1}\in(0,\;1).$ (4.15)
Similary, it is also shown by (4.9) that (4.15) holds for any $\beta\geq 1$.
Therefore, every trajectory of $x_{1}$ with its initial state in $(0,\ 1]$
converges to $1$ under (4.8) and (4.9). ∎
It is noted that $\bar{x}_{1}<\beta$ since $d/(m+n)<R^{*}$. So, (4.6) is
continuous if $\varepsilon=\bar{x}_{1}-x_{1}^{*}$.
#### 4.2.2 Performance evaluation
In this section, we provide the numerical analysis of the controller. We
consider the case where $m=n=2$, $d=100$, and $R^{*}=40$. Then, we have
$x_{1}^{*}=0.25$ from (3.1). Let the initial state of $x_{1}$ be
$x_{1}^{\mathrm{init}}=0.1$. We consider the following two cases where $K$ and
$\varepsilon$ satisfy (4.8) and (4.9).
Case 1)
$\bar{x}_{1}=0.26,\;\varepsilon=0.005,\;K=56.8125$.
Case 2)
$\bar{x}_{1}=1,\;\varepsilon=0.75,\;K=10.1$.
(a)
(b)
Figure 4: Trajectories of (a) the state $x_{1}$ and (b) the reward
$R_{2}(x_{1})$ with a feedback controller satisfying Proposition 1, when
$m=n=2$, $d=100$, $R^{*}=40$, $x_{1}^{*}=0.25$ from
$x_{1}^{\mathrm{init}}=0.1$, where
$\bar{x}_{1}=0.26,\varepsilon=0.005,K=56.8125$ (red) and
$\bar{x}_{1}=1,\varepsilon=0.75,K=10.1$ (blue).
Fig. 4 shows trajectories of the state and the reward. The red and blue lines
represent the trajectories of Cases 1) and 2), respectively. In Case 1), it
takes a longer time than Case 2) for the state $x_{1}$ to converge to $1$, but
the reward $R_{2}(x_{1})$ returns to the original value $R^{*}$ quickly. Note
that $R_{2}(x_{1})$ in Case 1) is not continuous because we switch the input
$\Delta R(x_{1})$ to $0$ when $x_{1}=x_{1}^{*}+\varepsilon$ (see (4.6)). In
Case 2), the state $x_{1}$ converges to $1$ quickly, but it takes longer time
than Case 1) for the reward $R_{2}(x_{1})$ to return to its original value
$R^{*}$. Thus, there is a trade-off in the choice of the design parameters
$\bar{x}_{1}$ and $\varepsilon$.
## 5 Conclusion
We proposed a dynamical model of the decision-making of miners in the
blockchain. The proposed model is described by the 1st-order differential
equation. So, it is simple but its theoretical analysis gives an insight into
the characteristics of the decision-making. We analyzed the stability of its
equilibrium points. We showed the occurrence of the transcritical bifurcations
and observed a hysteresis phenomenon. We also proposed a feedback controller
and showed that it can stabilize the state where all miners participate in the
mining from any non-zero initial participation ratio of the miners. Our future
work is to extend our model to the case where miners’ computational
performances are different from each other.
## Acknowledgements
This research was supported by JST ERATO JPMJER1603.
## Appendix A Transcritical bifurcation
We consider the following system.
$\displaystyle\dot{x}=f(x,\mu),\;\;x\in\mathbb{R},\;\;\mu\in\mathbb{R}.$ (A.1)
We assume that
$\displaystyle f(x,\mu)=xF(x,\mu),$ (A.2)
where $F:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ satisfies the following
condition.
$\displaystyle F(x,\mu)\coloneqq\begin{cases}\frac{f(x,\mu)}{x}&x\neq 0,\\\
\frac{\partial f(0,\mu)}{\partial x}&x=0.\end{cases}$ (A.3)
Then, it is shown in [24] that (A.1) undergoes a transcritical bifurcation at
$(x,\mu)=(0,0)$ if the following three conditions hold.
(T1)
$f(0,0)=0,\;\frac{\partial f(0,0)}{\partial x}=0$,
(T2)
$\frac{\partial f(0,0)}{\partial\mu}=0$,
(T3)
$\frac{\partial^{2}f(0,0)}{\partial x\partial\mu}\neq
0,\;\frac{\partial^{2}f(0,0)}{\partial x^{2}}\neq 0$.
Thus we will show that (2.6) satisfies the above three conditions at
$(x_{1},R)=(0,d/m),\ (1,d/(m+n))$.
### A.1 Case where $(x_{1},R)=(0,d/m)$
First, we consider the following coordination transformation by which
$(x_{1},R)=(0,d/m)$ is transformed to $(x,\mu)=(0,0)$.
$\displaystyle\left(\begin{array}[]{c}x_{1}\\\
R\end{array}\right)=\left(\begin{array}[]{c}x\\\
\mu\end{array}\right)+\left(\begin{array}[]{c}0\\\
\frac{d}{m}\end{array}\right).$ (A.10)
Then, we define
$\displaystyle f(x,\mu)$ $\displaystyle\coloneqq\varphi_{\mu+\frac{d}{m}}(x)$
$\displaystyle=\frac{x(1-x)}{m+nx}\left(\mu+\frac{d}{m}-\frac{d}{m+nx}\right)=xF(x,\mu),$
(A.11)
where the function $F$ is defined by
$\displaystyle
F(x,\mu)\coloneqq\frac{1-x}{m+nx}\left(\mu+\frac{d}{m}-\frac{d}{m+nx}\right).$
(A.12)
Then, we have
$\displaystyle\frac{\partial f(x,\mu)}{\partial x}$
$\displaystyle=F(x,\mu)+x\frac{\partial F(x,\mu)}{\partial x}$
$\displaystyle=\left(-\frac{x}{m+nx}+\frac{1-x}{m+nx}-\frac{nx(1-x)}{(m+nx)^{2}}\right)$
$\displaystyle\hskip
28.45274pt\times\left(\mu+\frac{d}{m}-\frac{d}{m+nx}\right)+\frac{dnx(1-x)}{(m+nx)^{3}}.$
(A.13)
It is obvious that
$\displaystyle F(x,\mu)=\frac{f(x,\mu)}{x}$ (A.14)
when $x\neq 0$ and
$\displaystyle F(0,\mu)=\frac{\mu}{m}=\frac{\partial f(0,\mu)}{\partial
x}\;\;(\because\eqref{eq:A_phi_deriv_x})$ (A.15)
when $x=0$. Thus, the function $f$ defined by (A.11) satisfies (A.2) and
(A.3).
Next, we show that $f(x,\mu)$ satisfies the conditions (T1) – (T3). We obtain
$\displaystyle\frac{\partial f(x,\mu)}{\partial\mu}=\frac{x(1-x)}{m+nx},$
(A.16) $\displaystyle\frac{\partial^{2}f(x,\mu)}{\partial
x\partial\mu}=-\frac{x}{m+nx}+\frac{1-x}{m+nx}-\frac{nx(1-x)}{(m+nx)^{2}},$
(A.17) $\displaystyle\frac{\partial^{2}f(x,\mu)}{\partial
x^{2}}=\left(\frac{n^{2}x(1-x)}{(m+nx)^{2}}-\frac{n(1-x)}{m+nx}+\frac{nx}{m+nx}-1\right)$
$\displaystyle\hskip
14.22636pt\times\frac{2}{m+nx}\left(\mu+\frac{d}{m}-\frac{d}{m+nx}\right)+\frac{2dn}{(m+nx)^{3}}$
$\displaystyle\hskip
28.45274pt\times\left(\frac{-2nx(1-x)}{m+nx}-x+(1-x)\right).$ (A.18)
Thus, $f(x,\mu)$ satisfies the conditions (T1) – (T3) because
$\displaystyle f(0,0)$ $\displaystyle=0,$ (A.19) $\displaystyle\frac{\partial
f(0,0)}{\partial x}$ $\displaystyle=0,$ (A.20) $\displaystyle\frac{\partial
f(0,0)}{\partial\mu}$ $\displaystyle=0,$ (A.21)
$\displaystyle\frac{\partial^{2}f(0,0)}{\partial x\partial\mu}$
$\displaystyle=\frac{1}{m}\neq 0,$ (A.22)
$\displaystyle\frac{\partial^{2}f(0,0)}{\partial x^{2}}$
$\displaystyle=\frac{2dn}{m^{3}}\neq 0.$ (A.23)
### A.2 Case where $(x_{1},R)=(1,d/(m+n))$
It is noted that the dynamics of $x_{0}$ is written as follows.
$\displaystyle\dot{x}_{0}$
$\displaystyle=-\frac{x_{0}(1-x_{0})}{m+n(1-x_{0})}\left(R-\frac{d}{m+n(1-x_{0})}\right)$
$\displaystyle=-\varphi_{R}(1-x_{0})$ (A.24)
because $x_{0}$ and $x_{1}$ satisfies (2.1). Thus, it is sufficient to show
that (A.24) undergoes a transcirtical bifurcation at $(x_{0},R)=(0,d/(m+n))$.
We consider the following coordination transformation by which
$(x_{0},R)=(0,d/(m+n))$ is transformed to $(x,\mu)=(0,0)$.
$\displaystyle\left(\begin{array}[]{c}x_{0}\\\
R\end{array}\right)=\left(\begin{array}[]{c}x\\\
\mu\end{array}\right)+\left(\begin{array}[]{c}0\\\
\frac{d}{m+n}\end{array}\right).$ (A.31)
Then, we define
$\displaystyle f(x,\mu)$
$\displaystyle\coloneqq-\varphi_{\mu+\frac{d}{m+n}}(1-x)$
$\displaystyle=-\frac{x(1-x)}{m+n(1-x)}$ $\displaystyle\hskip
28.45274pt\times\left(\mu+\frac{d}{m+n}-\frac{d}{m+n(1-x)}\right)$
$\displaystyle=xF(x,\mu),$ (A.32)
where the function $F$ is defined by
$\displaystyle F(x,\mu)$ $\displaystyle\coloneqq-\frac{1-x}{m+n(1-x)}$
$\displaystyle\hskip
28.45274pt\times\left(\mu+\frac{d}{m+n}-\frac{d}{m+n(1-x)}\right).$ (A.33)
Then, we have
$\displaystyle\frac{\partial f(x,\mu)}{\partial x}$
$\displaystyle=-\left(\frac{-x+(1-x)}{m+n(1-x)}+\frac{nx(1-x)}{(m+n(1-x))^{2}}\right)$
$\displaystyle\hskip
14.22636pt\times\left(\mu+\frac{d}{m+n}-\frac{d}{m+n(1-x)}\right)$
$\displaystyle\hskip 28.45274pt+\frac{dnx(1-x)}{(m+n(1-x))^{3}}.$ (A.34)
It is obvious that
$\displaystyle F(x,\mu)=\frac{f(x,\mu)}{x}$ (A.35)
when $x\neq 0$ and
$\displaystyle F(0,\mu)=-\frac{\mu}{m+n}=\frac{\partial f(0,\mu)}{\partial
x}\;\;(\because\eqref{eq:A_phi_1_deriv_x})$ (A.36)
when $x=0$. Thus, the function $f$ defined by (A.32) satisfies (A.2) and
(A.3).
In the same way as Appendix A.1, we obtain the partial derivatives of
$f(x,\mu)$ and show that $f(x,\mu)$ defined by (A.24) satisfies the conditions
(T1) – (T3) because
$\displaystyle f(0,0)$ $\displaystyle=0,$ (A.37) $\displaystyle\frac{\partial
f(0,0)}{\partial x}$ $\displaystyle=0,$ (A.38) $\displaystyle\frac{\partial
f(0,0)}{\partial\mu}$ $\displaystyle=0,$ (A.39)
$\displaystyle\frac{\partial^{2}f(0,0)}{\partial x\partial\mu}$
$\displaystyle=-\frac{1}{m+n}\neq 0,$ (A.40)
$\displaystyle\frac{\partial^{2}f(0,0)}{\partial x^{2}}$
$\displaystyle=\frac{2dn}{(m+n)^{3}}\neq 0.$ (A.41)
Therefore, (2.6) undergoes transcritical bifurcations at $(x_{1},R)=(0,d/m),\
(1,d/(m+n))$.
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|
11institutetext: Stellenbosch University, South Africa 22institutetext:
Praelexis, South Africa 33institutetext: KU Leuven, Belgium
# A derivation of variational message passing (VMP) for latent Dirichlet
allocation (LDA)
Rebecca M.C. Taylor 1122 Dirko Coetsee 112233 Johan A. du Preez 11
###### Abstract
Latent Dirichlet Allocation (LDA) is a probabilistic model used to uncover
latent topics in a corpus of documents. Inference is often performed using
variational Bayes (VB) algorithms, which calculate a lower bound to the
posterior distribution over the parameters. Deriving the variational update
equations for new models requires considerable manual effort; variational
message passing (VMP) has emerged as a "black-box" tool to expedite the
process of variational inference. But applying VMP in practice still presents
subtle challenges, and the existing literature does not contain the steps that
are necessary to implement VMP for the standard smoothed LDA model, nor are
available black-box probabilistic graphical modelling software able to do the
word-topic updates necessary to implement LDA. In this paper, we therefore
present a detailed derivation of the VMP update equations for LDA. We see this
as a first step to enabling other researchers to calculate the VMP updates for
similar graphical models.
###### keywords:
Latent Dirichlet Allocation, Variational, Graphical Model, Message Passing,
VMP, derivation
## 1 Introduction
Latent Dirichlet Allocation (LDA) [1] is an effective and popular
probabilistic document model with many applications [2]. These include,
banking and finance:
clustering banking clients based on their transactions and identification of
insurance fraud [3].
genomics:
using multilocus genotype data to learn about population structure and assign
individuals to populations [4], classification of gene expression in healthy
and diseased tissues, [5] and prediction of the functional effects of genetic
variation [6],
image processing:
image clustering and scene recognition [7, 8], and
medical:
feature extraction for rare and emerging diseases [9] and medical data set
clustering [10, 11].
While LDA can extract latent topics of any type from a wide range of inputs,
it is most commonly used to extract latent semantic information from text. The
scale at which LDA is applied has continued to grow with the increasing
availability of computing resources, large volumes of data [12], and improved
inference algorithms.
LDA is usually represented as a graphical model, as shown in Figure 1.
Figure 1: Plate model for LDA as a Bayesian network. Each node in the graph
represents a random variable and edges represent conditional dependencies.
This graph represents a probability distribution, where $M$ documents contain
$N$ words, and each word represents one of $K$ possible topics. Section 3.1
provides further details. LDA has been extended and modified to create many
new but similar graphical models such as Filtered-LDA [13], author topic
models [14], relational topic models [15], dynamic topic models [16], and
spatial LDA [17].
Exact inference is computationally intractable for many useful graphical
models, including LDA and its many variants [1, 18, 19], [20, p461]. A range
of approximate inference techniques is therefore used. Particle based
approaches such as Markov chain Monte Carlo (MCMC) [20, p462] have been used
but are still computationally expensive [21].
Because larger data sources are now readily available, faster and comparably
accurate methods to these particle-based approaches have gained popularity
[19]. These include variational Bayes (VB) [22, 23, 24] and expectation
propagation (EP) [25, 26, 27]. Variational Bayes in particular is notable for
achieving good run-time performance. By optimising a bound on the posterior
distribution, it provides guarantees on the inference quality that is not
always attainable with approximate methods. VB is usually available only for
exponential family distributions. In Section 3.2 we will therefore provide a
short overview of the exponential family and some of the notation we use later
on in this paper.
Constructing a new model that uses VB is labour-intensive since the modeller
must derive each of the variational update equations manually. This process
has been somewhat eased by the introduction of variational message passing
(VMP), a formulation of VB that promises to be general enough to apply to a
wide variety of graphical models but also a "black-box" inference engine that
automates the calculation of the update equations [19]. A general overview of
VMP is provided in Section 4.
The available VMP toolkits, however, do not cater for all possible conditional
probability distributions, either because a specific distribution is not
implemented yet or would be too costly in terms of computational or memory
resources.
The conditional Dirichlet-multinomial distribution in particular is necessary
to implement smoothed LDA, as discussed in Section 5 but, to the best of our
knowledge, is not found in any of the popular toolkits [28, 29]. As a first
step towards incorporating this distribution, and as an aid to other
researchers using similar models, we derive the full VMP update equations for
LDA in Section 5. This is the first time, to the best of our knowledge, that
these equations have been published.
## 2 Related work
A goal of probabilistic modelling is to differentiate between the model and
the inference algorithm to answer queries about the model. Concerning the
modelling aspect, two prominent LDA models exist, a non-smoothed and a
smoothed version [1]. The smoothed version, which is considered here, has
become the standard version for topic modelling, and we will only consider it
further here.
Most related to our work is the inference aspect of probabilistic modelling,
particularly the different inference techniques that have been proposed for
LDA. As far as inference is concerned, there is generally a trade-off between
inference quality, speed of execution, and guarantees such as whether converge
is guaranteed. Below we mention some of the inference techniques and how they
relate to VMP.
Collapsed Gibbs sampling (a type of MCMC technique) is often the inference
technique of choice for LDA because it is theoretically exact in the limit.
Although it provides high quality inference results, it often requires a
prolonged run-time for LDA.
Variational Bayesian inference is not exact, but is usually faster, is
guaranteed to converge, and provides some guarantees on the quality of the
result. The smoothed version of LDA was first introduced using variational
Bayesian inference [1]. An online version was later introduced in 2011 [30],
and a stochastic variant in 2013 [31]. Both of these later methods are
beneficial when there are larger amounts of data, but they do not improve LDA
performance as much as other later methods [30, 31, 23, 32].
Structured stochastic variational inference, introduced in 2015 [33], further
improved performance and scalability. Standard VB, however, is still a popular
technique for LDA due to its simplicity and many Python implementations.
Variational message passing (VMP) [34, 35] is the message passing equivalent
of the standard version of VB. It is a useful tool for constructing a
variational inference solution for a large variety of conjugate exponential
graphical models. A non-conjugate variant of VMP, namely non-conjugate VPM
(NCVMP) is also available for certain other models [36], but not applicable to
this work. An advantage of VMP is that it can speed up the process of deriving
a variational solution to a new graphical model [19].
There are software toolkits that implement VMP for general graphical models
[28, 29], but none, as far as we can tell, are able to do VMP for LDA-type
models at the moment. For Infer.NET, arguably the most well-known VMP toolkit,
the reason is that the current software implementation stores all intermediate
messages as separate objects in memory, and the VMP messages for a typical
Dirichlet-multinomial distribution used in LDA would take too much memory.
Although there are no immediate plans to change the implementation to allow
these updates [37], we believe that doing this could be fruitful future work.
Unfortunately, VMP update equations are therefore sometimes derived by hand,
but these results are usually not published [38, 12], in contrast to the
current work.
## 3 Background
In this section we present the graphical model for LDA, and also introduce the
exponential family.
### 3.1 The latent Dirichlet allocation (LDA) graphical model
Latent Dirichlet Allocation (LDA) is a hierarchical graphical model that can
be represented by the directed graphical model shown in Figure 1 [1] (see
Table 1 for details about the symbols).
The graphical model shown in Figure 1 allows us to visually identify the
conditional independence assumptions in the LDA model. Arrows in the graph
indicate the direction of dependence. From Figure 1, we can see that the
$n$’th word in document $m$ is $W_{m,n}$. The distribution over this word
depends on the topic $Z_{m,n}$ present in the document, which, selects the
Dirichlet random vector $\mathbf{\phi}_{k}$ that describes the words present
in each topic.
Figures 3 and 4 illustrate Dirichlet distributions of cardinality $K=3$ to
illustrate the effect of $\bm{\alpha}$ on the distribution. Low values of
$\alpha_{k}$ correspond to a low bias towards the corresponding $\theta_{k}$
parameter. These biases are also called pseudocounts.
Figure 2: Unrolled BN representation of LDA for a two document corpus with two words per document. A single branch is highlighted (where the document number is $m=1$ and the word number $n=2$). For the meanings of the symbols, refer to Table 1. Table 1: Symbols used for the LDA model shown in Figure 2. Symbol | Description
---|---
$M$ | Total number of documents
$m$ | Current document
$N$ | Number of words in current document
$n$ | Current word (in document)
$K$ | Total number of topics
$k$ | Current topic
$V$ | Total number words in the vocabulary
$v$ | Current word (in vocabulary)
$\mathsf{v}$ | Observed word (in vocabulary)
$\bm{\theta}_{m}$ | Topic-document Dirichlet for document $m$
$Z_{m,n}$ | Topic-document categorical for word $n$ in document $m$
$W_{m,n}$ | Word-topic conditional categorical for word $n$ in document $m$
$\bm{\phi}_{k}$ | Word-topic Dirichlet for topic $k$
In LDA, the topic-word Dirichlet distributions can range from as low as $K=3$
for a three topic model, up to very large values of $K$ for models containing
hundreds of topics. The word-topic Dirichlet distributions typically have a
much higher cardinality, typically in the thousands or hundreds of thousands
since it corresponds to the vocabulary size.
Figure 3: Dirichlet distributions of cardinality, $3$, visualised in 3D. (a)
Dirichlet with $\bm{\alpha}=\\{1,1,1\\}$. This is the non-informative
Dirichlet. (b) Dirichlet with $\bm{\alpha}=\\{1,1,5\\}$. There is more mass on
the corner of the Z-axis due to the higher pseudocount of $\alpha_{3}$.
Figure 4: Dirichlet distributions of cardinality, $3$, visualised in 3D. For
both (a) and (b), all $3$ $\bm{\alpha}$’s are equal. (a) Dirichlet with
$\bm{\alpha}=\\{0.5,0.5,0.5\\}$. There is more mass on the corners of all $3$
axes than in the middle of the hyperplane (sparse). (b) Dirichlet with
$\bm{\alpha}=\\{5,5,5\\}$. There is less mass on the corners of all $3$ axes
than in the middle of the hyperplane.
### 3.2 The exponential family (EF) of distributions
The standard VMP algorithm is limited to distributions in the exponential
family. The exponential family is the only family of distributions with
finite-sized sufficient statistics [39, 40, 41]. Many useful distributions,
including the Dirichlet and categorical distributions, fall into this family,
as do all the distributions involved in LDA.
The probability distribution of a random vector $\bm{x}$ with parameters
$\bm{\eta}$ can always be written in the following mathematically convenient
form if it falls within the exponential family [34],
$\displaystyle p(\bm{x};\bm{\eta})$
$\displaystyle=\frac{1}{Z(\bm{\eta})}h(\bm{x})\exp\left\\{\bm{\eta}^{T}\bm{T}(\bm{x})\right\\},$
(1)
where $\bm{\eta}$ is called the natural parameters and $\bm{T}(\bm{x})$ is the
sufficient statistics vector, called so since with a sufficiently large
sample, the probability of $\bm{x}$ under $\bm{\eta}$ only depends on $\bm{x}$
through $\bm{T}(\bm{x})$. The partition function, $Z(\bm{\eta})$, normalises
the distribution to unity volume i.e.
$\displaystyle Z(\bm{\eta})=\int
h(\bm{x})\exp\left\\{\bm{\eta}^{T}\bm{T}(\bm{x})\right\\}\text{d}\bm{x}.$ (2)
We can also formulate Equation 1 as,
$\displaystyle p(\bm{x};\bm{\eta})$
$\displaystyle=h(\bm{x})\exp\left\\{\bm{\eta}^{T}\bm{T}(\bm{x})-A(\bm{\eta})\right\\},$
$\displaystyle\text{with }A(\bm{\eta})\triangleq\log Z(\bm{\eta}).$ (3)
$A(\bm{\eta})$ is known as the log-partition or cumulant function since it can
be used to find the cumulants of a distribution. Below we show the first
cumulant (the mean) of exponential family distributions which will be used
later,
$\displaystyle\nabla_{\bm{\eta}}A(\bm{\eta})$
$\displaystyle=\nabla_{\bm{\eta}}\ln\left[\int
h(\bm{x})\exp\left\\{\bm{\eta}^{T}\bm{T}(\bm{x})\right\\}\text{d}\bm{x}\right]$
$\displaystyle=\frac{1}{\int
h(\bm{x})\exp\left\\{\bm{\eta}^{T}\bm{T}(\bm{x})\right\\}\text{d}\bm{x}}\nabla_{\bm{\eta}}\left[\int
h(\bm{x})\exp\left\\{\bm{\eta}^{T}\bm{T}(\bm{x})\right\\}\text{d}\bm{x}\right]$
$\displaystyle=\int\frac{1}{Z(\bm{\eta})}h(\bm{x})\exp\left\\{\bm{\eta}^{T}\bm{T}(\bm{x})\right\\}\bm{T}(\bm{x})\text{d}\bm{x}$
$\displaystyle=\int p(\bm{x})\bm{T}(\bm{x})\text{d}\bm{x}$
$\displaystyle=\left<\bm{T}(\bm{x})\right>_{p(\bm{x})}$ (4)
This shows that we can find the first expected moment of a distribution in the
exponential family by taking the derivative of its log-partition function.
This will always be the same as finding the expected value of the sufficient
statistics vector.
## 4 Variational message passing (VMP)
Variational Bayes (VB) is a framework for approximating the full posterior
distribution over a model’s parameters and latent variables in an iterative,
Expectation Maximization (EM)-like manner [22], since the true distribution
can often not be calculated efficiently for models of interest, such as LDA.
Variational message passing (VMP) is a way to derive the VB update equations
for a given model [34, 35]. A formulation of optimization in terms of local
computations is required to translate VB into its message passing variant,
VMP.
### 4.1 The generic VMP algorithm
Conjugate-exponential models, of which LDA is an example, are models where
conjugacy exists between all parent-child relationships and where all
distributions are in the exponential family. For these models we can perform
variational inference by performing local computations and message passing.
Based on the derivation in [34], these local computations depend only on the
variables within the Markov blanket of a node. The nodes in the Markov blanket
(shown in Figure 5) are nodes that are either parents, children, or co-parents
of the node. The co-parents of a node are the parents of its children and
excludes the node itself.
For Bayesian networks, the joint distribution can be expressed in terms of the
conditional distributions at each node $x_{i}$,
$\displaystyle p(\bm{x})=\prod_{i}p(x_{i}\mid\text{pa}_{x_{i}}),$ (5)
where $\text{pa}_{x_{i}}$ are the parents of node $x_{i}$ and $x_{i}$ the
variables associated with node $x_{i}$ [42]. These variables can be either
hidden, meaning we don’t know, or or observed, meaning we know their values at
inference.
Figure 5: The Markov blanket of a node ($x_{j}$). The nodes in the Markov
blanket are the shaded nodes and are defined by the set of parents (pa),
children (ch) and co-parents (cp) of a node. The variational update equation
for a node $x_{j}$ depends only on expectations over variables appearing in
the Markov blanket of that node [35].
In this section we review the general VMP message passing update equations and
apply them to the example graphical model shown in Figure 6 to explain the
broader principles involved in VMP.
Figure 6: The child and parent nodes in the example we will use to present the
message passing equations. In the example, $\bm{x}$ is the parent node of
$\bm{y}$, and $\bm{y}$ the child of $\bm{x}$; the node names correspond to the
random vectors.
The node $\bm{x}$, which represents a random vector is the parent of $\bm{y}$.
In exponential family form this is written as,
$\displaystyle p(\bm{x};\bm{\eta})$
$\displaystyle=\frac{1}{Z(\bm{\eta})}h(\bm{x})\exp\left\\{\bm{\eta}^{T}\bm{T}(\bm{x})\right\\},$
(6)
$\displaystyle=h(\bm{x})\exp\left\\{\bm{\eta}^{T}\bm{T}(\bm{x})-A(\bm{\eta})\right\\},$
$\displaystyle\text{with }A(\bm{\eta})\triangleq\ln Z(\bm{\eta}),$ (7)
where $\bm{\eta}$ are the natural parameters, $\bm{T}(\bm{x})$ the sufficient
statistics vector, and $A(\bm{\eta})$ the log-partition function.
If we limit ourselves to distributions in this family, the prior and posterior
distributions have the same form [34]. During inference, we therefore only
need to update the values of the parameters and do not have to change the
functional form [34].
#### 4.1.1 Message to a child node
Continuing with the description of the graphical model in Figure 6, the parent
to child node message for parent node $\bm{x}$ and child node $\bm{y}$ is the
expectation of the sufficient statistics vector [35],
$\bm{\mu}^{\text{p2c}}_{\bm{x}\rightarrow\bm{y}}=\left<\bm{T}(\bm{x})\right>_{p(\bm{x})}.$
(8)
We can calculate this by using the derivative of the log-partition function as
seen in Equation 4,
$\left<\bm{T}(\bm{x})\right>_{p(\bm{x})}=\nabla_{\bm{\eta}}A(\bm{\eta}).$ (9)
The parent to child message is therefore,
$\bm{\mu}^{\text{p2c}}_{\bm{x}\rightarrow\bm{y}}=\nabla_{\bm{\eta}}A(\bm{\eta}).$
(10)
This message which contains the expected values of the natural parameters of
$\bm{x}$ now becomes the new natural parameters of $\bm{y}$. We can therefore
write the child node’s distribution as
$p(\bm{y}\mid\left<\bm{T}(\bm{x})\right>)$ after receiving a message from node
$x$.
#### 4.1.2 Message to a parent node
Because we limit ourselves to conjugate-exponential models, the exponential
form of the child distribution can always be re-arranged to match that of the
parent distribution. This is due to the multi-linear properties of conjugate-
exponential models [34].
We define the re-arranged version of the sufficient statistics as
$\bm{\varphi}(\bm{y})$. This version is in the correct functional form to send
a child to parent message [34, 35]. A child to parent message can therefore be
written as,
$\bm{\mu}^{\text{c2p}}_{\bm{y}\rightarrow\bm{x}}=\left<\bm{\varphi}(\bm{y})\right>_{p(\bm{y})}.$
(11)
Note that if any node, ${a}$, is observed then the messages are as defined
above, but with $\left<\bm{\varphi}(\bm{a})\right>$ replaced by
$\bm{\varphi}(\bm{a})$. I.e. if we know the true values, we use them.
When a parent node has received all of its required messages, we can update
its belief by finding its updated natural parameter vector
$\bm{\eta}^{\prime}$. In the general case for a graphical model containing a
set of nodes $\bm{x}=\set{{x}_{1},{x}_{2},...,{x}_{U}}$, the update for parent
${x}_{j}$ becomes,
$\
\bm{\eta}^{\prime}_{{x_{j}}}=\left\\{\bm{\mu}_{{x}_{i}\rightarrow{x}_{j}}\right\\}_{{x}_{i}\in\text{pa}_{{x}_{j}}}+\sum_{s\in\text{ch}_{{x}_{j}}}\bm{\mu}_{{x}_{s}\rightarrow{x}_{j}}.$
(12)
Updating the parent node $\bm{x}$ from the graphical model in Figure 6 will
result in the update equation
$\bm{\eta}^{\prime}_{\bm{x}}=\bm{\mu}_{\bm{\theta}\rightarrow\bm{x}}+\bm{\mu}_{\bm{y}\rightarrow\bm{x}}$.
We now present the full VMP algorithm in Algorithm 1 as given by Winn [34] but
using our notation defined above.
Algorithm 1 Variational Message Passing (VMP) [34]
Initialization:
Initialize each factor distribution $\bm{q}_{j}$ by initializing the
corresponding moment vector $\left<\bm{T}_{j}(\bm{x}_{j})\right>$ with random
vector $\bm{x}_{j}$.
Iteration:
1. 1.
For each node $\bm{x}_{j}$ in turn:
1. (a)
Retrieve messages from all parent and child nodes, as defined in Equation 8
and Equation 11. This requires child nodes to retrieve messages from the co-
parents of $\bm{x}_{j}$ (Figure 5).
2. (b)
Compute updated natural parameter vector $\bm{\eta}^{\prime}_{j}$ using
Equation 12.
3. (c)
Compute updated moment vector $\left<\bm{T}_{j}(\bm{x}_{j})\right>$ given the
new setting of the parameter vector.
$\text{ELBO}(\bm{q})$ (optional).
Termination:
If the increase in the bound is negligible or a specified number of iterations
has been reached, stop. Otherwise, repeat from step 1.
## 5 The VMP algorithm for LDA
Here we describe the distribution at each node and also derive the child to
parent and parent to child messages (where applicable) for the LDA graph. We
keep the messages dependent only on the current round of message passing which
can be considered to be one epoch.
### 5.1 Topic-document Dirichlet nodes $\bm{\theta}_{m}$
In exponential family form we can write each topic-document Dirichlet as,
$\displaystyle\ln\text{Dir}(\bm{\theta}_{{m}};\bm{\alpha}_{{m}})$
$\displaystyle=\left[\begin{array}[]{c}\alpha_{{m},1}-1\\\ \alpha_{{m},2}-1\\\
\vdots\\\
\alpha_{{m},K}-1\end{array}\right]^{T}\left[\begin{array}[]{c}\ln\theta_{{m},1}\\\
\ln\theta_{{m},2}\\\ \vdots\\\
\ln\theta_{m,K}\end{array}\right]-\ln\frac{\Gamma(\sum_{k}\alpha_{{m},k})}{\prod_{k}\Gamma(\alpha_{{m},k})}.$
(21)
For each $m$ we can identify the natural parameters as,
$\bm{\eta}_{\bm{\theta}_{{m}}}=\left[\begin{array}[]{c}\alpha_{m,1}-1\\\
\alpha_{m,2}-1\\\ \vdots\\\ \alpha_{m,K}-1\end{array}\right],$ (22)
and the sufficient statistics as,
$\bm{T}(\bm{\theta}_{m})=\left[\begin{array}[]{c}\ln\theta_{m,1}\\\
\ln\theta_{m,2}\\\ \vdots\\\ \ln\theta_{m,K}\end{array}\right].$ (23)
#### 5.1.1 Message to a child node $Z_{m,n}$
The parent to child node message (for parent node $\bm{\theta}_{m}$ and child
node $Z_{m,n}$) is the expectation of the sufficient statistics vector
(Equation 8),
$\bm{\mu}^{\text{p2c}}_{\bm{\theta}_{m}\rightarrow
Z_{w,n}}=\left<\bm{T}(\bm{\theta}_{m})\right>_{p(\bm{\theta}_{m})}.$ (24)
Using Equation 9, we can calculate this expectation using the derivative of
the log-partition function. It is shown in [34, p128] to be,
$\displaystyle\left<\ln(\theta_{k})\right>_{p(\theta_{k})}$
$\displaystyle=\psi(\alpha_{k})-\psi(\sum_{k}\alpha_{k}),$ (25)
where $\psi$ is the digamma function. The parent to child message from each
topic-document Dirichlet node is therefore,
$\bm{\mu}^{\text{p2c}}_{\bm{\theta}_{m}\rightarrow
Z_{{m},n}}=\left[\begin{array}[]{c}\psi(\alpha_{{m},1})-\psi(\sum_{k}\alpha_{{m},k})\\\
\vdots\\\
\psi(\alpha_{{m},K})-\psi(\sum_{k}\alpha_{{m},k})\end{array}\right].$ (26)
We can now insert these expected sufficient statistics at the child node. The
natural parameter vector is then,
$\bm{\eta}^{\prime}_{Z_{{m},n}}=\left[\begin{array}[]{c}\psi(\alpha_{{m},1})-\psi(\sum_{k}\alpha_{{m},k})\\\
\vdots\\\
\psi(\alpha_{{m},K})-\psi(\sum_{k}\alpha_{{m},k})\end{array}\right]=\left[\begin{array}[]{c}\ln\theta^{{}^{\prime}}_{{m},1}\\\
\vdots\\\ \ln\theta^{{}^{\prime}}_{{m},K}\end{array}\right],$ (27)
with $\bm{\theta}^{\prime}$ denoting the updated topic proportions, and
$\bm{\eta}^{\prime}_{Z_{{m},n}}$ the updated natural parameter vector. Because
these are not inherently normalised, normalisation is required to represent
them as true probability distributions. Note also the conjugacy between the
Dirichlet and categorical distributions: this allows us to simply update the
natural parameters without changing the form of the distribution [12, 34].
### 5.2 Topic-document categorical nodes $Z_{m,n}$
In exponential form we can represent the topic-document categorical
distribution for a specific word in a specific document as,
$\displaystyle\ln\text{Cat}(Z_{{m},n}\mid\bm{\theta}_{{m}})$
$\displaystyle=\left[\begin{array}[]{c}\ln\theta_{{m},1}\\\
\ln\theta_{{m},2}\\\ \vdots\\\
\ln\theta_{m,k}\end{array}\right]^{T}\left[\begin{array}[]{c}\llbracket
Z_{{m},n}=1\rrbracket\\\ \llbracket Z_{{m},n}=2\rrbracket\\\ \vdots\\\
\llbracket Z_{{m},n}=K\rrbracket\end{array}\right]-\ln(\sum_{k}\theta_{m,k}),$
(36) $\displaystyle\therefore A(\bm{\theta}_{{m}})$
$\displaystyle=\ln(\sum_{k}\theta_{m,k}),$ (37)
by applying Equation 6.
For each branch (as defined in Figure 2) we can identify the natural
parameters as,
$\bm{\eta}_{Z_{{m},n}}=\left[\begin{array}[]{c}\ln\theta_{{m},1}\\\
\ln\theta_{{m},2}\\\ \vdots\\\ \ln\theta_{m,k}\end{array}\right],$ (38)
and the sufficient statistics as,
$\bm{T}(Z_{{m},n})=\left[\begin{array}[]{c}\llbracket Z_{{m},n}=1\rrbracket\\\
\llbracket Z_{{m},n}=2\rrbracket\\\ \vdots\\\ \llbracket
Z_{{m},n}=K\rrbracket\end{array}\right].$ (39)
#### 5.2.1 Message to a parent node $\bm{\theta}_{m}$
Before deriving the message from child node $Z_{{m},{n}}$ to parent node
$\bm{\theta}_{m}$, some discussion regarding the effect of the incoming
message from node $W_{{m},{n}}$ on node $Z_{{m},{n}}$ is in order. Because
this message to the topic-document node $Z_{{m},{n}}$ (from the respective
word-topic node $W_{{m},{n}}$) is a child to parent message, this message is
added to the natural parameter vector such that,
$\displaystyle\ln\bm{\tilde{\theta}}^{{}^{\prime}}_{{m}}$
$\displaystyle=\ln\bm{\theta}_{{m}}+\ln\bm{p}_{{m},{n}}$
$\displaystyle=\ln(\bm{\theta}_{{m}}\bm{p}_{{m},{n}}),$ (40)
with $\tilde{.}$ indicating that the factor is unnormalised. Re-normalizing so
that $\sum_{k}\theta^{{}^{\prime}}_{{m},k}=1$ gives,
$\displaystyle\ln\bm{{\theta}}^{{}^{\prime}}_{{m}}$
$\displaystyle=\ln(\frac{\bm{\theta}_{{m}}\bm{p}_{{m},{n}}}{\sum_{k}\theta_{{m},k}{p}_{{m},{n},k}}),$
(41)
where $\\{p_{{m},{n},1},...,p_{{m},{n},K}\\}$ are the topic probabilities for
a specific word in a specific document as given by the message from node
${W}_{{m},{n}}$. Once each topic-document node $Z_{{m},{n}}$ has received its
message from the corresponding word-topic node $W_{m,n}$, the natural
parameter vector at node $Z_{{m},{n}}$ (from Equation 36) will have been
modified to be $\bm{\eta}^{\prime}_{Z_{{m},n}}=\ln\bm{\theta}^{\prime}_{{m}}$.
In the case where no message has been received from node ${W}_{{m},{n}}$, then
$\ln\bm{\theta}^{\prime}_{{m}}=\ln\bm{\theta}_{{m}}$. In LDA, however, we will
only ever update the topic-document Dirichlet node $\bm{\theta}_{m}$ from the
topic-document node $Z_{{m},{n}}$ after receiving a message from the word-
topic side of the graph (except at initialisation).
The message from a topic-document node $Z_{{m},{n}}$ to a topic-document node
$\bm{\theta}_{m}$ is also a child to parent message. To send a child to parent
message we need to rearrange the exponential form of the child distribution to
match the parent distribution (as presented in Section 4.1). We can rearrange
Equation 36 in terms of $\bm{\theta}^{\prime}_{{m}}$ as follows,
$\displaystyle\ln\text{Cat}({Z}_{{m},{n}}\mid\bm{\theta}_{{m}})$
$\displaystyle=\left[\begin{array}[]{c}\llbracket Z_{{m},{n}}=1\rrbracket\\\
\llbracket Z_{{m},{n}}=2\rrbracket\\\ \vdots\\\ \llbracket
Z_{{m},{n}}=K\rrbracket\end{array}\right]^{T}\left[\begin{array}[]{c}\ln\theta^{{}^{\prime}}_{{m},1}\\\
\ln\theta^{{}^{\prime}}_{{m},2}\\\ \vdots\\\
\ln\theta^{{}^{\prime}}_{{m},K}\end{array}\right]-\ln(\sum_{k}\theta^{{}^{\prime}}_{m,k}).$
(50)
The message towards node $\bm{\theta}_{m}$ is therefore,
$\displaystyle\bm{\mu}^{\text{c2p}}_{{Z}_{{m},{n}}\rightarrow\bm{\theta}_{{m}}}$
$\displaystyle=\left<\varphi({Z}_{{m},{n}})\right>_{p({Z}_{{m},{n}})}$
$\displaystyle=\left<\left[\begin{array}[]{c}\llbracket
Z_{{m},{n}}=1\rrbracket\\\ \llbracket Z_{{m},{n}}=2\rrbracket\\\ \vdots\\\
\llbracket
Z_{{m},{n}}=K\rrbracket\end{array}\right]\right>_{p({Z}_{{m},{n}})}=\left[\begin{array}[]{c}\theta^{{}^{\prime}}_{{m},1}\\\
\theta^{{}^{\prime}}_{{m},2}\\\ \vdots\\\
\theta^{{}^{\prime}}_{{m},K}.\end{array}\right]=\left[\begin{array}[]{c}\frac{\theta_{{m,1}}{p}_{{m},{n},1}}{\sum_{k}\theta_{{m},k},{p}_{{m},{n},k}}\\\
\frac{\theta_{{m,2}}{p}_{{m},{n},2}}{\sum_{k}\theta_{{m},k}{p}_{{m},{n},k}}\\\
\vdots\\\
\frac{\theta_{{m,K}}{p}_{{m},{n},K}}{\sum_{k}\theta_{{m},k}{p}_{{m},{n},k}}\end{array}\right].$
(63)
For each topic-document node $\bm{\theta}_{m}$, for a single branch in the
graph, and for a single topic ${k}$, we have the update,
$\alpha^{{}^{\prime}}_{{m},{k}}=\frac{\theta_{{m,k}}{p}_{{m},{n},k}}{\sum_{j}\theta_{{m},j}{p}_{{m},{n},j}}+\alpha^{\text{prior}}_{{m},{k}},$
(64)
with $\alpha^{\text{prior}}_{{m},{k}}$ representing the initial hyperparameter
settings. Over all words in the document we therefore have,
$\alpha^{{}^{\prime}}_{{m},{k}}=\sum_{n}\frac{\theta_{{m,k}}{p}_{{m},{n},k}}{\sum_{j}\theta_{{m},j}{p}_{{m},{n},j}}+\alpha^{\text{prior}}_{{m},{k}},$
(65)
which can also be written as,
$\alpha^{{}^{\prime}}_{{m},{k}}=\sum_{n}\frac{\theta^{{}^{\prime}}_{{m,k}}}{\sum_{j}\theta^{{}^{\prime}}_{{m},j}}+\alpha^{\text{prior}}_{{m},{k}}.$
(66)
#### 5.2.2 Message to a child node $W_{m,n}$
The message from a topic-document node $Z_{m,n}$ to a word-topic node
$W_{{m},n}$ is a parent to child message. Based on Equation 8, the message is,
$\bm{\mu}^{\text{p2c}}_{{Z}_{{m},n}\rightarrow{W}_{{m},n}}=\left<\bm{T}({Z}_{{m},n})\right>_{p({Z}_{{m},n})}.$
(67)
Equation 6 defines the log-partition function that can be used to calculate
this moment using Equation 9. To do this we need to re-parameterise the
natural parameter vector from Equation 36. This is shown below for a single
word in a single topic $k$,
$\displaystyle\eta_{k}$ $\displaystyle\equiv\ln\theta^{\prime}_{k},$
$\displaystyle\therefore\theta^{{}^{\prime}}_{k}$
$\displaystyle=e^{\eta_{k}},$ $\displaystyle\therefore A(\bm{\eta})$
$\displaystyle=\ln(\sum_{k}e^{\eta_{k}}).$
From this we can calculate the expected sufficient statistics using Equation 4
for a specific topic ${k}$:
$\displaystyle<\llbracket Z_{{m},n}=k\rrbracket>_{p({Z}_{{m},n})}$
$\displaystyle=\frac{\delta}{\delta\eta_{k}}A(\bm{\eta})$
$\displaystyle=\frac{e^{\eta_{k}}}{\sum_{j}e^{\eta_{j}}}$
$\displaystyle=\frac{\theta^{{}^{\prime}}_{{m},{k}}}{\sum_{j}\theta^{{}^{\prime}}_{{m},{j}}}$
$\displaystyle=\theta^{*}_{{m},{{k}}},$ $\displaystyle\text{where the
$\theta^{*}$'s are normalised}.$ (68)
Each $\theta^{*}_{{m},{k}}$ is a normalised topic proportion for a single
topic. We can write the full parent to child message for a word within a topic
as,
$\bm{\mu}^{p2c}_{{Z}_{m,n}\rightarrow{W}_{{m},n}}=\left[\begin{array}[]{c}\theta^{*}_{{m},1}\\\
\vdots\\\ \theta^{*}_{{m},K}\end{array}\right],$ (69)
with,
$\theta^{*}_{{m},k}=\frac{\theta^{{}^{\prime}}_{{m},{k}}}{\sum_{j}\theta^{{}^{\prime}}_{{m},{j}}}.$
(70)
These updated topic proportions are then used at the word-topic node
${W}_{{m},n}$.
### 5.3 Word-topic conditional categorical nodes $W_{m,n}$
Initially, the $n$th word of a document is described by $K$ word-topic
distributions. We call this a conditional categorical distribution; for a
topic ${k}$ this reduces to a single categorical distribution. For all $K$ we
can write,
$\displaystyle\ln\text{Cat}({W}_{{m},n}\mid{Z}_{{m},n},\bm{\Phi})$
$\displaystyle=\left[\begin{array}[]{c}\sum_{k}\llbracket
Z_{{m},n}={k}\rrbracket\ln\phi_{k,1}\\\ \sum_{k}\llbracket
Z_{{m},n}={k}\rrbracket\ln\phi_{k,2}\\\ \vdots\\\ \sum_{k}\llbracket
Z_{{m},n}={k}\rrbracket\ln\phi_{k,V}\end{array}\right]^{T}\left[\begin{array}[]{c}\llbracket
W_{{m},n}=1\rrbracket\\\ \llbracket W_{{m},n}=2\rrbracket\\\ \vdots\\\
\llbracket W_{{m},n}=V\rrbracket\end{array}\right]$ (79)
$\displaystyle-\sum_{k}\llbracket
Z_{{m},n}={k}\rrbracket\ln(\sum_{v}\phi_{k,v}),$
where the vocabulary over all words ranges from $1$ to $V$.
#### 5.3.1 Message to a categorical parent node $Z_{m,n}$
Each word-topic node ${W}_{{m},{n}}$ is a child of a topic-document node
${Z}_{m,n}$; we therefore need to send child to parent messages between each
pair of nodes. We can rewrite Equation 79 in terms of ${Z}_{m,n}$ to give,
$\displaystyle\ln\text{Cat}({W}_{{m},{n}}\mid{Z}_{{m},{n}},\bm{\Phi})$
$\displaystyle=\left[\begin{array}[]{c}\sum_{v}\llbracket
W_{m,n}=v\rrbracket\ln\phi_{1,v}\\\ \vdots\\\ \sum_{v}\llbracket
W_{m,n}=v\rrbracket\ln\phi_{k,v}\\\ \vdots\\\ \sum_{v}\llbracket
W_{m,n}=v\rrbracket\ln\phi_{K,v}\end{array}\right]^{T}\left[\begin{array}[]{c}\llbracket{Z}_{m,n}=1\rrbracket\\\
\vdots\\\ \llbracket{Z}_{m,n}=k\rrbracket\\\ \vdots\\\
\llbracket{Z}_{m,n}=K\rrbracket\end{array}\right].$ (90)
After observing the word $\mathsf{v}$, Equation 90 reduces to a categorical
form (this is always the case in standard LDA). Using Equation 68, we can then
write the word-topic child to parent message as,
$\displaystyle\bm{\widetilde{\mu}}^{\text{c2p}}_{{W}_{m,n}\rightarrow{Z}_{m,n}}$
$\displaystyle=\left[\begin{array}[]{c}\ln{\phi}_{1,\mathsf{v}}\\\ \vdots\\\
\ln{\phi}_{k,\mathsf{v}}\\\ \vdots\\\
\ln{\phi}_{K,\mathsf{v}}\end{array}\right],$ (96)
where $\mathsf{v}$ is the observed word, and the message is unnormalised. This
is because we have taken a slice through the word-topic distributions for a
specific word, which means that the result is not a true distribution. We
normalise the message to obtain the topic proportions (for each word in each
document), which gives,
$\displaystyle\bm{{\mu}}^{\text{c2p}}_{{W}_{m,n}\rightarrow{Z}_{m,n}}$
$\displaystyle=\left[\begin{array}[]{c}\ln\phi^{*}_{1,\mathsf{v}}\\\ \vdots\\\
\ln\phi^{*}_{k,\mathsf{v}}\\\ \vdots\\\
\ln\phi^{*}_{K,\mathsf{v}}\end{array}\right],$ (102)
with,
$\phi^{*}_{k,\mathsf{v}}=\frac{\phi_{k,\mathsf{v}}}{\sum_{j}\phi_{k,\mathsf{v}}}.$
(103)
To determine the updated document topic proportions we update the natural
parameter vector by adding these topic weightings to the current document
topic proportions,
$\bm{\eta}^{\prime}_{{Z}_{m,n}}=\left[\begin{array}[]{c}\ln\theta_{{m},1}+\ln\phi^{*}_{1,\mathsf{v}}\\\
\vdots\\\ \ln\theta_{{m},K}+\ln\phi^{*}_{K,\mathsf{v}}\end{array}\right].$
(104)
#### 5.3.2 Message to a Dirichlet parent node $\bm{\phi}_{k}$
To send child to parent messages from the a word-topic node ${W}_{{m},n}$ to
each word-topic node $\bm{\phi}_{k}$, re-parameterisation is required.
After parameterisation in terms of $\phi_{k}$ we have,
$\displaystyle\ln\text{Cat}({W}_{{m},n}\mid{Z}_{{m},n},\bm{\phi}_{{k}})$
$\displaystyle=\left[\begin{array}[]{c}\llbracket{Z}_{{m},n}={k}\rrbracket\llbracket
W_{{m},n}=1\rrbracket\\\ \vdots\\\
\llbracket{Z}_{{m},n}={k}\rrbracket\llbracket W_{{m},n}=v\rrbracket\\\
\vdots\\\ \llbracket{Z}_{{m},n}={k}\rrbracket\llbracket
W_{{m},n}=V\rrbracket\end{array}\right]^{T}\left[\begin{array}[]{c}\ln\phi_{k,1}\\\
\vdots\\\ \ln\phi_{k,v}\\\ \vdots\\\ \ln\phi_{k,V}\end{array}\right]$ (115)
$\displaystyle+\text{ terms involving }\ln\bm{\phi}_{{Z}_{{m},n}\neq{k}},$
where $\phi_{k,\mathsf{v}}$ are the topic proportions for word $\mathsf{v}$ in
topic ${k}$. The messages from one of these categorical beliefs to
$\bm{\phi}_{k}$ can then be written as,
$\displaystyle\bm{\mu}^{\text{c2p}}_{W_{{m},{n}}\rightarrow\bm{\phi}_{k}}$
$\displaystyle=\left<\varphi({W_{{m},{n}},Z_{{m},n}=k})\right>_{p(W_{{m},{n}},Z_{m,n}=k)}$
$\displaystyle=\left<\llbracket
Z_{{m},n}=k\rrbracket\llbracket{W}_{{m},{n}}=v\rrbracket\right>_{p({W}_{{m},{n}},Z_{{m},n}=k)}$
$\displaystyle=\left<\llbracket
Z_{{m},n}=k\rrbracket\right>_{p({Z_{{m},n}=k)}},\text{ because $W_{{m},{n}}$
is observed}.$
Note that because ${W}_{m,n}$ is observed, the values in the vector for all
entries except for where ${W}_{{m},{n}}={v}$, are zero. To update
$\bm{\phi}_{{k}}$, we simply add all the incoming message to the respective
$\bm{\beta}_{k}$ values. For a specific word in the vocabulary ${v}$ this
would be,
$\displaystyle\beta^{{}^{\prime}}_{k,{v}}$
$\displaystyle=\sum_{m}\sum_{n}\theta^{*}_{{m},k}+\beta^{\text{prior}}_{k,{v}},$
(117)
with $\theta^{*}_{{m},k}$ denoting the normalised probability of topic $k$ for
document $m$. We can see that the scaled topic proportions for word
$\mathsf{v}$ are simply added to the respective word’s word-topic Dirichlet’s
parameters.
### 5.4 Word-topic Dirichlet nodes $\bm{\phi}_{k}$
The word-topic distribution factors are of Dirichlet form. For the entire
graph, we have: $\bm{\Phi}=\\{\bm{\phi}_{1},...,\bm{\phi}_{K}\\}$. For each
topic $k$ we write,
$\displaystyle\ln\text{Dir}(\bm{\phi}_{{k}};\bm{\beta}_{{k}})$
$\displaystyle=\left[\begin{array}[]{c}\beta_{{k},1}-1\\\ \beta_{{k},2}-1\\\
\vdots\\\
\beta_{{k},V}-1\end{array}\right]^{T}\left[\begin{array}[]{c}\ln\phi_{{k},1}\\\
\ln\phi_{{k},2}\\\ \vdots\\\
\ln\phi_{{k},V}\end{array}\right]-\ln\frac{\Gamma(\sum_{v}\beta_{k,v})}{\prod_{v}\Gamma(\beta_{{k},v})}.$
(126)
#### 5.4.1 Message to a child node $W_{m,n}$
These messages are very similar to the ones on the topic-document side of the
graph since they are also parent to child messages with each parent having a
Dirichlet form.
For each topic ${k}$ the messages sent to all word-topic nodes $W_{{m},{n}}$
(one for each word in each topic) will be identical,
$\bm{\mu}^{\text{p2c}}_{\bm{\phi}_{{k}}\rightarrow
W_{m,n}}=\left[\begin{array}[]{c}\psi(\beta_{{k},1})-\psi(\sum_{v}\beta_{{k},v})\\\
\vdots\\\ \psi(\beta_{{k},V})-\psi(\sum_{v}\beta_{{k},v})\end{array}\right].$
(127)
The additional complexity comes from the fact that the child nodes
$W_{{m},{n}}$ need to assimilate messages from $K$ Dirichlet distributions and
not only from one, as in the topic-document side of the graph.
We now perform a similar update to the update seen in Equation 27, except that
we have $K$ messages added instead of only one. For each ${k}$ we have,
$\bm{\eta}^{\prime}_{\bm{\phi}_{{k}}}=\left[\begin{array}[]{c}\psi(\beta_{k,1})-\psi(\sum_{v}\beta_{k,v})\\\
\vdots\\\
\psi(\beta_{{k},V})-\psi(\sum_{v}\beta_{{k},v})\end{array}\right]=\left[\begin{array}[]{c}\ln\phi^{{}^{\prime}}_{k,1}\\\
\vdots\\\ \ln\phi^{{}^{\prime}}_{k,V}\end{array}\right],$ (128)
where $\phi^{{}^{\prime}}$ denotes the updated values.
We have now presented the VMP message updates for each node in the LDA
graphical model.
Using these messages, VMP for LDA can be implemented using a range of message
passing schedules. In the next section we provide one such message passing
schedule.
## 6 Message passing schedule
Because LDA has a simple, known structure per document, it is sensible to
construct a fixed message passing schedule. This is not always the case for
graphical models, for example in some cases we chose rather to base the
schedule on message priority using divergence measures to prioritise messages
according to their expected impact [43, 44, 45, 46].
In Algorithm 2, we presented our proposed VMP message passing schedule for
LDA. It is based on the message passing schedule of the approximate loopy
belief update (ALBU) VMP implementation in [46] that uses a form of belief
propagation [47, 42][p364-366]. There are, of course, are many other variants
that one could use.
Algorithm 2 Message passing schedule for LDA
For each epoch:
* For each document $\bm{m}$:
* For each word $\bm{n}$ in document $\bm{m}$:
* –
send messages from each node $\bm{\phi}_{k}$ to node $W_{m,n}$
* –
observe word $W_{m,n}=\mathsf{v}$
* –
send message from node $W_{m,n}$ to node ${Z_{m,n}}$
* –
send message from node $Z_{m,n}$ to $\bm{\theta}_{m}$
For each word $\bm{n}$ in document $\bm{m}$:
* –
send message from node $\bm{\theta}_{m}$ to node $Z_{m,n}$
* –
send message from node $Z_{m,n}$ to node $W_{m,n}$
For each word $n$ in each document $m$:
* –
send messages from node ${W_{m,n}}$ to each $\bm{\phi}_{k}$
Based on this schedule, as well as the message passing equations provided in
4, VMP can be implemented for LDA.
## 7 Conclusion and future work
VMP, an elegant and tenable solution to inference problems, has not been
presented in detail for the standard, smoothed LDA graphical model, which is
surprising in view of its speed and ease of use.
In this article, we provided an introduction to variational message passing
(VMP), the message passing equivalent of VB. We present the generic VMP
algorithm and then applied VMP to the LDA graphical model. Finally we proposed
a message passing schedule for VMP for LDA. For future work, we recommend that
VMP and VB be compared in terms of execution time for LDA, and that
alternative message passing schedules be investigated to improve execution
time and convergence rate. We also recommend that the VMP equations for other,
similar graphical models be derived and published in a similar manner.
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|
# Using Social Dynamics to Make Individual Predictions:
Variational Inference with a Stochastic Kinetic Model
Zhen Xu, Wen Dong, and Sargur Srihari
Department of Computer Science and Engineering
University at Buffalo
<EMAIL_ADDRESS>
###### Abstract
Social dynamics is concerned primarily with interactions among individuals and
the resulting group behaviors, modeling the temporal evolution of social
systems via the interactions of individuals within these systems. In
particular, the availability of large-scale data from social networks and
sensor networks offers an unprecedented opportunity to predict state-changing
events at the individual level. Examples of such events include disease
transmission, opinion transition in elections, and rumor propagation. Unlike
previous research focusing on the collective effects of social systems, this
study makes efficient inferences at the individual level. In order to cope
with dynamic interactions among a large number of individuals, we introduce
the stochastic kinetic model to capture adaptive transition probabilities and
propose an efficient variational inference algorithm the complexity of which
grows _linearly_ — rather than exponentially— with the number of individuals.
To validate this method, we have performed epidemic-dynamics experiments on
wireless sensor network data collected from more than ten thousand people over
three years. The proposed algorithm was used to track disease transmission and
predict the probability of infection for each individual. Our results
demonstrate that this method is more efficient than sampling while nonetheless
achieving high accuracy.
## 1 Introduction
The field of social dynamics is concerned primarily with interactions among
individuals and the resulting group behaviors. Research in social dynamics
models the temporal evolution of social systems via the interactions of the
individuals within these systems [8]. For example, opinion dynamics can model
the opinion state transitions of an entire population in an election scenario
[3], and epidemic dynamics can predict disease outbreaks ahead of time [9].
While traditional social-dynamics models focus primarily on the macroscopic
effects of social systems, often we instead wish to know the answers to more
specific questions. Given the movement and behavior history of a subject with
Ebola, can we tell how many people should be tested or quarantined? City-size
quarantine is not necessary, but family-size quarantine is insufficient. We
aim to model a method to evaluate the paths of illness transmission and the
risks of infection for _individuals_ , so that limited medical resources can
be most efficiently distributed.
The rapid growth of both social networks and sensor networks offers an
unprecedented opportunity to collect abundant data at the individual level.
From these data we can extract temporal interactions among individuals, such
as meeting or taking the same class. To take advantage of this opportunity, we
model social dynamics from an individual perspective. Although such an
approach has considerable potential, in practice it is difficult to model the
dynamic interactions and handle the costly computations when a large number of
individuals are involved. In this paper, we introduce an event-based model
into social systems to characterize their temporal evolutions and make
tractable inferences on the individual level.
Our research on the temporal evolutions of social systems is related to
dynamic Bayesian networks and continuous time Bayesian networks [20, 12, 17].
Traditionally, a coupled hidden Markov model is used to capture the
interactions of components in a system [2], but this model does not consider
dynamic interactions. However, a stochastic kinetic model is capable of
successfully describing the interactions of molecules (such as collisions) in
chemical reactions [21, 11], and is widely used in many fields such as
chemistry and cell biology [1, 10]. We introduce this model into social
dynamics and use it to focus on individual behaviors.
A challenge in capturing the interactions of individuals is that in social
dynamics the state space grows exponentially with the number of individuals,
which makes exact inference intractable. To resolve this we must apply
approximate inference methods. One class of these involves sampling-based
methods. Rao and Teh introduce a Gibbs sampler based on local updates [19],
while Murphy and Russell introduce Rao-Blackwellized particle filtering for
dynamic Bayesian networks [16]. However, sampling-based methods sometimes mix
slowly and require a large number of samples/particles. To demonstrate this
issue, we offer empirical comparisons with two major sampling methods in
Section 4. An alternative class of approximations is based on variational
inference. Opper and Sanguinetti apply the variational mean field approach to
factor a Markov jump process [18], and Cohn and El-Hay further improve its
efficiency by exploiting the structure of the target network [4]. A problem is
that in an event-based model such as a stochastic kinetic model (SKM), the
variational mean field is not applicable when a single event changes the
states of two individuals simultaneously. Here, we use a general expectation
propagation principle [13] to design our algorithm.
This paper makes three contributions: First, we introduce the discrete event
model into social dynamics and make tractable inferences on both individual
behaviors and collective effects. To this end, we apply the stochastic kinetic
model to define adaptive transition probabilities that characterize the
dynamic interaction patterns in social systems. Second, we design an efficient
variational inference algorithm whose computation complexity grows linearly
with the number of individuals. As a result, it scales very well in large
social systems. Third, we conduct experiments on epidemic dynamics to
demonstrate that our algorithm can track the transmission of epidemics and
predict the probability of infection for each individual. Further, we
demonstrate that the proposed method is more efficient than sampling while
nonetheless achieving high accuracy.
The remainder of this paper is organized as follows. In Section 2, we briefly
review the coupled hidden Markov model and the stochastic kinetic model. In
Section 3, we propose applying a variational algorithm with the stochastic
kinetic model to make tractable inferences in social dynamics. In Section 4,
we detail empirical results from applying the proposed algorithm to our
epidemic data along with the proximity data collected from sensor networks.
Section 5 concludes.
## 2 Background
### 2.1 Coupled Hidden Markov Model
A coupled hidden Markov model (CHMM) captures the dynamics of a discrete time
Markov process that joins a number of distinct hidden Markov models (HMMs), as
shown in Figure 1(a). $\mathbf{x}_{t}=(x_{t}^{(1)},\dots,x_{t}^{(M)})$ defines
the hidden states of all HMMs at time $t$, and $x_{t}^{(m)}$ is the hidden
state of HMM $m$ at time $t$. $\mathbf{y}_{t}=(y_{t}^{(1)},\dots,y_{t}^{(M)})$
are observations of all HMMs at time $t$, and $y_{t}^{(m)}$ is the observation
of HMM $m$ at time $t$. $P(\mathbf{x}_{t}|\mathbf{x}_{t-1})$ are transition
probabilities, and $P(\mathbf{y}_{t}|\mathbf{x}_{t})$ are emission
probabilities for CHMM. Given hidden states, all observations are independent.
As such,
$P(\mathbf{y}_{t}|\mathbf{x}_{t})=\prod_{m}P(y_{t}^{(m)}|x_{t}^{(m)})$, where
$P(y_{t}^{(m)}|x_{t}^{(m)})$ is the emission probability for HMM $m$ at time
$t$. The joint probability of CHMM can be defined as follows:
$P\left(\mathbf{x}_{1,\dots,T},\mathbf{y}_{1,\dots,T}\right)=\prod_{t=1}^{T}P(\mathbf{x}_{t}|\mathbf{x}_{t-1})P(\mathbf{y}_{t}|\mathbf{x}_{t}).$
(1)
For a CHMM that contains $M$ HMMs in a binary state, the state space is
$2^{M}$, and the state transition kernel is a $2^{M}\times 2^{M}$ matrix. In
order to make exact inferences, the classic forward-backward algorithm sweeps
a forward/filtering pass to compute the forward statistics
$\alpha_{t}(\mathbf{x}_{t})=P(\mathbf{x}_{t}|\mathbf{y}_{1,\dots,t})$ and a
backward/smoothing pass to estimate the backward statistics
$\beta_{t}(\mathbf{x}_{t})=\frac{P(\mathbf{y}_{t+1,\dots,T}|\mathbf{x}_{t})}{P(\mathbf{y}_{t+1,\dots,T}|\mathbf{y}_{1,\dots,t})}$.
Then it can estimate the one-slice statistics
$\gamma_{t}(\mathbf{x}_{t})=P(\mathbf{x}_{t}|\mathbf{y}_{1,\dots,T})=\alpha_{t}(\mathbf{x}_{t})\beta_{t}(\mathbf{x}_{t})$
and two-slice statistics
$\xi_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t})=P(\mathbf{x}_{t-1},\mathbf{x}_{t}|\mathbf{y}_{1,\dots,T})=\frac{\alpha_{t-1}(\mathbf{x}_{t-1})P(\mathbf{x}_{t}|\mathbf{x}_{t-1})P(\mathbf{y}_{t}|\mathbf{x}_{t})\beta_{t}(\mathbf{x}_{t})}{P(\mathbf{y}_{t}|\mathbf{y}_{1,\dots,t-1})}$.
Its complexity grows exponentially with the number of HMM chains. In order to
make tractable inferences, certain factorizations and approximations must be
applied. In the next section, we introduce a stochastic kinetic model to lower
the dimensionality of transition probabilities.
Figure 1: Illustration of (a) Coupled Hidden Markov Model, (b) Stochastic
Kinetic Model.
### 2.2 The Stochastic Kinetic Model
A stochastic kinetic model describes the temporal evolution of a chemical
system with $M$ species $\mathcal{X}=\\{X_{1},X_{2},\cdots,X_{M}\\}$ driven by
$V$ events (or chemical reactions) parameterized by rate constants
$\mathbf{c}=(c_{1},\dots,c_{V})$. An event (chemical reaction) $k$ has a
general form as follows:
$\displaystyle
r_{1}X_{1}+\cdots+r_{M}X_{M}\overset{c_{k}}{\longrightarrow}p_{1}X_{1}+\cdots+p_{M}X_{M}.$
The species on the left are called _reactants_ , and $r_{m}$ is the number of
$m$th reactant molecules consumed during the reaction. The species on the
right are called _products_ , and $p_{m}$ is the number of $m$th product
molecules produced in the reaction. Species involved in the reaction
($r_{m}>0$) without consumption or production ($r_{m}=p_{m}$) are called
_catalysts_. At any specific time $t$, the populations of the species is
$\mathbf{x_{t}}=(x_{t}^{(1)},\dots,x_{t}^{(M)})$. An event $k$ happens with
rate $h_{k}(\mathbf{x_{t}},c_{k})$, determined by the rate constant and the
current population state [21]:
$\displaystyle h_{k}(\mathbf{x_{t}},c_{k})=$ $\displaystyle
c_{k}g_{k}(\mathbf{x_{t}})=c_{k}\prod_{m=1}^{M}g_{k}^{(m)}(x_{t}^{(m)}).$ (2)
The form of $g_{k}(\mathbf{x_{t}})$ depends on the reaction. In our case, we
adopt the product form $\prod_{m=1}^{M}g_{k}^{(m)}(x_{t}^{(m)})$, which
represents the total number of ways that reactant molecules can be selected to
trigger event $k$ [21]. Event $k$ changes the populations by
$\mathbf{\Delta_{k}}=\mathbf{x}_{t}-\mathbf{x}_{t-1}$. The probability that
event $k$ will occur during time interval $(t,t+dt]$ is
$h_{k}(\mathbf{x_{t}},c_{k})dt$. We assume at each discrete time step that no
more than one event will occur. This assumption follows the linearization
principle in the literature [17], and is valid when the discrete time step is
small. We treat each discrete time step as a unit of time, so that
$h_{k}(\mathbf{x_{t}},c_{k})$ represents the probability of an event.
In epidemic modeling, for example, an infection event $v_{i}$ has the form
$S+I\overset{c_{i}}{\longrightarrow}2I$, such that a susceptible individual
($S$) is infected by an infectious individual ($I$) with rate constant
$c_{i}$. If there is only one susceptible individual (type $m=1$) and one
infectious individual (type $m=2$) involved in this event,
$h_{i}(\mathbf{x_{t}},c_{i})=c_{i}$, $\mathbf{\Delta_{i}}=[-1~{}~{}1]^{T}$ and
$P(\mathbf{x}_{t}-\mathbf{x}_{t-1}=\mathbf{\Delta_{i}})=P(\mathbf{x}_{t}|\mathbf{x}_{t-1},v_{i})=c_{i}$.
In a traditional hidden Markov model, the transition kernel is typically
fixed. In comparison, SKM is better at capturing dynamic interactions in terms
of the events with rates dependent on reactant populations, as shown in
Eq.(2).
## 3 Variational Inference with the Stochastic Kinetic Model
In this section, we define the likelihood of the entire sequence of hidden
states and observations for an event-based model, and derive a variational
inference algorithm and parameter-learning algorithm.
### 3.1 Likelihood for Event-based Model
In social dynamics, we use a discrete time Markov model to describe the
temporal evolutions of a set of individuals $x^{(1)},\dots,x^{(M)}$ according
to a set of $V$ events. To cope with dynamic interactions, we introduce the
SKM and express the state transition probabilities in terms of event
probabilities, as shown in Figure 1(b). We assume at each discrete time step
that no more than one event will occur. Let $v_{1},\dots,v_{T}$ be a sequence
of events, $\mathbf{x_{1}},\dots,\mathbf{x_{T}}$ a sequence of hidden states,
and $\mathbf{y_{1}},\dots,\mathbf{y_{T}}$ a set of observations. Similar to
Eq.(1), the likelihood of the entire sequence is as follows:
$\displaystyle
P\left(\mathbf{x}_{1,\dots,T},\mathbf{y}_{1,\dots,T},v_{1,\dots,T}\right)=\prod_{t=1}^{T}P(\mathbf{x}_{t},v_{t}|\mathbf{x}_{t-1})P(\mathbf{y}_{t}|\mathbf{x}_{t}),\mbox{
where }$ (3) $\displaystyle
P(\mathbf{x}_{t},v_{t}|\mathbf{x}_{t-1})=\begin{cases}c_{k}\cdot
g_{k}\left(\mathbf{x}_{t-1}\right)\cdot\delta(\mathbf{x}_{t}-\mathbf{x}_{t-1}\equiv\mathbf{\Delta_{k}})&\mbox{if
}v_{t}=k\\\
(1-\sum_{k}c_{k}g_{k}\left(\mathbf{x}_{t-1}\right))\cdot\delta(\mathbf{x}_{t}-\mathbf{x}_{t-1}\equiv\mathbf{0})&\mbox{if
}v_{t}=\emptyset\end{cases}.$
$P(\mathbf{x}_{t},v_{t}|\mathbf{x}_{t-1})$ is the event-based transition
kernel. $\delta(\mathbf{x}_{t}-\mathbf{x}_{t-1}\equiv\mathbf{\Delta_{k}})$ is
1 if the previous state is $\mathbf{x}_{t-1}$ and the current state is
$\mathbf{x}_{t}=\mathbf{x}_{t-1}+\mathbf{\Delta_{k}}$, and 0 otherwise.
$\mathbf{\Delta_{k}}$ is the effect of event $v_{k}$. $\emptyset$ represents
an auxiliary event, meaning that there is no event. Substituting the product
form of $g_{k}$, the transition kernel can be written as follows:
$\displaystyle
P(\mathbf{x}_{t},v_{t}=k|\mathbf{x}_{t-1})=c_{k}\prod_{m}g_{k}^{(m)}(x_{t-1}^{(m)})\cdot\prod_{m}\delta(x_{t}^{(m)}-x_{t-1}^{(m)}\equiv\Delta_{k}^{(m)}),$
(4) $\displaystyle
P(\mathbf{x}_{t},v_{t}=\emptyset|\mathbf{x}_{t-1})=(1-\sum_{k}c_{k}\prod_{m}g_{k}^{(m)}(x_{t-1}^{(m)}))\cdot\prod_{m}\delta(x_{t}^{(m)}-x_{t-1}^{(m)}\equiv
0),$ (5)
where $\delta(x_{t}^{(m)}-x_{t-1}^{(m)}\equiv\Delta_{k}^{(m)})$ is 1 if the
previous state of an individual $m$ is $x_{t-1}^{(m)}$ and the current state
is $x_{t}^{(m)}=x_{t-1}^{(m)}+\Delta_{k}^{(m)}$, and 0 otherwise.
### 3.2 Variational Inference for Stochastic Kinetic Model
As noted in Section 2.1, exact inference in social dynamics is intractable due
to the formidable state space. However, we can approximate the posterior
distribution $P(\mathbf{x}_{1,...,T},v_{1,...,T}|\mathbf{y}_{1,...,T})$ using
an approximate distribution within the exponential family. The inference
algorithm minimizes the KL divergence between these two distributions, which
can be formulated as an optimization problem [13]:
$\displaystyle\mbox{Minimize:}\sum_{t,\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})\cdot\log\frac{\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})}{P(\mathbf{x}_{t},v_{t}|\mathbf{x}_{t-1})P(\mathbf{y}_{t}|\mathbf{x}_{t})}$
(6) $\displaystyle\hskip
180.00027pt-\sum_{t,\mathbf{x}_{t}}\prod_{m}\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})\log\prod_{m}\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})$
$\displaystyle\mbox{Subject to:
}\sum_{v_{t},\mathbf{x}_{t-1},\\{\mathbf{x}_{t}\backslash
x_{t}^{(m)}\\}}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})=\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})\mbox{,
for all }t,m,x_{t}^{(m)},$ $\displaystyle\hphantom{\mbox{Subject to:
}}\sum_{v_{t},\\{\mathbf{x}_{t-1}\backslash
x_{t-1}^{(m)}\\},\mathbf{x}_{t}}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})=\hat{\gamma}_{t-1}^{(m)}(x_{t-1}^{(m)})\mbox{,
for all }t,m,x_{t-1}^{(m)},~{}$ $\displaystyle\hphantom{\mbox{Subject to:
}}\sum_{x_{t}^{(m)}}\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})=1\mbox{, for all
}t,m.$
The objective function is the Bethe free energy, composed of average energy
and Bethe entropy approximation [22].
$\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})$ is the approximate two-
slice statistics and $\hat{\gamma}^{(m)}(x_{t}^{(m)})$ is the approximate one-
slice statistics for each individual $m$. They form the approximate
distribution over which to minimize the Bethe free energy. The
$\sum_{t,\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}}$ is an abbreviation for
summing over $t$, $\mathbf{x}_{t-1}$, $\mathbf{x}_{t}$, and $v_{t}$.
$\sum_{\\{\mathbf{x}_{t}\backslash x_{t}^{(m)}\\}}$ is the sum over all
individuals in $\mathbf{x_{t}}$ except $x_{t}^{(m)}$. We use similar
abbreviations below. The first two sets of constraints are marginalization
conditions, and the third is normalization conditions. To solve this
constrained optimization problem, we first define the Lagrange function using
Lagrange multipliers to weight constraints, then take the partial derivatives
with respect to $\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})$, and
$\hat{\gamma}^{(m)}(x_{t}^{(m)})$. The dual problem is to find the approximate
forward statistics $\hat{\alpha}_{t-1}^{(m)}(x_{t-1}^{(m)})$ and backward
statistics $\hat{\beta}_{t}^{(m)}(x_{t}^{(m)})$ in order to maximize the
pseudo-likelihood function. The duality is between minimizing Bethe free
energy and maximizing pseudo-likelihood. The fixed-point solution for the
primal problem is as follows111The derivations for the optimization problem
and its solution are shown in the Supplemental Material.:
$\displaystyle\hat{\xi}(x_{t-1}^{(m)},x_{t}^{(m)},v_{t})=\frac{1}{Z_{t}}\sum_{m^{\prime}\neq
m,x_{t-1}^{(m^{\prime})},x_{t}^{(m^{\prime})}}{\scriptstyle
P(\mathbf{x}_{t},v_{t}|\mathbf{x}_{t-1})\cdot\prod_{m}\hat{\alpha}_{t-1}^{(m)}(x_{t-1}^{(m)})\cdot\prod_{m}P(y_{t}^{(m)}|x_{t}^{(m)})\cdot\prod_{m}\hat{\beta}_{t}^{(m)}(x_{t}^{(m)})}.$
(7)
$\hat{\xi}(x_{t-1}^{(m)},x_{t}^{(m)},v_{t})$ is the two-slice statistics for
an individual $m$, and $Z_{t}$ is the normalization constant. Given the
factorized form of $P(\mathbf{x}_{t},v_{t}|\mathbf{x}_{t-1})$ in Eqs. (4) and
(5), everything in Eq. (7) can be written in a factorized form. After
reformulating the term relevant to the individual $m$,
$\hat{\xi}(x_{t-1}^{(m)},x_{t}^{(m)},v_{t})$ can be shown neatly as follows:
$\displaystyle\hat{\xi}_{t}(x_{t-1}^{(m)},x_{t}^{(m)},v_{t})=\frac{1}{Z_{t}}\hat{P}(x_{t}^{(m)},v_{t}|x_{t-1}^{(m)})\cdot\hat{\alpha}_{t-1}^{(m)}(x_{t-1}^{(m)})P(y_{t}^{(m)}|x_{t}^{(m)})\hat{\beta}_{t}^{(m)}(x_{t}^{(m)}),$
(8)
where the marginalized transition kernel
$\hat{P}(x_{t}^{(m)},v_{t}|x_{t-1}^{(m)})$ for the individual $m$ can be
defined as:
$\displaystyle\hat{P}(x_{t}^{(m)},v_{t}=k|x_{t-1}^{(m)})={\displaystyle
c_{k}g_{k}^{(m)}(x_{t-1}^{(m)})\prod\limits_{m^{\prime}\neq
m}\tilde{g}_{k,t-1}^{(m^{\prime})}\cdot\delta(x_{t}^{(m)}-x_{t-1}^{(m)}\equiv\Delta_{k}^{(m)})},$
(9)
$\displaystyle\hat{P}(x_{t}^{(m)},v_{t}=\emptyset|x_{t-1}^{(m)})={\scriptstyle{\displaystyle\left(1-\sum\limits_{k}c_{k}g_{k}^{(m)}(x_{t-1}^{(m)})\prod\limits_{m^{\prime}\neq
m}\hat{g}_{k,t-1}^{(m^{\prime})}\right)\delta(x_{t}^{(m)}-x_{t-1}^{(m)}\equiv
0),}}$ (10)
$\displaystyle{\scriptstyle\tilde{g}_{k,t-1}^{(m^{\prime})}=\sum\limits_{x_{t}^{(m^{\prime})}-x_{t-1}^{(m^{\prime})}\equiv\Delta_{k}^{(m^{\prime})}}\alpha_{t-1}^{(m^{\prime})}(x_{t-1}^{(m^{\prime})})P(y_{t}^{(m^{\prime})}|x_{t}^{(m^{\prime})})\beta_{t}^{(m^{\prime})}(x_{t}^{(m^{\prime})})g_{k}^{(m^{\prime})}(x_{t-1}^{(m^{\prime})})\big{/}\sum\limits_{x_{t}^{(m^{\prime})}-x_{t-1}^{(m^{\prime})}\equiv
0}\alpha_{t-1}^{(m^{\prime})}(x_{t-1}^{(m^{\prime})})P(y_{t}^{(m^{\prime})}|x_{t}^{(m^{\prime})})\beta_{t}^{(m^{\prime})}(x_{t}^{(m^{\prime})})},$
$\displaystyle{\scriptstyle\hat{g}_{k,t-1}^{(m^{\prime})}=\sum\limits_{x_{t}^{(m^{\prime})}-x_{t-1}^{(m^{\prime})}\equiv
0}\alpha(x_{t-1}^{(m^{\prime})})P(y_{t}^{(m^{\prime})}|x_{t}^{(m^{\prime})})\beta_{t}^{(m^{\prime})}(x_{t}^{(m^{\prime})})g_{k}^{(m^{\prime})}(x_{t-1}^{(m^{\prime})})\big{/}\sum\limits_{x_{t}^{(m^{\prime})}-x_{t-1}^{(m^{\prime})}\equiv
0}\alpha_{t-1}^{(m^{\prime})}(x_{t-1}^{(m^{\prime})})P(y_{t}^{(m^{\prime})}|x_{t}^{(m^{\prime})})\beta_{t}^{(m^{\prime})}(x_{t}^{(m^{\prime})})},$
In the above equations, we consider the mean field effect by summing over the
current and previous states of all the other individuals $m^{\prime}\neq m$.
The marginalized transition kernel considers the probability of event $k$ on
the individual $m$ given the context of the temporal evolutions of the other
individuals. Comparing Eqs. (9) and (10) with Eqs. (4) and (5), instead of
multiplying $g_{k}^{(m^{\prime})}(x_{t-1}^{(m^{\prime})})$ for individual
$m^{\prime}\neq m$, we use the expected value of $g_{k}^{(m^{\prime})}$ with
respect to the marginal probability distribution of $x_{t-1}^{(m^{\prime})}$.
Complexity Analysis: In our inference algorithm, the most computation-
intensive step is the marginalization in Eqs. (9)-(10). The complexity is
$O(MS^{2})$, where $M$ is the number of individuals and $S$ is the state space
of a single individual. The complexity of the entire algorithm is therefore
$O(MS^{2}TN)$, where $T$ is the number of time steps and $N$ is the number of
iterations until convergence. As such, the complexity of our algorithm grows
only linearly with the number of individuals; it offers excellent scalability
when the number of tracked individuals becomes large.
### 3.3 Parameter Learning
In order to learn the rate constant $c_{k}$, we maximize the expected log
likelihood. In a stochastic kinetic model, the probability of a sample path is
given in Eq. (3). The expected log likelihood over the posterior probability
conditioned on the observations $\mathbf{y}_{1},\dots,\mathbf{y}_{T}$ takes
the following form:
$\displaystyle\log
P\left(\mathbf{x}_{1,\dots,T},\mathbf{y}_{1,\dots,T},v_{1,\dots,T}\right)=\sum_{t,\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})\cdot\log(P(\mathbf{x}_{t},v_{t}|\mathbf{x}_{t-1})P(\mathbf{y}_{t}|\mathbf{x}_{t})).$
$\hat{\xi}_{t}\left(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}\right)$ is the
approximate two-slice statistics defined in Eq. (6). Maximizing this expected
log likelihood by setting its partial derivative over the rate constants to 0
gives the maximum expected log likelihood estimation of these rate constants.
$\displaystyle
c_{k}=\frac{\sum_{t,\mathbf{x}_{t-1},\mathbf{x}_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=k)}{\sum_{t,\mathbf{x}_{t-1},\mathbf{x}_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=\emptyset)g_{k}(\mathbf{x}_{t-1})}\approx\frac{\sum_{t}\
\sum_{\mathbf{x}_{t-1},\mathbf{x}_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=k)}{\sum_{t}\
\prod_{m}\sum_{x_{t-1}^{(m)}}\hat{\gamma}_{t-1}^{(m)}(x_{t-1}^{(m)})g_{k}^{(m)}(x_{t-1}^{(m)})}.$
(11)
As such, the rate constant for event $k$ is the expected number of times that
this event has occurred divided by the total expected number of times this
event could have occurred.
To summarize, we provide the variational inference algorithm below.
Algorithm: Variational Inference with a Stochastic Kinetic Model
Given the observations $y_{t}^{(m)}$ for $t=1,\dots,T$ and $m=1,\dots,M$, find
$x_{t}^{(m)}$, $v_{t}$ and rate constants $c_{k}$ for $k=1,\dots,V$.
Latent state inference. Iterate through the following forward and backward
passes until convergence, where $\hat{P}(x_{t}^{(m)},v_{t}|x_{t-1}^{(m)})$ is
given by Eqs. (9) and (10).
* •
Forward pass. For $t=1,\dots,T$ and $m=1,\dots,M$, update
$\hat{\alpha}_{t}^{(m)}(x_{t}^{(m)})$ according to
$\displaystyle\hat{\alpha}_{t}^{(m)}(x_{t}^{(m)})\leftarrow\frac{1}{Z_{t}}\sum_{x_{t-1}^{(m)},v_{t}}\hat{\alpha}_{t-1}^{(m)}(x_{t-1}^{(m)})\hat{P}(x_{t}^{(m)},v_{t}|x_{t-1}^{(m)})P(y_{t}^{(m)}|x_{t}^{(m)}).$
* •
Backward pass. For $t=T,\dots,1$ and $m=1,\dots,M$, update
$\hat{\beta}_{t-1}^{(m)}(x_{t-1}^{(m)})$ according to
$\displaystyle\hat{\beta}_{t-1}^{(m)}(x_{t-1}^{(m)})\leftarrow\frac{1}{Z_{t}}\sum_{x_{t}^{(m)},v_{t}}\hat{\beta}_{t}^{(m)}(x_{t}^{(m)})\hat{P}(x_{t}^{(m)},v_{t}|x_{t-1}^{(m)})P(y_{t}^{(m)}|x_{t}^{(m)}).$
Parameter estimation. Iterate through the latent state inference (above) and
rate constants estimate of $c_{k}$ according to Eq. (11), until convergence.
## 4 Experiments on Epidemic Applications
In this section, we evaluate the performance of variational inference with a
stochastic kinetic model (VISKM) algorithm of epidemic dynamics, with which we
predict the transmission of diseases and the health status of each individual
based on proximity data collected from sensor networks.
### 4.1 Epidemic Dynamics
In epidemic dynamics, $G_{t}=(\mathcal{M},E_{t})$ is a dynamic network, where
each node $m\in\mathcal{M}$ is an individual in the network, and
$E_{t}=\\{(m_{i},m_{j})\\}$ is a set of edges in $G_{t}$ representing that
individuals $m_{i}$ and $m_{j}$ have interacted at a specific time $t$. There
are two possible hidden states for each individual $m$ at time $t$,
$x_{t}^{(m)}\in\\{0,1\\}$, where 0 indicates the susceptible state and 1 the
infectious state. $y_{t}^{(m)}\in\\{0,1\\}$ represents the presence or absence
of symptoms for individual $m$ at time $t$. $P(y_{t}^{(m)}|x_{t}^{(m)})$
represents the observation probability. We define three types of events in
epidemic applications: (1) A previously infectious individual recovers and
becomes susceptible again: $I\overset{c_{1}}{\longrightarrow}S$. (2) An
infectious individual infects a susceptible individual in the network:
$S+I\overset{c_{2}}{\longrightarrow}2I$. (3) A susceptible individual in the
network is infected by an outside infectious individual:
$S\overset{c_{3}}{\longrightarrow}I$. Based on these events, the transition
kernel can be defined as follows:
$\displaystyle
P(x_{t}^{(m)}=0|x_{t-1}^{(m)}=1)=c_{1},~{}P(x_{t}^{(m)}=1|x_{t-1}^{(m)}=1)=1-c_{1},$
$\displaystyle P(x_{t}^{(m)}\negthinspace=\negthinspace
0|x_{t-1}^{(m)}\negthinspace=\negthinspace
0)=(1-c_{3})(1-c_{2})^{C_{m,t}},~{}P(x_{t}^{(m)}\negthinspace=\negthinspace
1|x_{t-1}^{(m)}\negthinspace=\negthinspace 0)=1-(1-c_{3})(1-c_{2})^{C_{m,t}},$
where $C_{m,t}=\sum_{m^{\prime}:(m^{\prime},m)\in
E_{t}}\delta(x_{t}^{(m^{\prime})}\equiv 1)$ is the number of possible
infectious sources for individual $m$ at time $t$. Intuitively, the
probability of a susceptible individual becoming infected is 1 minus the
probability that no infectious individuals (inside or outside the network)
infected him. When the probability of infection is very small, we can
approximate $P(x_{t}^{(m)}=1|x_{t-1}^{(m)}=0)\approx c_{3}+c_{2}\cdot
C_{m,t}$.
### 4.2 Experimental Results
Data Explanation: We employ two data sets of epidemic dynamics. The real data
set is collected from the Social Evolution experiment [5]. This study records
“common cold” symptoms of 65 students living in a university residence hall
from January 2009 to April 2009, tracking their locations and proximities
using mobile phones. In addition, the students took periodic surveys regarding
their health status and personal interactions. The synthetic data set was
collected on the Dartmouth College campus from April 2001 to June 2004, and
contains the movement history of 13,888 individuals [15]. We synthesized
disease transmission along a timeline using the popular susceptible-
infectious-susceptible (SIS) epidemiology model [14], then applied the VISKM
to calibrate performance. We selected this data set because we want to
demonstrate that our model works on data with a large number of people over a
long period of time.
Evaluation Metrics and Baseline Algorithms: We select the receiver operating
characteristic (ROC) curve as our performance metric because the
discrimination thresholds of diseases vary. We first compare the accuracy and
efficiency of VISKM with Gibbs sampling (Gibbs) and particle filtering (PF) on
the Social Evolution data set [6, 7].222Code and data are available at
http://cse.buffalo.edu/~wendong/. Both Gibbs sampling and particle filtering
iteratively sample the infectious and susceptible latent state sequences and
the infection and recovery events conditioned on these state sequences. Gibbs-
Prediction-10000 indicates 10,000 iterations of Gibbs sampling with 1000 burn-
in iterations for the prediction task. PF-Smoothing-1000 similarly refers to
1000 iterations of particle filtering for the smoothing task. All experiments
are performed on the same computer.
Individual State Inference: We infer the probabilities of a hidden infectious
state for each individual at different times under different scenarios. There
are three tasks: 1. _Prediction_ : Given an individual’s past health and
current interaction patterns, we predict the current infectious latent state.
Figure 2(a) compares prediction performance among the different approximate
inference methods. 2. _Smoothing:_ Given an individual’s interaction patterns
and past health with missing periods, we infer the infectious latent states
during these missing periods. Figure 2(b) compares the performance of the
three inference methods. 3. _Expansion_ : Given the health records of a
portion ($\sim 10\%$) of the population, we estimate the individual infectious
states of the entire population before medically inspecting them. For example,
given either a group of volunteers willing to report their symptoms or the
symptom data of patients who came to hospitals, we determine the probabilities
that the people near these individuals also became or will become infected.
This information helps the government or aid agencies to efficiently
distribute limited medical resources to those most in need. Figure 2(c)
compares the performance of the different methods. From the above three
graphs, we can see that all three methods identify the infectious states in an
accurate way. However, VISKM outperforms Gibbs sampling and particle filtering
in terms of area under the ROC curve for all three tasks. VISKM has an
advantage in the smoothing task because the backward pass helps to infer the
missing states using subsequent observations. In addition, the performance of
Gibbs and PF improves as the number of samples/particles increases.
(a) Prediction
(b) Smoothing
(c) Expansion
(d) Dartmouth
(e) Social Evolution Statistics
(f) Dartmouth Statistics
Figure 2: Experimental results. (a-c) show the prediction, smoothing, and
expansion performance comparisons for Social Evolution data, while (d) shows
performance of the three tasks for Dartmouth data. (e-f) represent the
statistical inferences for both data sets.
Figure 2(d) shows the performance of the three tasks on the Dartmouth data
set. We do not apply the same comparison because it takes too much time for
sampling. From the graph, we can see that VISKM infers most of the infectious
moments of individuals in an accurate way for a large social system. In
addition, the smoothing results are slightly better than the prediction
results because we can leverage observations from both directions. The
expansion case is _relatively_ poor, because we use only very limited
information to derive the results; however, even in this case the ROC curve
has good discriminating power to differentiate between infectious and
susceptible individuals.
Collective Statistics Inference: After determining the individual results, we
aggregate them to approximate the total number of infected individuals in the
social system as time evolves. This offers a collective statistical summary of
the spread of disease in one area as in traditional research, which typically
scales the sample statistics with respect to the sample ratio. Figures 2(e)
and (f) show that given $20\%$ of the Social Evolution data and $10\%$ of the
Dartmouth data, VISKM estimates the collective statistics better than the
other methods.
Efficiency and Scalability: Table 1 shows the running time of different
algorithms for the Social Evolution data on the same computer. From the table,
we can see that Gibbs sampling runs slightly longer than PF, but they are in
the same scale. However, VISKM requires much less computation time. In
addition, the computation time of VISKM grows linearly with the number of
individuals, which validates the complexity analysis in Section 3.2. Thus, it
offers excellent scalability for large social systems. In comparison, Gibbs
sampling and PF grow super linearly with the number of individuals, and
roughly linearly with the number of samples.
Summary: Our proposed VISKM achieves higher accuracy in terms of area under
ROC curve and collective statistics than Gibbs sampling or particle filtering
(within 10,000 iterations). More importantly, VISKM is more efficient than
sampling with much less computation time. Additionally, the computation time
of VISKM grows linearly with the number of individuals, demonstrating its
excellent scalability for large social systems.
Table 1: Running time for different approximate inference algorithms. Gibbs_10000 refers to Gibbs sampling for 10,000 iterations, and PF_1000 to particle filtering for 1000 iterations. Other entries follow the same pattern. All times are measured in seconds. | VISKM | Gibbs_1000 | Gibbs_10000 | PF_1000 | PF_10000
---|---|---|---|---|---
60 People | 0.78 | 771 | 7820 | 601 | 6100
30 People | 0.39 | 255 | 2556 | 166 | 1888
15 People | 0.19 | 101 | 1003 | 122 | 1435
## 5 Conclusions
In this paper, we leverage sensor network and social network data to capture
temporal evolution in social dynamics and infer individual behaviors. In order
to define the adaptive transition kernel, we introduce a stochastic dynamic
mode that captures the dynamics of complex interactions. In addition, in order
to make tractable inferences we propose a variational inference algorithm the
computation complexity of which grows linearly with the number of individuals.
Large-scale experiments on epidemic dynamics demonstrate that our method
effectively captures the evolution of social dynamics and accurately infers
individual behaviors. More accurate collective effects can be also derived
through the aggregated results. Potential applications for our algorithm
include the dynamics of emotion, opinion, rumor, collaboration, and
friendship.
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## 6 Appendix
### 6.1 Derivation of the optimization problem in Eq.(6)
Let $P(\mathbf{x}_{1,...,T},v_{1,...,T}|\mathbf{y}_{1,...,T})$ be the exact
posterior. Our goal is to approximate this posterior by a distribution
$Q(\mathbf{x}_{1,...,T},v_{1,...,T})$ in the exponential family that minimizes
the KL divergence between these two distributions:
$\displaystyle
KL(Q(\mathbf{x}_{1,...,T},v_{1,...,T})|P(\mathbf{x}_{1,...,T},v_{1,...,T}|\mathbf{y}_{1,...,T}))$
$\displaystyle=$
$\displaystyle\sum_{\mathbf{x}_{1,...,T},v_{1,...,T}}Q(\mathbf{x}_{1,...,T},v_{1,...,T})\log[\frac{Q(\mathbf{x}_{1,...,T},v_{1,...,T})\cdot
P(\mathbf{y}_{1,...,T})}{P(\mathbf{x}_{1,...,T},\mathbf{y}_{1,...,T},v_{1,...,T})}]$
$\displaystyle=$
$\displaystyle\sum_{\mathbf{x}_{1,...,T},v_{1,...,T}}Q(\mathbf{x}_{1,...,T},v_{1,...,T})\log
Q(\mathbf{x}_{1,...,T},v_{1,...,T})$
$\displaystyle-\sum_{t=1}^{T}\sum_{\mathbf{x}_{1,...,T},v_{1,...,T}}Q(\mathbf{x}_{1,...,T},v_{1,...,T})\log
P(\mathbf{x}_{t},\mathbf{y}_{t},v_{t}|\mathbf{x}_{t-1}).$ (12)
In the first step, we apply the definition of conditional probability and KL-
divergence. In the second, we omit $P(\mathbf{y}_{1,...,T})$ because it is a
constant in this optimization problem. In addition, we decompose
$P\left(\mathbf{x}_{1,\dots,T},\mathbf{y}_{1,\dots,T},v_{1,\dots,T}\right)=\prod_{t=1}^{T}P(\mathbf{x}_{t},\mathbf{y}_{t},v_{t}|\mathbf{x}_{t-1})$.
We then define the approximate two-slice statistics
$\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})$ and one-slice statistics
$\hat{\gamma}(\mathbf{x}_{t})$. Both are in the exponential family. In this
context, we have $M$ individuals in the system and the mean-field
approximation can be shown as
$\hat{\gamma}(\mathbf{x}_{t})=\prod_{m=1}^{N}\hat{\gamma}^{(m)}(x_{t}^{(m)})$,
where $\hat{\gamma}^{(m)}(x_{t}^{(m)})$ is the approximate one-slice
statistics for individual $m$. Given the observation that
$Q(\mathbf{x}_{1,...,T},v_{1,...,T})$ can be expressed as a product of two-
slice statistics divided by a product of one-slice statistics, then
$\displaystyle
Q(\mathbf{x}_{1,...,T},v_{1,...,T})=\frac{\prod_{t=1}^{T}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})}{\prod_{t=1}^{T-1}\hat{\gamma}(\mathbf{x}_{t})}=\frac{\prod_{t=1}^{T}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})}{\prod_{t=1}^{T-1}\prod_{m=1}^{M}\hat{\gamma}^{(m)}(x_{t}^{(m)})}.$
(13)
If we substitute Eq. (13) into Eq. (12), the objective function becomes the
following:
$\displaystyle\sum_{\mathbf{x}_{1,...,T},v_{1,...,T}}Q(\mathbf{x}_{1,...,T},v_{1,...,T})\log\frac{\prod_{t=1}^{T}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})}{\prod_{t=1}^{T-1}\prod_{m}\hat{\gamma}^{(m)}(x_{t}^{(m)})}$
$\displaystyle-\sum_{t=1}^{T}\sum_{\mathbf{x}_{1,...,T},v_{1,...,T}}Q(\mathbf{x}_{1,...,T},v_{1,...,T})\log
P(\mathbf{x}_{t},\mathbf{y}_{t},v_{t}|\mathbf{x}_{t-1})$
$\displaystyle=\sum_{t,\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})\log\frac{\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})}{P(\mathbf{x}_{t},\mathbf{y}_{t},v_{t}|\mathbf{x}_{t-1})}$
$\displaystyle-\sum_{t,\mathbf{x}_{t}}\prod_{m}\hat{\gamma}^{(m)}(x_{t}^{(m)})\log\prod_{m}\hat{\gamma}^{(m)}(x_{t}^{(m)}).$
(14)
This objective function is subject to marginalization and normalization
constraints:
$\displaystyle\sum_{v_{t},\mathbf{x}_{t-1},\\{\mathbf{x}_{t}\backslash
x_{t}^{(m)}\\}}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})=\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})\mbox{,
for all }t,m,x_{t}^{(m)},$
$\displaystyle\sum_{v_{t},\\{\mathbf{x}_{t-1}\backslash
x_{t-1}^{(m)}\\},\mathbf{x}_{t}}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})=\hat{\gamma}_{t-1}^{(m)}(x_{t-1}^{(m)})\mbox{,
for all }t,m,x_{t-1}^{(m)},$
$\displaystyle\sum_{x_{t}^{(m)}}\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})=1\mbox{,
for all }t,m.$
$\sum_{\\{\mathbf{x}_{t}\backslash x_{t}^{(m)}\\}}$ refers to the sum over all
values of $\mathbf{x}_{t}$ except $x_{t}^{(m)}$.
### 6.2 Derivation of the inference algorithm from Eq.(8) to Eq.(10)
The optimization problem derived from Eq. (14) along with the constraints can
be shown as follows:
$\displaystyle\sum\limits_{t,\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})\log\frac{\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})}{P(\mathbf{x}_{t},\mathbf{y}_{t},v_{t}|\mathbf{x}_{t-1})}-\sum\limits_{t,\mathbf{x}_{t}}\prod\limits_{m}\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})\log\prod\limits_{m}\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})$
(15) subject to:
$\displaystyle\sum_{v_{t},\mathbf{x}_{t-1},\\{\mathbf{x}_{t}\backslash
x_{t}^{(m)}\\}}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})=\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})\mbox{,
for all }t,m,x_{t}^{(m)},$
$\displaystyle\sum_{v_{t},\\{\mathbf{x}_{t-1}\backslash
x_{t-1}^{(m)}\\},\mathbf{x}_{t}}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})=\hat{\gamma}_{t-1}^{(m)}(x_{t-1}^{(m)})\mbox{,
for all }t,m,x_{t-1}^{(m)},$
$\displaystyle\sum_{x_{t}^{(m)}}\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})=1\mbox{,
for all }t,m.$
We apply the method of Lagrange multipliers to solve this, which begins with
forming the Lagrange function to be optimized:
$\displaystyle L$
$\displaystyle=\sum\limits_{t,\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})\log\frac{\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})}{P(\mathbf{x}_{t},\mathbf{y}_{t},v_{t}|\mathbf{x}_{t-1})}-\sum\limits_{t,\mathbf{x}_{\mathbf{t}}}\prod\limits_{m}\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})\log\prod\limits_{m}\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})$
(16)
$\displaystyle+\sum_{t,m,x_{t}^{(m)}}\lambda_{t}^{(m)}(x_{t}^{(m)})\left(\sum_{v_{t},\mathbf{x}_{t-1},\\{\mathbf{x}_{t}\backslash
x_{t}^{(m)}\\}}\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})-\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})\right)$
$\displaystyle+\sum_{t,m,x_{t-1}^{(m)}}\mu_{t-1}^{(m)}(x_{t-1}^{(m)})\left(\sum_{v_{t},\\{\mathbf{x}_{t-1}\backslash
x_{t-1}^{(m)}\\},\mathbf{x}_{t}}\hat{\gamma}_{t-1}^{(m)}(x_{t-1}^{(m)})-\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})\right).$
$\displaystyle+\sum_{t,m,x_{t}^{(m)}}\nu(x_{t}^{(m)})\left(\sum_{x_{t}^{(m)}}\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})-1\right)$
We then set the partial derivatives of Eq. (16) over
$\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})$ to 0, which results in the
following:
$\displaystyle\frac{\partial
L}{\partial\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})}=\log\frac{\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})}{P(\mathbf{x}_{t},\mathbf{y}_{t},v_{t}|\mathbf{x}_{t-1})}+1-\sum_{m}\lambda_{t}^{(m)}(x_{t}^{(m)})-\sum\limits_{m}\mu_{t-1}^{(m)}(x_{t-1}^{(m)})\stackrel{{\scriptstyle\mbox{set}}}{{=}}0$
$\displaystyle\Rightarrow\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})\propto\exp\left(\sum\limits_{m}\mu_{t-1}^{(m)}(x_{t-1}^{(m)})\right)P(\mathbf{x}_{t},\mathbf{y}_{t},v_{t}|\mathbf{x}_{t-1})\exp\left(\sum\limits_{m}\lambda_{t}^{(m)}(x_{t}^{(m)})\right),$
As such, we see that
${\hat{\alpha}_{t-1}^{(m)}(x_{t-1}^{(m)})=\exp(\mu_{t-1}^{(m)}(x_{t-1}^{(m)}))}$
is associated with the forward probabilities and
${\hat{\beta}_{t}^{(m)}(x_{t}^{(m)})=\exp(\lambda_{t}^{(m)}(x_{t}^{(m)}))}$
with the backward probabilities, with
$\hat{\gamma}_{t}^{(m)}(x_{t}^{(m)})=\hat{\alpha}_{t}^{(m)}(x_{t}^{(m)})\hat{\beta}_{t}^{(m)}(x_{t}^{(m)})$.
We can determine the two-slice statistics for an individual $m$ by
marginalizing the other individuals $m^{\prime}\neq m$:
$\displaystyle\hat{\xi}(x_{t-1}^{(m)},x_{t}^{(m)},v_{t})=\sum_{m^{\prime}\neq
m,x_{t-1}^{(m^{\prime})},x_{t}^{(m^{\prime})}}\hat{\xi}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})$
$\displaystyle\propto\sum_{m^{\prime}\neq
m,x_{t-1}^{(m^{\prime})},x_{t}^{(m^{\prime})}}P(\mathbf{x}_{t},v_{t}|\mathbf{x}_{t-1})\cdot\prod_{m}\hat{\alpha}_{t-1}^{(m)}(x_{t-1}^{(m)})\cdot\prod_{m}P(y_{t}^{(m)}|x_{t}^{(m)})\cdot\prod_{m}\hat{\beta}_{t}^{(m)}(x_{t}^{(m)}).$
The above is the same as in Eq. (7).
### 6.3 Derivation of the parameter-learning algorithm
From Eq.(3), the log-likelihood of the entire sequence can be shown as this:
$\displaystyle\log
P\left(\mathbf{x}_{1,\dots,T},\mathbf{y}_{1,\dots,T},v_{1,\dots,T}\right)=\sum_{t=1}^{T}\log
P(\mathbf{x}_{t},v_{t}|\mathbf{x}_{t-1})+\sum_{t=1}^{T}\log
P(\mathbf{y}_{t}|\mathbf{x}_{t}),\mbox{ where }$ (17) $\displaystyle
P(\mathbf{x}_{t},v_{t}|\mathbf{x}_{t-1})=\begin{cases}c_{k}\cdot
g_{k}\left(\mathbf{x}_{t-1}\right)\cdot\delta(\mathbf{x}_{t}-\mathbf{x}_{t-1}\equiv\mathbf{\Delta_{k}})&\mbox{if
}v_{t}=k\\\
(1-\sum_{k}c_{k}g_{k}\left(\mathbf{x}_{t-1}\right))\cdot\delta(\mathbf{x}_{t}-\mathbf{x}_{t-1}\equiv\mathbf{0})&\mbox{if
}v_{t}=\emptyset\end{cases}.$
The probabilities for state transition can be shown as the probabilities of a
set of events. The expected log likelihood over the posterior probability
conditioned on the observations $\mathbf{y}_{1},\dots,\mathbf{y}_{T}$ takes
the following form:
$\displaystyle\mathbf{E}_{P(\mathbf{x}_{1,...,T},v_{1,...,T}|\mathbf{y}_{1,...,T})}\left(\log
P\left(\mathbf{x}_{1,\dots,T},\mathbf{y}_{1,\dots,T},v_{1,\dots,T}\right)\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(18) $\displaystyle=$
$\displaystyle\sum_{t,\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t})\cdot\log\left(P(\mathbf{x}_{t},v_{t}|\mathbf{x}_{t-1})P(\mathbf{y}_{t}|\mathbf{x}_{t})\right)$
$\displaystyle=$
$\displaystyle\sum_{t,\mathbf{x}_{t-1},\mathbf{x}_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=v)\cdot\log\left(P(\mathbf{x}_{t},v_{t}=v|\mathbf{x}_{t-1})P(\mathbf{y}_{t}|\mathbf{x}_{t})\right)$
$\displaystyle+$
$\displaystyle\sum_{t,\mathbf{x}_{t-1},\mathbf{x}_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=\emptyset)\cdot\log\left(P(\mathbf{x}_{t},v_{t}=\emptyset|\mathbf{x}_{t-1})P(\mathbf{y}_{t}|\mathbf{x}_{t})\right)$
At a given time $t$, there are two possible cases: $v_{t}=v$, where
$v\in\\{1,\cdots,V\\}$, and $v_{t}=\emptyset$. The derivatives with respect to
$c_{k}$ can be shown as follows:
$\displaystyle\frac{\partial\log
P(\mathbf{x}_{t},v_{t}=k|\mathbf{x}_{t-1})}{\partial c_{k}}=\frac{1}{c_{k}}$
$\displaystyle\frac{\partial\log
P(\mathbf{x}_{t},v_{t}=\emptyset|\mathbf{x}_{t-1})}{\partial
c_{k}}=\frac{-g_{k}(\mathbf{x}_{t-1})}{1-\sum_{k}c_{k}g_{k}(\mathbf{x}_{t-1})}$
Note that here we do not detail
$\delta(\mathbf{x}_{t}-\mathbf{x}_{t-1}\equiv\mathbf{\Delta_{k}})$ and
$\delta(\mathbf{x}_{t}-\mathbf{x}_{t-1}\equiv\mathbf{0})$ explicitly, because
when calculating the derivatives of expected log likelihood in Eq.(18) these
terms will be contained in
$\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=k)$ and
$\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=\emptyset)$. Next we take
the derivative of expected log likelihood with respect to $c_{k}$:
$\displaystyle\frac{\mathbf{E}_{P(\mathbf{x}_{1,...,T},v_{1,...,T}|\mathbf{y}_{1,...,T})}\left(\log
P\left(\mathbf{x}_{1,\dots,T},\mathbf{y}_{1,\dots,T},v_{1,\dots,T}\right)\right)}{\partial
c_{k}}$ (19) $\displaystyle=$
$\displaystyle\sum_{t,\mathbf{x}_{t-1},\mathbf{x}_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=k)\frac{1}{c_{k}}-\sum_{t,\mathbf{x}_{t-1},\mathbf{x}_{t},}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=\emptyset)\frac{g_{k}(\mathbf{x}_{t-1})}{1-\sum_{k}c_{k}g_{k}(\mathbf{x}_{t-1})}$
Because we assume that the auxiliary event dominates when the time step is
small, we approximate $1-\sum_{k}c_{k}g_{k}(\mathbf{x}_{t})\approx 1$ and
$\sum_{\mathbf{x}_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=\emptyset)\approx\hat{\gamma}_{t-1}(\mathbf{x}_{t-1})$.
After applying this approximation and setting the derivative to $0$, the
result is as follows:
$\displaystyle c_{k}$ $\displaystyle=\frac{\sum_{t}\
\sum_{\mathbf{x}_{t-1},\mathbf{x}_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=k)}{\sum_{t}\
\sum_{\mathbf{x}_{t-1},\mathbf{x}_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=\emptyset)g_{k}(\mathbf{x}_{t-1})}$
(20) $\displaystyle\approx\frac{\sum_{t}\
\sum_{\mathbf{x}_{t-1},\mathbf{x}_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=k)}{\sum_{t}\
\sum_{\mathbf{x}_{t-1}}\hat{\gamma}_{t-1}(\mathbf{x}_{t-1})g_{k}(\mathbf{x}_{t-1})}$
$\displaystyle=\frac{\sum_{t}\
\sum_{\mathbf{x}_{t-1},\mathbf{x}_{t}}\hat{\xi}_{t}(\mathbf{x}_{t-1},\mathbf{x}_{t},v_{t}=k)}{\sum_{t}\
\prod_{m}\sum_{x_{t-1}^{(m)}}\hat{\gamma}_{t-1}^{(m)}(x_{t-1}^{(m)})g_{k}^{(m)}(x_{t-1}^{(m)})}.$
|
# Complex Momentum for Optimization in Games
Jonathan Lorraine$\vphantom{e}{}^{1,2}$ David Acuna$\vphantom{e}{}^{1,2,3}$
Paul Vicol$\vphantom{e}{}^{1,2}$ David Duvenaud$\vphantom{e}{}^{1,2}$
University of Toronto$\vphantom{e}{}^{1}$ Vector Institute$\vphantom{e}{}^{2}$
NVIDIA$\vphantom{e}{}^{3}$
{lorraine, davidj, pvicol<EMAIL_ADDRESS>
###### Abstract
We generalize gradient descent with momentum for optimization in
differentiable games to have complex-valued momentum. We give theoretical
motivation for our method by proving convergence on bilinear zero-sum games
for simultaneous and alternating updates. Our method gives real-valued
parameter updates, making it a drop-in replacement for standard optimizers. We
empirically demonstrate that complex-valued momentum can improve convergence
in realistic adversarial games—like generative adversarial networks—by showing
we can find better solutions with an almost identical computational cost. We
also show a practical generalization to a complex-valued Adam variant, which
we use to train BigGAN to better inception scores on CIFAR-10.
## 1 Introduction
Gradient-based optimization has been critical for the success of machine
learning, updating a single set of parameters to minimize a single loss. A
growing number of applications require learning in games, which generalize
single-objective optimization. Common examples are GANs [1], actor-critic
models [2], curriculum learning [3, 4, 5], hyperparameter optimization [6, 7,
8, 9], adversarial examples [10, 11], learning models [12, 13, 14], domain
adversarial adaptation [15], neural architecture search [16, 17], and meta-
learning [18, 19].
Games consist of multiple players, each with parameters and objectives. We
often want solutions where no player gains from changing their strategy
unilaterally, e.g., Nash equilibria [20] or Stackelberg equilibria [21].
Classical gradient-based learning often fails to find these equilibria due to
rotational dynamics [22]. Numerous saddle point finding algorithms for zero-
sum games have been proposed [23, 24]. Gidel et al. [25] generalizes GD with
momentum to games, showing we can use a negative momentum to converge if the
eigenvalues of the Jacobian of the gradient vector field have a large
imaginary part. We use the terminology in Gidel et al. [25] and say (purely)
cooperative or adversarial games for games with (purely) real or imaginary
eigenvalues. Setups like GANs are not purely adversarial, but rather have both
_purely cooperative and adversarial eigenspaces_ – i.e., eigenspaces with
purely real or imaginary eigenvalues. In cooperative eigenspaces, the players
do not interfere with each other.
We want solutions that converge with simultaneous and alternating updates in
purely adversarial games – a setup where existing momentum methods fail. Also,
we want solutions that are robust to different mixtures of adversarial and
cooperative eigenspaces, because this depends on the games eigendecomposition
which can be intractable. To solve this we unify and generalize existing
momentum methods [26, 25] to recurrently linked momentum – a setup with
multiple recurrently linked momentum buffers with potentially negative
coefficients shown in Figure 2(c).
We show that selecting two of these recurrently linked buffers with
appropriate momentum coefficients can be interpreted as the real and imaginary
parts of a single _complex buffer_ and complex momentum coefficient – see
Figure 2(d). This setup (a) allows us to converge in adversarial games with
simultaneous updates, (b) only introduces one new optimizer parameter – the
phase or $\arg$ of our momentum, (c) allows us to gain intuitions via complex
analysis, (d) is trivial to implement in libraries supporting complex
arithmetic, and (e) robustly converges for different eigenspace mixtures.
Intuitively, our complex buffer stores historical gradient information,
oscillating between adding or subtracting at a frequency dictated by the
momentum coefficient. Classical momentum only adds gradients, and negative
momentum changes between adding or subtracting each iteration, while we
oscillate at an arbitrary (fixed) frequency – see Figure 3.1. This reduces
rotational dynamics during training by canceling out opposing updates.
### Contributions
* •
We provide generalizations and variants of classical [27, 28, 29], negative
[25, 30], and aggregated [26] momentum for learning in differentiable games.
* •
We show our methods converges on adversarial games – including bilinear zero-
sum games and the Dirac-GAN – with simultaneous and alternating updates.
* •
We illustrate a robustness during optimization, converging faster and over a
larger range of mixtures of cooperative and adversarial games than existing
first-order methods.
* •
We give a practical extension of our method to a complex-valued Adam [31]
variant, which we use to train a BigGAN [32] on CIFAR-10, improving [32]’s
inception scores.
Actual JAX implementation: changes in green
⬇
mass = .8 + .3j
def momentum(step_size, mass):
...
def update(i, g, state):
x, velocity = state
velocity = mass * velocity + g
x=x-jnp.real(step_size(i)*velocity)
return x, velocity
...
Figure 1: How to modify JAX’s SGD with momentum here to use complex momentum.
The only changes are in green. jnp.real gets the real part of step_size times
the momentum buffer (called velocity here). We use a complex mass for our
method in this case
$\beta=|\beta|\exp(i\arg(\beta))=$0.9$\exp(i\nicefrac{{\pi}}{{$8$}})\approx.8+.3i$.
## 2 Background
Appendix Table 2 summarizes our notation. Consider the optimization problem:
$\boldsymbol{\theta}^{*}\vcentcolon=\textnormal{arg\,min}_{\boldsymbol{\theta}}\mathcal{L}(\boldsymbol{\theta})$
(1)
We can find local minima of loss $\mathcal{L}$ using (stochastic) gradient
descent with step size $\alpha$. We denote the loss gradient at parameters
$\boldsymbol{\theta}^{j}$ by
$\boldsymbol{g}^{j}\\!\\!\vcentcolon=\\!\\!\boldsymbol{g}(\boldsymbol{\theta}^{j})\\!\\!\vcentcolon=\\!\\!\smash{\left.\nabla_{\boldsymbol{\theta}}\mathcal{L}(\boldsymbol{\theta})\right|_{\boldsymbol{\theta}^{j}}}$.
$\smash{\boldsymbol{\theta}^{j\\!+\\!1}=\boldsymbol{\theta}^{j}-\alpha\boldsymbol{g}^{j}}$
(SGD)
Momentum can generalize SGD. For example, Polyak’s Heavy Ball [27]:
$\displaystyle\smash{\boldsymbol{\theta}^{j\\!+\\!1}=\boldsymbol{\theta}^{j}-\alpha\boldsymbol{g}^{j}+\beta(\boldsymbol{\theta}^{j}-\boldsymbol{\theta}^{j-1})}$
(2)
Which can be equivalently written with momentum buffer
$\smash{\boldsymbol{\mu}^{j}=\nicefrac{{(\boldsymbol{\theta}^{j}-\boldsymbol{\theta}^{j-1})}}{{\alpha}}}$.
$\smash{\boldsymbol{\mu}^{j\\!+\\!1}=\beta\boldsymbol{\mu}^{j}-\boldsymbol{g}^{j},\quad\quad\boldsymbol{\theta}^{j\\!+\\!1}=\boldsymbol{\theta}^{j}+\alpha\boldsymbol{\mu}^{j\\!+\\!1}}$
(SGDm)
We can also generalize SGDm to aggregated momentum [26], shown in Appendix
Algorithm 3.
### 2.1 Game Formulations
Another class of problems is learning in games, which includes problems like
generative adversarial networks (GANs) [1]. We focus on $2$-player games —with
players denoted by $A$ and $B$—where each player minimizes their loss
$\mathcal{L}_{A},\mathcal{L}_{B}$ with their parameters
$\boldsymbol{\theta}_{A}\in\mathbb{R}^{d_{A}}$,
$\boldsymbol{\theta}_{B}\in\mathbb{R}^{d_{B}}$. Solutions to $2$-player
games – which are assumed unique for simplicity – can be defined as:
$\displaystyle\smash{\boldsymbol{\theta}_{\\!A}^{*}\\!\vcentcolon=\\!\textnormal{arg\,min}_{\boldsymbol{\theta}_{\\!A}}\\!\mathcal{L}_{\\!A}(\boldsymbol{\theta}_{\\!A},\\!\boldsymbol{\theta}_{\\!B}^{*}),\,\,\,\,\,\boldsymbol{\theta}_{\\!B}^{*}\\!\vcentcolon=\\!\textnormal{arg\,min}_{\boldsymbol{\theta}_{\\!B}}\\!\mathcal{L}_{\\!B}(\boldsymbol{\theta}_{\\!A}^{*},\\!\boldsymbol{\theta}_{\\!B})}$
(3)
In deep learning, losses are non-convex with many parameters, so we often
focus on finding local solutions. If we have a player ordering, then we have a
Stackelberg game. For example, in GANs, the generator is the leader, and the
discriminator is the follower. In hyperparameter optimization, the
hyperparameters are the leader, and the network parameters are the follower.
If $\boldsymbol{\theta}_{\\!B}^{*}(\boldsymbol{\theta}_{\\!A})$ denotes player
$B$’s best-response function, then Stackelberg game solutions can be defined
as:
$\displaystyle\smash{\boldsymbol{\theta}_{\\!A}}^{*}\\!\vcentcolon=\textnormal{arg\,min}_{\boldsymbol{\theta}_{\\!A}}\\!\mathcal{L}_{\\!A}(\boldsymbol{\theta}_{\\!A},\\!\boldsymbol{\theta}_{\\!B}^{*}(\boldsymbol{\theta}_{\\!A})),\,\,\,\,\,\boldsymbol{\theta}_{\\!B}^{*}(\boldsymbol{\theta}_{\\!A})\\!\vcentcolon=\textnormal{arg\,min}_{\boldsymbol{\theta}_{\\!B}}\\!\mathcal{L}_{\\!B}(\boldsymbol{\theta}_{\\!A},\\!\boldsymbol{\theta}_{\\!B})$
(4)
If $\mathcal{L}_{\\!A}$ and $\mathcal{L}_{\\!B}$ are differentiable in
$\boldsymbol{\theta}_{\\!A}$ and $\boldsymbol{\theta}_{\\!B}$ we say the game
is differentiable. We may be able to approximately find
$\boldsymbol{\theta}_{\\!A}^{*}$ efficiently if we can do SGD on:
$\mathcal{L}_{\\!A}^{*}(\boldsymbol{\theta}_{\\!A})\vcentcolon=\mathcal{L}_{\\!A}(\boldsymbol{\theta}_{\\!A},\boldsymbol{\theta}_{\\!B}^{*}(\boldsymbol{\theta}_{\\!A}))$
(5)
Unfortunately, SGD would require computing
$\nicefrac{{d\mathcal{L}_{\\!A}^{*}}}{{d\boldsymbol{\theta}_{\\!A}}}$, which
often requires
$\nicefrac{{d\boldsymbol{\theta}_{\\!B}^{*}}}{{d\boldsymbol{\theta}_{\\!A}}}$,
but $\boldsymbol{\theta}_{\\!B}^{*}(\boldsymbol{\theta}_{\\!A})$ and its
Jacobian are typically intractable. A common optimization algorithm to analyze
for finding solutions is simultaneous SGD (SimSGD) – sometimes called gradient
descent ascent for zero-sum games – where
$\boldsymbol{g}_{\\!A}^{j}\vcentcolon=\boldsymbol{g}_{\\!A}(\boldsymbol{\theta}_{\\!A}^{j},\boldsymbol{\theta}_{\\!B}^{j})$
and
$\boldsymbol{g}_{\\!B}^{j}\vcentcolon=\boldsymbol{g}_{\\!B}(\boldsymbol{\theta}_{\\!A}^{j},\boldsymbol{\theta}_{\\!B}^{j})$
are estimators for
$\left.\smash{\nabla_{\boldsymbol{\theta}_{\\!A}}}\mathcal{L}_{\\!A}\right|_{\boldsymbol{\theta}_{\\!A}^{j},\boldsymbol{\theta}_{\\!B}^{j}}$
and
$\left.\smash{\nabla_{\boldsymbol{\theta}_{\\!B}}\mathcal{L}_{\\!B}}\right|_{\boldsymbol{\theta}_{\\!A}^{j},\boldsymbol{\theta}_{\\!B}^{j}}$:
$\displaystyle\smash{\boldsymbol{\theta}_{\\!A}^{j\\!+\\!1}=\boldsymbol{\theta}_{\\!A}^{j}-\alpha\boldsymbol{g}_{\\!A}^{j},\quad\boldsymbol{\theta}_{\\!B}^{j\\!+\\!1}=\boldsymbol{\theta}_{\\!B}^{j}-\alpha\boldsymbol{g}_{\\!B}^{j}}$
(SimSGD)
We simplify notation with the concatenated or joint-parameters
$\smash{\boldsymbol{\omega}\\!\vcentcolon=\\![\boldsymbol{\theta}_{\\!A},\boldsymbol{\theta}_{\\!B}]\\!\in\\!\mathbb{R}^{d}}$
and the joint-gradient vector field
$\hat{\boldsymbol{g}}:\mathbb{R}^{d}\to\mathbb{R}^{d}$, which at the $j^{th}$
iteration is the joint-gradient denoted:
$\hat{\boldsymbol{g}}^{j}\vcentcolon=\hat{\boldsymbol{g}}(\boldsymbol{\omega}^{j})\vcentcolon=[\boldsymbol{g}_{\\!A}(\boldsymbol{\omega}^{j}),\boldsymbol{g}_{\\!B}(\boldsymbol{\omega}^{j})]=[\boldsymbol{g}_{\\!A}^{j},\boldsymbol{g}_{\\!B}^{j}]$
(6)
We extend to $n$-player games by treating $\boldsymbol{\omega}$ and
$\hat{\boldsymbol{g}}$ as concatenations of the players’ parameters and loss
gradients, allowing for a concise expression of the SimSGD update with
momentum (SimSGDm):
$\displaystyle\smash{\boldsymbol{\mu}^{j\\!+\\!1}=\beta\boldsymbol{\mu}^{j}-\hat{\boldsymbol{g}}^{j},\quad\quad\boldsymbol{\omega}^{j\\!+\\!1}=\boldsymbol{\omega}^{j}+\alpha\boldsymbol{\mu}^{j\\!+\\!1}}$
(SimSGDm)
Gidel et al. [25] show classical momentum choices of $\beta\in[$0$,$1$)$ do
not improve solution speed over SimSGD in some games, while negative momentum
helps if the Jacobian of the joint-gradient vector field
$\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}$ has complex eigenvalues.
Thus, for purely adversarial games with imaginary eigenvalues, any non-
negative momentum and step size will not converge. For cooperative games –
i.e., minimization – $\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}$ has
strictly real eigenvalues because it is a losses Hessian, so classical
momentum works well.
gradient$\boldsymbol{\mu}$update$\alpha$$\beta$ (a) Classical [27]
gradient$\boldsymbol{\mu}_{(1)}$…$\boldsymbol{\mu}_{(n)}$update$\alpha_{(1)}$$\beta_{(1)}$$\alpha_{(n)}$$\beta_{(n)}$
(b) Aggregated [26]
gradient$\boldsymbol{\mu}_{(1)}$$\boldsymbol{\mu}_{(2)}$update$\beta_{(1,2)}$$\alpha_{(1)}$$\beta_{(1,1)}$$\beta_{(2,1)}$$\alpha_{(2)}$$\beta_{(2,2)}$
(c) Recurrently linked (new)
gradient$\Re(\boldsymbol{\mu}$)$\Im(\boldsymbol{\mu}$)update${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Im(\beta)}$$\alpha$$\Re(\beta)$$-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Im(\beta)}$$\Re(\beta)$
(d) Complex (ours)
Figure 2: We show computational diagrams for momentum variants simultaneously
updating all players parameters, which update the momentum buffers
$\boldsymbol{\mu}$ at iteration $j\\!+\\!1$ with coefficient $\beta$ via
$\boldsymbol{\mu}^{j\\!+\\!1}\\!=\\!(\beta\boldsymbol{\mu}^{j}-$gradient$)$.
Our parameter update is a linear combination of the momentum buffers weighted
by step sizes $\alpha$. _(a)_ Classical momentum [27, 29], with a single
buffer and coefficient $\beta\in[0,1)$. _(b)_ Aggregated momentum [26] which
adds multiple buffers with different coefficients. _(c)_ Recurrently linked
momentum, which adds cross-buffer coefficients and updates the buffers with
$\boldsymbol{\mu}_{(k)}^{j+1}\\!=\\!(\sum_{l}\beta_{(l,k)}\boldsymbol{\mu}_{(l)}^{j}-$gradient$)$.
We allow $\beta_{(l,k)}$ to be negative like negative momentum [25] for
solutions with simultaneous updates in adversarial games. _(d)_ Complex
momentum is a special case of recurrently linked momentum with two buffers and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\beta_{(1,1)}}\\!=\\!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\beta_{(2,2)}}\\!=\\!{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Re(\beta)}$,
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\beta_{(1,2)}}\\!=\\!-{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\beta_{(2,1)}}\\!=\\!{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\Im(\beta)}$.
Analyzing other recurrently linked momentum setups is an open problem.
### 2.2 Limitations of Existing Methods
Higher-order: Methods using higher-order gradients are often harder to
parallelize across GPUs, [33], get attracted to bad saddle points [34],
require estimators for inverse Hessians [35, 36], are complicated to
implement, have numerous optimizer parameters, and can be more expensive in
iteration and memory cost [37, 36, 35, 38, 39, 40]. Instead, we focus on
first-order methods.
First-order: Some first-order methods such as extragradient [41] require a
second, costly, gradient evaluation per step. Similarly, methods alternating
player updates are bottlenecked by waiting until after the first player’s
gradient is used to evaluate the second player’s gradient. But, many deep
learning setups can parallelize computation of both players’ gradients, making
alternating updates effectively cost another gradient evaluation. We want a
method which updates with the effective cost of one gradient evaluation. Also,
simultaneous updates are a standard choice in some settings [15].
Robust convergence: We want our method to converge in purely adversarial
game’s with simultaneous updates – a setup where existing momentum methods
fail [25]. Furthermore, computing a games eigendecomposition is often
infeasibly expensive, so we want methods that robustly converge over different
mixtures of adversarial and cooperative eigenspaces. We are particularly
interested in eigenspace mixtures that that are relevant during GAN training –
see Figure 7 and Appendix Figure 9.
### 2.3 Coming up with our Method
Combining existing methods: Given the preceding limitations, we would like a
robust first-order method using a single, simultaneous gradient evaluation. We
looked at combining aggregated [26] with negative [25] momentum by allowing
negative coefficients, because these methods are first-order and use a single
gradient evaluation – see Figure 2(b). Also, aggregated momentum provides
robustness during optimization by converging quickly on problems with wide
range of conditioning, while negative momentum works in adversarial setups. We
hoped to combine their benefits, gaining robustness to different mixtures of
adversarial and cooperative eigenspaces. However, with this setup we could not
find solutions that converge with simultaneous updates in purely adversarial
games.
Generalize to allow solutions: We generalized the setup to allow recurrent
connections between momentum buffers, with potentially negative coefficients –
see Figure 2(c) and Appendix Algorithm 4. There are optimizer parameters so
this converges with simultaneous updates in purely adversarial games, while
being first-order with a single gradient evaluation – see Corollary 1.
However, in general, this setup could introduce many optimizer parameters,
have unintuitive behavior, and not be amenable to analysis. So, we choose a
special case of this method to help solve these problems.
A simple solution: With two momentum buffers and correctly chosen recurrent
weights, we can interpret our buffers as the real and imaginary part of one
complex buffer – see Figure 2(d). This method is (a) capable of converging in
purely adversarial games with simultaneous updates – Corollary 1, (b) only
introduces one new optimizer parameter – the phase of the momentum
coefficient, (c) is tractable to analyze and have intuitions for with Euler’s
formula – ex., Eq. (8), (d) is trivial to implement in libraries supporting
complex arithmetic – see Figure 1, and (e) can be robust to games with
different mixtures of cooperative and adversarial eigenspaces – see Figure 5.
DiscriminatorGeneratorNorm of joint-gradient
$\|\hat{\boldsymbol{g}}\|$Distance to optimumIterations Figure 3: Complex
momentum helps correct rotational dynamics when training a Dirac-GAN [42].
_Left:_ Parameter trajectories with step size $\alpha\\!=\\!0.1$ and momentum
$\beta\\!=\\!0.9\exp(i\nicefrac{{\pi}}{{8}})$. We include the classical, real
and positive momentum which diverges for any step size. _Right:_ The distance
from optimum, which has a linear convergence rate matching our prediction with
Theorem 1 and (14).
## 3 Complex Momentum
We describe our proposed method, where the momentum coefficient
$\beta\in\mathbb{C}$, step size $\alpha\in\mathbb{R}$, momentum buffer
$\boldsymbol{\mu}\in\mathbb{C}^{d}$, and player parameters
$\boldsymbol{\omega}\in\mathbb{R}^{d}$. The simultaneous (or Jacobi) update
is:
$\displaystyle\smash{\boldsymbol{\mu}^{j\\!+\\!1}=\beta\boldsymbol{\mu}^{j}-\hat{\boldsymbol{g}}^{j},\,\,\,\,\,\boldsymbol{\omega}^{j\\!+\\!1}=\boldsymbol{\omega}^{j}+\Re(\alpha\boldsymbol{\mu}^{j\\!+\\!1})}$
(SimCM)
There are many ways to get a real-valued update from
$\boldsymbol{\mu}\in\mathbb{C}$, but we only consider updates equivalent to
classical momentum when $\beta\in\mathbb{R}$. Specifically, we simply update
the parameters using the real component of the momentum:
$\Re(\boldsymbol{\mu})$.
Algorithm 1 (SimCM) Momentum
1:$\smash{\beta,\alpha\in\mathbb{C},\boldsymbol{\mu}\in\mathbb{C}^{d},\boldsymbol{\omega}^{$0$}\in\mathbb{R}^{d}}$
2:for $i=$1$\dots N$ do
3:
$\smash{\boldsymbol{\mu}^{j\\!+\\!1}\\!=\\!\beta\boldsymbol{\mu}^{j}\\!-\\!\hat{\boldsymbol{g}}^{j}}$
4:
$\boldsymbol{\omega}^{j\\!+\\!1}\\!=\\!\boldsymbol{\omega}^{j}\\!+\\!\Re(\alpha\boldsymbol{\mu}^{j\\!+\\!1})$
return $\boldsymbol{\omega}^{N}$
We show the SimCM update in Algorithm 1 and visualize it in Figure 2(d). We
also show the alternating (or Gauss-Seidel) update, which is common for GAN
training:
$\displaystyle\boldsymbol{\mu}^{j\\!+\\!1}_{A}\\!$
$\displaystyle=\\!\beta\boldsymbol{\mu}^{j}_{A}\\!-\\!\hat{\boldsymbol{g}}_{A}(\boldsymbol{\omega}^{j}\\!),\boldsymbol{\theta}_{\\!A}^{j\\!+\\!1}\\!=\\!\boldsymbol{\theta}_{\\!A}^{j}\\!+\\!\Re(\alpha\boldsymbol{\mu}^{j\\!+\\!1}_{A}\\!)$
(AltCM) $\displaystyle\boldsymbol{\mu}^{j\\!+\\!1}_{B}\\!$
$\displaystyle=\\!\beta\boldsymbol{\mu}^{j}_{B}\\!-\\!\hat{\boldsymbol{g}}_{B}(\boldsymbol{\theta}_{\\!A}^{j\\!+\\!1}\\!\\!,\boldsymbol{\theta}_{\\!B}^{j}\\!),\boldsymbol{\theta}_{\\!B}^{j\\!+\\!1}\\!=\\!\boldsymbol{\theta}_{\\!B}^{j}\\!+\\!\Re(\alpha\boldsymbol{\mu}^{j\\!+\\!1}_{B}\\!)$
#### Generalizing negative momentum:
Consider the negative momentum from Gidel et al. [25]:
$\boldsymbol{\omega}^{j\\!+\\!1}=\boldsymbol{\omega}^{j}-\alpha\hat{\boldsymbol{g}}^{j}+\beta(\boldsymbol{\omega}^{j}-\boldsymbol{\omega}^{j-1})$.
Expanding (SimCM) with
$\boldsymbol{\mu}^{j}=\nicefrac{{(\boldsymbol{\omega}^{j}-\boldsymbol{\omega}^{j-1})}}{{\alpha}}$
for real momentum shows the negative momentum method of Gidel et al. [25] is a
special case of our method:
$\displaystyle\boldsymbol{\omega}^{j\\!+\\!1}=\boldsymbol{\omega}^{j}+\Re(\alpha(\beta\nicefrac{{(\boldsymbol{\omega}^{j}-\boldsymbol{\omega}^{j-1})}}{{\alpha}}-\hat{\boldsymbol{g}}^{j}))=\boldsymbol{\omega}^{j}-\alpha\hat{\boldsymbol{g}}^{j}+\beta(\boldsymbol{\omega}^{j}-\boldsymbol{\omega}^{j-1})$
(7)
### 3.1 Dynamics of Complex Momentum
For simplicity, we assume Numpy-style [43] component-wise broadcasting for
operations like taking the real-part $\Re(\boldsymbol{z})$ of vector
$\boldsymbol{z}=[z_{1},\dots,z_{n}]\in\mathbb{C}^{n}$, with proofs in the
Appendix.
Expanding the buffer updates with the polar components of $\beta$ gives
intuition for complex momentum:
$\displaystyle\begin{split}\vphantom{A^{A^{A^{A}}}}\boldsymbol{\mu}^{j\\!+\\!1}\\!=\\!\beta\boldsymbol{\mu}^{j}-\hat{\boldsymbol{g}}^{j}&\iff\boldsymbol{\mu}^{j\\!+\\!1}\\!=\\!\beta(\beta(\cdots)-\hat{\boldsymbol{g}}^{j-1})-\hat{\boldsymbol{g}}^{j}\iff\boldsymbol{\mu}^{j\\!+\\!1}\\!=\\!\smash{-\sum_{k=0}^{k=j}\beta^{k}\hat{\boldsymbol{g}}^{j-k}}\iff\\\
\Re(\boldsymbol{\mu}^{j\\!+\\!1})\\!=\\!-&\sum_{k=0}^{k=j}|\beta|^{k}\cos(k\arg(\beta))\hat{\boldsymbol{g}}^{j-k},\,\,\,\,\,\Im(\boldsymbol{\mu}^{j\\!+\\!1})\\!=\\!-\sum_{k=0}^{k=j}|\beta|^{k}\sin(k\arg(\beta))\hat{\boldsymbol{g}}^{j-k}\end{split}$
(8)
The final line is simply by Euler’s formula (26). From (8) we can see $\beta$
controls the momentum buffer $\boldsymbol{\mu}$ by having $|\beta|$ dictate
prior gradient decay rates, while $\arg(\beta)$ controls oscillation frequency
between adding and subtracting prior gradients, which we visualize in Figure
3.1.
Dependence on gradient $k$Iteration $k$
Figure 4(()): We show the real part of our momentum buffer – which dictates
the parameter update – at the $50^{th}$ iteration $\Re(\boldsymbol{\mu}^{50})$
dependence on past gradients $\hat{\boldsymbol{g}}^{k}$ for $k\\!=\\!1\dots
50$. The momentum magnitude is fixed to $|\beta|\\!=\\!$0.9$$ as in Figure 3.
Euler’s formula is used in (8) to for finding dependence or coefficient of
$\hat{\boldsymbol{g}}^{k}$ via
$\Re(\boldsymbol{\mu}^{50})\\!=\\!-\sum_{k=0}^{k=50}|\beta|^{k}\cos(k\arg(\beta))\hat{\boldsymbol{g}}^{j-k}$.
Complex momentum allows smooth changes in the buffers dependence on past
gradients.
Momentum phase $\arg(\beta)$Momentum magnitude $|\beta|$Number of steps to
converge
Figure 4(()): How many steps simultaneous complex momentum on a Dirac-GAN
takes for a set solution distance. We fix step size $\alpha\\!=\\!$0.1$$ as in
Figure 3, while varying the phase and magnitude of our momentum
$\beta\\!=\\!|\beta|\exp(i\arg(\beta))$. There is a red star at the optima,
dashed red lines at real $\beta$, and a dashed magenta line for simultaneous
gradient descent. There are no real-valued $\beta$ that converge for this – or
any – $\alpha$ with simultaneous updates [25]. Appendix Figure 8 compares this
with alternating updates (AltCM).
Expanding the parameter updates with the Cartesian components of $\alpha$ and
$\beta$ is key for Theorem 1, which characterizes the convergence rate:
$\displaystyle\begin{split}\smash{\boldsymbol{\mu}^{j\\!+\\!1}}\\!=\\!\smash{\beta\boldsymbol{\mu}^{j}-\hat{\boldsymbol{g}}^{j}\iff}\\\
\smash{\Re(\boldsymbol{\mu}^{j\\!+\\!1})}\\!=\\!\smash{\Re(\beta)\\!\Re(\boldsymbol{\mu}^{j})\\!-\\!\Im(\beta)\\!\Im(\boldsymbol{\mu}^{j})\\!-\\!\Re(\hat{\boldsymbol{g}}^{j})},\,\,\,&\Im(\boldsymbol{\mu}^{j\\!+\\!1})\\!=\\!\Im(\beta)\\!\Re(\boldsymbol{\mu}^{j})\\!+\\!\Re(\beta)\\!\Im(\boldsymbol{\mu}^{j})\end{split}$
(9)
$\displaystyle\boldsymbol{\omega}^{j\\!+\\!1}\\!=\\!\boldsymbol{\omega}^{j}\\!+\\!\Re(\alpha\boldsymbol{\mu}^{j\\!+\\!1})\iff\boldsymbol{\omega}^{j\\!+\\!1}\\!=\\!\boldsymbol{\omega}^{j}\\!-\\!\alpha\hat{\boldsymbol{g}}^{j}\\!+\\!\Re(\alpha\beta)\\!\Re(\boldsymbol{\mu}^{j})\\!-\\!\Im(\alpha\beta)\\!\Im(\boldsymbol{\mu}^{j})$
(10)
So, we can write the next iterate with a fixed-point operator:
$\smash{[\Re(\boldsymbol{\mu}^{j\\!+\\!1}),\\!\Im(\boldsymbol{\mu}^{j\\!+\\!1}),\\!\boldsymbol{\omega}^{j\\!+\\!1}]\\!\\!=\\!\boldsymbol{F}_{\alpha,\beta}([\Re(\boldsymbol{\mu}^{j}),\\!\Im(\boldsymbol{\mu}^{j}),\\!\boldsymbol{\omega}^{j}]\\!)}$
(11)
(9) and (10) allow us to write the Jacobian of $\boldsymbol{F}_{\alpha,\beta}$
which can be used to bound convergence rates near fixed points, which we name
the Jacobian of the augmented dynamics of buffer $\boldsymbol{\mu}$ and joint-
parameters $\boldsymbol{\omega}$ and denote with:
$\boldsymbol{R}\\!\vcentcolon=\\!\nabla_{[\boldsymbol{\mu},\boldsymbol{\omega}]}\boldsymbol{F}_{\alpha,\beta}=\begin{bmatrix}\Re(\beta)\boldsymbol{I}&-\Im(\beta)\boldsymbol{I}&-\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}\\\
\Im(\beta)\boldsymbol{I}&\Re(\beta)\boldsymbol{I}&0\\\
\Re(\alpha\beta)\boldsymbol{I}&-\Im(\alpha\beta)\boldsymbol{I}&\boldsymbol{I}\\!-\\!\alpha\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}\\\
\end{bmatrix}$ (12)
So, for quadratic losses our parameters evolve via:
$\smash{[\Re(\boldsymbol{\mu}^{j\\!+\\!1}),\\!\Im(\boldsymbol{\mu}^{j\\!+\\!1}),\\!\boldsymbol{\omega}^{j\\!+\\!1}]^{\top}\\!\\!=\\!\boldsymbol{R}\,[\Re(\boldsymbol{\mu}^{j}),\\!\Im(\boldsymbol{\mu}^{j}),\\!\boldsymbol{\omega}^{j}]^{\top}\\!}$
(13)
We can bound convergence rates by looking at the spectrum of $\boldsymbol{R}$
with Theorem 1.
###### Theorem 1 (Consequence of Prop. 4.4.1 Bertsekas [44]).
Convergence rate of complex momentum: If the spectral radius
$\rho(\boldsymbol{R})\\!=\\!\rho(\nabla_{[\boldsymbol{\mu},\boldsymbol{\omega}]}\boldsymbol{F}_{\alpha,\beta})\\!<\\!1$,
then, for $[\boldsymbol{\mu},\boldsymbol{\omega}]$ in a neighborhood of
$[\boldsymbol{\mu}^{*}\\!,\boldsymbol{\omega}^{*}]$, the distance of
$[\boldsymbol{\mu}^{j}\\!,\boldsymbol{\omega}^{j}]$ to the stationary point
$[\boldsymbol{\mu}^{*}\\!,\boldsymbol{\omega}^{*}]$ converges at a linear rate
$\mathcal{O}((\rho(\boldsymbol{R})+\epsilon)^{j}),\forall\epsilon\\!>\\!0$.
Here, linear convergence means
$\lim_{j\to\infty}\\!\\!\nicefrac{{\|\boldsymbol{\omega}^{j\\!+\\!1}-\boldsymbol{\omega}^{*}\|}}{{\|\boldsymbol{\omega}^{j}-\boldsymbol{\omega}^{*}\|}}\\!\in\\!($0$,$1$)$,
where $\boldsymbol{\omega}^{*}$ is a fixed point. We should select
optimization parameters $\alpha,\beta$ so that the augmented dynamics spectral
radius $\operatorname*{Sp}(\boldsymbol{R}(\alpha,\beta))\\!<\\!1$—with the
dependence on $\alpha$ and $\beta$ now explicit. We may want to express
$\operatorname*{Sp}(\boldsymbol{R}(\alpha,\beta))$ in terms of the spectrum
$\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$, as in
Theorem 3 in Gidel et al. [25]:
$\smash{\boldsymbol{f}(\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}),\alpha,\beta)\\!=\\!\operatorname*{Sp}(\boldsymbol{R}(\alpha,\beta))}$
(14)
We provide a Mathematica command in Appendix A.2 for a cubic polynomial $p$
characterizing $\boldsymbol{f}$ with coefficients that are functions of
$\alpha,\beta$ &
$\lambda\in\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$,
whose roots are eigenvalues of $\boldsymbol{R}$, which we use in subsequent
results. O’donoghue and Candes [45], Lucas et al. [26] mention that in
practice we do not know the condition number, eigenvalues – or the mixture of
cooperative and adversarial eigenspaces – of a set of functions that we are
optimizing, so we try to design algorithms which work over a large range.
Sharing this motivation, we consider convergence behavior on games ranging
from purely adversarial to cooperative.
In Section 4.2 at every non-real $\beta$ we could select $\alpha$ and
$|\beta|$ so Algorithm 1 converges. We define _almost-positive_ to mean
$\arg(\beta)\\!=\\!\epsilon$ for small $\epsilon$, and show there are almost-
positive $\beta$ which converge.
###### Corollary 1 (Convergence of Complex Momentum).
There exist $\alpha\in\mathbb{R},\beta\in\mathbb{C}$ so Algorithm 1 converges
for bilinear zero-sum games. More-so, for small $\epsilon$ (we show for
$\epsilon=\frac{\pi}{$16$}$), if $\arg(\beta)=\epsilon$ (i.e., almost-
positive) or $\arg(\beta)=\pi-\epsilon$ (i.e., almost-negative), then we can
select $\alpha,|\beta|$ to converge.
Why show this? Our result complements Gidel et al. [25] who show that for all
real $\alpha,\beta$ Algorithm 1 _does not_ converge. We include the proof for
bilinear zero-sum games, but the result generalizes to some games that are
purely adversarial near fixed points, like Dirac GANs [34]. The result’s
second part shows evidence there is a sense in which the only $\beta$ that do
not converge are real (with simultaneous updates on purely adversarial games).
It also suggests a form of robustness, because almost-positive $\beta$ can
approach acceleration in cooperative eigenspaces, while converging in
adversarial eigenspaces, so almost-positive $\beta$ may be desirable when we
have games with an uncertain or variable mixtures of real and imaginary
eigenvalues like GANs. Sections 4.2, 4.3, and 4.4 investigate this further.
### 3.2 What about Acceleration?
With classical momentum, finding the step size $\alpha$ and momentum $\beta$
to optimize the convergence rate tractable if $$0$\\!<\\!l\\!\leq\\!L$ and
$\smash{\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})\\!\in\\![l,L]^{d}}$
[46] – i.e., we have an $l$-strongly convex and $L$-Lipschitz loss. The
conditioning $\kappa\\!=\\!\nicefrac{{L}}{{l}}$ can characterize problem
difficulty. Gradient descent with an appropriate $\alpha$ can achieve a
convergence rate of $\smash{\frac{\kappa-$1$}{\kappa+$1$}}$, but using
momentum with appropriate $(\alpha^{*}\\!,\beta^{*})$ can achieve an
_accelerated_ rate of
$\smash{\rho^{*}\\!=\\!\frac{\sqrt{\kappa}-$1$}{\sqrt{\kappa}+$1$}}$. However,
there is no consensus for constraining
$\smash{\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})}$
in games for tractable and useful results. Candidate constraints include
monotonic vector fields generalizing notions of convexity, or vector fields
with bounded eigenvalue norms capturing a kind of sensitivity [47]. Figure 7
shows
$\smash{\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})}$
for a GAN – we can attribute some eigenvectors to a single player’s
parameters. The discriminator can be responsible for the largest and smallest
norm eigenvalues, suggesting we may benefit from varying $\alpha$ and $\beta$
for each player as done in Section 4.4.
### 3.3 Implementing Complex Momentum
Complex momentum is trivial to implement with libraries supporting complex
arithmetic like JAX [48] or Pytorch [49]. Given an SGD implementation, we
often only need to change a few lines of code – see Figure 1. Also, (9) and
(10) can be easily used to implement Algorithm 1 in a library without complex
arithmetic. More sophisticated optimizers like Adam can trivially support
complex optimizer parameters with real-valued updates, which we explore in
Section 4.4.
### 3.4 Scope and Limitations
For some games, we need higher than first-order information to converge – ex.,
pure-response games [7] – because the first-order information for a player is
identically zero. So, momentum methods only using first-order info will not
converge in general. However, we can combine methods with second-order
information and momentum algorithms [7, 9]. Complex momentum’s computational
cost is almost identical to classical and negative momentum, except we now
have a buffer with twice as many real parameters. We require one more
optimization hyperparameter than classical momentum, which we provide an
initial guess for in Section 4.5.
## 4 Experiments
We investigate complex momentum’s performance in training GANs and games with
different mixtures of cooperative and adversarial eigenspaces, showing
improvements over standard baselines. Code for experiments will be available
on publication, with reproducibility details in Appendix C.
Overview: We start with a purely adversarial Dirac-GAN and zero-sum games,
which have known solutions
$\boldsymbol{\omega}^{*}\\!=\\!(\boldsymbol{\theta}_{\\!A}^{*},\boldsymbol{\theta}_{\\!B}^{*})$
and spectrums
$\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$, so we
can assess convergence rates. Next, we evaluate GANs generating $2$D
distributions, because they are simple enough to train with a plain,
alternating SGD. Finally, we look at scaling to larger-scale GANs on images
which have brittle optimization, and require optimizers like Adam. Complex
momentum provides benefits in each setup.
We only compare to first-order optimization methods, despite there being
various second-order methods due limitations discussed in Section 2.2.
### 4.1 Optimization in Purely Adversarial Games
Here, we consider the optimizing the Dirac-GAN objective, which is
surprisingly hard and where many classical optimization methods fail, because
$\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$ is
imaginary near solutions:
$\smash{\min_{x}\max_{y}-\log(1+\exp(-xy))-\log(2)}$ (15)
Figure 3 empirically verifies convergence rates given by Theorem 1 with (14),
by showing the optimization trajectories with simultaneous updates.
Figure 3.1 investigates how the components of the momentum $\beta$ affect
convergence rates with simultaneous updates and a fixed step size. The best
$\beta$ was almost-positive (i.e., $\arg(\beta)\\!=\\!\epsilon$ for small
$\epsilon$). We repeat this experiment with alternating updates in Appendix
Figure 8, which are standard in GAN training. There, almost-positive momentum
is best (but negative momentum also converges), and the benefit of alternating
updates can depend on if we can parallelize player gradient evaluations.
### 4.2 How Adversarialness Affects Convergence Rates
Here, we compare optimization with first-order methods for purely adversarial,
cooperative, and mixed games. We use the following game, allowing us to easily
interpolate between these regimes:
$\smash{\min_{\boldsymbol{x}}\max_{\boldsymbol{y}}\boldsymbol{x}^{\top}(\boldsymbol{\gamma}\boldsymbol{A})\boldsymbol{y}+\boldsymbol{x}^{\top}((\boldsymbol{I}-\boldsymbol{\gamma})\boldsymbol{B}_{1})\boldsymbol{x}-\boldsymbol{y}^{\top}((\boldsymbol{I}-\boldsymbol{\gamma})\boldsymbol{B}_{2})\boldsymbol{y}}$
(16)
# grad. eval. to convergeMax adversarialness $\gamma_{max}$ Figure 5: We
compare first-order methods convergence rates on the game in (16), with
$\boldsymbol{A}\\!=\\!\boldsymbol{B}_{1}\\!=\\!\boldsymbol{B}_{2}$ diagonal
and entries linearly spaced in $[\nicefrac{{$1$}}{{$4$}},$4$]$. We interpolate
from purely cooperative to a mixture of purely cooperative and adversarial
eigenspaces in
$\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$ by
making $\boldsymbol{\gamma}$ diagonal with
$\gamma_{j}\\!\sim\\!U[0,\gamma_{max}]$, inducing $j^{th}$ eigenvalue pair to
have $\arg(\lambda_{j})\\!\approx\\!\pm\gamma_{j}\frac{\pi}{2}$. So,
$\gamma_{max}$ controls the largest possible eigenvalue $\arg$ or _max
adversarialness_. Every method generalizes gradient descent-ascent (GDA) by
adding an optimizer parameter, tuned via grid search. Positive momentum and
negative momentum do not converge if there are purely adversarial eigenspaces
(i.e., $\gamma_{max}\\!=\\!1$). Almost-positive momentum
$\arg(\beta)\\!=\\!\epsilon$ $>\\!0$ like
${\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}\nicefrac{{\pi}}{{8}}}$
allows us to approach the acceleration of positive momentum if sufficiently
cooperative (i.e., $\gamma_{max}\\!<\\!$0.5$$), while still converging if
there are purely adversarial eigenspaces (i.e., $\gamma_{max}\\!=\\!1$).
Tuning $\arg(\beta)$ with complex momentum performs competitively with
extragradient (EG), optimistic gradient (OG) for any adversarialness – ex.,
${\color[rgb]{.75,.5,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.5,.25}\arg(\beta)\\!=\\!\nicefrac{{\pi}}{{2}}}$
does well if there are purely adversarial eigenspaces (i.e.,
$\gamma_{max}\\!=\\!1$).
If $\boldsymbol{\gamma}\\!=\\!\boldsymbol{I}$ the game is purely adversarial,
while if the $\boldsymbol{\gamma}\\!=\\!\boldsymbol{0}$ the game is purely
cooperative.
Figure 6 explores $\operatorname*{Sp}(\boldsymbol{R})$ in purely adversarial
games for a range of $\alpha,\beta$, generalizing Figure 4 in Gidel et al.
[25]. At every non-real $\beta$—i.e., $\arg(\beta)\\!\neq\\!\pi$ or $0$—we
could select $\alpha,|\beta|$ that converge.
$\Im(\operatorname*{Sp}(\boldsymbol{R}))$$\arg(\beta)=$0$$$\arg(\beta)=\frac{\pi}{$4$}$The
real component of the spectrum of the augmented learning dynamics
$\Re(\operatorname*{Sp}(\boldsymbol{R}))$$\arg(\beta)=\frac{\pi}{$2$}$$\arg(\beta)=\
\frac{$3$\pi}{$4$}$$\arg(\beta)=\pi$$|\beta|$ Figure 6: The spectrum of the
augmented learning dynamics $\boldsymbol{R}$ is shown, whose spectral norm is
the convergence rate in Theorem 1. Each image is a different momentum phase
$\arg(\beta)$ for a range of $\alpha,\\!|\beta|\\!\in\\![$0$,\\!$1$]$. The
opacity of an eigenvalue (eig) is the step size $\alpha$ and the color
corresponds to momentum magnitude $|\beta|$. A red unit circle shows where all
eigs must lie to converge for a fixed $\alpha,\beta$. If the max eig norm
$<\\!$1$$, we draw a green circle whose radius is our convergence rate and a
green star at the associated eig. Notably, at every non-real $\beta$ we can
select $\alpha,\\!|\beta|$ for convergence. The eigs are symmetric over the
$x$-axis, and eigs near $\Re(\lambda)\\!=\\!$1$$ dictate convergence rate.
Eigs near the center are due to state augmentation, have small magnitudes, and
do not impact convergence rate. Simultaneous gradient descent corresponds to
the magenta values where $|\beta|\\!=\\!$0$$.
Figure 5 compares first-order algorithms as we interpolate from the purely
cooperative games (i.e., minimization) to mixtures of purely adversarial and
cooperative eigenspaces, because this setup range can occur during GAN
training – see Figure 7. Our baselines are simultaneous SGD (or gradient
descent-ascent (GDA)), extragradient (EG) [41], optimistic gradient (OG) [50,
51, 52], and momentum variants. We added extrapolation parameters for EG and
OG so they are competitive with momentum – see Appendix Section C.3. We show
how many gradient evaluations for a set solution distance, and EG costs two
evaluations per update. We optimize convergence rates for each game and method
by grid search, as is common for optimization parameters in deep learning.
Takeaway: In the cooperative regime – i.e., $\gamma_{max}\\!<\\!.5$ or
$\max_{\lambda\in\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})}|\arg(\lambda)|\\!<\\!\nicefrac{{\pi}}{{4}}$
– the best method is classical, positive momentum, otherwise we benefit from a
method for learning in games. If we have purely adversarial eigenspaces then
GDA, positive and negative momentum fail to converge, while EG, OG, and
complex momentum can converge. In games like GANs, our eigendecomposition is
infeasible to compute and changes during training – see Appendix Figure 9 – so
we want an optimizer that converges robustly. Choosing any non-real momentum
$\beta$ allows robust convergence for every eigenspace mixture. More so,
almost-positive momentum $\beta$ allows us to approach acceleration when
cooperative, while still converging if there are purely adversarial
eigenspaces.
### 4.3 Training GANs on $2$D Distributions
Here, we investigate improving GAN training using alternating gradient descent
updates with complex momentum. We look at alternating updates, because they
are standard in GAN training [1, 32, 53]. It is not clear how EG and OG
generalize to alternating updates, so we use positive and negative momentum as
our baselines. We train to generate a $2$D mixture of Gaussians, because more
complicated distribution require more complicated optimizers than SGD. Figure
1 shows all changes necessary to use the JAX momentum optimizer for our
updates, with full details in Appendix C.4. We evaluate the log-likelihood of
GAN samples under the mixture as an imperfect proxy for matching.
Appendix Figure 10 shows heatmaps for tuning $\arg(\beta)$ and $|\beta|$ with
select step sizes. Takeaway: The best momentum was found at the almost-
positive $\beta\approx$0.7$\exp(i\nicefrac{{\pi}}{{$8$}})$ with step size
$\alpha\\!=\\!0.03$, and for each $\alpha$ we tested a broad range of non-real
$\beta$ outperformed any real $\beta$. This suggests we may be able to often
improve GAN training with alternating updates and complex momentum.
### 4.4 Training BigGAN with a Complex Adam
Here, we investigate improving larger-scale GAN training with complex
momentum. However, larger-scale GANs train with more complicated optimizers
than gradient descent – like Adam [31] – and have notoriously brittle
optimization. We look at training BigGAN [32] on CIFAR-10 [54], but were
unable to succeed with optimizers other than [32]-supplied setups, due to
brittle optimization. So, we attempted to change procedure minimally by taking
[32]-supplied code here which was trained with Adam, and making the
$\beta_{1}$ parameter – analogous to momentum – complex. The modified complex
Adam is shown in Algorithm 2, where the momentum bias correction is removed to
better match our theory. It is an open question on how to best carry over the
design of Adam (or other optimizers) to the complex setting. Training each
BigGAN took $10$ hours on an NVIDIA T4 GPU, so Figure C.4 and Table 1 took
about $1000$ and $600$ GPU hours respectively.
Figure C.4 shows a grid search over $\arg(\beta_{1})$ and $|\beta_{1}|$ for a
BigGAN trained with Algorithm 2. We only changed $\beta_{1}$ for the
discriminator’s optimizer. Takeaway: The best momentum was at the almost-
positive $\beta_{1}\\!\approx\\!$0.8$\exp(i\nicefrac{{\pi}}{{$8$}})$, whose
samples are in Appendix Figure C.4.
Algorithm 2 Complex Adam variant without momentum bias-correction
1:$\beta_{1}\\!\in\\!\mathbb{C},\beta_{2}\\!\in\\![$0$,\\!$1$)$
2:$\alpha\\!\in\\!\mathbb{R}^{+},\epsilon\\!\in\\!\mathbb{R}^{+}$
3:for $j=1\dots N$ do
4:
$\\!\\!\\!\\!\\!\boldsymbol{\mu}^{j\\!+\\!1}\\!\\!=\\!\beta_{1}\boldsymbol{\mu}^{j}-\boldsymbol{g}^{j}$
5:
$\\!\\!\\!\\!\\!\boldsymbol{v}^{j\\!+\\!1}\\!\\!=\\!\beta_{2}\boldsymbol{v}^{j}\\!+\\!(1\\!-\\!\beta_{2})(\boldsymbol{g}^{j})^{\\!$2$}$
6:
$\\!\\!\\!\\!\\!\hat{\boldsymbol{v}}^{j\\!+\\!1}\\!\\!=\\!\frac{\boldsymbol{v}^{j\\!+\\!1}}{1-(\beta_{2})^{j}}$
7:
$\\!\\!\\!\\!\\!\boldsymbol{\omega}^{j\\!+\\!1}\\!\\!=\\!\boldsymbol{\omega}^{j}+\alpha\frac{\Re(\boldsymbol{\mu}^{j})}{\sqrt{\hat{\boldsymbol{v}}^{j\\!+\\!1}}+\epsilon}$
return $\boldsymbol{\omega}^{N}$
CIFAR-10 BigGAN | Best IS for $10$ seeds
---|---
Discriminator $\beta_{1}$ | Min | Max
$0$ – [32]’s default | $8.9$ | $9.1$
$$0.8$\exp(i\nicefrac{{\pi}}{{$8$}})$ – ours | $$8.96$({\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}+.06})$ | $$9.25$({\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}+.15})$
$0.8$ | $$3.12$({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-5.78})$ | $$9.05$({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-0.05})$
Table 1: We display the best inception scores (IS) found over $10$ runs for
training BigGAN on CIFAR-10 with various optimizer settings. We use a complex
Adam variant outlined in Algorithm 2, where we only tuned $\beta_{1}$ for the
discriminator. The best parameters found in Figure C.4 were
$\beta_{1}=$0.8$\exp(i\nicefrac{{\pi}}{{$8$}})$, which improved the min and
max IS from our runs of the BigGAN authors baseline, which was the SoTA
optimizer in this setting to best of our knowledge. We tested
$\beta_{1}=$0.8$$ to see if the gain was solely from tuning $|\beta_{1}|$,
which occasionally failed and decreased the best IS.
We tested the best momentum value over $10$ seeds against the author-provided
baseline in Appendix Figure C.4, with the results summarized in Table 1. [32]
reported a single inception score (IS) on CIFAR-10 of $9.22$, but the best we
could reproduce over the seeds with the provided PyTorch code and settings was
$9.10$. Complex momentum improves the best IS found with
$$9.25$({\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}+.15}\text{
over author
code},{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}+.03}\text{
author reported})$. We trained a real momentum $|\beta_{1}|\\!=\\!$0.8$$ to
see if the improvement was solely from tuning the momentum magnitude. This
occasionally failed to train and decreased the best IS over re-runs, showing
we benefit from a non-zero $\arg(\beta_{1})$.
### 4.5 A Practical Initial Guess for Optimizer Parameter $\arg(\beta)$
Here, we propose a practical initial guess for our new hyperparameter
$\arg(\beta)$. Corollary 1 shows we can use almost-real momentum coefficients
(i.e., $\arg(\beta)$ is close to $0$). Figure 5 shows almost-positive $\beta$
approach acceleration in cooperative eigenspaces, while converging in all
eigenspaces. Figure 7 shows GANs can have both cooperative and adversarial
eigenspaces. Figures 10 and C.4 do a grid search over $\arg(\beta)$ for GANs,
finding that almost-positive $\arg(\beta)\approx\nicefrac{{\pi}}{{$8$}}$ works
in both cases. Also, by minimally changing $\arg(\beta)$ from $0$ to a small
$\epsilon$, we can minimally change other hyperparameters in our model, which
is useful to adapt existing, brittle setups like in GANs. Based on this, we
propose an initial guess of $\arg(\beta)=\epsilon$ for a small $\epsilon>0$,
where $\epsilon=\nicefrac{{\pi}}{{$8$}}$ worked in our GAN experiments.
-$\pi$ -$\frac{\pi}{2}$ $0$ -$\frac{\pi}{2}$ -$\pi$Phase of eigenvalue $\arg(\lambda)$Spectrum of Jacobian of joint-grad $\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}^{j})$ for GANLog-magnitude of eigenvalue $\log(|\lambda|)$disc. unsure gen.Does eigenvector point at a player? Figure 7: A log-polar coordinate visualization reveals structure in the spectrum for a GAN at the end of the training on a $2$D mixture of Gaussians with a $1$-layer (disc)riminator and (gen)erator, so the joint-parameters $\boldsymbol{\omega}\\!\in\\!\mathbb{R}^{$723$}$. It is difficult to see structure by graphing the Cartesian (i.e., $\Re$ and $\Im$) parts of eigenvalues, because they span orders of magnitude, while being positive and negative. Appendix Figure 9 shows the spectrum through training.
There is a mixture of many cooperative (i.e., real or
$\arg(\lambda)\\!\approx\\!$0$,\pm\pi$) and some adversarial (i.e., imaginary
or $\arg(\lambda)\\!\approx\\!\pm\frac{\pi}{$2$}$) eigenvalues, so – contrary
to what the name may suggest – generative adversarial networks are not purely
adversarial. We may benefit from optimizers leveraging this structure like
complex momentum.
Eigenvalues are colored if the associated eigenvector is mostly in one
player’s part of the joint-parameter space – see Appendix Figure 9 for details
on this. Many eigenvectors lie mostly in the the space of (or point at) a one
player. The structure of the set of eigenvalues for the disc. (green) is
different than the gen. (red), but further investigation of this is an open
problem. Notably, this may motivate separate optimizer choices for each player
as in Section 4.4.
## 5 Related Work
Accelerated first-order methods: A broad body of work exists using momentum-
type methods [27, 28, 55, 56], with a recent focus on deep learning [29, 57,
58, 59, 60]. But, these works focus on momentum for minimization as opposed to
in games.
Learning in games: Various works approximate response-gradients - some by
differentiating through optimization [61, 34, 62]. Multiple works try to
leverage game eigenstructure during optimization [63, 64, 65, 39, 66, 67].
First-order methods in games: In some games, we can get away with using only
first-order methods – Zhang et al. [68, 69], Ibrahim et al. [70], Bailey et
al. [71], Jin et al. [72], Azizian et al. [47], Nouiehed et al. [73], Zhang et
al. [74] discuss when and how these methods work. Gidel et al. [25] is the
closest work to ours, showing a negative momentum can help in some games.
Zhang and Wang [30] note the suboptimality of negative momentum in a class of
games. Azizian et al. [75], Domingo-Enrich et al. [76] investigate
acceleration in some games.
Bilinear zero-sum games: Zhang and Yu [77] study the convergence of gradient
methods in bilinear zero-sum games. Their analysis extends Gidel et al. [25],
showing that we can achieve faster convergence by having separate step sizes
and momentum for each player or tuning the extragradient step size. Loizou et
al. [78] provide convergence guarantees for games satisfying a _sufficiently
bilinear_ condition.
Learning in GANs: Various works try to make GAN training easier with methods
leveraging the game structure [79, 80, 81, 53, 82]. Metz et al. [83]
approximate the discriminator’s response function by differentiating through
optimization. Mescheder et al. [34] find solutions by minimizing the norm of
the players’ updates. Both of these methods and various others [84, 85, 86]
require higher-order information. Daskalakis et al. [52], Gidel et al. [87],
Chavdarova et al. [88] look at first-order methods. Mescheder et al. [42]
explore problems for GAN training convergence and Berard et al. [22] show that
GANs have significant rotations affecting learning.
## 6 Conclusion
In this paper we provided a generalization of existing momentum methods for
learning in differentiable games by allowing a complex-valued momentum with
real-valued updates. We showed that our method robustly converges in games
with a different range of mixtures of cooperative and adversarial eigenspaces
than current first-order methods. We also presented a practical generalization
of our method to the Adam optimizer, which we used to improve BigGAN training.
More generally, we highlight and lay groundwork for investigating optimizers
which work well with various mixtures of cooperative and competitive dynamics
in games.
### Societal Impact
Our main contribution in this work is methodological – specifically, a
scalable algorithm for optimizing in games. Since our focus is on improving
optimization methods, we do not expect there to be direct negative societal
impacts from this contribution.
### Acknowledgements
Resources used in preparing this research were provided, in part, by the
Province of Ontario, the Government of Canada through CIFAR, and companies
sponsoring the Vector Institute. Paul Vicol was supported by an NSERC PGS-D
Scholarship. We thank Guodong Zhang, Guojun Zhang, James Lucas, Romina Abachi,
Jonah Phillion, Will Grathwohl, Jakob Foerster, Murat Erdogdu, Ken Jackson,
and Ioannis Mitliagkis for feedback and helpful discussion.
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Appendix: Complex Momentum for Optimization in Games
Table 2: Notation SGD | Stochastic Gradient Descent
---|---
CM | Complex Momentum
SGDm, SimSGDm, … | …with momentum
SimSGD, SimCM | Simultaneous …
AltSGD, AltCM | Alternating …
GAN | Generative Adversarial Network [1]
EG | Extragradient [41]
OG | Optimistic Gradient [52]
IS | Inception Score [89]
$\vcentcolon=$ | Defined to be equal to
$x,y,z,\dots\in\mathbb{C}$ | Scalars
$\boldsymbol{x},\boldsymbol{y},\boldsymbol{z},\dots\in\mathbb{C}^{n}$ | Vectors
$\boldsymbol{X},\boldsymbol{Y},\boldsymbol{Z},\dots\in\mathbb{C}^{n\times n}$ | Matrices
$\boldsymbol{X}^{\top}$ | The transpose of matrix $\boldsymbol{X}$
$\boldsymbol{I}$ | The identity matrix
$\Re(z),\Im(z)$ | The real or imaginary component of $z\in\mathbb{C}$
$i$ | The imaginary unit. $z\in\mathbb{C}\implies z=\Re(z)+i\Im(z)$
$\widebar{z}$ | The complex conjugate of $z\in\mathbb{C}$
$|z|\vcentcolon=\sqrt{z\widebar{z}}$ | The magnitude or modulus of $z\in\mathbb{C}$
$\arg(z)$ | The argument or phase of $z\in\mathbb{C}\implies z=|z|\exp(i\arg(z))$
$z\\!\in\\!\mathbb{C}$ is _almost-positive_ | $\arg(z)\\!=\\!\epsilon$ for small $\epsilon$ respectively
$A,B$ | A symbol for the outer/inner players
$d_{A},d_{B}\in\mathbb{N}$ | The number of weights for the outer/inner players
$\boldsymbol{\theta}$ | A symbol for the parameters or weights of a player
$\boldsymbol{\theta}_{\\!A}\in\mathbb{R}^{d_{A}},\boldsymbol{\theta}_{\\!B}\in\mathbb{R}^{d_{B}}$ | The outer/inner parameters or weights
$\mathcal{L}:\mathbb{R}^{n}\to\mathbb{R}$ | A symbol for a loss
$\mathcal{L}_{\\!A}(\boldsymbol{\theta}_{\\!A},\boldsymbol{\theta}_{\\!B}),\mathcal{L}_{\\!B}(\boldsymbol{\theta}_{\\!A},\boldsymbol{\theta}_{\\!B})$ | The outer/inner losses – $\mathbb{R}^{d_{A}+d_{B}}\mapsto\mathbb{R}$
$\boldsymbol{g}_{\\!A}(\boldsymbol{\theta}_{\\!A},\boldsymbol{\theta}_{\\!B}),\boldsymbol{g}_{\\!B}(\boldsymbol{\theta}_{\\!A},\boldsymbol{\theta}_{\\!B})$ | Gradient of outer/inner losses w.r.t. their weights in $\mathbb{R}^{d_{A}/d_{B}}$
$\boldsymbol{\theta}_{\\!B}^{*}\\!(\boldsymbol{\theta}_{\\!A}\\!)\\!\vcentcolon=\\!\operatorname*{arg\,min}\limits_{\boldsymbol{\theta}_{\\!B}}\\!\\!\mathcal{L}_{\\!B}\\!(\\!\boldsymbol{\theta}_{\\!A}\\!,\\!\boldsymbol{\theta}_{\\!B}\\!)$ | The best-response of the inner player to the outer player
$\mathcal{L}_{\\!A}^{*}(\boldsymbol{\theta}_{\\!A})\\!\vcentcolon=\\!\mathcal{L}_{\\!A}\\!(\\!\boldsymbol{\theta}_{\\!A}\\!,\\!\boldsymbol{\theta}_{\\!B}^{*}(\boldsymbol{\theta}_{\\!A})\\!)$ | The outer loss with a best-responding inner player
$\boldsymbol{\theta}_{\\!A}^{*}\\!\vcentcolon=\\!\operatorname*{arg\,min}\limits_{\boldsymbol{\theta}_{\\!A}}\mathcal{L}_{\\!A}^{*}(\boldsymbol{\theta}_{\\!A}\\!)$ | Outer optimal weights with a best-responding inner player
$d\vcentcolon=d_{A}+d_{B}$ | The combined number of weights for both players
$\boldsymbol{\omega}\vcentcolon=[\boldsymbol{\theta}_{\\!A},\boldsymbol{\theta}_{\\!B}]\in\mathbb{R}^{d}$ | A concatenation of the outer/inner weights
$\hat{\boldsymbol{g}}(\boldsymbol{\omega})\\!\vcentcolon=\\![\boldsymbol{g}_{\\!A}(\boldsymbol{\omega}),\boldsymbol{g}_{\\!B}(\boldsymbol{\omega})]\in\mathbb{R}^{d}$ | A concatenation of the outer/inner gradients
$\boldsymbol{\omega}^{$0$}=[\boldsymbol{\theta}_{\\!A}^{$0$},\boldsymbol{\theta}_{\\!B}^{$0$}]\in\mathbb{R}^{d}$ | The initial parameter values
$j$ | An iteration number
$\hat{\boldsymbol{g}}^{j}\vcentcolon=\hat{\boldsymbol{g}}(\boldsymbol{\omega}^{j})\in\mathbb{R}^{d}$ | The joint-gradient vector field at weights $\boldsymbol{\omega}^{j}$
$\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}^{j}\vcentcolon=\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}|_{\boldsymbol{\omega}^{j}}\in\mathbb{R}^{d\times d}$ | The Jacobian of the joint-gradient $\hat{\boldsymbol{g}}$ at weights $\boldsymbol{\omega}^{j}$
$\alpha\in\mathbb{C}$ | The step size or learning rate
$\beta\in\mathbb{C}$ | The momentum coefficient
$\beta_{1}\in\mathbb{C}$ | The first momentum parameter for Adam
$\boldsymbol{\mu}\in\mathbb{C}^{d}$ | The momentum buffer
$\lambda\in\mathbb{C}$ | Notation for an arbitrary eigenvalue
$\operatorname*{Sp}(\boldsymbol{M})\in\mathbb{C}^{n}$ | The spectrum – or set of eigenvalues – of $\boldsymbol{M}\in\mathbb{R}^{n\times n}$
_Purely adversarial/cooperative_ game | $\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$ is purely real/imaginary
$\rho(\boldsymbol{M})\\!\vcentcolon=\\!\max_{z\in\operatorname*{Sp}(\boldsymbol{M})}|z|$ | The spectral radius in $\mathbb{R}^{+}$ of $\boldsymbol{M}\in\mathbb{R}^{n\times n}$
$\boldsymbol{F}_{\alpha,\beta}([\boldsymbol{\mu},\boldsymbol{\omega}])$ | Fixed point op. for CM, or augmented learning dynamics
$\boldsymbol{R}\vcentcolon=\nabla_{[\boldsymbol{\mu},\boldsymbol{\omega}]}\boldsymbol{F}_{\alpha,\beta}\in\mathbb{R}^{$3$d\times$3$d}$ | Jacobian of the augmented learning dynamics in Corollary 1
$\alpha^{*}\\!,\beta^{*}\\!\vcentcolon=\\!\operatorname*{arg\,min}\limits_{\alpha,\beta}\rho(\boldsymbol{R}(\alpha,\\!\beta)\\!)$ | The optimal step size and momentum coefficient
$\rho^{*}\vcentcolon=\rho(R(\alpha^{*},\beta^{*}))$ | The optimal spectral radius or convergence rate
$\kappa\vcentcolon=\frac{\max\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\boldsymbol{g})}{\min\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\boldsymbol{g})}$ | Condition number, for convex single-objective optimization
$\sigma_{min}^{2}(\boldsymbol{M})\\!\vcentcolon=\\!\max\operatorname*{Sp}(\boldsymbol{M}^{\top}\boldsymbol{M})$ | The minimum singular value of a matrix $\boldsymbol{M}$
## Appendix A Supporting Results
First, some basic results about complex numbers that are used:
$z=\Re(z)+i\Im(z)=|z|\exp(i\arg(z))$ (17)
$\widebar{z}=\Re(z)-i\Im(z)=|z|\exp(-i\arg(z))$ (18)
$\exp(iz)+\exp(-iz)=2\cos(z)$ (19)
$\widebar{z_{1}z_{2}}=\widebar{z}_{1}\widebar{z}_{2}$ (20)
$\nicefrac{{1}}{{2}}(z+\widebar{z})=\Re(z)$ (21)
$\Re(z_{1}z_{2})=\Re(z_{1})\Re(z_{2})-\Im(z_{1})\Im(z_{2})$ (22)
$z_{1}+z_{2}=(\Re(z_{1})+\Re(z_{2}))+i(\Im(z_{1})+\Im(z_{2}))$ (23)
$z_{1}z_{2}=(\Re(z_{1})\Re(z_{2})-\Im(z_{1})\Im(z_{2}))+i(\Im(z_{1})\Re(z_{2})+\Re(z_{1})\Im(z_{2}))$
(24) $z_{1}z_{2}=|z_{1}||z_{2}|\exp(i(\arg(z_{1})+\arg(z_{2})))$ (25)
$z^{k}=|z|^{k}\exp(i\arg(z)k)=|z|^{k}(\cos(k\arg(z))+i\sin(k\arg(z))$ (26)
This Lemma shows how we expand the complex-valued momentum buffer
$\boldsymbol{\mu}$ into its Cartesian components as in (9).
###### Lemma 1.
$\displaystyle\boldsymbol{\mu}^{j\\!+\\!1}$
$\displaystyle=\beta\boldsymbol{\mu}^{j}-\hat{\boldsymbol{g}}^{j}\iff$
$\displaystyle\Re(\boldsymbol{\mu}^{j\\!+\\!1})$
$\displaystyle=\Re(\beta)\Re(\boldsymbol{\mu}^{j})-\Im(\beta)\Im(\boldsymbol{\mu}^{j})-\Re(\hat{\boldsymbol{g}}^{j}),\Im(\boldsymbol{\mu}^{j\\!+\\!1})=\Im(\beta)\Re(\boldsymbol{\mu}^{j})+\Re(\beta)\Im(\boldsymbol{\mu}^{j})-\Im(\hat{\boldsymbol{g}}^{j})$
###### Proof.
$\displaystyle\boldsymbol{\mu}^{j\\!+\\!1}$
$\displaystyle=\beta\boldsymbol{\mu}^{j}-\hat{\boldsymbol{g}}^{j}$
$\displaystyle\iff\boldsymbol{\mu}^{j\\!+\\!1}$
$\displaystyle=\left(\Re(\beta)+i\Im(\beta)\right)\left(\Re(\boldsymbol{\mu}^{j})+i\Im(\boldsymbol{\mu}^{j})\right)-\left(\Re(\hat{\boldsymbol{g}}^{j})+i\Im(\hat{\boldsymbol{g}}^{j})\right)$
$\displaystyle\iff\boldsymbol{\mu}^{j\\!+\\!1}$
$\displaystyle=\left(\Re(\beta)\Re(\boldsymbol{\mu}^{j})-\Im(\beta)\Im(\boldsymbol{\mu}^{j})\right)+$
$\displaystyle\hskip
108.405pti\left(\Im(\beta)\Re(\boldsymbol{\mu}^{j})+\Re(\beta)\Im(\boldsymbol{\mu}^{j})\right)-\left(\Re(\hat{\boldsymbol{g}}^{j})+i\Im(\hat{\boldsymbol{g}}^{j})\right)$
$\displaystyle\iff\boldsymbol{\mu}^{j\\!+\\!1}$
$\displaystyle=\left(\Re(\beta)\Re(\boldsymbol{\mu}^{j})-\Im(\beta)\Im(\boldsymbol{\mu}^{j})-\Re(\hat{\boldsymbol{g}}^{j})\right)+$
$\displaystyle\hskip
108.405pti\left(\Im(\beta)\Re(\boldsymbol{\mu}^{j})+\Re(\beta)\Im(\boldsymbol{\mu}^{j})-\Im(\hat{\boldsymbol{g}}^{j})\right)$
$\displaystyle\iff\Re(\boldsymbol{\mu}^{j\\!+\\!1})\\!$
$\displaystyle=\\!\Re(\beta)\Re(\boldsymbol{\mu}^{j})\\!-\\!\Im(\beta)\Im(\boldsymbol{\mu}^{j})\\!-\\!\Re(\hat{\boldsymbol{g}}^{j}),\Im(\boldsymbol{\mu}^{j\\!+\\!1})\\!=\\!\Im(\beta)\Re(\boldsymbol{\mu}^{j})\\!+\\!\Re(\beta)\Im(\boldsymbol{\mu}^{j})\\!-\\!\Im(\hat{\boldsymbol{g}}^{j})$
∎
We further assume $\Im(\hat{\boldsymbol{g}}^{j})$ is $0$ \- i.e., our
gradients are real-valued. This Lemma shows how we can decompose the joint-
parameters $\boldsymbol{\omega}$ at the next iterate as a linear combination
of the joint-parameters, joint-gradient, and Cartesian components of the
momentum-buffer at the current iterate as in (10).
###### Lemma 2.
$\displaystyle\boldsymbol{\omega}^{j\\!+\\!1}=\boldsymbol{\omega}^{j}+\Re(\alpha\boldsymbol{\mu}^{j\\!+\\!1})\iff\boldsymbol{\omega}^{j\\!+\\!1}=\boldsymbol{\omega}^{j}-\Re(\alpha)\hat{\boldsymbol{g}}^{j}+\Re(\alpha\beta)\Re(\boldsymbol{\mu}^{j})-\Im(\alpha\beta)\Im(\boldsymbol{\mu}^{j})$
###### Proof.
$\displaystyle\Re(\alpha\boldsymbol{\mu}^{j\\!+\\!1})$ $\displaystyle=$
$\displaystyle\left(\Re(\alpha)\Re(\boldsymbol{\mu}^{j\\!+\\!1})-\Im(\alpha)\Im(\boldsymbol{\mu}^{j\\!+\\!1})\right)$
$\displaystyle=$
$\displaystyle\left(\Re(\alpha)\left(\Re(\beta)\Re(\boldsymbol{\mu}^{j})-\Im(\beta)\Im(\boldsymbol{\mu}^{j})-\hat{\boldsymbol{g}}^{j}\right)-\Im(\alpha)\left(\Im(\beta)\Re(\boldsymbol{\mu}^{j})+\Re(\beta)\Im(\boldsymbol{\mu}^{j})\right)\right)$
$\displaystyle=$
$\displaystyle-\Re(\alpha)\hat{\boldsymbol{g}}^{j}+\left(\Re(\alpha)\left(\Re(\beta)\Re(\boldsymbol{\mu}^{j})-\Im(\beta)\Im(\boldsymbol{\mu}^{j})\right)-\Im(\alpha)\left(\Im(\beta)\Re(\boldsymbol{\mu}^{j})+\Re(\beta)\Im(\boldsymbol{\mu}^{j})\right)\right)$
$\displaystyle=$
$\displaystyle-\Re(\alpha)\hat{\boldsymbol{g}}^{j}+\left(\Re(\alpha)\Re(\beta)-\Im(\alpha)\Im(\beta)\right)\Re(\boldsymbol{\mu}^{j})-\left(\Re(\alpha)\Im(\beta)+\Im(\alpha)\Re(\beta)\right)\Im(\boldsymbol{\mu}^{j})$
$\displaystyle=$
$\displaystyle-\Re(\alpha)\hat{\boldsymbol{g}}^{j}+\Re(\alpha\beta)\Re(\boldsymbol{\mu}^{j})-\Im(\alpha\beta)\Im(\boldsymbol{\mu}^{j})$
Thus,
$\displaystyle\boldsymbol{\omega}^{j\\!+\\!1}=\boldsymbol{\omega}^{j}+\Re(\alpha\boldsymbol{\mu}^{j\\!+\\!1})\iff\boldsymbol{\omega}^{j\\!+\\!1}=\boldsymbol{\omega}^{j}-\Re(\alpha)\hat{\boldsymbol{g}}^{j}+\Re(\alpha\beta)\Re(\boldsymbol{\mu}^{j})-\Im(\alpha\beta)\Im(\boldsymbol{\mu}^{j})$
∎
### A.1 Theorem 1 Proof Sketch
See 1
###### Proof.
We reproduce the proof for a simpler case of quadratic games, which is simple
case of Polyak [27]’s well-known method for analyzing the convergence of
iterative methods. Bertsekas [44] generalizes this result from quadratic games
to when we are sufficiently close to any stationary point.
For quadratic games, we have that
$\hat{\boldsymbol{g}}^{j}=\left(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}\right)^{\top}\boldsymbol{\omega}^{j}$.
Well, by Lemma 1 and Lemma 2 we have:
$\begin{pmatrix}\Re(\boldsymbol{\mu}^{j\\!+\\!1})\\\
\Im(\boldsymbol{\mu}^{j\\!+\\!1})\\\
\boldsymbol{\omega}^{j\\!+\\!1}\end{pmatrix}=\boldsymbol{R}\begin{pmatrix}\Re(\boldsymbol{\mu}^{j})\\\
\Im(\boldsymbol{\mu}^{j})\\\ \boldsymbol{\omega}^{j}\end{pmatrix}$ (27)
By telescoping the recurrence for the $j^{th}$ augmented parameters:
$\begin{pmatrix}\Re(\boldsymbol{\mu}^{j})\\\ \Im(\boldsymbol{\mu}^{j})\\\
\boldsymbol{\omega}^{j}\end{pmatrix}=\boldsymbol{R}^{j}\begin{pmatrix}\Re(\boldsymbol{\mu}^{0})\\\
\Im(\boldsymbol{\mu}^{0})\\\ \boldsymbol{\omega}^{0}\end{pmatrix}$ (28)
We can compare $\boldsymbol{\mu}^{j}$ with the value it converges to
$\boldsymbol{\mu}^{*}$ which exists if $\boldsymbol{R}$ is contractive. We do
the same with $\boldsymbol{\omega}$. Because
$\boldsymbol{\mu}^{*}=\boldsymbol{R}\boldsymbol{\mu}^{*}=\boldsymbol{R}^{j}\boldsymbol{\mu}^{*}$:
$\begin{pmatrix}\Re(\boldsymbol{\mu}^{j})-\Re(\boldsymbol{\mu}^{*})\\\
\Im(\boldsymbol{\mu}^{j})-\Im(\boldsymbol{\mu}^{*})\\\
\boldsymbol{\omega}^{j}-\boldsymbol{\omega}^{*}\end{pmatrix}=\boldsymbol{R}^{j}\begin{pmatrix}\Re(\boldsymbol{\mu}^{0})-\Re(\boldsymbol{\mu}^{*})\\\
\Im(\boldsymbol{\mu}^{0})-\Im(\boldsymbol{\mu}^{*})\\\
\boldsymbol{\omega}^{0}-\boldsymbol{\omega}^{*}\end{pmatrix}$ (29)
By taking norms:
$\displaystyle\left\|\begin{pmatrix}\Re(\boldsymbol{\mu}^{j})-\Re(\boldsymbol{\mu}^{*})\\\
\Im(\boldsymbol{\mu}^{j})-\Im(\boldsymbol{\mu}^{*})\\\
\boldsymbol{\omega}^{j}-\boldsymbol{\omega}^{*}\end{pmatrix}\right\|_{2}$
$\displaystyle=\left\|\boldsymbol{R}^{j}\begin{pmatrix}\Re(\boldsymbol{\mu}^{0})-\Re(\boldsymbol{\mu}^{*})\\\
\Im(\boldsymbol{\mu}^{0})-\Im(\boldsymbol{\mu}^{*})\\\
\boldsymbol{\omega}^{0}-\boldsymbol{\omega}^{*}\end{pmatrix}\right\|_{2}$ (30)
$\displaystyle\implies\left\|\begin{pmatrix}\Re(\boldsymbol{\mu}^{j})-\Re(\boldsymbol{\mu}^{*})\\\
\Im(\boldsymbol{\mu}^{j})-\Im(\boldsymbol{\mu}^{*})\\\
\boldsymbol{\omega}^{j}-\boldsymbol{\omega}^{*}\end{pmatrix}\right\|_{2}$
$\displaystyle\leq\left\|\boldsymbol{R}^{j}\right\|_{2}\left\|\begin{pmatrix}\Re(\boldsymbol{\mu}^{0})-\Re(\boldsymbol{\mu}^{*})\\\
\Im(\boldsymbol{\mu}^{0})-\Im(\boldsymbol{\mu}^{*})\\\
\boldsymbol{\omega}^{0}-\boldsymbol{\omega}^{*}\end{pmatrix}\right\|_{2}$ (31)
With Lemma 11 from Foucart [90], we have there exists a matrix norm
$\forall\epsilon>0$ such that:
$\|\boldsymbol{R}^{j}\|\leq\left(\rho\left(\boldsymbol{R}\right)+\epsilon\right)^{j}$
(32)
We also have an equivalence of norms in finite-dimensional spaces. So for all
norms $\|\cdot\|$, $\exists C\geq B>0$ such that:
$B\|\boldsymbol{R}^{j}\|\leq\|\boldsymbol{R}^{j}\|_{2}\leq
C\|\boldsymbol{R}^{j}\|$ (33)
Combining (32) and (33) we have:
$\left\|\begin{pmatrix}\Re(\boldsymbol{\mu}^{j})-\Re(\boldsymbol{\mu}^{*})\\\
\Im(\boldsymbol{\mu}^{j})-\Im(\boldsymbol{\mu}^{*})\\\
\boldsymbol{\omega}^{j}-\boldsymbol{\omega}^{*}\end{pmatrix}\right\|_{2}\leq
C\left(\rho\left(\boldsymbol{R}\right)+\epsilon\right)^{j}\left\|\begin{pmatrix}\Re(\boldsymbol{\mu}^{0})-\Re(\boldsymbol{\mu}^{*})\\\
\Im(\boldsymbol{\mu}^{0})-\Im(\boldsymbol{\mu}^{*})\\\
\boldsymbol{\omega}^{0}-\boldsymbol{\omega}^{*}\end{pmatrix}\right\|_{2}$ (34)
So, we have:
$\left\|\begin{pmatrix}\Re(\boldsymbol{\mu}^{j})-\Re(\boldsymbol{\mu}^{*})\\\
\Im(\boldsymbol{\mu}^{j})-\Im(\boldsymbol{\mu}^{*})\\\
\boldsymbol{\omega}^{j}-\boldsymbol{\omega}^{*}\end{pmatrix}\right\|_{2}=\mathcal{O}((\rho(\boldsymbol{R})+\epsilon)^{j})$
(35)
Thus, we converge linearly with a rate of
$\mathcal{O}(\rho(\boldsymbol{R})+\epsilon)$. ∎
### A.2 Characterizing the Augmented Dynamics Eigenvalues
Here, we present polynomials whose roots are the eigenvalues of our the
Jacobian of our augmented dynamics $\operatorname*{Sp}(\boldsymbol{R})$, given
the eigenvalues of the Jacobian of the joint-gradient vector field
$\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$. We use
a similar decomposition as Gidel et al. [25].
We can expand $\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}=PTP^{-1}$
where $T$ is an upper-triangular matrix and $\lambda_{i}$ is an eigenvalue of
$\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}$.
$T=\begin{bmatrix}\lambda_{1}&*&\dots&*\\\ $0$&\dots&\dots&\dots\\\
\dots&\dots&\dots&*\\\ $0$&\dots&$0$&\lambda_{d}\\\ \end{bmatrix}$ (36)
We then break up into components for each eigenvalue, giving us submatrices
$\mathbf{R}_{k}\in\mathbb{C}^{$3$\times$3$}$:
$\boldsymbol{R}_{k}\vcentcolon=\begin{bmatrix}\Re(\beta)&-\Im(\beta)&-\lambda_{k}\\\
\Im(\beta)&\Re(\beta)&0\\\
\Re(\alpha\beta)&-\Im(\alpha\beta)&1-\Re(\alpha)\lambda_{k}\\\ \end{bmatrix}$
(37)
We can get the characteristic polynomial of $\boldsymbol{R}_{k}$ with the
following Mathematica command, where we use substitute the symbols
$r+iu=\lambda_{k}$, $a=\Re(\beta)$, $b=\Im(\beta)$, $c=\Re(\alpha)$, and
$d=\Im(\alpha)$.
CharacteristicPolynomial[{{a, -b, -(r + u I)}, {b, a, 0}, {a c - b d, -(b c +
a d), 1 - c (r + u I)}}, x]
The command gives us the polynomial associated with eigenvalue
$\lambda_{k}=r+iu$:
$p_{k}(x)=-a^{2}x+a^{2}+acrx+iacux+2ax^{2}-2ax-b^{2}x+b^{2}+bdrx+ibdux-
crx^{2}-icux^{2}-x^{3}+x^{2}$ (38)
Consider the case where $\lambda_{k}$ is imaginary – i.e, $r=0$ – which is
true in all purely adversarial and bilinear zero-sum games. Then (38)
simplifies to:
$p_{k}(x)=-a^{2}x+a^{2}+iacux+2ax^{2}-2ax-b^{2}x+b^{2}+ibdux-
icux^{2}-x^{3}+x^{2}$ (39)
Our complex $\lambda_{k}$ come in conjugate pairs where $\lambda_{k}=u_{k}i$
and $\bar{\lambda}_{k}=-u_{k}i$. (39) has the same roots for $\lambda_{k}$ and
$\bar{\lambda}_{k}$, which can be verified by writing the roots with the cubic
formula. This corresponds to spiraling around the solution in either a
clockwise or counterclockwise direction. Thus, we restrict to analyzing
$\lambda_{k}$ where $u_{k}$ is positive without loss of generality.
If we make the step size $\alpha$ real – i.e., $d=0$ – then (39) simplifies
to:
$p_{k}(x)=x(-a^{2}+iacu-2a-b^{2})+a^{2}+x^{2}(2a-icu+1)+b^{2}-x^{3}$ (40)
Using a heuristic from single-objective optimization, we look at making step
size proportional to the inverse of the magnitude of eigenvalue $k$ – i.e.,
$\alpha_{k}=\frac{\alpha^{\prime}}{|\lambda_{k}|}=\frac{\alpha^{\prime}}{u_{k}}$.
With this, (40) simplifies to:
$p_{k}(x)=x(-a^{2}+ia\alpha^{\prime}-2a-b^{2})+a^{2}+x^{2}(2a-i\alpha^{\prime}+1)+b^{2}-x^{3}$
(41)
Notably, in (41) there is no dependence on the components of imaginary
eigenvalue $\lambda_{k}=r+iu=$0$+iu$, by selecting a $\alpha$ that is
proportional to the eigenvalues inverse magnitude. We can simplify further
with $a^{2}+b^{2}=|\beta|^{2}$:
$p_{k}(x)=x(\Re(\beta)(i\alpha^{\prime}-2)-|\beta|^{2})+x^{2}(2\Re(\beta)-i\alpha^{\prime}+1)+|\beta|^{2}-x^{3}$
(42)
We could expand this in polar form for $\beta$ by noting
$\Re(\beta)=|\beta|\cos(\arg(\beta))$:
$p_{k}(x)=x(|\beta|\cos(\arg(\beta))(i\alpha^{\prime}-2)-|\beta|^{2})+x^{2}(2|\beta|\cos(\arg(\beta))-i\alpha^{\prime}+1)+|\beta|^{2}-x^{3}$
(43)
We can simplify further by considering an imaginary $\beta$ – i.e.,
$\Re(\beta)=0$ or $\cos(\arg(\beta))=0$:
$p_{k}(x)=|\beta|^{2}-x|\beta|^{2}-x^{2}(i\alpha^{\prime}-1)-x^{3}$ (44)
The roots of these polynomials can be trivially evaluated numerically or
symbolically with the by plugging in $\beta,\alpha,$ and $\lambda_{k}$ then
using the cubic formula. This section can be easily modified for the
eigenvalues of the augmented dynamics for variants of complex momentum by
defining the appropriate $\boldsymbol{R}$ and modifying the Mathematica
command to get the characteristic polynomial for each component, which can be
evaluated if it is a sufficiently low degree using known formulas.
### A.3 Convergence Bounds
See 1
###### Proof.
Note that Theorem 1 bounds the convergence rate of Algorithm 1 by
$\operatorname*{Sp}(\boldsymbol{R})$. Also, (40) gives a formula for 3
eigenvalues in $\operatorname*{Sp}(\boldsymbol{R})$ given $\alpha,\beta,$ and
an eigenvalue
$\lambda\in\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$.
The formula works by giving outputting a cubic polynomial whose roots are
eigenvalues of $\operatorname*{Sp}(\boldsymbol{R})$, which can be trivially
evaluated with the cubic formula.
We denote the $k^{th}$ eigenspace of
$\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$ with
eigenvalue $\lambda_{k}=ic_{k}$ and $|c_{1}|\leq\dots\leq|c_{n}|$, because
bilinear zero-sum games have purely imaginary eigenvalues due to
$\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}$ being antisymmetric.
Eigenvalues come in a conjugate pairs, where $\bar{\lambda_{k}}=i(-c_{k})$
If we select momentum coefficient $\beta=|\beta|\exp(i\arg(\beta))$ and step
size $\alpha_{k}=\frac{\alpha_{k}^{\prime}}{|c_{k}|}$, and use that
$\lambda\in\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$
are imaginary, then – as shown in Appendix Section A.2 – (40) simplifies to:
$p_{k}(x)=x(|\beta|\cos(\arg(\beta))(i\alpha_{k}^{\prime}-2)-|\beta|^{2})+x^{2}(2|\beta|\cos(\arg(\beta))-i\alpha_{k}^{\prime}+1)+|\beta|^{2}-x^{3}$
(45)
So, with these parameter selections, the convergence rate of Algorithm 1 in
the $k^{th}$ eigenspace is bounded by the largest root of (45).
First, consider $\arg(\beta)=\pi-\epsilon$, where $\epsilon=\frac{\pi}{16}$.
We select $\alpha_{k}^{\prime}=$0.75$$ (equivalently,
$\alpha_{k}=\frac{$0.75$}{|c_{k}|}$) and $|\beta|=$0.986$$ via grid search.
Using the cubic formula on the associated $p(x)$ from (45) the maximum
magnitude root has size $\approx$0.9998$<1$, so this selection converges in
the $k^{th}$ eigenspace. So, selecting:
$\displaystyle\hat{\alpha}$ $\displaystyle\leq\min_{k}\alpha_{k}$ (46)
$\displaystyle=\min_{k}\frac{$0.75$}{c_{k}}$ (47)
$\displaystyle=\frac{$0.75$}{\max_{k}c_{k}}$ (48)
$\displaystyle=\frac{$0.75$}{\|\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}\|_{2}}$
(49)
with $\beta=$0.986$\exp(i(\pi-\epsilon))$ will converge in each eigenspace.
Now, consider $\arg(\beta)=\epsilon=\frac{\pi}{16}$ with
$\alpha_{k}^{\prime}=$0.025$$ and $|\beta|=$0.9$$. Using the cubic formula on
the associated $p(x)$ from (45) the maximum magnitude root has size
$\approx$0.973$<1$, so this selection converges in the $k^{th}$ eigenspace.
So, selecting:
$\displaystyle\hat{\alpha}$ $\displaystyle\leq\min_{k}\alpha_{k}$ (50)
$\displaystyle=\min_{k}\frac{$0.025$}{c_{k}}$ (51)
$\displaystyle=\frac{$0.025$}{\max_{k}c_{k}}$ (52)
$\displaystyle=\frac{$0.025$}{\|\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}\|_{2}}$
(53)
with $\beta=$0.9$\exp(i\epsilon)$ will converge in each eigenspace.
Thus, for any of the choices of $\arg(\beta)$ we can select
$\hat{\alpha},|\beta|$ that converges in every eigenspace, and thus converges.
∎
In the preceding proof, our prescribed selection of $\hat{\alpha}$ depends on
knowing the largest norm eigenvalue of
$\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$,
because our selections of
$\hat{\alpha}\,\,\propto\,\,\frac{1}{\|\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}\|_{2}}$.
We may not have access to largest norm eigenvalue of
$\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$ in-
practice. Nonetheless, this shows that a parameter selection exists to
converge, even if it may be difficult to find. Often, in convex optimization
we describe choices of $\alpha,\beta$ in terms of the largest and smallest
norm eigenvalues of
$\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}})$ (i.e.
the Hessian of the loss) [91].
## Appendix B Algorithms
Here, we include additional algorithms, which may be of use to some readers.
Algorithm 3 show aggregated momentum [26]. Algorithm 4 shows the recurrently
linked momentum that generalizes and unifies aggregated momentum with negative
momentum [25]. Algorithm 5 shows our algorithm with alternating updates, which
we use for training GANs. Algorithm 6 shows our method with all real-valued
objects, if one wants to implement complex momentum in a library that does not
support complex arithmetic.
Algorithm 3 Aggregated Momentum
1:Select number of buffers $K\in\mathbb{N}$
2:Select $\beta_{(k)}\in[0,1)$ for $k=1\dots K$
3:Select $\alpha_{(k)}\in\mathbb{R}^{+}$ for $k=1\dots K$
4:Initialize $\boldsymbol{\mu}_{(k)}^{0}$ for $k=1\dots K$
5:for $j=$1$\dots N$ do
6: for $k=$1$\dots K$ do
7:
$\boldsymbol{\mu}_{(k)}^{j\\!+\\!$1$}=\beta_{(k)}\boldsymbol{\mu}_{(k)}^{j}-\hat{\boldsymbol{g}}^{j}$
8:
$\boldsymbol{\omega}^{j\\!+\\!$1$}=\boldsymbol{\omega}^{j}+\sum_{k=1}^{K}\alpha_{(k)}\boldsymbol{\mu}^{j\\!+\\!1}_{(k)}$
return $\boldsymbol{\omega}_{N}$
Algorithm 4 Recurrently Linked Momentum
1:Select number of buffers $K\in\mathbb{N}$
2:Select $\beta_{(l,k)}\in\mathbb{R}$ for $l=1\dots K$ and $k=1\dots K$
3:Select $\alpha_{(k)}\in\mathbb{R}^{+}$ for $k=1\dots K$
4:Initialize $\boldsymbol{\mu}_{(k)}^{0}$ for $k=1\dots K$
5:for $j=$1$\dots N$ do
6: for $k=$1$\dots K$ do
7:
$\boldsymbol{\mu}_{(k)}^{j+1}\\!=\\!\sum_{l}\beta_{(l,k)}\boldsymbol{\mu}_{(l)}^{j}-\hat{\boldsymbol{g}}^{j}$
8:
$\boldsymbol{\omega}^{j\\!+\\!$1$}=\boldsymbol{\omega}^{j}+\sum_{k=1}^{K}\alpha_{(k)}\boldsymbol{\mu}^{j\\!+\\!1}_{(k)}$
return $\boldsymbol{\omega}_{N}$
Algorithm 5 (AltCM) Momentum
1:Select $\beta\in\mathbb{C},\alpha\in\mathbb{R}^{+}$
2:Initialize $\boldsymbol{\mu}_{A}^{0},\boldsymbol{\mu}_{B}^{0}$
3:for $j=$1$\dots N$ do
4:
$\boldsymbol{\mu}_{A}^{j\\!+\\!$1$}=\beta\boldsymbol{\mu}_{A}^{j}-\boldsymbol{g}_{\\!A}^{j}$
5:
$\boldsymbol{\theta}_{\\!A}^{j\\!+\\!$1$}=\boldsymbol{\theta}_{\\!A}^{j}+\Re(\alpha\boldsymbol{\mu}^{j\\!+\\!$1$}_{A})$
6:
$\boldsymbol{\mu}_{B}^{j\\!+\\!$1$}=\beta\boldsymbol{\mu}_{B}^{j}-\boldsymbol{g}_{\\!B}(\boldsymbol{\theta}_{\\!A}^{j\\!+\\!1}\\!\\!\\!,\boldsymbol{\theta}_{\\!B}^{j})$
7:
$\boldsymbol{\theta}_{\\!B}^{j\\!+\\!$1$}=\boldsymbol{\theta}_{\\!B}^{j}+\Re(\alpha\boldsymbol{\mu}^{j\\!+\\!1}_{B})$
return $\boldsymbol{\omega}_{N}$
Algorithm 6 (SimCM) Complex Momentum - $\mathbb{R}$ valued
1:Select $\Re(\beta),\Im(\beta),\Re(\alpha),\Im(\alpha)\in\mathbb{R}$
2:Select $\Re(\beta),\Im(\beta),\Re(\alpha),\Im(\alpha)\in\mathbb{R}$
3:Initialize $\Re(\boldsymbol{\mu})^{0},\Im(\boldsymbol{\mu})^{0}$
4:for $j=1\dots N$ do
5:
$\Re(\boldsymbol{\mu}^{j\\!+\\!1})=\Re(\beta)\Re(\boldsymbol{\mu}^{j})-\Im(\beta)\Im(\boldsymbol{\mu}^{j})-\hat{\boldsymbol{g}}^{j}$
6:
$\Im(\boldsymbol{\mu}^{j\\!+\\!1})=\Re(\beta)\Im(\boldsymbol{\mu}^{j})+\Im(\beta)\Re(\boldsymbol{\mu}^{j})$
7:
$\boldsymbol{\omega}^{j\\!+\\!1}\\!\\!=\\!\boldsymbol{\omega}^{j}\\!-\\!\Re(\alpha\\!)\\!\hat{\boldsymbol{g}}^{j}\\!+\\!\Re(\alpha\beta\\!)\\!\Re(\boldsymbol{\mu}^{j}\\!)\\!-\\!\Im(\alpha\beta\\!)\\!\Im(\boldsymbol{\mu}^{j}\\!)$
return $\boldsymbol{\omega}_{N}$
### B.1 Complex Momentum in PyTorch
Our method can be easily implemented in PyTorch 1.6+ by using complex tensors.
The only necessary change to the SGD with momentum optimizer is extracting the
real-component from momentum buffer as with JAX – see here.
In older versions of Pytorch, we can use a tensor to represent the momentum
buffer $\boldsymbol{\mu}$, step size $\alpha$, and momentum coefficient
$\beta$. Specifically, we represent the real and imaginary components of the
complex number independently. Then, we redefine the operations __add__ and
__mult__ to satisfy the rules of complex arithmetic – i.e., equations (23) and
(24).
## Appendix C Experiments
### C.1 Computing Infrastructure and Runtime
For the purely adversarial experiments in Sections 4.1 and 4.2, we do our
computing in CPU. Training each $2$D GAN in Section 4.3 takes $2$ hours and we
can train $10$ simultaneously on an NVIDIA T$4$ GPU. Training each CIFAR GAN
in Section 4.4 takes $10$ hours and we can only train $1$ model per NVIDIA T4
GPU.
### C.2 Optimization in Purely Adversarial Games
We include the alternating update version of Figure 3.1 in Figure 8, which
allows us to contrast simultaneous and alternating updates. With alternating
updates on a Dirac-GAN for $\alpha\\!=\\!$0.1$$ the best value for the
momentum coefficient $\beta$ was complex, but we could converge with real,
negative momentum. Simultaneous updates may be a competitive choice with
alternating updates, only if alternating updates cost two gradient evaluations
per step, which is common in deep learning setups.
### C.3 How Adversarialness Affects Convergence Rates
We include the extragradient (EG) update with extrapolation parameter
$\alpha^{\prime}$ and step size $\alpha$:
$\displaystyle\boldsymbol{\omega}^{j\\!+\\!\frac{1}{2}}\\!=\\!\boldsymbol{\omega}^{j}\\!-\\!\alpha^{\prime}\hat{\boldsymbol{g}}^{j}$
(EG)
$\displaystyle\boldsymbol{\omega}^{j\\!+\\!1}\\!=\\!\boldsymbol{\omega}^{j}\\!-\\!\alpha\hat{\boldsymbol{g}}^{j\\!+\\!\frac{1}{2}}\\!$
and the optimistic gradient (OG) update with extrapolation parameter
$\alpha^{\prime}$ and step size $\alpha$:
$\displaystyle\boldsymbol{\omega}^{j\\!+\\!1}\\!=\\!\boldsymbol{\omega}^{j}\\!-\\!2\alpha\hat{\boldsymbol{g}}^{j}\\!+\\!\alpha^{\prime}\hat{\boldsymbol{g}}^{j-1}$
(OG)
Often, EG and OG are used with $\alpha=\alpha^{\prime}$, however we found that
this constraint crippled these methods in cooperative games (i.e.,
minimization). As such, we tuned the extrapolation parameter $\alpha^{\prime}$
separately from the step size $\alpha$, so EG and OG were competitive
baselines.
We include Figure 9 which investigates a GANs spectrum throughout training,
and elaborates on the information that is shown in Figure 7. This shows that
there are many real and imaginary eigenvalues, so GAN training is neither
purely cooperative or purely adversarial. Also, the structure of the set of
eigenvalues for the discriminator is different than the generator, which may
motivate separate optimizer choices. The structure between the players
persists through training, but the eigenvalues grow in magnitude and spread
out their phases. This indicates how adversarial the game is can change during
training.
Momentum phase $\arg(\beta)$SimCM for # grad. eval. = # stepsMomentum
Magnitude $|\beta|$AltCM for # grad. eval.# grad. eval. to convergeAltCM for #
steps# steps to converge Figure 8: We show many steps and gradient
evaluations, both simultaneous and alternating complex momentum on a Dirac-GAN
take for a set solution distance. We fix step size $\alpha\\!=\\!$0.1$$ as in
Figure 3, while varying the phase and magnitude of our momentum
$\beta\\!=\\!|\beta|\exp(i\arg(\beta))$. There is a red star at the optima,
dashed red lines at real $\beta$, and a dashed magenta line for simultaneous
or alternating gradient descent. We only display color for convergent setups.
_Left:_ Simultaneous complex momentum (SimCM). This is the same as Figure 3.1,
which we repeat to contrast with alternating updates. There are no real-valued
$\beta$ that converge for this – or any – $\alpha$ with simultaneous updates
[25]. Simultaneous updates can parallelize gradient computation for all
players at each step, thus costing only one gradient evaluation per step for
many deep learning setups. The best rate of convergence per step and gradient
evaluation is
$\approx{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$0.955$}$.
_Middle:_ Alternating complex momentum (AltCM), where we show how many
gradient evaluations – as opposed to steps – to reach a set solution distance.
Alternating updates are bottlenecked by waiting for first player’s update to
compute the second players update, effectively costing two gradient
evaluations per step for many deep learning setups. Negative momentum can
converge here, as shown by Gidel et al. [25], but the best momentum is still
complex. Also, Alternating updates can make the momentum phase $\arg(\beta)$
choice less sensitive to our convergence. The best rate of convergence per
gradient evaluation is
$\approx{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$0.965$}$.
_Right:_ AltCM, where we show how many steps to reach a set solution distance.
The best rate of convergence per step is
$\approx{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$0.931$}$.
Takeaway: If we can parallelize computation of both players gradients we can
benefit from SimCM, however if we can not then AltCM can converge more quickly
and for a broader set of optimizer parameters. In any case, the best solution
uses a complex momentum $\beta$ for this $\alpha$. Joint-gradient
$\hat{\boldsymbol{g}}$ index $l$$\log(|a_{lk}|+\epsilon)$ for $a_{lk}$ in
$\nabla_{\\!\boldsymbol{\omega}}\hat{\boldsymbol{g}}^{j}$Joint-parameter
$\boldsymbol{\omega}$ index $k$Jacobian of joint-gradient
$\nabla_{\\!\boldsymbol{\omega}}\hat{\boldsymbol{g}}^{j}$ for GANAbs of
eigenvector componentIndex into $\mathbf{v}$ – first $337$ are for D’s
paramsEigenvector $\mathbf{v}$ components for eigenvalues
$\lambda\in\operatorname*{Sp}(\nabla_{\\!\boldsymbol{\omega}}\hat{\boldsymbol{g}})$-$\pi$
-$\frac{\pi}{2}$ $0$ -$\frac{\pi}{2}$ -$\pi$Phase of eigenvalue
$\arg(\lambda)$disc. unsure gen.Does eigenvector point at a player?Log-
magnitude of eigenvalue $\log(|\lambda|)$Start of trainingEnd of trainingLog-
polar graph of the spectrum of Jacobian of the joint-gradient
$\operatorname*{Sp}(\nabla_{\\!\boldsymbol{\omega}}\hat{\boldsymbol{g}}^{j})$
throughout training Figure 9: These plots investigate the spectrum of the
Jacobian of the joint-gradient for the GAN in Figure 7 through training. The
spectrum is key for bounding convergence rates in learning algorithms.
_Top left:_ The Jacobian $\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}$
for a GAN on a $2$D mixture of Gaussians with a two-layer, fully-connected
$16$ hidden unit discriminator (D) and generator (G) at the end of training.
In the concatenated parameters $\boldsymbol{\omega}\in\mathbb{R}^{723}$, the
first $337$ are for D, while the last $386$ are for G. We display the $\log$
of the absolute value of each component plus $\epsilon=10^{-10}$. The upper
left and lower right quadrants are the Hessian of D and G’s losses
respectively.
_Top Right:_ We visualize two randomly sampled eigenvectors from
$\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}$. The first part of the
parameters is for the discriminator, while the second part is for the
generator. Given an eigenvalue with eigenvector $\mathbf{v}$, we roughly
approximate attributing eigenvectors to players by calculating how much of it
lies in D’s parameter space with
$\frac{\|\mathbf{v}_{1:|\textnormal{D}|}\|_{1}}{\|\mathbf{v}\|_{1}}=\frac{\|\mathbf{v}_{1:337}\|_{1}}{\|\mathbf{v}\|_{1}}$.
If this ratio is near $1$ (or $0$) and say _the eigenvector mostly points at D
(or G)_. The blue eigenvector mostly points at G, while the orange eigenvector
is unclear. Finding useful ways to attribute eigenvalues to players is an open
problem.
_Bottom:_ The spectrum of the Jacobian of the joint-gradient
$\operatorname*{Sp}(\nabla_{\boldsymbol{\omega}}\hat{\boldsymbol{g}}^{j})$ is
shown in log-polar coordinates, because it is difficult to see structure when
graphing in Cartesian (i.e., $\Re$ and $\Im$) coordinates, due to eigenvalues
spanning orders of magnitude, while being positive and negative. The end of
training is when we stop making progress on the log-likelihood. We have
imaginary eigenvalues at $\arg(\lambda)=\pm\nicefrac{{\pi}}{{$2$}}$, positive
eigenvalues at $\arg(\lambda)=$0$$, and negative eigenvalues at
$\arg(\lambda)=\pm\pi$.
_Takeaway:_ There is a banded structure for the coloring of the eigenvalues
that persists through training. We may want different optimizer parameters for
the discriminator and generator, due to asymmetry in their associated
eigenvalues. Also, the magnitude of the eigenvalues grows during training, and
the $\arg$s spread out indicating the game can change eigenstructure near
solutions.
### C.4 Training GANs on $2$D Distributions
For $2$D distributions, the data is generated by sampling from a mixture of
$8$ Gaussian distributions, which are distributed uniformly around the unit
circle.
For the GAN, we use a fully-connected network with $4$ hidden ReLU [92] layers
with $256$ hidden units. We chose this architecture to be the same as Gidel et
al. [25]. Our noise source for the generator is a $4$D Gaussian. We trained
the models for $100\,000$ iterations. The performance of the optimizer
settings is evaluated by computing the negative log-likelihood of a batch of
$100\,000$ generated $2$D samples.
Momentum phase $\arg(\beta)$Step size $\alpha=$0.001$$Momentum Magnitude
$|\beta|$Step Size $\alpha=$0.003$$NLL, lower = better Figure 10: Heatmaps of
the negative log-likelihood (NLL) for tuning $\arg(\beta),|\beta|$ with
various fixed $\alpha$ on a $2$D mixture of Gaussians GAN. We highlight the
best performing cell in red, which had
$\arg(\beta)\approx\nicefrac{{\pi}}{{$8$}}$. Runs equivalent to alternating
SGD are shown in a magenta box. We compare to negative momentum with
alternating updates as in Gidel et al. [25] in the top row with
$\arg(\beta)=\pi$. _Left_ : Tuning the momentum with $\alpha=$0.001$$. _Right_
: Tuning the momentum with $\alpha=$0.003$$.
Figure 11(()): Mixture of Gaussian samples from GAN with the best
hyperparameters from the heatmaps in Appendix Figure 10
Figure 11(()): Class-conditional CIFAR-10 samples from GAN with the best
hyperparameters from the heatmap in C.4
Momentum phase $\arg(\beta_{1})$Momentum magnitude $|\beta_{1}|$Inception
score, higher = better
Figure 12(()): The inception score (IS) for a grid search on $\arg(\beta_{1})$
and $|\beta_{1}|$ for training BigGAN on CIFAR-10 with the Adam variant in
Algorithm 2. The $\beta_{1}$ is complex for the discriminator, while the
generator’s optimizer is fixed to author-supplied defaults. Red points are
runs that failed to train to the minimum IS in the color bar. The vertical
magenta line denotes runs equivalent to alternating SGD. Negative momentum
failed to train for any momentum magnitude $|\beta_{1}|>.5$, so we do not
display it for more resolution near values of interest.
Inception score (IS)Iteration
Figure 12(()): We compare the best optimization parameters from grid search
Figure C.4 for our complex Adam variant (i.e., Algorithm 2) shown in green,
with the author provided values shown in red for the CIFAR-10 BigGAN over $10$
seeds. A star is displayed at the best IS over all runs, a cross is displayed
at the worst IS over all runs, while a circle is shown at the best IS for each
run. Dashed lines are shown at the max/min IS over all runs at each iteration,
low-alpha lines are shown for each runs IS, while solid lines are shown for
the average IS over all seeds at each iteration. The results are summarized in
Table 1.
|
# An Improved and More Accurate Expression for a PDF Related to Eigenvalue-
Based Spectrum Sensing
Fuhui Zhou, _Member, IEEE_ , and Norman C. Beaulieu, _Fellow, IEEE_ Fuhui Zhou
is with the School of Information Engineering, Nanchang University, P. R.
China, 330031 (e-mail: [email protected]). Norman C. Beaulieu is with
Beijing University of of Posts and Telecommunications and the Beijing Key
Laboratory of Network System Architecture and Convergence, Beijing, China
100876 (e-mail: [email protected]). The research was supported by the National
Natural Science Foundation of China (61701214), the Young Natural Science
Foundation of Jiangxi Province (20171BAB212002), and the China Postdoctoral
Science Foundation (2017M610400).
###### Abstract
Cooperative spectrum sensing based on the limiting eigenvalue ratio of the
covariance matrix offers superior detection performance and overcomes the
noise uncertainty problem. While an exact expression exists, it is complex and
multiple useful approximate expressions have been published in the literature.
An improved, more accurate, integral solution for the probability density
function of the ratio is derived using order statistical analysis to remove
the simplifying, but incorrect, independence assumption. Thereby, the letter
makes an advance in the rigorous theory of eigenvalue-based spectrum sensing.
###### Index Terms:
Cooperative spectrum sensing, eigenvalue ratio analysis, order statistics,
probability distribution.
## I Introduction
THE Eigenvalue-based detection schemes, using the eigenvalues of the
covariance matrix to construct a test statistic, are considered to be one of
the most effective methods to test for the presence of a primary user (PU)
signal in cognitive radio systems [1]. The maximum-to-minimum eigenvalue (MME)
detector is based on the ratio of the largest eigenvalue to the smallest
eigenvalue of the covariance matrix. However, theoretical results for
eigenvalue ratio schemes usually depend on asymptotic assumptions, since the
distribution of the ratio of two extreme eigenvalues is difficult to compute
[1]-[5].
The probability of false alarm (PFA) is the probability that the PU is absent
but is detected to be present. PFA is one of the most important performance
metrics in spectrum sensing for cognitive radio. Accurate determination of the
PFA improves the accuracy of the decision threshold of a detector, and the
efficiency of spectrum utilization. Note that the derivation of the PFA is
dependent on the probability density function (PDF) of the test statistic of
the detector under the hypothesis that the PU is absent. Meanwhile, this PDF
is not known exactly in a tractable form suitable for further theoretical
analysis, and the derivation of a popular approximation necessarily employs
assumptions that are not rigorously correct mathematically. This letter makes
a contribution to the theory of eigenvalue-based spectrum sensing in two ways,
by removing invalid assumption, and by deriving a more accurate mathematically
tractable approximation to this important PDF after removing the invalid
assumption.
The distribution of the ratio of the two extreme eigenvalues for the MME
scheme is commonly approximated by the Tracy-Widom distribution based on the
Tracy-Widom law [1], [2]. However, this approximation is based on the
unrealistic assumption that the number of the received signal samples as well
as the number of the cooperating secondary users are infinite. It has been
shown that this approximation is poor when the number of the signal samples is
small [5], [6]. An improved approximation to the PDF of the ratio of the two
extreme eigenvalues is derived in [5] and [6] with the assumption that the two
extreme eigenvalues are independent and Gaussian distributed. In [7] and [8],
an exact PDF for the ratio of the two extreme eigenvalues has been derived.
The derived exact expression for the PDF is quite complex, and other authors
have chosen to use the approximation over the exact solution in [5], [6],
[9]-[11].
In this paper, an improved PDF approximation of the ratio of the two extreme
eigenvalues is derived by using the order statistic theory and the Gaussian
distribution assumption. It is shown that the derived PDF is more accurate
than the commonly employed PDFs given in [2] and [5], and is simpler than the
exact PDF given in [7].
The rest of this paper is organized as follows. Section II presents the system
model and MME spectrum sensing. An improved solution for the PDF of the ratio
of two extreme eigenvalues is presented in Section III. Section IV presents
simulation results. The paper concludes with Section V.
## II System Model and MME Spectrum Sensing
Figure 1: The system model.
As shown in Fig. 1, a cognitive radio network with $M$ SUs that cooperatively
detect one PU is considered. During the sensing time, each SU collects $N$
samples of received signal, denoted by ${{x_{i}}\left(n\right)}$, where
$n=1,2,\cdots,N$ and $i=1,2,\cdots,M$. Note that all the SUs receive the
signal at the same time. To achieve synchronous sampling, each SU has the
center frequency derived from the local oscillator and the same digital clock
[1], [2], [5], [6]. This system model for cooperative spectrum sensing has
been widely applied in these works. Then, the samples are transmitted to the
fusion center. The aim of cooperative spectrum sensing is to construct a test
statistic and make a decision between the two hypotheses $\left({H_{0}}\
\textrm{and}\ {H_{1}}\right)$ based on those collected samples, where $H_{0}$
denotes the absence of the PU, and $H_{1}$ represents the presence of the PU.
Thus, the samples from each SU under the two hypotheses are given as
$\displaystyle{{H_{0}}:}\ {{x_{i}}\left(n\right)={w_{i}}\left(n\right)}$ (1a)
$\displaystyle{{H_{1}}:}\
{{x_{i}}\left(n\right)={h_{i}}\left(n\right)\sqrt{P}_{s}{s}\left(n\right)+{w_{i}}\left(n\right)}$
(1b)
where ${w_{i}}\left(n\right)\sim{\mathcal{CN}}\left({0,\sigma_{w}^{2}}\right)$
is complex Gaussian noise and $\mathcal{CN}\left({0,\sigma_{w}^{2}}\right)$
denotes the complex Gaussian normal distribution with mean zero and variance
$\sigma_{w}^{2}$. ${s}\left(n\right)$ is the primary user signal and
${h_{i}}\left(n\right)$ are the channel coefficients. $P_{s}$ is the
transmitted power of the primary user. The distribution of the PU signal is
unknown and independent of the noise. Based on the collected samples from $M$
SUs, a data matrix is defined as
$\mathbf{X}=\left[{X_{1}^{T},X_{2}^{T},\cdots,X_{M}^{T}}\right]$, where
${X_{m}}=\left[{{x_{m}}\left(1\right)},\ {{x_{m}}\left(2\right)},\ \cdots,\
{{x_{m}}\left(N\right)}\right]$ with $m=1,2,\cdots,M$. The sample covariance
matrix is defined as
${\mathbf{R}_{x}}=\left({1/N}\right)\mathbf{X}{\mathbf{X}^{H}}$, where
${\left(\cdot\right)^{H}}$ represents the Hermitian transpose operator. Let
${\lambda_{1}}\geq{\lambda_{2}}\geq\cdots\geq{\lambda_{M}}$ denote the ordered
eigenvalues of the matrix ${\mathbf{R}_{x}}$. The test statistic for the MME
spectrum sensing scheme was formulated in [5]. It is denoted by $T_{\xi}$,
given as
$\displaystyle\
\begin{split}{T_{\xi}}=\frac{{\lambda_{1}}}{{\lambda_{M}}}\mathop{\mathbin{\lower
1.72218pt\hbox{$\buildrel\mathbin{\buildrel\scriptstyle<\over{\smash{\scriptstyle\relbar}\vphantom{{}_{\scriptstyle
x}}}}\over{\smash{\scriptstyle>}\vphantom{{}_{x}}}$}}}\limits_{H_{1}}^{H_{0}}{\gamma_{\xi}}\end{split}$
(2)
where ${\gamma_{\xi}}$ is the decision threshold of the MME spectrum sensing
scheme.
## III Improved Solution for The PDF of The Ratio of Two Extreme Eigenvalues
Under Hypothesis $H_{0}$
In [1], [2], the PDF of the ratio of two limiting eigenvalues under the
hypothesis $H_{0}$ is approximated by the PDF of the Tracy-Widom distribution.
This approximation is poor when the number of samples is small or moderate.
The PDF of the ratio is derived based on the assumption that the two limiting
eigenvalues are independent normal random variables [5]. The assumption that
the two extreme eigenvalues are independent is not correct and is removed in
this paper. In [5], when there only exist Gaussian noises, the largest
eigenvalue and the smallest eigenvalue of the covariance matrix are assumed to
be normal random variables, namely
$\displaystyle{\lambda_{1}}\sim{\mathcal{N}}\left({{u_{{\lambda_{1}}}},\sigma_{{\lambda_{1}}}^{2}}\right)$
(3a)
$\displaystyle{\lambda_{M}}\sim{\mathcal{N}}\left({{u_{{\lambda_{M}}}},\sigma_{{\lambda_{M}}}^{2}}\right)$
(3b)
where $u_{{\lambda_{1}}}$ and $u_{{\lambda_{M}}}$ are the means of the largest
eigenvalue and of the smallest eigenvalue, respectively. The variances of the
largest eigenvalue and of the smallest eigenvalue are denoted by
$\sigma_{{\lambda_{1}}}^{2}$ and $\sigma_{{\lambda_{M}}}^{2}$, respectively.
According to [5], the means and variances of the largest eigenvalue and of the
smallest eigenvalue are given as
$\displaystyle{u_{{\lambda_{\rm K}}}}=\mathbb{E}\left({{\lambda_{\rm
K}}}\right)$ (4a) $\displaystyle\sigma_{{\lambda_{\rm
K}}}^{2}=\mathbb{E}\left({\lambda_{\rm K}^{2}}\right)-u_{{\lambda_{\rm
K}}}^{2}$ (4b)
$\displaystyle\mathbb{E}\left({\lambda_{1}^{p}}\right)=C_{0}^{-1}{\beta_{{\lambda_{1}}}}\left(p\right)$
(4c)
$\displaystyle\mathbb{E}\left({\lambda_{M}^{p}}\right)=C_{0}^{-1}{\beta_{{\lambda_{M}}}}\left(p\right)$
(4d)
where
${C_{0}}=\mathop{\Pi}\limits_{i=1}^{M}\left({N-i}\right)!\mathop{\Pi}\limits_{j=1}^{M}\left({M-j}\right)!$;
$\mathbb{E}[\cdot]$ denotes the expectation operator; $\rm
K=\left\\{1,M\right\\}$; ${\beta_{{\lambda_{1}}}}\left(p\right)$ and
${\beta_{{\lambda_{M}}}}\left(p\right)$ are given by eq. $\left(5\right)$ at
the top of the next page. In eq. $\left(5\right)$, $\rm sgn\left(\cdot\right)$
denotes the Signum function; $\alpha_{m}$ is the $m$th element of the $\alpha$
and $\alpha$ is the permutation of $\left\\{1,2,\cdots,M-1\right\\}$;
${p_{i,j}}=p+N-M+i+j$; $\sum{l_{1}^{M-1}}=\sum\limits_{i=1}^{M-1}{{l_{i}}}$;
$l_{1}^{M-1}!=\prod\limits_{i=1}^{M-1}{{l_{i}}!}$; $\sum\nolimits_{{l_{1\sim
M-1}}}^{{L_{1\sim
M-1}}}{}=\sum\nolimits_{{l_{1}}=0}^{{L_{1}}}{\sum\nolimits_{{l_{2}}=0}^{{L_{2}}}\cdots}\sum\nolimits_{{l_{M-1}}=0}^{{L_{M-1}}}$;
$\mathcal{S}$ is any subset of the set
$\left\\{l_{1},l_{2},\cdots,l_{M-1}\right\\}$ and $l_{m}$ is from $0$ to
${L_{{\alpha_{m}},m}}-1$; ${\left|{\mathcal{S}}\right|}$ represents the
cardinality of subset $\mathcal{S}$.
$\displaystyle{\beta_{{\lambda_{M}}}}\left(p\right)=\sum\limits_{i,j}^{M}{{{\left({-1}\right)}^{i+j}}}\sum\limits_{\alpha}{{\mathop{\rm
sgn}}\left(\alpha\right)\prod\limits_{m=1}^{M-1}{\Gamma\left({{L_{{\alpha_{m}},m}}}\right)}}\left({\sum\nolimits_{{l_{1\sim
M-1}}}^{{L_{1\sim
M-1}}}{\frac{{\Gamma\left({\sum{l_{1}^{M-1}}+{p_{i,j}}-1}\right)}}{{l_{1}^{M-1}!{M^{\sum{l_{1}^{M-1}}+{p_{i,j}}-1}}}}}}\right)$
(5a)
$\displaystyle{\beta_{{\lambda_{1}}}}\left(p\right)=\sum\limits_{i,j}^{M}{{{\left({-1}\right)}^{i+j}}}\sum\limits_{\alpha}{{\mathop{\rm
sgn}}\left(\alpha\right)\prod\limits_{m=1}^{M-1}{\Gamma\left({{L_{{\alpha_{m}},m}}}\right)}}\left({\sum\limits_{\mathcal{S}}{{{\left({-1}\right)}^{\left|{\mathcal{S}}\right|}}}\frac{{\Gamma\left({\sum\mathcal{S}+{p_{i,j}}-1}\right)}}{{\prod{l_{1}^{M-1}!{M^{\sum{l_{1}^{M-1}}+{p_{i,j}}-1}}}}}}\right)$
(5b)
$\displaystyle{L_{{\alpha_{m}},m}}=\left\\{\begin{array}[]{l}\begin{array}[]{*{20}{c}}{N-M+m+{\alpha_{m}}-1}&{\text{if}\
{\alpha_{m}}<i\ \text{and}\ m<j}\end{array}\\\
\begin{array}[]{*{20}{c}}{N-M+m+{\alpha_{m}}+1}&{\text{if}\ {\alpha_{m}}\geq
i\ \text{and}\ m\geq j}\end{array}\\\
\begin{array}[]{*{20}{c}}{N-M+m+{\alpha_{m}}}&{\text{otherwise}}\end{array}\end{array}\right.$
(5i)
Since the largest eigenvalue and the smallest eigenvalue are order statistics,
the joint PDF, $f_{r,s}\left({x,y}\right)$, of ${X_{r}}$ and ${X_{s}}$, $1\leq
r<s\leq M$, ${X_{r}}\leq{X_{s}}$, for $x\leq y$, is [12],
$\displaystyle{f_{r,s}}\left({x,y}\right)=$
$\displaystyle\frac{{M!\left[{F_{r}}\left(x\right)\right]^{r-1}{f_{r}}\left(x\right)}{f_{s}}\left(y\right)}{{\left({r-1}\right)!\left({s-r-1}\right)!\left({M-s}\right)!}}$
$\displaystyle\times{\left[{{F_{s}}\left(y\right)-{F_{r}}\left(x\right)}\right]^{s-r-1}}{\left[{1-{F_{s}}\left(y\right)}\right]^{M-s}}$
(6)
where ${F_{r}}\left(x\right)$ and ${F_{s}}\left(y\right)$ are the cumulative
marginal distribution functions (CMDFs) for ${X_{r}}$ and ${X_{s}}$,
respectively, and ${f_{r}}\left(x\right)$ and ${f_{s}}\left(y\right)$ denote
the marginal PDFs for ${X_{r}}$ and ${X_{s}}$. Thus, the joint PDF for the
largest eigenvalue and the smallest eigenvalue,
${\widetilde{f}_{{\lambda_{1}},{\lambda_{M}}}}\left({x,y}\right)$, is given by
$\displaystyle{\widetilde{f}_{{\lambda_{1}},{\lambda_{M}}}}\left({x,y}\right)=$
$\displaystyle
M\left({M-1}\right){f_{{\lambda_{1}}}}\left(x\right){f_{{\lambda_{M}}}}\left(y\right)$
$\displaystyle\times{\left[{{F_{{\lambda_{1}}}}\left(x\right)-{F_{{\lambda_{M}}}}\left(y\right)}\right]^{M-2}}$
(7)
where $F_{{\lambda_{1}}}\left(x\right)$ and $F_{{\lambda_{M}}}\left(y\right)$
are the cumulative distribution functions (CDF) of the two extreme
eigenvalues, respectively, and $f_{{\lambda_{1}}}\left(x\right)$ and
$f_{{\lambda_{M}}}\left(y\right)$ are their corresponding marginal PDFs. $M$
is the number of secondary users. Therefore, the improved PDF for the ratio of
the two extreme eigenvalues, ${\widetilde{f}_{Z}}\left(z\right)$ is derived as
$\displaystyle{\widetilde{f}_{Z}}\left(z\right)=\int_{-\infty}^{\infty}{{{\widetilde{f}}_{{\lambda_{1}},{\lambda_{M}}}}\left({yz,y}\right)}\left|y\right|dy$
(8)
where $\left|\cdot\right|$ is the magnitude operator. After substituting eq.
$\left(7\right)$ into eq. $\left(8\right)$ and some algebraic manipulations,
the improved PDF, ${\widetilde{f}_{Z}}\left(z\right)$, is given by
$\displaystyle{\widetilde{f}_{Z}}\left(z\right)=M\left({M-1}\right)\sum\limits_{i=1}^{M-2}{\left(\begin{array}[]{l}M-2\\\
i\end{array}\right)}{\left({-1}\right)^{M-2-i}}$ (11)
$\displaystyle\times\int_{-\infty}^{\infty}\left[F_{{\lambda_{1}}}\left({yz}\right)\right]^{i}\left[F_{{\lambda_{M}}}\left(y\right)\right]^{M-2-i}{f_{{\lambda_{1}}}}\left({yz}\right){f_{{\lambda_{M}}}}\left(y\right)\left|y\right|dy.$
(12)
Therefore, the improved PDF for the ratio of the extreme eigenvalues is
derived, based on the assumption that the two extreme eigenvalues follow
normal distributions [5], as
$\displaystyle{\widetilde{f}_{Z}}\left(z\right)=$ $\displaystyle
M\left({M-1}\right)\sum\limits_{i=1}^{M-2}{\left(\begin{array}[]{l}M-2\\\
i\end{array}\right)}{\left({-1}\right)^{M-2-i}}$ (15)
$\displaystyle\times\int_{-\infty}^{\infty}\Bigg{\\{}\left[{\Phi}\left({\frac{{yz-{u_{1}}}}{{{\sigma_{1}}}}}\right)\right]^{i}\left[{\Phi}\left({\frac{{z-{u_{2}}}}{{{\sigma_{2}}}}}\right)\right]^{M-2-i}$
$\displaystyle\times\frac{{\left|y\right|{e^{-\left[{\frac{{{{\left({yz-{u_{1}}}\right)}^{2}}}}{{2{\sigma_{1}}}}+\frac{{{{\left({y-{u_{2}}}\right)}^{2}}}}{{2{\sigma_{2}}}}}\right]}}}}{{2\pi{\sigma_{1}}{\sigma_{2}}}}\Bigg{\\}}dy$
(16)
where $\Phi\left(x\right)$ is the CDF of the standard normal distribution.
Although the solution for the ratio of the two limiting eigenvalues given in
eq. $\left(10\right)$ is in integral form, the integral is well behaved,
having a stictly positive integrand, and it can be evaluated readily by using
standard numerical computation or commonly available mathematical softwares.
It is seen that the proposed solution for the PDF of the ratio of the two
limiting eigenvalues is simpler than the solution given in [8, eq.
$\left(7\right)$]. It is seen from eq. $\left(10\right)$ and eq.
$\left(7\right)$ given in [8] that the complexity of these two expressions
mainly depends on the multiple integration. Moreover, double integrations are
required. In [8], there are ${\rm O}\left({{N^{2}}}\right)$ additions due to
the permutation operation and ${\rm O}\left({{NM^{4}}}\right)$ multiplications
in the integrals, where ${\rm O}$ is the big ${\rm O}$ notation [13]. In eq.
$\left(10\right)$, $M-1$ additions and ${\rm O}\left({{M}}\right)$
multiplications in the integrals are required. In the simulation, a comparison
of the required time for these two expressions is given to further clarify the
superiority of our proposed PDF in term of the complexity.
## IV Simulation Evaluations And Discussion
In this section, simulation results are given to contrast the proposed simple
form for the PDF of the ratio of the two limiting eigenvalues and the
expressions for the PDF of that ratio given in [2], [5] and [7]. We also
present some example results that compare the accuracies of the new and
previous theoretical approximations for the PDF of the ratio of the
eigenvalues to the exact PDF obtained by simulation. The noises are
independent identically distributed Gaussian real noises with mean zero and
unit variance. All the simulation results are achieved by using $10^{6}$ Monte
Carlo simulations. The number of samples and the number of secondary users are
set as $N=50$ and $M=10$ or $N=50$ and $M=20$, respectively.
Figure 2: Comparison of the improved PDF approximation for the ratio of the
two limiting eigenvalues of the covariance matrix with the known approximate
PDFs for $N=50$ and $M=10$.
Fig. 2 shows the commonly employed approximations to the PDF of the ratio of
the two limiting eigenvalues of the covariance matrix obtained by using
existing methods and our proposed approximation which is obtained without the
assumption that the two eigenvalues are independent. In Fig. 2, the empirical
PDF curve is the empirical PDF of the ratio of the two liming eigenvalues
while the TW PDF is the PDF approximation obtained by using the Tracy-Widom
distribution of order $2$ [2]. The PDF curve labeled eq. (30) [5] is the
approximate solution given in [5, eq. $\left(30\right)$]. The PDF curve
labeled PDF [7] is the exact PDF given in [7]. It is observed that the PDF of
the ratio test statistic obtained by using the new solution matches better
with the empirical PDF than the PDFs obtained from the other two methods. It
is also seen that the PDF given by the exact solution in [7] matches well with
the empirical PDF. The result consists with the result obtained in [7]. The
PDF from [5, eq. $\left(30\right)$] and the PDF used to approximate the Tracy-
Widom PDF in [2] are both approximate PDFs, and both are inferior
approximations to the new approximation. The Tracy-Widom PDF of order $2$ is
not a good approximation to the precise PDF obtained by simulation. The reason
is that the Tracy-Widom PDF of order $2$ for the ratio of the two limiting
eigenvalues is valid when $\mathop{\lim}\limits_{N\to\infty}\frac{M}{N}=c$,
where $c$ is a constant. However, in the simulation, $N$ and $M$ are set as
$50$ and $10$, respectively, and these values do not satisfy the limiting
condition. It is seen that the new solution provides very high accuracy. The
reason is that our new solution is derived without the independence assumption
that the samples are independent, and the only source of discrepancy is the
Gaussian distribution assumption, which causes only small discrepancies when
the number of samples is even moderately large. These results show that the
major source of error in the previous approximations is the independence
assumption, and not the Gaussian assumption.
Figure 3: Comparison of the improved PDF approximation for the ratio of the
two limiting eigenvalues of the covariance matrix with the known approximate
PDFs, for $N=50$ and $M=10$ or $N=50$ and $M=20$.
Fig. 3 shows the comparison of the improved PDF approximation for the ratio of
the two limiting eigenvalues of the covariance matrix with the known
approximate PDF given by eq. (30) [5] with different $M$. It is seen that the
accuracy of both our proposed solution and the form given by [5] increases
with $M$. The reason is that the accuracy of the mean and variance of the two
limiting eigenvalues increases with $M$.
TABLE I: Comparison of the required computation times (s) Schemes $\left(N,M\right)$ | $\left(50,5\right)$ | $\left(100,5\right)$ | $\left(100,10\right)$ | $\left(100,20\right)$
---|---|---|---|---
Eq. $\left(7\right)$ in [8] | 10.248 | 14.835 | 27.482 | 43.498
Eq. $\left(10\right)$ | 6.529 | 6.572 | 10.593 | 22.179
In order to compare the complexity of our proposed PDF form with that of the
form given by eq. $\left(7\right)$ in [8], the computation times for different
parameters $\left(N,M\right)$ are given in Table 1. The results are obtained
by using a computer with 64-bit Intel(R) Core(TM) i7-4790 CPU, $8$ GB RAM. It
is seen from Table 1 that the required time for calculating the PDF given by
eq. $\left(7\right)$ in [8] is larger than that for our proposed PDF. This
indicates the complexity of our proposed expression is lower than that
presented in [8]. It further verifies that our proposed PDF is simpler than
that proposed in [8].
## V Conclusion
A new approximation for the PDF of the ratio of the two limiting eigenvalues
of the covariance matrix in eigenvalue-based spectrum sensing was derived
based on order statistic analysis. The new approximate solution is the most
accurate approximation known, and its derivation does not rely on an invalid
independence assumption used to derive a popular previous approximation. The
precise new approximation was used to show that the major source of error in
previous approximation is the independence assumption and not Gaussian
approximation. The relative poorness of the Tracy-Widom approximation was
clarified and explained.
## References
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* [2] F. Penna, R. Garello, and M. A. Spirito, “Cooperative spectrum sensing based on the limiting eigenvalue ratio distribution in Wishart matrices,” _IEEE Commun. Lett._ , vol. 13, no. 7, pp. 507-509, Jul. 2009.
* [3] A. Kortun, T. Ratnarajah, M. Sellathurai, C. Zhong, and C. B. Papadias,“On the performance of eigenvalue-based cooperative spectrum sensing for cognitive radio,” _IEEE J. Sel. Topics Signal Process._ , vol. 5, no. 2, pp. 49-55, Feb. 2011.
* [4] P. Zhang and R. C. Qiu, “GLRT-based spectrum sensing with blindly learned feature under rank-1 assumption,” _IEEE Trans. Commun._ , vol. 61, no. 1, pp. 87-96, Jan. 2013.
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* [6] F. F. Gao, C. Qian, H. Qian, and, T. Zhao, “Sensing and recognition for multiple-primary-power level scenario with noise uncertainty,” _IEEE Trans. Veh. Technol._ , vol. 66, no. 3, pp. 2289-2300, Mar. 2017.
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|
11institutetext: European Southern Observatory, Av. Alonso de Córdova 3107,
Vitacura, Santiago, Chile ,
11email<EMAIL_ADDRESS>22institutetext: INAF, Osservatorio Astronomico di
Roma, via Frascati 33, I-00078 Monteporzio Catone, Italy 33institutetext: Aix
Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR
7326, 13388, Marseille, France 44institutetext: INAF - Osservatorio di
Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129,
Bologna, Italy 55institutetext: Observatoire de Genève, Université de Genève,
51 Ch. des Maillettes, 1290 Versoix, Switzerland 66institutetext: Instituto de
Investigación Multidisciplinar en Ciencia y Tecnología, Universidad de La
Serena, Raúl Bitrán 1305, La Serena, Chile 77institutetext: Departamento de
Física y Astronomía, Universidad de La Serena, Norte, Av. Juan Cisternas 1200,
La Serena, Chile 88institutetext: SUPA, Institute for Astronomy, University of
Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, UK 99institutetext: INAF-
Astronomical Observatory of Trieste, via G.B.Tiepolo 11, 34143 Trieste, Italy
1010institutetext: Department of Astronomy, The University of Texas at Austin,
Austin, TX, 78712 1111institutetext: Instituto de Astrofísica, Universidad
Católica de Chile, Vicuña Mackenna 4860, Santiago, Chile 1212institutetext:
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD,
21218, USA 1313institutetext: The Cosmic Dawn Center, Niels Bohr Institute,
Copenhagen University, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark
1414institutetext: University of Bologna, Department of Physics and Astronomy
(DIFA), Via Gobetti 93/2, I-40129, Bologna, Italy
# The Intergalactic medium transmission towards z$\gtrsim$4 galaxies with
VANDELS and the impact of dust attenuation††thanks: Based on observations made
with ESO Telescopes at the La Silla or Paranal Observatories under programme
ID(s) 194.A-2003
R. Thomas 11 L. Pentericci The Intergalactic medium transmission towards
z$\gtrsim$4 galaxies with VANDELS and the impact of dust attenuation††thanks:
Based on observations made with ESO Telescopes at the La Silla or Paranal
Observatories under programme ID(s) 194.A-2003The Intergalactic medium
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with VANDELS and the impact of dust attenuation††thanks: Based on observations
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194.A-2003 F. Marchi The Intergalactic medium transmission towards z$\gtrsim$4
galaxies with VANDELS and the impact of dust attenuation††thanks: Based on
observations made with ESO Telescopes at the La Silla or Paranal Observatories
under programme ID(s) 194.A-2003The Intergalactic medium transmission towards
z$\gtrsim$4 galaxies with VANDELS and the impact of dust attenuation††thanks:
Based on observations made with ESO Telescopes at the La Silla or Paranal
Observatories under programme ID(s) 194.A-2003 L. Pozzetti The Intergalactic
medium transmission towards z$\gtrsim$4 galaxies with VANDELS and the impact
of dust attenuation††thanks: Based on observations made with ESO Telescopes at
the La Silla or Paranal Observatories under programme ID(s) 194.A-2003The
Intergalactic medium transmission towards z$\gtrsim$4 galaxies with VANDELS
and the impact of dust attenuation††thanks: Based on observations made with
ESO Telescopes at the La Silla or Paranal Observatories under programme ID(s)
194.A-2003 A. Saxena The Intergalactic medium transmission towards z$\gtrsim$4
galaxies with VANDELS and the impact of dust attenuation††thanks: Based on
observations made with ESO Telescopes at the La Silla or Paranal Observatories
under programme ID(s) 194.A-2003The Intergalactic medium transmission towards
z$\gtrsim$4 galaxies with VANDELS and the impact of dust attenuation††thanks:
Based on observations made with ESO Telescopes at the La Silla or Paranal
Observatories under programme ID(s) 194.A-2003 M. Talia The Intergalactic
medium transmission towards z$\gtrsim$4 galaxies with VANDELS and the impact
of dust attenuation††thanks: Based on observations made with ESO Telescopes at
the La Silla or Paranal Observatories under programme ID(s) 194.A-2003The
Intergalactic medium transmission towards z$\gtrsim$4 galaxies with VANDELS
and the impact of dust attenuation††thanks: Based on observations made with
ESO Telescopes at the La Silla or Paranal Observatories under programme ID(s)
194.A-2003The Intergalactic medium transmission towards z$\gtrsim$4 galaxies
with VANDELS and the impact of dust attenuation††thanks: Based on observations
made with ESO Telescopes at the La Silla or Paranal Observatories under
programme ID(s) 194.A-2003The Intergalactic medium transmission towards
z$\gtrsim$4 galaxies with VANDELS and the impact of dust attenuation††thanks:
Based on observations made with ESO Telescopes at the La Silla or Paranal
Observatories under programme ID(s) 194.A-2003 M. Bolzonella The Intergalactic
medium transmission towards z$\gtrsim$4 galaxies with VANDELS and the impact
of dust attenuation††thanks: Based on observations made with ESO Telescopes at
the La Silla or Paranal Observatories under programme ID(s) 194.A-2003The
Intergalactic medium transmission towards z$\gtrsim$4 galaxies with VANDELS
and the impact of dust attenuation††thanks: Based on observations made with
ESO Telescopes at the La Silla or Paranal Observatories under programme ID(s)
194.A-2003
(;)
###### Abstract
Aims. Our aim is to estimate the intergalactic medium transmission towards UV-
selected star-forming galaxies at redshift 4 and above and study the effect of
the dust attenuation on these measurements.
Methods. The ultra-violet spectrum of high redshift galaxies is a combination
of their intrinsic emission and the effect of the Inter-Galactic medium (IGM)
absorption along their line of sight. Using data coming from the unprecedented
deep spectroscopy from the VANDELS ESO public survey carried out with the
VIMOS instrument we compute both the dust extinction and the mean transmission
of the IGM as well as its scatter from a set of 281 galaxies at z¿3.87.
Because of a degeneracy between the dust content of the galaxy and the IGM, we
first estimate the stellar dust extinction parameter E(B-V) and study the
result as a function of the dust prescription. Using these measurements as
constraint for the spectral fit we estimate the IGM transmission
Tr(Ly$\alpha$). Both photometric and spectroscopic SED fitting are done using
the SPectroscopy And photometRy fiTting tool for Astronomical aNalysis
(SPARTAN) that is able to fit the spectral continuum of the galaxies as well
as photometric data.
Results. Using the classical Calzetti’s attenuation law we find that E(B-V)
goes from 0.11 at z=3.99 to 0.08 at z=5.15. These results are in very good
agreement with previous measurements from the literature. We estimate the IGM
transmission and find that the transmission is decreasing with increasing
redshift from Tr(Ly$\alpha$)=0.53 at z=3.99 to 0.28 at z=5.15. We also find a
large standard deviation around the average transmission that is more than 0.1
at every redshift. Our results are in very good agreement with both previous
measurements from AGN studies and with theoretical models.
###### Key Words.:
Extragalactic astronomy – Spectroscopy – High redshift – Intergalactic medium
## 1 Introduction
The observation of distant galaxies necessarily include the effect of the
Inter-Galactic Medium (IGM) along the line of sight (LOS), and its associated
extinction. The light coming from those sources is travelling through clouds
that are lying along the line of sight. As the redshift of the source
increases, the clouds along the LOS can be so numerous that all the light
below the Lyman $\alpha$ line (at 1216Å, hereafter, Ly$\alpha$) can be
absorbed. Numerous authors have studied this phenomenon and it is thought that
it is a natural result of the hierarchical formation of structure (e.g. Cen et
al. 1994).
More than two decades ago, shortly after a work on the effect of the
intergalactic medium on galaxy emission by Yoshii & Peterson (1994), Madau
(1995) (hereafter M95) simulated the average IGM transmission as a function of
redshift and found that it strongly decreases with increasing redshift.
Moreover, the IGM leads to a very specific stair-like pattern where each step
corresponds to a line of the Lyman series of the Hydrogen atom. In addition, a
large scatter was expected and, for instance, the average transmission at z =
3.5 was estimated to range from 20% to 70% with an average of 40% (M95). A
decade later Meiksin (2006) (hereafter M06) updated this model producing a new
IGM prescription using the $\Lambda$-CDM model of Meiksin & White (2004). It
was found that the IGM transmission is higher than the one of M95, mainly
because of differences in the estimates of the contributions of resonant
absorption. More recently, Inoue et al. (2014) developed a new model of
transmission. Their model predicts a weaker absorption in the range z=3-5 than
the M95 models while it becomes stronger at z¿6.
For years, the average transmission (noted Tr(Ly$\alpha$)) has been estimated
from the Ly$\alpha$ forest measurements on the LOS of QSOs. It is often
referred to as the HI optical depth $\tau_{eff}$ with Tr(Ly$\alpha$) =
$\exp(-\tau_{eff})$ and its measurements are used to constrain the intensity
of the ionizing background (Haardt & Madau 1996; Rauch et al. 1997; Bolton et
al. 2005) and to investigate the sources responsible for the ionizing
background. Surprisingly, only a few reports have been published on the
observed dispersion in Tr(Ly$\alpha$) as a function of redshift. Faucher-
Giguère et al. (2008b) used 86 high-resolution quasar spectra with a high
signal-to-noise ratio to provide reference measurements of the dispersion in
Tr(Ly$\alpha$) over 2.2 ¡ z ¡ 4.6.
Until few years ago, no observational study had been made of the evolution of
the IGM transmission from galaxy samples mainly because of the lack of large
spectroscopic samples with high signal-to-noise ratios at high redshift that
would probe a wavelength range significantly bluer than Ly$\alpha$. Hence, the
comparison of IGM transmission towards extended galaxies with point-like QSOs
had not yet been performed. In a recent paper (Thomas et al. 2017a) we were
able to compute for the first time the IGM transmission towards a set of more
than 2000 galaxies (with $\sim$120 of them at z¿4) provided by the VIMOS Ultra
Deep Survey (VUDS; Le Fèvre et al. 2015). This study allowed us to show that,
(i) the IGM transmission towards galaxies was a measurable parameter, (ii) the
IGM transmission at $z<4$ was in very good agreement with the one computed
towards QSO data in terms of both absolute measurements and also scatter
around the mean values), (iii) at $z>4$ there might be a possible departure of
the observational data from the theoretical prediction. This observed
difference was interpreted as a signature of degeneracy between the dust and
IGM models.
In this paper we perform a study of 281 galaxies at z¿4 from the very deep
VANDELS survey (McLure et al., 2018b; Pentericci et al., 2018) to compute the
IGM properties. We therefore have more than twice the number of galaxies we
had for the VUDS sample and with much deeper observation (ranging from 20 to
80 hours, instead of 14h). We also focus on the impact of different dust
attenuation prescription on the IGM measurements. We describe the VANDELS
galaxy sample and selection in Sect. 2 . The fitting method with the SPARTAN
tool and the range of IGM templates used in the spectral fitting is described
in Sect. 3 along the definition of the Ly$\alpha$ transmission we use in this
paper. The estimation of the dust extinction and IGM transmission are
described in Sect. 4 and 5, respectively. We look at stacked spectra of
different population in Sect.6. Finally, we discuss the robustness of our
results in section 7. All magnitudes are given in the AB system (Oke & Gunn,
1983) and we use a cosmology with $\Omega_{M}$ = 0.3, $\Omega_{\Lambda}$ = 0.7
and h = 0.7.
## 2 Data and sample selection
Our study is based on galaxies from the VANDELS survey. The data sample
selection is described in McLure et al. (2018b) while the data reduction and
redshift measurements and validation are described in Pentericci et al.
(2018). We briefly present an overview of the survey in this section.
VANDELS is a public spectroscopic survey carried out with the VIMOS instrument
(Le Fèvre et al., 2003) located at the NASMYTH focus of the Unit Telescope 3
Melipal of the Very Large Telescope (VLT). It made use of the medium
resolution grism spanning a wavelength window from 4800 to 10000Å with a
spectral resolution of R=580. It targeted $\sim$2100 objects in a wide
redshift range (1.0¡z¡7). Targets were selected in the two widely observed UDS
and CDFS fields covering a total area of 0.2 deg2. The primary target
selection was performed using the photometric redshift technique. The
reduction of the raw data was carried out using the EASYLIFE package (Garilli
et al., 2012) and all redshifts were estimated using the EZ software (Garilli
et al., 2010). A redshift flag has been assigned to each redshift measurement.
This flag corresponds to the probability of the redshift to be correct. The
quality scheme is composed of six values. Flags 2, 3, 4 and 9 (for objects
with a single emission line) are the most reliable flags with a probability to
be correct of 75%, 95%, 100% and 80%, respectively. A quality flag of 1
indicates a 50% probability of being correct, while a quality flag of 0
indicates that no redshift could be assigned. At the moment of writing, the
internal VANDELS database provides 1527 unique sources (with more than 1300
available from the DR2). It gives access to 1-dimensional and 2-dimenstional
spectra. Photometric data are available for each of the VANDELS galaxies from
different ground-based or space-based observatory. Both fields are partially
covered by optical and infrared photometric observations coming from the
CANDELS survey with ACS and WFC3/IR and SPITZER/IRAC instruments (Galametz et
al., 2013; Guo et al., 2013). Ground based data are also available with
optical bands from the Subaru/Suprime-Cam instrument (Furusawa et al., 2008;
Cardamone et al., 2010; Furusawa et al., 2016; Sobral et al., 2012), near
infrared bands from the VIRCAM instrument from the VLT (Jarvis et al., 2013)
and near infrared bands from the WIRcam camera of the CFHT (Hsieh et al.,
2012). We refer the reader to McLure et al. (2018b) for further details.
The aim of this paper is to study the IGM towards high redshift galaxies. As
presented in Sect.1, the IGM signature in the spectra of distant galaxies is a
stair-like pattern below the Ly$\alpha$ line. The Ly$\alpha$ transmission that
we want to estimate is computed between the Ly$\alpha$ position, at 1216Å, and
the Ly$\beta$ position at 1025Å. Therefore we must be able to observe this
wavelength domain for our analysis. As the reduction process is sometimes not
very efficient in extracting the edges of the spectra, we take a lower limit
for our observed windows at 5000Å (instead of the nominal 4800Å limit of the
medium resolution grism of VIMOS). This leads to a minimum redshift of z =
3.87. We do not impose, a-priori, any threshold on the signal to noise (SNR)
nor the redshift flag for our working sample but we show the distribution
ofSNR per spectral pixel measured with the recipe from Stoehr et al. 2008 (the
dispersion of VANDELS spectra is $\sim$2.55Å/pix) along the distribution of
apparent magnitude in the i-band in Fig.1. This leads to a selected sample of
281 galaxies. In our sample 25 galaxies have redshift flag of 1, 69 have a
redshift flag of 2 or 9, and 185 have a redshift flag of 3 or 4. Therefore 2/3
of our selected sample has an assigned redshift with a probability to be
correct higher than 95%. The stability of our results with respect to the
choice of redshift flag is discussed in Sect.7.
Figure 1: Observed properties of our selected sample of galaxies and
comparison to the global released VANDELS data. Left: Apparent magnitude
(sextractor MAGAUTO) of our data in the i-band. Right: Signal-to-Noise
measurements; at 1070-1170Å restframe in red and at a fixed observed
wavelength (6000-7400Å) in purple. In the former case the median SNR is 0.97.
## 3 Method
### 3.1 The SPARTAN tool
To estimate the IGM transmission toward our galaxies we use the SPARTAN tool
which is able to fit both photometry and spectroscopic data. In this paper we
use the capability of SPARTAN to fit spectroscopic dataset and photometric
dataset separately. This single data type fitting follows the same recipe as
other codes used in the literature (e.g. Salim et al. 2007, Thomas et al.
2017b). For a given object and a single template the $\chi^{2}$ and associated
probability are estimated with:
$\chi^{2}=\sum_{i=1}^{N}\frac{(F_{obs,i}-A_{i}F_{syn,i})^{2}}{\sigma_{i}^{2}};\;\;P=\exp\left[-\frac{1}{2}(\chi^{2}-\chi^{2}_{min})\right]$
(1)
where N, Fobs,i, Fsyn,i, $\sigma_{i}$ ,Ai and $\chi^{2}_{min}$ stand for the
number of observed data points, the flux of the data point itself, the
synthetic template value at the same wavelength, the observed error associated
to Fobs,i, the normalization factor applied to the template, and the minimum
$\chi^{2}$ of the library of template, respectively. The latter is used to set
the maximum of the probability distribution to unity. From the properties of
the exponential function this is only a normalization factor and does not
change the values of the parameters’ estimation nor their errors. The set of
probability values (second part of equation 1) are then used to create the
probability distribution function (PDF, whose integral is normalized to unity)
for each parameter to be estimated. From the PDF we create the cumulative
distribution function (CDF) where the measured value of the parameter is taken
where CDF(X)=0.5 and the errors on this measurement correspond to the value of
the parameter for which the CDF=0.05 and 0.95.
The photometric fitting process is performed as follows. The set of synthetic
templates is redshifted to the redshift of the fitted galaxies and then
normalized in one pre-defined band. For the photometric fitting we performed
in this paper this normalization is applied in the i-band. Once this
normalization is done, SPARTAN convolves the normalized template with all the
photometric bandpasses available for the observed galaxy. Finally, the
relations in Eq.1 are applied to estimate the physical parameters of the
observed galaxy and their associated errors.
When dealing with spectroscopic data, the general principle of the fitting
process is similar. Nevertheless, this type of data allows for a different
normalisation method. SPARTAN has to normalize the redshifted template to the
observed spectrum. As for the photometry, we can consider a photometric filter
and estimate the magnitude in the same filter from the spectrum itself. This
magnitude serves as a normalization to all the templates. This approach,
widely used in the literature with photometric dataset, uses normalization
that is always done in a given photometric band (e.g., the i-band). As a
result, each galaxy is normalized to the template in a different rest-frame
region and all the galaxies are not treated in a similar manner. The
spectroscopy opens a new redshift-dependent method of normalization. This
method uses an emission line free region available in the spectrum. In the UV
spectrum of distant galaxies, a region free from strong spectral line is
between 1070 and 1170 Å (rest-frame). When fitting a UV spectrum at z=4.5,
this spectral region will be shifted at 5885-6435Å. SPARTAN will compute a
spectro-photometric point in this region directly in the template and in the
data using a box filter of the size of this region. This box-magnitude will
then be used to normalize the template to the observed spectrum. At higher
redshift, i.e. z=5.0, this spectral region will be at a redder wavelength
(6420-7020Å) and this observed-frame region will be used to perform the
normalization as well. This new method of normalization has the merit of being
consistent from one object to another. Moreover, as it is used in emission
line free region, it relies less on the emission line physics of the
templates. We use in this paper the latter redshift dependent normalisation
method and we make the normalisation in the region 1070-1170Å (restframe).
This choice is supported by; (i) Once redshifted this region provides a wide
window for SNR estimation ($\sim$500Å @ z=3.87 and $\sim$750Å @ z=6.5), (ii)
It is free of strong emission line, (iii) considering the VIMOS wavelength
window, it is one of the only wide-enough wavelength range available across
the redshift range we consider.
### 3.2 IGM models and Tr(Ly$\alpha$) definition
To estimate the IGM transmission we must be able to fit it. For years, the IGM
transmission was fixed at a single value at a given redshift most often using
the M95 model that provides a single transmission curve at a given redshift.
Therefore, it was assumed that at a given redshift, the lines of sight of
objects observed at a different position in the sky are populated by hydrogen
clouds with the same properties. In M95, the author provides an estimate of
the 1 $\sigma$ dispersion and, as mentioned in Sect. 1, it can vary from 20%
to 70% at z=3.5. Additionally, it was shown that this dispersion around the
mean IGM could produce better photometric redshift (Furusawa et al., 2000).
Therefore, we proposed in our previous paper to use a set of empirical models
that can reproduce this dispersion in the IGM transmission (Thomas et al.
2017a, hereafter T17). We summarize here how these templates were constructed.
To test different line of sights during the SED-fitting we constructed 6
additional templates around the mean of M06. This additional models were built
considering the $\pm 1$ sigma variation of M95 IGM models (see Fig.3a in M95
paper at $z=3.5$) that we propagated at any redshift. Finally, to explore more
possibilities, we created, still from this $\pm 1$ sigma variation, the $\pm
0.5\sigma$ and $\pm 1.5\sigma$. As a result, the IGM can be chosen from the
set of 7 discrete values at any redshift and this allows us to use the IGM as
a free parameter in out fitting procedure and explore a larger range of IGM
transmission. At z=3.0 it ranges from 20% to 100% while at z=5.0, it ranges
from 5% to 50%. As an example, we show in Fig.2 the set of extinction curves
at z=4.0.
Figure 2: Example of IGM transmission curves at z=4.0. The red curve is from
M06 Prescription while the black curves represent the augmented prescription
from Thomas et al. (2017a). The latter allows, at this redshift to span
possible transmission from $\sim$15% to $\sim$90%. The grey area shows where
we compute the Ly$\alpha$ transmission.
In this paper we aim at computing the Ly$\alpha$ Transmission, Tr(Lyα) which
is determined as the mean transmission between 1070Å and 1170Å computed on the
transmission curve itself, shown in Fig.2 by the grey area. In the case of
SPARTAN we use the PDF of Tr(Lyα) to estimate the value of the parameter, as
described in Sect. 3.
Finally, we emphasize that the IGM models we use here, while based on
simulation, are also empirical (in the additional curves we use). More recent
models include more components in the simulations such as the inclusion of the
CGM (Steidel et al., 2018; Kakiichi & Dijkstra, 2018). We will compare these
different prescriptions in a near-future paper. It is also worth noting that
the general shape of the curves is the same from one model to another while it
can vary from one line-of-sight to another, depending on the presence of
absorbers. In T17 we compared the results of the fit using the templates
presented here and real Lyman $\alpha$ forest simulation (Bautista et al.,
2015) and found that there was a very good agreement in the resulting
measurement of the Ly$\alpha$ Transmission.
## 4 Dust content of z¿4 galaxies
Figure 3: Dust evolution from our 281 galaxies. Top Panel: The four different
dust prescriptions used to estimate the dust extinction in our sample. In red
we show the prescription from Calzetti et al. (2000), in black the
prescription from Prevot et al. (1984) for the SMC, in violet the prescription
from Fitzpatrick (1986) for the LMC and in green for the Milky Way
prescription by Allen (1976). Bottom Panel: Evolution with redshift of the
dust attenuation in our selected sample of 281 galaxies from the photometric
fitting for the four dust prescription shown in the Top Panel. Measurements
report the mean and median absolute deviation for both the redshift and the
E(B-V) values. We compare our results with previous measurements found in the
literature at similar redshifts. The empty black diamonds are estimation from
Bouwens et al. (2009), black triangles from Bouwens et al. (2007) and light
blue cross from Ouchi et al. (2004). The dashed black line shows a fit from
Hayes et al. (2011). Note: All the VANDELS estimation are at the same
redshifts, violet ones have been slightly shifted for clarity.
In Thomas et al. (2017a) we identified a potential strong degeneracy between
the estimates of the dust content of the galaxy and the IGM transmission
prescription. This degeneracy is more prominent at z¿4. In other words, the
same data can be fitted with high values of both dust extinction and IGM
transmission or lower values for both parameters. This is due to the small
wavelength range available from UV spectra for fitting that is not able to
constrain the dust content of the galaxy. In order to address this problem in
the present work, we measure the IGM transmission in a two step process.
First, we estimate the dust content of each galaxy in our sample using the
photometric data presented in Sect. 2. We fit the SED over a broader
wavelength range than the spectra including NIR data, providing robust
constraints on dust extinction. Then, we estimate Tr(Ly$\alpha$), keeping the
E(B-V) value fixed to the one measured during the photometric-fitting process.
For this two-step fitting process we use the following parameter space. We use
Bruzual & Charlot (2003) models with a Chabrier (2003) initial mass function.
The stellar-phase metallicity ranges from sub-solar (0.2$Z_{\odot}$ and
0.4$Z_{\odot}$) to solar (1.0$Z_{\odot}$). We assume a star formation history
prescription that is exponentially delayed with a timescale parameter, $\tau$,
ranging from 0.1 Gyr to 1.0 Gyr. The ages range from 1 Myr to 3 Gyr in 24
steps. It is worth noting that this range of age is further limited by the age
of the Universe at the redshift that is considered during the fit. The
emission lines are added to the template following the work of Schaerer & de
Barros (2009) that adds nebular continuum and emission using the conversion
from ionizing photon into H$\beta$ luminosity. Then other emission lines are
added using line ratio from Anders & Fritze-v. Alvensleben (2003). This first
step is made to estimate the dust extinction. Therefore, we use in this
section four different dust extinction curves111 http://webast.ast.obs-
mip.fr/hyperz/hyperz$\\_$manual1/node10.html: the classical starburst galaxy
prescription from Calzetti et al. (2000) (hereafter SB) with an extrapolation,
the Small Magellanic Cloud from (Prevot et al. 1984, hereafter SMC), the
prescription for the Large Magellanic Cloud (Fitzpatrick 1986, hereafter LMC)
and finally the prescription for the Milky Way (Allen 1976, hereafter MW). All
the curves are presented in Fig. 3 (top panel). For the photometric fitting,
the E(B-V) parameter can vary from 0.0 to 0.39 (in 0.03 steps). Finally, the
IGM prescription is using the models developed in Thomas et al. (2017a) based
on the M06 models (see previous section). The redshift used for this fitting
is the spectroscopic redshift, $z_{spec}$. Finally, it is worth noting that
during this fitting process the IGM is estimated. These IGM measurements from
the pure photometric fitting are discussed in Sect.7.
Table 1: Measurements of E(B-V) from the fit of photometry only using four different dust prescriptions. Each value represents the mean in each redshfit bins and the errors is the median absolute deviation. ¡z¿ | MW | LMC | SMC | SB
---|---|---|---|---
3.99 | 0.14$\pm$0.06 | 0.11$\pm$0.06 | 0.04$\pm$0.03 | 0.11$\pm$0.06
4.23 | 0.11$\pm$0.06 | 0.09$\pm$0.06 | 0.03$\pm$0.03 | 0.10$\pm$0.05
4.59 | 0.12$\pm$0.06 | 0.08$\pm$0.03 | 0.04$\pm$0.03 | 0.11$\pm$0.06
5.15 | 0.08$\pm$0.06 | 0.05$\pm$0.03 | 0.03$\pm$0.03 | 0.08$\pm$0.06
The dust extinction measurements can be seen in Fig. 3 (bottom panel) that
shows the evolution of the dust attenuation with redshift for our 281 selected
galaxies and in Tab. 1 we report the measurements. Using the SB prescription
we report mean values of E(B-V)=0.11; 0.10; 0.10 and 0.08 at z=3.99; 4.23;
4.59 and 5.15, respectively. Using the LMC curve the measurements are very
similar and we obtain 0.11, 0.09, 0.08 and 0.05 at the same redshifts. The
prescription of the SMC leads to a much weaker extinction at any redshift with
0.04, 0.03, 0.03 and 0.02 while the Milky-Way extinction from MW leads to
slightly higher values of E(B-V with 0.14, 0.11, 0.12 and 0.08. This is easily
explained by the fact that for a fixed value of E(B-V) the curve of the SMC
will lead to a much stronger extinction when studying UV-restframe galaxies
(below 2000Å) while the curve from the MW will give lower extinctions. The
measurements obtained using SB, LMC and MW are in good agreement with previous
estimation at similar redshifts. Studying dropout selected galaxies Bouwens et
al. (2009) reports an E(B-V) measurement of 0.14 at z$\sim$3.8 while
E(B-V)=0.095 at z=5.0. At similar redshift, Ouchi et al. (2004) used Lyman
break galaxies between 3.5¡z¡5.2 and measured E(B-V)=0.075 at z=4.7. It is
worth mentioning that SB-like laws have been supported by other studies in the
literature (e-g McLure et al. 2018a). On the contrary, the measurements using
SMC are in strong disagreement with previous measurement from the literature,
as reported in Scoville et al. (2015) and Fudamoto et al. (2017). We will use,
in the rest of the paper, both SB and MW models (LMC being very close to the
SB) and see how this influences the measurements of the IGM.
## 5 Tr(Ly$\alpha$) towards z¿4 galaxies
Figure 4: Example of fits of VANDELS galaxies at redshift $z>4$. In each plot
the spectrum is shown by the black line while the best fit, produced by
SPARTAN, is given in red. We indicate the position of the Ly$\alpha$ line and
the redshift for each spectrum.
We measured in the last section the dust content of our galaxies. We now move
towards the estimation of the IGM transmission with the spectral fit. In order
to estimate it we constrain the spectral fit of our VANDELS spectroscopic data
fixing the E(B-V) to the one measured during the fit of the photometric data.
We consider individual E(B-V) value for each of our galaxies and do not use
the average values presented in Fig. 3. The other parameters, such as age or
metallicity are still free to vary and the parameter ranges correspond to the
ones of the fit of Sect.4. Examples of spectral fit of VANDELS galaxies are
presented in Fig. 4 at various redshift and they show that SPARTAN reproduces
very well the UV continuum of our galaxies at all wavelengths. It is worth
mentioning that we can see that the Ly$\alpha$ is poorly reproduced. As
presented in the previous section, the emission lines are added using line
ratios and therefore not fitted individually. Our results remain if we mask
out the line during the fit.
Table 2: Measurements of the Tr(Ly$\alpha$) from our study. We report in this table the redshift, the Tr(Ly$\alpha$), the standard deviation, the standard error, the Tr(Ly$\alpha$) for redshift flag 2, 3 and 4 only (with the number of galaxies with these flags in parenthesis) and the measurements of the same quantity from the pure photometric fitting. Redshift | Ngal | Tr(Ly$\alpha$) | Std deviation | Standard error | Flag 2,3&4 | Photometry
---|---|---|---|---|---|---
3.99 | 81 | 0.55 | 0.14 | 0.015 | 0.54 (58) | 0.41
4.23 | 74 | 0.49 | 0.14 | 0.016 | 0.48 (50) | 0.24
4.59 | 72 | 0.42 | 0.13 | 0.015 | 0.41 (54) | 0.34
5.15 | 54 | 0.29 | 0.11 | 0.015 | 0.30 (23) | 0.26
Results on the measurement of Tr(Ly$\alpha$) are presented in Fig. 5 where we
show the distributions of the Lyman $\alpha$ transmission in four redshift
bins: 3.85¡z¡4.1; 4.1¡z¡4.4, 4.4¡z¡4.8 and at z¿4.8. We also display the
evolution of this quantity with redshift as compared with previous
measurements in the literature (we take uneven binning to ensure a maximum
number of galaxies in each bins). Table. 2 provides the measurements for each
bin.
Figure 5: Lyman $\alpha$ transmission (Tr(Ly$\alpha$)) as function of
redshift. The four top first small plots show the distributions of the
transmission in four redshift bins 3.85¡z¡4.1, 4.1¡z¡4.4, 4.4¡z¡4.8 and z¿4.8
for each dust prescription. We indicate in each of these plot, the number of
galaxies entering in the distribution. The bottom plot display the evolution
the transmission with redshift. Our measurements are indicated in red for SB
and green for MW and in violet for LMC. We show the measurements from QSO in
blue from Becker et al. (2013), from the VIMOS Ultra Deep Survey (VUDS, Thomas
et al. (2017a), in black) and the theoretical prediction from Meiksin (2006)
represented with the black dashed line.
We estimated the Ly$\alpha$ transmission in these four redshift bins. Using
the SB dust attenuation, this quantity goes from Tr(Ly$\alpha$)=0.55 at z=3.99
with a large standard deviation of 0.14 to Tr(Ly$\alpha$)=0.29 at z=5.15 with
a standard deviation of 0.11. The standard error of the mean is small and is
below 0.02 at any redshift. The measurements using the MW dust curve are
giving similar measurements. This measurements are similar to measurements
done with QSOs at similar redshift. Becker et al. (2013) measured
Tr(Ly$\alpha$)=0.59 at z=3.70 and Tr(Ly$\alpha$)$\sim$0.35 at z$\sim$4.8. This
shows that even at high redshift we are able to reproduce equivalent
measurements with galaxies. Comparing our results to theoretical prediction,
we find that we are in good agreement with the M06 models that predicts
Tr(Ly$\alpha$)=0.39 at z=4.6 and Tr(Ly$\alpha$)=0.25 at z=5.15. We note that
our measurements are in partial disagreement with our previous measurements at
z¿4 from the VUDS galaxies. At z=4.23, the difference from our previous
measurement is more than 10% and reaches 20% at z¿4.5. Nevertheless, as
reported in Thomas et al. (2017a), this high values of Tr(Ly$\alpha$) could
actually be corrected limiting the E(B-V) to low values. Thus, the method we
employed in the present paper with an estimation of the E(B-V) value before
the spectral fit seems to correct for this degeneracy.
More importantly we report a large standard deviation of Tr(Ly$\alpha$) for
all our measured points. It goes from 0.14 at z=3.99 to 0.11 at z=5.15. This
is in good agreement with our previous study and confirms that IGM should be
treated as free parameter during fitting process. Surprisingly, we find that
the measurement using the LMC prescription are above the other ones with a
difference that peaks at +0.08 at z=4.23 while the highest redshift point is
in good agreement with the other dust solutions. We try to investigate these
differences in the next section.
## 6 Averaged spectra
Finally, we have a look at averaged spectra of the population of our selected
galaxies. We build two stack spectra that are constructed based on the IGM
transmission as measured in our VANDELS galaxies: one where we select all the
galaxies with a transmission higher than the mean curve given by M06, and one
where all the galaxies have a transmission lower than the mean curve. Each
stack spectrum is constructed using the
specstack222https://specstack.readthedocs.io/en/latest/ program (Thomas,
2019b) that works as follows. For a given stack we de-redshift all the
individual spectra and normalise them in a region red-ward of the Ly$\alpha$
line free of emission or absorption lines, in this case between the
SiII($\lambda 1260$) and OI($\lambda 1303$) absorption line. Then we re-grid
all the spectra in a common wavelength grid. Finally, at a given wavelength,
we compute the mean of all the fluxes using a sigma-clipping method (at
3$\sigma$).
Figure 6: Averaged spectra. We display two stack spectra. In blue we show the
average of all the spectra (79) with IGM transmission lower than the mean. The
mean redshift is these spectrum $\sim$4.36. In red we show the average of all
the spectra (101) with IGM transmission higher than the mean, with a mean
redshift of $\sim$4.60. The stack spectra have been made with the specstack
program (Thomas, 2019b).
The averaged spectra are presented in Fig. 6 (using the fits made with SB).
This figure shows that below Ly$\alpha$ at 1215$\mathrm{\AA}$, there is a non-
negligible variation in flux. The low-transmission stacked spectra are on
average $\sim$30% dimmer than the stacked spectra with high IGM transmission.
This means that at z$>$4, the standard deviation of the IGM is very important,
and that IGM transmission should be treated as a free parameter when studying
galaxy at such high redshift. It is also worth mentioning that the average
redshift of the two stacks is slightly different. For the stacks with IGM
below than the mean, the average redshift is $z\sim 4.36$ while it is $z\sim
4.60$ for the galaxies with an IGM higher than the mean. Consequently, the
difference might be even higher than what we measure here. Finally, the figure
shows that the spectra beyond the Ly$\alpha$ line are very similar. Few
absorption lines are stronger in the case of high IGM transmission (e.g. OI
and SiIV) but others present similar strength (e.g. SiII and CII) it is
therefore delicate to conclude on this aspect.
## 7 Discussion
### 7.1 Flag system and flux calibration
As mentioned in Sect. 2 we did not take into account the flag system when
selecting our galaxies. However we checked if our result are impacted by the
presence of lower quality redshift measurements that could potentially
indicate presence of low-redshift interlopers. We removed the redshift flags 1
and 9 and results are reported in Table 2 and displayed in Fig. 7. The only
notable difference is the last point that is at a slightly lower redshift at
z=4.98 instead of z=5.15, indicating that the highest measured redshift have a
lower quality than the lower redshift sample. This is because the redshift is
often measured due to the presence of the Ly$\alpha$ which leads to a redshift
quality flag of 9. For the other points the changes in Tr(Ly$\alpha$) is less
than 0.01, which represent less than 2% of difference. We conclude that
including redshift flag 1, 2 and 9 have almost no impact on the global result
on our study.
As reported in Pentericci et al. (2018) the very bluest part of the VANDELS
spectra was suffering from a systematic mismatch with the broad-band
photometry available for the sources. The underlying cause for this is still
under investigation but for the moment (and at the time of DR1 and 2) we have
implemented an empirically derived correction to the spectra. This effect
could in principle be relevant for the objects belonging to the first redshift
bin. For this reason we repeated the same measurements in the two first bins
using uncorrected spectra and measure an IGM transmission of 0.55 at z=3.99
with a standard deviation of 0.16 and 0.51 at z=4.23 with a standard deviation
0.16 as well. This shows that the spectral corrections effects are negligible
and lower than 5%.
Figure 7: Intergalactic medium transmission Tr(Ly$\alpha$) as function of
redshift from different estimation. In red we show our final results, as
presented in Fig.5, in blue we show the evolution of Tr(Ly$\alpha$) for
galaxies with a redshift flag of 3 or 4 only and in black we show the results
out of the photometric fit only.
### 7.2 Discussion on the method
Another measurement we have done is the measurement of the IGM transmission
from the first pass photometric fitting performed in Sect. 4 (with free dust
extinction parameter). Results are reported in Fig. 7 and Tab. 2. As expected
the measurements are in strong disagreement with the results from the spectral
fit. The difference is on average -0.11 towards low Tr(Ly$\alpha$) values.
This difference can reach up to 0.24 at z=4.23. This shows that the use of
photometric data alone to constrain the IGM transmission is not efficient.
Photometric data are less numerous and we have access to less bands to
constrain the fit. Indeed, photometry provides us with a data-point every 500
or 1000Å . Spectroscopy, to the contrary, brings much more constraints with
one data-point every $\sim$4Å.
We finally test the difference in dust extinction estimation (E(B-V), using
SB) and Tr(Ly$\alpha$) if we do not fix the dust extinction during the
spectral fit. Leaving it free, the dust extinction that we measure increases
with respect to the photometric fitting of Sect.4. The measurement of E(B-V)
give 0.12, 0.13, 0.15 and 0.12 at z=3.99, 4.23, 4.59 and z=5.15. While we are
still in the dispersion, this corresponds to a change of 10% at z=4.05 and
more than 50% at z=5.15. When looking at Tr(Ly$\alpha$), the measurements is
slightly different from the main results of our paper. The first point at
z=3.99 remains the same while the second and third measurements are higher
when leaving the dust free with Tr(Ly$\alpha$)=0.51 at z=4.23 and
Tr(Ly$\alpha$)=0.44 at z=4.59. This makes a difference of between 0.03 and
0.02, respectively. The strongest difference is for the last point where the
$\Delta$Tr(Ly$\alpha$)=0.05. These behavior is expected. If the dust content
goes toward higher E(B-V) values (i.e. more extinction), the IGM transmission
must compensate this extinction going toward higher Tr(Ly$\alpha$) values
(higher transmission). This behaviour was already noted in our previous study.
### 7.3 IGM template resampling
In the work presented previously we used 7 IGM transmission curves at any
redshift. We want to know now if the sampling of our prescription has an
influence on the final measurements. To this aim, we created a new IGM
prescription, not composed of 7 possible transmissions, but of 31 transmission
templates. We keep the same range but add intermediate curves, each at
multiples of 0.1$\sigma$ from 0.1 to 1.5$\sigma$. The transmission curves at
z=4.0 can be seen in Fig.8 (top).
Figure 8: Top: Intergalactic medium transmission at z=4 in the case where we
finely sample the IGM templates and consider 31 transmission curves instead of
7. The blue curve shows the average M06 prescription and the grey region
locates the region where we measure Tr(Lyα). Bottom: Comparison of the
Tr(Ly$\alpha$) in the case where we use 7 curves or 31 curves.
Using this fine-sampled prescription we recompute the dust and the IGM
transmission using all the three dust prescription, the results are displayed
in Fig. 8 where we compare the results from the fit with the 7-curves
prescription and the ones where we use the 31-curves version of the IGM
prescription. This comparison shows that the difference is minimal. Using the
SB dust attenuation the use of both prescription has no effect and the
difference is less than 0.1%. For the two other prescription the main
difference is for the point at the lowest redshift where the difference
reaches 4% for the LMC and 5% for the MW prescription. We can conclude that
the prescription with 7-curves seems to be detailed enough and adding more
curves does not substantially change the results.
## 8 Conclusion
This paper reports the study of the intergalactic medium transmission
Tr(Ly$\alpha$) at z¿4. We measured the IGM transmission from the spectra of
281 galaxies coming from the VANDELS public survey carried out by the VIMOS
instrument at the VLT. Galaxies have been observed up to $\sim$80h, thus
providing unprecedented spectral depth. Using a previously published IGM
transmission prescription for template fitting studies we used the SPARTAN
fitting tool to compute the IGM transmission. We summarize our results below:
* •
In order to tackle the dust-IGM degeneracy that was discovered in a previous
study, we estimated first the dust content in our galaxies with a pure
photometric fitting technique. We estimated the mean E(B-V) at z¿4 and found
that it ranges from 0.11 at z=3.99 to 0.08 at z=5.15. These measurements are
similar to previous measurements reported in the literature.
* •
Using individual measurements of E(B-V) as constraint, we use the SPARTAN
software to perform the spectral fitting of our galaxies. From this fitting we
extract the values of Tr(Ly$\alpha$) at various redshift. It decreases from
Tr(Ly$\alpha$)=0.53 at z=3.99 to Tr(Ly$\alpha$)=0.28 at z=5.15. These results
match very well the measurements from QSOs study and from theoretical
predictions. This reinforces the fact that high redshift galaxies can be used
to estimate the IGM.
* •
Even more importantly the $1\sigma$ scatter of Tr(Ly$\alpha$) is large at any
redshift. It is higher than 0.1 and equivalent to the standard deviation
reported from QSOs data.
* •
As expected we find that the IGM transmission measurements are sensitive to
the choice of dust attenuation prescription.
* •
We test whether our results are sensitive to the redshift flag system in place
in VANDELS and find that the differences are minimal.
* •
Due to a lack of observational constraints, the measurements coming from a
pure photometric fitting are not able to reproduce the results from the
spectral fitting and from the literature.
* •
Finally, we compute the IGM transmission leaving the dust extinction as a free
parameter and we confirm the presence of dust/IGM degeneracy. In that case,
the dust extinction goes toward higher value that are in tension with measure
from the literature. This is then compensated by an higher IGM transmission.
Finally, it is worth reminding that they are multiple IGM models in the
literature and they should be tested against real data. High-redshift data
samples are getting big enough to make statistically significant tests of
these models. This will be studied in a paper in preparation.
###### Acknowledgements.
The authors wish to thank the referee we provided us with very insightful
comments that improved a lot the paper.
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## Appendix A Reproducibility
The reproducibility aspect had become a crucial aspect of modern research with
the use of softwares and codes. Sharing codes and method in paper is as
important as sharing results. In this appendix we aim at answering this
aspect. Table 3 lists the availability of the data-related and technique-
related aspects of our work. Each point is detailed in the next paragraph.
Table 3: Summary of the reproducibility of this work | Public | Partial | Private
---|---|---|---
VANDELS Data | $\surd$ | $\chi$ | $\chi$
SPARTAN-tool | $\surd$ | $\chi$ | $\chi$
Spectral measurements | $\surd$ | $\chi$ | $\chi$
Results | $\surd$ | $\chi$ | $\chi$
Plotting tool: Photon | $\surd$ | $\chi$ | $\chi$
fits file library: dfitspy | $\surd$ | $\chi$ | $\chi$
* •
As presented in Sec. 2, the VANDELS survey is a public spectroscopic survey.
As such all the data are already publicly available and freely available from
the ESO archive facility333http://archive.eso.org/cms.html.
* •
The SPARTAN-tool is available on GITHUB and comes with all the inputs needed
to make the code run. The version released at this moment allows separate fit
on the photometry and on the spectroscopy, as used in this paper. The version
used in this paper is version 0.4.4444https://astrom-
tom.github.io/SPARTAN/build/html/index.html. The final version will be
presented in a paper in preparation (Thomas et al, in prep).
* •
In Addition, the main python packages used during this work are public:
catalog query module catscii (v1.2, Thomas 2019b), catalogue matching
algorithm catmatch (v1.3 Thomas 2019a), our fits display library dfitspy
(v19.3.4, Thomas 2019), the spectrum stacking program specstack (v19.4, Thomas
2019b) and our plotting tool, Photon (v0.3.2, Thomas 2019a). They are all
available in the main python package index repository (pypi).
|
# Series expansions for the Riemann zeta function
Alexey Kuznetsov Department of Mathematics and Statistics, York University,
Toronto, Ontario, M3J1P3, Canada<EMAIL_ADDRESS>
###### Abstract.
We prove a general result on representing the Riemann zeta function as a
convergent infinite series in a complex vertical strip containing the critical
line. This result can be used to re-derive known expansions as well as to
discover new series representations of the Riemann zeta function in terms of
the incomplete gamma functions, generalized hypergeometric functions and
Meijer $G$-functions.
Research supported by the Natural Sciences and Engineering Research Council of
Canada.
_To the memory of Richard Paris_
Keywords: Riemann zeta function, series expansion, Mellin transform,
incomplete gamma function, generalized hypergeometric function, Meijer
$G$-function 2020 Mathematics Subject Classification : 11M06, 41A58
## 1\. Introduction
The asymptotics of the Riemann zeta function on the critical line was a topic
that greatly interested Richard Paris. He wrote the first paper [8] on this
subject in 1994. This was followed by a series of papers [9, 10, 13, 14] in
the 1990s, two of them co-authored with his PhD student Shuang Cang. Richard
returned to this topic in 2009 [11] and, very recently, in 2022 [12]. To
explain the origin of the motivation for his work on the Riemann zeta
function, we need to start with the Riemann-Siegel formula (see [15][Section
4.17]):
(1) $\displaystyle
Z(t):=e^{{\textnormal{i}}\theta(t)}\zeta\big{(}\tfrac{1}{2}+{\textnormal{i}}t\big{)}$
$\displaystyle=2\sum\limits_{n=1}^{N_{t}}\frac{\cos(\theta(t)-t\ln(n))}{\sqrt{n}}$
$\displaystyle+(-1)^{N_{t}-1}\Big{(}\frac{2\pi}{t}\Big{)}^{\frac{1}{4}}\frac{\cos(\pi\tau^{2}/2+3\pi/8)}{\cos(\pi\tau)}+O(t^{-3/4}),\;\;\;t\to+\infty.$
Here $\zeta(s)$ is the Riemann zeta function, $\theta$ is defined as
$\theta(t):=\Big{[}\frac{\Gamma(1/4+{\textnormal{i}}t/2)}{\Gamma(1/4-{\textnormal{i}}t/2)}\Big{]}^{1/2}\pi^{-{\textnormal{i}}t/2},$
and $N_{t}:=\lfloor\sqrt{t/(2\pi)}\rfloor$ and
$\tau:=1+2(N_{t}-\sqrt{t/(2\pi)})$. The function $Z(t)$ is even, it is real
for real values of $t$ and satisfies $|Z(t)|=|\zeta(1/2+{\textnormal{i}}t)|$,
which makes it a convenient tool for detecting zeros of $\zeta(s)$ on the
critical line $\textnormal{Re}(s)=1/2$. The sum in (1) is the leading term in
the asymptotic expansion of $Z(t)$ (it is often called “the main sum”). We
also included in (1) the first order correction term, which has asymptotic
order $O(t^{-1/4})$. The higher order correction terms can also be given
explicitly, see [15][Theorem 4.16]. In applications we are usually interested
to compute $Z(t)$ for $t$ very large, where most computational effort is spent
in evaluating the main sum in (1). See [2] for results of such computations
for $t$ as large as $10^{34}$. For such extremely large values of $t$ the
error term $O(t^{-3/4})$ is already very small. Gabcke in his PhD Thesis [3]
proved that this error term is in fact less than $0.127t^{-3/4}$ for all
$t\geq 200$ and he also provided explicit bounds for higher order error terms.
An important feature of the Riemann-Siegel formula (1) is that the main sum,
which is the leading asymptotic term, is a discontinuous function of $t$. The
discontinuity in the main sum in (1) carries over to a discontinuity in the
asymptotic correction term. One would hope that finding asymptotic
approximations to $Z(t)$ where the leading term is a smooth function of $t$
will result in smaller correction terms over wider range of values of $t$.
Berry and Keating [1] in 1992 were the first to develop a “smoothed” version
of the Riemann-Siegel formula. They derived an asymptotic approximation to
$Z(t)$, where the leading term is given by a convergent series
(2) $Z_{0}(t,K)=2\textnormal{Re}\sum\limits_{n\geq
1}\frac{e^{{\textnormal{i}}(\theta(t)-t\ln(n))}}{\sqrt{n}}\times\frac{1}{2}{\textnormal{erfc}}\Big{(}\frac{\xi(n,t)}{q(K,t)}\sqrt{\frac{t}{2}}\Big{)}.$
In the above formula $\xi(n,t):=\ln(n)-\theta^{\prime}(t)$,
$q^{2}(K,t):=K^{2}-{\textnormal{i}}t\theta^{\prime\prime}(t)$ and
${\textnormal{erfc}}(x)$ is the complementary error function. The parameter
$K>0$ in (2) can be chosen freely. For large $t$ we have
$\xi(n,t)\approx\ln(n/N_{t})$ and $q^{2}(K,t)\approx
K^{2}-{\textnormal{i}}/2$, thus $Z_{0}(t,K)$ resembles the main sum in (1) but
with the complementary error function smoothing the sharp cut-off at
$n=N_{t}$. As Berry and Keating show in [1], in the infinite series in (2)
only the terms with $n\leq N_{t}+AK$ are important (the terms with larger $n$
are much smaller). This clarifies the meaning of the parameter $K$: it is
proportional to the number of terms by which the truncation in the Riemann-
Siegel formula has been smoothed. Berry and Keating also show that the leading
approximation term $Z_{0}(t,K)$ contains the first correction term shown in
(1) and that numerically $Z_{0}(t,K)$ gives a better approximation to $Z(t)$
for large $t$ when compared with the main sum in (1).
Following the paper by Berry and Keating, Richard Paris (together with his PhD
student Shuang Cang) developed several other asymptotic approximations to
$Z(t)$ and investigated their properties. In [8] an asymptotic formula for
$Z(t)$ was derived by applying the Poisson summation to the tail of the
Dirichlet series of $\zeta(s)$ for $\textnormal{Re}(s)>1$. In [13] Paris and
Cang developed another approximation to $Z(t)$, starting with the following
formula
(3)
$Z(t)=2\textnormal{Re}\;e^{{\textnormal{i}}\theta(t)}\Big{[}\sum\limits_{n\geq
1}n^{-s}Q(s/2,\pi{\textnormal{i}}n^{2})-\frac{\pi^{s/2}e^{\pi{\textnormal{i}}s/4}}{s\Gamma(s/2)}\Big{]},\;\;\;s=1/2+{\textnormal{i}}t,$
where $Q(s,x):=\Gamma(s,x)/\Gamma(s)$ is the normalized incomplete gamma
function. Using the uniform asymptotic results for $Q(s,x)$, they found that
this new approximation was more accurate than the Riemann-Siegel formula, and
that it required little additional computational effort. Two generalizations
of this approximation were studied in [9, 11, 14].
Our goal in this paper is to state and prove a general result (Theorem 1
below), which allows one to derive convergent series representations for the
Riemann zeta function in a vertical strip containing the critical line. This
result and its proof were inspired by the method used by Berry and Keating in
[1]. We show that formula (2) is a special case of our general theorem. We
also derive several new series representations for the Riemann zeta function,
given in in terms of the incomplete Gamma functions, generalized
hypergeometric functions and Meijer $G$-functions.
## 2\. Results
For $a\in{\mathbb{R}}$ and $b<\pi/2$ we denote
(4)
${\mathcal{D}}_{a,b}:=\Big{\\{}(s,x)\in{\mathbb{C}}^{2}\;:\;\textnormal{Re}(s)>-a,\;x\neq
0,\;|{\textnormal{arg}}(x)|<\frac{\pi}{2}-b\Big{\\}}.$
The following theorem is our main result.
###### Theorem 1.
Let $g(z)$ be an odd function that is analytic everywhere in the vertical
strip $|\textnormal{Re}(z)|<a$ for some $a>1/2$, except for a simple pole at
$z=0$ with residue equal to one. We also assume that for some $C>0$ and
$b<\frac{\pi}{2}$ we have
(5) $|g(z)|<Ce^{b|\textnormal{Im}(z)|/2}\;\;{\textnormal{ for all $z$ with
}}\;|\textnormal{Re}(z)|<a\;{\textnormal{ and }}\;|\textnormal{Im}(z)|>1.$
Then the following statements are true:
* (i)
There exists a function $h(s,x)$, analytic in the domain ${\mathcal{D}}_{a,b}$
and having Mellin transform
(6) $\int_{0}^{\infty}h(s,x)x^{w-1}{\textnormal{d}}x=\Gamma(w)g(2w-s),$
where
$\frac{1}{2}\max(0,\textnormal{Re}(s))<\textnormal{Re}(w)<\frac{1}{2}(a+\textnormal{Re}(s))$.
* (ii)
For any $\tau\in{\mathbb{C}}\setminus\\{0\\}$ with
$|{\textnormal{arg}}(\tau)|<\frac{\pi}{2}-b$ and $s$ in the vertical strip
$1-a<\textnormal{Re}(s)<a$ we have
(7) $\displaystyle\pi^{-s/2}\Gamma(s/2)\zeta(s)$
$\displaystyle=\tau^{s/2}\Big{[}-g(s)+2\sum\limits_{n\geq 1}h(s,\pi\tau
n^{2})\Big{]}$ $\displaystyle+\tau^{(s-1)/2}\Big{[}-g(1-s)+2\sum\limits_{n\geq
1}h(1-s,\pi\tau^{-1}n^{2})\Big{]}.$
Both infinite series in (7) congerve absolutely and uniformly in the strip
$1-a<\textnormal{Re}(s)<a$.
###### Proof.
For $s$ with $\textnormal{Re}(s)>-a$ we denote by $I_{s}$ the interval
$(\frac{1}{2}\max(0,\textnormal{Re}(s)),\frac{1}{2}(a+\textnormal{Re}(s))\subset{\mathbb{R}}$.
For $(s,x)\in{\mathcal{D}}_{a,b}$ and $\gamma\in I_{s}$ we define
(8)
$h_{\gamma}(s,x):=\frac{1}{2\pi{\textnormal{i}}}\int_{\gamma-{\textnormal{i}}\infty}^{\gamma+{\textnormal{i}}\infty}\Gamma(w)g(2w-s)x^{-w}{\textnormal{d}}w.$
First, let us prove that the above integral converges, so that $h(s,x)$ is
indeed well defined. The condition $\gamma\in I_{s}$ implies $\gamma>0$ and
$2\gamma-\textnormal{Re}(s)\in(0,a)$. This fact combined with our assumption
that $g(z)$ is analytic in the strip $0<\textnormal{Re}(z)<a$ show that the
integrand $w\mapsto\Gamma(w)g(2w-s)x^{-w}$ has no singularities on the line of
integration $w\in\gamma+{\textnormal{i}}{\mathbb{R}}$. From formula 8.328.1 in
[4] we have the following asymptotic result for the gamma function: for
$z=x+{\textnormal{i}}y$
(9) $\Gamma(z)\approx\sqrt{2\pi}|y|^{x-1/2}e^{-\pi|y|/2},\;\;\;y\to\infty.$
This result and our assumption (5) imply that there exists a constant
$C_{1}=C_{1}(\gamma,s)>0$ such that for $w=\gamma+{\textnormal{i}}y$ and
$y\in{\mathbb{R}}$ we have
$|\Gamma(w)g(2w-s)|<C_{1}e^{-(\pi/2-b)|y|}(1+|y|)^{\gamma-1/2},\;\;\;y\to\infty.$
From the above asymptotic result (10) and the identity
$|x^{-w}|=|x|^{-\gamma}e^{\arg(x)y},\;\;\;w=\gamma+{\textnormal{i}}y,$
we conclude that the integrand in (8) is bounded by
(10) $|\Gamma(w)g(2w-s)x^{-w}|\leq
C_{1}|x|^{-\gamma}e^{-(\pi/2-b)|y|+\arg(x)y}(1+|y|)^{\gamma-1/2},$
and this (as a function of $y\in{\mathbb{R}}$) is integrable as long as
$(s,x)\in{\mathcal{D}}_{a,b}$. Thus $h_{\gamma}(s,x)$ is well-defined for
$(s,x)\in{\mathcal{D}}_{a,b}$ and $\gamma\in I_{s}$. The estimate (10) also
implies the following result, which we will need later: for any small
$\epsilon>0$, $s$ in the half-plane $\textnormal{Re}(s)>-a$ and $\gamma\in
I_{s}$, there exists $C_{2}=C_{2}(\epsilon,s,\gamma)$ such that
$|h_{\gamma}(s,x)|<C_{2}|x|^{-\gamma}$ for all $x$ with
$|\arg(x)|<\pi/2-b-\epsilon$.
Next, let us fix $(s,x)\in{\mathcal{D}}_{a,b}$. The estimate (10) shows that
the integrand decays exponentially fast as $\textnormal{Im}(w)\to\infty$,
uniformly in the vertical strip $\textnormal{Re}(w)\in I_{s}$. Thus, by
shifting the contour of integration in this strip, we see that the function
$\gamma\mapsto h_{\gamma}(s,x)$ is constant in the interval $I_{s}$ and the
functions $\\{h_{\gamma}(s,x)\\}_{\gamma\in I_{s}}$ are analytic continuations
of a single function $h(s,x)$. For any fixed $\gamma>0$, the formula (8)
defines this function $h(s,x)$ in a domain
$\Big{\\{}(s,x)\in{\mathbb{C}}^{2}\;:\;2\gamma-a<\textnormal{Re}(s)<2\gamma,\;x\neq
0,\;|{\textnormal{arg}}(x)|<\frac{\pi}{2}-b\Big{\\}}\subset{\mathcal{D}}_{a,b}.$
This is true since the condition $\gamma\in I_{s}$ is equivalent to
$2\gamma-a<\textnormal{Re}(s)<2\gamma$. As $\gamma$ ranges through all
positive numbers, the union of these above smaller domains gives us the entire
domain ${\mathcal{D}}_{a,b}$, therefore the function $h(s,x)$ is analytic in
${\mathcal{D}}_{a,b}$.
As we have established above, for any fixed $s$ with $\textnormal{Re}(s)>-a$
and any $\gamma\in I_{s}$, there exists $C_{2}=C_{2}(s,\gamma)$ such that
$|h(s,x)|<C_{2}|x|^{-\gamma}$ for all $x\in(0,\infty)$. This implies that the
integral in the left-hand side of (6) converges for all $w$ in the vertical
strip
$\frac{1}{2}\max(0,\textnormal{Re}(s))<\textnormal{Re}(w)<\frac{1}{2}(a+\textnormal{Re}(s))$,
and now identity (6) follows by Mellin inversion formula. This ends the proof
of part (i).
Next, we denote $G(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$. It is well known [15]
that $G$ is analytic in ${\mathbb{C}}$, except for two simple poles at $s=0$
and $s=1$ with the corresponding residues $-1$ and $1$ and that $G$ satisfies
the functional equation $G(s)=G(1-s)$. It is also known that as
$\textnormal{Im}(s)\to\infty$ in any vertical strip
$|\textnormal{Re}(s-1/2)|<A$ the function $|\zeta(s)|$ is bounded by $|s|^{B}$
for some $B=B(A)>0$ (in fact, one can take $B=1/6+A$, see [15][Chapter V]).
Combining this estimate with (9) we conclude that for any $A>0$ there exists
$B_{2}=B_{2}(A)>0$ such that
(11)
$|G(s)|=O\Big{(}|s|^{B_{2}}e^{-\frac{\pi}{4}|\textnormal{Im}(s)|}\Big{)},\;\;\;\textnormal{Im}(s)\to\infty,$
uniformly in the strip $|\textnormal{Re}(s)-1/2|<A$.
Let us fix $s\notin\\{0,1\\}$ and $\tau\in{\mathbb{C}}\setminus\\{0\\}$,
satisfying $1-a<\textnormal{Re}(s)<a$ and $|\arg(\tau)|<\pi/2-b$, and take any
$u>|\textnormal{Im}(s)|$ and $c>0$ such that
$\max(\textnormal{Re}(s),1-\textnormal{Re}(s))<c<a$. We define a contour,
traversed counter-clockwise
$R_{c,u}:=(c-{\textnormal{i}}u,c+{\textnormal{i}}u]\cup(c+{\textnormal{i}}u,-c+{\textnormal{i}}u]\cup(-c+{\textnormal{i}}u,-c-{\textnormal{i}}u]\cup(-c-{\textnormal{i}}u,c-{\textnormal{i}}u].$
It is clear that $R_{c,u}$ is the boundary of a rectangle built on vertices
$\pm c\pm{\textnormal{i}}u$. Now we consider the following integral
(12)
$F(s):=\frac{1}{2\pi{\textnormal{i}}}\int_{R_{c,u}}G(s+z)g(z)\tau^{-\frac{z}{2}}{\textnormal{d}}z.$
By construction, the contour $R_{c,u}$ contains the three points $0$, $-s$ and
$1-s$, which are the simple poles of the integrand. Applying Cauchy’s Theorem
we obtain
(13) $F(s)=G(s)-\tau^{s/2}g(-s)+\tau^{(s-1)/2}g(1-s).$
Using our assumptions on $\arg(\tau)$ and $g(z)$ (see (5)) and upper bound
(11) we conclude that the function
$z\mapsto G(s+z)g(z)\tau^{-\frac{z}{2}}$
converges to zero exponentially fast as $\textnormal{Im}(z)\to\pm\infty$,
uniformly in the strip $\textnormal{Re}(z)<a$. Thus we can take the limit in
(12) and (13) as $u\to+\infty$, and the integrals over the upper and lower
sides of the rectangle $R_{c,u}$ will converge to zero, giving us the
following result:
(14) $\displaystyle G(s)$ $\displaystyle=-\tau^{s/2}g(s)-\tau^{(1-s)/2}g(1-s)$
$\displaystyle+\frac{1}{2\pi{\textnormal{i}}}\int_{c-{\textnormal{i}}\infty}^{c+{\textnormal{i}}\infty}G(s+z)g(z)\tau^{-z/2}{\textnormal{d}}z+\frac{1}{2\pi{\textnormal{i}}}\int_{-c+{\textnormal{i}}\infty}^{-c-{\textnormal{i}}\infty}G(s+z)g(z)\tau^{-z/2}{\textnormal{d}}z.$
When deriving the above formula we also used the fact that $g(-s)=-g(s)$.
Since $G(s+z)=G(1-s-z)$, by changing the variable of integration $z\mapsto-z$
in the second integral in the right-hand side of (14) we obtain
(15) $G(s)=-\tau^{s/2}g(s)-\tau^{(1-s)/2}g(1-s)+H(s,\tau)+H(1-s,\tau^{-1}),$
where we denoted
$H(s,\tau):=\frac{1}{2\pi{\textnormal{i}}}\int_{c-{\textnormal{i}}\infty}^{c+{\textnormal{i}}\infty}f(s+z)g(z)\tau^{-z/2}{\textnormal{d}}z.$
The condition $c>1-\textnormal{Re}(s)$ (that we imposed above) allows us to
expand $\zeta(s+z)=\sum\limits_{n\geq 1}n^{-s-z}$ as an absolutely convergent
series (this is valid for all $z$ on the vertical line
$\textnormal{Re}(z)=c$). Applying Fubini’s theorem we have
(16) $\displaystyle H(s,\tau)$ $\displaystyle=\sum\limits_{n\geq
1}\frac{1}{2\pi{\textnormal{i}}}\int_{c-{\textnormal{i}}\infty}^{c+{\textnormal{i}}\infty}\pi^{-(s+z)/2}\Gamma((s+z)/2)g(z)n^{-s-z}\tau^{-z/2}{\textnormal{d}}z$
$\displaystyle=2\tau^{s/2}\sum\limits_{n\geq
1}\frac{1}{2\pi{\textnormal{i}}}\int_{c_{1}-{\textnormal{i}}\infty}^{c_{1}+{\textnormal{i}}\infty}\Gamma(w)g(2w-s)(\pi\tau
n^{2})^{-w}{\textnormal{d}}w=2\tau^{s/2}\sum\limits_{n\geq 1}h(s,\pi\tau
n^{2}).$
In the second step in the above computation we changed the variable of
integration $z\mapsto w=(s+z)/2$ and denoted
$c_{1}:=(c+\textnormal{Re}(s))/2$. Note that
$\max(\textnormal{Re}(s),1/2)<c_{1}<(a+\textnormal{Re}(s))/2$, which implies
$c_{1}\in I_{s}$ and justifies the last step in (16). Formulas (15) and (16)
give us the desired result (7).
$\sqcap\kern-8.0pt\hbox{$\sqcup$}$
As we show next, under stronger assumption on the function $g(s)$, the
function $h(s,x)$ can be given as an integral of the incomplete gamma
function.
###### Proposition 1.
Assume that the function $g$ satisfies the assumptions of Theorem 1 with some
$b<0$. Then there exists a function $f:(0,\infty)\mapsto{\mathbb{C}}$, defined
via the Mellin transform
(17)
$\int_{0}^{\infty}f(x)x^{w-1}{\textnormal{d}}x=wg(2w),\;\;\;-a/2<\textnormal{Re}(w)<a/2,$
which is analytic in the sector $|\arg(x)|<|b|$ and satisfies an identity
$f(x)=f(1/x)$ in this sector. We have an integral representation
(18)
$h(s,x)=x^{-s/2}\int_{0}^{\infty}\Gamma(s/2,v)f(x/v)v^{-1}{\textnormal{d}}v,\;\;\;\textnormal{Re}(s)>-a,\;|\arg(x)|<|b|.$
###### Proof.
For $x>0$ we define
(19)
$f(x)=\frac{1}{2\pi{\textnormal{i}}}\int_{\gamma-{\textnormal{i}}\infty}^{\gamma+{\textnormal{i}}\infty}wg(2w)x^{-w}{\textnormal{d}}w,$
where $|\gamma|<a/2$. Since we assumed that $b<0$, formula (5) tells us that
$|g(2w)|=O\big{(}e^{-|b|\times|\textnormal{Im}(w)|}\big{)}$
as $w\to\infty$ in the strip $|\textnormal{Re}(w)|<a/2$. Thus the integral in
(19) converges and $f(x)$ is well defined for $x>0$. Due to the exponential
decay of $g(2w)$, the function $f(x)$ is analytic in the sector
$|\arg(x)|<|b|$. The fact that the function $wg(2w)$ is even implies that
$f(x)=f(1/x)$ for all $x$ in the sector $|\arg(x)|<|b|$, and the fact that
$wg(2w)$ is analytic in the strip $|\textnormal{Re}(w)|<a/2$ implies that for
every $\epsilon>0$
(20) $f(x)=\begin{cases}O(x^{a/2-\epsilon}),\;\;\;&x\to 0,\\\
O(x^{-a/2+\epsilon}),\;\;\;&x\to\infty,\end{cases}$
and this asymptotics holds in the sector $|\arg(x)|<|b|$.
Now, according to (6), the function $x\mapsto h(s,x)$ has Mellin transform
$\Phi_{1}(w)\Phi_{2}(w)$, where we denoted
$\Phi_{1}(w):=\frac{\Gamma(w)}{w-s/2},\;\;\;\Phi_{2}(w):=(w-s/2)g(2w-s).$
Formula (6.455.1) from [4] implies
(21)
$\int_{0}^{\infty}x^{-\nu}\Gamma(\nu,x)x^{z-1}{\textnormal{d}}x=\frac{\Gamma(z)}{z-\nu},\;\;\;\textnormal{Re}(z)>0,\textnormal{Re}(z-\nu)>0.$
Therefore, the function $\Phi_{1}(w)$ is the Mellin transform of the function
$x\mapsto x^{-s/2}\Gamma(s/2,x)$, and this holds for
$\textnormal{Re}(w)>\max(0,\textnormal{Re}(s)/2)$. The function $\Phi_{2}(w)$
is the Mellin transform of $x^{-s/2}f(x)$, for $|\textnormal{Re}(w-s/2)|<a/2$.
The reader may check that the half-plane
$\textnormal{Re}(w)>\max(0,\textnormal{Re}(s)/2)$ intersectst with the
vertical strip $|\textnormal{Re}(w-s/2)|<a/2$ as long as
$\textnormal{Re}(s)>-a$. Thus we can use the Mellin convolution identity and
conclude that for $x>0$ and $\textnormal{Re}(s)>-a$
$h(s,x)=\int_{0}^{\infty}t^{-s/2-1}\big{(}x/t)^{-s/2}\Gamma(s/2,x/t)f(t){\textnormal{d}}t,$
and from this formula we obtain the desired result (18) by change of variables
$t=x/v$. Thus we have established formula (18) for $\textnormal{Re}(s)>-a$ and
$x>0$, and we can extend it to the sector $|\arg(x)|<|b|$ using the
analyticity and asymptotic properties of $f(x)$ that we established above.
$\sqcap\kern-8.0pt\hbox{$\sqcup$}$
Next we present five applications of the above results. Theorem 1 is intended
to be used in the following way. We start with a function $g$ satisfying the
necessary assumptions and then we try to identify a function $h(s,x)$ via its
Mellin transform (6). Sometimes we can identify $h(s,x)$ by finding the
desired Mellin transform pair in a table of integral transforms, such as [4]
or [7]. This is the method we use in examples 1, 3 and 5 below. When the
desired Mellin transform pair can not be found, we may attempt to find an
integral or a series representation for $h(s,x)$ – this is the approach we
follow in examples 2 and 4.
### Example 1
Take $g(z)=1/z$. This function satisfies conditions of Theorem 1 with $b=0$
and $a=+\infty$. Comparing formula (21) with the Mellin transform equation (6)
that defines $h(s,x)$, we find
$h(s,x)=\frac{1}{2}x^{-s/2}\Gamma(s/2,x),$
and applying Theorem 1 we obtain the following result, valid for for
$s\in{\mathbb{C}}$ and $|\textnormal{arg}(\tau)|<\pi/2$:
(22) $\displaystyle\pi^{-s/2}\Gamma(s/2)\zeta(s)$
$\displaystyle=-\frac{\tau^{s/2}}{s}+\pi^{-s/2}\sum\limits_{n\geq
1}n^{-s}\Gamma(s/2,\pi\tau n^{2})$
$\displaystyle+\frac{\tau^{(s-1)/2}}{s-1}+\pi^{(s-1)/2}\sum\limits_{n\geq
1}n^{s-1}\Gamma((1-s)/2,\pi\tau^{-1}n^{2}).$
The above formula is well known. A special case of this formula with $\tau=1$
was used by Riemann to prove the functional equation for $\zeta(s)$ (see
[15][Section 2.6]). In the limit as $\tau\to{\textnormal{i}}$ this formula
gives us (3).
### Example 2
Now we take $g(z)=e^{\alpha z^{2}}/z$. Conditions of Theorem 1 are satisfied
with $a=+\infty$ and any $b<0$. Berry and Keating [1] used this function when
deriving the asymptotic term $Z_{0}(t,K)$ in (2), though their method was
slightly different. The difference lies in formula (12), where we used a
function $G(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$ and Berry and Keating used
$Z({\textnormal{i}}(s-1/2))$. For this choice of $g(z)$ it seems that there
are no explicit representations for $h(s,x)$, however Proposition 1 provides a
simple integral representation. Performing a change of variables $x=e^{y}$ and
using the Gaussian integral one can easily check that the function $f$ defined
by the Mellin transform identity (19) is given by
$f(x)=\frac{1}{8\sqrt{\pi\alpha}}\exp\Big{(}-\frac{1}{16\alpha}\ln(x)^{2}\Big{)},\;\;\;x>0.$
Note that the function $2f(x)/x$ is the probability density function of a
random variable $\xi$ having lognormal distribution with parameters $\mu=0$
and $\sigma^{2}=8\alpha$. Therefore, formula (18) gives us the following
expression
$h(s,x)=\frac{1}{2}x^{-s/2}{\mathbb{E}}\big{[}\Gamma\big{(}s/2,xe^{\sqrt{8\alpha}X}\big{)}\big{]},$
where $X$ is a standard normal random variable.
### Example 3
For our third example we choose
$g(z)=\frac{\pi/2}{\sin(\pi z/2)}.$
This function satisfies conditions of Theorem 1 with $b=-\pi$ and $a=2$.
Formula 5.26 from [7] states that
$\int_{0}^{\infty}e^{x}\Gamma(\nu,x)x^{z-1}{\textnormal{d}}x=\frac{\pi\Gamma(z)}{\Gamma(1-\nu)\sin(\pi(\nu+z))},\;\;\;0<\textnormal{Re}(z)<1-\textnormal{Re}(\nu).$
From the above formula and the Mellin transform identity (6) we find
$h(s,x)=\frac{1}{2}\Gamma(1+s/2)e^{x}\Gamma(-s/2,x).$
We recall that the normalized incomplete gamma function is
$Q(s,x):=\Gamma(s,x)/\Gamma(s)$. Applying the reflection formula for the gamma
function we can write
$h(s,x)=-\frac{\pi/2}{\sin(\pi s/2)}e^{x}Q(-s/2,x).$
Now Theorem 1 gives us the following result, which seems to be new:
###### Corollary 1.
For $-1<\textnormal{Re}(s)<2$ and $|\textnormal{arg}(\tau)|<3\pi/2$
$\displaystyle\pi^{-s/2}\Gamma(s/2)\zeta(s)$
$\displaystyle=-\frac{\pi\tau^{s/2}}{\sin(\pi
s/2)}\Big{[}\frac{1}{2}+\sum\limits_{n\geq 1}e^{\pi\tau n^{2}}Q(-s/2,\pi\tau
n^{2})\Big{]}$ $\displaystyle-\frac{\pi\tau^{(s-1)/2}}{\cos(\pi
s/2)}\Big{[}\frac{1}{2}+\sum\limits_{n\geq
1}e^{\pi\tau^{-1}n^{2}}Q((s-1)/2,\pi\tau^{-1}n^{2})\Big{]}.$
### Example 4
Example 3 can be extended in the following way. Let $r=p/q$ be a positive
rational number and define
$g(z)=\frac{\pi r}{\sin(\pi rz)}.$
Now conditions of Theorem 1 are satisfied with $b=-2\pi r$ and $a=1/r$. We
claim that in this case $h(s,x)$ can be computed as a sum of two infinite
power series:
(23) $\displaystyle h(s,x)$ $\displaystyle=-\pi r\sum\limits_{n\geq
0}\frac{(-1)^{n}}{\sin(\pi r(2n+s)}\times\frac{x^{n}}{n!}$
$\displaystyle+\frac{1}{2}\pi x^{-s/2}\sum\limits_{m\geq
0}\frac{(-1)^{m}}{\sin(\pi(s-m/r)/2)}\times\frac{x^{m/(2r)}}{\Gamma(1-s/2+m/(2r))}.$
We will sketch the proof of formula (23) (the main ideas and details are very
similar to the proof of Corollary 3.16 in [5]). To establish formula (23), we
start by writing $h(s,x)$ in the form (8) :
(24)
$h(s,x)=\frac{r}{2{\textnormal{i}}}\int_{\gamma-{\textnormal{i}}\infty}^{\gamma+{\textnormal{i}}\infty}\Phi(w){\textnormal{d}}w,\;\;\;{\textnormal{
where }}\;\Phi(w)=\frac{\Gamma(w)x^{-w}}{\sin(\pi r(2w-s))}.$
We take $s$ to be a fixed number in the vertical strip
$-1/(2p)<\textnormal{Re}(s)<0$ and $\gamma$ any number in the interval
$(0,1/(4p))$. The reader may want to recall the definition of the interval
$I_{s}$ in the proof of Theorem 1, to see that this choice of $\gamma$ is
legitimate. The gamma function in the integrand in (24) has simple poles at
points $-n$, $n=0,1,2,\dots$ and the function $\sin(\pi r(2w-s))^{-1}$ has
simple poles at points $(1/2)(s-m/r)$, $m\in{\mathbb{Z}}$. Due to our
restriction on $s$, these two sets do not intersect, thus all the poles of the
function $\Phi(w)$ are simple. The periodic structure of the poles of $\Phi$
implies that we can find a sequence of $\\{\gamma_{k}\\}_{k\geq 1}$ decreasing
to $-\infty$ such that each point $\gamma_{k}$ satisfies $\gamma_{k}<\gamma$
and has distance at least $1/(8p)$ from each pole of $\Phi$.
Now we perform the standard operation of shifting the contour of integration
in (24) from $\gamma+{\textnormal{i}}{\mathbb{R}}$ to
$\gamma_{k}+{\textnormal{i}}{\mathbb{R}}$:
(25) $\displaystyle h(s,x)$ $\displaystyle=\pi
r\sum{\textnormal{Res}}\big{(}\Phi(w),w=-n\big{)}$ $\displaystyle+\pi
r\sum{\textnormal{Res}}\Big{(}\Phi(w),w=\frac{1}{2}\Big{(}s-\frac{m}{r}\Big{)}\Big{)}+\frac{r}{2{\textnormal{i}}}\int_{\gamma_{k}-{\textnormal{i}}\infty}^{\gamma_{k}+{\textnormal{i}}\infty}\Phi(w){\textnormal{d}}w,$
where the summation in the first sum is over all $n$ such that
$\gamma_{k}\leq-n\leq 0$ and the summation in the second sum is over all $m$
such that $\gamma_{k}\leq(s-m/r)/2\leq s/2$. The residues in (25) give us the
coefficients in the two infinite series in (23). The integral in the right-
hand side of the (25) converges to zero as $k\to+\infty$. This is true since
the function $x^{-w}\Gamma(w)$ resricted to the vertical line
$\textnormal{Re}(w)=\gamma_{k}$ converges to zero uniformly in
$\textnormal{Im}(w)$ as $k\to+\infty$ and the term $1/\sin(\pi r(2w-s))$ is
bounded on line $\textnormal{Re}(w)=\gamma_{k}$. Here our choice of
$\gamma_{k}$ is crucial – we need to ensure that the line of integration
$w\in\gamma_{k}+{\textnormal{i}}{\mathbb{R}}$ stays away from the poles of the
integrand. To summarize, in the limit as $k\to+\infty$, formula (25) gives us
the desired result (23).
The infinite series representation (23) is in fact valid for almost all
irrational $r$. This can be established in the same way as Corollary 3.16 in
[5]. However, for rational $r=p/q$ the convergence of both series in (23) is
obvious and for small $p$ and $q$ these series can be simplified further. We
already saw that in the case when $r=1/2$ (our example 3) we could express
$h(s,x)$ in terms of a normalized incomplete gamma function. Let us consider
now the case with $r=1/4$. By separating the terms with $n$ even from the
terms with $n$ odd in the first sum in (23) and applying reflection formula
for the gamma function in the second sum we obtain
(26) $\displaystyle h(s,x)$
$\displaystyle=-\frac{\pi}{4}\frac{\cos(x)}{\sin(\pi
s/4)}+\frac{\pi}{4}\frac{\sin(x)}{\cos(\pi s/4)}$
$\displaystyle+\frac{1}{2}\Gamma(s/2)x^{-s/2}{}_{1}F_{2}(1;1/2-s/4,1-s/4;-x^{2}/4).$
### Example 5
There is another form of $g(z)$ that can lead to somewhat explicit expressions
for $h(s,x)$. We take
(27)
$g(z)=\frac{A}{z}\times\frac{\prod\limits_{j=1}^{n}\Gamma(\alpha_{j}+z/2)\Gamma(\alpha_{j}-z/2)}{\prod\limits_{j=1}^{m}\Gamma(\beta_{j}+z/2)\Gamma(\beta_{j}-z/2)}$
where the normalizing constant $C$ is given by
$A=\prod\limits_{j=1}^{n}\Gamma(\alpha_{j})^{-2}\prod\limits_{j=1}^{m}\Gamma(\beta_{j})^{2}.$
If $n\geq m$ and the constants $\beta_{j}$ have positive real part and
$\alpha_{j}$ satisfy $\textnormal{Re}(\alpha_{j})>1/4$, then $g(z)$ given in
(27) satisfies the assumptions in Theorem 1 with
$a=2\times\min\\{\textnormal{Re}(\alpha_{j})\;:\;1\leq j\leq
n\\}\;\;{\textnormal{ and }}\;\;b=\pi(m-n).$
For function $g$ given by (27), the function $h(s,x)$ defined via (8) can be
expressed in terms of the Meijer $G$-function (see [6])
(28) $h(s,x)=A\times G_{p,q}^{n+2,n}\Big{(}\begin{matrix}{\bf a}-s/2\\\ 0,{\bf
b}-s/2\end{matrix}\Big{|}x\Big{)},$
where $p=m+n+1$, $q=m+n+2$ and
${\bf
a}:=[1-\alpha_{1},\dots,1-\alpha_{n},\beta_{1},\dots,\beta_{m},0],\;\;\;{\bf
b}:=[1,\alpha_{1},\dots,\alpha_{n},1-\beta_{1},\dots,1-\beta_{m}].$
Note that our example 3, where $h(s,x)$ is given in terms of the incomplete
gamma function, can be obtained from (27) by setting $n=\alpha_{1}=1$ and
$m=0$. It is possible that other choices of parameters may also lead to simple
expressions for $h(s,x)$, however we did not pursue this further.
## References
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* [3] W. Gabcke. Neue herleitung und explizite restabschätzung der Riemann-Siegel-formel. PhD Thesis, Georg-August-Universität zu Göttingen, 1979. http://dx.doi.org/10.53846/goediss-5113.
* [4] I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, seventh edition, 2007.
* [5] A. Kuznetsov, A. Kyprianou, J. C. Pardo, and A. Watson. The hitting time of zero for a stable process. Electronic Journal of Probability, 19:1 – 26, 2014. https://doi.org/10.1214/EJP.v19-2647.
* [6] A. M. Mathai and R. K. Saxena. Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Lecture Notes in Mathematics. Springer Berlin, Heidelberg, 1973. https://doi.org/10.1007/BFb0060468.
* [7] F. Oberhettinger. Tables of Mellin transform. Springer-Verlag, 1974.
* [8] R. B. Paris. An asymptotic representation for the Riemann zeta function on the critical line. Proc. R. Soc. London, 446:565–587, 1994. https://doi.org/10.1098/rspa.1994.0121.
* [9] R. B. Paris. A generalisation of Lavrik’s expansion for the Riemann zeta function. Technical Report MACS 94:01, University of Abertay Dundee, 1994\.
* [10] R. B. Paris. New asymptotic formulas for the Riemann zeta function on the critical line. Special Functions, proceedings of the international workshop, editors C. Dunkl, M. Ismail and R. Wong, pages 247–261, 2000. https://doi.org/10.1142/9789812792303_0020.
* [11] R. B. Paris. A generalisation of an expansion for the Riemann zeta function involving incomplete gamma functions. Applied Mathematical Sciences, 3(60):2973–2984, 2009.
* [12] R. B. Paris. An asymptotic approximation for the Riemann zeta function revisited. preprint, 2022. https://arxiv.org/abs/2203.07863.
* [13] R. B. Paris and S. Cang. An asymptotic representation for $\zeta(\frac{1}{2}+it)$. Methods and Applications of Analysis, 4(4):449–470, 1997.
* [14] R. B. Paris and S. Cang. An exponentially-smoothed Gram-type formula for the Riemann zeta function. Methods and Applications of Analysis, 4(3):326–338, 1997.
* [15] E. C. Titchmarsh. The theory of the Riemann zeta-function. Oxford University Press, second edition, 1987.
|
# Targeted and Troublesome: Tracking and Advertising on Children’s Websites
WARNING: Contains potentially NSFW images
Zahra Moti Radboud University Asuman Senol imec-COSIC, KU Leuven Hamid
Bostani Radboud University Frederik Zuiderveen Borgesius Radboud University
Veelasha Moonsamy Ruhr University Bochum Arunesh Mathur Independent
Researcher Gunes Acar Radboud University
###### Abstract
On the modern web, trackers and advertisers frequently construct and monetize
users’ detailed behavioral profiles without consent. Despite various studies
on web tracking mechanisms and advertisements, there has been no rigorous
study focusing on websites targeted at children. To address this gap, we
present a measurement of tracking and (targeted) advertising on websites
directed at children. Motivated by the lack of a comprehensive list of child-
directed (i.e., targeted at children) websites, we first build a multilingual
classifier based on web page titles and descriptions. Applying this classifier
to over two million pages from the Common Crawl dataset, we compile a list of
two thousand child-directed websites. Crawling these sites from five vantage
points, we measure the prevalence of trackers, fingerprinting scripts, and
advertisements. Our crawler detects ads displayed on child-directed websites
and determines if ad targeting is enabled by scraping ad disclosure pages
whenever available. Our results show that around 90% of child-directed
websites embed one or more trackers, and about 27% contain targeted
advertisements—a practice that should require verifiable parental consent.
Next, we identify improper ads on child-directed websites by developing an ML
pipeline that processes both images and text extracted from ads. The pipeline
allows us to run semantic similarity queries for arbitrary search terms,
revealing ads that promote services related to dating, weight loss, and mental
health, as well as ads for sex toys and flirting chat services. Some of these
ads feature repulsive, sexually-explicit and highly-inappropriate imagery. In
summary, our findings indicate a trend of non-compliance with privacy
regulations and troubling ad safety practices among many advertisers and
child-directed websites. To ensure the protection of children and create a
safer online environment, regulators and stakeholders must adopt and enforce
more stringent measures. Keywords – online tracking, advertising, children,
privacy
## 1 Introduction
The proliferation of online tracking for analytics, behavioral advertising,
and marketing has resulted in over a decade’s worth of research into this
ecosystem. Prior research has shown that not only is online tracking rampant
on the web [1] but that trackers use increasingly-invasive tracking
mechanisms—e.g., third-party cookies, tracking pixels, evercookies, and
browser fingerprinting [2, 3, 4, 1]—to relentlessly build detailed profiles of
users across the web without any consent for targeted advertising.
Such privacy concerns aside, online advertising has shown to be problematic in
other ways. Ads and ad networks are a vector for distributing ransomware,
malicious programs, and cryptojackers—posing a serious security threat to
users [5, 6, 7, 8, 9, 10, 11, 12, 13]. Ad networks also suffer from click
fraud, which is estimated to reach $100 billion in 2023 [14, 15]. Finally,
online ads often contain clickbait, untrustworthy, or distasteful content that
peddle software downloads, listicles, and health supplements—all of which
users find problematic to their online experience [16].
While online tracking and targeted advertising pose a threat to users of all
ages, children especially bear an acute cost. Children may not fully
understand the consequences of online tracking and revealing their personal
data online [17, 18], but they yield immense “pester power” to influence their
parents’ purchase decisions [19]. Thus, children are an attractive target
audience for advertisers and marketers alike [19, 20], as they are more
vulnerable to persuasive advertising [21, 22, 23], and susceptible to harmful
content [24, 25].
Despite the aforementioned evidence that suggests a differential impact on
children, there is little empirical research on online tracking and
advertising practices on children’s websites. The lack of a comprehensive and
updated list of websites directed at children poses a major challenge for
studying children’s websites. Previous large-scale internet measurement
studies have relied on popular website lists such as Tranco [26] and Alexa
[27] (before it shut down in 2021 [28]), but these lists may not specify
website categories, and even when they do, the website categories may not be
reliable and comprehensive [29, 30]. As a result, prior work [31, 23] has only
examined online tracking on at most a hundred children’s websites and has been
restricted in scope and methods—lacking a comprehensive investigation of both
online tracking and advertising. To overcome this limitation, we built our own
repository of child-directed websites. We trained a text-based classifier that
detects children’s websites using HTML metadata fields such as <title> and
<description>. The classifier is based on a pre-trained multilingual model
that we fine-tuned for our binary classification task. Applying the classifier
to the Common Crawl dataset [32], we compiled a list of 2K manually verified
child-directed websites.
Figure 1: A sample of improper ads found on child-directed websites in our
crawls.
To study several online tracking, ad targeting, and problematic ad practices,
we crawl our list of 2K child-directed websites—varying the location (five
vantage points) and form factors (desktop & mobile). Starting with ad
targeting, we study the extent to which ads that appear on children’s websites
are targeted—a practice that has come under increasing scrutiny both in the EU
and the US [33, 34, 35]. We then present an exploratory analysis of ads from
categories deemed problematic for children, such as dating, weight loss,
mental health, and ads that contain racy content. Next, we turn to online
tracking, which is a necessary ingredient of behavioral advertising. We study
the ecosystem of trackers and specifically quantify the prevalence of
trackers, cookies, and use of browser fingerprinting techniques such as
Canvas, Canvas Font, AudioContext Fingerprinting, and WebRTC local IP
discovery [1].
Our work is especially pertinent in light of impending regulatory changes. In
the US, there have been calls [35] to update the Children’s Online Privacy
Protection Act (COPPA) [36] in order to prohibit “internet companies from
collecting personal information from users who are 13 to 16 years old without
their consent” and to “ban targeted advertising to children and teens.” The
current US President Joe Biden has called for a ban on collecting data on and
serving targeted ads to children [34]; whereas in the EU, the upcoming Digital
Services Act (DSA) will specifically prohibit ads targeted at children [33].
Our research seeks to offer empirical evidence on advertising and tracking
practices employed on children’s websites by making the following
contributions:
* •
Using a lightweight classifier based on web page metadata fields, we build a
repository of child-directed websites and crawl them to measure tracking and
advertising practices using multiple vantage points and form factors (desktop
& mobile).
* •
We measure targeted ads using two ad vendors’ (Google and Criteo) ad
disclosure statements, and find that targeting is enabled for 73% of the ads
we could measure.
* •
Using text and images extracted from the ads, we detect racy ads, and ads
about weight loss, dating, and mental health using semantic similarity search
based on a lightweight, multilingual language model. While this content
analysis is exploratory, our method enables human-in-the-loop investigations
with arbitrary queries, and it paves the way for the automatic content
analysis of ads.
* •
We also find ads linking to malicious content, improper ads of sex toys,
dating services, and ads containing sexually suggestive images (Figure 1).
All the data and software from our study will be made available to
researchers.111We share the list of identified child-directed websites and a
sample of advertisement disclosures on https://github.com/targeted-and-
troublesome/.
## 2 Related Work
### 2.1 Web tracking measurements
Over the past decade, several web privacy measurements have shown the scale
and complexity of online tracking [37, 1, 38, 39, 40]. Research on stateful
tracking has examined how unique tracking identifiers are stored on the client
side [41] using cookies [42, 43], localStorage [2], cache (ETags) [2], or
other client-side storage mechanisms. On the other hand, research on stateless
tracking has examined the use of fingerprinting, a mechanism that exploits
differences in browsers and devices to obtain a likely unique identifier [44].
Past research has shown that there are various fingerprinting vectors,
including fonts, clock skew, GPUs, audio hardware, installed writing scripts
and browser extensions, among others [45, 46, 47, 1, 48, 49, 50]. Research on
defense against fingerprinting has contributed methods to detect
fingerprinting, tracking and advertising [51, 4, 1, 38, 52, 53]. Our study
borrows heuristics from prior work [1, 38] to detect fingerprinting scripts,
and we use existing filter lists to identify trackers and advertisers.
### 2.2 Tracking & ads on children’s media
Motivated by the challenges posed by ads to children, Cai and Zhao [23]
manually labeled ads displayed on 117 children’s websites. They found that 68%
of the websites featured ads, and less than half complied with COPPA. The
authors also argued that children are unlikely to distinguish many ads from
the website’s original content. Vlajic et al. [31] investigated online
tracking on twenty websites from Alexa’s “Kids and Teens” category [27] from
two vantage points (EU & US). The authors manually analyzed the HTTP headers
and quantified hidden images (i.e., likely tracking pixels) loaded from ads
and analytics domains. Compared to this past work, we study orders of
magnitude more websites, follow more rigorous tracking measurement methods,
and compare results across different vantage points. Additionally, we
automatically detect targeted ads using ad disclosure pages and present an
exploratory analysis of the content of ads that appear on children’s websites.
Focusing on mobile platforms, Reyes et al. [54] dynamically analyzed around
6,000 free children’s Android apps and found that most apps violate COPPA due
to their use of third-party SDKs.
### 2.3 Improper and malicious ads
A recent line of research has investigated the content of online ads. Zeng et
al. [16] conducted a survey with 1,000 participants to determine the type of
advertising content (e.g., chumboxes, clickbait, political, and low-quality
content) that makes people dislike ads. In [55], the same authors also studied
problematic ads in the news and misinformation websites, where they found
problematic ads served by native ad platforms. In a study leading to the 2020
US elections, Zeng et al. [56] found that ads for misleading political polls
that aim to hoover email addresses are widely used in online political
advertising. Subramani et al. [7] studied the role of web push notifications
in ad delivery, especially malicious ads. Through a large-scale study of
malicious ads, Zarras et al. [5] showed that some ad exchanges are more prone
to serving malicious ads due to inadequate detection. Akgul et al. [57]
examined influencer VPN ads on YouTube and found advertisements disseminating
misleading claims about online safety. Ali et al. [58] measured how the
distribution of potentially harmful ads on Facebook varies across users.
Venkatadri et al. [59] used Facebook’s advertiser interface to study how
Facebook obtains personal identifiers used in advertising. In a concurrent
work, Zhao et al. [60] analyzed mobile ads aimed at children, uncovering
inappropriate advertisements by certified mobile ad SDKs. Medjkoune et al.
[61] showed that advertisers can target ads to children by placing their ads
in children-focused videos—bypassing YouTube’s age restrictions.
### 2.4 Ad transparency
In response to concerns about targeted advertising, ad networks and platforms
have offered ad transparency interfaces that allow users to ascertain when and
how they are being targeted. Andreou et al. [62] investigated Facebook’s ad
explanations and found that they are often incomplete or misleading.
Researchers have also argued that ad networks should provide users with
interpretable explanations and increase the visibility of disclosure
mechanisms [63].
Bin Musa and Nithyanand [64] developed ATOM, a technique for determining data
sharing between online trackers and advertisers. They used simulated personas
to crawl websites, collect ad images, and conduct statistical analyses to
identify correlations between tracker presence and advertiser behavior. Liu et
al. [65] developed a framework called AdReveal to investigate different
mechanisms used in online targeted advertising. Vallina et al. [66] used
statements found in Google’s ad disclosure pages in their crowdsourced
measurement of online behavioral advertisements. To detect stealthy ads that
aim to bypass adblockers, Storey et al. [67] developed an extension that
detects the AdChoices icon using perceptual hashing. While we considered
applying Storey et al.’s method, we found URL-based detection of ad disclosure
links (§4.6) to be more reliable and efficient.
### 2.5 Website categorization
The majority of studies on web categorization have focused on text-based
classifiers because most web content and metadata are text-based [68, 69].
Various studies used machine learning models such as BERT and recurrent neural
networks to learn contextual representations and features of web pages using
meta tags and body content [70, 69, 68]. Other researchers proposed image-
based web page classification techniques using pre-trained convolutional
neural networks and using Google image search results [71, 72]. In our work,
we built a lightweight classifier by fine-tuning an existing distilled
language model and using text-based website metadata to detect child-directed
websites.
## 3 Building a list of child-directed websites
Figure 2: Pipeline for building a list of child-directed websites.
It is estimated that there are more than one billion websites on the Internet
[73], but only a small fraction are targeted at children. A central challenge,
therefore, is identifying the websites that contain content directed to
children. We initially searched and found three curated lists of children’s
websites: kidSAFE Seal Program [74], CommonSense (filtered for children below
the age of 13) [75], and a list compiled by Kaspersky [76]. Unfortunately,
these lists contained only a total of 355 websites, some of which were no
longer online.
To expand our limited list, we experimented with web categorization services
such as McAfee, WebShrinker, and SimilarWeb, but decided to use VirusTotal’s
(academic) API because other services were either not freely available or did
not let us query in bulk. VirusTotal aggregates category labels from third-
party scanners and categorization services, including BitDefender and
TrendMicro [77]. We used the VirusTotal API to retrieve web category data for
the top one million websites from the Chrome User Experience Report (CrUX)
list from May 2022 [78]. We observed VirusTotal’s rate limits (20K/day/per
academic license) during the process, which took roughly four weeks. By
searching for substrings “kid” and “child” in the returned category labels and
removing false positives (such as “Child abuse”), we obtained 1,264 websites
categorized as related to children. However, our manual verification of these
websites following the criteria presented in Appendix A revealed that 68.6% of
them were false positives, yielding only 396 child-directed websites. Note
that the low accuracy and inconsistency of domain
classification/categorization services align with findings from prior work
[30]. Combining our initial 355 websites with our verified list of 396
websites and removing all inaccessible (5) and duplicate (164) websites, we
obtained a total of 582 child-directed websites.
Motivated by the lack of accurate, up-to-date, and comprehensive sources of
child-directed websites, we built a classifier to detect child-directed
websites using the list of 582 websites as labeled data. Figure 2 illustrates
the training and fine-tuning process. We define “child-directed websites” as
websites that are primarily intended for use by children, and contain content,
activities, or other features that are likely to be appealing to children (see
Appendix A, for our criteria). Note that our labeling criteria do not fully
overlap with COPPA’s definition [36]; for example, we do not require that
sites have the actual knowledge of collecting children’s data. Thus, we do not
claim to measure compliance with COPPA or other relevant laws. In our study,
children refer to individuals under 13, aligning with US and EU regulations
(§6.1).
### 3.1 Labeled data for ML classifier
Many web page classification methods use the entire text of the page [70] and
its images [72], which can be resource-intensive and time-consuming.
Alternatively, researchers have explored web page classification on metadata
fields such as <title>, <description>, and <keywords>, which tend to be
shorter and shown to have strong correlations with the topic of web pages
[68]. We followed the latter approach for its computational efficiency and
reasonable accuracy. Our preliminary analysis of over 500K web pages from the
most popular one million websites in the Common Crawl dataset [32] showed that
more than 97% of the websites have a title, 63% of the websites include a
description, and 24% contain a keywords meta tag. Based on these availability
statistics, we used the titles and descriptions for classification, leaving
out the keywords. To extract the titles and descriptions, we used the
following HTML tags: title, description, og:[title|description], and
twitter:[title|description]. Applying this method to the WAT metadata files
from the June-July 2022 Common Crawl snapshot [32], we extracted the titles
and descriptions, limiting ourselves to the top million websites in the Tranco
[26] or the CrUX [78] list. We further refined our data by keeping a single
page with the shortest URL from each hostname, which is more likely to be the
home page. This resulted in metadata from 2.28 million pages, which included
pages from the subdomains of the top million Tranco and CrUX websites. We also
extracted the same title and metadata information from the 582 known child-
directed websites using a simple script based on Playwright [79]. In both
instances, when the page had more than one description or title available, we
picked the longest one.
After completing the data collection process, we constructed a training set
for our classifier. For negative samples, we randomly selected 2,500 of the
2.28 million pages and manually checked to remove children’s websites. Our
positive samples consisted of 576 title-description pairs after filtering out
websites with titles shorter than ten characters.
### 3.2 Building the ML classifier
Our training data contained a limited number of labeled samples and our input
consisted of text-based meta fields, potentially in multiple languages. This
made designing naive classifiers such as bag-of-words and TF-IDF less suitable
for our task. Instead, we employed a pre-trained and multi-lingual language
model. Pre-trained models have proven to be adequate for general text
classification tasks, but they need to be fine-tuned for more specific
downstream tasks [70]. In particular, we decided to use the Paraphrase-
Multilingual-MPNet-base-v2 (PM-MPNet-v2) model from the SentenceTransformers
[80, 81] library, which is a pre-trained multilingual and distilled model
based on the MPNet method [82]. The distillation process [80, 83] involves
training a smaller model (student) to mimic the behavior of a larger model
(teacher). In particular, PM-MPNet-v2 is fine-tuned with a large number of
paraphrased sentences in more than 50 languages [80]. However, PM-MPNet-v2
cannot be directly used for text classification since it only produces
embeddings that are useful for semantic similarity-related tasks. Thus, we
used HuggingFace’s Trainer API [84] and AutoModelForSequenceClassification
[85] class to fine-tune the model and add a binary classification layer on top
of the PM-MPNet-v2’s embedding outputs. As input to the classifier, we used
the concatenation of title and descriptions since this combination gave the
best accuracy compared to using title or description alone. In particular, we
fine-tuned the model to detect child-directed websites using the training set
explained in §3.1. We used the HuggingFace Transformers [86] and Ray Tune
libraries’ Population Based Training (PBT) algorithm [87, 88] to find the
best-performing hyperparameters (batch size=12, epochs=2, and learning
rate=4.2e-05). The fine-tuning process took roughly five minutes on a
consumer-grade GPU (GeForce RTX 3080 Ti).
To reduce false positives, we employed the modified “Classify-Verify”
technique [89], which involves setting an acceptance threshold $t$, and
accepting a prediction only if it is above $t$. Following Juarez et al. [90],
we choose the threshold that maximizes $F_{\beta=0.5}$, which gives more
weight to precision to reduce false positives. $F_{\beta}$ is a weighted
harmonic mean of precision and recall, adjustable to emphasize either metric
according to the classification task’s needs [91]. A grid search of different
threshold values shows that the maximum $F_{\beta=0.5}$ is achieved when
$t=0.93$, which reduces the false positive by $50\%$. Ultimately, our
classifier achieved a precision of 86% and a recall of 70% using 10-fold
cross-validation, as detailed in Table I.
TABLE I: Classification results before and after applying threshold (with 10-fold cross-validation). | Precision | Recall | F-beta | TP | FP
---|---|---|---|---|---
Without threshold | 0.79 | 0.81 | 0.79 | 47 | 12
With threshold | 0.86 | 0.70 | 0.82 | 40 | 6
### 3.3 The list of 2K children’s websites
Using the fine-tuned classifier, we calculated the label and probability score
for 2.28M web pages from Common Crawl, excluding websites used in the training
process. This process took roughly 5 hours. Our classifier identified 53,092
web pages as children’s websites. Due to time constraints, we focused on
manually verifying the top 2,500 websites sorted by classifier probability,
starting with websites that are most likely to be child-directed. An
evaluation of our classifier and the details of our manual verification
process can be found in Appendix A.1.
Our final list contained 2,004 websites in 48 distinct languages after
eliminating false positives and deduplicating websites by their registrable
domain (TLD+1). English was the most prevalent, accounting for 63% of all
websites. The prevalence of other prominent languages, including Russian,
Spanish, French, German, and Portuguese, ranging between 3% and 6%. The list
included 582 websites from the training data and 1,422 websites identified by
the classifier.
Website ranks: 1,422 of the 2,004 websites were ranked in the top 1 million
Tranco list (median rank 304K). While over a quarter of the websites are in
the top 200K ranks, websites from all popularity levels are captured in our
list. 404 of the 582 websites that are not ranked by Tranco were ranked in the
top one million by the CrUX list. Only 163 (8%) websites were not ranked
either by Crux or Tranco in the top one million.
DNS0 Kids filter check: DNS0 Kids [92] is a domain name resolver that detects
and filters out content that is not suitable for children such as adult,
dating, and copyright-infringing content. To find out the status of the
websites in our list, we compared DNS0 Kids with CloudFlare’s DNS resolver. If
a website can be resolved by CloudFlare, but not by DNS0, we treated it as
blocked. We found that only ten (0.5%) of the 2,004 websites in our list were
blocked by DNS0. Reviewing these ten websites, we found six of them to contain
pirated videos, including cartoons. The remaining four websites contained
activities for children but it was not obvious to us why they were blocked.
## 4 Web Tracking and Advertising Measurements
To assess the prevalence of trackers, fingerprinting scripts, and (targeted)
advertisements on child-directed websites, we extended Tracker Radar Collector
(TRC) [93]. TRC is a Puppeteer-based [94] web crawler, which consists of
modules called collectors that record different types of data during a crawl,
such as HTTP requests/responses, cookies, screenshots, and JavaScript API
calls. New collector modules can be easily added to collect data necessary to
perform different web measurements such as ours. Specifically, we added the
following collectors to TRC:
* •
FingerprintCollector (§4.1): detects fingerprinting related function calls and
property accesses
* •
LinkCollector (§4.3): extracts inner page links
* •
VideoCollector (§4.5): captures the crawl video
* •
AdCollector (§4.6): detects ads and scrapes ad disclosures
We also used the existing TRC collectors, including RequestCollector to
capture request/response details and detect tracking-related requests (§4.2),
TargetCollector to detect newly-opened tabs in §4.6, CookieCollector to
analyze cookies, and finally CMPCollector (§4.4) to interact with the consent
dialogs and consent management platforms (CMP). We used TRC’s anti-bot
measures [93], which thwarts bot detection to a certain extent by overwriting
artifacts typically probed by anti-bot scripts (e.g., navigator.plugins,
Notification.permission) [95]. For the mobile crawls, we emulated a mobile
browser using TRC’s built-in features to spoof viewport dimensions, touch
support, and the user-agent string.
### 4.1 Identifying fingerprinting attempts
Identifying fingerprinting scripts can be challenging due to obfuscation and
potential false positives. For example, scripts may use Canvas API for both
drawing images or fingerprinting the user’s browser [47]. We draw on well-
established methods to distinguish between fingerprinting and benign use of
fingerprinting vectors [38, 1]. Specifically, we focused on Canvas, WebRTC,
Canvas Font, or AudioContext fingerprinting and detected them using the
heuristics presented by Iqbal et al. [38]. To detect fingerprinting attempts,
we modified the getter and setter methods of the several Web APIs such as
CanvasRenderingContext2D.fillText and HTMLCanvasElement.toDataURL to intercept
potentially fingerprinting-related function calls and property accesses.
Although TRC can intercept JavaScript API calls, we implemented a separate
collector (FingerprintCollector) to avoid a known issue that prevented TRC
from intercepting early function calls [96]. FingerprintCollector simply
injects the instrumentation script into each page and its descendant frames as
soon as they are created. We verified that our collector captures calls missed
by TRC on both our custom-developed fingerprinting test pages and external
demo pages such as BrowserLeaks [97].
### 4.2 Identifying tracking-related requests
To identify tracking-related requests, we utilized the uBlock Origin Core [98]
npm package, which mimics the tracking protection of the widely-used uBlock
Origin extension [99]. We used uBlock Origin’s default filter lists, which
include EasyList and EasyPrivacy, among others [100]. To accurately detect
tracking-related requests, we provided uBlock Origin Core with the request’s
resource type (e.g., image or script) and the page and request URL derived
from the HTTP request/response details recorded by the crawler. Then, we
mapped the tracker domains to their owner entities (i.e.,
organizations/companies) using DuckDuckGo’s entity map [101]. Using entities
to quantify tracker prevalence reduces overcounting as multiple domains can be
owned by the same business (e.g., googleanalytics.com and doubleclick.net are
both owned by Google).
### 4.3 Discovering inner pages
We refrained from only focusing on homepages as prior work found that
websites’ inner pages tend to contain more trackers and cookies [102, 103].
Thus, we also gathered five inner links from each of the 2,004 websites by
conducting four separate link-collection crawls (desktop and mobile crawls
from Frankfurt and NYC). We preferred to crawl sites from two vantage points
to reduce the time and effort required for the link collection process. We
excluded external domain links and documents such as PDFs and images, and we
preferred links near the viewport’s center to avoid unrelated links from
footers or less visible page areas. After gathering these inner links, we
merged them with the homepage URLs to form the final set.
### 4.4 Interacting with consent dialogs
Since the GDPR came into effect, websites typically show consent dialogs when
viewed from the EU and to some extent even from the US [104]. Ignoring these
dialogs may lead to undermeasurement of the tracking and advertising
practices. We decided to provide affirmative consent to all data processing
request options (accept all) in our crawls to measure the full extent of
advertisements and tracking a child could experience. To handle consent
dialogs in an accurate and automated manner, we used DuckDuckGo’s autoconsent
library [105], which comes bundled with TRC [106]. Autoconsent incorporates
rules from Consent-O-Matic [107, 108], allowing programmatic interactions with
the CMPs.
### 4.5 Video screen captures
To detect ads and scrape their disclosures, our crawler performed a series of
interactions with the page, including dismissing popup dialogs, interacting
with CMPs, and clicking on visible ad disclosure links(§ 4.6). To monitor
these interactions, we added a video capture functionality to the crawler
(VideoCollector). We used videos of the crawler’s interactions to troubleshoot
potential issues with the crawler process as well as to label animated ads and
other crawl artifacts manually.
### 4.6 Identifying ads and ad targeting criteria
The AdCollector performed three main functions: 1) detecting ads, 2) scraping
ads —including its screenshots, links, iframes, scripts, videos, and
background images, and 3) detecting and scraping ad disclosure pages to
determine whether an ad is targeted or not.
Detecting ads: To detect ads, we built on Zeng et al.’s [16] approach to use
EasyList’s rules [109]. EasyList rules are commonly employed by popular
adblockers to block or hide ads. For each detected ad element, the crawler
recorded a set of attributes, including its position on the page, its
dimensions, class, ID, and name, in addition to the complete HTML source and a
screenshot. If the ad element contained any child elements, which was mostly
true, the crawler recursively recorded their details, including all links,
images, video, script, and iframe elements. Small elements ($<30px$ in either
dimension) and elements lacking any link, image, background image, or video
were excluded.
The crawler not only took screenshots of each ad but also downloaded image and
video elements that were descendants of the ad element. These media were
utilized in the ML-based ad content analysis pipeline 4.6 alongside the ad
screenshots. The crawler sent a single HTTP request during the page visit with
the appropriate HTTP headers—such as the HTTP Referer [sic.] set to the
current address-bar URL—when downloading these files. Finally, the crawler
saved data-URL images found within the ad’s subtree. Bin Musa and Nithyanand
[64] also utilized EasyList’s rules for ad identification in their study on
tracker-advertiser relationships, but their implementation differs from ours.
While they focus on detecting image-containing HTTP responses using the
EasyList filter set, we query the DOM to detect ad elements, such as div
elements, and their relevant descendant elements, such as images, iframes,
links (a), and videos. Operating at the DOM level also allows us to detect and
scrape ad disclosure pages to detect targeted ads. To verify how accurately
our crawler detects ads, we performed a sanity check on a random sample of 105
ads (15 ads from each crawl). The crawler correctly detected ads in 85% of
cases, misidentified non-ads in 7.5%, and captured blank or empty ads in 7.5%.
Some ad screenshots also included multiple (2.8%) or only part (4.5%) of the
ads. However, the overall accuracy and quality of our ads appear to be higher
than prior work by Zeng et al. [55], which reported 34% unrendered
(blank/unreadable) ads. We attribute this difference in data quality to two
potential reasons. First, we use a more realistic crawler equipped with anti-
bot measures; and second, unlike Zeng et al., we opted not to click the
ads—which may trigger more stringent anti-bot, anti-fraud protections that
prevent the delivery or rendering of the ads. Further, to evaluate whether our
EasyList-based detector missed any ads, we manually reviewed 50 random pages
where no ads were detected by the crawler. Our review did not reveal any false
negatives, suggesting that our ad detection was robust. We also verified the
accuracy of the ad images separately downloaded by the crawler, finding all of
them to be present in the ads shown on the page.
Determining targeting criteria: To measure the prevalence of targeted
advertisements at scale, we automated the process of scraping ad disclosure
(e.g., “Why this ad”) pages. While the content of ad disclosure pages may vary
by ad platform, they generally explain in broad terms why a specific
advertisement was shown to a user. The reasons may include, for instance,
Google’s estimation of your interests or Websites you’ve visited. The
disclosure pages may also contain information about the website and the
advertiser, and whether ad targeting is turned off for the website or a
specific ad. Two example disclosure pages for a targeted and non-targeted ad
are shown in Figure 3.
(a) Targeted ad
(b) Non-targeted ad
Figure 3: Google’s ad disclosure pages indicating whether an ad is targeted or
not. The top figure belongs to a targeted ad (indicated by Google’s estimation
of your interests), while the bottom one is for a non-targeted ad (indicated
by Ad personalization is turned off)
Ad disclosure pages are reachable by clicking the AdChoices icon and the “Why
this ad” button for Google ads [110] and other ad providers. Initially, we
attempted to detect the ad disclosure links using fuzzy image matching based
on the AdChoices icon. However, we found that the icon’s shape and visibility
substantially vary across different ad vendors, and sometimes the icon can be
hidden, making it unclickable. We chose to identify ad disclosure links
through their URLs, focusing on a fixed set of providers we could detect
reliably and deterministically. By analyzing ad disclosure pages in our pilot
crawls, we compiled a list of hostnames (i.e., adssettings.google.com,
privacy.us.criteo.com and privacy.eu.criteo.com) that appear in the ad
disclosure links, and explain whether an ad is targeted or not. We limited our
investigation to ad disclosure pages from these two providers because other
providers we encountered in our pilot crawls did not offer useful information
about the targeting criteria of the ads.
Once the crawler detects and clicks on the AdChoices link, the ad disclosure
page opens in a new tab. We intercepted this new tab, stored its URL,
screenshot and text contents (via document.innerText) for analysis. The
scraped text contents are then used to detect whether ad targeting is enabled
or not. Specifically, we searched in the ad disclosure texts for specific
disclosure statements indicating whether and how an ad was targeted. The
disclosure statements include, for instance, Google’s estimation of your
interests (targeted), Websites you’ve visited (targeted) and Ad
personalization is turned off (non-targeted). If one or more statements
indicating targeted ads occur in an ad disclosure text, we label the ad as
targeted. Otherwise, we label the ad as non-targeted. Note that we count
behavioral or retargeted ads also in the targeted category. We compiled a list
of 18 statements (Appendix A.2) incrementally, using over 40K ad disclosure
texts extracted during the crawls. We ensured that all ad disclosures contain
at least one of these statements, to make sure our analysis is exhaustive.
Interacting with the page and ads: Upon loading a page, our crawler waited for
2.5 seconds and dismissed any popup dialogs using heuristics from prior work
[29]. We dismissed these dialogs to prevent them from blocking our crawler’s
interactions with the webpage. The crawler then waited for another second
before scrolling through the page in 10 steps, taking strides of about 500–600
pixels each interlaced with a random delay of 500–600 milliseconds. Finally,
after waiting for another second, it scrolled up to the beginning of the page
using the same scrolling behavior. We engineered this up-and-down scrolling
behavior to allow the webpage to load any ad slots that are lazily loaded as
the user scrolls the page below the landing fold.
The crawler then identified all ads on the page. It set the border color of
each ad to red to visually mark the ads for manual review. The crawler then
took a screenshot of the entire page and then scraped each ad in a top-down
fashion. To ensure that an advertisement is fully seen, it scrolled down to
each ad before taking its screenshot. Finally, the crawler detected ad
disclosure links and clicked each one individually to capture all ad
disclosure texts and screenshots. We limited the number of scraped ads per
page visit to ten, which limits over-representation by a few websites with
many ads.
Analyzing advertisement content: We identified and measured four kinds of ads
in our corpus: weight loss ads, mental health ads, dating services ads, and
ads that contain clickbait racy content. While our dataset of ads can be used
to perform fuller content analysis, we focused on these four categories since
prior work [111, 112] and regulatory reports [113] have argued that these can
be especially harmful to children. In fact, many ad networks’ moderation
policies [114, 115] explicitly restrict these categories of ads from appearing
on children’s websites. We note that the categories we focused on are not
exhaustive, and our ads analysis is exploratory, serving as a preliminary
investigation into this critical problem.
An overview of the ad content analysis pipeline is shown in Figure 4. To
identify ads containing click-bait racy content, we employed Google Cloud
Vision API’s SafeSearch Detection [116], which is a service that uses deep
learning to analyze images and identify potentially unsafe content. It
evaluates images against categories such as adult, violent, racy, and medical
content and returns likelihood scores for each category, ranging from
‘VERY_UNLIKELY’ to ‘VERY_LIKELY.’ Upon manually evaluating the output
generated by the algorithm, we focused on the ‘racy’ category with a
likelihood of ‘VERY_LIKELY’. We also tested Microsoft’s Adult Content
Detection [117], part of Azure Cognitive Services, to identify racy images.
However, due to more false positives compared to Google Cloud Vision API, we
chose the latter for our study.
We used the Google Cloud Vision API to extract text from ad images following a
similar approach to Bin Musa and Nithyanand [64]. The text in each image was
extracted using the Optical Character Recognition (OCR) feature of the API,
specifically by employing the fullTextAnnotation attribute of the API
response. This allowed us to extract text data at different levels, such as
page, paragraph, and word. We opted to use the paragraph level since it gives
the best separation in ads promoting multiple unrelated products. Despite
their name, paragraphs returned by the API were relatively short and akin to
sentences (21 characters, on average).
We then employed semantic similarity to identify the most similar ad texts
(paragraphs) corresponding to a given search query, which in our case were
“weight loss”, “mental health”, and “dating”. This approach is versatile and
can be used to retrieve ads related to any arbitrary words or phrases. To
compute the embeddings of the queries and ad paragraphs, we used the
“paraphrase-multilingual-mpnet-base-v2” model, the distilled multilingual
model we used to classify web pages (§3.2). To find the most similar results,
we calculated the cosine similarity between the embeddings of the search query
and each ad paragraph and sorted them accordingly. Next, we manually reviewed
the 100 most similar distinct paragraphs and their associated images,
including ad screenshots or background ad images, to identify those that
pertained to the three categories of interest. We also experimented with
BERTopic [118] to create topic models and searched for clusters similar to our
chosen categories. While this resulted in well-grouped texts, it required
manual verification of numerous (several thousand) clusters. Sorting based on
semantic similarity proved to be faster, more flexible, and easier to
implement and evaluate, making it the preferred approach for manual reviewing.
Figure 4: Overview of the ad content analysis pipeline.
### 4.7 List of crawls
The main dataset used in our study consists of seven crawls (Table II), all of
which were run in April 2023 using cloud-based servers on Digital Ocean. The
crawls include five desktop and two mobile crawls from five and two vantage
points, respectively. We limited the mobile browser crawls to two vantage
points because we do not focus on mobile-desktop comparison, which we leave to
future work. The vantage points used for crawls consisted of Frankfurt,
Amsterdam, London, San Francisco, and New York City (NYC), to capture ads and
tracking from different jurisdictions. Crawls were run in parallel using
separate servers with moderate resources (8 vCPU cores, 16GB RAM). Each crawl
took between 13 and 32 hours to complete. A clean browser profile is used for
each page visit to prevent ads targeted to our browsing history. During each
crawl, we visited both landing and inner pages, following the process
described in Section 4.3. When applicable, we accepted all personal data
processing on consent (cookie) dialogs. The order of visited pages was
randomized within each vantage point. Note that the San Francisco crawl used
inner links extracted from the NYC crawl, while the London and Amsterdam crawl
used inner links from the Frankfurt crawl. While this constraint did not
appear to impact the success rate of visits across these vantage points,
future research could explore identifying inner pages during the crawling
process.
## 5 Measurement Results
Table II summarizes the overall statistics for measurement crawls. A total of
71,567 pages were loaded successfully across all crawls. The success rate of
our crawler was over 93%, according to the successful visit criteria we
developed and applied (Appendix A.3). For simplicity, certain comparative
results presented below are based on desktop crawls from NYC and Frankfurt,
representing one location each in the US and the EU.
TABLE II: Crawl statistics based on different vantage points. *: Avg./Sum of all sites visited across all crawls. | Form
---
factor
| Vantage
---
point
| Successfully
---
loaded pages
| Successful
---
crawling rate
Desk. | NYC | 10,310 | 95%
SF | 10,301 | 95%
LON | 10,270 | 95%
FRA | 10,221 | 95%
AMS | 10,014 | 93%
Mobile | NYC | 10,168 | 94%
FRA | 10,283 | 96%
Avg./Sum | | 71,567 | 95%
### 5.1 Ad targeting and content analysis
Our crawler scraped 70,303 ads from 804 of the 2,004 distinct websites across
seven crawls. An average of 36% of the pages contained one or more ads, and we
detected targeted ads on 27% of the pages we crawled. The crawler scraped
10,839 and 9,447 ads on average in the crawls from the US and Europe,
respectively.
#### 5.1.1 Over 70% of ads with disclosures are targeted in nature
Our crawler captured a total of 40,281 ad disclosure pages, which we used to
determine the advertiser’s identity and whether ad targeting is enabled or
not. There are fewer disclosure pages than ads due to ads without disclosure
links and failures in detecting or opening those links. In fact, we only
consider ad disclosures from two ad providers: Google (97.8%) and Criteo
(2.2%), since ad disclosure pages of other providers did not reveal the
targeted status of the ad or the advertisers’ identity. Limiting our analysis
to 40,281 ads with disclosure pages, we found that targeting was enabled for
73% of the ads with disclosures. Comparing across different privacy
jurisdictions, we find 68% of the ads on average were targeted in the EU
crawls, compared to 76% in the UK and the US crawls. Comparing the crawls from
the two US cities (SF & NYC), we find that 67% of the ads were targeted in the
SF desktop crawl, compared to 79% and 82% in the NYC-based desktop and mobile
crawls, respectively. Although these variations might be attributed to
stricter privacy regulations like CCPA and GDPR, our available data and
methods do not permit us to make this attribution. Comparing the Tranco ranks
of the 689 websites that contain at least one targeted ad to 59 websites that
only contain non-targeted ads, we find a tendency for popular websites to
disable ad targeting (Figure 5). Sites with targeted ads had a median rank of
$\sim 340K$, while those with only non-targeted ads had a median rank of $\sim
128K$. Note that we only include 40,281 ads for which we can determine the
targeted status in this analysis.
Figure 5: Tranco rank (x-axis) distribution of sites that use targeted vs.
non-targeted ads. Popular websites (below) appear to be more prone to
disabling ad targeting. TABLE III: Number of visits and scraped ads, along
with percentages of ads/targeted ads per crawl. *: Percentage of targeted ads
is only based on ads with disclosures. In the rightmost two columns, we
include a site if we scraped at least one ad/targeted ad from one of its
pages.
| Form
---
factor
| Vantage
---
point
# ads | | % sites
---
with
ads
| % sites with
---
targeted
ads
| % targeted
---
ads
Desk. | NYC | 11,288 | 38% | 30% | 79%
| SF | 10,950 | 38% | 28% | 67%
| LON | 9,702 | 36% | 27% | 76%
| FRA | 9,700 | 36% | 26% | 68%
| AMS | 9,250 | 35% | 26% | 67%
Mobile | NYC | 10,278 | 36% | 29% | 82%
FRA | 9,135 | 33% | 26% | 70%
Avg./Sum | | 70,303 | 36% | 27% | 73%
#### 5.1.2 Ads can be targeted from anywhere
The “About the advertiser” section in Google’s ad disclosures shows the name
and location (country) of the advertisers. This information is only available
in 70% of the ad disclosures in our dataset. Extracting these fields from the
ad disclosure texts, we identified 1,685 distinct advertisers from 81
different countries. Advertisers with the most ads in our data are displayed
in Table IV. We note that due to the transient, targeted and localized nature
of ad campaigns, the list in Table IV may not represent the most common
advertisers on child-directed websites in general. Further, in certain cases
(e.g., Gloworld LLC and Marketism), an advertising or marketing agency is
listed on the ad disclosure page instead of the company offering the
advertised products or services.
The top ten advertisers are located in seven different countries and three
continents. We observed that many of those advertisers are located far from
our crawl vantage points, thus indicating that children visiting websites in
our list can be targeted with ads from anywhere in the world. Reviewing a
sample of 100 ads from each advertiser, we marked in the rightmost column
their predominant ad theme. Five of the ten advertisers display ads for search
results about various products—such as depression tests, belly fat reduction,
senior meals, and electronic payments—on lesser-known search engines such as
IngoSearch [119]. Ads from Betterme [120], a “behavioral healthcare app” with
more than 100M installations, featured plans for weight loss, muscle gain, and
intermittent fasting (e.g., Figure 1 ⓗ).222We note that Better.me’s data
sharing practices with third parties were investigated by Privacy
International, but the company reportedly took corrective action[121]. Brain
Metrics Initiative displays ads for IQ tests, an example for which is given in
Figure 1 ⓒ. Alibaba Hong Kong, on the other hand, displays ads featuring racy
and disturbing images of products sold on alibaba.com. For instance, the ad on
the top left (ⓐ) in Figure 1 features recurring images in Alibaba ads: a naked
baby model (leftmost), rabbit meat (rightmost), and a semi-transparent
underwear ad in the middle. We investigate similar racy clickbait ads and
other improper ads in the following subsection.
TABLE IV: Top ten advertisers by the number of ads across all crawls.
Advertiser | Location | # ads | | %
---
targeted
Type of ads
Vinden.nl B.V. | Netherlands | 4,707 | 86% | Search results
EXPLORADS | Cyprus | 3,265 | 73% | Search results
| All Response
---
UK | 2,453 | 68% | Search results
Gloworld LLC | USA | 2,365 | 55% | Online learning
| Amomama M.
---
Cyprus | 921 | 72% | | Workout muscle
---
gain, weight loss
| Media Quest
---
UAE | 910 | 79% | Search results
| Brain Metrics I.
---
Cyprus | 814 | 50% | IQ tests
BetterMe | Cyprus | 731 | 85% | Weight loss
Marketism | Israel | 645 | 49% | Search results
| Alibaba.com HK
---
| Hong Kong
---
541 | 86% | | Products sold
---
on Alibaba.com
#### 5.1.3 Improper ads on child-directed sites
TABLE V: Number of improper ads identified for each crawl.
| Form
---
factor
| Vantage
---
point
Dating | | Mental
---
health
| Weight
---
loss
Racy | | Some-
---
what
racy
Total
Desk. | NYC | 4 | 21 | 16 | 21 | 26 | 88
SF | 7 | 9 | 15 | 6 | 25 | 62
LON | 10 | 17 | 48 | 12 | 31 | 118
FRA | 1 | 0 | 48 | 19 | 25 | 93
AMS | 8 | 4 | 82 | 10 | 33 | 137
Mobile | NYC | 22 | 25 | 113 | 98 | 17 | 275
FRA | 18 | 5 | 190 | 11 | 6 | 230
Total | | 70 | 81 | 512 | 177 | 163 | 1003
In total, our crawler collected 199,935 screenshots and images from the 70,303
scraped ads. After deduplicating the images, we queried the Cloud Vision API
to obtain the category and OCR texts of the resulting 98,264 distinct images.
We manually reviewed 741 images classified as ‘VERY_LIKELY’ racy by the API.
Separately, we reviewed 1,136 ad images with OCR text semantically most
similar to our search terms (mental health, dating, and weight loss). Due to
study limitations, we only examined the ads related to the top 100 distinct
texts for each term. Since each distinct text may appear in multiple ads
differently, we labeled the images separately and used videos captured by the
crawler when the ad was animated or the ad screenshot was obscured. Table V
shows the number of improper ads identified in each crawl, amounting to 1,003
across 311 distinct websites. A notable finding is the higher prevalence of
such ads on mobile devices compared to desktops in general.
Racy images. We found 177 racy ads and 163 somewhat racy ads, considered edge
cases due to their potential inappropriateness for child-directed websites.
These ads were identified across 80 distinct websites mostly ranked within the
top one million according to the Tranco list, with a median rank of 426K.
Figure 1 ⓐ, ⓖ, ⓚ are examples of some of these ads. Notably, the majority of
the racy ads were encountered in mobile crawls; especially within the NYC
crawl (98/177). Out of 177 racy ads, only 38 had disclosure pages. Among
these, targeting was enabled for 35 ads.
Mental health. By manually labeling 236 ad images, we identified 81 ads
offering mental health services on 48 distinct websites. Examples of ads in
this category contained “take a depression test” (Figure 1 ⓕ), “online
psychiatrists,” “how to get over depression,” and a “mental health chatbot
which helps people with depression.” We excluded false positives that were not
mental health service offerings, such as ads for “mental health counselor
salaries,” “online psychology courses,” and “psychology books.”
Dating. Manually labeling 231 ad images, we identified 70 dating platform ads
on 48 distinct websites, most of which targeted mobile users. The ads promoted
dating platforms such as “dating.com,” and “Live Me,” a live streaming app
with ads featuring suggestive imagery (Figure 1, ⓙ, ⓚ). Another ad for
DateMyAge.com featured a call to “[m]eet your mature singles” (ⓔ). False
positives removed during the manual labeling of this category included ads for
customer relationship tools, romantic holiday tours, and online appointment
services.
Weight loss. We identified 512 weight loss-related ads (plans, apps, products)
on 170 distinct websites by labeling 669 ad images. Notably, there was a
higher number of weight loss ads on mobile devices, indicating campaigns
targeting mobile users. Examples of text featured in these ads included
“intermittent fasting for weight loss,” “keto weight loss plan,” and “eating
plan to lose weight” (Figure 1 ⓗ).
In Figure 1, we provide additional examples of advertisements that are likely
not suitable for children. Examples of these included an ad for a test called
“Am I Gay Test” ⓓ, for a sex toy ⓘ and a sex toy shop ⓑ333Reportedly Germany’s
largest online adult retailer [122] featuring an image of ice cream that could
be appealing to children, and ads featuring clickbait and sexually suggestive
images. The ads were found on websites related to K-12 e-learning, kids games,
coloring books and worksheets, among others. The ads in Figure 1 do not
necessarily fit our four investigated categories but showcase the diversity of
improper ads.
Malicious ad links. Finally, we present an exploratory analysis of whether ads
on child-directed websites link to malicious pages. We submitted a sample of
links extracted from the ad elements to the VirusTotal API in August 2023.
Specifically, we removed links with duplicate hostnames, and for Google ads,
we extracted a direct link to the ad landing page using the ‘adurl’ parameter
[123]. While the overwhelming majority of the links were classified as benign,
149 of the nearly 3,940 scanned links were flagged as malicious or phishing by
at least one scan engine. Notably, the word “taboola” was mentioned in 78 of
the 149 detected links as a URL parameter that seems to indicate the ad
network (network=taboola).
### 5.2 Tracking and fingerprinting analysis
TABLE VI: Average number of third-party and tracker domains, and the
prevalence of tracking and fingerprinting on child-directed websites from five
vantage points.
| Form
---
factor
| Vantage
---
point
| 3rd-Party
---
domains
| Tracker
---
domains
| Tracker
---
entities
| % sites
---
with
3rd
Parties
| % sites
---
with
trackers
| % site
---
with
FP
Desk. | NYC | 31.6 | 23.4 | 20.0 | 95% | 90% | 9%
SF | 29.3 | 21.3 | 17.8 | 95% | 91% | 9%
LON | 21.3 | 14.3 | 10.6 | 96% | 91% | 7%
FRA | 23.2 | 15.6 | 11.7 | 95% | 90% | 10%
AMS | 21.4 | 14.3 | 10.6 | 93% | 89% | 7%
Mobile | NYC | 29.8 | 21.8 | 18.4 | 95% | 91% | 9%
FRA | 22.6 | 15.2 | 11.5 | 95% | 90% | 11%
Table VI shows the prevalence of third-party trackers detected across
different crawls. We find that around 90% of the websites have at least one
tracker domain, and over 93% embed at least one third-party domain.
TABLE VII: Prevalence of tracker entities in terms of number of distinct websites in Frankfurt and NYC desktop crawls. FRA | NYC
---|---
Entity | # Sites | Entity | # Sites
Google | 1,702 | Google | 1,718
Facebook | 458 | Microsoft | 549
Index Exchange | 424 | Adobe | 543
Xandr | 416 | Xandr | 516
Adform | 412 | The Trade Desk | 501
The Trade Desk | 390 | Index Exchange | 495
OpenX | 378 | IPONWEB | 467
Adobe | 366 | Facebook | 456
Quantcast | 361 | Magnite | 446
PubMatic | 359 | OpenX | 426
Third-party trackers. The average number of tracker domains per site differs
significantly, e.g., 15.6 and 23.4 in Frankfurt and NYC crawls, respectively,
while the median is 15 and 16, respectively. The difference in averages can be
attributed to websites with the high number of trackers in the NYC crawl. This
explanation is in line with the results displayed in Table VIII, which shows
the top five websites with the most trackers in Frankfurt and NYC crawls. Most
of these websites are among the top one million, which means they likely
receive substantial traffic. Notably, all of these sites displayed ads that
were targeted. The numbers shown in the table - number of trackers, requests,
and cookies - reflect averages across the web pages. In the NYC crawl,
visiting mathfunworksheets.com triggered a total of 1,547 requests involving
161 unique third-party tracker entities (i.e., organizations/companies).
Another website, woojr.com found to contain 148 distinct third-party tracker
entities when visited from NYC. This website includes resources for children’s
activities and educational materials, including printable worksheets and fun
activity pages. When visited from Frankfurt, www.wowescape.com, a website
offering various games for children and teenagers, triggered requests to 95
distinct third-party tracker entities.
Most prevalent trackers. Table VII shows the tracker entities with the most
prevalence in Frankfurt and NYC desktop crawls. We found a tracking-related
request to Google domains, including its analytics, advertising and tag
management scripts on $\sim$84% of the 2,004 child-directed websites in both
crawls. Facebook is the second most prevalent entity in the Frankfurt crawl
mostly due to Facebook Pixel (on 427 websites), which facilitates ad
retargeting and conversion measurement, among others [124]. Largely thanks to
Linked Insight Tag (px.ads.linkedin.com, 466 websites), Microsoft is the
second most prevalent entity in the NYC crawl. Linked Insight Tag serves
multiple purposes, including retargeting, conversion measurement, and
providing demographic insights about website visitors [125].
Regional differences. To explore the differences in tracker entities across
vantage points, we compared the tracker entities from Frankfurt and NYC
desktop crawls. Despite a considerable overlap among the detected tracker
entities (Jaccard index=$0.85$), we also identified variations. Specifically,
our investigation unveiled 47 tracker entities exclusive to the Frankfurt
crawl and 118 tracker entities that were only found in the NYC crawl. For
instance, tracking related requests to advanced STORE [126] (ad4m.at &
ad4mat.net, 236 websites) exclusively appear in the crawl from Frankfurt,
whereas Throtle, a company that provides an identity graph to marketers and
advertisers, only appears on 171 websites in the NYC crawl [127]. Furthermore,
we find that the majority of the websites in both Frankfurt and NYC crawls
(∼70% and ∼72%, respectively) contain third-party trackers that set at least
one cookie with the SameSite=None attribute and a lifespan of over three
months. Primarily through doubleclick.net domain, Google set these cookies on
over 51% of the websites. While identifying the individual purposes of these
cookies is out of scope, this combination of cookie attributes (esp. setting
SameSite=None) makes it possible to track users across websites.
Sites with and without ads. As part of our investigation, we conducted an
additional analysis to compare how the number of third parties and trackers
change between websites with and without ads. Figure 6 shows that websites
with ads tend to have substantially more third-party and tracker domains. More
specifically, the figure shows websites with ads tend to contain two to four
times more third-party and tracker domains.
Figure 6: Comparison of the average number of third-party and tracker
domains/entities on websites containing ads vs. not containing ads. TABLE
VIII: websites with the most distinct tracker entities. The table shows the
websites’ distinct third-party tracker entities, the number of requests,
cookies, and Tranco rank.
Loc. | Website | | # Trackers
---
| # Requests
---
| # Cookies
---
Rank
NYC | mathfunworksheets.com | 161 | 1,547 | 395 | 669K
woojr.com | 148 | 2,181 | 391 | 83K
innerchildfun.com | 139 | 1,235 | 336 | 308K
kidzfeed.com | 138 | 1,050 | 272 | 797K
thecolor.com | 138 | 1,068 | 260 | 192K
FRA | wowescape.com | 95 | 392 | 55 | 258K
webgames.io | 94 | 564 | 92 | 155K
coloriages-pour-enfants.net | 90 | 401 | 66 | 919K
theschoolrun.com | 87 | 417 | 91 | 760K
testsworld.net | 86 | 478 | 138 | -
Browser Fingerprinting. We now discuss our findings on fingerprinting scripts
on child-directed websites. Table VI shows that we detect fingerprinting
scripts on 176 (9%) and 218 (10%) websites in Frankfurt and NYC crawls,
respectively. The overall prevalence of fingerprinting aligns with the recent
research by Iqbal et al., which finds fingerprinting on 10.18% of the top-100K
websites [38]. One of the most prevalent fingerprinters in both crawls is an
online payment company (Stripe; 66, 67 sites on Frankfurt and NYC crawls,
respectively). According to their help pages [128] Stripe primarily employs
fingerprinting for fraud prevention purposes. Webgains (82 sites in the
Frankfurt crawl), an affiliate marketing company, also mentions fingerprinting
in their Data Processing Agreement with Merchants [129], but without
specifying its purpose. The most commonly used fingerprinting method is Canvas
fingerprinting, present on about 208 sites in the Frankfurt crawl and about
172 sites in the NYC crawl.
We found one or more trackers to be present on more than 90% of mobile
websites (Table VI), which is similar to our finding for the desktop websites.
NYC and Frankfurt crawls differ slightly in the number of ads: we scraped
10,278 ads in the NYC crawl and only 9,135 in the Frankfurt crawl—the latter
is the crawl with the least amount of ads. Slightly more (29 vs 26%) websites
in the NYC mobile crawl have targeted ads; and NYC mobile crawl has the
highest proportion of targeted ads (82%) across all crawls. We also discovered
that improper ads, particularly racy and weight loss ads, were more prevalent
on mobile devices compared to desktops.
## 6 Discussion
Our research paints a troubling picture of tracking and inappropriate
advertising practices on child-directed websites. Advertisements featuring
sexually suggestive imagery and ads about weight loss, dating, and mental
health may pose potential risks to children’s emotional and psychological
welfare. We discuss the legal implications, ethical considerations and
limitations of our study below.
### 6.1 Legal implications
In this section, we discuss what the law says about tracking and advertising
practices uncovered in our research. We focus on the EU General Data
Protection Regulation (GDPR) and the US Children’s Online Privacy Protection
Act (COPPA)444We do not analyze whether specific companies breach the law. For
such an analysis, each case would have to be examined separately, considering
all the circumstances of that specific case. Rather, we discuss legal
requirements in general terms..
The GDPR and the ePrivacy Directive. Under the GDPR, companies are only
allowed to process personal data if they have a ‘legal basis’ for such
processing. The GDPR provides six possible legal bases (article 6 GDPR).
However, generally, the data subject’s consent is the only possible legal
basis for online tracking and behavioral (targeted) advertising [130].
Moreover, the ePrivacy Directive [131] requires, in short, companies to ask
the internet user for consent before they use tracking cookies or similar
tracking technologies (Article 5(3)).
The GDPR’s requirements for valid consent are strict. Consent is only valid if
it is really voluntary (‘freely given’), and ‘specific’ and ‘informed’. The
data controllers (the website owner and the companies involved in tracking and
targeted advertising) must ‘be able to demonstrate that the data subject has
consented to process of his or her personal data’ (Article 7(1) GDPR). The
GDPR’s requirements for valid consent also apply to consent (for cookies etc.)
as prescribed by the ePrivacy Directive. The GDPR has specific rules for
consent by children. Roughly summarized, children cannot give valid consent;
the parent should give consent instead (Article 8 GDPR). EU member states have
set different minimum consent ages, ranging from 13 to 16 years [132]. Hence,
only parental consent can legitimize tracking on a children’s website. Observe
that a parent clicking a consent dialog (as done by our crawler) does not
constitute parental consent under GDPR. Even in low-risk cases, verification
of parental responsibility via email may be necessary [133].
The EU Digital Services Act. The rules for tracking and targeting children
will become stricter in the EU. From 17 February 2024 on, the EU Digital
Services Act [134] applies. Article 28 says, roughly summarized, that online
platforms must not use behavioral advertising ‘when they are aware with
reasonable certainty that the recipient of the service is a minor’ [134]. This
prohibition cannot be overridden with the consent of the child or the parent.
The DSA also requires “very large online platforms” [135] (with more than 45
million users in the EU) to publish the advertisements that it presented to
users in an online repository, together with information about, for instance,
the targeting criteria (Article 33, 39 DSA). The methods that we used in this
paper could be used to check the completeness and accuracy of data published
in those repositories.
COPPA. COPPA regulates companies offering a website or online service directed
to children under the age of 13. Specifically, COPPA applies to companies
using children’s ‘personal information,’ which includes ‘persistent
identifiers such as cookies and device fingerprints’ (COPPA §312.2) [136]. The
website owner is responsible for data collection by third parties through its
site. Such third parties must also comply with COPPA. Companies based outside
the US must also comply with COPPA if their services are directed to children
in the US [136].
Our results showed that 27% of the child-directed websites use targeted
advertising. Under COPPA, data collection for targeted advertising on these
websites is only allowed after getting parents’ Verifiable Parental Consent
(VPC). VPC entails utilizing stringent verification methods, such as credit
card verification, face recognition, or government ID checks [137]. This makes
VPC much more complex than simply clicking an accept button on a dialog. We
note that our crawler cannot simply give VPC.
### 6.2 Research Ethics
Our crawler visited over 166K pages and it triggered many ad impressions that
could be viewed by a real visitor (likely a child). Given the huge scale of
the digital ad market (projected to reach US$700bn in 2023 [138]) we believe
these ad impressions are a negligible cost for raising the transparency around
tracking and ads targeted to children. Furthermore, we took several measures
to limit our footprint on the crawled websites. For instance, we only crawled
five inner pages from each site in a crawl, and we randomly shuffled the
target URLs to avoid concurrently visiting the inner pages of a website. We
also took appropriate measures to ensure that no harm was done to
collaborators involved in the project, especially when dealing with explicitly
graphic images.
Disclosures and outreach In July 2023, we reached out to five companies that
we found to serve racy ads. One company invited us to a call and explained the
likely reason for showing inappropriate ads (websites failing to label
themselves as child-directed and a false negative in the ad company’s
automated child-directed site detection tool). They added the sites in
question to the child-directed list and pledged further investigation. Another
thanked us and began an internal review. The third redirected our query to the
relevant department. Moreover, we disclosed 34 racy ads to Google by manually
visiting the ad disclosure URLs of each racy (Google) ad; and using the Report
this ad button. Note that, to identify the ad vendors involved in serving the
ad, we used a combination of ad images, and src/href attributes of the ads’
descendant iframe, image and link elements (§4.6).
In addition, we shared our preliminary results with a European data protection
agency (DPA), and a consumer protection agency. Both showed interest; the DPA
asked if there were any websites from their country containing improper ads.
The consumer protection agency invited us to present our work. We also shared
our findings with civil society and industry organizations including the
5Rights Foundation [139]. We plan to further share our study with regulators
and other relevant stakeholders.
While using the VirusTotal API, we found and reported three porn websites
miscategorized as kids-related to the respective third-party categorization
service. Although no response was received, the categories were later
rectified.
### 6.3 Limitations
Our study predominantly covers websites targeting younger children, as we
define ‘children’ as under 13, aligning with both US and EU regulations. While
our classifier detected child-directed sites in 48 languages, it may have a
potential bias towards English content, which is overrepresented in the
training data. The classifier may prefer sites with descriptive titles and
content or may carry biases related to website age, design, or accessibility.
Moreover, the classifier favors precision over recall to reduce the manual
labeling workload.
Our research is the first attempt to build a large list of child-directed
websites. Classifying websites as child-directed or not is challenging,
primarily due to the existence of gray areas that complicate the labeling
process. For instance, many websites offer content that appeals to both
children and adults. While we tried to exclude websites that can only be used
by teachers or parents, certain websites included pages such as online
exercises, videos or games that can be consumed by children. To validate our
list, we performed further analysis on two subsets of websites: two senior
researchers relabeled a random 100-website sample; one of the senior
researchers relabeled a random subset of 100 websites with targeted (50) and
inappropriate ads (50). In both cases, we identified less than 9% of websites
that are related to children, but mainly catered to teachers or parents (8.6%,
8.3%, respectively). While this impurity could be avoided with more
conservative labeling, our analysis of 200 websites strongly suggests these
cases do not skew our primary findings.
While we found fewer targeted ads in the EU than in the US, we cannot directly
attribute this to differences in privacy regulation or another specific
factor. Failure to detect and interact with consent dialogs may be a
confounding factor, among others.
When detecting targeted ads, we only used ad disclosure pages from two
providers (Google and Criteo) due to the unavailability of useful ad
disclosures from other vendors. Thus, our targeted ad detection method depends
on the accuracy, completeness and precision of Google and Criteo’s ad
disclosures.
Websites may treat cloud-based IP addresses or automated browsers differently
[140, 141, 39]. To curb such effects, we used the anti-bot detection features
of TRC [93]. Reviewing the screenshots captured during the visits, we observed
very few blocked visits.
We conducted four sets of inner link collection crawls: two from NYC and two
from Frankfurt, encompassing both desktop and mobile crawls. This constraint
does not appear to impact the success rate of visits across these vantage
points; nonetheless, future research could explore the possibility of
identifying inner pages during the crawling process.
Since we use a fresh profile for each page visit, we may not capture re-
targeted or other personalized ads that are only shown to users with a
behavioral profile. Future work could extend our method to incorporate
personas and warm-up crawls to study such ads. Overall we do not claim that
our findings are representative of tracking and advertising practices on
child-directed websites. Our focus in this study is not on how ads are
targeted, but simply on whether the targeting is enabled or not.
## 7 Conclusion
We presented an empirical study of online tracking and advertisements on over
2,000 child-directed websites. Building a lightweight and versatile ML
pipeline to analyze ad content, we identify hundreds of cases of improper ads,
including weight loss and mental health ads, and ads featuring dating
services, racy and sexually suggestive imagery. Our study reveals several
notable trends: websites featuring advertisements tend to contain two to four
times more trackers, mobile websites exhibit a greater prevalence of
inappropriate ads, and popular websites are less likely to deploy targeted
advertisements. Our findings provide concrete evidence of troublesome
practices that are likely illegal, unethical, or simply careless. We call for
more research, regulation and enforcement to limit the ongoing violation of
children’s privacy and well-being.
## 8 Acknowledgments
Asuman Senol was funded by the Cyber-Defence (CYD) Campus of armasuisse
Science and Technology. Veelasha Moonsamy was funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy - EXC 2092 CASA - 390781972.
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## Appendix A Criteria for labeling child-directed websites
To identify a child-directed website, we manually visit and review its design,
content, and policies, including necessary translations to English. A site is
labeled as child-directed if any of the following conditions are met:
* •
Does the website include content, activities, or games that can be used by
children?
* •
Does the website promote products (e.g., apps, sites, books, videos,
workshops, animations, etc.) designed for and usable by children online?
* •
Does the website include content or promote products whose end users are
children, but children’s parents must first subscribe or register?
A site is not child-directed if one of the following is true:
* •
The website redirects to another page that is not child-directed.
* •
The website features children-related products intended for adult use, such as
parents or teachers.
* •
The website is generally appealing to adults (e.g., news or academic
websites).
### A.1 Manual Verification of the Classifier Output
Two researchers manually labeled a total of 2,500 websites detected as child-
directed by the classifier. Initially, both researchers jointly labeled a
50-website sample, reaching agreement on 45 of these decisions (Cohen’s
Kappa=0.79 [142]). The remaining websites were divided into two equal batches
and labeled individually by the two researchers. This process took
approximately one person-week to complete. We followed the criteria for
identifying child-directed websites (Appendix A) and considered four potential
labels for each website: child-directed, child-related, non-children and
unknown. The majority of the websites (64%) were labeled as child-directed, ,
while 23% were identified as child-related, indicating they are relevant to
children but primarily intended for parents or teachers. About 4.5% of the
websites were inaccurately classified as non-children’s sites, such as
entertainment bands, academic forums, and movie streaming services, mostly due
to vague or brief titles and descriptions. In some cases, discerning whether
websites were targeted at children, parents, or teachers was challenging,
leading to 5.5% being labeled as ‘unknown’ due to uncertainty about their
target audience. Of the misclassified websites, we found that four were adult
entertainment websites (0.16%) that had very short metadata fields mentioning
words such as “teens”, “cartoon”, “animations” which likely caused the
misclassification.
### A.2 Ad Transparency Statements
The following ad transparency statements are used to classify advertisements
as targeted or non-targeted. Note that targeted categories also include
retargeting and behavioral ads. The statements are compiled from Google’s and
Criteo’s ad disclosure interfaces, reached via the AdChoices icon.
When searching for the statements, we use exact, case-insensitive search.
Targeted:
* •
Google’s estimation of your interests
* •
Websites you’ve visited
* •
Your similarity to groups of people the advertiser is trying to reach
* •
Your activity on Google on this device
* •
According to your activity on this device
* •
You have enabled ad personalization
* •
Information collected by the publisher. The publisher partners with Google to
show ads
* •
Google’s estimation of the languages you know, based on your activity on this
device
* •
Your visit to the advertiser’s website or app
* •
The advertiser’s interest in reaching new customers who haven’t bought
something from them before
Non-Targeted:
* •
Ad personalization is turned off
* •
You have turned off ad personalization
* •
Ad personalization is off
* •
The time of day or your general location
* •
Google shows ads based on general factors like the time of day and the info on
a page, our policies, and your ad personalization settings
* •
The information on the website you were viewing
* •
General factors about the placement of the ad
### A.3 Detecting failed or errored visits
We classified a visit as failed under the following conditions: if the first
request elicited a 4XX or 5XX error; if the size of the first non-3XX response
(root document) was less than 512 bytes, following Le Pochat et al. [26]; or
if there was no successful (200 OK) response.
|
# L2 PROFICIENCY ASSESSMENT USING SELF-SUPERVISED SPEECH REPRESENTATIONS
###### Abstract
There has been a growing demand for automated spoken language assessment
systems in recent years. A standard pipeline for this process is to start with
a speech recognition system and derive features, either hand-crafted or based
on deep-learning, that exploit the transcription and audio. Though these
approaches can yield high performance systems, they require speech recognition
systems that can be used for L2 speakers, and preferably tuned to the specific
form of test being deployed. Recently a self-supervised speech representation
based scheme, requiring no speech recognition, was proposed. This work extends
the initial analysis conducted on this approach to a large scale proficiency
test, Linguaskill, that comprises multiple parts, each designed to assess
different attributes of a candidate’s speaking proficiency. The performance of
the self-supervised, wav2vec 2.0, system is compared to a high performance
hand-crafted assessment system and a BERT-based text system both of which use
speech transcriptions. Though the wav2vec 2.0 based system is found to be
sensitive to the nature of the response, it can be configured to yield
comparable performance to systems requiring a speech transcription, and yields
gains when appropriately combined with standard approaches.
Index Terms— automatic assessment of spoken language proficiency, computer-
assisted language learning
## 1 Introduction
In recent years, the growing number of learners of English as a second
language (L2) on a global scale has created an increasing demand for automated
spoken language assessment systems for applications in Computer-Assisted
Language Learning (CALL) in private practice and classroom situations and to
certify language exams. One of the compelling reasons for automatic assessment
is the need to evaluate and provide feedback to increasing numbers of learners
and return results in a timely manner. Secondly, compared to human graders,
not only can automatic systems ensure greater speed, but they can do it at a
lower cost, since the recruitment and training of new human experts is
expensive and can offer only a small increase in performance. Finally, the use
of automatic assessment methods can enhance consistency, reliability, and
objectivity of scoring, since machines are not susceptible to rater effects
and - more simply - to tiredness [1].
Automated systems for assessing L2 speaking proficiency typically receive
sequential data from a learner as input to predict their grade or level with
respect to overall proficiency or specific facets of proficiency. This input
data may consist - as needed - of phones, recognised words, acoustic features,
or other information derived directly from audio or from automatic speech
recognition (ASR) transcriptions. In most cases, sets of hand-crafted features
related to specific aspects of proficiency, such as fluency [2], pronunciation
[3], prosody [4] and text complexity [5], are extracted and fed into graders
to predict analytic grades related to those specific aspects. An alternative
but similar approach consists of concatenating multiple hand-crafted features
related to multiple aspects of proficiency in order to obtain overall feature
sets, which are subsequently projected through graders to predict holistic
grades, as shown in [6, 7, 8, 9]. The efficacy of hand-crafted features for
scoring either overall proficiency or individual aspects heavily relies on
their specific underlying assumptions, and salient information about
proficiency may risk being discarded. For holistic grading, this limitation
has been addressed by substituting hand-crafted features with automatically
derived features for holistic grading prediction, either through end-to-end
systems [10] or in multiple stages [11, 12]. Other studies have investigated
the use of graders that are trained on holistic scores but are defined with
both their inputs and structure adapted to target particular aspects of
proficiency, such as pronunciation [13], rhythm [14] and text [15, 16]. In
these cases, a possible limitation might be the absence of information
concerning facets of proficiency that are not present in the input data fed to
the grader, although we have shown that it is possible to combine multiple
graders targeting different aspects of proficiency in a previous study [17].
Such a limitation is particularly evident in systems using ASR transcriptions,
since a) they will contain errors and so will not provide faithful verbatim
transcriptions of a learner’s performance, thus not rendering its content
appropriately; b) transcriptions do not contain any information relating to
the message realisation, such as fluency, pronunciation, intonation, rhythm,
or prosody111Some information about pronunciation can be obtained from the
performance of the ASR system and associated confidence scores, whereas
hesitation markers, truncated words, and other disfluencies (if available from
the ASR transcript) might be considered proxies of fluency.. Instead, they
represent a precious resource for highly specific tasks in CALL applications,
e.g., for spoken grammatical error correction and feedback, as we have shown
in [18].
In this paper, in order to tackle these issues and address these limitations,
we propose an approach based on self-supervised speech representations using
wav2vec 2.0 [19]. Recent studies have shown that self-supervised learning
(SSL) is an effective approach in various downstream tasks of speech
processing applications, such as ASR, keyword spotting, emotion recognition,
intent classification, speaker identification, and speaker diarisation [19,
20]. In these studies, contextual representations were applied by means of
pre-trained models. In particular, it has been demonstrated that these models
are able to capture a vast range of speech-related features and linguistic
information, such as audio, fluency, suprasegmental pronunciation, and even
semantic and syntactic text-based features for L1, L2, read and spontaneous
speech [21]. In the field of CALL, SSL has been investigated for
mispronunciation detection and diagnosis [22, 23, 24] and automatic
pronunciation assessment [25].
To the best of our knowledge, our previous work [26] was the first to propose
the use of SSL for holistic and analytic proficiency assessment. Nevertheless,
this study had two limitations: a) the relatively small amount of data used in
the experiments and b) the comparison with a BERT-based [27] baseline only,
which, although fed with manual transcriptions containing hesitations, false
starts, and truncated words, did not consider purely acoustic features, thus
potentially missing strictly speech-related aspects of proficiency. To address
these limitations, in the present contribution, we conduct our experiments of
proficiency assessment using a large amount of L2 learner data and comparing
the performance of wav2vec 2.0 to two types of grader: a BERT-based grader and
a standard grader fed with a set of hand-crafted features across different
facets of proficiency (see Figure 1). In addition to this, we test the
effectiveness of wav2vec 2.0 on a multi-part examination predicting both the
overall grades and the individual grades of each part of the exam.
Furthermore, we investigate various combinations between the wav2vec2-based,
the BERT-based, and the standard graders.
In Section 2, we describe the data used in our experiments. Section 3 shows
the model architectures and the combinations considered in our study. Section
4 illustrates the results of our experiments. Finally, in Section 5, we
discuss and analyse the results.
Fig. 1: The three systems considered in this study: a) standard grader, b)
BERT-based grader, and c) wav2vec2-based grader.
## 2 DATA
The data used in our study are obtained from candidate responses to the spoken
parts of the Linguaskill examinations for L2 learners of English, provided by
Cambridge English Language Assessment [28]. The Linguaskill speaking exam is
divided into five parts. In Part 1, the candidates should answer eight
questions about themselves, of which the first two are not marked. The answers
last about 10 or 20 seconds. Part 2 consists of a reading aloud activity
including eight sentences of 10 seconds each. Part 3 and Part 4 test the
candidates’ ability to deliver a long turn, speaking for up to one minute. In
the first, the candidates should talk about a given topic, whereas in the
latter they should describe one or more graphics, such as charts, diagrams, or
information sheets. Finally, in Part 5, test-takers should give their opinions
in form of responses of about 20 seconds to five questions related to a given
topic. Since each part contributes 20% to the speaking exam, the overall grade
is calculated as the average of the grades assigned to the five parts.
Each speaker is graded on a scale from 1 to 6 based on the proficiency scales
of the Common European Framework of Reference (CEFR) for languages [29], i.e.,
A1, A2, B1, B2, C1, and C2.
Non-overlapping datasets of 31475 and 1033 speakers are used as the training
and development/calibration set, respectively. For evaluation, we consider two
test sets, LinGen and LinBus, of 1049 and 712 speakers, respectively. LinGen
includes learners’ responses to questions on General English, while LinBus
contains answers to questions on Business English. Each test set features
around 30 L1s and is balanced for gender and proficiency level.
The text transcriptions for training and test used by the standard and BERT-
based graders are generated using a non-native English ASR system. A Kaldi-
based system with a TDNN-F acoustic model and a trigram language model is used
similar to that in [30] with average WER of $\sim$20%.
## 3 MODEL ARCHITECTURES
Wav2vec2-based graders: in wav2vec 2.0, speech audio is encoded through a
multilayer convolutional neural network (CNN). After encoding, masking is
applied to spans of the resulting latent representations. Subsequently, these
are fed into a transformer to build contextualised representations. Gumbel
softmax is used to calculate the contrastive loss on which the model is
trained, and speech representations are learned from this training.
For our experiments, we initialised the model configuration and processor from
a version provided by HuggingFace
[31]222huggingface.co/patrickvonplaten/wav2vec2-base. After the learners’
answers are fed into the model, wav2vec 2.0 provides contextualised
representations. To handle representations of various audio lengths, we
employed a mean pooling method to concatenate 3D representations into 2D
representations, which are finally passed through a regression head [26].
Since we trained a grader for each part of the exam, after trying different
architectures, for Part 1 and Part 5 we used a regression head composed of a
layer of 768 units, a Dropout layer, and the output layer, whereas, for Part
2, Part 3 and Part 4, we used a deeper architecture, consisting of a stack of
three layers of 768 units, a Dropout layer, a layer of 128 units, and,
finally, the output layer. The graders use mean squared error (MSE) as the
loss function. Training uses the AdamW optimiser [32] and hyperparameters vary
depending on each part. For Part 1, we used batch size 16, gradient
accumulation step 2, dropout rate 0.1, and learning rate 5e-5, and we trained
the grader for 2 epochs. For Part 2, we used batch size 16, gradient
accumulation step 2, dropout rate 0.5, and learning rate 1e-6, and we trained
the grader for 3 epochs. For Part 3 and Part 4, the hyperparameters are the
same: we set batch size to 8, gradient accumulation step to 4, dropout rate to
0.5, and learning rate to 1e-5, and we trained the grader for 2 epochs.
Finally, the grader for Part 5 has batch size set to 8, gradient accumulation
step to 2, dropout rate to 0.1, and learning rate to 5e-5, and we trained it
for 1 epoch.
As has been said, the first part of wav2vec 2.0 consists of a stack of CNN
layers that are used to obtain acoustically meaningful - but contextually
independent - features from the raw speech audio signal. This part of the
model has already been sufficiently trained during the pre-training stage and
does not need fine-tuning. Therefore, for our experiments we froze the
parameters of the feature extractor.
BERT-based graders: for comparison, we use the text grader presented in [16],
which consists of an LSTM with attention over its hidden representation. The
inputs are word embeddings obtained by passing the words of each utterance
through a trained BERT language model [27].
Standard graders: we also compare our SSL approach to a standard grader, which
is a Deep Density Network (DDN) trained on a set of hand-crafted features
designed to cover all the different aspects of proficiency and is described in
[33]. These features include: grade dependent language model and word level
statistics; statistics of phone duration; statistics to capture rhythm;
fluency metrics; and fundamental frequency statistics are used to represent
intonation. More detailed information about the features employed can be found
in [8].
For all graders, their predictions are the result of ensembles. Further
information about the ensemble approach can be found in [34]. Systems are
calibrated and in a final set of experiments combined using linear
combination:
${\hat{y}}^{(n)}=\beta_{0}+\sum_{p\in{\cal P}}\beta_{p}{\hat{y}}^{(n)}_{p}$
(1)
where ${\cal P}$ is the set of parts to combine, which may come from multiple
systems, and $\beta_{p}$ are the coefficients associated with the parts. For
the baseline submission performance the values of $\beta_{p},p>0$ are all
constrained to be the same, and equal to 0.2, to yield simple averaging
consistent with the combination of operational examiner scores. Ordinary Least
Squares (OLS) estimation using the development/calibration set is used to find
the values of $\beta_{p}$ when unequal weighting is used.
For the evaluation of the grading systems at the per-part level, we use root-
mean-square error (RMSE), whilst further comparisons also include Pearson’s
correlation coefficient (PCC), Spearman’s rank correlation coefficient (SRC),
and the percentage of the predicted scores that are equal to or lie within 0.5
(i.e., within half a grade) ($\%\leq{}0.5$), and within 1.0 (i.e., one grade)
($\%\leq{}1.0$) of the actual score.
## 4 EXPERIMENTAL RESULTS AND ANALYSIS
Part-Level Performance: we start our series of experiments with grading each
of the five parts of the exam. For this part of our analysis we only consider
LinGen. Table 1 reports the results of the three grading systems in terms of
RMSE.
Model | P1 | P2 | P3 | P4 | P5
---|---|---|---|---|---
std (${\tt s_{d}}$) | 0.625 | 0.662 | 0.671 | 0.686 | 0.633
BERT (${\tt b_{t}}$) | 0.628 | 0.683 | 0.681 | 0.694 | 0.629
wav2vec2 (${\tt w_{v}}$) | 0.601 | 0.827 | 0.845 | 0.845 | 0.674
Table 1: RMSE results on the five parts of the LinGen exam.
We can see that the wav2vec 2.0 performance varies across the parts, with
close or better RMSE to the other two graders for Parts 1 and 5 and lower
performance on Parts 2, 3 and 5. This appears to be due to the nature of the
responses required for different parts. Parts 1 and 5 consist of several short
spontaneous answers, whereas Parts 3 and 4 are also composed of spontaneous
speech but a single longer and more complex response in each case. The lower
wav2vec 2.0 performance may be due to our use of a mean pooling method, which
may be giving too compressed a representation for longer utterances.
Part 2, by contrast, is similar in length to Part 1 but consists of read
speech responses. This part mainly targets pronunciation and fluency skills at
the expense of content-related aspects of proficiency so the wav2vec 2.0
system might have been expected to do well. Its higher RMSE might be due to
the absence of information related to the reference text read by the test-
takers, which is present in the other two grading systems. It is noticeable
that the standard grader, which covers all aspects of a candidate’s speech,
performs the best, with the BERT grader which cannot measure pronunciation or
prosody slightly behind.
Submission-Level Performance: the second part of our experiments is focused on
the overall grades of the Linguaskill exam, i.e., the average of the grades
assigned to the five parts. Table 2 shows the results of the three grading
systems both on LinGen and LinBus.
| LinGen
---|---
Model | PCC | SRC | RMSE | %$\leq$0.5 | %$\leq$1.0
${\tt s_{d}}$ | 0.932 | 0.937 | 0.383 | 81.5 | 98.6
${\tt b_{t}}$ | 0.929 | 0.934 | 0.395 | 80.3 | 98.5
${\tt w_{v}}$ | 0.908 | 0.931 | 0.455 | 73.3 | 97.3
| LinBus
---|---
Model | PCC | SRC | RMSE | %$\leq$0.5 | %$\leq$1.0
${\tt s_{d}}$ | 0.911 | 0.918 | 0.416 | 76.5 | 98.3
${\tt b_{t}}$ | 0.920 | 0.925 | 0.398 | 80.1 | 99.2
${\tt w_{v}}$ | 0.893 | 0.911 | 0.446 | 72.1 | 97.9
Table 2: Submission-level performance on LinGen and LinBus.
They all achieve good results across all metrics, with the standard grader and
the BERT-based grader performing moderately better than the wav2vec2-based
grading system. As regards the BERT-based grader specifically, we have already
observed analogous trends in [16] and [17], and this fact is quite
significant, since it highlights the importance of content-related aspects of
speaking proficiency, which is far from being a mere surrogate of fluency and
pronunciation. Furthermore, it appears that the standard grading system
outperforms the other two graders on LinGen, but has a slightly worse
performance than the BERT-based grader on LinBus. This might be ascribable to
the different language models, since the first test set contains questions on
General English, whereas the latter includes questions on Business English,
which is typically more specific and complex.
Figure 2 shows that the wav2vec2-based graders can discriminate between lower
levels of proficiency, but we can clearly see that it is not able to
distinguish between the highest levels as its maximum prediction is 4.6, i.e.,
between grades B2 and C1 (the plots for LinBus show a similar trend). Our
hypothesis is that higher-level assessment tends to be more dependent on
message construction (what is said) rather than message realisation (how it is
said), and wav2vec 2.0 does not have actual knowledge of words.
Combinations: as a preliminary step, we investigated combinations of the
grading systems by calculating their simple average and using a multiple
linear regression model fit with the submission-level predictions, but they
did not provide significant gains. Therefore, we investigated the application
of a multiple linear regression model using the per-part predictions as
predictors for each individual grading system and for four combinations of
them. The results on LinGen and LinBus are reported in Table 4. The
combinations show performances that are aligned to or better than the
individual models across all metrics, with the combination including all three
grading systems achieving the best results. This combination also overcomes
the issue of the wav2vec2-based grader related to scoring higher levels, as
can be seen in Figure 3a. Table 3 reports the $\beta$ coefficients of the
individual models described in Table 4 and the combination of all three. In
the standard grader, as well as in the BERT-based grader, Parts 1, 2 and 5
affect the linear model most, whereas in the wav2vec2-based grader Part 1 and
5 are the most influential. In their combination, it appears that the highest
$\beta$ coefficients are Parts 1 and 5 of the wav2vec2-based grader and Part 2
of the BERT-based and the standard graders. These values seem to confirm the
RMSE results shown in Table 1.
(a)
std
(b)
wav2vec 2.0
Fig. 2: Reference vs predicted scores for standard and wav2vec2-based graders
on LinGen. (a) Fig. 3: Reference vs predicted scores for combined system
(${\tt s_{d}}$$\otimes$${\tt b_{t}}$$\otimes$${\tt w_{v}}$)
per-part combination ${\tt s_{d}}$$\otimes$${\tt b_{t}}$$\otimes$${\tt w_{v}}$
Model | P1 | P2 | P3 | P4 | P5 | $\beta_{0}$
---|---|---|---|---|---|---
${\tt s^{\otimes}_{d}}$ | 0.23 | 0.25 | 0.14 | 0.15 | 0.22 | -0.11
${\tt b^{\otimes}_{t}}$ | 0.20 | 0.26 | 0.13 | 0.17 | 0.23 | -0.13
${\tt w^{\otimes}_{v}}$ | 0.29 | 0.05 | 0.01 | 0.01 | 0.45 | 0.76
${\tt s_{d}}$$\otimes$${\tt b_{t}}$$\otimes$${\tt w_{v}}$ | ${\tt s_{d}}$ | -0.01 | 0.12 | 0.06 | 0.01 | -0.04 | 0.20
${\tt b_{t}}$ | 0.06 | 0.16 | 0.05 | 0.09 | 0.09
${\tt w_{v}}$ | 0.20 | -0.08 | -0.02 | 0.02 | 0.20
Table 3: $\beta$ coefficients of per-part linear regression model for the
standard (${\tt s^{\otimes}_{d}}$), BERT (${\tt b^{\otimes}_{t}}$), wav2vec2
(${\tt w^{\otimes}_{v}}$), and combination (${\tt s_{d}}$$\otimes$${\tt
b_{t}}$$\otimes$${\tt w_{v}}$) estimated on the calibration data.
| LinGen
---|---
Model | PCC | SRC | RMSE | %$\leq$0.5 | %$\leq$1.0
${\tt s^{\otimes}_{d}}$ | 0.932 | 0.937 | 0.382 | 82.3 | 98.7
${\tt b^{\otimes}_{t}}$ | 0.930 | 0.935 | 0.393 | 80.3 | 98.6
${\tt w^{\otimes}_{v}}$ | 0.933 | 0.937 | 0.393 | 79.7 | 99.0
${\tt s_{d}}$$\otimes$${\tt w_{v}}$ | 0.941 | 0.945 | 0.363 | 84.5 | 99.3
${\tt s_{d}}$$\otimes$${\tt b_{t}}$ | 0.936 | 0.940 | 0.373 | 81.9 | 98.8
${\tt b_{t}}$$\otimes$${\tt w_{v}}$ | 0.943 | 0.947 | 0.359 | 84.3 | 99.2
${\tt s_{d}}$$\otimes$${\tt b_{t}}$$\otimes$${\tt w_{v}}$ | 0.943 | 0.947 | 0.356 | 85.0 | 99.1
| LinBus
---|---
Model | PCC | SRC | RMSE | %$\leq$0.5 | %$\leq$1.0
${\tt s^{\otimes}_{d}}$ | 0.912 | 0.920 | 0.415 | 77.0 | 99.0
${\tt b^{\otimes}_{t}}$ | 0.920 | 0.924 | 0.400 | 80.1 | 99.0
${\tt w^{\otimes}_{v}}$ | 0.916 | 0.919 | 0.394 | 79.1 | 99.0
${\tt s_{d}}$$\otimes$${\tt w_{v}}$ | 0.925 | 0.928 | 0.378 | 82.0 | 99.4
${\tt s_{d}}$$\otimes$${\tt b_{t}}$ | 0.925 | 0.929 | 0.391 | 80.8 | 99.4
${\tt b_{t}}$$\otimes$${\tt w_{v}}$ | 0.930 | 0.932 | 0.368 | 82.7 | 99.3
${\tt s_{d}}$$\otimes$${\tt b_{t}}$$\otimes$${\tt w_{v}}$ | 0.931 | 0.933 | 0.366 | 82.5 | 99.4
Table 4: Results on overall grades on LinGen and LinBus using per-part linear
regression estimated on the calibration data.
## 5 CONCLUSIONS
In this study, we have extended our recent novel approach to proficiency
assessment using a wav2vec2-based grader applying it to a large quantity of L2
learner data. First, we compared its performance on the five parts of the
Linguaskill exam to a high performing standard grader fed with hand-crafted
features and a BERT-based grader. We found that our proposed approach appears
to be sensitive to the nature of the responses for a part with good
performance for parts consisting of short spontaneous answers. Secondly, we
found that the three grading systems have comparable performances on overall
grades, with the wav2vec2-based grader showing some difficulties in assessing
higher levels. Finally, we combined the standard and the wav2vec2-based
graders by means of various linear combinations and we found interesting
improvements. A concern with the wav2vec2 and BERT-based graders is that they
are not fully valid alone since neither considers all aspects of the
assessment construct and their results are not interpretable to provide
feedback to a learner. As well as boosting the assessment performance,
combination with the standard hand-crafted feature grader removes these
concerns.
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|
# Non-abelian extensions and Wells exact sequences of Lie-Yamaguti algebras
Qinxiu Sun Department of Mathematics, Zhejiang University of Science and
Technology, Hangzhou, 310023<EMAIL_ADDRESS>and Zhen Li Department of
Mathematics, Zhejiang University of Science and Technology, Hangzhou, 310023
<EMAIL_ADDRESS>
###### Abstract.
The goal of the present paper is to investigate non-abelian extensions of Lie-
Yamaguti algebras and explore extensibility of a pair of automorphisms about a
non-abelian extension of Lie-Yamaguti algebras. First, we study non-abelian
extensions of Lie-Yamaguti algebras and classify the non-abelian extensions in
terms of non-abelian cohomology groups. Next, we characterize the non-abelian
extensions in terms of Maurer-Cartan elements. Moreover, we discuss the
equivalent conditions of the extensibility of a pair of automorphisms about a
non-abelian extension of Lie-Yamaguti algebras, and derive the fundamental
sequences of Wells in the context of Lie-Yamaguti algebras. Finally, we
discuss the previous results in the case of abelian extensions of Lie-Yamaguti
algebras.
###### Key words and phrases:
Lie-Yamaguti algebra, non-abelian extension, Maurer-Cartan element,
extensibility, Wells exact sequence
###### 2010 Mathematics Subject Classification:
17A30, 17A36, 17A40, 17B99
###### Contents
1. 1 Introduction
2. 2 Preliminaries on Lie-Yamaguti algebras
3. 3 Non-abelian extensions and non-abelian (2,3)-cocycles of Lie-Yamaguti algebras
4. 4 Non-abelian extensions in terms of Maurer-Cartan elements
5. 5 Extensibility of a pair of Lie-Yamaguti algebra automorphisms
6. 6 Wells exact sequences for Lie-Yamaguti algebras
7. 7 Particular case: abelian extensions of Lie-Yamaguti algebras
## 1\. Introduction
A Lie-Yamaguti algebra first appeared in Nomizu’s work on the affine invariant
connections on homogeneous spaces in 1950’s [23], which was a generalization
of Lie algebras and Lie triple systems. In 1960’s, Yamaguti introduced an
algebraic structure and named it a general Lie triple system or a Lie triple
algebra [34, 33]. Later, the cohomology theory of this object was investigated
in [34] by Yamaguti. In the study of Courant algebroids, Kinyon and Weinstein
named general Lie triple systems by Lie–Yamaguti algebras [19]. Since then,
Lie-Yamaguti algebras have attracted much attention and are widely explored.
For example, the irreducible modules of Lie–Yamaguti algebras were considered
in [3, 2], deformations and extensions of Lie-Yamaguti algebras were
investigated in [20, 35]. The relative Rota-Bxter operators on Lie-Yamaguti
algebras and their cohomologies were considered in [25, 31, 32].
Extensions are useful mathematical objects to understand the underlying
structures. The non-abelian extension is a relatively general one among
various extensions (e.g. central extensions, abelian extensions, non-abelian
extensions etc.). Non-abelian extensions were first developed by Eilenberg and
Maclane [10], which induced to the low dimensional non-abelian cohomology
group. Then numerous works have been devoted to non-abelian extensions of
various kinds of algebras, such as Lie (super)algebras, Leibniz algebras, Lie
2-algebras, Lie Yagamuti algebras, associative conformal algebras, Rota-Baxter
groups, Rota-Baxter Lie algebras and Rota-Baxter Leibniz algebras, see [5, 7,
11, 13, 14, 17, 21, 22, 16] and their references. The abelian extensions of
Lie Yagamuti algebras were considered in [13]. But little is known about the
non-abelian extensions of Lie Yagamuti algebras. This is the first motivation
for writing this paper.
Another interesting study related to extensions of algebraic structures is
given by the extensibility or inducibility of a pair of automorphisms. When a
pair of automorphisms is inducible? This problem was first considered by Wells
[29] for abstract groups and further studied in [18, 24]. Since then, several
authors have studied this subject further, see [16, 13, 14, 15, 22] and
references therein. The extensibility problem of a pair of derivations on
abelian extensions was investigated in [6, 30]. Recently, the extensibility
problem of a pair of derivations and automorphisms was extended to the context
of abelian extensions of Lie coalgebras [9]. As byproducts, the Wells short
exact sequences were obtained for various kinds of algebras [7, 8, 13, 14, 15,
18, 22, 16], which connected the relative automorphism groups and the non-
abelian second cohomology groups. Inspired by these results, we investigate
extensibility of a pair of automorphisms on a non-abelian extension of Lie
Yagamuti algebras. This is another motivation for writing the present paper.
Moreover, we give necessary and sufficient conditions for a pair of
automorphisms to be extensible, and derive the analogue of the Wells short
exact sequences in the context of non-abelian extensions of Lie Yagamuti
algebras.
The paper is organized as follows. In Section 2, we recall the definition of
Lie-Yamaguti algebras and their representations. We also recall some basic
information about the cohomology groups of Lie-Yamaguti algebras. In Section
3, we investigate non-abelian extensions and classify the non-abelian
extensions using the non-abelian cohomology groups. In Section 4, we
characterize equivalent non-abelian extensions using Maurer-Cartan elements.
In Section 5, we study the extensibility problem of a pair of automorphisms
about a non-abelian extension of Lie-Yamaguti algebras. In Section 6, we
derive Wells short exact sequences in the context of non-abelian extensions of
Lie-Yamaguti algebras. Finally, we discuss the previous results in the case of
abelian extensions of Lie-Yamaguti algebras.
Throughout the paper, let $k$ be a field. Unless otherwise specified, all
vector spaces and algebras are over $k$.
## 2\. Preliminaries on Lie-Yamaguti algebras
We recall the notions of Lie-Yamaguti algebras, representations and their
cohomology theory. For the details see [34, 33].
###### Definition 2.1.
A Lie-Yamaguti algebra is a vector space $\mathfrak{g}$ with a bilinear map
$[\cdot,\cdot]:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$ and
a trilinear map $\\{\ ,\ ,\
\\}:\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$,
satisfying
(1)
$[x_{1},x_{2}]+[x_{2},x_{1}]=0,~{}~{}\\{x_{1},x_{2},x_{3}\\}+\\{x_{2},x_{1},x_{3}\\}=0,$
(2)
$[[x_{1},x_{2}],x_{3}]+[[x_{2},x_{3}],x_{1}]+[[x_{3},x_{1}],x_{2}]+\\{x_{1},x_{2},x_{3}\\}+\\{x_{2},x_{3},x_{1}\\}+\\{x_{3},x_{1},x_{2}\\}=0,$
(3)
$\\{[x_{1},x_{2}],x_{3},y_{1}\\}+\\{[x_{2},x_{3}],x_{1},y_{1}\\}+\\{[x_{3},x_{1}],x_{2},y_{1}\\}=0,$
(4)
$\\{x_{1},x_{2},[y_{1},y_{2}]\\}=[\\{x_{1},x_{2},y_{1}\\},y_{2}]+[y_{1},\\{x_{1},x_{2},y_{2}\\}],$
(5)
$\\{x_{1},x_{2},\\{y_{1},y_{2},y_{3}\\}\\}=\\{\\{x_{1},x_{2},y_{1}\\},y_{2},y_{3}\\}+\\{y_{1},\\{x_{1},x_{2},y_{2}\\},y_{3}\\}+\\{y_{1},y_{2},\\{x_{1},x_{2},y_{3}\\}\\},$
for all $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}\in\mathfrak{g}$. Denote it by
$(\mathfrak{g},[\ ,\ ],\\{\ ,\ ,\ \\})$ or simply by $\mathfrak{g}$.
A subspace $L$ of $\mathfrak{g}$ is an ideal of $\mathfrak{g}$ if
$[L,\mathfrak{g}]\subseteq L,\\{L,\mathfrak{g},\mathfrak{g}\\}\subseteq L$ and
$\\{\mathfrak{g},\mathfrak{g},L\\}\subseteq L$. An ideal $L$ of $\mathfrak{g}$
is said to be an abelian ideal of $\mathfrak{g}$ if $[L,L]=0$ and
$\\{L,L,\mathfrak{g}\\}=\\{\mathfrak{g},L,L\\}=\\{L,\mathfrak{g},L\\}=0$.
###### Definition 2.2.
A representation of a Lie-Yamaguti algebra $\mathfrak{g}$ consists of a vector
space $V$ together with a linear map $\mu:\mathfrak{g}\longrightarrow{gl}(V)$
and bilinear maps
$\theta,D:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow{gl}(V)$ satisfying
(6)
$\theta([x_{1},x_{2}],x_{3})=\theta(x_{1},x_{3})\mu(x_{2})-\theta(x_{2},x_{3})\mu(x_{1}),$
(7) $[D(x_{1},x_{2}),\mu(y_{1})]=\mu(\\{x_{1},x_{2},y_{1}\\}),$ (8)
$\theta(x_{1},[y_{1},y_{2}])=\mu(y_{1})\theta(x_{1},y_{2})-\mu(y_{2})\theta(x_{1},y_{1}),$
(9)
$[D(x_{1},x_{2}),\theta(y_{1},y_{2})]=\theta(\\{x_{1},x_{2},y_{1}\\},y_{2})+\theta(y_{1},\\{x_{1},x_{2},y_{2}\\}),$
(10)
$\theta(x_{1},\\{y_{1},y_{2},y_{3}\\})=\theta(y_{2},y_{3})\theta(x_{1},y_{1})-\theta(y_{1},y_{3})\theta(x_{1},y_{2})+D(y_{1},y_{2})\theta(x_{1},y_{3}),$
(11)
$D(x_{1},x_{2})-\theta(x_{2},x_{1})+\theta(x_{1},x_{2})+\mu([x_{1},x_{2}])-[\mu(x_{1}),\mu(x_{2})]=0,$
for all $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}\in\mathfrak{g}$. Denote the
representation of $\mathfrak{g}$ by $(V,\mu,\theta,D)$ or simply by $V$.
When $(V,\mu,\theta,D)$ is a representation of $\mathfrak{g}$, by a direct
computation, the following conditions are also satisfied:
(12) $D([x_{1},x_{2}],x_{3})+D([x_{2},x_{3}],x_{1})+D([x_{3},x_{1}],x_{2})=0,$
(13)
$D(\\{x_{1},x_{2},x_{3}\\},x_{4})+D(x_{3},\\{x_{1},x_{2},x_{4}\\})=[D(x_{1},x_{2}),D(x_{3},x_{4})],$
(14)
$\theta(\\{y_{1},y_{2},y_{3}\\},x_{1})=\theta(y_{1},x_{1})\theta(y_{3},y_{2})-\theta(y_{2},x_{1})\theta(y_{3},y_{1})-\theta(y_{3},x_{1})D(y_{1},y_{2}).$
###### Proposition 2.3.
Let $(\mathfrak{g},[\ ,\ ]_{\mathfrak{g}},\\{\ ,\ ,\ \\}_{\mathfrak{g}})$ be a
Lie-Yamaguti algebra and $V$ a vector space. Assume that
$\mu:\mathfrak{g}\longrightarrow{gl}(V)$ is a linear map and
$\theta,D:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow{gl}(V)$ are bilinear
maps. Then $(V,\mu,\theta,D)$ is a representation of $\mathfrak{g}$ if and
only if $(\mathfrak{g}\oplus V,[\ ,\ ],\\{\ ,\ ,\ \\})$ is a Lie-Yamaguti
algebra, where
$\\{x+u,y+v,z+w\\}=\\{x,y,z\\}_{\mathfrak{g}}+\theta(y,z)u-\theta(x,z)v+D(x,y)w,$
and
$[x+u,y+v]=[x,y]_{\mathfrak{g}}+\mu(x)v-\mu(y)u,$
for all $x,y,z\in T,u,v,w\in V$. The Lie-Yamaguti algebra $(\mathfrak{g}\oplus
V,[\ ,\ ],\\{\ ,\ ,\ \\})$ is called the semidirect product Lie-Yamaguti
algebra. Denote it simply by $\mathfrak{g}\ltimes V$.
###### Example 2.4.
Let $\mathfrak{g}$ be a Lie-Yamaguti algebra. Define
${ad}:\mathfrak{g}\longrightarrow{gl}(\mathfrak{g}),~{}R,L:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow{gl}(\mathfrak{g})$
respectively by ${ad}(x)(y)=[x,y],R(x,y)(z)=\\{z,x,y\\}$ and
$L(x,y)(z)=\\{x,y,z\\}$. Then $(\mathfrak{g},{ad},R,L)$ is a representation of
$\mathfrak{g}$, which is called the adjoint representation.
Next, we recall the cohomology of Lie-Yamaguti algebras following [33]. Let
$\mathfrak{g}$ be a Lie-Yamaguti algebra and $(V,\mu,\theta,D)$ be its
representation. The cochain complex is given as follows:
* •
Set $C^{1}(\mathfrak{g},V)=\mathrm{Hom}(\mathfrak{g},V)$ and
$C^{0}(\mathfrak{g},V)$ be the subspace spanned by the diagonal elements
$(f,f)\in C^{1}(\mathfrak{g},V)\times C^{1}(\mathfrak{g},V)$.
* •
(For $n\geq 2$) Set
$C^{n}(\mathfrak{g},V)=\mathrm{Hom}(\otimes^{n}\mathfrak{g},V)$, where the
space of all $n$-linear maps $f\in C^{n}(\mathfrak{g},V)$ satisfying
$f(x_{1},\cdot\cdot\cdot,x_{2i-1},x_{2i},\cdot\cdot\cdot,x_{n})=0,~{}~{}\hbox{if}~{}x_{2i-1}=x_{2i},~{}\forall~{}i=1,2,\cdot\cdot\cdot,[\frac{n}{2}].$
Then, for all $n\geq 1$, put the $(2n,2n+1)$-cochain groups
$C^{(2n,2n+1)}(\mathfrak{g},V)=C^{2n}(\mathfrak{g},V)\times
C^{2n+1}(\mathfrak{g},V)$
and
$C^{(3,4)}(\mathfrak{g},V)=C^{3}(\mathfrak{g},V)\times C^{4}(\mathfrak{g},V).$
Define the coboundary operator $\delta=(\delta_{I},\delta_{II})$ in the
following cochain complex of $\mathfrak{g}$ with coefficents in $V$:
$\textstyle{C^{0}(\mathfrak{g},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{C^{(2,3)}(\mathfrak{g},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta^{*}}$$\scriptstyle{\delta}$$\textstyle{C^{(4,5)}(\mathfrak{g},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{C^{(6,7)}(\mathfrak{g},V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{\cdot\cdot\cdot}$$\textstyle{C^{(3,4)}(\mathfrak{g},V)}$.
For $n\geq 1$, the coboundary operator
$\delta=(\delta_{I},\delta_{II}):C^{(2n,2n+1)}(\mathfrak{g},V)\longrightarrow
C^{(2n+2,2n+3)}(\mathfrak{g},V)$ is defined by
$\displaystyle(\delta_{I}f)(x_{1},\cdot\cdot\cdot,x_{2n+2})$ $\displaystyle=$
$\displaystyle\mu(x_{2n+1})g(x_{1},\cdot\cdot\cdot,x_{2n},x_{2n+2})-\mu(x_{2n+2})g(x_{1},\cdot\cdot\cdot,x_{2n+1})-g(x_{1},\cdot\cdot\cdot,x_{2n},[x_{2n+1},x_{2n+2}])$
$\displaystyle+\sum_{k=1}^{n}(-1)^{n+k+1}D(x_{2k-1},x_{2k})f(x_{1},\cdot\cdot\cdot,x_{2k-2},x_{2k+1},\cdot\cdot\cdot,x_{2n+2})$
$\displaystyle+\sum_{k=1}^{n}\sum_{j=2k+1}^{2n+2}(-1)^{n+k}f(x_{1},\cdot\cdot\cdot,x_{2k-2},x_{2k+1},\cdot\cdot\cdot,\\{x_{2k-1},x_{2k},x_{j}\\},\cdot\cdot\cdot,x_{2n+2})$
and
$\displaystyle(\delta_{II}g)(x_{1},\cdot\cdot\cdot,x_{2n+3})$ $\displaystyle=$
$\displaystyle\theta(x_{2n+2},x_{2n+3})g(x_{1},\cdot\cdot\cdot,x_{2n+1})-\theta(x_{2n+1},x_{2n+3})g(x_{1},\cdot\cdot\cdot,x_{2n},x_{2n+2})$
$\displaystyle+\sum_{k=1}^{n+1}(-1)^{n+k+1}D(x_{2k-1},x_{2k})g(x_{1},\cdot\cdot\cdot,x_{2k+1},\cdot\cdot\cdot,x_{2n+3})$
$\displaystyle+\sum_{k=1}^{n+1}\sum_{j=2k+1}^{2n+3}(-1)^{n+k}g(x_{1},\cdot\cdot\cdot,x_{2k+1},\cdot\cdot\cdot,\\{x_{2k-1},x_{2k},x_{j}\\},\cdot\cdot\cdot,x_{2n+3})$
for any pair $(f,g)\in C^{(2n,2n+1)}(\mathfrak{g},V)$ and
$x_{1},\cdot\cdot\cdot,x_{2n+3}\in\mathfrak{g}$. And when $n=0$, the
coboundary operator
$\delta=(\delta_{I},\delta_{II}):C^{0}(\mathfrak{g},V)\longrightarrow
C^{2}(\mathfrak{g},V)\times C^{3}(\mathfrak{g},V)$
is defined as follows:
$(\delta_{I}f)(x_{1},x_{2})=\mu(x_{1})f(x_{2})-\mu(x_{2})f(x_{1})-f([x_{1},x_{2}]),$
$(\delta_{II}f)(x_{1},x_{2},x_{3})=\theta(x_{2},x_{3})f(x_{1})-\theta(x_{1},x_{3})f(x_{2})+D(x_{1},x_{2})f(x_{3})-f(\\{x_{1},x_{2},x_{3}\\})$
for any $f\in C^{1}(\mathfrak{g},V)$ and $x_{1},x_{2},x_{3}\in\mathfrak{g}$.
Define
$\delta^{*}=(\delta_{I}^{*},\delta_{II}^{*}):C^{(2,3)}(\mathfrak{g},V)\longrightarrow
C^{(3,4)}(\mathfrak{g},V)$ by
$\displaystyle\delta_{I}^{*}f(x,y,z)=$
$\displaystyle-\mu(x)f(y,z)-\mu(y)f(z,x)-\mu(z)f(x,y)+f([x,y],z)+f([y,z],x)$
$\displaystyle+f([z,x],y)+g(x,y,z)+g(y,z,x)+g(z,x,y),$
$\displaystyle\delta_{II}^{*}f(x,y,z,w)=$
$\displaystyle\theta(x,w)f(y,z)+\theta(y,w)f(z,x)+\theta(z,w)f(x,y)+g([x,y],z,w)$
$\displaystyle+g([y,z],x,w)+g([z,x],y,w),$
for all $(f,g)\in C^{(2,3)}(\mathfrak{g},V)$ and $x,y,z,w\in\mathfrak{g}$.
Denote the set of all the $(2n,2n+1)$-cocycles and the
$(2n,2n+1)$-coboundaries, respectively by $Z^{(2n,2n+1)}(\mathfrak{g},V)$ and
$B^{(2n,2n+1)}(\mathfrak{g},V)$. In particular,
$Z^{(2,3)}(\mathfrak{g},V)=\\{(f,g)\in
C^{(2,3)}(\mathfrak{g},V)|\delta(f,g)=0,\delta^{*}(f,g)=0\\},$
$B^{(2,3)}(\mathfrak{g},V)=\\{(\delta_{I}(f),\delta_{II}(f))|f\in
C^{1}(\mathfrak{g},V)\\}.$
Define $H^{1}(\mathfrak{g},V)=\\{f\in
C^{1}(\mathfrak{g},V)|\delta_{I}(f)=0,\delta_{II}(f)=0\\}$, which is called
the first cohomology group of $\mathfrak{g}$ with coefficients in the
representation $V$ and define
$H^{(2n,2n+1)}(\mathfrak{g},V)=Z^{(2n,2n+1)}(\mathfrak{g},V)/B^{(2n,2n+1)}(\mathfrak{g},V),~{}~{}n\geq
1$
which is called the (2n,2n+1)-cohomology group of $\mathfrak{g}$ with
coefficients in the representation $V$.
## 3\. Non-abelian extensions and non-abelian (2,3)-cocycles of Lie-Yamaguti
algebras
In this section, we are devoted to considering non-abelian extensions and non-
abelian (2,3)-cocycles of Lie-Yamaguti algebras.
###### Definition 3.1.
Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie-Yamaguti algebras. A non-
abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$ is a Lie-Yamaguti
algebra $\hat{\mathfrak{g}}$, which fits into a short exact sequence of Lie-
Yamaguti algebras
$\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0.$
When $\mathfrak{h}$ is an abelian ideal of $\hat{\mathfrak{g}}$, the extension
$\mathcal{E}$ is called an abelian extension of $\mathfrak{g}$ by
$\mathfrak{h}$. Denote an extension as above simply by $\hat{\mathfrak{g}}$ or
$\mathcal{E}$. A section of $p$ is a linear map
$s:\mathfrak{g}\longrightarrow\hat{\mathfrak{g}}$ such that
$ps=I_{\mathfrak{g}}$.
###### Definition 3.2.
Let $\hat{\mathfrak{g}}_{1}$ and $\hat{\mathfrak{g}}_{2}$ be two non-abelian
extensions of $\mathfrak{g}$ by $\mathfrak{h}$. They are said to be equivalent
if there is a homomorphism of Lie-Yamaguti algebras
$f:\hat{\mathfrak{g}}_{1}\longrightarrow\hat{\mathfrak{g}}_{2}$ such that the
following commutative diagram holds:
(15)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\textstyle{\hat{\mathfrak{g}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{p_{1}}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{2}}$$\textstyle{\hat{\mathfrak{g}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{2}}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
Denote by $\mathcal{E}_{nab}(\mathfrak{g},\mathfrak{h})$ the set of all non-
abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$.
Next, we define a non-abelian cohomology group and show that the non-abelian
extensions are classified by the non-abelian cohomology groups.
###### Definition 3.3.
Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie-Yamaguti algebras. A non-
abelian (2,3)-cocycle on $\mathfrak{g}$ with values in $\mathfrak{h}$ is a
septuple $(\chi,\omega,\mu,\theta,D,\rho,T)$ of maps such that
$\omega:\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h}$
is trilinear,
$\chi:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h},~{}\theta,D:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h})$
are bilinear and
$\mu:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h}),~{}\rho,T:\mathfrak{g}\longrightarrow\mathrm{Hom}(\mathfrak{h}\wedge\mathfrak{h},\mathfrak{h})$
are linear, and the following five parts of identities are satisfied for all
$x_{i},y_{i},x,y,z\in\mathfrak{g}~{}(i=1,2,3),a,b,c\in\mathfrak{h}$,
* •
Those resembling Eq. (1):
(16)
$\chi([x,y]_{\mathfrak{g}}+\chi(y,x)=0,~{}~{}\omega(x,y,z)+\omega(y,x,z)=0,$
(17) $D(x,y)a+D(y,x)a=0,~{}T(x)(a,b)+T(x)(b,a)=0.$
* •
Those resembling Eq. (2):
$\displaystyle\chi([x,y]_{\mathfrak{g}},z)-\mu(z)\chi(x,y)+\omega(x,y,z)+\chi([y,z]_{\mathfrak{g}},x)-\mu(x)\chi(y,z)+\omega(y,z,x)$
(18)
$\displaystyle+\chi([z,x]_{\mathfrak{g}},y)-\mu(y)\chi(z,x)+\omega(z,x,y)=0,$
(19)
$\mu([x,y]_{\mathfrak{g}})a+[\chi(x,y),a]_{\mathfrak{h}}+D(x,y)a-\mu(x)\mu(y)a-\theta(y,x)a+\mu(y)\mu(x)a+\theta(x,y)a=0,$
(20)
$[\mu(x)a,b]_{\mathfrak{h}}+\rho(x)(a,b)-\mu(x)[a,b]_{\mathfrak{h}}+T(x)(a,b)-[\mu(x)b,a]_{\mathfrak{h}}-\rho(x)(b,a)=0.$
* •
Those resembling Eq. (3):
(21) $\theta(z,w)\chi(x,y)+\theta(x,w)\chi(y,z)+\theta(y,w)\chi(z,x)=0,$
$\displaystyle
D([x,y]_{\mathfrak{g}},z)a-\rho(z)(\chi(x,y),a)+D([y,z]_{\mathfrak{g}},x)a-\rho(x)(\chi(y,z),a)$
(22) $\displaystyle+D([z,x]_{\mathfrak{g}},y)a-\rho(y)(\chi(z,x),a)=0,$ (23)
$\theta([x,y]_{\mathfrak{g}},z)a-T(z)(\chi(x,y),a)-\theta(x,z)\mu(y)a+\theta(y,z)\mu(x)a=0,$
(24)
$\rho([x,y]_{\mathfrak{g}})(a,b)+\\{\chi(x,y),a,b\\}_{\mathfrak{h}}-\rho(x)(\mu(y)a,b)+\rho(y)(\mu(x)a,b)=0,$
(25) $T(y)(\mu(x)a,b)+\theta(x,y)[a,b]_{\mathfrak{h}}-T(y)(\mu(x)b,a)=0,$ (26)
$\\{\mu(x)a,b,c\\}_{\mathfrak{h}}-\rho(x)([a,b]_{\mathfrak{h}},c)-\\{\mu(x)b,a,c\\}_{\mathfrak{h}}=0,$
(27)
$T(x)([a,b]_{\mathfrak{h}},c)+T(x)([b,c]_{\mathfrak{h}},a)+T(x)([c,a]_{\mathfrak{h}},b)=0.$
* •
Those resembling Eq. (4):
(28)
$D(x,y)\chi(z,w)=\chi(\\{x,y,z\\}_{\mathfrak{g}},w)-\mu(w)\omega(x,y,z)+\mu(z)\omega(x,y,w)+\chi(z,\\{x,y,w\\}_{\mathfrak{g}}),$
(29)
$D(x,y)\mu(z)a=\mu(\\{x,y,z\\}_{\mathfrak{g}})a+[\omega(x,y,z),a]_{\mathfrak{h}}+\mu(z)D(x,y)a,$
(30)
$\theta(x,[y,z]_{\mathfrak{g}})a-\rho(x)(a,\chi(y,z))=\mu(y)\theta(x,z)a-\mu(z)\theta(x,y)a,$
(31)
$D(x,y)[a,b]_{\mathfrak{h}}=[D(x,y)a,b]_{\mathfrak{h}}+[a,D(x,y)b]_{\mathfrak{h}},$
(32)
$\rho(x)(a,\mu(y)b)+[\theta(x,y)a,b]_{\mathfrak{h}}-\mu(y)\rho(x)(a,b)=0,$
(33)
$T([x,y]_{\mathfrak{g}})(a,b)+\\{a,b,\chi(x,y)\\}_{\mathfrak{h}}-\mu(x)T(y)(a,b)+\mu(y)T(x)(a,b)=0,$
(34)
$\\{a,b,\mu(x)c\\}_{\mathfrak{h}}=\mu(x)\\{a,b,c\\}_{\mathfrak{h}}-[c,T(x)(a,b)]_{\mathfrak{h}},$
(35)
$\rho(x)(a,[b,c]_{\mathfrak{h}})=[\rho(x)(a,b),c]_{\mathfrak{h}}+[b,\rho(x)(a,c)]_{\mathfrak{h}}.$
* •
Those resembling Eq. (5):
$\displaystyle
D(x_{1},x_{2})\omega(y_{1},y_{2},y_{3})+\omega(x_{1},x_{2},\\{y_{1},y_{2},y_{3}\\}_{\mathfrak{g}})=\omega(\\{x_{1},x_{2},y_{1}\\}_{\mathfrak{g}},y_{2},y_{3})$
$\displaystyle+\theta(y_{2},y_{3})\omega(x_{1},x_{2},y_{1})+\omega(y_{1},\\{x_{1},x_{2},y_{2}\\}_{\mathfrak{g}},y_{3})-\theta(y_{1},y_{3})\omega(x_{1},x_{2},y_{2})$
(36)
$\displaystyle+\omega(y_{1},y_{2},\\{x_{1},x_{2},y_{3}\\}_{\mathfrak{g}})+D(y_{1},y_{2})\omega(x_{1},x_{2},y_{3}),$
$\displaystyle D(x,y)\theta(z,w)a-\theta(z,w)D(x,y)a$ (37) $\displaystyle=$
$\displaystyle\theta(\\{x,y,z\\}_{\mathfrak{g}},w)a+\theta(z,\\{x,y,w\\}_{\mathfrak{g}})a-T(w)(\omega(x,y,z),a)-\rho(z)(a,\omega(x,y,w)),$
(38)
$\theta(x,\\{y,z,w\\}_{\mathfrak{g}})a-\rho(x)(a,\omega(y,z,w))=\theta(z,w)\theta(x,y)a-\theta(y,w)\theta(x,z)a+D(y,z)\theta(x,w)a,$
$\displaystyle D(x,y)D(z,w)a-D(z,w)D(x,y)a$ (39) $\displaystyle=$
$\displaystyle
D(\\{x,y,z\\}_{\mathfrak{g}},w)a+D(z,\\{x,y,w\\}_{\mathfrak{g}})a-\rho(w)(\omega(x,y,z),a)+\rho(z)(\omega(x,y,w),a),$
(40) $\displaystyle(D(x,y)\rho(z)-\rho(\\{x,y,z\\}_{\mathfrak{g}}))(a,b)$
$\displaystyle=\rho(z)((D(x,y)a,b)+(a,D(x,y)b))+\\{\omega(x,y,z),a,b\\}_{\mathfrak{h}},$
(41)
$D(y,z)(\rho(x)(a,b))=\rho(x)(a,D(y,z)b)-\rho(z)(\theta(x,y)a,b)+\rho(y)(\theta(x,z)a,b),$
(42)
$\theta(y,z)(\rho(x)(a,b))=\rho(x)(a,\theta(y,z)b)-T(z)(\theta(x,y)a,b)-\rho(y)(b,\theta(x,z)a),$
(43)
$\displaystyle(D(x,y)T(z)-T(\\{x,y,z\\}_{\mathfrak{g}}))(a,b)=T(z)((D(x,y)a,b)+(a,D(x,y)b))+\\{a,b,\omega(x,y,z)\\}_{\mathfrak{h}},$
(44)
$\displaystyle(T(\\{x,y,z\\}_{\mathfrak{g}})-D(x,y)T(z))(a,b)+\\{a,b,\omega(x,y,z)\\}_{\mathfrak{h}}=(\theta(y,z)T(x)-\theta(x,z)T(y))(a,b),$
(45)
$D(x,y)(\\{a,b,c\\}_{\mathfrak{h}})=\\{D(x,y)a,b,c\\}_{\mathfrak{h}}+\\{a,D(x,y)b,c\\}_{\mathfrak{h}}+\\{a,b,D(x,y)c\\}_{\mathfrak{h}},$
(46)
$\rho(x)(a,\rho(y)(b,c))=\rho(y)(\rho(x)(a,b),c)+\rho(y)(b,\rho(x)(a,c))-\\{\theta(x,y)a,b,c\\}_{\mathfrak{h}},$
(47)
$\\{a,b,\theta(x,y)c\\}_{\mathfrak{h}}=\theta(x,y)\\{a,b,c\\}_{\mathfrak{h}}-T(y)(T(x)(a,b),c)-\rho(x)(c,T(y)(a,b)),$
(48)
$\rho(x)(a,T(y)(b,c))=T(y)(\rho(x)(a,b),c)+T(y)(b,\rho(x)(a,c))-\\{a,c,\theta(x,y)a\\}_{\mathfrak{h}},$
(49)
$\\{a,b,D(x,y)c\\}_{\mathfrak{h}}=D(x,y)\\{a,b,c\\}_{\mathfrak{h}}-\rho(y)(T(x)(a,b),c),+\rho(x)(T(y)(a,b),c),$
(50)
$\\{a,b,\rho(x)(c,d)\\}_{\mathfrak{h}}=\rho(x)(\\{a,b,c\\}_{\mathfrak{h}},d)-\\{c,T(x)(a,b),d\\}_{\mathfrak{h}}+\rho(x)(c,\\{a,b,d\\}_{\mathfrak{h}}),$
(51)
$\rho(x)(a,\\{b,c,d\\}_{\mathfrak{h}})=\\{\rho(x)(a,b),c,d\\}_{\mathfrak{h}}+\\{b,\rho(x)(a,c),d\\}_{\mathfrak{h}}+\\{b,c,\rho(x)(a,d)\\}_{\mathfrak{h}}.$
(52)
$\\{a,b,T(x)(c,d)\\}_{\mathfrak{h}}=T(x)(\\{a,b,c\\}_{\mathfrak{h}},d)+T(x)(c,\\{a,b,d\\}_{\mathfrak{h}})+\\{c,d,T(x)(a,b)\\}_{\mathfrak{h}}.$
###### Definition 3.4.
Let $(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})$ and
$(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$ be two non-
abelian (2,3)-cocycles on $\mathfrak{g}$ with values in $\mathfrak{h}$. They
are said to be equivalent if there exists a linear map
$\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ such that for all
$x,y,z\in\mathfrak{g}$ and $a,b\in\mathfrak{h}$, the following equalities
hold:
(53)
$\chi_{1}(x,y)-\chi_{2}(x,y)=[\varphi(x),\varphi(y)]_{\mathfrak{h}}+\varphi[x,y]_{\mathfrak{g}}-\mu_{2}(x)\varphi(y)+\mu_{2}(y)\varphi(x),$
$\displaystyle\omega_{1}(x,y,z)-\omega_{2}(x,y,z)=\theta_{2}(x,z)\varphi(y)-D_{2}(x,y)\varphi(z)+\rho_{2}(x)(\varphi(y),\varphi(z))-\theta_{2}(y,z)\varphi(x)$
(54)
$\displaystyle+T_{2}(z)(\varphi(x),\varphi(y))-\rho_{2}(y)(\varphi(x),\varphi(z))-\\{\varphi(x),\varphi(y),\varphi(z)\\}_{\mathfrak{h}}+\varphi\\{x,y,z\\}_{\mathfrak{g}},$
(55) $\mu_{1}(x)a-\mu_{2}(x)a=[a,\varphi(x)]_{\mathfrak{h}},$ (56)
$\theta_{1}(x,y)a-\theta_{2}(x,y)a=\rho_{2}(x)(a,\varphi(y))-T_{2}(y)(a,\varphi(x))+\\{a,\varphi(x),\varphi(y)\\}_{\mathfrak{h}},$
(57)
$D_{1}(x,y)a-D_{2}(x,y)a=\rho_{2}(y)(\varphi(x),a)-\rho_{2}(x)(\varphi(y),a)+\\{\varphi(x),\varphi(y),a\\}_{\mathfrak{h}},$
(58)
$\rho_{1}(x)(a,b)-\rho_{2}(x)(a,b)=\\{a,\varphi(x),b\\}_{\mathfrak{h}},~{}~{}T_{1}(x)(a,b)-T_{2}(x)(a,b)=\\{b,a,\varphi(x)\\}_{\mathfrak{h}}.$
For convenience, we abbreviate a non-abelian (2,3)-cocycle
$(\chi,\omega,\mu,\theta,D,\rho,T)$ as $(\chi,\omega)$, denote the equivalent
class of a non-abelian (2,3)-cocycle $(\chi,\omega,\mu,\theta,D,\rho,T)$
simply by $[(\chi,\omega)]$. Furthermore, we denote the set of equivalent
classes of non-abelian (2,3)-cocycles by
$H_{nab}^{(2,3)}(\mathfrak{g},\mathfrak{h})$.
Using the above notations, we define multilinear maps $[\ ,\ ]_{\chi}$ and $[\
,\ ,\ ]_{\omega}$ on $\mathfrak{g}\oplus\mathfrak{h}$ by
(59) $\displaystyle[x+a,y+b]_{\chi}$
$\displaystyle=[x,y]_{\mathfrak{g}}+\chi(x,y)+\mu(x)b-\mu(y)a+[a,b]_{\mathfrak{h}},$
$\displaystyle\\{x+a,y+b,z+c\\}_{\omega}=$
$\displaystyle\\{x,y,z\\}_{\mathfrak{g}}+\omega(x,y,z)+D(x,y)c+\theta(y,z)a-\theta(x,z)b$
(60)
$\displaystyle+T(z)(a,b)+\rho(x)(b,c)-\rho(y)(a,c)+\\{a,b,c\\}_{\mathfrak{h}}$
for all $x,y,z\in\mathfrak{g}$ and $a,b,c\in\mathfrak{h}$.
###### Proposition 3.5.
With the above notions, $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\\{\
,\ ,\ \\}_{\omega})$ is a Lie-Yamaguti algebra if and only if the septuple
$(\chi,\omega,\mu,\theta,D,\rho,T)$ is a non-abelian (2,3)-cocycle. Denote
this Lie-Yamaguti algebra $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\\{\
,\ ,\ \\}_{\omega})$ simply by
$\mathfrak{g}\oplus_{(\chi,\omega)}\mathfrak{h}$.
###### Proof.
$(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\\{\ ,\ ,\ \\}_{\omega})$ is a
Lie-Yamaguti algebra if and only if Eqs. (1)-(5) hold for $[\ ,\ ]_{\chi},\\{\
,\ ,\ \\}_{\omega}$. In fact, it is easy to check that Eq. (1) holds for $[\
,\ ]_{\chi},\\{\ ,\ ,\ \\}_{\omega}$ if and only if (16)-(17) hold. In the
following, we always assume that $x,y,z,w\in\mathfrak{g}$ and
$a,b,c,d\in\mathfrak{h}$.
For the Eq. (2), we discuss it for the following cases: for all
$x_{1},x_{2},x_{3}\in\mathfrak{g}\oplus\mathfrak{h}$,
1. $(I)$
when all of the three elements $x_{1},x_{2},x_{3}$ belong to $\mathfrak{g}$,
(2) holds if and only if (• ‣ 3.3) holds.
2. $(II)$
when $x_{1},x_{2},x_{3}$ equal to:
1. $(i)$
$x,y,a$ or $a,x,y$ or $x,a,y$ respectively, (2) holds if and only if (19)
holds.
2. $(ii)$
$a,b,x$ or $a,x,b$ or $x,a,b$ respectively, (2) holds for if and only if (20)
holds.
For the Eq. (3), we discuss it for the following cases: for all
$x_{1},x_{2},x_{3},y_{1}\in\mathfrak{g}\oplus\mathfrak{h}$,
1. $(I)$
when all of the four elements $x_{1},x_{2},x_{3},y_{1}$ belong to
$\mathfrak{g}$, (3) holds if and only if (21) holds.
2. $(II)$
when $x_{1},x_{2},x_{3},y_{1}$ equal to:
1. $(i)$
$x,y,z,a$, (3) holds if and only if (• ‣ 3.3) holds.
2. $(ii)$
$x,y,a,z$ or $x,a,y,z$ or $a,x,y,z$ respectively, (3) holds if and only if
(23) holds.
3. $(iii)$
$x,y,a,b$ or $x,a,y,b$ or $a,x,y,b$ respectively, (3) holds if and only if
(24) holds.
4. $(iv)$
$x,a,b,y$ or $a,x,b,y$ or $a,b,x,y$ respectively, (3) holds if and only if
(25) holds.
5. $(v)$
$x,a,b,c$ or $a,x,b,c$ or $a,b,x,y,c$ respectively, (3) holds if and only if
(26) holds.
6. $(vi)$
$a,b,c,x$, (3) holds if and only if (27) holds.
For the Eq. (4), we discuss it for the following cases: for all
$x_{1},x_{2},y_{1},y_{2}\in\mathfrak{g}\oplus\mathfrak{h}$,
1. $(I)$
when all of the four elements $x_{1},x_{2},y_{1},y_{2}$ belong to
$\mathfrak{g}$, (4) holds if and only if (28) holds.
2. $(II)$
when $x_{1},x_{2},y_{1},y_{2}$ equal to:
1. $(i)$
$x,y,z,a$ or $x,y,a,z$, (4) holds if and only if (29) holds.
2. $(ii)$
$x,a,y,z$ or $a,x,y,z$ respectively, (4) holds if and only if (30) holds.
3. $(iii)$
$x,y,a,b$, (4) holds if and only if (31) holds.
4. $(iv)$
$x,a,y,b$ or $a,x,y,b$ or $x,a,b,y$ or $a,x,b,y$ respectively, (4) holds if
and only if (32) holds.
5. $(v)$
$a,b,x,y$, (4) holds if and only if (33) holds.
6. $(vi)$
$a,b,c,x$ or $a,b,x,c$ respectively, (4) holds if and only if (34) holds.
7. $(vii)$
$a,x,b,c$ or $x,a,b,c$ respectively, (4) holds if and only if (35) holds.
For the Eq. (5), we discuss it for the following cases: for all
$x_{1},x_{2},y_{1},y_{2},y_{3}\in\mathfrak{g}\oplus\mathfrak{h}$,
1. $(I)$
when all of the five elements $x_{1},x_{2},y_{1},y_{2},y_{3}$ belong to
$\mathfrak{g}$, (5) holds if and only if (• ‣ 3.3) holds.
2. $(II)$
when $x_{1},x_{2},y_{1},y_{2},y_{3}$ equal to:
1. $(i)$
$x,y,z,w,a$, (5) holds if and only if (• ‣ 3.3) holds.
2. $(ii)$
$x,y,z,a,w$ or $x,y,a,z,w$ respectively, (5) holds if and only if (• ‣ 3.3)
holds.
3. $(iii)$
$x,a,y,z,w$ or $a,x,y,z,w$ respectively, (5) holds if and only if (38) holds.
4. $(iv)$
$x,y,z,a,b$ or $x,y,a,y,b$, (5) holds if and only if (40) holds.
5. $(v)$
$x,a,y,z,b$ or $a,x,y,z,b$ respectively, (5) holds if and only if (41) holds.
6. $(vi)$
$x,y,a,b,z$, (5) holds if and only if (43) holds.
7. $(vii)$
$x,a,y,b,z$ or $a,x,y,b,z$ or $x,a,b,y,z$ or $a,x,b,y,z$ respectively, (4)
holds if and only if (42) holds.
8. $(viii)$
$a,b,x,y,z$, (5) holds if and only if (44) holds.
9. $(ix)$
$x,y,a,b,c$, (5) holds if and only if (45) holds.
10. $(x)$
$x,a,y,b,c$ or $a,x,y,b,c$ or $a,x,b,y,c$ or $x,a,b,y,c$ respectively, (5)
holds if and only if (46) holds.
11. $(xi)$
$x,a,b,c,y$ or $a,x,b,c,y$ respectively, (5) holds if and only if (48) holds.
12. $(xii)$
$a,b,x,c,y$ or $a,b,c,x,y$ respectively, (5) holds if and only if (47) holds.
13. $(xiii)$
$a,b,x,y,c$, (5) holds if and only if (49) holds.
14. $(xiv)$
$a,b,c,d,x$, (5) holds if and only if (52) holds.
15. $(xv)$
$a,b,c,x,d$ or $a,b,x,c,d$ respectively, (5) holds if and only if (50) holds.
16. $(xvi)$
$a,x,b,c,d$ or $x,a,b,c,d$ respectively, (5) holds if and only if (51) holds.
This completes the proof. ∎
Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian
extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$.
Define
$\chi_{s}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h},~{}\omega_{s}:\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h},~{}\mu_{s}:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h}),~{}\theta_{s},D_{s}:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h}),~{}\rho_{s},T_{s}:\mathfrak{g}\longrightarrow\mathrm{Hom}(\mathfrak{h}\wedge\mathfrak{h},\mathfrak{h})$
respectively by
(61) $\chi_{s}(x,y)=[s(x),s(y)]_{\hat{\mathfrak{g}}}-s[x,y]_{\mathfrak{g}},$
(62)
$\omega_{s}(x,y,z)=\\{s(x),s(y),s(z)\\}_{\hat{\mathfrak{g}}}-s\\{x,y,z\\}_{\mathfrak{g}},$
(63)
$\theta_{s}(x,y)a=\\{a,s(x),s(y)\\}_{\hat{\mathfrak{g}}},~{}~{}~{}~{}\rho_{s}(x)(a,b)=\\{s(x),a,b\\}_{\hat{\mathfrak{g}}},$
(64)
$D_{s}(x,y)a=\\{s(x),s(y),a\\}_{\hat{\mathfrak{g}}},~{}~{}~{}~{}T_{s}(x)(a,b)=\\{a,b,s(x)\\}_{\hat{\mathfrak{g}}}$
for any $x,y,z\in\mathfrak{g},a,b\in\mathfrak{h}$.
By direct computations, we have
###### Proposition 3.6.
With the above notions,
$(\chi_{s},\omega_{s},\mu_{s},\theta_{s},D_{s},\rho_{s},T_{s})$ is a non-
abelian (2,3)-cocycle on $\mathfrak{g}$ with values in $\mathfrak{h}$. We call
it the non-abelian (2,3)-cocycle corresponding to the extension $\mathcal{E}$
induced by $s$. Naturally, $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\
]_{\chi_{s}},\\{\ ,\ ,\ \\}_{\omega_{s}})$ is a Lie-Yamaguti algebra. Denote
this Lie-Yamaguti algebra simply by
$\mathfrak{g}\oplus_{(\chi_{s},\omega_{s})}\mathfrak{h}$.
In the following, we denote
$(\chi_{s},\omega_{s},\mu_{s},\theta_{s},D_{s},\rho_{s},T_{s})$ by
$(\chi,\omega,\mu,\theta,D,\rho,T)$ without ambiguity.
###### Lemma 3.7.
Let $(\chi_{i},\omega_{i},\mu_{i},\theta_{i},D_{i},\rho_{i},T_{i})$ be the
non-abelian (2,3)-cocycle corresponding to the extension $\mathcal{E}$ induced
by $s_{i}$ (i=1,2). Then
$(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})$ and
$(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$ are
equivalent, that is, the equivalent classes of non-abelian (2,3)-cocycles
corresponding to a non-abelian extension induced by a section are independent
on the choice of sections.
###### Proof.
Let $\hat{\mathfrak{g}}$ be a non-abelian extension of $\mathfrak{g}$ by
$\mathfrak{h}$. Assume that $s_{1}$ and $s_{2}$ are two different sections of
$p$, $(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})$ and
$(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$ are the
corresponding non-abelian (2,3)-cocycles. Define a linear map
$\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ by
$\varphi(x)=s_{2}(x)-s_{1}(x)$. Since $p\varphi(x)=ps_{2}(x)-ps_{1}(x)=0$,
$\varphi$ is well defined. Thanks to Eqs. (62)-(64), we get
$\displaystyle
w_{1}(x,y,z)=\\{s_{1}(x),s_{1}(y),s_{1}(z)\\}_{\hat{\mathfrak{g}}}-s_{1}\\{x,y,z\\}_{\mathfrak{g}}$
$\displaystyle=$
$\displaystyle\\{s_{2}(x)-\varphi(x),s_{2}(y)-\varphi(y),s_{2}(z)-\varphi(z)\\}_{\hat{\mathfrak{g}}}-(s_{2}\\{x,y,z\\}_{\mathfrak{g}}-\varphi(\\{x,y,z\\}_{\mathfrak{g}})$
$\displaystyle=$
$\displaystyle\\{s_{2}(x),s_{2}(y),s_{2}(z)\\}_{\hat{\mathfrak{g}}}-\\{s_{2}(x),\varphi(y),s_{2}(z)\\}_{\hat{\mathfrak{g}}}-\\{s_{2}(x),s_{2}(y),\varphi(z)\\}_{\hat{\mathfrak{g}}}+\\{s_{2}(x),\varphi(y),\varphi(z)\\}_{\hat{\mathfrak{g}}}$
$\displaystyle-\\{\varphi(x),s_{2}(y),s_{2}(z)\\}_{\hat{\mathfrak{g}}}+\\{\varphi(x),\varphi(y),s_{2}(z)\\}_{\hat{\mathfrak{g}}}+\\{\varphi(x),s_{2}(y),\varphi(z)\\}_{\hat{\mathfrak{g}}}-\\{\varphi(x),\varphi(y),\varphi(z)\\}_{\hat{\mathfrak{g}}}$
$\displaystyle-
s_{2}\\{x,y,z\\}_{\mathfrak{g}}+\varphi\\{x,y,z\\}_{\mathfrak{g}}$
$\displaystyle=$ $\displaystyle
w_{2}(x,y,z)+\theta_{2}(x,z)\varphi(y)-D_{2}(x,y)\varphi(z)+\rho_{2}(x)(\varphi(y),\varphi(z))-\theta_{2}(y,z)\varphi(x)+T_{2}(z)(\varphi(x),\varphi(y))$
$\displaystyle-\rho_{2}(y)(\varphi(x),\varphi(z))-\\{\varphi(x),\varphi(y),\varphi(z)\\}_{\hat{\mathfrak{g}}}+\varphi\\{x,y,z\\}_{\mathfrak{g}},$
which yields that Eq. (3.4) holds. Similarly, Eqs. (53) and (55)-(58) hold.
This finishes the proof. ∎
According to Proposition 3.5 and Proposition 3.6, given a non-abelian
extension $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ of $\mathfrak{g}$ by
$\mathfrak{h}$ with a section $s$ of $p$, we have a non-abelian (2,3)-cocycle
$(\chi_{s},\omega_{s},\mu_{s},\theta_{s},D_{s},\rho_{s},T_{s})$ and a Lie-
Yamaguti algebra $\mathfrak{g}\oplus_{(\chi_{s},\omega_{s})}\mathfrak{h}$. It
follows that
$\mathcal{E}_{(\chi_{s},\omega_{s})}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\mathfrak{g}\oplus_{(\chi_{s},\omega_{s})}\mathfrak{h}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathfrak{g}\longrightarrow
0$ is a non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$. Since any
element $\hat{w}\in\hat{\mathfrak{g}}$ can be written as $\hat{w}=a+s(x)$ with
$a\in\mathfrak{h},x\in\mathfrak{g}$, define a linear map
$f:\hat{\mathfrak{g}}\longrightarrow\mathfrak{g}\oplus_{(\chi_{s},\omega_{s})}\mathfrak{h},~{}f(\hat{w})=f(a+s(x))=a+x.$
It is easy to check that $f$ is an isomorphism of Lie-Yamaguti algebras such
that the following commutative diagram holds:
$\textstyle{\mathcal{E}:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\hat{\mathfrak{g}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{p}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\mathcal{E}_{(\chi_{s},\omega_{s})}:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathfrak{g}\oplus_{(\chi_{s},\omega_{s})}\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$
which indicates that the non-abelian extensions $\mathcal{E}$ and
$\mathcal{E}_{(\chi_{s},\omega_{s})}$ of $\mathfrak{g}$ by $\mathfrak{h}$ are
equivalent. On the other hand, if $(\chi,\omega,\mu,\theta,D,\rho,T)$ is a
non-abelian (2,3)-cocycle on $\mathfrak{g}$ with values in $\mathfrak{h}$,
there is a Lie-Yamaguti algebra
$\mathfrak{g}\oplus_{(\chi,\omega)}\mathfrak{h}$, which yields the following
non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$:
$\mathcal{E}_{(\chi,\omega)}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\mathfrak{g}\oplus_{(\chi,\omega)}\mathfrak{h}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathfrak{g}\longrightarrow
0,$
where $i$ is the inclusion and $\pi$ is the projection.
In the following, we focus on the relationship between non-abelian
(2,3)-cocycles and extensions.
###### Proposition 3.8.
Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie-Yamaguti algebras. Then the
equivalent classes of non-abelian extensions of $\mathfrak{g}$ by
$\mathfrak{h}$ are classified by the non-abelian cohomology group, that is,
$\mathcal{E}_{nab}(\mathfrak{g},\mathfrak{h})\simeq
H_{nab}^{(2,3)}(\mathfrak{g},\mathfrak{h})$.
###### Proof.
Define a linear map
$\Theta:\mathcal{E}_{nab}(\mathfrak{g},\mathfrak{h})\rightarrow
H_{nab}^{(2,3)}(\mathfrak{g},\mathfrak{h}),~{}$
where $\Theta$ assigns an equivalent class of non-abelian extensions to the
class of corresponding non-abelian (2,3)-cocycles. First, we check that
$\Theta$ is well-defined. Assume that $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$
are two equivalent non-abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$
via the map $f$, that is, the commutative diagram (15) holds. Let
$s_{1}:\mathfrak{g}\rightarrow\hat{\mathfrak{g}}_{1}$ be a section of $p_{1}$.
Then $p_{2}fs_{1}=p_{1}s_{1}=I_{\mathfrak{g}}$, which follows that
$s_{2}=fs_{1}$ is a section of $p_{2}$. Let
$(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})$ and
$(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$ be two non-
abelian (2,3)-cocycles induced by the sections $s_{1},s_{2}$ respectively.
Then we have,
$\displaystyle\theta_{1}(x,y)a$ $\displaystyle=$ $\displaystyle
f(\theta_{1}(x,y)a)=f(\\{a,s_{1}(x),s_{1}(y)\\}_{\hat{\mathfrak{g}}_{1}})$
$\displaystyle=$
$\displaystyle\\{f(a),fs_{1}(x),fs_{1}(y)\\}_{\hat{\mathfrak{g}}_{2}}$
$\displaystyle=$
$\displaystyle\\{a,s_{2}(x),s_{2}(y)\\}_{\hat{\mathfrak{g}}_{2}}$
$\displaystyle=$ $\displaystyle\theta_{2}(x,y)a.$
By the same token, we have
$D_{1}(x,y)a=D_{2}(x,y)a,\chi_{1}(x,y)=\chi_{2}(x,y),\omega_{1}(x,y,z)=\omega_{2}(x,y,z),$
$\rho_{1}(x)(a,b)=\rho_{2}(x)(a,b),T_{1}(x)(a,b)=T_{2}(x)(a,b).$
Thus,
$(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})=(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$,
which means that $\Theta$ is well-defined.
Next, we verify that $\Theta$ is injective. Indeed, suppose that
$\Theta([\mathcal{E}_{1}])=[(\chi_{1},\omega_{1})]$ and
$\Theta([\mathcal{E}_{2}])=[(\chi_{2},\omega_{2})]$. If the equivalent classes
$[(\chi_{1},\omega_{1})]=[(\chi_{2},\omega_{2})]$, we obtain that the non-
abelian (2,3)-cocycles
$(\chi_{1},\omega_{1},\mu_{1},\theta_{1},D_{1},\rho_{1},T_{1})$ and
$(\chi_{2},\omega_{2},\mu_{2},\theta_{2},D_{2},\rho_{2},T_{2})$ are equivalent
via the linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$,
satisfying Eqs. (53)-(58). Define a linear map
$f:\mathfrak{g}\oplus_{(\chi_{1},\omega_{1})}\mathfrak{h}\longrightarrow\mathfrak{g}\oplus_{(\chi_{2},\omega_{2})}\mathfrak{h}$
by
$f(x+a)=x-\varphi(x)+a,~{}~{}\forall~{}x\in\mathfrak{g},a\in\mathfrak{h}.$
According to Eq. (3), for all $x,y,z\in\mathfrak{g},a,b,c\in\mathfrak{h}$, we
get
$\displaystyle f(\\{x+a,y+b,z+c\\}_{\omega_{1}})$ $\displaystyle=$
$\displaystyle
f(\\{x,y,z\\}_{\mathfrak{g}}+\omega_{1}(x,y,z)+D_{1}(x,y)c+\theta_{1}(y,z)a-\theta_{1}(x,z)b$
$\displaystyle+T_{1}(z)(a,b)+\rho_{1}(x)(b,c)-\rho_{1}(y)(a,c)+\\{a,b,c\\}_{\mathfrak{h}})$
$\displaystyle=$
$\displaystyle\\{x,y,z\\}_{\mathfrak{g}}-\varphi(\\{x,y,z\\}_{\mathfrak{g}})+\omega_{1}(x,y,z)+D_{1}(x,y)c+\theta_{1}(y,z)a-\theta_{1}(x,z)b$
$\displaystyle+T_{1}(z)(a,b)+\rho_{1}(x)(b,c)-\rho_{1}(y)(a,c)+\\{a,b,c\\}_{\mathfrak{h}},$
and
$\displaystyle\\{f(x+a),f(y+b),f(z+c)\\}_{\omega_{2}}$ $\displaystyle=$
$\displaystyle\\{x-\varphi(x)+a,y-\varphi(y)+b,z-\varphi(z)+c\\}_{\omega_{2}}$
$\displaystyle=$
$\displaystyle\\{x,y,z\\}_{\mathfrak{g}}+\omega_{2}(x,y,z)+D_{2}(x,y)(c-\varphi(z))+\theta_{2}(y,z)(a-\varphi(x))-\theta_{2}(x,z)(b-\varphi(y))$
$\displaystyle+T_{2}(z)(a-\varphi(x),b-\varphi(y))+\rho_{2}(x)(b-\varphi(y),c-\varphi(z))-\rho_{2}(y)(a-\varphi(x),c-\varphi(z))$
$\displaystyle+\\{a-\varphi(x),b-\varphi(y),c-\varphi(z)\\}_{\mathfrak{h}})$
$\displaystyle=$
$\displaystyle\\{x,y,z\\}_{\mathfrak{g}}+\omega_{2}(x,y,z)+D_{2}(x,y)(c-\varphi(z))+\theta_{2}(y,z)(a-\varphi(x))-\theta_{2}(x,z)(b-\varphi(y))$
$\displaystyle+T_{2}(z)(\varphi(x),\varphi(y))-T_{2}(z)(a,\varphi(y))-T_{2}(z)(\varphi(x),b)+T_{2}(z)(a,b)$
$\displaystyle+\rho_{2}(x)(\varphi(y),\varphi(z))-\rho_{2}(x)(\varphi(y),c)-\rho_{2}(x)(b,\varphi(z))+\rho_{2}(x)(b,c)-\rho_{2}(y)(\varphi(x),\varphi(z))$
$\displaystyle+\rho_{2}(y)(\varphi(x),c)+\rho_{2}(y)(a,\varphi(z))-\rho_{2}(y)(a,c)-\\{\varphi(x),\varphi(y),\varphi(z)\\}_{\mathfrak{h}}+\\{\varphi(x),\varphi(y),c\\}_{\mathfrak{h}}$
$\displaystyle+\\{\varphi(x),b,\varphi(z)\\}_{\mathfrak{h}}-\\{\varphi(x),b,c\\}_{\mathfrak{h}}+\\{a,\varphi(y),\varphi(z)\\}_{\mathfrak{h}}-\\{a,b,\varphi(z)\\}_{\mathfrak{h}}-\\{a,\varphi(y),c\\}_{\mathfrak{h}}+\\{a,b,c\\}_{\mathfrak{h}}.$
In view of Eqs. (53)-(58), we have
$f(\\{x+a,y+b,z+c\\}_{\omega_{1}})=\\{f(x+a),f(y+b),f(z+c)\\}_{\omega_{2}}$.
Similarly, $f([x+a,y+b]_{\chi_{1}})=[f(x+a),f(y+b)]_{\chi_{2}}$. Hence, $f$ is
a homomorphism of Lie-Yamaguti algebras. Clearly, the following commutative
diagram holds:
(65)
$\textstyle{\mathcal{E}_{(\chi_{1},\omega_{1})}:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathfrak{g}\oplus_{(\chi_{1},\omega_{1})}\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\mathcal{E}_{(\chi_{2},\omega_{2})}:0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathfrak{g}\oplus_{(\chi_{2},\omega_{2})}\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
Thus $\mathcal{E}_{(\chi_{1},\omega_{1})}$ and
$\mathcal{E}_{(\chi_{2},\omega_{2})}$ are equivalent non-abelian extensions of
$\mathfrak{g}$ by $\mathfrak{h}$, which means that
$[\mathcal{E}_{(\chi_{1},\omega_{1})}]=[\mathcal{E}_{(\chi_{2},\omega_{2})}]$.
Thus, $\Theta$ is injective.
Finally, we claim that $\Theta$ is surjective. For any equivalent class of
non-abelian (2,3)-cocycles $[(\chi,\omega)]$, by Proposition 3.5, there is a
non-abelian extension of $\mathfrak{g}$ by $\mathfrak{h}$:
$\mathcal{E}_{(\chi,\omega)}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\mathfrak{g}\oplus_{(\chi,\omega)}\mathfrak{h}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\mathfrak{g}\longrightarrow
0.$
Therefore, $\Theta([\mathcal{E}_{(\chi,\omega)}])=[(\chi,\omega)]$, which
follows that $\Theta$ is surjective. In all, $\Theta$ is bijective. This
finishes the proof.
∎
## 4\. Non-abelian extensions in terms of Maurer-Cartan elements
In this section, we classify the non-abelian extensions using Maurer-Cartan
elements. We start with recalling the Maurer-Cartan elements from [12].
Let $(L=\oplus_{i}L_{i},[\ ,\ ],d)$ be a differential graded Lie algebra. The
set $\mathrm{MC}(L)$ of Maurer-Cartan elements of $(L,[\ ,\ ],d)$ is defined
by
$\mathrm{MC}(L)=\\{\eta\in L_{1}|d\eta+\frac{1}{2}[\eta,\eta]=0\\}.$
Moreover, $\eta_{0},\eta_{1}\in\mathrm{MC}(L)$ are called gauge equivalent if
and only if there exists an element $\varphi\in L_{0}$ such that
$\eta_{1}=e^{ad_{\varphi}}\eta_{0}-\frac{e^{ad_{\varphi}}-1}{ad_{\varphi}}d\varphi.$
Let $\mathfrak{g}$ be a vector space. Denote by
$\mathbb{C}(\mathfrak{g},\mathfrak{g})=\mathrm{Hom}(\wedge^{2}\mathfrak{g}\otimes\mathfrak{g},\mathfrak{g})\times\mathrm{Hom}(\wedge^{2}\mathfrak{g}\otimes\wedge^{2}\mathfrak{g},\mathfrak{g})$
and
(66) $\mathcal{C}^{n}(\mathfrak{g},\mathfrak{g})=\left\\{\begin{aligned}
&\mathrm{Hom}(\mathfrak{g},\mathfrak{g}),&n=0,\\\
&\mathrm{Hom}(\underbrace{\wedge^{2}\mathfrak{g}\otimes\cdot\cdot\cdot\otimes\wedge^{2}\mathfrak{g}}_{n},\mathfrak{g})\times\mathrm{Hom}(\underbrace{\wedge^{2}\mathfrak{g}\otimes\cdot\cdot\cdot\otimes\wedge^{2}\mathfrak{g}}_{n}\otimes\mathfrak{g},\mathfrak{g}),&n\geq
1,\end{aligned}\right.$
Then
$\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g})=\mathcal{C}^{*}(\mathfrak{g},\mathfrak{g})\oplus\mathbb{C}(\mathfrak{g},\mathfrak{g})=\oplus_{n\geq
0}\mathcal{C}^{n}(\mathfrak{g},\mathfrak{g})\oplus\mathbb{C}(\mathfrak{g},\mathfrak{g})$,
where the degree of elements in $\mathcal{C}^{n}(\mathfrak{g},\mathfrak{g})$
is $n$, the degree of elements in $\mathbb{C}(\mathfrak{g},\mathfrak{g})$ is
$1$ and $f\in\mathrm{Hom}(\otimes^{2n+1}\mathfrak{g},\mathfrak{g})$ satisfying
(67) $\displaystyle
f(x_{1},\cdot\cdot\cdot,x_{2i-1},x_{2i},\cdot\cdot\cdot,x_{n})=0,~{}~{}\hbox{if}~{}x_{2i-1}=x_{2i},~{}\forall~{}i=1,2,\cdot\cdot\cdot,[\frac{n}{2}].$
For all
$P=(P_{I},P_{II})\in\mathcal{C}^{p}(\mathfrak{g},\mathfrak{g}),Q=(Q_{I},Q_{II})\in\mathcal{C}^{q}(\mathfrak{g},\mathfrak{g})~{}(p,q\geq
1)$, denote by
$P\circ Q=((P\circ Q)_{I},(P\circ
Q)_{II})\in\mathcal{C}^{p+q}(\mathfrak{g},\mathfrak{g}).$
The definition of $P\circ Q$ is given in [31]. In detail,
$\displaystyle(P\circ Q)_{I}(X_{1},\cdot\cdot\cdot,X_{p+q})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}\sigma\in sh(p,q),\\\
\sigma(p+q)=p+q\end{subarray}}(-1)^{pq}sgn(\sigma)P_{II}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(p)},Q_{I}(X_{\sigma(p+1)},\cdot\cdot\cdot,X_{\sigma(p+q)})$
$\displaystyle+\sum_{\begin{subarray}{c}k=1,\\\ \sigma\in
sh(k-1,q)\end{subarray}}^{p}(-1)^{q(k-1)}sgn(\sigma)P_{I}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(k-1)},x_{q+k}\wedge
Q_{II}(X_{\sigma(k)},\cdot\cdot\cdot,X_{\sigma(k+q-1)},y_{k+q}),X_{k+q+1},\cdot\cdot\cdot,X_{p+q})$
$\displaystyle+\sum_{\begin{subarray}{c}k=1,\\\ \sigma\in
sh(k-1,q)\end{subarray}}^{p}(-1)^{q(k-1)}sgn(\sigma)P_{I}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(k-1)},Q_{II}(X_{\sigma(k)},\cdot\cdot\cdot,X_{\sigma(k+q-1)},x_{k+q})\wedge
y_{q+k},X_{k+q+1},\cdot\cdot\cdot,X_{p+q}),$
and
$\displaystyle(P\circ Q)_{II}(X_{1},\cdot\cdot\cdot,X_{p+q},z)$
$\displaystyle=$ $\displaystyle\sum_{\sigma\in
sh(p,q)}(-1)^{pq}sgn(\sigma)P_{II}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(p)},Q_{II}(X_{\sigma(p+1)},\cdot\cdot\cdot,X_{\sigma(p+q)},z)$
$\displaystyle+\sum_{\begin{subarray}{c}k=1,\\\ \sigma\in
sh(k-1,q)\end{subarray}}^{p}(-1)^{q(k-1)}sgn(\sigma)P_{II}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(k-1)},x_{q+k}\wedge
Q_{II}(X_{\sigma(k)},\cdot\cdot\cdot,X_{\sigma(k+q-1)},y_{k+q}),X_{k+q+1},\cdot\cdot\cdot,X_{p+q},z)$
$\displaystyle+\sum_{\begin{subarray}{c}k=1,\\\ \sigma\in
sh(k-1,q)\end{subarray}}^{p}(-1)^{q(k-1)}sgn(\sigma)P_{II}(X_{\sigma(1)},\cdot\cdot\cdot,X_{\sigma(k-1)},Q_{II}(X_{\sigma(k)},\cdot\cdot\cdot,X_{\sigma(k+q-1)},x_{k+q})\wedge
y_{q+k},X_{k+q+1},\cdot\cdot\cdot,X_{p+q},z).$
In particular, for
$f\in\mathcal{C}^{0}(\mathfrak{g},\mathfrak{g})=\mathrm{Hom}(\mathfrak{g},\mathfrak{g})$
and $P=(P_{I},P_{II})\in\mathcal{C}^{p}(\mathfrak{g},\mathfrak{g})$, define
$\displaystyle(P\circ f)_{I}(X_{1},\cdot\cdot\cdot,X_{p})=$
$\displaystyle\sum_{k=1}^{p}P_{I}(X_{1},\cdot\cdot\cdot,X_{k-1},x_{k}\wedge
f(y_{k}),X_{k+1},\cdot\cdot\cdot,X_{p})$ (68)
$\displaystyle+\sum_{k=1}^{p}P_{I}(X_{1},\cdot\cdot\cdot,X_{k-1},f(x_{k})\wedge
y_{k},X_{k+1},\cdot\cdot\cdot,X_{p}),$ (69) $(f\circ
P)_{I}(X_{1},\cdot\cdot\cdot,X_{p})=f(P_{I}(X_{1},\cdot\cdot\cdot,X_{p})),$
$\displaystyle(P\circ f)_{II}(X_{1},\cdot\cdot\cdot,X_{p},z)=$
$\displaystyle\sum_{k=1}^{p}P_{II}(X_{1},\cdot\cdot\cdot,X_{k-1},x_{k}\wedge
f(y_{k}),X_{k+1},\cdot\cdot\cdot,X_{p},z)$ (70)
$\displaystyle+\sum_{k=1}^{p}P_{I}(X_{1},\cdot\cdot\cdot,X_{k-1},f(x_{k})\wedge
y_{k},X_{k+1},\cdot\cdot\cdot,X_{p},z),$ (71) $(f\circ
P)_{II}(X_{1},\cdot\cdot\cdot,X_{p},z)=f(P_{II}(X_{1},\cdot\cdot\cdot,X_{p},z)).$
In order to derive sufficient and necessary conditions of Lie-Yamaguti
algebras in terms of Maurer-Cartan element of some graded Lie algebras, we
define $P\bullet Q$ as follows: for all
$P=(P_{I},P_{II})\in\mathcal{C}^{n}(\mathfrak{g},\mathfrak{g}),~{}Q=(Q_{I},Q_{II})\in\mathcal{C}^{n}(\mathfrak{g},\mathfrak{g})$,
(72) $P\bullet Q=\left\\{\begin{aligned} &0,&p,q\neq 1,\\\ &((P\bullet
Q)_{I},(P\bullet Q)_{II}),&p=q=1,\end{aligned}\right.$
where
$\displaystyle(P\bullet Q)_{I}(x_{1},x_{2},x_{3})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\sigma\in
S_{3}}sgn(\sigma)P_{I}(Q_{I}(x_{\sigma(1)},x_{\sigma(2)}),x_{\sigma(3)})+\frac{1}{2}\sum_{\sigma\in
S_{3}}sgn(\sigma)Q_{II}(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)}),$
$\displaystyle(P\bullet
Q)_{II}(x_{1},x_{2},x_{3},x_{4})=\frac{1}{2}\sum_{\sigma\in
S_{3}}sgn(\sigma)P_{II}(Q_{I}(x_{\sigma(1)},x_{\sigma(2)}),x_{\sigma(3)},x_{4}).$
Let $\mathfrak{g}$ and $V$ be vector spaces. For any
$(\chi,\omega),(\chi^{\prime},\omega^{\prime})\in\mathcal{C}^{1}(\mathfrak{g},\mathfrak{g})$,
put $\Pi=(\chi,\omega),~{}\Pi^{\prime}=(\chi^{\prime},\omega^{\prime})$ and
$\Pi+\Pi^{\prime}=(\chi+\chi^{\prime},\omega+\omega^{\prime})$.
###### Proposition 4.1.
With the above notations, $(\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g}),[\ ,\
]_{LY})$ is a graded Lie algebra, where
(73) $[P,Q]_{LY}=\left\\{\begin{aligned} &P\bullet Q+Q\bullet P+P\circ
Q+Q\circ P,&p=q=1,\\\ &P\circ Q-(-1)^{pq}Q\circ
P,&otherwise,\end{aligned}\right.$
for all
$P\in\mathcal{C}^{p}(\mathfrak{g},\mathfrak{g}),Q\in\mathcal{C}^{q}(\mathfrak{g},\mathfrak{g}).$
Furthermore, $\Pi=(\chi,\omega)\in\mathcal{C}^{1}(\mathfrak{g},\mathfrak{g})$
defines a Lie-Yamaguti algebra structure on $\mathfrak{g}$ if and only if $[\
,\ ]_{LY}=0$, that is, $\Pi$ is a Maurer-Cartan element of the graded Lie
algebra $(\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g}),[\ ,\ ]_{LY})$. We write
$[P,Q]_{LY}=([P,Q]_{I},[P,Q]_{II}).$
###### Proof.
Take the same procedure of the proof of Proposition 4.1 [31], we can check
that Eq. (67) holds. Clearly, Eq. (67) implies that Eq. (1) holds. For all
$\Pi=(\chi,\omega)\in\mathcal{C}^{1}(\mathfrak{g},\mathfrak{g})$,
$[\Pi,\Pi]_{LY}=2\Pi\circ\Pi+2\Pi\bullet\Pi$ and for all
$x_{1},x_{2},x_{3},x_{4}\in\mathfrak{g}$, we have
$\displaystyle(\Pi\bullet\Pi)_{I}(x_{1},x_{2},x_{3})=$
$\displaystyle\chi(\chi(x_{1},x_{2}),x_{3})+\chi(\chi(x_{2},x_{3}),x_{1})+\chi(\chi(x_{3},x_{1}),x_{2})$
$\displaystyle+\omega(x_{1},x_{2},x_{3})+\omega(x_{2},x_{3},x_{1})+\omega(x_{3},x_{1},x_{2}),$
and
$\displaystyle(\Pi\bullet\Pi)_{II}(x_{1},x_{2},x_{3},x_{4})$ $\displaystyle=$
$\displaystyle\omega(\chi(x_{1},x_{2}),x_{3},x_{4})+\omega(\chi(x_{2},x_{3}),x_{1},x_{4})+\omega(\chi(x_{3},x_{1}),x_{2},x_{4}).$
Combining Theorem 3.1 [31],
$\Pi=(\chi,\omega)\in\mathcal{C}^{1}(\mathfrak{g},\mathfrak{g})$ defines a
Lie-Yamaguti algebra structure on $\mathfrak{g}$ if and only if $\Pi$ is a
Maurer-Cartan element of the graded Lie algebra
$(\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g}),[\ ,\ ]_{LY})$. ∎
By Proposition 4.1, we rewrite Theorem 3.3 [31] as follows:
###### Theorem 4.2.
Let $(\mathfrak{g},\chi_{\mathfrak{g}},\omega_{\mathfrak{g}})$ be a Lie-
Yamaguti algebra. Then $(\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g}),[\ ,\
]_{LY},d_{\Pi})$ is a differential graded Lie algebra, where $d_{\Pi}$ with
$\Pi=(\chi_{\mathfrak{g}},\omega_{\mathfrak{g}})$ is given by
(74)
$d_{\Pi}(\nu)=[\Pi,\nu]_{LY},~{}~{}\forall~{}\nu\in\mathcal{C}^{n-1}(\mathfrak{g},\mathfrak{g}).$
Moreover, $\Pi+\Pi^{\prime}$ with
$\Pi^{\prime}\in\mathcal{C}^{1}(\mathfrak{g},\mathfrak{g})$ defines a Lie-
Yamaguti algebra structure on $\mathfrak{g}$ if and only if $\Pi^{\prime}$ is
a Maurer-Cartan element of the differential graded Lie algebra
$(\mathcal{L}^{*}(\mathfrak{g},\mathfrak{g}),[\ ,\ ]_{LY},d_{\Pi})$.
Denote
$\bar{\mu}(x+a,y+b)=\mu(x)b-\mu(y)a$
and
$\bar{\theta}(x+a,y+b,z+c)=D(x,y)c+\theta(y,z)a-\theta(x,z)b$
for all $x,y,z\in\mathfrak{g}$ and $a,b,c\in V$.
###### Proposition 4.3.
With the above notations, $(V,\mu,\theta,D)$ is a representation of Lie-
Yamaguti algebra $(\mathfrak{g},\chi_{\mathfrak{g}},\omega_{\mathfrak{g}})$ if
and only if $\bar{\Pi}\in\mathcal{L}^{*}(\mathfrak{g}\ltimes
V,\mathfrak{g}\ltimes V)$ is a Maurer-Cartan element of the differential
graded Lie algebra $(\mathcal{L}^{*}(\mathfrak{g}\ltimes V,\mathfrak{g}\ltimes
V),[\ ,\ ]_{LY},d_{\bar{\Pi}})$, where $\bar{\Pi}=(\bar{\mu},\bar{\theta}).$
###### Proof.
It follows from Theorem 4.2. ∎
Let $(\mathfrak{g},\chi_{\mathfrak{g}},\omega_{\mathfrak{g}})$ and
$(\mathfrak{h},\chi_{\mathfrak{h}},\omega_{\mathfrak{h}})$ be two Lie-Yamaguti
algebras. Then
$(\mathfrak{g}\oplus\mathfrak{h},\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})$
is a Lie-Yamaguti algebra, where
$\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}$
are defined by
$\chi_{\mathfrak{g}\oplus\mathfrak{h}}(x+a,y+b)=\chi_{\mathfrak{g}}(x,y)+\chi_{\mathfrak{h}}(a,b),~{}~{}\omega_{\mathfrak{g}\oplus\mathfrak{h}}(x+a,y+b,z+c)=\omega_{\mathfrak{g}}(x,y,z)+\omega_{\mathfrak{h}}(a,b,c)$
for all $x,y,z\in\mathfrak{g},a,b,c\in\mathfrak{h}$.
In view of Theorem 4.2,
$(\mathcal{L}^{*}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{g}\oplus\mathfrak{h}),[\
,\
]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$
is a differential graded Lie algebra. Define
$\mathcal{C}_{>}^{n}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})\subset\mathcal{C}^{n}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h}),~{}\mathbb{C}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})\subset\mathbb{C}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})$
respectively by
$\mathcal{C}^{n}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})=\mathcal{C}_{>}^{n}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})\oplus\mathcal{C}^{n}(\mathfrak{h},\mathfrak{h}),~{}\mathbb{C}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})=\mathbb{C}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})\oplus\mathbb{C}(\mathfrak{h},\mathfrak{h}).$
Denote by
$\mathcal{C}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})=\oplus_{n}\mathcal{C}_{>}^{n}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})$
and
$\mathcal{L}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})=\mathcal{C}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})\oplus\mathbb{C}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h})$.
Similar to the case of $3$-Lie algebras [27], we have
###### Proposition 4.4.
With the above notations,
$(\mathcal{L}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h}),[\ ,\
]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$
is a differential graded Lie subalgebra of
$(\mathcal{L}^{*}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{g}\oplus\mathfrak{h}),[\
,\
]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$.
###### Proposition 4.5.
The following conditions are equivalent:
1. $(i)$
$(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\\{\ ,\ ,\ \\}_{\omega})$ is a
Lie-Yamaguti algebra, which is a non-abelian extension of $\mathfrak{g}$ by
$\mathfrak{h}$.
2. $(ii)$
$\Pi=(\bar{\chi},\bar{\omega})$ is a Maurer-Cartan element of the differential
graded Lie algebra $(C_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h}),[\ ,\
]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$,
where
$\bar{\chi}(x+a,y+b)=\chi(x,y)+\mu(x)b-\mu(y)a,$
$\displaystyle\bar{\omega}(x+a,y+b,z+c)=$
$\displaystyle\omega(x,y,z)+D(x,y)c+\theta(y,z)a-\theta(x,z)b$
$\displaystyle+T(z)(a,b)+\rho(x)(b,c)-\rho(y)(a,c),$
for all $x,y,z\in\mathfrak{g},a,b,c\in\mathfrak{h}$.
###### Proof.
In view of the definition of Maurer-Cartan element,
$\Pi=(\bar{\chi},\bar{\omega})$ is a Maurer-Cartan element of the differential
graded Lie algebra
$(\mathcal{L}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h}),[\ ,\
]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$
if and only if
$d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\Pi+\frac{1}{2}[\Pi,\Pi]_{LY}=0,$
that is,
(75)
$[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),(\bar{\chi},\bar{\omega})]_{LY}+\frac{1}{2}[(\bar{\chi},\bar{\omega}),(\bar{\chi},\bar{\omega})]_{LY}=0.$
On the other hand, by Proposition 4.1, we know that
$(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\\{\ ,\ ,\ \\}_{\omega})$ is a
Lie-Yamaguti algebra if and only if
(76)
$[(\chi_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\chi},\omega_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\omega}),(\chi_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\chi},\omega_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\omega})]_{LY}=0.$
Since
$(\mathfrak{g}\oplus\mathfrak{h},\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})$
is a Lie-Yamaguti algebra, by computations, we have
$\displaystyle[(\chi_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\chi},\omega_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\omega}),(\chi_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\chi},\omega_{\mathfrak{g}\oplus\mathfrak{h}}+\bar{\omega})]_{LY}$
$\displaystyle=$
$\displaystyle[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})]_{LY}+[(\bar{\chi},\bar{\omega}),(\bar{\chi},\bar{\omega})]_{LY}+2[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),(\bar{\chi},\bar{\omega})]_{LY}$
$\displaystyle=$
$\displaystyle[(\bar{\chi},\bar{\omega}),(\bar{\chi},\bar{\omega})]_{LY}+2[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),(\bar{\chi},\bar{\omega})]_{LY},$
which yields that Eq. (75) holds if and only if Eq. (76) holds. This completes
the proof. ∎
###### Proposition 4.6.
Two non-abelian extensions $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\
]_{\chi_{0}},\\{\ ,\ ,\ \\}_{\omega_{0}})$ and
$(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\\{\ ,\ ,\ \\}_{\omega})$ are
equivalent if and only if the Maurer-Cartan elements
$\Pi_{0}=(\bar{\chi}_{0},\bar{\omega}_{0})$ and
$\Pi=(\bar{\chi},\bar{\omega})$ are gauge equivalent.
###### Proof.
Let $\Pi_{0},\Pi$ be two Maurer-Cartan elements of the differential graded Lie
algebra $(\mathcal{L}_{>}(\mathfrak{g}\oplus\mathfrak{h},\mathfrak{h}),[\ ,\
]_{LY},d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})})$.
$\Pi_{0},\Pi$ are gauge equivalent if and only if there is a linear map
$\varphi\in\mathrm{Hom}(\mathfrak{g},\mathfrak{h})$ such that
$\displaystyle\Pi_{0}=$ $\displaystyle
e^{ad_{\varphi}}\Pi-\frac{e^{ad_{\varphi}}-1}{ad_{\varphi}}d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi$
$\displaystyle=$
$\displaystyle(id+ad_{\varphi}+\frac{1}{2!}ad_{\varphi}^{2}+\cdot\cdot\cdot++\frac{1}{n!}ad_{\varphi}^{n}+\cdot\cdot\cdot)\Pi$
$\displaystyle-(id+\frac{1}{2!}ad_{\varphi}+\frac{1}{3!}ad_{\varphi}^{2}+\cdot\cdot\cdot+\frac{1}{n!}ad_{\varphi}^{n-1}+\cdot\cdot\cdot)d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi.$
In the following, we denote by
$ad_{\varphi}\Pi=[\varphi,\Pi]_{LY}=([\varphi,\Pi]_{I},[\varphi,\Pi]_{II}),~{}~{}ad_{\varphi}^{2}\Pi=[\varphi,[\varphi,\Pi]_{LY}]_{LY}=([\varphi,[\varphi,\Pi]_{LY}]_{I},[\varphi,[\varphi,\Pi]_{LY}]_{II}),$
$d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi=[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{LY}=([(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{I},[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}),$
and
$[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{LY}=([\varphi,[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{LY}]_{I},[\varphi,[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{LY}]_{II}).$
Using Eqs. (68)-(71), for all
$w_{i}=x_{i}+a_{i}\in\mathfrak{g}\oplus\mathfrak{h}~{}(i=1,2,3)$, we get
$\displaystyle[\varphi,\Pi]_{I}(w_{1},w_{2})=$
$\displaystyle\varphi\bar{\chi}(w_{1},w_{2})-\bar{\chi}(w_{1},\varphi(x_{2}))-\bar{\chi}(\varphi(x_{1}),w_{2})$
$\displaystyle=$
$\displaystyle-\mu(x_{1})\varphi(x_{2})+\mu(x_{2})\varphi(x_{1}),$
$\displaystyle[\varphi,\Pi]_{II}(w_{1},w_{2},w_{3})=$
$\displaystyle\varphi\bar{\omega}(w_{1},w_{2},w_{3})-\bar{\omega}(\varphi(x_{1}),w_{2},w_{3})-\bar{\omega}(w_{1},w_{2},\varphi(x_{3}))-\bar{\omega}(w_{1},\varphi(x_{2}),w_{3})$
$\displaystyle=$
$\displaystyle\theta(x_{1},x_{3})\varphi(x_{2})-T(x_{3})(a_{1},\varphi(x_{2}))-\rho(x_{1})(\varphi(x_{2}),a_{3})$
$\displaystyle-\theta(x_{2},x_{3})\varphi(x_{1})-T(x_{3})(\varphi(x_{1}),a_{2})+\rho(x_{2})(\varphi(x_{1}),a_{3})$
$\displaystyle+\rho(x_{2})(a_{1},\varphi(x_{3}))-D(x_{1},x_{2})\varphi(x_{3})-\rho(x_{1})(a_{2},\varphi(x_{3})),$
$[\varphi,[\varphi,\Pi]_{LY}]_{I}(w_{1},w_{2})=\varphi[\varphi,\Pi]_{I}(w_{1},w_{2})-[\varphi,\chi]_{I}(w_{1},\varphi(x_{2}))-[\varphi,\Pi]_{I}(\varphi(x_{1}),w_{2})=0,$
$\displaystyle[\varphi,[\varphi,\Pi]_{LY}]_{II}(w_{1},w_{2},w_{3})=$
$\displaystyle\varphi[\varphi,\Pi]_{II}(w_{1},w_{2},w_{3})-[\varphi,\Pi]_{II}(w_{1},\varphi(x_{2}),w_{3})$
$\displaystyle-[\varphi,\Pi]_{II}(\varphi(x_{1}),w_{2},w_{3})-[\varphi,\Pi]_{II}(w_{1},w_{2},\varphi(x_{3}))$
$\displaystyle=$ $\displaystyle
2T(x_{3})(\varphi(x_{1}),\varphi(x_{2}))+2\rho(x_{1})(\varphi(x_{2}),\varphi(x_{3}))-2\rho(x_{2})(\varphi(x_{1}),\varphi(x_{3})),$
$\displaystyle[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{I}(w_{1},w_{2})=$
$\displaystyle\chi_{\mathfrak{g}\oplus\mathfrak{h}}(w_{1},\varphi(x_{2}))+\chi_{\mathfrak{g}\oplus\mathfrak{h}}(\varphi(x_{1}),w_{2})-\varphi\chi_{\mathfrak{g}\oplus\mathfrak{h}}(w_{1},w_{2})$
$\displaystyle=$
$\displaystyle[a_{1},\varphi(x_{2})]_{\mathfrak{h}}+[\varphi(x_{1}),a_{2}]_{\mathfrak{h}}-\varphi([x_{1},x_{2}]_{\mathfrak{g}}),$
$\displaystyle[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}(w_{1},w_{2},w_{3})$
$\displaystyle=$
$\displaystyle\omega_{\mathfrak{g}\oplus\mathfrak{h}}(w_{1},\varphi(x_{2}),w_{3})+\omega_{\mathfrak{g}\oplus\mathfrak{h}}(\varphi(x_{1}),w_{2},w_{3})+\omega_{\mathfrak{g}\oplus\mathfrak{h}}(w_{1},w_{2},\varphi(x_{3}))-\varphi\omega_{\mathfrak{g}\oplus\mathfrak{h}}(w_{1},w_{2},w_{3})$
$\displaystyle=$
$\displaystyle\\{a_{1},\varphi(x_{2}),a_{3}\\}_{\mathfrak{h}}+\\{\varphi(x_{1}),a_{2},a_{3}\\}_{\mathfrak{h}}+\\{a_{1},a_{2},\varphi(x_{3})\\}_{\mathfrak{h}}-\varphi(\\{x_{1},x_{2},x_{3}\\}_{\mathfrak{g}}),$
$\displaystyle[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{I}(w_{1},w_{2})$
$\displaystyle=$
$\displaystyle\varphi[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{I}(w_{1},w_{2})-[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{I}(w_{1},\varphi(x_{2}))-[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{I}(\varphi(x_{1}),w_{2})$
$\displaystyle=$
$\displaystyle-2[\varphi(x_{1}),\varphi(x_{2})]_{\mathfrak{h}},$
$\displaystyle[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{II}(w_{1},w_{2},w_{3})$
$\displaystyle=$
$\displaystyle\varphi[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}(w_{1},w_{2},w_{3})-[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}(w_{1},\varphi(x_{2}),w_{3})$
$\displaystyle-[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}(\varphi(x_{1}),w_{2},w_{3})-[(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}}),\varphi]_{II}(w_{1},w_{2},\varphi(x_{3}))$
$\displaystyle=$
$\displaystyle-\\{\varphi(x_{1}),\varphi(x_{2}),a_{3}\\}_{\mathfrak{h}}-\\{a_{1},\varphi(x_{2}),\varphi(x_{3})\\}_{\mathfrak{h}}-\\{\varphi(x_{1}),\varphi(x_{2}),a_{3}\\}_{\mathfrak{h}}$
$\displaystyle-\\{\varphi(x_{1}),a_{2},\varphi(x_{3})\\}_{\mathfrak{h}}-\\{a_{1},\varphi(x_{2}),\varphi(x_{3})\\}_{\mathfrak{h}}-\\{\varphi(x_{1}),a_{2},\varphi(x_{3})\\}_{\mathfrak{h}}$
$\displaystyle=$
$\displaystyle-2\\{\varphi(x_{1}),\varphi(x_{2}),a_{3}\\}_{\mathfrak{h}}-2\\{a_{1},\varphi(x_{2}),\varphi(x_{3})\\}_{\mathfrak{h}}-2\\{\varphi(x_{1}),a_{2},\varphi(x_{3})\\}_{\mathfrak{h}},$
$\displaystyle[\varphi,[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{LY}]_{I}(w_{1},w_{2})$
$\displaystyle=$
$\displaystyle\varphi[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{I}(w_{1},w_{2})-[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{I}(x_{1}+a_{1},\varphi(x_{2}))-[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{I}(\varphi(x_{1}),w_{2})$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle[\varphi,[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{LY}]_{II}(w_{1},w_{2},w_{3})$
$\displaystyle=$
$\displaystyle[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{II}(x_{1}+a_{1},\varphi(x_{2}),w_{3})+[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{II}(\varphi(x_{1}),w_{2},w_{3})$
$\displaystyle+[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{II}(w_{1},w_{2},\varphi(x_{3}))-\varphi[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}(\varphi)]_{II}(w_{1},w_{2},w_{3})$
$\displaystyle=$ $\displaystyle
6\\{\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3})\\}_{\mathfrak{h}},$
and
$ad_{\varphi}^{n}\Pi=0,~{}~{}ad_{\varphi}^{n}(d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\Pi)=0,~{}~{}\forall~{}~{}n\geq
3.$
Thus, $\Pi$ and $\Pi_{0}$ are gauge equivalent Maurer-Cartan elements if and
only if
(77)
$\chi_{0}=\chi+[\varphi,\Pi]_{I}-d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi-\frac{1}{2!}[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi]_{I},$
$\displaystyle\omega_{0}=$
$\displaystyle\omega+[\varphi,\Pi]_{II}+\frac{1}{2!}[\varphi,[\varphi,\Pi]_{LY}]_{II}-d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi$
(78)
$\displaystyle-\frac{1}{2!}[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi]_{II}-\frac{1}{3!}[\varphi,[\varphi,d_{(\chi_{\mathfrak{g}\oplus\mathfrak{h}},\omega_{\mathfrak{g}\oplus\mathfrak{h}})}\varphi]_{LY}]_{II}.$
Therefore, Eqs. (77) and (78) hold if and only if Eqs. (53)-(58) hold. This
finishes the proof.
∎
## 5\. Extensibility of a pair of Lie-Yamaguti algebra automorphisms
In this section, we study extensibility of pairs of Lie-Yamaguti algebra
automorphisms and characterize them by equivalent conditions.
Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian
extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$.
Denote
$\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})=\\{\gamma\in\mathrm{Aut}(\hat{\mathfrak{g}})\mid\gamma(\mathfrak{h})=\mathfrak{h}\\}.$
###### Definition 5.1.
A pair of automorphisms
$(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$
is said to be extensible with respect to a non-abelian extension
$\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$
if there is an automorphism
$\gamma\in\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})$ such that
$i\beta=\gamma i,~{}p\gamma=\alpha p$, that is, the following commutative
diagram holds:
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{i}$$\textstyle{\hat{\mathfrak{g}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{p}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\hat{\mathfrak{g}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
It is natural to ask: when is a pair of automorphisms
$(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$
extensible? We discuss this problem in the following.
###### Theorem 5.2.
Let $0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian
extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$ and
$(\chi,\omega,\mu,\theta,D,\rho,T)$ the corresponding non-abelian
(2,3)-cocycle induced by $s$. A pair
$(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$
is extensible if and only if there is a linear map
$\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ satisfying the following
conditions:
$\displaystyle\beta\omega(x,y,z)-\omega(\alpha(x),\alpha(y),\alpha(z))=T(\alpha(z))(\varphi(x),\varphi(y))-\rho(\alpha(y))(\varphi(x),\varphi(z))-\theta(\alpha(y),\alpha(z))\varphi(x)$
(79) $\displaystyle+$
$\displaystyle\rho(\alpha(x))(\varphi(y),\varphi(z))+\theta(\alpha(x),\alpha(z))\varphi(y)-D(\alpha(x),\alpha(y))\varphi(z)+\varphi(\\{x,y,z\\}_{\mathfrak{h}})-\\{\varphi(x),\varphi(y),\varphi(z)\\}_{\mathfrak{h}},$
(80)
$\beta\chi(x,y)-\chi(\alpha(x),\alpha(y))=[\varphi(x),\varphi(y)]_{\mathfrak{h}}+\varphi([x,y]_{\mathfrak{g}})-\mu(\alpha(x))\varphi(y)+\mu(\alpha(y))\varphi(x),$
(81)
$\beta(\theta(x,y)a)-\theta(\alpha(x),\alpha(y))\beta(a)=\\{\beta(a),\varphi(x),\varphi(y)\\}_{\mathfrak{h}}-T(\alpha(y))(\beta(a),\varphi(x))+\rho(\alpha(x))(\beta(a),\varphi(y)),$
(82) $\beta
D(x,y)a-D(\alpha(x),\alpha(y))\beta(a)=\\{\varphi(x),\varphi(y),\beta(a)\\}_{\mathfrak{h}}-\rho(\alpha(x))(\varphi(y),\beta(a))+\rho(\alpha(y))(\varphi(x),\beta(a)),$
(83)
$\beta(\rho(x)(a,b))-\rho(\alpha(x))(\beta(a),\beta(b))=\\{\beta(a),\varphi(x),\beta(b)\\}_{\mathfrak{h}},$
(84) $\beta
T(x)(a,b)-T(\alpha(x))(\beta(a),\beta(b))=\\{\beta(b),\beta(a),\varphi(x)\\}_{\mathfrak{h}},$
(85)
$\beta\mu(x)a-\mu(\alpha(x))\beta(a)=[\beta(a),\varphi(x)]_{\mathfrak{h}},$
for all $x,y,z\in\mathfrak{g}$ and $a\in\mathfrak{h}$.
###### Proof.
Assume that
$(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$
is extensible, that is, there is an automorphism
$\gamma\in\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})$ such that $\gamma
i=i\beta$ and $p\gamma=\alpha p$. Due to $s$ being a section of $p$, for all
$x\in\mathfrak{g}$,
$p(s\alpha-\gamma s)(x)=\alpha(x)-\alpha(x)=0,$
which implies that $(s\alpha-\gamma s)(x)\in\mathrm{ker}p=\mathfrak{h}$. So we
can define a linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ by
$\varphi(x)=(s\alpha-\gamma s)(x),~{}~{}\forall~{}x\in\mathfrak{g}.$
Using Eqs. (63) and (64), for $x,y\in\mathfrak{g},a\in\mathfrak{h}$, we get
$\displaystyle\beta(\theta(x,y)a)-\theta(\alpha(x),\alpha(y))\beta(a)$
$\displaystyle=$
$\displaystyle\beta\\{a,s(x),s(y)\\}_{\hat{\mathfrak{g}}}-\\{\beta(a),s\alpha(x),s\alpha(y)\\}_{\hat{\mathfrak{g}}}$
$\displaystyle=$ $\displaystyle\\{\beta(a),\beta s(x),\beta
s(y)\\}_{\hat{\mathfrak{g}}}-\\{\beta(a),s\alpha(x),s\alpha(y)\\}_{\hat{\mathfrak{g}}}$
$\displaystyle=$ $\displaystyle\\{\beta(a),\beta s(x)-s\alpha(x),\beta
s(y)\\}_{\hat{\mathfrak{g}}}+\\{\beta(a),s\alpha(x),\beta
s(y)\\}_{\hat{\mathfrak{g}}}-\\{\beta(a),s\alpha(x),s\alpha(y)\\}_{\hat{\mathfrak{g}}}$
$\displaystyle=$ $\displaystyle\\{\beta(a),-\varphi(x),\beta
s(y)\\}_{\hat{\mathfrak{g}}}+\\{\beta(a),s\alpha(x),-\varphi(y)\\}_{\hat{\mathfrak{g}}}$
$\displaystyle=$ $\displaystyle-\\{\beta(a),\varphi(x),\beta
s(y)-s\alpha(y)\\}_{\hat{\mathfrak{g}}}-\\{\beta(a),\varphi(x),s\alpha(y)\\}_{\hat{\mathfrak{g}}}-\\{\beta(a),s\alpha(x),\varphi(y)\\}_{\hat{\mathfrak{g}}}$
$\displaystyle=$
$\displaystyle\\{\beta(a),\varphi(x),\varphi(y)\\}_{\hat{\mathfrak{g}}}-\\{\beta(a),\varphi(x),s\alpha(y)\\}_{\hat{\mathfrak{g}}}+\\{s\alpha(x),\beta(a),\varphi(y)\\}_{\hat{\mathfrak{g}}}$
$\displaystyle=$
$\displaystyle\\{\beta(a),\varphi(x),\varphi(y)\\}_{\mathfrak{h}}-T(\alpha(y))(\beta(a),\varphi(x))+\rho(\alpha(x))(\beta(a),\varphi(y)),$
which indicates that Eq. (81) holds. Take the same procedure, we can prove
that Eqs. (5.2), (80) and (82)-(85) hold.
Conversely, suppose that
$(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$
and there is a linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$
satisfying Eqs. (5.2)-(85). Since $s$ is a section of $p$, all
$\hat{w}\in\hat{\mathfrak{g}}$ can be written as $\hat{w}=a+s(x)$ for some
$a\in\mathfrak{h},x\in\mathfrak{g}.$ Define a linear map
$\gamma:\hat{\mathfrak{g}}\longrightarrow\hat{\mathfrak{g}}$ by
$\gamma(\hat{w})=\gamma(a+s(x))=\beta(a)-\varphi(x)+s\alpha(x).$
It is easy to check that $i\beta=\gamma i,~{}p\gamma=\alpha p$ and
$\gamma(\mathfrak{h})=\mathfrak{h}$. In the sequel, firstly, we prove that
$\gamma$ is bijective. Indeed if $\gamma(\hat{w})=0,$ we have $s\alpha(x)=0$
and $\beta(a)-\varphi(x)=0$. In view of $s$ and $\alpha$ being injective, we
get $x=0$, which follows that $a=0$. Thus, $\hat{w}=a+s(x)=0$, that is
$\gamma$ is injective. For any $\hat{w}=a+s(x)\in\hat{\mathfrak{g}}$,
$\gamma(\beta^{-1}(a)+\beta^{-1}\varphi\alpha^{-1}(x)+s\alpha^{-1}(x))=a+s(x)=\hat{w},$
which yields that $\gamma$ is surjective. In all, $\gamma$ is bijective.
Secondly, we show that $\gamma$ is a homomorphism of the Lie-Yamaguti algebra
$\hat{\mathfrak{g}}$. In fact, for all
$\hat{w}_{i}=a_{i}+s(x_{i})\in\hat{\mathfrak{g}}~{}(i=1,2,3)$,
$\displaystyle\\{\gamma(\hat{w}_{1}),\gamma(\hat{w}_{2}),\gamma(\hat{w}_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle=$
$\displaystyle\\{\beta(a_{1})-\varphi(x_{1})+s\alpha(x_{1}),\beta(a_{2})-\varphi(x_{2})+s\alpha(x_{2}),\beta(a_{3})-\varphi(x_{3})+s\alpha(x_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle=$
$\displaystyle\\{\beta(a_{1}),\beta(a_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}-\\{\beta(a_{1}),\beta(a_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}+\\{\beta(a_{1}),\beta(a_{2}),s\alpha(x_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle-\\{\beta(a_{1}),\varphi(x_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}+\\{\beta(a_{1}),\varphi(x_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}-\\{\beta(a_{1}),\varphi(x_{2}),s\alpha(x_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle+\\{\beta(a_{1}),s\alpha(x_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}-\\{\beta(a_{1}),s\alpha(x_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}+\\{\beta(a_{1}),s\alpha(x_{2}),s\alpha(x_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle-\\{\varphi(x_{1}),\beta(a_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}+\\{\varphi(x_{1}),\beta(a_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}-\\{\varphi(x_{1}),\beta(a_{2}),s\alpha(x_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle+\\{\varphi(x_{1}),\varphi(x_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}-\\{\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}+\\{\varphi(x_{1}),\varphi(x_{2}),s\alpha(x_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle-\\{\varphi(x_{1}),s\alpha(x_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}+\\{\varphi(x_{1}),s\alpha(x_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}-\\{\varphi(x_{1}),s\alpha(x_{2}),s\alpha(x_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle+\\{s\alpha(x_{1}),\beta(a_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}-\\{s\alpha(x_{1}),\beta(a_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}+\\{s\alpha(x_{1}),\beta(a_{2}),s\alpha(x_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle-\\{s\alpha(x_{1}),\varphi(x_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}+\\{s\alpha(x_{1}),\varphi(x_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}-\\{s\alpha(x_{1}),\varphi(x_{2}),s\alpha(x_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle+\\{s\alpha(x_{1}),s\alpha(x_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}-\\{s\alpha(x_{1}),s\alpha(x_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}+\\{s\alpha(x_{1}),s\alpha(x_{2}),s\alpha(x_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle=$
$\displaystyle\\{\beta(a_{1}),\beta(a_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}-\\{\beta(a_{1}),\beta(a_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}+T(\alpha(x_{3}))(\beta(a_{1}),\beta(a_{2}))$
$\displaystyle-\\{\beta(a_{1}),\varphi(x_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}+\\{\beta(a_{1}),\varphi(x_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}-T(\alpha(x_{3}))(\beta(a_{1}),\varphi(x_{2}))$
$\displaystyle-\rho(\alpha(x_{2}))(\beta(a_{1}),\beta(a_{3}))+\rho(\alpha(x_{2}))(\beta(a_{1}),\varphi(x_{3}))+\theta(\alpha(x_{2}),\alpha(x_{3}))\beta(a_{1})$
$\displaystyle-\\{\varphi(x_{1}),\beta(a_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}+\\{\varphi(x_{1}),\beta(a_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}-T(\alpha(x_{3}))(\varphi(x_{1}),\beta(a_{2}))$
$\displaystyle+\\{\varphi(x_{1}),\varphi(x_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}-\\{\varphi(x_{1}),\varphi(x_{2}),\varphi(x_{3})\\}_{\hat{\mathfrak{g}}}+T(\alpha(x_{3}))(\varphi(x_{1}),\varphi(x_{2}))$
$\displaystyle+\rho(\alpha(x_{2}))(\varphi(x_{1}),\beta(a_{3}))-\rho(\alpha(x_{2}))(\varphi(x_{1}),\varphi(x_{3}))-\theta(\alpha(x_{2}),\alpha(x_{3}))\varphi(x_{1})$
$\displaystyle+\rho(\alpha(x_{1}))(\beta(a_{2}),\beta(a_{3}))-\rho(\alpha(x_{1}))(\beta(a_{2}),\varphi(x_{3}))-\theta(\alpha(x_{1}),\alpha(x_{3}))\beta(a_{2})$
$\displaystyle-\rho(\alpha(x_{1}))(\varphi(x_{2}),\beta(a_{3}))+\rho(\alpha(x_{1}))(\varphi(x_{2}),\varphi(x_{3}))+\theta(\alpha(x_{1}),\alpha(x_{3}))\varphi(x_{2})$
$\displaystyle+D(\alpha(x_{1}),\alpha(x_{2}))\beta(a_{3})-D(\alpha(x_{1}),\alpha(x_{2}))\varphi(x_{3})+\omega(\alpha(x_{1}),\alpha(x_{2}),\alpha(x_{3}))+s\alpha\\{x_{1},x_{2},x_{3}\\}_{\mathfrak{g}}$
and
$\displaystyle\gamma(\\{\hat{w}_{1},\hat{w}_{2},\hat{w}_{3}\\}_{\hat{\mathfrak{g}}})$
$\displaystyle=$
$\displaystyle\gamma(\\{a_{1},a_{2},a_{3}\\}_{\hat{\mathfrak{g}}}+\\{a_{1},a_{2},s(x_{3})\\}_{\hat{\mathfrak{g}}}+\\{a_{1},s(x_{2}),a_{3}\\}_{\hat{\mathfrak{g}}}+\\{a_{1},s(x_{2}),s(x_{3})\\}_{\hat{\mathfrak{g}}}$
$\displaystyle+\\{s(x_{1}),a_{2},s(x_{3})\\}_{\hat{\mathfrak{g}}}+\\{s(x_{1}),s(x_{2}),a_{3}\\}_{\hat{\mathfrak{g}}}+[s(x_{1}),a_{2},a_{3}]_{\hat{\mathfrak{g}}}+\omega(x_{1},x_{2},x_{3})+s\\{x_{1},x_{2},x_{3}\\}_{\mathfrak{g}})$
$\displaystyle=$
$\displaystyle\\{\beta(a_{1}),\beta(a_{2}),\beta(a_{3})\\}_{\hat{\mathfrak{g}}}+\beta(T(x_{3})(a_{1},a_{2})-\rho(x_{2})(a_{1},a_{3})+\theta(x_{2},x_{3})a_{1}-\theta(x_{1},x_{3})a_{2}$
$\displaystyle+D(x_{1},x_{2})a_{3}+\rho(x_{1})(a_{2},a_{3}))+\beta(\\{s(x_{1}),s(x_{2}),s(x_{3})\\}_{\hat{\mathfrak{g}}})+\beta\omega(x_{1},x_{2},x_{3})$
$\displaystyle+s\alpha\\{x_{1},x_{2},x_{3}\\}_{\mathfrak{g}}-\varphi(\\{x_{1},x_{2},x_{3}\\}_{\mathfrak{g}}).$
Thanks to Eqs. (5.2)-(83), we have
$\gamma(\\{\hat{w}_{1},\hat{w}_{2},\hat{w}_{3}\\}_{\hat{\mathfrak{g}}})=\\{\gamma(\hat{w}_{1}),\gamma(\hat{w}_{2}),\gamma(\hat{w}_{3})\\}_{\hat{\mathfrak{g}}}.$
By the same token,
$\gamma([\hat{w}_{1},\hat{w}_{2}]_{\hat{\mathfrak{g}}})=[\gamma(\hat{w}_{1}),\gamma(\hat{w}_{2})]_{\hat{\mathfrak{g}}}.$
Hence, $\gamma\in\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})$. This
completes the proof. ∎
Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian
extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$ and
$(\chi,\omega,\mu,\theta,D,\rho,T)$ be the corresponding non-abelian
(2,3)-cocycle induced by $s$. For all
$(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$,
define maps
$\chi_{(\alpha,\beta)}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h},~{}\omega_{(\alpha,\beta)}:\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{h},~{}\mu_{(\alpha,\beta)}:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h}),~{}\theta_{(\alpha,\beta)},D_{(\alpha,\beta)}:\mathfrak{g}\wedge\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{h}),~{}\rho_{(\alpha,\beta)},T_{(\alpha,\beta)}:\mathfrak{g}\longrightarrow\mathrm{Hom}(\mathfrak{h}\wedge\mathfrak{h},\mathfrak{h})$
respectively by
(86)
$\omega_{(\alpha,\beta)}(x,y,z)=\beta\omega(\alpha^{-1}(x),\alpha^{-1}(y),\alpha^{-1}(z)),~{}~{}\chi_{(\alpha,\beta)}(x,y)=\beta\chi(\alpha^{-1}(x),\alpha^{-1}(y)),$
(87)
$\theta_{(\alpha,\beta)}(x,y)a=\beta(\theta(\alpha^{-1}(x),\alpha^{-1}(y))\beta^{-1}(a)),~{}~{}D_{(\alpha,\beta)}(x,y)a=\beta
D(\alpha^{-1}(x),\alpha^{-1}(y))\beta^{-1}(a),$ (88)
$\rho_{(\alpha,\beta)}(x)(a,b)=\beta\rho(\alpha^{-1}(x))(\beta^{-1}(a),\beta^{-1}(b)),~{}~{}T_{(\alpha,\beta)}(x)(a,b)=\beta
T(\alpha^{-1}(x))(\beta^{-1}(a),\beta^{-1}(b)),$ (89)
$\mu_{(\alpha,\beta)}(x)a=\beta\mu(\alpha^{-1}(x))\beta^{-1}(a),$
for all $x,y,z\in\mathfrak{g},a,b\in\mathfrak{h}.$
###### Proposition 5.3.
With the above notations,
$(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})$
is a non-abelian (2,3)-cocycle.
###### Proof.
By Eqs. (• ‣ 3.3), (86) and (87), for all
$x_{1},x_{2},y_{1},y_{2},y_{3}\in\mathfrak{g}$, we get
$\displaystyle
D_{(\alpha,\beta)}(x_{1},x_{2})\omega_{(\alpha,\beta)}(y_{1},y_{2},y_{3})+\omega_{(\alpha,\beta)}(x_{1},x_{2},\\{y_{1},y_{2},y_{3}\\}_{\mathfrak{g}})$
$\displaystyle=$ $\displaystyle\beta
D(\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}))\beta^{-1}\beta\omega_{(\alpha,\beta)}(\alpha^{-1}(y_{1}),\alpha^{-1}(y_{2}),\alpha^{-1}(y_{3}))$
$\displaystyle-\beta\omega_{(\alpha,\beta)}(\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(\\{y_{1},y_{2},y_{3}\\}_{\mathfrak{g}}))$
$\displaystyle=$
$\displaystyle\beta\omega(\\{\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{1})\\}_{\mathfrak{g}},\alpha^{-1}(y_{2}),\alpha^{-1}(y_{3}))$
$\displaystyle+\beta\theta(\alpha^{-1}(y_{2}),\alpha^{-1}(y_{3}))\omega(\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{1}))$
$\displaystyle+\beta\omega(\alpha^{-1}(y_{1}),\\{\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{2})\\}_{\mathfrak{g}},\alpha^{-1}(y_{3}))$
$\displaystyle-\beta\theta(\alpha^{-1}(y_{1}),\alpha^{-1}(y_{3}))\omega(\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{2}))$
$\displaystyle+\beta\omega(\alpha^{-1}(y_{1}),\alpha^{-1}(y_{2}),\\{\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{3})\\}_{\mathfrak{g}})$
$\displaystyle+\beta
D(\alpha^{-1}(y_{1}),\alpha^{-1}(y_{2}))\omega(\alpha^{-1}(x_{1}),\alpha^{-1}(x_{2}),\alpha^{-1}(y_{3}))$
$\displaystyle=$
$\displaystyle\omega_{(\alpha,\beta)}(\\{x_{1},x_{2},y_{1}\\}_{\mathfrak{g}},y_{2},y_{3})+\theta_{(\alpha,\beta)}(y_{2},y_{3})\omega_{(\alpha,\beta)}(x_{1},x_{2},y_{1})+\omega_{(\alpha,\beta)}(y_{1},\\{x_{1},x_{2},y_{2}\\}_{\mathfrak{g}},y_{3})$
$\displaystyle-\theta_{(\alpha,\beta)}(y_{1},y_{3})\omega_{(\alpha,\beta)}(x_{1},x_{2},y_{2})+\omega_{(\alpha,\beta)}(y_{1},y_{2},\\{x_{1},x_{2},y_{3}\\}_{\mathfrak{g}})+D_{(\alpha,\beta)}(y_{1},y_{2})\omega_{(\alpha,\beta)}(x_{1},x_{2},y_{3}),$
which implies that Eq. (• ‣ 3.3) holds for
$(\omega_{(\alpha,\beta)},\theta_{(\alpha,\beta)})$. Similarly, we can check
that Eqs. (• ‣ 3.3)-(52) hold. This completes the proof. ∎
###### Theorem 5.4.
Let $0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian
extension of a Lie-Yamaguti algebra $\mathfrak{g}$ by $\mathfrak{h}$ with a
section $s$ of $p$ and $(\chi,\omega,\mu,\theta,D,\rho,T)$ be the
corresponding non-abelian (2,3)-cocycle induced by $s$. A pair
$(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$
is extensible if and only if the non-abelian (2,3)-cocycles
$(\chi,\omega,\mu,\theta,D,\rho,T)$ and
$(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})$
are equivalent.
###### Proof.
Suppose
$(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$
is extensible, by Theorem 5.2, there is a linear map
$\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ satisfying Eqs. (5.2)-(85).
For all $x,y\in\mathfrak{g},a\in\mathfrak{h}$, there exist
$x_{0},y_{0}\in\mathfrak{g},a_{0}\in\mathfrak{h}$ such that
$x=\alpha(x_{0}),y=\alpha(y_{0}),a=\beta(a_{0})$. Thus, by Eqs. (81), (87) and
(88), we have
$\displaystyle\theta_{(\alpha,\beta)}(x,y)a-\theta(x,y)a$ $\displaystyle=$
$\displaystyle\beta(\theta(\alpha^{-1}(x),\alpha^{-1}(y))\beta^{-1}(a))-\theta(x,y)a$
$\displaystyle=$
$\displaystyle\beta(\theta(x_{0},y_{0})a_{0})-\theta(\alpha(x_{0}),\alpha(y_{0}))\beta(a_{0})$
$\displaystyle=$
$\displaystyle\\{\beta(a_{0}),\varphi(x_{0}),\varphi(y_{0})\\}_{\mathfrak{h}}-T(\alpha(y_{0}))(\beta(a_{0}),\varphi(x_{0}))+\rho(\alpha(x_{0}))(\beta(a_{0}),\varphi(y_{0}))$
$\displaystyle=$
$\displaystyle\\{a,\varphi\alpha^{-1}(x),\varphi\alpha^{-1}(y)\\}_{\mathfrak{h}}-T(y)(a,\varphi\alpha^{-1}(x))+\rho(x)(a,\varphi\alpha^{-1}(y)),$
which indicates that Eq. (56) holds. Analogously, Eqs. (53)-(55) and (57)-(58)
hold. Thus, $(\chi,\omega,\mu,\theta,D,\rho,T)$ and
$(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})$
are equivalent via the linear map
$\varphi\alpha^{-1}:\mathfrak{g}\longrightarrow\mathfrak{h}$.
The converse part can be obtained analogously.
∎
## 6\. Wells exact sequences for Lie-Yamaguti algebras
In this section, we consider the Wells map associated with non-abelian
extensions of Lie-Yamaguti algebras. Then we interpret the results gained in
Section 5 in terms of the Wells map.
Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian
extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$. Then
there is a linear map $t:\hat{\mathfrak{g}}\longrightarrow\mathfrak{h}$, such
that
(90) $it+sp=I_{\hat{\mathfrak{g}}}.$
Assume that $(\chi,\omega,\mu,\theta,D,\rho,T)$ is the corresponding non-
abelian (2,3)-cocycle induced by $s$. Define a linear map
$W:\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})\longrightarrow
H^{(2,3)}_{nab}(\mathfrak{g},\mathfrak{h})$ by
(91)
$W(\alpha,\beta)=[(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})-(\chi,\omega,\mu,\theta,D,\rho,T)].$
The map $W$ is called the Wells map.
###### Proposition 6.1.
The Wells map $W$ does not depend on the choice of sections.
###### Proof.
For all $x,y,z\in\mathfrak{g}$, there are elements
$x_{0},y_{0},z_{0}\in\mathfrak{g}$ such that
$x=\alpha(x_{0}),y=\alpha(y_{0}),z=\alpha(z_{0})$. Assume that
$(\chi^{\prime},\omega^{\prime},\mu^{\prime},\theta^{\prime},D^{{}^{\prime}},\rho^{{}^{\prime}},T^{{}^{\prime}})$
is another non-abelian (2,3)-cocycle corresponding to the non-abelian
extension $\mathcal{E}$. By Lemma 3.7, we know that
$(\chi^{\prime},\omega^{\prime},\mu^{\prime},\theta^{\prime},D^{{}^{\prime}},\rho^{{}^{\prime}},T^{{}^{\prime}})$
and $(\chi,\omega,\mu,\theta,D,\rho,T)$ are equivalent non-abelian
(2,3)-cocycles via a linear map
$\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$ According to Eqs. (3.4) and
(86)-(88), denote $\psi=\beta\varphi\alpha^{-1}$, which follows that
$\displaystyle\omega^{{}^{\prime}}_{(\alpha,\beta)}(x,y,z)-\omega_{(\alpha,\beta)}(x,y,z)$
$\displaystyle=$
$\displaystyle\beta\omega^{{}^{\prime}}(\alpha^{-1}(x),\alpha^{-1}(y),\alpha^{-1}(z))-\beta\omega(\alpha^{-1}(x),\alpha^{-1}(y),\alpha^{-1}(z))$
$\displaystyle=$
$\displaystyle\beta\omega^{{}^{\prime}}(x_{0},y_{0},z_{0})-\beta\omega(x_{0},y_{0},z_{0})$
$\displaystyle=$
$\displaystyle\beta\Big{(}\theta(x_{0},z_{0})\varphi(y_{0})-D(x_{0},y_{0})\varphi(z_{0})+\rho(x_{0})(\varphi(y_{0}),\varphi(z_{0}))-\theta(y_{0},z_{0})\varphi(x_{0})$
$\displaystyle+T(z_{0})(\varphi(x_{0}),\varphi(y_{0}))-\rho(y_{0})(\varphi(x_{0}),\varphi(z_{0}))-\\{\varphi(x_{0}),\varphi(y_{0}),\varphi(z_{0})\\}_{\mathfrak{h}}+\varphi\\{x_{0},y_{0},z_{0}\\}_{\mathfrak{g}}\Big{)}$
$\displaystyle=$
$\displaystyle\beta\Big{(}\theta(\alpha^{-1}(x),\alpha^{-1}(z))\beta^{-1}\psi(y)-D(\alpha^{-1}(x),\alpha^{-1}(y))\beta^{-1}\psi(z)+\rho(\alpha^{-1}(x))(\beta^{-1}\psi(y),\beta^{-1}\psi(z))$
$\displaystyle-\theta(\alpha^{-1}(y),\alpha^{-1}(z))\beta^{-1}\psi(x)+T(\alpha^{-1}(z))(\beta^{-1}\psi(x),\beta^{-1}\psi(y))-\rho(\alpha^{-1}(y))(\beta^{-1}\psi(x),\beta^{-1}\psi(z))\Big{)}$
$\displaystyle-\\{\psi(x),\psi(y),\psi(z)\\}_{\mathfrak{h}}+\psi(\\{x,y,z\\}_{\mathfrak{g}})$
$\displaystyle=$
$\displaystyle\theta_{(\alpha,\beta)}(x,z)\psi(y)-D_{(\alpha,\beta)}(x,y)\psi(z)+\rho_{(\alpha,\beta)}(x)(\psi(y),\psi(z))-\theta_{(\alpha,\beta)}(y,z)\psi(x)$
$\displaystyle+T_{(\alpha,\beta)}(z)(\psi(x),\psi(y))-\rho_{(\alpha,\beta)}(y)(\psi(x),\psi(z))-\\{\psi(x),\psi(y),\psi(z)\\}_{\mathfrak{h}}+\psi(\\{x,y,z\\}_{\mathfrak{g}}).$
By the same token,
$\displaystyle\chi^{{}^{\prime}}_{(\alpha,\beta)}(x,y)-\chi_{(\alpha,\beta)}(x,y)=[\psi(x),\psi(y)]_{\mathfrak{h}}+\psi([x,y]_{\mathfrak{g}})-\mu_{(\alpha,\beta)}(x)\psi(y)+\mu_{(\alpha,\beta)}(y)\psi(x),$
$\displaystyle\theta^{{}^{\prime}}_{(\alpha,\beta)}(x,y)a-\theta_{(\alpha,\beta)}(x,y)a=\rho_{(\alpha,\beta)}(x)(a,\psi(y))-T(y)(a,\psi(x))+\\{a,\psi(x),\psi(y)\\}_{\mathfrak{h}},$
$\displaystyle
D^{{}^{\prime}}_{(\alpha,\beta)}(x,y)a-D_{(\alpha,\beta)}(x,y)a=\rho_{(\alpha,\beta)}(y)(\psi(x),a)-\rho_{(\alpha,\beta)}(x)(\psi(y),a)+\\{\psi(x),\psi(y),a\\}_{\mathfrak{h}},$
$\displaystyle\mu^{{}^{\prime}}_{(\alpha,\beta)}(x)a-\mu_{(\alpha,\beta)}(x)a=[a,\psi(x)]_{\mathfrak{h}},~{}~{}~{}~{}\rho^{{}^{\prime}}_{(\alpha,\beta)}(x)(a,b)-\rho_{(\alpha,\beta)}(x)(a,b)=\\{a,\psi(x),b\\}_{\mathfrak{h}},$
$\displaystyle
T^{{}^{\prime}}_{(\alpha,\beta)}(x)(a,b)-T_{(\alpha,\beta)}(x)(a,b)=\\{b,a,\psi(x)\\}_{\mathfrak{h}}.$
So,
$(\chi^{\prime}_{(\alpha,\beta)},\omega^{\prime}_{(\alpha,\beta)},\mu^{\prime}_{(\alpha,\beta)},\theta^{\prime}_{(\alpha,\beta)},D^{\prime}_{(\alpha,\beta)},\rho^{\prime}_{(\alpha,\beta)},T^{\prime}_{(\alpha,\beta)})$
and
$(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})$
are equivalent non-abelian (2,3)-cocycles via the linear map
$\psi=\beta\varphi\alpha^{-1}$. Combining Lemma 3.7, we know that
$\displaystyle(\chi^{\prime}_{(\alpha,\beta)},\omega^{\prime}_{(\alpha,\beta)},\mu^{\prime}_{(\alpha,\beta)},\theta^{\prime}_{(\alpha,\beta)},D^{\prime}_{(\alpha,\beta)},\rho^{\prime}_{(\alpha,\beta)},T^{\prime}_{(\alpha,\beta)})-(\chi^{\prime},\omega^{\prime},\mu^{\prime},\theta^{\prime},D^{{}^{\prime}},\rho^{{}^{\prime}},T^{{}^{\prime}})$
and
$\displaystyle(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)},\mu_{(\alpha,\beta)},\theta_{(\alpha,\beta)},D_{(\alpha,\beta)},\rho_{(\alpha,\beta)},T_{(\alpha,\beta)})-(\chi,\omega,\mu,\theta,D,\rho,T)$
are equivalent via the linear map $\beta\varphi\alpha^{-1}-\varphi$. ∎
###### Proposition 6.2.
Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian
extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$.
Define a map
(92)
$K:\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})\longrightarrow\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{g}),~{}~{}K(\gamma)=(p\gamma
s,\gamma|_{\mathfrak{h}}),~{}\forall~{}\gamma\in\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}}).$
Then $K$ is a homomorphism of groups.
###### Proof.
One can take the same procedure of Lie algebras, see [1]. ∎
###### Theorem 6.3.
Assume that $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ is a non-abelian
extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of
$\hat{\mathfrak{g}}$. Then there is an exact sequence:
$1\longrightarrow\mathrm{Aut}_{\mathfrak{h}}^{\mathfrak{g}}(\hat{\mathfrak{g}})\stackrel{{\scriptstyle
H}}{{\longrightarrow}}\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})\stackrel{{\scriptstyle
K}}{{\longrightarrow}}\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})\stackrel{{\scriptstyle
W}}{{\longrightarrow}}H^{(2,3)}_{nab}(\mathfrak{g},\mathfrak{h}),$
where
$\mathrm{Aut}_{\mathfrak{h}}^{\mathfrak{g}}(\hat{\mathfrak{g}})=\\{\gamma\in\mathrm{Aut}(\hat{\mathfrak{g}})|K(\gamma)=(I_{\mathfrak{g}},I_{\mathfrak{h}})\\}$.
###### Proof.
Obviously, $\mathrm{Ker}K=\mathrm{Im}H$ and $H$ is injective. We only need to
prove that $\mathrm{Ker}W=\mathrm{Im}K$. By Theorem 5.4, for all
$(\alpha,\beta)\in\mathrm{Ker}W$, we get that $(\alpha,\beta)$ is extensible
with respect to the non-abelian extension $\mathcal{E}$, that is, there is a
$\gamma\in\mathrm{Aut}_{\mathfrak{h}}^{\mathfrak{g}}(\hat{\mathfrak{g}})$,
such that $i\beta=\gamma i,~{}p\gamma=\alpha p$, which follows that
$\alpha=\alpha ps=p\gamma s,~{}\beta=\gamma|_{\mathfrak{h}}.$ Thus,
$(\alpha,\beta)\in\mathrm{Im}K$. On the other hand, for any
$(\alpha,\beta)\in\mathrm{Im}K$, there is an isomorphism
$\gamma\in\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})$, such that Eq. (92)
holds. Combining Eq. (90) and $\mathrm{Im}i=\mathrm{Ker}p$, we obtain $\alpha
p=p\gamma sp=p\gamma(I_{\hat{\mathfrak{g}}}-it)=p\gamma$ and $i\beta=\gamma
i$. Hence, $(\alpha,\beta)$ is extensible with respect to the non-abelian
extension $\mathcal{E}$. According to Theorem 5.4,
$(\alpha,\beta)\in\mathrm{Ker}W$. In all, $\mathrm{Ker}W=\mathrm{Im}K$. ∎
Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian
extension of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$.
Suppose that $(\chi,\omega,\mu,\theta,D,\rho,T)$ is a non-abelian
(2,3)-cocycle induced by the section $s$.
Denote
(93) $\displaystyle Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})=$
$\displaystyle\left\\{\varphi:\mathfrak{g}\rightarrow\mathfrak{h}\left|\begin{aligned}
&[a,\varphi(x)]_{\mathfrak{h}}=\\{a,b,\varphi(x)\\}_{\mathfrak{h}}=\\{\varphi(x),a,b\\}_{\mathfrak{h}}=0,\\\
&\\{\varphi(x),\varphi(y),a\\}_{\mathfrak{h}}-\rho(x)(\varphi(y),a)+\rho(y)(\varphi(x),a)=0,\\\
&\\{a,\varphi(x),\varphi(y)\\}_{\mathfrak{h}}-T(y)(a,\varphi(x))+\rho(x)(a,\varphi(y))=0,\\\
&\mu(x)\varphi(y)-\mu(y)\varphi(x)=\varphi([x,y]_{\mathfrak{g}})+[\varphi(x),\varphi(y)]_{\mathfrak{h}},~{}\\\
&T(z)(\varphi(x),\varphi(y))-\rho(y)(\varphi(x),\varphi(z))-\theta(y,z)\varphi(x)\\\
&+\rho(x)(\varphi(y),\varphi(z))+\theta(x,z)\varphi(y)-D(x,y)\varphi(z)\\\
&=\\{\varphi(x),\varphi(y),\varphi(z)\\}_{\mathfrak{h}}-\varphi(\\{x,y,z\\}_{\mathfrak{h}}),~{}\forall~{}x,y,z\in{\mathfrak{g}},a,b\in{\mathfrak{h}}\end{aligned}\right.\right\\}.$
It is easy to check that $Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$ is an
abelian group, which is called a non-abelian 1-cocycle on $\mathfrak{g}$ with
values in $\mathfrak{h}$.
###### Proposition 6.4.
With the above notations, we have
1. $(i)$
The linear map $S:\mathrm{Ker}K\longrightarrow
Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$ defined by
(94) $S(\gamma)(x)=\varphi_{\gamma}(x)=s(x)-\gamma
s(x),~{}\forall~{}~{}\gamma\in\mathrm{Ker}K,~{}x\in\mathfrak{g}$
is a homomorphism of groups.
2. $(ii)$
$S$ is an isomorphism, that is, $\mathrm{KerK}\simeq
Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$.
###### Proof.
1. $(i)$
By Eqs. (62)-(64), (92) and (94), for all $x,y,z\in\mathfrak{g}$, we have,
$\displaystyle\\{\varphi_{\gamma}(x),\varphi_{\gamma}(y),\varphi_{\gamma}(z)\\}_{\mathfrak{h}}-T(z)(\varphi_{\gamma}(x),\varphi_{\gamma}(y))+\rho(y)(\varphi_{\gamma}(x),\varphi_{\gamma}(z))+\theta(y,z)\varphi_{\gamma}(x)$
$\displaystyle-\rho(x)(\varphi_{\gamma}(y),\varphi_{\gamma}(z))-\theta(x,z)\varphi_{\gamma}(y)+D(x,y)\varphi_{\gamma}(z)-\varphi_{\gamma}(\\{x,y,z\\}_{\mathfrak{g}})$
$\displaystyle=$ $\displaystyle\\{s(x)-\gamma s(x),s(y)-\gamma
s(y),s(z)-\gamma s(z)\\}_{\hat{\mathfrak{g}}}-\\{s(x)-\gamma s(x),s(y)-\gamma
s(y),s(z)\\}_{\hat{\mathfrak{g}}}$ $\displaystyle+\\{s(y),s(x)-\gamma
s(x),s(z)-\gamma s(z)\\}_{\hat{\mathfrak{g}}}+\\{s(x)-\gamma
s(x),s(y),s(z)\\}_{\hat{\mathfrak{g}}}-\\{s(x),s(y)-\gamma s(y),s(z)-\gamma
s(z)\\}_{\hat{\mathfrak{g}}}$ $\displaystyle-\\{s(y)-\gamma
s(y),s(x),s(z)\\}_{\hat{\mathfrak{g}}}+\\{s(x),s(y),s(z)-\gamma
s(z)\\}_{\hat{\mathfrak{g}}}+\gamma
s(\\{x,y,z\\}_{\mathfrak{g}})-s(\\{x,y,z\\}_{\mathfrak{g}})$ $\displaystyle=$
$\displaystyle\gamma s(\\{x,y,z\\}_{\mathfrak{g}})-\\{\gamma s(x),\gamma
s(y),\gamma
s(z)\\}_{\hat{\mathfrak{g}}}+\\{s(x),s(y),s(z)\\}_{\hat{\mathfrak{g}}}-s(\\{x,y,z\\}_{\mathfrak{g}})$
$\displaystyle=$ $\displaystyle\omega(x,y,z)-\gamma\omega(x,y,z)$
$\displaystyle=$ $\displaystyle 0.$
Analogously, we can check that $\varphi_{\gamma}$ satisfies the other
identities in $Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$. Thus, $S$ is well-
defined. For any $\gamma_{1},\gamma_{2}\in\mathrm{Ker}K$ and
$x\in\mathfrak{g}$, suppose $S(\gamma_{1})=\varphi_{\gamma_{1}}$ and
$S(\gamma_{2})=\varphi_{\gamma_{2}}$. By Eqs. (92) and (94), we have
$\displaystyle S(\gamma_{1}\gamma_{2})(x)$
$\displaystyle=s(x)-\gamma_{1}\gamma_{2}s(x)$
$\displaystyle=s(x)-\gamma_{1}(s(x)-\varphi_{\gamma_{2}}(x))$
$\displaystyle=s(x)-\gamma_{1}s(x)+\gamma_{1}\varphi_{\gamma_{2}}(x)$
$\displaystyle=\varphi_{\gamma_{1}}(x)+\varphi_{\gamma_{2}}(x)$
which means that $S(\gamma_{1}\gamma_{2})=S(\gamma_{1})+S(\gamma_{2})$ is a
homomorphism of groups.
2. $(ii)$
For all $\gamma\in\mathrm{Ker}K$, we obtain that $K(\gamma)=(p\gamma
s,\gamma|_{\mathfrak{h}})=(I_{\mathfrak{g}},I_{\mathfrak{h}})$. If
$S(\gamma)=\varphi_{\gamma}=0$, we can get $\varphi_{\gamma}(x)=s(x)-\gamma
s(x)=0$, that is, $\gamma=I_{\hat{\mathfrak{g}}}$, which indicates that $S$ is
injective. Secondly, we prove that $S$ is surjective. Since $s$ is a section
of $p$, all $\hat{x}\in\hat{\mathfrak{g}}$ can be written as $a+s(x)$ for some
$a\in\mathfrak{h},x\in\mathfrak{g}$. For any $\varphi\in
Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$, define a linear map
$\gamma:\hat{\mathfrak{g}}\rightarrow\hat{\mathfrak{g}}$ by
(95)
$\gamma(\hat{x})=\gamma(a+s(x))=s(x)-\varphi(x)+a,~{}\forall~{}\hat{x}\in\hat{\mathfrak{g}}.$
It is obviously that $(p\gamma
s,\gamma|_{\mathfrak{h}})=(I_{\mathfrak{g}},I_{\mathfrak{h}})$. We need to
verify that $\gamma$ is an automorphism of Lie-Yamaguti algebra
$\hat{\mathfrak{g}}$. One can take the same procedure as the proof of the
converse part of Theorem 5.2. It follows that $\gamma\in\mathrm{Ker}K$. Thus,
$S$ is surjective. In all, $S$ is bijective. So $\mathrm{Ker}K\simeq
Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})$.
∎
Combining Theorem 6.3 and Proposition 6.4, we have
###### Theorem 6.5.
Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be a non-abelian
extension of $\mathfrak{g}$ by $\mathfrak{h}$. There is an exact sequence:
$0\longrightarrow
Z_{nab}^{1}(\mathfrak{g},\mathfrak{h})\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})\stackrel{{\scriptstyle
K}}{{\longrightarrow}}\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})\stackrel{{\scriptstyle
W}}{{\longrightarrow}}H^{(2,3)}_{nab}(\mathfrak{g},\mathfrak{h}).$
## 7\. Particular case: abelian extensions of Lie-Yamaguti algebras
In this section, we discuss the results of previous section in particular
case.
We fix the abelian extension
$\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ of $\mathfrak{g}$ by
$\mathfrak{h}$ with a section $s$ of $p$. Assume that $(\chi,\omega)$ is a
(2,3)-cocycle corresponding to $\mathcal{E}$. In the case of abelian
extensions $\mathcal{E}$, the maps $\rho,T$ defined by (64) become to zero.
Then the quadruple $(\mathfrak{h},\mu,\theta,D)$ given by Eq. (63) is a
representation of $\mathfrak{g}$ [35]. Moreover,
###### Theorem 7.1 ([35]).
1. $(i)$
The triple $(\mathfrak{g}\oplus\mathfrak{h},[\ ,\ ]_{\chi},\\{\ ,\ ,\
\\}_{\omega})$ is a Lie-Yamaguti algebra if and only if $(\chi,\omega)$ is a
(2,3)-cocycle of $\mathfrak{g}$ with coefficients in the representation
$(\mathfrak{h},\mu,\theta,D)$.
2. $(ii)$
Abelian extensions of $\mathfrak{g}$ by $\mathfrak{h}$ are classified by the
cohomology group $H^{(2,3)}(\mathfrak{g},\mathfrak{h})$ of $\mathfrak{g}$ with
coefficients in $(\mathfrak{h},\mu,\theta,D)$.
###### Theorem 7.2.
Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be an abelian extension
of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$. Assume that
$(\chi,\omega)$ is a (2,3)-cocycle and $(\mathfrak{h},\mu,\theta,D)$ is a
representation of $\mathfrak{g}$ associated to $\mathcal{E}$. A pair
$(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$
is extensible with respect to the abelian extension $\mathcal{E}$ if and only
if there is a linear map $\varphi:\mathfrak{g}\longrightarrow\mathfrak{h}$
satisfying the following conditions:
$\displaystyle\beta\omega(x,y,z)-\omega(\alpha(x),\alpha(y),\alpha(z))=$
$\displaystyle\theta(\alpha(x),\alpha(z))\varphi(y)-\theta(\alpha(y),\alpha(z))\varphi(x)$
(96)
$\displaystyle-D(\alpha(x),\alpha(y))\varphi(z)+\varphi(\\{x,y,z\\}_{\mathfrak{h}}),$
(97)
$\beta\chi(x,y)-\chi(\alpha(x),\alpha(y))=\mu(\alpha(y))\varphi(x)-\mu(\alpha(x))\varphi(y)+\varphi([x,y]_{\mathfrak{g}}),$
(98)
$\beta(\theta(x,y)a)=\theta(\alpha(x),\alpha(y))\beta(a),~{}~{}\beta\mu(x)a=\mu(\alpha(x))\beta(a).$
###### Proof.
It can be obtained directly from Theorem 5.2. ∎
By Eqs. (10) and (98), we obtain
$\beta D(x,y)a=D(\alpha(x),\alpha(y))\beta(a).$
For all
$(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})$,
$(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)})$ may not be a (2,3)-cocycle.
Indeed, $(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)})$ is a (2,3)-cocycle
if Eq. (98) holds. Thus, it is natural to introduce the space of compatible
pairs of automorphisms:
$\displaystyle C_{(\mu,\theta)}=$
$\displaystyle\left\\{(\alpha,\beta)\in\mathrm{Aut}(\mathfrak{g})\times\mathrm{Aut}(\mathfrak{h})\left|\begin{aligned}
&\beta(\theta(x,y)a)=\theta(\alpha(x),\alpha(y))\beta(a),\\\
&\beta\mu(x)a=\mu(\alpha(x))\beta(a),~{}\forall~{}x,y\in{\mathfrak{g}},a\in{\mathfrak{h}}\end{aligned}\right.\right\\}.$
More detail on the space of compatible pairs of automorphisms can be found in
[13].
Analogous to Theorem 5.4, we get
###### Theorem 7.3 ([13]).
Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be an abelian extension
of $\mathfrak{g}$ by $\mathfrak{h}$ with a section $s$ of $p$ and
$(\chi,\omega)$ be a (2,3)-cocycle associated to $\mathcal{E}$. A pair
$(\alpha,\beta)\in C_{(\mu,\theta)}$ is extensible with respect to the abelian
extension $\mathcal{E}$ if and only if $(\chi,\omega)$ and
$(\chi_{(\alpha,\beta)},\omega_{(\alpha,\beta)})$ are in the same
cohomological class.
In the case of abelian extensions, $Z^{1}_{nab}(\mathfrak{g},\mathfrak{h})$
defined by (93) becomes to ${H}^{1}(\mathfrak{g},\mathfrak{h})$ given in
Section 2. In the light of Theorem 6.5 and Theorem 7.3, we have the following
exact sequence:
###### Theorem 7.4 ([13]).
Let $\mathcal{E}:0\longrightarrow\mathfrak{h}\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\hat{\mathfrak{g}}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\mathfrak{g}\longrightarrow 0$ be an abelian extension
of $\mathfrak{g}$ by $\mathfrak{h}$. There is an exact sequence:
$0\longrightarrow H^{1}(\mathfrak{g},\mathfrak{h})\stackrel{{\scriptstyle
i}}{{\longrightarrow}}\mathrm{Aut}_{\mathfrak{h}}(\hat{\mathfrak{g}})\stackrel{{\scriptstyle
K}}{{\longrightarrow}}C_{(\mu,\theta)}\stackrel{{\scriptstyle
W}}{{\longrightarrow}}H^{(2,3)}(\mathfrak{g},\mathfrak{h}).$
Acknowledgments
This work was supported by the National Natural Science Foundation of China
(11871421); Natural Science Foundation of Zhejiang Province of China
(LY19A010001); and Science and Technology Planning Project of Zhejiang
Province (2022C01118).
Statements and Declarations
All datasets underlying the conclusions of the paper are available to readers.
No conflict of interest exits in the submission of this manuscript.
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|
# Synthetic photometry of OB star clusters with stochastically sampled IMFs:
analysis of models and HST observations.
Rogelio Orozco-Duarte1, Aida Wofford1, Alba Vidal-García2,3, Gustavo Bruzual4,
Stephane Charlot4, Mark R. Krumholz5,6, Stephen Hannon7,8, Janice Lee9,7,
Timothy Wofford10, Michele Fumagalli11, Daniel Dale12, Matteo Messa13,14, Eva
K. Grebel15, Linda Smith16, Kathryn Grasha5, David Cook7
1Universidad Nacional Autónoma de México, Instituto de Astronomía, AP 106,
Ensenada 22800, BC, México 2Sorbonne Université, UPMC-CNRS, UMR7095, Institut
d’Astrophysique de Paris, F-75014 Paris, France 3LPENS, Ecole Normale
Supérieure, Université PSL, CNRS, Sorbonne Université, Université Paris-
Diderot, Paris, France 4Instituto de Radioastronomía y Astrofísica, UNAM,
Campus Morelia, Michoacán, Mé́xico, C.P. 58089, Mé́xico 5Research School of
Astronomy and Astrophysics, Australian National University, Canberra, ACT
2611, Australia 6ARC Centre of Excellence for Astronomy in Three Dimensionns
(ASTRO-3D), Canberra, ACT 2611, Australia 7Caltech-IPAC, 1200 E. California
Blvd. Pasadena, CA 91125, USA 8Department of Physics $\And$ Astronomy,
University of California, Riverside, CA, USA 9Gemini Observatory/NSF’s
NOIRLab, 950 N. Cherry Avenue, Tucson, AZ, 85719, USA 10Facultad de Ciencias,
Universidad Autónoma de Baja California, AP 1880, Ensenada 22800, BC, México
11Dipartimento di Fisica G. Occhialini, Università degli Studi di Milano
Bicocca, Piazza della Scienza 3, 20126 Milano, Italy 12Department of Physics &
Astronomy, University of Wyoming, Laramie WY 13Observatoire de Genève,
Université de Genève, Chemin Pegasi 51, Versoix CH-1290, Switzerland
14Department of Astronomy, Oscar Klein Centre, Stockholm University, AlbaNova,
Stockholm SE-106 91, Sweden 15Astronomisches Rechen-Institut, Zentrum für
Astronomie der Universität Heidelberg, Mönchhofstraße 12-14, 69120 Heidelberg,
Germany 15Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia 16European Space Agency (ESA), ESA
Office, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore,
MD 21218, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We present a pilot library of synthetic NUV, U, B, V, and I photometry of star
clusters with stochastically sampled IMFs and ionized gas for initial masses,
$M_{i}=10^{3}$, $10^{4}$, and $10^{5}$ $M_{\odot}$; $t=1$, 3, 4, and 8 Myr;
$Z=0.014$ and $Z=0.002$; and log(US) =-2 and -3. We compare the library with
predictions from deterministic models and observations of isolated low-mass
($<10^{4}$ $M_{\odot}$) star clusters with co-spatial compact H ii regions.
The clusters are located in NGC 7793, one of the nearest galaxies observed as
part of the HST LEGUS and H$\alpha$-LEGUS surveys. 1) For model magnitudes
that only account for the stars: a) the residual |deterministic mag - median
stochastic mag| can be $\geq 0.5$ mag, even for $M_{i}=10^{5}$ $M_{\odot}$;
and b) the largest spread in stochastic magnitudes occurs when Wolf-Rayet
stars are present. 2) For $M_{i}=10^{5}$ $M_{\odot}$: a) the median stochastic
mag with gas can be $>$1.0 mag more luminous than the median stochastic
magnitude without gas; and b) nebular emission lines can contribute with
$>50\%$ and $>30\%$ to the total emission in the V and I bands, respectively.
3) Age-dating OB-star clusters via deterministic tracks in the U-B vs. V-I
plane is highly uncertain at $Z=0.014$ for $M_{i}\sim 10^{3}$ $M_{\odot}$ and
$Z=0.002$ for $M_{i}\sim 10^{3}-10^{5}$ $M_{\odot}$. 4) For low-mass clusters,
the V-band extinction derived with stochastic models significantly depends on
the value of log(US). 5) The youngest clusters tend to have higher extinction.
6) The majority of clusters have multi-peaked age PDFs. 7) Finally, we discuss
the importance of characterising the true variance in the number of stars per
mass bin in nature.
###### keywords:
stars: luminosity function, mass function – galaxies: stellar content –
galaxies: ISM – (ISM:) HII regions – methods: statistical
††pubyear: 2015††pagerange: Synthetic photometry of OB star clusters with
stochastically sampled IMFs: analysis of models and HST
observations.–Synthetic photometry of OB star clusters with stochastically
sampled IMFs: analysis of models and HST observations.
## 1 Introduction
Star clusters and cluster mass function. Star clusters are groupings of stars
that are born from the same molecular cloud and are gravitationally bound.
They can contain anywhere between millions of stars to less than a few hundred
members. Observations of star clusters with $<10^{4}\,$$M_{\odot}$ exist for
the Milky Way, M31, NGC 4214, and other comparatively-nearby galaxies. In
particular, the PHAT survey (Panchromatic Hubble Andromeda Treasury, PI
Dalcanton) covered approximately 1/3 of M31’s star forming disk from the near
ultraviolet (NUV) to the near infrared (NIR) at the high spatial resolution of
the Hubble Space Telescope (HST), and provided a robust distance measurement
and high quality data for M31. Studies of the above three galaxies show that
the cluster mass function (CMF) in the range $<10^{4}\,$$M_{\odot}$ appears to
be similar to the distribution at higher masses, down to the sensitivity
limit, i.e., it is consistent with $dN/dM\sim M^{-2}$, where $N$ is the number
of clusters and $M$ is the mass of the cluster (Johnson et al., 2017; Krumholz
et al., 2019). The shape of the CMF has been confirmed by the studies of Adamo
et al. (2017), Messa et al. (2018), and Cook et al. (2019) that target
galaxies NGC 628, M51 and the dwarf galaxies from HST’s Legacy Extra Galactic
Ultraviolet Survey (LEGUS, Calzetti et al. 2015), respectively. In their study
of 25 LEGUS galaxies Hannon et al. 2019 (hereafter H19) do not measure the CMF
but find more clusters in the interval $10^{3}-10^{4}\,$$M_{\odot}$ than with
$>10^{4}\,$$M_{\odot}$.
Importance of low-mass star clusters. Observations of the Small Magellanic
Cloud analysed by Lamb et al. (2010) seem to suggest that a distribution of
the type $dN/dM\sim M^{-2}$ describes systems with masses as low as
$10\,$$M_{\odot}$. This means that the mass range between 10 and 1000
$M_{\odot}$ contains the same total mass in stars as the range between 1000
and $10^{5}\,$$M_{\odot}$. Thus, one cannot obtain a complete understanding of
stellar populations without studying low-mass ($<10^{4}\,$$M_{\odot}$) star
clusters. In addition, studying a statistically significant number of low-mass
clusters in a diversity of environments is necessary in order to understand
how cluster properties depend on their environment.
The IMF of star clusters. The stellar initial mass function (IMF) describes
the stellar mass distribution of the cluster at birth, when the cluster is
still embedded. It is a main ingredient of population synthesis models (e.g.
Bruzual & Charlot 2003; Leitherer et al. 1999, 2014; Eldridge et al. 2017),
which aim to predict the radiative, mechanical, and chemical feedback of
stellar populations; and are used as input to cosmological simulations (e.g.,
Hirschmann et al. 2019). The IMF has a universal form for conditions as they
are found at the present time throughout galaxies of the Local Group
(Salpeter, 1955; Chabrier, 2003; Kroupa, 2012); and it is stochastically
sampled (Cerviño & Luridiana, 2003; Fouesneau & Lançon, 2010; Fumagalli et
al., 2011; Krumholz et al., 2015a). For star clusters with total initial
masses ($M_{i}$) of 103, 104, 105, and 10${}^{6}\,$$M_{\odot}$, and a
stochastically-sampled IMF, Figure 1 shows the mean and variation in the
number of stars in each mass bin. For generating the figure, we assume a
Chabrier (2003) IMF shape and stop adding stars when the next star makes the
total mass be $>M_{i}$. Since most stars have a low mass, the final total mass
is rarely much different from $M_{i}$. The stars are collected in 49 bins with
boundaries evenly spaced on a logarithmic scale from 0.1 to 100 $M_{\odot}$.
The figure shows the following. 1) The standard deviation around the mean
increases as the mass of the star increases, i.e., as the probability of a
star belonging to the bin decreases. 2) In addition, as the value of $M_{i}$
decreases the standard deviation decreases while the relative standard
deviation increases. These two results (1 and 2) are because the number of
stars in each bin follows an approximately binomial distribution111For a fixed
number of stars the collection of bin counts would follow a multinomial
distribution. Fixing the total mass removes some of the independence, so a
multinomial distribution is only an approximation. with a probability given by
the IMF. For example, if the IMF says that the probability in bin "j" is
$P_{j}$ and we generate one realization of the IMF with N stars, then we
expect to find $n_{j}=NP_{j}$ stars in that bin with a standard deviation of
$\sigma_{j}=\sqrt{NP_{j}(1-P_{j})}$ stars. The relative standard deviation
$n_{j}/\sigma_{j}$ decreases as $N$ increases, i.e., as $M_{i}$ increases.
Figure 1: Mean number of stars in each mass bin resulting from stochastically
sampling the Chabrier (2003) IMF (his equation 1) 1000 times (filled circles),
and standard deviation around the mean (error bars). We show this for clusters
with initial masses of 103, 104, 105, and 106 M⊙, as indicated by the legend.
The horizontal dashed line corresponds to a number of stars equal to 1.
The need for improved models. Beyond the Local Group of galaxies, star
clusters can only be resolved into individual stars in comparatively nearby
galaxies, and one requires model spectro-photometry generated with population
synthesis codes to infer the extinction, mass, and age of a star cluster from
its integrated light. Although in general, population synthesis codes do not
account for the stochastic sampling of the IMF, SLUG (da Silva et al., 2012)
does. The latter code has been used to study broad-band HST NUV to NIR
observations of large samples of star clusters. Krumholz et al. (2015a) find
that the stochastic SLUG models are generally a better fit to such
observations compared to the deterministic Yggdrasil models of Zackrisson et
al. (2011), but that the overall properties of the star clusters recovered by
both codes are qualitatively similar. Ashworth et al. (2017) find that
including H$\alpha$ photometry in the SLUG stochastic models significantly
improves the age determination of young clusters. However, in the latter two
works, the stochastic models do not include the effect of the stochastic
variation in the shape of the ionizing continuum, on the nebular emission. As
we show in this work, properly accounting for the emission of the ionized gas
is important when fitting models to observations that include the light from
OB-star clusters and nebular emission, as is the case for compact H ii
regions, where the ionized gas is co-spatial with the stars.
This work. In this work, we present a pilot library of synthetic photometry of
young ($1-8$ Myr) star clusters that accounts for the stochastic sampling of
the IMF and where the contribution of the ionized gas is fully modelled. The
library includes the HST F275W (UV), F336W (U), F438W (B), F555W (V), and
F814W (I) broad band filters and is based on the model GALAXEV-C spectra that
are presented in Vidal-García et al. (in prep.). Our specific objectives are:
i) for clusters with different initial masses, ages, and metallicities, and
for different values of the ionization parameter, to determine the spread in
predicted magnitudes due to a) the stochastic sampling of the IMF and b) the
inclusion of the ionized gas; ii) to quantify the relative contributions of
the stellar continuum, nebular continuum, and emission lines to the total
emission in the photometric bands; iii) to compare the location of the
stochastic models relative to the deterministic predictions in the U - B
versus V - I colour - colour magnitude diagram (this diagram is used for age-
dating clusters); and iv) for a sample of observed star clusters with a) low
masses according to deterministic models and b) compact H ii regions, to
obtain the probability distribution functions (PDFs) of the extinction, mass,
and age corresponding to the independent stochastic GALAXEV-C and SLUG. We use
cluster observations from the HST LEGUS (Calzetti et al., 2015) and
H$\alpha$-LEGUS (PID 13773, PI Chandar) surveys. The clusters are located in
NGC 7793, which is one of the nearest galaxies in these surveys.
In Section 2, we present our pilot library; in Section 3, we analyse the
predictions of our pilot-library; in Section 4, we present the observations;
in Section 5, we describe BAYESPHOT (Krumholz et al., 2015b), which is the
photometric interpretation tool that we use in this paper, and we discuss the
derived cluster properties; in Appendix A, we discuss what happens if we
change the number of realizations of the IMF; in Appendices B and C we present
complementary colour-colour magnitude diagrams and PDF plots; finally, in
Section 7, we summarise and conclude.
## 2 Models
In this Section, we present a pilot library of synthetic HST-equivalent UV, U,
B, V and I photometry, which accounts for the stochastic sampling of the IMF
and the contribution of the ionized gas. The specific bands for which
photometry was computed are: WFC3/UVIS F275W, F336W, and F438W; and ACS/WFC
F555W and F814W, where WFC3 is the Wide Field Camera 3, ACS is the Advanced
Camera for Surveys, and UVIS and WFC (Wide Field Channel) are the channels of
the instruments. This is the filter set that was used for observing the west
field of galaxy NGC 7793, whose star clusters are analysed in Section 5. Our
pilot library covers sufficient parameter space to i) determine if there is a
significant difference between deterministic models and the median of the
stochastic models; ii) quantify the impact of the nebular emission in the
filters; and iii) show the limitations of colour-colour diagrams for age-
dating low-mass star clusters.
### 2.1 Stars
The star clusters are assumed to be simple stellar populations (SSPs), i.e.,
populations where all stars are born simultaneously from the same molecular
cloud. We compute models for four ages (1, 3, 4, and 8 Myr); three initial
cluster masses ($10^{3}$, $10^{4}$, and $10^{5}$ $M_{\odot}$); and two
metallicities ($Z=0.014$ and $Z=0.002$). The highest metallicity, $Z=0.014$,
is the solar reference value of Asplund et al. (2009) and the closest
metallicity of the star clusters (see Pilyugin et al. 2014 and Section 4).
However, we also compute models at $Z=0.002$ for comparison with the higher
metallicity the models. The IMF of the star clusters is stochastically sampled
and assumes a Chabrier (2003) form and a mass range for the individual stars
of 0.1 to 100 $M_{\odot}$. When populating the IMF, we keep adding stars until
we reach at least 0.05 $M_{\odot}$ above the required mass of the cluster.
Since most of the stars are low mass, typically, our cluster masses are within
0.1 $M_{\odot}$ of the target mass. For each combination of the above
parameters, we generated 220 realizations. This number is set by GALAXEV (Plat
et al., 2019), which is the code that we use to model the stellar population
spectra. In its deterministic version, 220 corresponds to the maximum number
of time steps and spectra that are generated in one run. In its stochastic
version, the time steps are replaced by IMF realizations. In Appendix A, we
show that using a larger library of 1000 realisations does not significantly
alter the mean to standard deviation or the main conclusions of this work. We
compute deterministic models and corresponding stochastic models. In the
deterministic models, the shape of the spectrum remains unchanged and the
luminosity is simply scaled in proportion to the value of $M_{i}$, which
effectively assumes that the IMF is fully sampled. The models were computed
with the latest version of the population synthesis code GALAXEV (Plat et al.
2019; Charlot & Bruzual, in prep.), which only includes the contribution from
the stars. The models assume that the stars evolve as single stars with zero
rotation. These models are useful for comparing with published results based
on the same assumptions and more comprehensive models that account for the
evolution of massive stars in close binary systems. In the latter systems the
stars exchange mass and rotate.
### 2.2 Ionized gas
In order to compute the contribution of the gas ionized by the massive stars,
we use the above stellar models as input to photoionization models generated
with CLOUDY (Ferland et al., 2017) and the approach presented in Vidal-García
et al. (2017), i.e., we use: spherical geometry, a covering factor of 1, and a
filling factor of 1. The H ii regions are ionisation-bounded. The code used
for this purpose is GALAXEV-C (C for Cloudy, Vidal-Garcia et al., in prep.).
For the pilot library, we compute models for the following parameters:
hydrogen number density, n(H)=100 cm-3; metallicities, $Z=0.002$ and
$Z=0.014$; ionisation parameters at the Strömgren radius (as defined in Gutkin
et al. 2016), log(US) = -2 and -3; and C/O = (C/O)⊙.
### 2.3 Extinction due to dust
In order to account for the effect of dust mixed with the ionized gas, we
include dust grains in CLOUDY and for consistency deplete the refractory
elements in the ionized gas by using a dust-to-metal gas ratio of
$\xi_{d}$=0.3. In order to account for dust in the intervening neutral medium
when deriving the cluster properties, we apply an extinction law to the CLOUDY
output. In the process of finding the extinction in the V-band, $A_{\rm V}$,
of observed star clusters, we try $A_{\rm V}$ values in the range $0-3$ mag,
in steps of 0.01 mag.
## 3 Analysis of Models
For all combinations of photometric band (NUV, U, B, V, and I ), initial mass
($10^{3}$, $10^{4}$, and $10^{5}\,$$M_{\odot}$), metallicity (Z=0.002 and
0.014), and log(US) value (-2 and -3), Figure 2 shows the magnitudes predicted
by the deterministic (small squares) and stochastic (violin plots)
predictions. The photomtric bands are those used to observe field NGC 7793-W
(see Section 4). The left side of the violin corresponds to stars only
(GALAXEV output), while the right side corresponds to stars + ionized gas +
dust mixed with the ionized gas (GALAXEV-C output).
### 3.1 Models with just stars
In this subsection, we analyse the behaviour of the models that account for
the stochastic sampling of the IMF and the contribution of the stars alone
(left violin halves of Figure 2).
Figure 2: How accounting for the stochastic sampling of the IMF and including
the contributions of the ionized gas + dust mixed with the ionized gas affects
the magnitude predictions. Each violin diagram includes 220 realizations of
the IMF. The left half of the violin represents the models with just stars and
the right half the models that account for stars + ionized gas + dust. We show
models for: five LEGUS bands (given by the column titles); initial cluster
masses of $M_{i}=10^{3}$, 104, and 105 $M_{\odot}$ (bottom-blue, middle-red,
and top-magenta violins, respectively, as given by the legend); ages of $t=1$,
3, 4, and 8 Myr (given by the x-axis label); and all combinations of
metallicity (Z=0.002 or 0.014) and ionization parameter (log(US)=-2 or -3, as
given by the row titles). The shape of the half violin indicates the frequency
of models at a given magnitude. The solid horizontal lines within the half
violins give the median of the stochastic models while the 25th and 75th
quartiles are given by the dashed lines. The squares on each side of the
violin give the deterministic magnitude without gas (square on the left) and
with gas (square on the right) at each value of $M_{i}$. We number the panels
from 1 to 20 for facilitating the discussion in Table 4.
Mass effect. As the initial cluster mass, $M_{i}$, increases from bottom to
top in each panel, the spread in magnitudes decreases. This behaviour is
observed in all LEGUS bands and at all ages and metallicities. This is because
massive stars are significantly more luminous than lower mass stars. Thus, the
presence or absence of massive stars in the stellar population severely
impacts its integrated luminosity.
Age effect. In general, the spread in magnitudes is larger at 3 and 4 Myr than
at 1 and 8 Myr, specially for $M_{i}=10^{3}$ $M_{\odot}$. This is due to the
presence of classical Wolf-Rayet stars at 3 and 4 Myr. These stars, which are
in a phase of helium burning and have lost their hydrogen envelope, are the
evolved descendants of the most massive stars ($\geq 25\,M_{\odot}$, depending
on metallicity, Massey et al. 2000). Their presence or absence in the
population significant impacts the integrated luminosity.
Metallicity effect. Stars of different chemical compositions evolve on
different time-scales. For $M_{i}=10^{5}$ $M_{\odot}$, and each broad band and
age, Table 1 gives the difference between the median magnitude of the
stochastic models at $Z=0.014$ and $Z=0.002$. At $t=1$ Myr, the $Z=0.014$
models are more luminous in all bands, whereas at $t=8$ Myr, the $Z=0.002$
models are more luminous in all bands except F814W.
Age | F275W | F336W | F438W | F555W | F814W
---|---|---|---|---|---
(Myr) | M(Z=0.002) - M(Z=0.014)
1 | 0.38 | 0.40 | 0.40 | 0.40 | 0.39
3 | 0.42 | 0.37 | 0.11 | -0.01 | -0.30
4 | 0.28 | 0.00 | -0.77 | -1.19 | -1.80
8 | -0.15 | -0.23 | -0.28 | -0.02 | 1.29
Table 1: For the GALAXEV models with $M_{i}=10^{5}$ M⊙, difference in median
stochastic absolute magnitude, M(Z=0.002) - M(Z=0.014).
### 3.2 Models with stars, gas, and dust
In this subsection, we analyse the behaviour of models that account for the
stochastic sampling of the IMF and the added contributions of the ionized gas
and dust mixed with the ionized gas (right violin halves of Figure 2).
Gas effect. Adding the ionized gas significantly increases the luminosity for
certain combinations of age, Z, and log(US). This can be seen in Table 2,
which for models with $M_{i}=10^{5}$ $M_{\odot}$ shows the difference in
median magnitude with and without gas. The effect of adding the gas is largest
for the F555W (V) and F814W (I) bands, where at $t=1$ Myr, the difference is
$>1$ mag for both values of $Z$ and log(US). At $t=1$ Myr, significant
differences also occur in other bands. At 4 Myr, for $Z=0.014$ and both values
of log(US), the difference is $\sim 0.6$ mag in the F555W band. However, as
expected, at 8 Myr, when the ionising flux from the most massive stars is
greatly diminished, including the gas does not significantly change the
synthetic magnitudes. The contributions of the nebular continuum and emission
lines to the V and I bands are discussed next.
Age | F275W | F336W | F438W | F555W | F814W
---|---|---|---|---|---
(Myr) | M(stars) - M(stars+gas)
$Z=0.002$, log(U)=-2
1 | 0.68 | 1.00 | 1.02 | 2.61 | 1.69
3 | 0.33 | 0.50 | 0.37 | 1.26 | 0.57
4 | 0.22 | 0.29 | 0.09 | 0.35 | 0.06
8 | -0.02 | 0.03 | 0.014 | 0.08 | 0.12
$Z=0.002$, log(U)=-3
1 | 0.88 | 1.13 | 1.11 | 2.10 | 1.83
3 | 0.48 | 0.62 | 0.45 | 0.90 | 0.65
4 | 0.35 | 0.38 | 0.15 | 0.22 | 0.09
8 | 0.06 | 0.09 | 0.06 | 0.10 | 0.14
$Z=0.014$, log(U)=-2
1 | -0.03 | 0.35 | 0.28 | 1.70 | 1.23
3 | -0.27 | -0.06 | -0.07 | 0.22 | 0.39
4 | -0.27 | -0.08 | -0.09 | 0.58 | 0.18
8 | -0.38 | -0.29 | -0.24 | -0.19 | -0.13
$Z=0.014$, log(U)=-3
1 | 0.51 | 0.86 | 0.76 | 1.53 | 1.77
3 | 0.11 | 0.31 | 0.23 | 0.37 | 0.63
4 | 0.15 | 0.29 | 0.21 | 0.57 | 0.47
8 | -0.06 | -0.03 | -0.04 | -0.03 | -0.03
Table 2: For the GALAXEV and GALAXEV-C models with $M_{i}=10^{5}$ M⊙,
difference in median stochastic absolute magnitude, M(stars) - M(stars+gas).
### 3.3 Contributions of stellar continuum, nebular continuum, and emission
lines to total in band.
When gas is present in the models, the total emission includes the
contributions from the nebular continuum and the emission lines. The
importance of including nebular emission lines in spectral synthesis models
has been shown by Bruzual & Charlot (2003), Zackrisson et al. (2011), and
Schaerer & de Barros (2009). For $M_{i}=10^{5}$ $M_{\odot}$, $Z=0.014$, and
log(US)=-3, Figure 3 shows the strongest spectral features that contribute to
the LEGUS bands used to image two overlapping fields of galaxy NGC 7793. NGC
7793-E, was imaged with WFC3/UVIS in the HST-equivalent NUV, U, B, V, and I
bands; while NGC 7793-W used ACS/WFC for the V and I bands. The strongest
emission lines in each filter are: Mg II $\lambda$2798 (F275W); H$\gamma$
$\lambda$4102, Ar I $\lambda$4300, and H$\delta$ $\lambda$4341 (F438W); [O
iii] $\lambda\lambda$4859, 5007, H$\beta$ $\lambda 4861$, and/or H$\alpha$
$\lambda 6563$ \+ [N ii] $\lambda\lambda$6548, 6584 (F555W, depending on the
instrument/channel combination, and [S iii] $\lambda$9069 and $\lambda$9532
(F814W). Note that WFC3/UVIS F555W contains H$\alpha$ $\lambda 6563$ \+ [N ii]
$\lambda\lambda$6548, 6584 because it cuts out at 7000 Å and even though the
throughput at 6563 is low (0.05 compared to 0.28 at peak) the strength of
H$\alpha$ can dominate depending on the strength of [O iii]. ACS/WFC F555W is
different and cuts off at a shorter wavelength. The contribution of the
nebular continuum is strongest in the F275W and F336W bands. Note the presence
of a strong Balmer break around $\sim$3800 Å. Such Balmer breaks have been
observed, see for example Guseva et al. (2006).
For $M_{i}=10^{5}$ $M_{\odot}$ and the full range of parameters of our pilot
library, Figure 4 shows the contributions to the total luminosity in the V and
I bands of the stellar continuum, nebular continuum, and strongest emission
lines. The 220 realizations of the IMF are included. We select the V and I
bands because they have contributions from the emission lines alone of $\geq
30$%. The Figure shows that the stellar continuum dominates at 8 Myr. This is
because the ionizing flux from the massive stars in not significant at this
age. The stellar continuum is also dominant in some cases at ages of 3 and 4
Myr (panels 5 to 12). The Figure also shows that the nebular continuum
dominates in one case (panel 2). Finally, the Figure shows that the nebular
emission lines can contribute with $>50\%$ to the total emission in the V-band
(panels 1, 3, 5 and 11) and $>30\%$ to the I-bands (panel 4). The ranges
corresponding to the y-axis of Figure 4 are given in Table 3. The Table and
Figure illustrate the importance of accounting for the contribution of the
ionized gas in the LEGUS bands.
Figure 3: Top-panel–. Throughputs of the LEGUS filters + WFC3/UVIS (filled curves) or ACS/WFC (dashed curves) instrument/channel. Middle-panel–. The strongest spectral features that contribute to the different LEGUS bands according to models that include the contributions of the stars the stars + ionized gas + dust mixed with the ionized gas. The stellar spectra used as input correspond to a cluster of initial mass = 105 M⊙, $Z=0.014$, and age = 1 Myr. The gas parameters are: $Z=0.014$ and log(US)=-3. We use FWHM=130 km s-1 for the width of the emission lines, which is the typical value obtained from VLT MUSE observations of H ii regions in the galaxy under study (NGC 7793, Wofford et al. 2020). Bottom-panel–. Enlargement of the middle panel to show the continua from the stars alone (blue curve) and stars + ionized gas + dust in the ionized gas (black curve). Figure 4: For the GALAXEV and GALAXEV-C models with $M_{i}=10^{5}$ M⊙, contributions to the total luminosities in the V and I bands of the stellar continuum (SteCon, magenta background), nebular continuum (NebCon, white background), and strongest emission lines (EmLin, blue background). The cluster age is given by the y-axis label the log(US) value in the legend, and the metallicity at the top. The horizontal lines indicate contributions of 30 and 50% to the total luminosity in the band. The panels are numbered from 1 to 16. | F555W | F814W
---|---|---
Age | SteCon | NebCon | EmLin | SteCon | NebCon | EmLin
| Z=0.002, log(US)=-2
1 | $0.08-0.10$ | $0.10-0.10$ | $0.80-0.82$ | $0.20-0.25$ | $0.59-0.63$ | $0.16-0.17$
3 | $0.17-0.45$ | $0.05-0.10$ | $0.49-0.74$ | $0.34-0.80$ | $0.15-0.51$ | $0.05-0.15$
4 | $0.53-0.81$ | $0.00-0.05$ | $0.19-0.43$ | $0.85-0.98$ | $0.01-0.11$ | $0.01-0.04$
8 | $0.89-0.93$ | $0.00-0.01$ | $0.06-0.10$ | $0.88-0.94$ | $0.04-0.09$ | $0.02-0.03$
| Z=0.002, log(US)=-3
1 | $0.12-0.16$ | $0.18-0.19$ | $0.66-0.69$ | $0.19-0.23$ | $0.61-0.65$ | $0.15-0.16$
3 | $0.26-0.59$ | $0.10-0.18$ | $0.31-0.56$ | $0.32-0.78$ | $0.18-0.54$ | $0.05-0.13$
4 | $0.66-0.87$ | $0.03-0.09$ | $0.10-0.25$ | $0.83-0.95$ | $0.04-0.14$ | $0.01-0.03$
8 | $0.89-0.94$ | $0.02-0.04$ | $0.04-0.07$ | $0.86-0.92$ | $0.06-0.12$ | $0.01-0.02$
| Z=0.014, log(US)=-2
1 | $0.18-0.23$ | $0.03-0.04$ | $0.74-0.78$ | $0.32-0.38$ | $0.29-0.32$ | $0.33-0.36$
3 | $0.67-0.77$ | $0.04-0.09$ | $0.14-0.28$ | $0.68-0.90$ | $0.01-0.13$ | $0.10-0.19$
4 | $0.36-0.65$ | $0.02-0.08$ | $0.27-0.62$ | $0.69-0.93$ | $0.00-0.14$ | $0.05-0.18$
8 | $0.84-0.85$ | $0.15-0.15$ | $0.01-0.01$ | $0.89-0.89$ | $0.11-0.11$ | $0.00-0.00$
| Z=0.014, log(US)=-3
1 | $0.22-0.27$ | $0.14-0.14$ | $0.59-0.64$ | $0.21-0.25$ | $0.40-0.42$ | $0.35-0.37$
3 | $0.66-0.81$ | $0.05-0.11$ | $0.14-0.24$ | $0.53-0.75$ | $0.15-0.29$ | $0.10-0.18$
4 | $0.42-0.73$ | $0.04-0.07$ | $0.23-0.50$ | $0.50-0.81$ | $0.11-0.27$ | $0.08-0.22$
8 | $0.94-0.95$ | $0.04-0.04$ | $0.01-0.02$ | $0.96-0.97$ | $0.03-0.04$ | $0.00-0.00$
Table 3: For the GALAXEV and GALAXEV-C models with $M_{i}=10^{5}$ M⊙, ranges
of the contributions to the total luminosities in the V and I bands of the
stellar continuum (SteCon), nebular continuum (NebCon), and the strongest
emission lines in the band (EmLin, see Figure 4).
### 3.4 Stochastic versus deterministic models.
In Figure 2, the filled squares represent the deterministic magnitudes for the
cases with just stars (left square) and stars + gas + dust (right square) at
each value of $M_{i}$. How the deterministic magnitude compares to the median
of the stochastic models depends on whether the models include just stars or
stars + gas + dust, and on the combination of $M_{i}$, age, $Z$, log(US), and
photometric band. Differences of less than 0.5 mag between the median and
deterministic magnitude can be seen in the Figure for combinations of the
parameters in the following cases: i) {stars} or {stars + gas + dust},
$M_{i}/M_{\odot}=10^{4}$ or $10^{5}$, $Z=0.014$, and log(U${}_{\rm S}=-3$);
and ii) {stars}, $M_{i}/M_{\odot}=10^{3}$, $t=1$ Myr, both values of $Z$, and
both values of log(US). Examples of cases where $|$Det - <Sto>$|$> 0.5 mag are
provided in Table 4. Note that $|$Det - <Sto>$|$ can be > 0.5 mag for
$M_{i}/M_{\odot}$ as high as $10^{5}$, and that although the deterministic
magnitude tends to be more luminous, this is not systematically the case.
Gas? | Mi | Age | Panels | Most
---|---|---|---|---
| (M⊙) | (Myr) | | Luminous
No | 1E3 - 1E5 | 1 | none | -
No | 1E3 - 1E5 | 3 | 5 & 10 | Det
No | 1E4 | 4 | 15 & 20 | Det
No | 1E3 | 4 | 4, 5, 9 & 10 | Det
No | 1E3 | 8 | 15 & 20 | Det
Yes | 1E3 | 1, 3, 4 | 12 | <Sto>
Yes | 1E3 | 8 | 4 & 5 | Det
Yes | 1E4 | 3 | 5 & 10 | Det
Table 4: Examples of cases where the absolute value of the residual
(deterministic mag - median stochastic mag) is > 0.5 mag. We consider the
GALAXEV and GALAXEV-C models. Column 1 indicates if gas is included in the
models. Columns 2 and 3 give the initial mass and age of the cluster,
respectively (or their ranges). Column 4 gives the IDs of the panels in Figure
2 where examples can be found. Column 5 says which of the two magnitudes is
the most luminous (Det=deterministic, <Sto>=median stochastic mag).
For star-only broad-band magnitudes: a) the absolute value of the residual
(deterministic prediction - median of stochastic models) can be $\geq 0.5$
mag, even for $M_{i}=10^{5}$ $M_{\odot}$
### 3.5 Models in the U - B versus V - I diagram
We now analyse the positions of the models in the U - B versus V - I diagram,
which has been used to age-date star clusters by (Chandar et al., 2004, 2016).
Although Chandar et al. (2016) conclude that using UBVIH$\alpha$ photometry
yields better agreement between photometric and spectroscopic ages,
determining accurate H$\alpha$ photometry is very difficult in practice,
particularly, when star clusters are not isolated and the H$\alpha$ morphology
is complex, as we discuss in Section 6.4. We note that other diagrams such as
(U - B) versus (B - V), which is not discussed here, have also been used to
age-date star clusters (e.g., Bica et al. 1991).
Let D0 and S0 be the deterministic and stochastic GALAXEV models,
respectively. Similarly, let (D2, S2) and (D3, S3) be the GALAXEV-C pairs for
log(Us)=-2 and -3, respectively. For $Z=0.014$, which is the closest
metallicity to the observations (see Section 4), Figures 5 \- 7 show
comparisons of the D0 and S0, D3 and S3, and D2 and S2 predictions in the U -
B vs. V - I diagram, respectively. Appendix Figures 14 \- 15 are similar to
Figures 5 \- 7 but for $Z=0.002$.
Let us start by discussing Figure 5, which corresponds to the $Z=0.014$ models
that only account for the stars. In this and similar figures, the age and
initial mass of the cluster are given by the y-axis label and the column
title, respectively. In Figure 5, the clouds of magenta filled-symbols are the
stochastic S0 models and the magenta curves are the corresponding
deterministic track. The comparison between the S0 and D0 predictions yields
the following results, which are organised in order of increasing cluster age
and decreasing cluster mass.
1 Myr (top row of panels).– For $M_{i}/M_{\odot}=10^{5}$, the S0 models are
tightly grouped on top of the 1 Myr D0 point that is located at the
intersection of the vertical and horizontal dashed lines. As mass decreases,
the spread of the S0 models increases. In particular, for
$M_{i}/M_{\odot}=10^{3}$, the spread is mostly in the vertical direction and
towards the red.
3 Myr (second row of panels from the top).– For $M_{i}/M_{\odot}=10^{5}$, the
S0 models break into three clouds, one tightly grouped on top of the 3 Myr D0
point, one bluer and closer to the 1 Myr D0 point, and the last one redder and
closer to the 4 Myr D0 point. For $M_{i}/M_{\odot}=10^{4}$, a small fraction
of the S0 models are located far away from the 3 Myr D0 point, towards the
red, and approach the 100 Myr D0 point. Finally, for $M_{i}/M_{\odot}=10^{4}$
and $M_{i}/M_{\odot}=10^{3}$, most of the S0 models are bluer than the 3 Myr
D0 point.
4 Myr (third row of panels from the top).– For $M_{i}/M_{\odot}=10^{5}$, the
spread in S0 models is larger than at 1 Myr, and it is both towards the red
and the blue. For $M_{i}/M_{\odot}=10^{4}$, a larger but still small fraction
of S0 models is found significantly offset to the red. For
$M_{i}/M_{\odot}=10^{3}$, a small fraction of S0 models is significantly
redder than the 4 Myr D0 point and is located between the 100 Myr and 1 Gyr
markers along the deterministic track (however, the main S0 cloud is spread
closer to the 1-4 Myr D0 predictions.
8 Myr (bottom row of panels.– For $M_{i}/M_{\odot}=10^{5}$, the spread is
larger than at 1 Myr, and it is both towards the red and the blue, similar to
the behaviours at 3 and 4 Myr, but the spread is mostly horizontal. For
$M_{i}/M_{\odot}=10^{4}$, the S0 models are spread almost horizontally towards
the blue and the red and a few S0 models are significantly bluer than the D0
prediction. For $M_{i}/M_{\odot}=10^{3}$, the S0 models are divided into two
clouds, one bluer in V - I and close to the 3 - 4 Myr D0 predictions and
another starting at the D0 point and spreading towards significantly redder
colours.
Figure 5: U-B vs. V-I diagrams with $Z=0.014$ star-only GALAXEV deterministic
tracks (magenta solid curves) and stochastic models (magenta small filled
symbols) overlaid. Ages along the D0 track are marked with orange-filled
triangles and labelled using M=Myr and G=Gyr. The dashed vertical and
horizontal lines mark the position of the deterministic prediction at the age
given by the y-axis label. For clarity, for the stochastic models, each panel
shows a different combination of cluster initial mass (given by the column
title) and age (given by the y-axis label).
Let us now discuss Figures 6 and 7, i.e., the $Z=0.014$ models with gas. The
S2 and S3 models yield some of the same trends that are observed in the star-
only case but also some additional behaviours. At 1 Myr, the S3 and S2 models
are bluer in V - I relative to their respective deterministic predictions. At
3 and 4 Myr, for $M_{i}/M_{\odot}>10^{5}$, a significant fraction of the S2
and S3 models are offset towards bluer U - B relative to the corresponding
deterministic models. Finally, a major conclusion from the figures is that the
$Z=0.014$ D2 and D3 tracks are not useful for age-dating clusters with
$M_{i}/M_{\odot}\sim 10^{3}$. This is because at 8 Myr, for this mass, the
number of stochastic models is similar in the $\sim$4 Myr cloud and the 8 M yr
cloud.
Figure 6: Similar to Figure 5 but for log(Us)=-3 deterministic (dark blue
curves) and stochastic (dark blue symbols) models that include the
contributions of the ionized gas + dust mixed with the ionized gas. Figure 7:
Similar to Figure 6 but for log(Us)=-2.
We proceed with the analysis of the $Z=0.002$ models in the U - B vs. V - I
plane. By looking at the star-only tracks of Figure 14, one can see that the 4
Myr D0 marker is almost coincident with the 100 Myr marker and that the 8 Myr
marker is between the 1 and 3 Myr markers. When gas is included (Figures 15
and 16), the 3 and 8 Myr deterministic markers are close to each other and
they are almost coincident for log(Us)=-3. On the other hand, the 4 and 100
Myr deterministic markers become more separated than in the star-only case. In
conclusion, age-dating star clusters with $Z=0.002$ using their positions in
the U - B vs. V - I diagram along the GALAXEV-C deterministic tracks is highly
uncertain at any mass.
## 4 Sample and Observations
In this work, we use HST images of two partially overlapping fields of galaxy
NGC 7793, NGC 7793-E and NGC 7793-W. The images were obtained as part of the
LEGUS and H$\alpha$ LEGUS programs. We fit models to the LEGUS broad-band
photometry and investigate how close the H$\alpha$ equivalent widths of the
clusters that were obtained by Hannon et al. from the H$\alpha$-LEGUS images
are to the values predicted by our best-fitting models. In this section, we
describe the observations, the galaxy, and how we selected the sample of star
clusters.
### 4.1 HST LEGUS and H$\alpha$ LEGUS
LEGUS (Calzetti et al., 2015, PID 13364) is a Cycle 21 HST treasury program
that obtained high spatial resolution ($\sim 0.07"$) images of portions of 50
nearby ($\leq 16$ Mpc) galaxies, using the UVIS channel of the Wide Field
Camera Three (WFC3), and the broad band filters F275W (2704 Å), F336W (3355
Å), F438W (4325 Å), F555W (5308 Å), and F814W (8024 Å), which roughly
correspond to the photometric bands NUV, U, B, V, and I, respectively. The
survey includes galaxies of different morphological types and spans a factor
of $\sim 10^{3}$ in both star formation rate (SFR) and specific star formation
rate (sSFR), $\sim 10^{4}$ in stellar mass ($\sim
10^{7}-10^{11}\,\mathrm{M_{\odot}}$), and $\sim 10^{2}$ in oxygen abundance
($12+\mathrm{log\,O/H}=7.2-9.2$). Some of the targets in the survey have high
quality archival images in bandpasses similar to those required by LEGUS, most
of them from the Wide Field Channel of HST’s Advanced Camera for Surveys
(ACS), and fewer of them from ACS’s High Resolution Channel (HRC). For the
latter targets, LEGUS completed the five band coverage. The choice of filters
was dictated by the desire to distinguish young massive bright stars from
faint star clusters, to derive accurate star formation histories for the stars
in the field from their CMDs, and to obtain extinction-corrected estimates of
age and mass for the star clusters. Star and star-cluster catalogues have been
released for the LEGUS sample and are described in Sabbi et al. (2018) and
Adamo et al. (2017) (hereafter A17), respectively.
H$\alpha$ LEGUS (PI Chandar, PID 13773) is a Cycle 22 HST program that
obtained narrow-band, H$\alpha$ (F657N) and medium band, continuum (F547M)
images for the 25 LEGUS galaxies with the highest star formation rates, using
the WFC3. The corresponding H$\alpha$ observations reveal thousands of
previously undetected H ii regions, including those ionized by stellar
clusters and “single" massive stars. We note that the LEGUS data do not have
the spatial resolution to visually resolve massive stars in close binary
systems.
### 4.2 NGC 7793
We used the observations of NGC 7793 obtained by LEGUS and H$\alpha$ LEGUS
which are summarised in Table 2 of Wofford et al. (2020). NGC 7793 is a
Southern SAd flocculent spiral galaxy that is part of the Sculptor group and
is located at a Cepheid distance of 3.44 Mpc (Pietrzyński et al., 2010). It
has a small bulge and a spiral filamentary morphology, and the following
additional properties: an inclination of 47°; a colour excess due to the
Galaxy of E(B - V) = 0.017 mag (Schlafly & Finkbeiner, 2011); a stellar mass
determined from the extinction-corrected B-band luminosity and colour
information of M∗ = $3\text{\times}{10}^{9}$ M⊙ (Bothwell et al., 2009);
lastly, a galaxy-wide star formation rate calculated from dust-attenuation
corrected GALEX far-UV, adopting a distance of 3.44 Mpc, SFR = 0.52 M⊙yr-1
(Lee et al., 2009). According to Pilyugin et al. (2014), the ionized-gas
oxygen abundance at the centre of the galaxy is 12 + log(O/H)=$8.50\pm 0.02$,
and the O/H gradient is $-0.066\pm 0.0104$ dex kpc-1.
### 4.3 Sample of star clusters
We select a sample of 17 isolated, low-mass ($<10^{4}$ $M_{\odot}$), young
($<10$ Myr) star clusters from the catalogue of clusters with compact
H$\alpha$ morphologies of H19. H19 use the LEGUS datasets, which are aligned
to the F438W image, and the LEGUS photometry, which uses an aperture with a
radius of 5 pixels or 0.2” (3 pc), which was selected based on a curve of
growth analysis. For H$\alpha$, H19 extracted their own photometry using
apertures with different radii for each cluster, also based on a curve of
growth analysis (see H21 in prep. for more details). The masses and ages used
for the selection come from deterministic models, i.e., models where the
luminosity is scaled in proportion to the initial cluster mass.
Figure 8 shows the location of the star clusters within the galaxy. The figure
shows that the star clusters are located within a radius of $\sim 3$ arcmin
from the centre of the galaxy. Adopting Z${}_{\odot}=0.014$ as the reference
solar metallicity (Asplund et al., 2009) and using O/H as a gauge of
metallicity, we find that the metallicity range of the clusters in our sample
is Z=0.006 to 0.009. Thus, their metallicity is close to Z⊙.
Table 5 lists the J2000 coordinates and apparent Vega magnitudes of the star
clusters in the LEGUS and H$\alpha$ LEGUS bands. Note that clusters 93 and
383, and 417 and 1252 have different IDs but very similar coordinates and
photometry. This is because the clusters are the same, but their measurements
come from field NGC 7793E in one case and NGC 7793W in the other. The clusters
are in the overlapping region between these two fields. This constitutes a
good check on repeatability of the cluster-finding process. When comparing
models to observations, we keep the repeated clusters in order to check if
small differences in the photometry due to the different pointings affect the
derived properties of the clusters. Figure 9 shows postage stamps of the
clusters in our sample with the 5 pixel aperture overlaid.
Figure 8: Composite RGB images of the partially-overlapping fields NGC 7793E (left panel) and NGC 7793W (righ panel), where red = continuum-subtracted H$\alpha$, green = F555W, and blue = F438W. We use white circles to indicate the positions of the star clusters in our sample and give their LEGUS ID. In both images, the centre of the galaxy appears as a cyan knot. We overlay cyan circles centered on this knot, of radii equal to 0, 1, 2, and 3 arcmin. The value of 12+log(O/H) in the ionized gas at each of these radii is shown with cyan characters. At 3.44 Mpc (Pietrzyński et al., 2010), 1 armin is $\sim$ 1 kpc. Note that clusters 93 and 383, and 417 and 1252 have similar coordinates and photometry but have different IDs (see Section 4.3 for a discussion of these clusters). Also note that cluster 1576 appears in both fields and is at the edge of the H$\alpha$ image for field NGC 7793W. Figure 9: Postage stamps of the star clusters in our sample using the same colour scheme as in Figure 8. The images are 12" (200 pc) on the side. The small white circle represents the aperture used for obtaining the photometry in the five LEGUS broad-bands. Star clusters from fields NGC 7793-E and NGC 7793-W are shown in the top-two and bottom-two rows show, respectively. Note that clusters 93 and 383, and 417 and 1252 have similar coordinates and photometry but have different IDs (see Section 4.3 for a discussion of these clusters). Also note that cluster 1576 appears in both fields and at the edge of the H$\alpha$ image in field NGC 7793W. ID | RA | Dec | F275W | F336W | F438W | F547M | F555W | F657N | F814W | Rad
---|---|---|---|---|---|---|---|---|---|---
| J2000 | J2000 | mag | mag | mag | mag | mag | mag | mag | pix
0011-E | 23:57:54.2424 | -32:33:59.148 | 20.97$\pm$0.09 | 20.81$\pm$0.09 | 22.16$\pm$0.08 | 20.70 | 21.07$\pm$0.07 | 17.39 | 20.71$\pm$0.06 | 30
0090-E | 23:57:48.3768 | -32:34:33.672 | 18.80$\pm$0.09 | 19.32$\pm$0.08 | 20.82$\pm$0.07 | 21.06 | 20.99$\pm$0.05 | 19.80 | 21.06$\pm$0.05 | 10
0093-E* | 23:57:45.6384 | -32:34:33.888 | 19.03$\pm$0.09 | 19.50$\pm$0.08 | 20.92$\pm$0.07 | 20.96 | 20.96$\pm$0.05 | 17.70 | 20.97$\pm$0.06 | 30
0383-W* | 23:57:45.6384 | -32:34:33.852 | 18.95$\pm$0.07 | 19.40$\pm$0.07 | 20.88$\pm$0.06 | 21.06 | 20.97$\pm$0.05 | 17.62 | 21.06$\pm$0.07 | 40
0417-E* | 23:57:49.0584 | -32:35:23.028 | 19.70$\pm$0.09 | 19.88$\pm$0.08 | 21.38$\pm$0.07 | 20.59 | 21.01$\pm$0.05 | 18.81 | 20.59$\pm$0.07 | 10
0451-W | 23:57:35.1792 | -32:34:38.820 | 22.44$\pm$0.14 | 21.88$\pm$0.09 | 23.21$\pm$0.09 | 21.81 | 21.95$\pm$0.06 | 19.71 | 21.81$\pm$0.07 | 10
0531-E | 23:57:46.2240 | -32:35:33.036 | 18.86$\pm$0.09 | 19.30$\pm$0.08 | 20.76$\pm$0.07 | 20.75 | 20.86$\pm$0.05 | 17.26 | 20.76$\pm$0.06 | 50
0534-E | 23:57:47.1456 | -32:35:33.144 | 19.81$\pm$0.09 | 20.17$\pm$0.08 | 21.56$\pm$0.07 | 20.97 | 21.55$\pm$0.05 | 17.84 | 20.98$\pm$0.06 | 40
0589-E | 23:57:56.1552 | -32:35:39.804 | 19.77$\pm$0.09 | 19.90$\pm$0.08 | 21.16$\pm$0.07 | 20.39 | 20.93$\pm$0.06 | 17.49 | 20.39$\pm$0.05 | 30
0816-E | 23:57:48.2376 | -32:36:14.796 | 17.33$\pm$0.09 | 17.69$\pm$0.08 | 19.16$\pm$0.07 | 18.94 | 18.96$\pm$0.05 | 16.58 | 18.94$\pm$0.05 | 50
0894-E | 23:57:51.2400 | -32:36:48.456 | 20.57$\pm$0.09 | 20.54$\pm$0.08 | 21.93$\pm$0.07 | 20.64 | 20.81$\pm$0.05 | 16.96 | 20.65$\pm$0.05 | 30
1252-W* | 23:57:49.0656 | -32:35:23.028 | 19.69$\pm$0.07 | 19.91$\pm$0.07 | 21.43$\pm$0.07 | 20.70 | 20.96$\pm$0.05 | 18.83 | 20.71$\pm$0.07 | 10
1381-W | 23:57:40.4448 | -32:35:27.888 | 21.77$\pm$0.09 | 21.79$\pm$0.08 | 23.27$\pm$0.09 | 22.49 | 22.02$\pm$0.06 | 20.15 | 22.50$\pm$0.07 | 10
1564-W | 23:57:41.4264 | -32:35:34.368 | 20.10$\pm$0.07 | 20.18$\pm$0.07 | 21.53$\pm$0.07 | 20.42 | 20.80$\pm$0.05 | 18.84 | 20.43$\pm$0.06 | 10
1576-W | 23:57:48.7728 | -32:35:34.656 | 18.73$\pm$0.07 | 19.24$\pm$0.06 | 20.80$\pm$0.06 | 20.74 | 20.73$\pm$0.05 | 18.85 | 20.75$\pm$0.07 | 10
1949-W | 23:57:40.0344 | -32:35:47.436 | 18.93$\pm$0.07 | 19.38$\pm$0.06 | 20.97$\pm$0.06 | 20.96 | 20.71$\pm$0.05 | 19.22 | 20.96$\pm$0.06 | 10
2449-W | 23:57:38.6064 | -32:36:12.312 | 19.67$\pm$0.07 | 19.84$\pm$0.07 | 21.19$\pm$0.07 | 20.47 | 20.89$\pm$0.06 | 17.08 | 20.47$\pm$0.06 | 50
2732-W | 23:57:40.3200 | -32:36:41.328 | 19.32$\pm$0.07 | 19.61$\pm$0.07 | 21.06$\pm$0.06 | 20.68 | 20.59$\pm$0.06 | 18.80 | 20.69$\pm$0.06 | 10
2740-W | 23:57:40.1136 | -32:36:43.452 | 20.39$\pm$0.08 | 20.78$\pm$0.08 | 22.10$\pm$0.07 | 21.77 | 21.83$\pm$0.06 | 19.15 | 21.77$\pm$0.06 | 10
Table 5: Column (1): ID of star cluster and field (NGC 7793-E or -W). Columns
(2)-(3): Right Ascension and Declination. Columns (4)-(10): Apparent
magnitudes from A17 (LEGUS-bands) and H19 (H$\alpha$-LEGUS bands), based on
PSF-photometry. We use Vega and AB magnitudes for the LEGUS and
H$\alpha$-LEGUS bands, respectively. The photometry is corrected for
foreground extinction as explained in the text. Column (11): Radius in pixels
used for the H$\alpha$ photometry. Note that clusters 93 and 383, and 417 and
1252, which are marked with an asterisk in the first column, have similar
coordinates and photometry but have different IDs (see Section 4.3 for a
discussion of these clusters).
Figure 10 combines Figures 5, 6, and 7 in one. This helps to see where the
$Z=0.014$ S0, S2, and S3 models fall relative to the corresponding D0, D2, and
D3 predictions. The corresponding figure for $Z=0.002$ is Figure 17.
In addition, the panels in the right column of Figure 10 include the LEGUS
observations. The red error bars represent the observations corrected for
reddening due to dust in the MW (using Av=0.053 mag) and uncorrected for
intrinsic reddening while the black error bars are the observations also
corrected for dust in NGC 7793, using the Av values of column 5 in Table 8.
The black error bars include the propagation of the uncertainties in the
intrinsic Av values that are given in column 5 of Table 8. For both the
foreground and intrinsic reddening corrections, we use the MW-extinction law
of Cardelli et al. (1989) for R(V)=3.1. As expected, after the full correction
for reddening, the observations move towards bluer colours and the size of the
error bars increases.
Note that none of our observations are found near the location of the
“outlier" stochastic models, which are the few models located far from their
corresponding deterministic prediction. This is expected since our observed
sample is small and according to stochastic models, “outlier" clusters have a
low probability of being created in nature, and observed.
Several works of the LEGUS collaboration use deterministic Yggdrasil tracks of
Zackrisson et al. (2011). The bottom-left panel of Figure 10 shows the
Yggdrasil track corresponding to log(U)=-3 and $Z=0.020$ (dashed-black curve
with ages marked using black-filled triangles). $Z=0.020$ is the closest track
to $Z=0.014$ that is available. The Yggdrasil track for log(U)=-3 and
$Z=0.004$ is shown in Figure 17. Note that along the $Z=0.004$ Yggdrasil, the
8 Myr marker is redder in U - B than the 4 Myr marker, contrary to what
happens in the GALAXEV-C $Z=0.002$ track.
Figure 10: Combination of Figures 5 \- 7, as indicated by the legend on the
top-left panel. Some ages along the log(Us)=-3 (dark-blue) track are marked
with orange-filled triangles and labelled using M=Myr and G=Gyr. The bottom-
left panel shows the Yggdrasil track corresponding to log(U)=-3 and $Z=0.020$
(dashed-black curve with black-filled triangles). The last column of panels
includes: i) LEGUS observations corrected for reddening due to dust in the MW
that are: uncorrected for dust in NGC 7793 (red error bars) and corrected for
dust in NGC 7793 (black error bars), as given by the legend in the top-right
panel; and ii) a reddening vector corresponding to an extinction of AV=1 mag.
## 5 Method for interpreting the observations
In this section, we use our models in order to derive the dust extinctions,
masses, and ages of the star clusters in our sample. The derived properties
are the physical parameters of the models that best fit the observations. In
order to find the model that best fits the observations, one can use
$\chi^{2}$ minimization (Popescu, 2010); construct probability maps in the
parameter space to explore the nodes of the grid and then select the more
probable solutions (Fouesneau & Lançon, 2010); or use a Bayesian inference
method (Krumholz et al., 2015a; Wofford et al., 2016; Fouesneau & Lancon,
2010). We use the latter method, which is explained in the following section.
In order to find the probability distribution function of the physical
properties, we use the method of conditional regression, coupled with a kernel
density estimation, which is presented in (Krumholz et al., 2015a). In
summary, if we let $p(x|y_{obs};\sigma_{y})$ be the probability distribution
of the physical parameters, $x$, given $N$ photometric observations,
$y_{obs}$, with error $\sigma_{y}$, the probability distribution of the
physical properties, given a set of photometric observations can be written
as:
$p(x|y_{obs})\equiv\sum_{i}\omega_{i}G((x-x_{i},y_{obs}-y_{i}),h^{{}^{\prime}}),$
(1)
where $h^{{}^{\prime}}$ is a new bandwidth that depends on both, the bandwidth
of the physical properties of the models $h_{x}$ and the photometric
properties ($h_{y}$). This also allows us to find an expression for
calculating the marginal probability distribution of each physical parameter,
which we will call $x_{1}$.
$p(x_{1}|y_{obs})\propto\sum_{i}\omega_{i}G((x_{1}-x_{1,i},h_{1})G(y_{obs}-y_{i}),\sqrt{\sigma_{y}^{2}+h_{y}^{2}}).$
(2)
This procedure can be followed for each of the physical parameters.
In order to derive the physical properties of the clusters in our sample, we
adapted our pilot-library based on the models presented in Vidal-García et
al.(, in prep.) to the tool BAYESPHOT, which uses Bayesian inference to
estimate joint and marginal PDFs by following the approach which was just
described. In addition to the synthetic photometry, BAYESPHOT requires
additional output from population synthesis, such as the time step, birth mass
of the SSP, current mass of all stars in the cluster (accounting for the
effects of mass loss and supernovae), number of living stars in the cluster at
the present time, visual extinction, to name a few. In this process, we used
the python module SLUGPY, which is presented in Krumholz et al. (2015a). This
module is a series of functions that allows to handle spectro-photometric data
generated with the population synthesis code SLUG. Since NGC 7793 is a spiral
galaxy and the clusters have solar metallicity, in order to obtain the
extinction due to dust mixed with the neutral gas, in the V-band, we adopted
the Milky Way law of Mathis (1990), with $R(V)=3.1$.
## 6 RESULTS
In this section, we test our models using the observations which were
presented in Section 4.
### 6.1 S3 models versus LEGUS photometry
In order to illustrate how well one can fit the LEGUS observations with our
models, which are only available for three values of the initial mass, we use
the S3 models, which have Z = 0.014 and log(US) = -3.
For each cluster in our sample, Figure 11 shows the comparison of the best-
fitting S3 models (black-dashed curves) and the observations. In order to find
the best-fitting model, we use equation (29) from Krumholz et al. (2015a). The
figure shows observations (apparent Vega magnitudes) with and without a
correction for dust intrinsic to NGC 7793. The red error bars are the
observations corrected for reddening due to dust in the Milky Way (using
Av=0.017 mag), while the blue error bars include the additional correction for
dust in NGC 7793 (using the median V-band extinctions of column 5 in Table 8).
For both corrections we use the Milky Way extinction law of Cardelli et al.
(1989) for R(V)=3.1. Since in the figure, the models are uncorrected for
reddening in the neutral gas, they should be compared to the red error bars.
In Figure 11, the panels are arranged in order of increasing AV value. This is
why in the right-side panels there is a larger offset between the red and blue
curves. As can be seen in the figure, the F275W magnitude is more affected by
the reddening correction than the F814W magnitude, which is in agreement with
the shape of the Milky Way extinction law. Note that there are differences in
the $A_{\rm V}$ values of 0.08 and 0.61 mag between the clusters that are
repeated and discussed above, i.e., 93 and 383; and 417 and 1252,
respectively.
For each cluster, Table 6 gives the AV residual (observation - best-fitting
model) in each LEGUS band. For the clusters in our sample, the median residual
in each LEGUS band is within the observational error which is reported in
Table 5. The observations of cluster 1381-W are not well reproduced in the
bands which are redder than U. This is likely because it is least massive
cluster in our sample according to the K15 stochastic models which are
presented in Table 9. The mass of the latter cluster is 219 $M_{\odot}$ while
the minimum mass in our models is M=$10^{3}$ $M_{\odot}$.
We find models that fit the observations reasonably well in spite of the poor
sampling in cluster mass and age of our pilot library thanks to the fine
sampling in V-band extinction values.
Figure 11: Observed apparent Vega magnitudes (red error bars, corrected for reddening in the Milky Way) versus best-fitting model with $Z=0.014$ and log($U_{\rm S}=-3$ (black dashed lines). For comparison, we also show the observations corrected for reddening in the Milky Way and NGC 7793 (blue error bars). The panels are arranged in order of increasing AV value, which is why for the panels on the right, a larger offset between the red and blue curves can be observed. Note how the bluest magnitude (F275W) is more affected by the reddening correction compared to the reddest magnitude (F814W), which is expected from the shape of the extinction law. As discussed in Section 4.3, cluster 93 is a copy of 383, and cluster 417 is a copy of 1252. We mark the IDs of these clusters with asterisks. The clusters have different AV values due their slightly different observed magnitudes (see Table 5). | Residual of observed - model magnitude
---|---
ID | F275W | F336W | F438W | F555W | F814W
(1) | (2) | (3) | (4) | (5) | (6)
0011-E | -0.05 | 0.07 | 0.08 | -0.14 | 0.07
0090-E | -0.06 | -0.02 | 0.02 | 0.07 | -0.04
0093-E | -0.04 | 0.01 | 0.04 | 0.02 | -0.07
0383-W | 0.01 | -0.00 | 0.03 | 0.03 | -0.07
0417-E | -0.09 | 0.05 | 0.00 | 0.04 | -0.04
0451-W | 0.26 | 0.05 | 0.18 | -0.32 | 0.29
0531-E | -0.07 | 0.09 | -0.04 | 0.03 | -0.01
0534-E | -0.18 | 0.08 | 0.02 | 0.13 | -0.13
0589-E | -0.12 | 0.09 | 0.01 | 0.01 | -0.02
0816-E | -0.05 | 0.04 | 0.08 | -0.08 | 0.01
0894-E | -0.04 | 0.13 | 0.12 | -0.17 | 0.12
01252-W | -0.09 | 0.05 | 0.02 | 0.01 | -0.03
01381-W | 0.08 | 0.08 | 0.52 | -0.51 | 0.31
01564-W | -0.07 | 0.11 | -0.02 | -0.03 | -0.05
01576-W | -0.11 | 0.05 | -0.02 | -0.05 | -0.01
01949-W | -0.14 | 0.01 | -0.04 | -0.11 | 0.14
02449-W | -0.05 | 0.11 | -0.04 | -0.04 | -0.03
02732-W | 0.05 | 0.11 | -0.05 | -0.13 | -0.01
02740-W | -0.09 | 0.05 | 0.15 | -0.06 | -0.02
RANGE | -0.18 - 0.26 | -0.02 - 0.13 | -0.05 - 0.52 | -0.51 - 0.13 | -0.13 - 0.31
MEDIAN | -0.04 | 0.06 | 0.06 | -0.07 | 0.02
Table 6: Residuals in each photometric band. Column (1) shows the observed
cluster ID. Columns (2) - (6) shows the difference between the observed
magnitudes of the clusters in NGC 7793 and the best-fitting models
corresponding to $Z=0.014$, log(Us) = -3, and cluster mass = 103 $M_{\odot}$.
For each LEGUS filter, the last two rows give the range and median of the
values in the column.
### 6.2 Equivalent width of H$\alpha$
The F656N filter includes the H$\alpha$ and [N ii] lines. We compare the
equivalent width of the combined H$\alpha$ \+ [N ii] emission from the best-
fitting S2 and S3 models with $Z=0.014$, against the value measured by Hannon
et al. (in prep., hereafter H21) using the H$\alpha$-LEGUS observations. In
order to catch all of the H$\alpha$ emission due to ionisation by the cluster,
the size of the aperture used by H21 to obtain EW(H$\alpha$\+ [N ii]) was
selected based on the curve of growth of each cluster. The radius of the
aperture is provided in the last column of Table 5. We present the observed
and best-fit model EWs in Table 7. We find that for two clusters (816-E and
1564-W) the S2 models are within $100$ Åof the observed value, while the S3
models are within $100$ Åof the observed value only in one case (531-E). Note
that the observed EW(H$\alpha$\+ [N ii]) value is highly uncertain. We also
find that for the S2 models, the mean of EW(H$\alpha$\+ [N ii]) is a factor of
$\sim 4$ lower than the mean of the observations, and that the mean of the S3
models is closer to the mean of the observations. Finally, we find that in
general, the youngest clusters have the largest model value of EW(H$\alpha$\+
[N ii]) (see Table 10).
ID | | EW(H$\alpha$)
---
Å
| $\Delta$EW(H$\alpha$)
---
Å
| (1)
---
| H21
---
(2)
| S2
---
(3)
| S3
---
(4)
| (3) - (2)
---
(5)
| (4) - (2)
---
(6)
0011-E | 3038 | 1919 | 2097 | -1119 | -841
0090-E | 1489 | 171 | 235 | -1318 | -1254
0093-E | 1232 | 128 | 1987 | -1104 | 755
0383-W | 1043 | 128 | 325 | -915 | -718
0417-E | 1640 | 102 | 2365 | -1538 | 725
0451-W | 3288 | 102 | 2299 | -3186 | -989
0531-E | 364 | 54 | 293 | -310 | -71
0534-E | 444 | 143 | 2080 | -301 | 1636
0589-E | 2423 | 41 | 2080 | -2382 | -415
0816-E | 1988 | 1919 | 803 | -69 | -1185
0894-E | 2043 | 1919 | 1913 | -124 | -130
1252-W | 1823 | 41 | 3121 | -1782 | 1298
1381-W | 789 | 63 | 14 | -726 | -775
1564-W | 1197 | 1290 | 3852 | 93 | 2655
1576-W | 2472 | 91 | 2177 | -2381 | -295
1949-W | 1053 | 41 | 2048 | -1012 | 995
2449-W | 358 | 80 | 3187 | -278 | 2829
2732-W | 796 | 41 | 577 | -755 | -219
2740-W | 826 | 153 | 325 | -673 | -501
RANGE | 358 - 3288 | 41 - 1919 | 14 - 3852 | \- 3186 - 93 | -1254 - 2829
MEAN | 1489 | 387 | 1674 | -1046 | -184
Table 7: Observed versus predicted H$\alpha$ equivalent widths. Column (1) -
Cluster ID. Column (2) - Observed EW(H$\alpha$) from H21. Columns (3) and (4)
- value from best-fitting S2 and S3 models, respectively ($Z=0.014$). Columns
(5) and (6) give the differences between values in the columns which are
indicated in the header of the table.
### 6.3 Physical properties of the star clusters
We use the Bayesian inference tool BAYESPHOT and the formalism presented in
Section 5 in order to determine the extinction, mass and age of the star
clusters in our sample.
V-band extinction. Table 8 gives V-band extinctions (AV) from the literature
and our work. Column (2) gives the value from A17, which is derived via
deterministic models and $\chi^{2}$ minimization. Column (3) gives the median
value that is derived with BAYESPHOT and the Z=0.020 stochastic models of
Krumholz et al. (2015a, hereafter K15). Columns (4) and (5) give the median
values that are derived with BAYESPHOT and the Z=0.014 S2 and S3 GALAXEV-C
models, respectively. We find that the S2 models yield lower extinctions than
the S3 models. Thus, the extinction depends on the log(US) value of the
models. We also find good general agreement between the A17, K15 and S3
extinctions (within the error bars), and that the A17 extinctions tend to be
the largest.
Mass. Table 9 gives masses from the literature and our work. Although the
models in our pilot library use a coarse grid of cluster masses, we find that
the observed clusters are low mass, in agreement with previous results. The
mean cluster mass is 103 $M_{\odot}$ using the A17, K15 S2, and S3 models.
Thus, the value of log(US) does not affect the estimated mass value.
Age. Table 10 gives ages from the literature and our work. We find that the S2
models yield an older mean cluster age relative to the S3 models and that the
A17 models yield the youngest mean age (2 Myr). We also find that the
oldest/youngest clusters using K15 are the oldest/youngest clusters from S3 as
well. Finally, according to the S3 models, four clusters in our sample are 1
Myr. This is an age when nebular emission lines contribute significantly to
the V-band luminosity and the nebular continuum to the I-band luminosity.
ID | Av | $\Delta$Av
---|---|---
| A17 | K15 | S2 | S3 | (3)-(2) | (3)-(5) | (5)-(4)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
0011-E | 1.89 ${}^{+1.80}_{-1.98}$ | 1.25 ${}^{+0.24}_{-0.19}$ | 0.54 ${}^{+0.10}_{-0.09}$ | 1.32 ${}^{+0.16}_{-0.14}$ | -0.64 | -0.07 | 0.78
0090-E | 0.12 ${}^{+0.00}_{-0.19}$ | 0.17 ${}^{+0.11}_{-0.12}$ | 0.07 ${}^{+0.07}_{-0.05}$ | 0.12 ${}^{+0.14}_{-0.07}$ | 0.05 | 0.05 | 0.05
0093-E | 0.37 ${}^{+0.00}_{-0.46}$ | 0.31 ${}^{+0.16}_{-0.17}$ | 0.09 ${}^{+0.1}_{-0.07}$ | 0.24 ${}^{+0.16}_{-0.15}$ | -0.06 | 0.07 | 0.15
0383-W | 0.25 ${}^{+0.12}_{-0.31}$ | 0.21 ${}^{+0.14}_{-0.14}$ | 0.07 ${}^{+0.09}_{-0.05}$ | 0.16 ${}^{+0.15}_{-0.11}$ | -0.04 | 0.05 | 0.09
0417-E | 1.05 ${}^{+0.93}_{-1.15}$ | 0.83 ${}^{+0.23}_{-0.26}$ | 0.28 ${}^{+0.10}_{-0.12}$ | 0.73 ${}^{+0.19}_{-0.19}$ | -0.22 | 0.10 | 0.45
0451-W | 2.08 ${}^{+1.98}_{-2.17}$ | 1.39 ${}^{+0.26}_{-0.26}$ | 0.64 ${}^{+0.11}_{-0.12}$ | 1.72 ${}^{+0.28}_{-0.17}$ | -0.69 | -0.33 | 1.08
0531-E | 0.37 ${}^{+0.22}_{-0.46}$ | 0.33 ${}^{+0.17}_{-0.19}$ | 0.09 ${}^{+0.10}_{-0.07}$ | 0.19 ${}^{+0.16}_{-0.12}$ | -0.04 | 0.14 | 0.10
0534-E | 0.81 ${}^{+0.00}_{-0.90}$ | 0.71 ${}^{+0.21}_{-0.21}$ | 0.21 ${}^{+0.12}_{-0.09}$ | 0.61 ${}^{+0.17}_{-0.16}$ | -0.10 | 0.10 | 0.40
0589-E | 1.21 ${}^{+1.12}_{-1.36}$ | 0.99 ${}^{+0.24}_{-0.64}$ | 0.31 ${}^{+0.09}_{-0.12}$ | 0.97 ${}^{+0.23}_{-0.22}$ | -0.22 | 0.02 | 0.66
0816-E | 0.62 ${}^{+0.50}_{-0.71}$ | 0.38 ${}^{+0.19}_{-0.21}$ | 0.12 ${}^{+0.09}_{-0.07}$ | 0.35 ${}^{+0.17}_{-0.19}$ | -0.24 | 0.03 | 0.23
0894-E | 1.71 ${}^{+1.61}_{-1.86}$ | 1.02 ${}^{+0.18}_{-0.19}$ | 0.47 ${}^{+0.12}_{-0.09}$ | 1.11 ${}^{+0.16}_{-0.14}$ | -0.69 | -0.09 | 0.64
1381-W | 1.21 ${}^{+1.15}_{-1.30}$ | 0.33 ${}^{+0.40}_{-0.24}$ | 0.49 ${}^{+1.58}_{-0.18}$ | 1.34 ${}^{+0.21}_{-0.21}$ | -0.88 | -1.01 | 0.85
1252-W | 0.96 ${}^{+0.87}_{-1.05}$ | 0.66 ${}^{+0.28}_{-0.23}$ | 0.26 ${}^{+0.12}_{-0.10}$ | 0.64 ${}^{+0.21}_{-0.17}$ | -0.30 | 0.02 | 0.38
1564-W | 1.46 ${}^{+1.40}_{-1.61}$ | 1.04 ${}^{+0.21}_{-0.24}$ | 0.42 ${}^{+0.10}_{-0.09}$ | 0.94 ${}^{+0.24}_{-0.16}$ | -0.42 | 0.10 | 0.52
1576-W | 0.34 ${}^{+0.19}_{-0.40}$ | 0.24 ${}^{+0.16}_{-0.15}$ | 0.09 ${}^{+0.07}_{-0.07}$ | 0.16 ${}^{+0.15}_{-0.11}$ | -0.10 | 0.08 | 0.07
1949-W | 0.37 ${}^{+0.03}_{-0.43}$ | 0.24 ${}^{+0.16}_{-0.15}$ | 0.09 ${}^{+0.10}_{-0.07}$ | 0.19 ${}^{+0.16}_{-0.12}$ | -0.13 | 0.05 | 0.10
2449-W | 1.08 ${}^{+0.99}_{-1.24}$ | 0.94 ${}^{+0.19}_{-0.25}$ | 0.28 ${}^{+0.10}_{-0.12}$ | 0.80 ${}^{+0.24}_{-0.19}$ | -0.14 | 0.14 | 0.52
2732-W | 0.74 ${}^{+0.46}_{-0.90}$ | 0.54 ${}^{+0.26}_{-0.28}$ | 0.21 ${}^{+0.10}_{-0.12}$ | 0.42 ${}^{+0.29}_{-0.18}$ | -0.20 | 0.12 | 0.21
2740-W | 0.50 ${}^{+0.00}_{-0.78}$ | 0.66 ${}^{+0.21}_{-0.21}$ | 0.19 ${}^{+0.12}_{-0.10}$ | 0.71 ${}^{+0.16}_{-0.17}$ | 0.16 | -0.05 | 0.52
RANGE | 0.1 - 2.1 | 0.2 - 1.4 | 0.1 - 0.6 | 0.1 - 1.7 | -0.9 - 0.2 | -1.0 - 0.1 | 0.1 - 1.1
MEAN | 0.9 | 0.6 | 0.3 | 0.7 | -0.3 | -0.0 | 0.4
Table 8: V-band extinctions of star-clusters derived with different models, all of which include gas. The difference between values given by the different models is also given. Column (1) - Cluster ID. Column (2) - Deterministic models with Z=0.020 of A17. Column (3) - Stochastic models with Z=0.020 of Krumholz et al. (2015a). Columns (4) and (5) - Stochastic models with Z=0.014 and log(US)=-2 and log(US)=-3, respectively, presented in this work. Columns (6) to (8) - Differences between values in the columns which are indicated. Column (9) Comment. ID | log(M / M⊙) | $\Delta$log(M / M⊙)
---|---|---
| A17 | K15 | S2 | S3 | (3)-(2)
(1) | (2) | (3) | (4) | (5) | (6)
0011-E | 3.43${}^{+3.47}_{-3.39}$ | 2.59${}^{+0.40}_{-0.35}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | -0.44
0090-E | 2.77${}^{+2.86}_{-2.63}$ | 2.89${}^{+0.35}_{-0.55}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.47
0093-E | 2.88${}^{+2.95}_{-2.72}$ | 2.94${}^{+0.40}_{-0.55}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.46
0383-W | 2.71${}^{+2.85}_{-2.68}$ | 2.94${}^{+0.35}_{-0.55}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.58
0417-E | 3.25${}^{+3.30}_{-3.19}$ | 2.69${}^{+0.40}_{-0.40}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | -0.16
0451-W | 2.96${}^{+3.00}_{-2.92}$ | 2.54${}^{+0.45}_{-0.35}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.03
0531-E | 2.94${}^{+3.04}_{-2.82}$ | 2.89${}^{+0.45}_{-0.55}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.40
0534-E | 2.99${}^{+3.03}_{-2.90}$ | 2.64${}^{+0.40}_{-0.40}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.05
0589-E | 3.37${}^{+3.41}_{-3.28}$ | 3.04${}^{+0.50}_{-0.60}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.17
0816-E | 3.65${}^{+3.83}_{-3.62}$ | 3.49${}^{+0.25}_{-0.25}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.09
0894-E | 3.43${}^{+3.46}_{-3.25}$ | 2.54${}^{+0.40}_{-0.35}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | -0.49
1252-W | 3.19${}^{+3.24}_{-3.14}$ | 2.69${}^{+0.35}_{-0.40}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | -0.15
1381-W | 2.48${}^{+2.52}_{-2.45}$ | 2.34${}^{+0.30}_{-0.20}$ | 3.02${}^{+1.27}_{-0.02}$ | 3.0${}^{+0.16}_{-0.00}$ | 0.16
1564-W | 3.43${}^{+3.47}_{-3.28}$ | 2.74${}^{+0.40}_{-0.40}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | -0.29
1576-W | 2.95${}^{+3.06}_{-2.81}$ | 2.79${}^{+0.40}_{-0.50}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.24
1949-W | 2.81${}^{+2.91}_{-2.76}$ | 2.79${}^{+0.35}_{-0.45}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.33
2449-W | 3.31${}^{+3.34}_{-3.21}$ | 2.84${}^{+0.45}_{-0.50}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | -0.02
2732-W | 3.05${}^{+3.13}_{-3.00}$ | 2.89${}^{+0.35}_{-0.45}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.19
2740-W | 2.58${}^{+2.62}_{-2.26}$ | 2.69${}^{+0.40}_{-0.40}$ | 3.0${}^{+0.02}_{-0.00}$ | 3.0${}^{+0.02}_{-0.00}$ | 0.51
RANGE | 2.48 - 3.65 | 2.34 - 3.49 | 3.00 - 3.00 | 3.00 - 3.00 | -0.49 - 0.58
MEAN | 3.06 | 2.79 | 3.00 | 3.00 | 0.11
Table 9: Total mass in stars of star-cluster derived with different models, all of which include gas. The difference between values given by the different models is also given. Column (1) - Cluster ID. Column (2) - deterministic models with Z=0.020 A17. Column (3) - Stochastic models with Z=0.020 of Krumholz et al. (2015a). Columns (4) and (5) - stochastic models with (Z=0.014, log(US)=-2) and (Z=0.014, log(US)=-3) from this work, respectively. Columns (6) Differences between values in the columns which are indicated. ID | t / Myr | $\Delta$t / Myr
---|---|---
| A17 | K15 | S2 | S3 | (3)-(2) | (3)-(5) | (5)-(4)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
0011-E | 1.0${}^{+1.0}_{-1.0}$ | 0.7${}^{+1.9}_{-0.5}$ | 3.0${}^{+0.1}_{-0.0}$ | 1.0${}^{+0.0}_{-0.0}$ | -0.3 | -0.4 | -2.0
0090-E | 2.0${}^{+3.0}_{-1.0}$ | 5.6${}^{+1.6}_{-0.6}$ | 7.8${}^{+0.1}_{-0.1}$ | 7.8${}^{+0.1}_{-3.8}$ | 3.7 | -2.2 | 0.0
0093-E | 2.0${}^{+5.0}_{-1.0}$ | 5.1${}^{+1.9}_{-0.5}$ | 7.8${}^{+0.1}_{-0.1}$ | 4.0${}^{+3.8}_{-0.2}$ | 3.2 | 1.1 | -3.8
0383-W | 3.0${}^{+4.0}_{-2.0}$ | 5.6${}^{+1.8}_{-0.5}$ | 7.8${}^{+0.1}_{-0.1}$ | 7.6${}^{+0.4}_{-3.6}$ | 2.6 | -1.9 | -0.3
0417-E | 1.0${}^{+1.0}_{-1.0}$ | 2.0${}^{+1.5}_{-0.6}$ | 7.8${}^{+0.1}_{-0.1}$ | 3.0${}^{+0.9}_{-0.1}$ | 1.0 | -1.0 | -4.8
0451-W | 4.0${}^{+4.0}_{-4.0}$ | 0.7${}^{+7.1}_{-0.3}$ | 7.8${}^{+0.1}_{-0.1}$ | 1.0${}^{+3.0}_{-0.0}$ | -3.3 | -0.4 | -6.8
0531-E | 2.0${}^{+3.0}_{-1.0}$ | 4.3${}^{+1.9}_{-0.5}$ | 7.8${}^{+0.1}_{-0.1}$ | 3.9${}^{+0.1}_{-0.2}$ | 2.3 | 0.3 | -3.9
0534-E | 1.0${}^{+15.0}_{-1.0}$ | 2.2${}^{+1.6}_{-0.8}$ | 7.8${}^{+0.1}_{-0.1}$ | 3.9${}^{+0.1}_{-1.0}$ | 1.2 | -1.7 | -3.9
0589-E | 1.0${}^{+2.0}_{-1.0}$ | 2.7${}^{+8.5}_{-0.6}$ | 7.8${}^{+0.1}_{-0.1}$ | 3.0${}^{+1.0}_{-0.1}$ | 1.7 | -0.4 | -4.8
0816-E | 3.0${}^{+3.0}_{-1.0}$ | 2.5${}^{+1.6}_{-0.6}$ | 3.0${}^{+0.8}_{-0.1}$ | 3.0${}^{+0.1}_{-0.1}$ | -0.6 | -0.6 | 0.0
0894-E | 1.0${}^{+3.0}_{-1.0}$ | 0.5${}^{+1.9}_{-0.5}$ | 3.0${}^{+0.1}_{-0.1}$ | 1.0${}^{+0.0}_{-0.0}$ | -0.5 | -0.5 | -2.0
1252-W | 1.0${}^{+1.0}_{-1.0}$ | 1.7${}^{+1.8}_{-0.6}$ | 7.8${}^{+0.1}_{-0.1}$ | 3.0${}^{+0.9}_{-2.0}$ | 0.7 | -1.4 | -4.8
1381-W | 4.0${}^{+4.0}_{-4.0}$ | 4.3${}^{+1.3}_{-0.8}$ | 7.8${}^{+0.1}_{-5.1}$ | 7.8${}^{+0.1}_{-0.3}$ | 0.3 | -3.5 | 0.0
1564-W | 1.0${}^{+3.0}_{-1.0}$ | 1.3${}^{+2.1}_{-0.6}$ | 3.0${}^{+0.1}_{-0.0}$ | 1.0${}^{+2.0}_{-0.0}$ | 0.3 | 0.2 | -2.0
1576-W | 2.0${}^{+3.0}_{-1.0}$ | 3.0${}^{+1.9}_{-0.6}$ | 7.8${}^{+0.1}_{-0.1}$ | 3.9${}^{+0.1}_{-0.9}$ | 0.9 | -1.0 | -3.9
1949-W | 3.0${}^{+5.0}_{-2.0}$ | 3.5${}^{+1.6}_{-0.6}$ | 7.8${}^{+0.1}_{-0.1}$ | 4.0${}^{+3.8}_{-1.0}$ | 0.5 | -0.5 | -3.8
2449-W | 1.0${}^{+2.0}_{-1.0}$ | 2.2${}^{+1.8}_{-0.7}$ | 7.8${}^{+0.1}_{-0.1}$ | 3.0${}^{+1.0}_{-0.1}$ | 1.2 | -0.8 | -4.8
2732-W | 4.0${}^{+5.0}_{-2.0}$ | 2.2${}^{+1.9}_{-0.4}$ | 7.8${}^{+0.1}_{-0.1}$ | 3.0${}^{+1.0}_{-2.0}$ | -1.8 | -0.8 | -4.8
2740-W | 5.0${}^{+6.0}_{-4.0}$ | 3.9${}^{+2.3}_{-0.4}$ | 7.8${}^{+0.1}_{-0.1}$ | 7.8${}^{+0.1}_{-3.9}$ | -1.1 | -3.9 | 0.0
RANGE | 1 - 5 | 0.5 - 5.6 | 2.9 - 7.8 | 1.0 - 7.8 | -3.3 - 3.6 | -3.9 - 1.0 | -6.8 - 0.0
MEAN | 2.2 | 2.8 | 6.79 | 3.8 | 0.6 | -1.0 | -3.0
Table 10: Ages of star-cluster derived with different models, all of which
include gas. The difference between values given by the different models is
also given. Column (1) - Cluster ID. Column (2) Deterministic models with
Z=0.020 (A17). Column (2) - Stochastic models with Z=0.020 of Krumholz et al.
(2015a). Columns (4) and (5) - Stochastic models with Z=0.014 and log(US)=-2
and log(US)=-3, respectively, presented in this work. Columns (6) to (8) -
Differences between values in the columns which are indicated.
### 6.4 Well versus poorly constrained solutions
For clusters 816-W (top row) and 2732-W (bottom row), Figure 12 shows the
extinction PDFs corresponding to the Z = 0.014 / S3 / GALAXEV-C models (left
column) and the Z=0.020 / log(Us)=-3 / SLUG models (middle column). For
2732-W, note that GALAXEV-C and SLUG yield single- and multiple-valued
extinction PDFs, respectively. That one PDF is single-peaked and the other one
is not is attributed to differences between the GALAXEV-C and SLUG libraries.
In particular, as explained in the introduction, the SLUG models do not
include the effect of the stochastic variation in the shape of the ionizing
continuum, on the nebular emission.
The right column of Figure 12 shows the SLUG age PDFs corresponding the the
above two clusters. We only show the SLUG results because the pilot GALAXEV-C
age grid is very coarse. The age PDF is single-peaked for cluster 816-W and
multi-peaked for cluster 2732-W. A discussion of the PDFs for the whole sample
of clusters is provided in Appendix C.
Figure 12: Left column–. The Z=0.014/log(Us)=-3/GALAXEV-C V-band extinction
PDFs of clusters 816-E (top panel) and 2732-W (bottom panel). The blue, red,
and green lines give the 16th, 50th and 84th percentiles, respectively. Middle
column–. Similar to the left column but we show the Z=0.020/log(Us)=-3/SLUG
V-band extinction PDFs. Right column–. Z=0.020/log(Us)=-3/SLUG age PDFs.
## 7 Summary and Conclusions
1. 1.
We present a pilot GALAXEV-C library of synthetic HST-equivalent NUV, U, B, V,
and I photometry of star clusters that accounts for the stochastic sampling of
the stellar IMF and the contribution of the ionized gas and dust mixed with
the ionized gas (Section 2). The library uses the spectra that are presented
in Vidal-García et al. (in prep.) and includes models for clusters with
initial masses, $M_{i}=10^{3}$, $10^{4}$, and $10^{5}$ $M_{\odot}$; ages,
$t=1$, 3, 4, and 8 Myr; metallicities, $Z=0.002$ and $Z=0.014$ (solar); and
ionisation parameters, log(U${}_{\rm S})=-2$ (S2 models) and -3 (S3 models).
We compare the stochastic models to corresponding deterministic models
(Section 3.4); and to HST LEGUS and H$\alpha$-LEGUS observations of star
clusters in galaxy NGC 7793 that are isolated, have compact H$\alpha$
morphologies, $Z\sim 0.014$ (Figure 8), and deterministic masses and ages of
$<10^{4}$ $M_{\odot}$ and $\leq 10$ Myr, respectively. We determine the V-band
extinctions, masses, and ages of these clusters using the stochastic models
(Section 5). We compare the cluster properties derived with deterministic
models that are published in A17, and derived with independent GALAXEV-C and
SLUG (K15) stochastic models (Section 6.3).
2. 2.
For GALAXEV magnitudes that only account for the stars: a) the absolute value
of the residual, deterministic mag - median stochastic mag, can be $\geq 0.5$
mag, even for $M_{i}=10^{5}$ $M_{\odot}$ (Table 4); and b) the largest spread
of the stochastic models occurs at 3 and 4 Myr, when Wolf-Rayet stars are
present (Figure 2).
3. 3.
For $M_{i}=10^{5}$ $M_{\odot}$: a) the median stochastic mag with gas can be
$>$1.0 mag more luminous than the median stochastic mag without gas (Table 2);
and b) the nebular emission lines can contribute with $>50\%$ and $>30\%$ to
the total emission in the V and I bands, respectively (Figure 4).
4. 4.
Regarding age-dating OB clusters via deterministic tracks in the U - B vs. V -
I diagram, we find that this method leads to highly uncertain ages at
$Z=0.014$ for $M_{i}\sim 10^{3}$ $M_{\odot}$ (Figures 5 \- 7) and $Z=0.002$
for all masses in the stochastic library (Figures 14 \- 16).
5. 5.
Also regarding the U - B vs. V - I diagram, we find that at $Z=0.014$, a small
fraction of models with $M_{i}\sim 10^{3}$ and $M_{i}\sim 10^{4}$ $M_{\odot}$
are located far from their corresponding deterministic predictions and none of
our observations are found near these outlier models. This is expected given
that our observational sample is small and according to stochastic models the
corresponding outlier clusters have a low probability of being created.
6. 6.
Regarding the SED fitting, we find good agreement between the best-fitting S3
model and the observations (Figure 11, Table 6).
7. 7.
We derive the extinctions in the V-band, masses, and ages of the star clusters
in our sample using two independent libraries of stochastic models with gas,
the K15 library and our pilot library. We compare the results with those of
A17, which are based on deterministic models (Tables 8 to 10).
8. 8.
Regarding the extinction, we find that the GALAXEV-C AV value is
systematically lower for log(US)=-2 than for log(US)=-3. We also find that the
A17, K15, and S3 extinctions are in general agreement (within the errors), and
that the A17 extinctions tend to be the largest. Finally, we find that for a
given cluster, the extinction PDF can be single-peaked for GALAXEV-C and
multi-peaked for SLUG and vise versa, which is attributed to differences in
the stochastic libraries.
9. 9.
Regarding the masses, we find that the observed clusters are low mass, in
agreement with the deterministic predictions (Table 9).
10. 10.
Regarding the ages, we find that the S2 models tend to yield ages relative to
the S3 models, and that the A17 models yield the youngest mean age (2 Myr;
Table 10). We also find that in several cases, the age PDF presents multiple
peaks.
11. 11.
Regarding models versus nature, we recall that for a multinomial distribution,
the standard deviations of the different mass bins are not independent from
each other, whereas we have no reason to believe that the same correlation
between the standard deviations is true in nature. This is why it is important
to observationally characterize the true variance in nature.
12. 12.
An extension of the pilot library to other metallicities and ages is near
completion and can be made available upon request to AVG, who is a co-author
of this work.
## Appendix A Is 220 realizations enough?
The left-panel of Figure 13 is similar to Figure 1 but includes only the
number of realizations that are used in this work. These Figures show that
reducing the number of realizations from 1000 to 220 has no significant effect
on the estimated mean and standard deviation of the number of stars in each
mass bin. Increasing the number of realizations only changes the uncertainty
of the standard deviation estimates.
Figure 13: Left-panel—Same as Figure 1 but for 220 realizations. Right-
panel—Expected mean and standard deviations for a number of stars that is
fixed such that the expected mass is $M_{i}$.
If we fix the total number of stars rather than the total mass, then the
distribution of bin counts will follow a multinomial distribution with the
desired expected total mass, as shown in the right panel of Figure 13. We can
use the known statistics of this multinomial distribution to approximate the
standard deviations for the case in which the total mass is strictly fixed
(compare the standard deviations in the left and right panels). The standard
deviations estimated from the stochastically-generated distributions (left
panel) show a somewhat larger standard deviation due to having a variable
total number of stars. An in depth discussion about mass-limited sampling
versus other sampling procedures can be found in Cerviño et al. (2013).
The importance of computing 220 realizations of the IMF is thus, not to
characterize the spread in observables predicted by the stochastic models
(because in principle, that can be calculated analytically) but to fill in the
space between random realizations in diagrams such as the colour-colour
diagram, which is useful for comparison to the observations.
## Appendix B Additional colour-colour diagrams.
In Figures 5 \- 7 and Figure 10, we compare Z=0.014 deterministic and
stochastic predictions in the U - B vs. V - I diagram for cases: only-stars,
log(Us)=-3, log(Us)=-2, and the three previous cases combined, respectively.
Figures 14 \- 17 are similar but for Z=0.002. Figures 14 \- 16 are discussed
in Section 3, while Figure 17 is discussed in Section 5.
Figure 14: Similar to Figure 5 but for $Z=0.002$. Figure 15: Similar to Figure
6 but for $Z=0.002$. Figure 16: Similar to Figure 6 but for $Z=0.002$. Figure
17: Similar to Figure 10 but for $Z=0.002$. In this case, the Yggdrasil
predictions are for $Z=0.004$.
## Appendix C Extinction and age PDF.
For all clusters in our sample, the left-three panels of Figures 18 \- 21 show
the V-band extinction PDFs from GALAXEV-C for models with log(U${}_{\rm
S})=-3$ and $Z=0.014$ (left column) and from SLUG for models with
log(U${}_{\rm S})=-3$ and $Z=0.020$ (middle column); and the age PDFs from
SLUG (right column). The code and cluster ID is indicated in the column
titles. The blue, red, and green lines give the 16th, 50th and 84th
percentiles respectively. In each figure, the clusters are arranged in order
of increasing age. The right-five panels of Figures 18 \- 21 show the NUV, U,
B, V, and I postage stamps of the clusters, from left to right.
Although the top three clusters of Figure 18 have the youngest median SLUG
ages, their age PDFs show a second peak at an older age. These three clusters
also have high Av values (for the sample), as expected if the surrounding dust
has not been as affected by the ionising photons from massive stars compared
to other clusters. The high extinction of the youngest clusters leads to
NUV/U/B-band postage stamps with low number of counts because reddening due to
dust increases as wavelength decreases.
Figures 18 \- 21 also show that for some clusters the extinction PDF is multi-
peaked according to one code but single-peaked according to the other. For
instance, for cluster 451-W in Figure 18 GALAXEV-C yields two peaks while for
cluster 589-E in Figure 20 it is the opposite. Cases where both codes yield
single-peaked PDFs (e.g., cluster 534-E) or multiple-peaked PDFs (e.g.,
cluster 2732-W) also occur. The different shapes of the extinction PDFs is
attributed to the different ways of modelling the ionising continuum in the
GALAXEV- and SLUG libraries (see the introduction of this paper).
Figure 18: Left–. PDFs of V-band extinction (left and middle columns) and age
(right column) for five clusters in our sample. The left and middle columns
show the GALAXEV-C and SLUG PDFs, respectively, while the right column shows
the SLUG age PDF. The clusters are arranged in order of increasing SLUG age.
In each PDF sub-panel, we give the median value (50th percentile) of the
extinction or age. Right–. Postage stamps of the clusters in the NUV, U, B, V
and I LEGUS bands, from left to right. We use a logarithmic scale from 0 to 10
and SAO-ds9’s colour scale "b", such that blue corresponds to the lowest
number of counts and yellow corresponds to pixels with 10 counts or more.
Figure 19: Similar to Figure 18 but for five different clusters.
Figure 20: Similar to Figure 18 but for five different clusters. Note that
cluster 1381-W, which is $\sim$4 Myr and has a high Av value (for the sample)
according to GALEX-C and the second PDF peak of SLUG, has postage stamps that
follow a similar pattern to that of the youngest ($<1$ Myr) high Av clusters
of Figure 18.
Figure 21: Similar to Figure 18 but for five different clusters.
## Acknowledgements
We thank the jury of ROD’s M.S. thesis (E. Terlevich, S. Sánchez, L. Aguilar,
S. Srinivasan) as well as M. Cerviño and B. Elmegreen for comments and
suggestions which have greatly improved the quality of this paper. ROD and AW
acknowledge the support of UNAM via grant agreement PAPIIT no. IA-102120. AVG,
SC and GB acknowledge support from the ERC via an Advanced Grant under grant
agreement no. 321323-NEOGAL. AVG also aknowledges support from the ERC
Advanced Grant MIST (FP7/2017-2022, No 742719). MRK acknowledges support from
the Australian Research Council’s Future Fellowship funding scheme, award
FT180100375, and from resources and services provided by the National
Computational Infrastructure (NCI), which is supported by the Australian
Government.
## Appendix D Data availability
The HST data underlying this article are available in the Mikulski Archive for
Space Telescopes at
https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html, and can be
accessed with the dataset identifiers 13364 and 13773. LEGUS high level
science products can be found at
https://archive.stsci.edu/prepds/legus/dataproducts-public.html
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|
# CoNO: Complex Neural Operator for Continous Dynamical Physical Systems
Karn Tiwari
Department of Electrical Communication Engineering
Indian Institute of Science, Bangalore
Bengaluru, 560012, India
<EMAIL_ADDRESS>
&N M Anoop Krishnan
Yardi School of Artificial Intelligence
Indian Institute of Technology, Delhi
New Delhi, 110016, India
<EMAIL_ADDRESS>
&Prathosh A P
Department of Electrical Communication Engineering
Indian Institute of Science, Bangalore
Bengaluru, 560012, India
<EMAIL_ADDRESS>
###### Abstract
Neural operators extend data-driven models to map between infinite-dimensional
functional spaces. While these operators perform effectively in either the
time or frequency domain, their performance may be limited when applied to
non-stationary spatial or temporal signals whose frequency characteristics
change with time. Here, we introduce a Complex Neural Operator (CoNO) that
parameterizes the integral kernel using Fractional Fourier Transform (FrFT),
better representing non-stationary signals in a complex-valued domain.
Theoretically, we prove the universal approximation capability of CoNO. We
perform an extensive empirical evaluation of CoNO on seven challenging partial
differential equations (PDEs), including regular grids, structured meshes, and
point clouds. Empirically, CoNO consistently attains state-of-the-art
performance, showcasing an average relative gain of 10.9%. Further, CoNO
exhibits superior performance, outperforming all other models in additional
tasks such as zero-shot super-resolution and robustness to noise. CoNO also
exhibits the ability to learn from small amounts of data—giving the same
performance as the next best model with just 60% of the training data.
Altogether, CoNO presents a robust and superior model for modeling continuous
dynamical systems, providing a fillip to scientific machine learning.
## 1 Introduction
Continuum systems span various scientific and engineering fields, such as
robotics, biological systems, climate modeling, and fluid dynamics, among
others [15]. These systems are represented using Partial Differential
Equations (PDEs), the solution of which provides the system’s time evolution.
The solution of PDEs necessitates the identification of an optimal solution
operator, which maps across functional spaces while including the initial
conditions and coefficients. Traditionally, numerical methods, such as finite
element and spectral methods, have been employed to approximate the solution
operator for PDEs. However, these approaches often incur high computational
costs and exhibit limited adaptability to arbitrary resolutions and geometries
[56]. Such high computational costs of these numerical methods inhibit the
real-time prediction crucial in weather forecasting and robotics.
Figure 1: FrFT heatmaps illustrating the temporal-frequency characteristics
of the 2D Navier-Stokes equation for varying values of $\alpha$. Each subplot
represents the magnitude of the transformed frequency content over time,
obtained by applying the FrFT and then flattened 2D frequency map of the
Navier-Stokes equation. Different subplots correspond to fractional orders
$\alpha$, highlighting the diverse spectral behaviors captured by the FrFT
across both the temporal and frequency domains. Note that $\alpha=0$
represents the time domain while $\alpha=1$ represents the frequency domain.
Recently, in the realm of scientific machine learning, neural networks present
a promising alternative to solve PDEs through a data-driven approach.
Specifically, neural operators represent an extension of neural networks,
facilitating the mapping between infinite-dimensional functional spaces and
serving as a universal approximation of the operator. Notably, these operators
learn the functionals without any prior knowledge of the underlying PDE,
relying solely on data-driven training, leading to faster inference times than
the traditional methods [41, 35].
Among different neural operators, the Fourier Neural Operator (FNO) [41] has
gained widespread recognition due to its ability to navigate the infinite-
dimensional functional space via kernel integral operations in the Fourier
domain. Renowned for its versatility, FNO has found successful applications
across diverse domains, including weather forecasting [36], biomedical
surrogate modeling [24], and expediting sampling processes in diffusion models
[72]. Nevertheless, recent investigations have brought to light some specific
challenges associated with FNO, including aliasing errors [18], a departure
from continuous-discrete equivalence [6], susceptibility to vanishing gradient
problems with an increasing number of layers [58] and susceptibility to poor
performance when presented with noisy data [8]. FNO also exhibits suboptimal
performance on time-dependent PDEs [40, 8] exemplified by the turbulent
Navier-Stokes equations (chaotic flow). Notably, FNO encounters difficulties
in making predictions over extended time horizons, constrained by the
limitations of the Fourier Transform (FT), which is tailored for stationary
signals and needs more time-frequency characteristics—as it decomposes the
function based on monochromatic basis function and is unable to detect
oscillatory patterns present in chirps signal [57]. For instance, the change
of frequency with time in the Navier-Stokes equation is non-stationary and
nonlinear (see Fig. 1 and App. Fig. 6), leading to the concentration of
spectrum around low frequency in the Fourier Transform. Note that the Navier-
Stokes equation holds significant relevance across diverse practical domains,
including but not limited to aerodynamics, weather forecasting, and
oceanography. Addressing these challenges through a data-driven methodology
underscores the pressing need for alternative operator formulation that has
ability to learn non-stationary signals.
A generalization of the FT for non-stationary signals, where spectra evolve
with time, by employing fractional FT (FrFT) can potentially improve the
prediction of long-horizon dynamics, particularly in handling highly nonlinear
and rapidly changing time-frequency characteristics [11]. FrFT improves the
efficiency of the classical FT by using a general parameterized adjustable
orthonormal basis. This basis allows us to break down a chirp function into
simpler parts. The adjustable parameter helps efficiently reconstruct the
chirp functions [57]. The FrFT generalizes FT by rotating the signal between
the time and frequency domains, transitioning from real to complex domain, and
incorporating phase information [51], resulting in a complex representation of
the signal. However, the complex-valued representations have remained
unexplored in the operator learning paradigm, although a large literature
exists on complex-valued neural networks (CVNNs) [48]. CVNNs offer easier
optimization [48], faster convergence during training [4, 14, 66], better
generalization [30], data efficiency, and resilience to noise [14, 66]. Thus,
combining the operator paradigm with CVNNs can potentially result in an
architecture that exploits both best, leading to a model that provides
improved long-horizon prediction.
Our Contributions. Motivated by the above observations, we introduce the
Complex Neural Operator (CoNO). The major contributions of the present work
are as follows.
1. 1.
Novel architecture. CoNO represents the first instance that performs operator
learning employing a CVNN that parameterizes the integral kernel through FrFT,
enabling richer information through phase details learnable by CVNNs suitable
for non-stationary signals. Table 1 shows the comprehensive comparison of CoNO
with current SOTA operators.
2. 2.
Universal approximation. We prove theoretically that CoNO follows the
universal approximation theorem for operators (see Thm 4.3).
3. 3.
Superior performance. We show that CoNO consistently exhibits superior
performance compared to SOTA operators, always ranking among the top two on
all the datasets with an average gain of 10.9% as presented in Table 2.
4. 4.
Data efficiency and robustness to noise. CoNO demonstrates superior
performance even with limited samples, data instances, and training epochs.
CoNO provides more robustness when noise is injected into the training data
compared to the existing methods and performs better than SOTA methods even
under 0.1% data noise.
## 2 Related Work
Table 1: A comprehensive comparative analysis of features of state-of-the-art
operators with CoNO. "*" denotes not applicable.
Features | FNO | LSM | CoNO (Ours)
---|---|---|---
Integral Kernel | Frequency | Spatial | Spatial-Frequency
Elementary function | Sine | * | Linear Frequency Modulation
Representation | Real | Real | Complex
Pertinent Signal | Stationary | * | Time Varying Signal
$\alpha$ Parameter | Fixed ($90^{o}$) | Fixed ($0^{o}$) | Not Fixed (Learnable)
Applicability | Individual | Individual | Unified
Neural Operators (NO): Neural operators have shown promise in solving PDEs in
a data-driven fashion [35]. Lu et al. [45] introduced DeepOnet, theoretically
establishing its universal approximation capabilities. The DeepOnet
architecture comprises a branch network and a trunk network, with the former
dedicated to learning the input function operator and the latter tasked with
learning the function space onto which it is projected. Another famous
architecture, the FNO, proposed by Li et al. [41], utilizes a frequency domain
method. FNO incorporates Fourier kernel-based integral transformations through
fast Fourier transform and projection blocks. An enhanced version, F-FNO [58],
improves upon the original FNO by integrating distinct spectral mixing and
residual connections. Subsequently, various kernel integral neural operators
based on transformations have emerged. For example, Fanaskov and Oseledets
[18] introduced spectral methods, such as Chebyshev and Fourier series, to
mitigate aliasing errors and enhance clarity in FNO outputs. Li et al. [42]
incorporated specialized learnable transforms to facilitate operator learning
on irregular domains, achieved by transforming them into uniform latent
meshes.
Kovachki et al. [35] demonstrated that the well-known attention mechanism can
be seen as a particular case of neural operator learning the integral kernel
specifically applied to address irregular meshes for solving PDEs, a
characteristic managed by Geo-FNO [42] differently. Cao [10] employed two
self-attention mechanism-based operators without utilizing a softmax layer,
accompanied by a theoretical interpretation. Recently, Hao et al. [26]
introduced the GNOT operator, featuring a linear cross-attention block
designed to enhance the encoding of irregular geometries. However,
transformer-based operators are susceptible to issues arising from limited
data samples, displaying a tendency to overfit easily on training data without
exhibiting robust generalization. In addressing the challenges posed by
multiscale PDEs, Liu et al. [43] proposed a hierarchical attention-based
operator.
Fractional Fourier Transform (FrFT): The FrFT represents a generalization of
the classical Fourier Transform (FT), providing robust capabilities for
spectral analysis. It achieves this by facilitating the transformation of
input signals into an intermediate domain between the time and frequency
domains, thereby establishing a time-frequency representation [51]. FrFT is
particularly effective in processing non-stationary signals, commonly called
"chirp signals" or signals exhibiting frequency derivatives over time. App.
Fig. 7 illustrates the efficacy of employing the FrFT for noise filtering
within the signal spectrum through the rotation of the fractional order axis.
In contrast to FrFT, alternative signal processing techniques such as wavelet
or Gabor transformation don’t provide a joint signal energy distribution
across both time and frequency domains. Additionally, these alternatives often
grapple with challenges related to high computational complexity, the
selection of wavelet functions, and sensitivity to signal noise. FrFT, with
its distinct advantages, has found applications in various domains, ranging
from solving differential equations [46], wireless communication [39],
biomedical signal processing [23], image encryption and image compression [47]
etc. Yu et al. [71] demonstrated the benefits of employing FrFT over
conventional convolution across various tasks in computer vision, including
segmentation, object detection, classification, super-resolution, etc.
Complex Valued Neural Networks (CVNNs): A CVNN incorporates complex-valued
parameters and variables within its architecture, enabling the representation
of both magnitude and phase information in the neural networks [37, 13]. The
utilization of CVNNs encompasses diverse advantages, extending from biological
to signal processing applications. Danihelka et al. [14] has demonstrated that
complex numbers enhance efficiency and stability in information retrieval
processes. Additionally, Arjovsky et al. [4] have introduced complex recurrent
neural networks (RNNs), highlighting that unitary matrices offer a more
intricate representation, thereby mitigating issues associated with vanishing
and exploding gradient problems. In image processing, phase information is a
critical descriptor, offering detailed insights about image shape,
orientation, and edges. Oppenheim and Lim [50] showed the information
encapsulated in the phase of an image proves sufficient for the recovery of a
substantial proportion of the encoded magnitude information.
CVNNs have been a research focus for a long time [20, 31, 29, 49]. Recently,
Geuchen et al. [22] established the universal approximation capabilities of
CVNNs for deep, narrow architectures, significantly contributing to
understanding their expressive power. Prior works [54, 60, 4, 14, 12, 21] have
made noteworthy strides in both experimental and theoretical aspects of CVNNs.
In the domain of computer vision, the utilization of scattering
transformation-based complex neural networks has demonstrated considerable
promise, showcasing their ability to achieve performance on par with real-
valued counterparts while employing significantly fewer parameters [33, 67,
53]. In NLP, complex embeddings have been incorporated for semantic and
phonetic processing of natural languages [59, 16]. In contrast, Yang et al.
[70] and Dong et al. [17] showcased the advantages of employing CVNNs
transformers for the NLP community. Despite these notable applications across
various domains, exploring the applicability of CVNNs within the SciML
community is still limited.
## 3 Preliminaries
Problem Setting: We have followed and adopted the notations in Li et al. [41].
Let $D$ denote a bounded open set as $D\subset\mathbb{R}^{d}$, with
$A=A(D;\mathbb{R}^{d_{a}})$ and $U=U(D;\mathbb{R}^{d_{u}})$ as separable
Banach spaces of functions representing elements in $\mathbb{R}^{d_{a}}$ and
$\mathbb{R}^{d_{u}}$, respectively. Consider
$\mathcal{G}^{\dagger}:A\rightarrow U$ to be a nonlinear surrogate mapping
arising from the solution operator for a parametric PDE. It is assumed that
there is access to i.i.d. observations ${(a_{j},u_{j})}_{j=1}^{N}$, where
$a_{j}\sim\mu$, drawn from the underlying probability measure $\mu$ supported
on $A$, and $u_{j}=\mathcal{G}^{\dagger}(a_{j})$.
The objective of operator learning is to construct an approximation for
$\mathcal{G}^{\dagger}$ via a parametric mapping
$\mathcal{G}:A\times\Theta\rightarrow U$, or equivalently,
$\mathcal{G}_{\theta}:A\rightarrow U,\theta\in\Theta$, within a finite-
dimensional parameter space $\Theta$. The aim is to select
$\theta^{\dagger}\in\Theta$ such that
$\mathcal{G}(\cdot,\theta^{\dagger})=\mathcal{G}_{\theta}^{\dagger}\approx\mathcal{G}^{\dagger}$.
This framework facilitates learning in infinite dimensional spaces as the
solution to the optimization problem in Eq. 1 constructed using a loss
function $\mathcal{L}:U\times U\rightarrow\mathbb{R}$.
$\min_{\theta\in\Theta}\mathbb{E}_{a\sim\mu}\left[\mathcal{L}(\mathcal{G}(a,\theta),\mathcal{G}^{\dagger}(a))\right],$
(1)
The optimization problem is solved in operator learning frameworks using a
data-driven empirical approximation of the loss function akin to the regular
supervised learning approach using train-test observations. Usually,
$\mathcal{G}_{\theta}$ is parameterized using deep neural networks.
Fractional Fourier Transform (FrFT): Inspired by the kernel formulation for
solving linear PDEs using Green’s function, we construct the model
$\mathcal{G}_{\theta}$ employing an iterative approach to map an input
function $a$ to an output function $u$ within the CoNO framework as detailed
in Sec. 4. In CoNO, the kernel integral is formulated using the FrFT with a
learnable order. The fractional transformation of order $\alpha$
($\alpha\in\mathbb{R}$) is a parameter of the Fractional Fourier Transform
(FrFT) that corresponds to the $\alpha^{th}$ power of the Fourier Transform
(FT) (denote by $\mathcal{F}^{\alpha}$).
###### Definition 3.1 (FrFT).
The fractional Fourier transform with angle $\alpha$ of a signal $f(y)$ is
defined as:
$\mathcal{F}^{\alpha}(f)(m)=\int_{-\infty}^{\infty}f(y)\mathcal{K}_{\alpha}(m,y)\,dy,$
(2)
where,
$\mathcal{K}_{\alpha}(m,y)=\begin{cases}c(\alpha)\exp\\{j\pi
a(\alpha)[(m^{2}+y^{2})-2b(\alpha)my]\\}&\text{if }\alpha\neq
n\pi\mathbb{Z},\\\ \delta(m-y)&\text{if }\alpha=2\pi\mathbb{Z},\\\
\delta(m+y)&\text{if }\alpha+\pi=2\pi\mathbb{Z}\end{cases}$
where,
$a(\alpha)=\cot\alpha,\>b(\alpha)=\csc\alpha,\>\text{and}\>c(\alpha)=\sqrt{1-j\cot\alpha}$
and $\mathcal{F}^{\alpha}(f)(m)$ denotes the $m^{th}$ fractional fourier
coefficient of order $\alpha$ of $f$.
###### Remark 3.2.
For Eq. 2, it reduces to Standard Fourier Transform (FT) when
$\alpha=\frac{\pi}{2}$, then $\cot{\alpha}=0$ and $\csc{\alpha}=1$.
Complex Valued Neural Networks (CVNNs): In the CoNO framework, the integration
of kernels in the operator $\mathcal{G}_{\theta}$ is performed within the
complex-valued domain using CVNNs. A CVNN is modeled as real and imaginary
parts or magnitude and phases as follows [59]:
$z=x+jy=|z|e^{j\angle z}$ (3)
where $j=\sqrt{-1}$ is imaginary unit, $x$ and $y$ are the real and imaginary
parts of $z$, and $|z|$ and $\angle z$ are the magnitude and phase of $z$.
###### Definition 3.3 (Complex Valued Activation).
Let $z\in\mathbb{C}$ be a complex number with real part $\text{Re}(z)$ and
imaginary part $\text{Im}(z)$. The Complex GeLU (CGeLU) activation function is
defined as follows:
$\text{CGeLU}(z)=\text{GeLU}(\text{Re}(z))+j\cdot\text{GeLU}(\text{Im}(z)),$
(4)
where $\text{GeLU}(\cdot)$ is the Gaussian Error Linear Unit activation
function [28]. The CGeLU activation function satisfies the Cauchy-Riemann
equations when the real and imaginary parts of $z$ are strictly positive or
negative.
Complex Valued Back-propagation: Complex-valued back-propagation involves
extending traditional back-propagation algorithms to handle complex numbers,
utilizing mathematical tools like Wirtinger calculus [3] which enable training
neural networks with complex-valued weights and activation, allowing for the
modeling of intricate relationships within complex data domains [5, 13].
## 4 Complex Neural Operator (CoNO)
### 4.1 Proposed Method
Here, we introduce our framework, named the Complex Neural Operator (CoNO)
depicted in Fig. 2.
Figure 2: CoNO Architecture Overview. (Top) (1) The input function $a(x)$
undergoes a deformation $\phi^{-1}$ to convert an irregular mesh into a
uniform mesh. (2) The deformed input is then lifted to a higher dimension in
the channel space using a neural network. (3) Apply iterative CoNO layer in
complex domain consisting of fractional integral kernel. (4) Then, the output
is projected back on the lower dimension in channel space using neural
network. (5) Solution $u(x)$ is obtained by passing through the deformation
$\phi$. (Bottom) Zoomed version of FrFT integral kernel defined in Eq. 6 with
learnable parameters $R^{\alpha}$, $R^{\alpha^{\prime}}$, $W$ and fractional
order $\alpha$ and $\alpha^{\prime}$.
Overview: Suppose $\mathcal{G}_{\theta}$ represents an iterative sequence of
functions. The operator $\mathcal{G}_{\theta}:A\rightarrow U$, where
$A=A(D;\mathbb{R}^{d_{a}})$ and $U=U(D;\mathbb{R}^{d_{u}})$ are separable
Banach spaces of functions, representing elements in $\mathbb{R}^{d_{a}}$ and
$\mathbb{R}^{d_{u}}$, respectively. Our goal is to construct the operator in a
structure-preserving manner where we band-limit a function over a given
spectrum [62], thus preserving complex continuous-discrete equivalence such
that Shannon-Whittaker-Kotel’nikov theorem is obeyed for all continuous
operations [61]. With this, our operator CoNO, denoted by
$\mathcal{G}_{\theta}$ is defined as follows:
$\mathcal{G}_{\theta}=\phi\circ\mathcal{Q}\circ\mathcal{L}_{l}\circ\ldots\circ\
\mathcal{L}_{1}\circ\mathcal{P}\circ\phi^{-1},$ (5)
where $\circ$ denotes composition. On irregular geometries, $\phi$ represents
the function modeled by a neural network that maps the irregular domain into a
regular latent mesh for operator learning. The operators
$\mathcal{P}:\mathbb{R}^{d_{a}}\rightarrow\mathbb{R}^{d_{1}}$ and
$\mathcal{Q}:\mathbb{R}^{d_{l}}\rightarrow\mathbb{R}^{d_{u}}$ correspond to
the lifting and projection operations, encoding lower-dimensional spaces into
higher-dimensional spaces or vice versa which helps in converting the non-
linear dynamics into linear dynamics inspired by Koopman Operator [7]. The
operator consists of $l$ layers of non-linear integral operators
$\sigma(\mathcal{L}_{l})$ (Eq. 6), where $\sigma$ is an activation function
which is applied from layer $\mathcal{L}_{i}$, where $i$ belongs to $1$ to
$l-1$ to introduce non-linearity in operators, akin to standard neural
networks, to learn highly nonlinear operators, and $\theta$ denotes all the
learnable parameters of the operator.
Non-linear Operator Layer, $\mathcal{L}$: In CoNO, the complex kernel operator
is defined as follows:
$v^{l+1}(x)=\sigma\left(Wv^{l}(x)+b^{l}+\mathcal{K}(a,\alpha)v^{l}(x)\right)\quad\forall
x\in D$ (6)
where: $v^{l}$ is the representation at layer $l$ and
$v^{l}\in\mathbb{C}^{d_{l}}$,
$\mathcal{K}:V^{l}\times\Theta_{K}\to\mathcal{L}(V^{l+1}(D;\mathbb{C}^{d_{l}}),V^{l+1}(D;\mathbb{C}^{d_{l+1}}))$
maps to bounded linear operators on $V^{l+1}(D;\mathbb{R}^{d_{l+1}})$ as
integral operator is linear and bounded operator as shown in App. section B,
$\sigma$ is a non-linear complex activation function, $W$ is a linear
transformation, $W:\mathbb{C}^{d_{l}}\rightarrow\mathbb{C}^{d_{l+1}}$,
$\mathcal{K}(a,\phi)$ is an integral kernel parameterized by CVNN (Eq. 7), $a$
is a complex input function, and $\alpha$ belongs to the parameter space of
$\mathcal{G}_{\theta}$ and $b$ is the bias.
Fractional Integral Operator, $\mathcal{K}(a,\alpha)$: In CoNO, the integral
operator is defined as, $\mathcal{K}(a,\alpha):V^{l}(D)\rightarrow V^{l}(D)$
as follows:
$\mathcal{K}(a,\alpha)v^{l}(x)=\sum_{\alpha,\alpha^{\prime}}\mathcal{F}^{-\alpha}(R^{\alpha}\cdot(\mathcal{F}^{\alpha}v^{l}))(x)\quad\forall
x\in D$ (7)
where, $\mathcal{F}^{\alpha}$ denotes the FrFT with order $\alpha$,
$\mathcal{F}^{-\alpha}$ denotes the inverse FrFT, $R^{\alpha}$ is the
learnable function which we learn from data, $a$ is a complex input function,
$\alpha$ is the learnable fractional order which we learn with data as well
and $\alpha$ belongs to the parameter space of $\mathcal{G}_{\theta}$. Note
that although $R^{\alpha}$ can be parameterized using a neural network, in the
present work, we observed that linear transformation provided slightly better
performance than deep neural networks and hence used the same. In the
subsequent subsection, we theoretically prove that the proposed operator can
map between infinite-dimensional spaces, leading to the learning of non-linear
operators.
### 4.2 Theoretical Analysis
In this subsection, we present a theoretical analysis of several components of
the proposed method. Specifically, we prove the following. (i) The
$N$-dimensional FrFT can be decomposed into one-dimensional FrFT. This
decomposition is crucial for CoNO, as it relies on multidimensional FrFT (Thm.
4.1). (ii) Multiplication in the fractional domain is equivalent to
convolution in the spatial domain for the proposed operator CoNO (Thm. 4.2).
(iii) The universal approximation capability for the CoNO (Thm. 4.3).
###### Theorem 4.1 (Product Rule for FrFT).
Suppose $\mathcal{K}_{\alpha}(m_{1},x_{1},m_{2},x_{2}...,m_{n},x_{n})$ denote
the fractional integral kernel of the $N$-dimensional FrFT of a signal
$f(x_{1},x_{2}...x_{n})$ with $m_{1},m_{2}...m_{n}$ denoting the
multidimensional FrFT coefficients, then the following fractional kernel
property holds:
$\mathcal{K}_{\alpha}(m_{1},x_{1},m_{2},x_{2}...,m_{n},x_{n})=\prod_{i=1}^{N}\mathcal{K}_{\alpha}(m_{i},x_{i})$
(8)
Proof. Refer to the App. B.9 for the proof. Thm. 4.1, show that an
$N$-dimensional fractional integral kernel is a product of one-dimensional
fractional integral kernels along each dimension.
###### Theorem 4.2 (Convolution Theorem for FrFT).
Suppose $f$ and $g$ are square-integrable functions. Define
$e_{\alpha}(m)=e^{i\pi\mid m\mid^{2}cot(\alpha)}$ and $h(x)=(f*g)(x)$ where
$*$ denotes the usual convolution integral with $\mathcal{F}^{\alpha}$,
$\mathcal{G}^{\alpha}$, $\mathcal{H}^{\alpha}$ respectively denoting the FrFT
of $f$, $g$ and $h$, respectively. Then,
$\mathcal{H}^{\alpha}(m)=\mathcal{F}^{\alpha}(m)\mathcal{G}^{\alpha}(m)e_{-\alpha}(m).$
(9)
Proof. Refer to App. B.12 for the proof. Thm. 4.2 states that the convolution
of two functions in the spatial domain is equivalent to the product of their
respective fractional Fourier transforms, multiplied by a function dependent
on the fractional order.
Finally, we present the universal approximation theorem of CoNO below, similar
to that for FNO [34].
###### Theorem 4.3 (Universal Approximation).
Let $s,s^{\prime}>0$ and $\alpha\in\mathbb{R}$; Suppose
$\mathcal{G}:H^{s}(\mathcal{T}_{\alpha}^{d};\mathbb{R}^{d_{\text{a}}})\rightarrow
H^{s^{\prime}}(\mathcal{T}_{\alpha}^{d};\mathbb{R}^{d_{\text{u}}})$ represent
a continuous operator between Sobolev spaces where $\mathcal{T}_{\alpha}^{d}$
denotes the fractional order torus and $d_{a},d_{u}\in\mathbb{N}$; and
$K\subset H^{s}(\mathcal{T}_{\alpha}^{d};\mathbb{R}^{d_{\text{a}}})$ is a
compact subset. Then, for any $\varepsilon>0$, there exists CoNO layers
$\mathcal{N}:H^{s}(\mathcal{T}_{\alpha}^{d};\mathbb{R}^{d_{\text{a}}})\rightarrow
H^{s^{\prime}}(\mathcal{T}_{\alpha}^{d};\mathbb{R}^{d_{\text{u}}})$
satisfying:
$\sup_{v\in K}\|\mathcal{G}(v)-\mathcal{N}(v)\|_{L^{2}}\leq\varepsilon$ (10)
Proof. Refer to App. B.15 for the proof.
## 5 Numerical Experiments
This section provides a thorough empirical investigation of CoNO in contrast
to multiple vision models and neural operator baselines. We conduct extensive
experiments on a diverse set of challenging benchmarks spanning various
domains to demonstrate the efficacy of our proposed method.
Table 2: The main result with sixteen baselines on all benchmarks datasets:
Mean Relative $\ell_{2}$ Error (Equation 11) is reported as the evaluation
metric, where a smaller $\ell_{2}$ Error indicates superior performance.
"INCREMENT %" refers to the relative error reduction concerning the second-
best model on each benchmark. Specifically focusing on the 2D Navier–Stokes
benchmark, a detailed comparison is conducted with KNO [69], and TF-Net [64],
as they are designed for auto-regressive (time-dependent) tasks. Instances
marked with ’*’ indicate that the baseline cannot handle the benchmark. In the
color legend, blue represents the best performance, green indicates the
second-best performance, and orange signifies the third-best performance among
the baselines.
MODEL | Elasticity-P | Elasticity-G | Plasticity | Navier-Stokes | Darcy | Airfoil | Pipe
---|---|---|---|---|---|---|---
U-NET [2015] | 0.0235 | 0.0531 | 0.0051 | 0.1982 | 0.0080 | 0.0079 | 0.0065
RESNET [2016] | 0.0262 | 0.0843 | 0.0233 | 0.2753 | 0.0587 | 0.0391 | 0.0120
TF-NET [2020] | \ | \ | \ | 0.1801 | \ | \ | \
SWIN [2021] | 0.0283 | 0.0819 | 0.0170 | 0.2248 | 0.0397 | 0.0270 | 0.0109
DEEPONET [2021] | 0.0965 | 0.0900 | 0.0135 | 0.2972 | 0.0588 | 0.0385 | 0.0097
FNO [2020] | 0.0229 | 0.0508 | 0.0074 | 0.1556 | 0.0108 | 0.0138 | 0.0067
U-FNO [2022] | 0.0239 | 0.0480 | 0.0039 | 0.2231 | 0.0183 | 0.0269 | 0.0056
WMT [2021] | 0.0359 | 0.0520 | 0.0076 | 0.1541 | 0.0082 | 0.0075 | 0.0077
GALERKIN [2021] | 0.0240 | 0.1681 | 0.0120 | 0.2684 | 0.0170 | 0.0118 | 0.0098
SNO [2022] | 0.0390 | 0.0987 | 0.0070 | 0.2568 | 0.0495 | 0.0893 | 0.0294
U-NO [2022] | 0.0258 | 0.0469 | 0.0034 | 0.1713 | 0.0113 | 0.0078 | 0.0100
HT-NET [2022] | 0.0372 | 0.0472 | 0.0333 | 0.1847 | 0.0079 | 0.0065 | 0.0059
F-FNO [2021] | 0.0263 | 0.0475 | 0.0047 | 0.2322 | 0.0077 | 0.0078 | 0.0070
KNO [2023] | \ | \ | \ | 0.2023 | \ | \ | \
GNOT [2023] | 0.0315 | 0.0494 | * | 0.1670 | 0.0105 | 0.0081 | *
LSM [2023] | 0.0218 | 0.0408 | 0.0025 | 0.1535 | 0.0065 | 0.0062 | 0.0050
CoNO (Ours) | 0.0210 | 0.0436 | 0.0019 | 0.1287 | 0.0051 | 0.0057 | 0.0054
INCREMENT % | 3.8% | -6.8% | 31.6% | 19.3% | 27.5% | 8.7% | -8.0%
### 5.1 Experiments Details and Main Result
Benchmarks: We assess the performance of our model on Darcy and Navier Stokes
[41] benchmarks to gauge its proficiency on regular grids. Subsequently, we
extend our experimentation to benchmarks featuring irregular geometries, such
as Airfoil, Plasticity, and Pipe [42], modeled using structured meshes and
Elasticity [42], represented in point clouds. Refer to App. section C for more
details about benchmarks and tasks.
Baselines: We assess CoNO by comparing it against sixteen established models
across seven benchmarks, which include baselines from vision models (U-Net
[55], ResNet [27], SwinTransformer [44]) and thirteen baselines specifically
designed for PDEs (DeepONet [45], TF-Net [64], FNO [41], U-FNO [65], WMT [25],
GalerkinTransformer [10], SNO [18], U-NO [52], HT-Net [43], F-FNO [58], KNO
[69], GNOT [26], LSM [68]). Notably, for the Elasticity-P benchmark in the
point cloud, we incorporate the specialized transformation proposed by geo-FNO
[42] at both the start and end of these models. This transformation
facilitates the conversion of irregular input domains into or back from a
uniform mesh.
Evaluation Metric: Mean relative $\ell_{2}$ error is used throughout the
experiments.
$\mathcal{L}=\frac{1}{N}\sum_{i=1}^{N}\frac{\|\mathcal{G}_{\theta}(a_{i})-\mathcal{G}^{\dagger}(a_{i})\|_{2}}{\|\mathcal{G}^{\dagger}(a_{i})\|_{2}}$
(11)
the regular mean-squared error (MSE) is enhanced with a normalizer
$\|\mathcal{G}^{\dagger}(a_{i})\|_{2}$ to take account for discrepancies in
absolute resolution scale across different benchmarks as described in [41].
Implementation Details: We have used mean relative $\ell_{2}$ error (Eq. 11)
as the training and evaluation metric. We train all the models for 500 epochs
using the Adam optimizer [32]. Comprehensive details are provided in the App.
section D. All the experiments are conducted on a Linux machine running Ubuntu
20.04.3 LTS on an Intel(R) Core(TM) i9-10900X processor and a single NVIDIA
RTX A6000 GPU with 48 GB RAM.
Empirical Results: As illustrated in Table 2, CoNO consistently demonstrates
superior performance on all the datasets in comparison to all baseline models,
ranking among the top two. This performance superiority is evident across
benchmark datasets characterized by diverse geometries and dimensions,
exhibiting an average improvement of 10.9%. With the second-best performance
observed in the Pipe and Elasticity-G benchmarks, our findings suggest that
architectures resembling UNET demonstrate superior efficacy in capturing the
underlying solution compared to the CoNO model. When applied to time-dependent
PDEs, CoNO surpasses all previously established baselines, achieving an
average improvement of 25.5%. This result underscores the efficacy of
incorporating the change in frequency derivative captured by the FrFT in
complex domains, thereby showcasing the promise of our approach in handling
temporal dynamics in PDEs.
### 5.2 Ablation Study and Additional Results
Table 3: Comprehensive ablation study on CoNO: investigating the impact of
individual component removal on Navier-Stokes and Darcy Flow benchmark (w/o
denotes the performance without that component).
DESIGN | Navier-Stokes | Darcy
---|---|---
w/o Bias | 0.1425 | 0.0080
w/o FrFT | 0.1535 | 0.0086
w/o Complex NN | 0.1390 | 0.0072
w/o Alias Free Activation | 0.1295 | 0.0052
CoNO | 0.1287 | 0.0050
Figure 3: Depiction of results for different methods on fluid datasets, Darcy
Flow (Left) and Navier Stokes (Right). We plotted the heatmap of the absolute
difference value between ground truth and prediction to compare the predicted
output. See the App. Fig. 10 for more solid physics and fluid physics
benchmarks showcases.
To assess the efficacy of each component in the CoNO operator, we conducted a
comprehensive ablation study by systematically excluding individual
components. The results presented in Table 3 indicate that all components are
crucial for the effectiveness of the CoNO operator, as evidenced by a notable
change in the $\ell_{2}$ error with the addition or deletion of an element.
Specifically, removing the FrFT block results in a substantial degradation in
performance, underscoring its effectiveness in capturing non-stationary
signals. Similarly, the absence of the CVNN leads to comparable adverse
effects. Notably, our analysis reveals that including bias is instrumental in
introducing high-frequency components, further emphasizing its importance.
Interestingly, CoNO performance degraded only slightly after removing the
alias-free activation function as defined. It raises an intriguing question
regarding its necessity for optimizing the operator’s efficiency.
Visual Demonstrations: For a concise representation of intuitive performance,
Fig. 3 presents a comparative analysis between FNO, LSM, and CoNO. Notably,
CoNO exhibits remarkable proficiency in addressing time-dependent PDEs,
including Navier-Stokes and Plasticity, as depicted in App. Fig. 10. Moreover,
CoNO outperforms LSM and FNO significantly in the case of Darcy by 27% and
having fewer artifacts present in prediction. Additionally, CoNO excels in
capturing singularities around corners, as illustrated in the elasticity
dataset in the App. Fig. 10, emphasizing its robust and superior performance.
Performance across various Resolutions: In Fig. 4 (Left), the operator CoNO
consistently exhibits superior performance compared to other operators on
darcy flow PDEs at various resolutions. Notably, CoNO demonstrates stability
across different resolutions, adhering to the principle of discrete-continuous
equivalence as shown in App. section F.1. It contrasts HT-Net, which
experiences degradation in very high dimensions. Furthermore, FNO and CoNO
represent the exclusive class of operators capable of zero-shot super-
resolution without requiring explicit training.
Out of Distribution Generalization: We conducted experiments on the Navier-
Stokes dataset in this investigation, training our model with a viscosity
coefficient of $10^{-5}$. Subsequently, we assessed the out-of-distribution
generalization capabilities by evaluating the trained model on a viscosity
coefficient of $10^{-4}$. Our findings consistently reveal that CoNO
demonstrates a significantly superior generalization performance, exhibiting
an increment of $64.3\%$ compared to FNO. It also highlights the significance
of capturing latent variable information or the UNET architecture, as achieved
by LSM, which outperforms all other operators even CoNO as shown in App.
section F.3.
Data Efficiency: As demonstrated in Fig. 4 (Middle), CoNO exhibits comparable
performance to the second-best operator LSM when trained on 60% of the data.
Furthermore, across various training dataset ratios, CoNO consistently
outperforms all other operators, underscoring its superior data efficiency
compared to SOTA operators as demonstrated in App. section F.6.
Robustness to Noise: In this study, we performed experiments introducing
different noise levels into the training using Gaussian noise. The noise
addition process follows: For each input sample denoted as $x(n)$ within the
dataset $D$, we modified it by adding Gaussian noise with parameters $\gamma
N(0,\sigma^{2}_{D})$. Here, $\sigma^{2}_{D}$ represents the variance of the
entire dataset, and $\gamma$ indicates the specified noise intensity level.
Our investigation yielded notable results as in Fig. 4 (Right), particularly
when evaluating the performance of CoNO in the presence of noise within the
training dataset; specifically, the noisy training with 0.1% yielded a better
result than the LSM operator without noisy training, confirming the robustness
of CoNO to the noisy dataset shown in App. section F.2.
Figure 4: (Left) Models performance under different resolutions on Darcy.
(Middle) Models performance under different training datasets the ratio on
Darcy. (Right) Models performance under the presence of noise on Darcy. The
lower $l2$ loss indicates better performance. Figure 5: Learning Curve for
Darcy flow (Left) and Navier Stokes (Right) where the x-axis denotes epochs
and the y-axis $l2$ error.
Training Stability: The performance of CoNO is more stable during training, as
visually observed in Fig. 5. The model exhibits reduced oscillations and
consistently performs better than FNO and LSM. Remarkably, CoNO attains an
equivalent performance compared with the best LSM performance after training
within the initial 200 epochs, demonstrating its efficient faster and better
convergence while training.
Long Time Prediction on Navier Stokes: We evaluated the long-term behavior of
the proposed operator CoNO by training it on the Navier-Stokes equation
(viscosity coefficient of $10^{-4}$), as shown in the App. section F.5 shows
that CoNO excels in extrapolating beyond the prediction horizon compared to
LSM and FNO.
## 6 Conclusion
Altogether, we introduce a new operator learning paradigm called the CoNO,
which capitalizes on CVNNs and the FrFT as the integral operator. We
theoretically prove that CoNO follows the universal approximation theorem. We
demonstrate the effectiveness of leveraging the expressive power of CVNNs
within the operator learning framework to construct resilient, data-efficient,
and superior neural operators capable of improving the learning of function-
to-function mappings. Empirically, we show that CoNO presents the SOTA results
in terms of performance, zero-shot super-resolution, out-of-distribution
generalization, and noise robustness. This advancement makes CoNO a promising
method for developing efficient operators for real-time PDE inference,
offering new tools for the SciML community.
Limitations and future work. Although CoNO shows improved empirical
performance, the specific features that enable this superiority remain
unclear. Understanding the loss landscape and learning mechanisms behind this
performance is crucial. Additionally, making CoNO computationally efficient is
essential to accelerate inference. Additional future work and limitations are
discussed in App. section G. Further, the broader impacts of the work are
discussed in App. section H.
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## Appendix
## Appendix A Fractional Fourier Transform
Figure 6: The figure illustrates the FrFT of the Navier-Stokes Partial
Differential Equation (PDE) across varying $\alpha$ values. The (Top) displays
heatmaps representing the magnitude of the transformed data, while the
(Bottom) showcases corresponding heatmaps depicting the phase angle. Each
column corresponds to a distinct $\alpha$ value, highlighting the impact of
fractional Fourier transform parameters on the learned representation of the
Navier-Stokes PDE. It is observed that in FT, most spectrum is concentrated at
low frequency for $\alpha=1$. Figure 7: The figure illustrates the inherent
optimality of FrFT in signal filtering for non-stationary image signals,
showcasing its ability to separate effects along the fractional order axis and
examining its relationship with spatial-frequency plane rotation.
Fig. 7 is an elucidative representation of the intrinsic optimality associated
with the Fractional Fourier Transform (FrFT) in signal filtering for non-
stationary image signals. It notably underscores FrFT’s distinctive capability
to separate various signal effects along the fractional order axis
effectively. Furthermore, the figure provides insightful scrutiny into the
intricate relationship between FrFT and the rotation of the spatial-frequency
plane. This visualization contributes to a nuanced understanding of FrFT’s
prowess in handling non-stationary image signals. It offers valuable insights
into its utility and potential applications in the broader signal-processing
domain. Fig. 6 shows that as the parameter $\alpha$ increases, the spectral
distribution becomes increasingly concentrated around the origin. This
concentration results in a loss of spatial information while emphasizing
frequency information. Conversely, lower values of $\alpha$ reveal the
preservation of information in the phase domain, which, unfortunately, becomes
increasingly noisy when transformed into the frequency domain. This behavior
highlights the delicate balance between spatial and frequency information,
emphasizing the parameter $\alpha$’s critical role in shaping the signal’s
characteristics.
## Appendix B Mathematical Analysis
In this section, we will first present the necessary definitions and then
prove the theorem as stated in the main content of the paper. Subsequently, we
present a theoretical analysis of several components of the proposed method.
###### Definition B.1 (Fractional Torus).
[19] Let consider $\alpha\in\mathbb{R}$, and $\alpha\neq n\mathbb{Z}$. Define
fractional torus of order $\alpha$, denoted as $\mathcal{T}_{\alpha}^{n}$ is
the cube $[0,\mid\sin{\alpha}\mid]^{n}$ with opposite sides identified. We
will define FrFT on $\mathcal{T}_{\alpha}^{n}$ for the rest of the analysis.
We will define Fractional convolution and fractional approximation in
$L^{p}(\mathcal{T}_{\alpha}^{n})$ for $1\leq p\leq\infty$.
###### Definition B.2.
Let $\alpha\in\mathcal{R}$ and $\alpha\neq n$. Define
$e_{\alpha}(x)=e^{i\pi\mid x\mid^{2}cot\alpha}$ and
$e_{\alpha}f(x)=e_{\alpha}(x)f(x)$ for function on $\mathcal{T}_{\alpha}^{n}$.
###### Definition B.3.
Let $1\leq p\leq\infty$. A function $f$ on $\mathcal{T}_{\alpha}^{n}$ lies in
the space $e_{-\alpha}L^{p}(\mathcal{T}_{\alpha}^{n})$ if
$f(x)=e_{-\alpha}g(x),\>\>g\in L^{p}(\mathcal{T}_{\alpha}^{n})$ (12)
and $\mid\mid f\mid\mid_{L^{p}(\mathcal{T}_{\alpha}^{n})}<\infty$.
###### Definition B.4 (Fractional Fourier Transform on Fractional Torus Fu et
al. [19]).
For complex-valued functions $f\in
e_{-\alpha}L^{1}(\mathcal{T}_{\alpha}^{n})$, $\alpha\in\mathbb{R}$ and
$m\in\mathbb{Z}^{n}$, we define,
$\mathcal{F}^{\alpha}(f)(m)=\begin{cases}\int_{\mathcal{T}_{\alpha}^{n}}f(x)K_{\alpha}(m,x)dx&\alpha\neq\pi\mathbb{(}Z)\\\
f(m)&\alpha=2\pi\mathbb{(}Z)\\\ f(-m)&\alpha=2\pi\mathbb{(}Z)+\pi\end{cases}$
(13)
where,
$K_{\alpha}(m,x)=A_{\alpha}^{n}e_{\alpha}(x)e_{\alpha}(m,x)e_{\alpha}(m)$ (14)
here, $A_{\alpha}=\sqrt{1-i\cot{\alpha}}$ and
$e_{\alpha}(m,x)=e^{-i2\pi(m.x)\csc{\alpha}}$. where
$\mathcal{F}^{\alpha}(f)(m)$ denotes the $m^{th}$ fractional fourier
coefficient of order $\alpha$ for $f\in
e_{-\alpha}L^{1}(\mathcal{T}_{\alpha}^{n})$.
###### Remark B.5.
For Eq. 13, it reduces to Fourier Transform (FT) when $\alpha=\frac{\pi}{2}$,
$\cot{\alpha}=0$ and $\csc{\alpha}=1$, where
$K_{\alpha}(m,x)=e^{-i2\pi(m.x)}$ (15)
###### Proposition B.6 (Linear and Bounded Operator).
Let $f$, $g$ be in $e_{-\alpha}L^{1}(\mathcal{T}_{\alpha}^{n})$. Then for
$m,k\in\mathbb{Z}^{n}$, $\lambda\in\mathbb{C}$, $y\in\mathcal{T}_{\alpha}^{n}$
then following holds true.
1. 1.
$\mathcal{F}^{\alpha}(f+g)(m)$ =
$\mathcal{F}^{\alpha}(f)(m)+\mathcal{F}^{\alpha}(g)(m)$.
2. 2.
$\mathcal{F}^{\alpha}(\lambda f)(m)=\lambda\mathcal{F}^{\alpha}(f)(m)$.
3. 3.
$\sup_{m\in\mathbb{Z}^{n}}|\mathcal{F}^{\alpha}(f)(m)|\leq|\csc{\alpha}|^{\frac{n}{2}}\mid\mid
f\mid\mid_{L^{1}(\mathcal{T}_{\alpha}^{n})}$
Proof: For proof refer Fu et al. [19].
###### Proposition B.7 (Uniqueness of Fractional Fourier Transform).
If $f,g\in e_{-\alpha}L^{1}(\mathcal{T}_{\alpha}^{n})$ satisfy
$\mathcal{F}^{\alpha}(f)(m)=\mathcal{F}^{\alpha}(g)(m)$ for all
$m\in\mathbb{Z}^{n}$, then $f=g$ a.e.
Proof: For proof refer Fu et al. [19].
###### Corollary B.8 (Fractional Fourier Inversion).
If $f\in e_{-\alpha}L^{1}(\mathcal{T}_{\alpha}^{n})$ and
$\sum_{m\in\mathbb{Z}^{n}}|\mathcal{F}^{\alpha}(f)(m)|<\infty.$ (16)
Then,
$f(x)=\sum_{m\in\mathbb{Z}^{n}}\mathcal{F}^{\alpha}(f)(m)K_{-\alpha}(m,x)\>\text{a.e.}$
(17)
and therefore, $f$ is almost everywhere, equal to a continuous function.
###### Theorem B.9 (Product Rule).
Suppose $\mathcal{K}_{\alpha}(m_{1},x_{1},m_{2},x_{2}...,m_{n},x_{n})$ denote
the fractional integral kernel of the $N$-dimensional FrFT of a signal
$f(x_{1},x_{2}...x_{n})$ with $m_{1},m_{2}...m_{n}$ denoting the
multidimensional FrFT coefficients, then the following fractional kernel
property holds:
$\mathcal{K}_{\alpha}(m_{1},x_{1},m_{2},x_{2}...,m_{n},x_{n})=\prod_{i=1}^{N}\mathcal{K}_{\alpha}(m_{i},x_{i})$
(18)
Proof:
Let consider the $N$-dimensional signal $f(x_{1},x_{2},...,x_{n})$,
Then, let’s define the FrFT of the signal as follows:
$\mathcal{F}^{\alpha}(f)(m_{1},m_{2},...,m_{N})=\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}f(x_{1},x_{2},...,x_{N})\prod_{i=1}^{N}\mathcal{K}_{\alpha}(m_{i},x_{i})dx_{i},$
(19)
From Eq. 19, we can easily prove for $\alpha=0$,
$\mathcal{F}^{\alpha}(f)(m_{1},m_{2},...,m_{N})=f(x_{1},x_{2},...,x_{n})$ (20)
and for $\alpha=1$, it reduces to Fourier Transform (FT).
Therefore, if we can show that
$\mathcal{F}^{\alpha}.\mathcal{F}^{\beta}=\mathcal{F}^{\alpha+\beta}$ then
obviously $\mathcal{F}^{\alpha}$ satisfy all the FrFT properties.
Then,
$\mathcal{F}^{\alpha}.\mathcal{F}^{\beta}(f)(m_{1},m_{2},...,m_{n})=\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}\prod_{i=1}^{N}\mathcal{K}_{\alpha}(m_{i},x_{i})\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}f(y_{1},y_{2},...,y_{N})\prod_{i=1}^{N}\mathcal{K}_{\beta}(x_{i},y_{i})dy_{i}dx_{i}$
(21)
$\mathcal{F}^{\alpha}.\mathcal{F}^{\beta}(f)(m_{1},m_{2},...,m_{n})=\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}f(y_{1},y_{2},...,y_{N})\\{\prod_{i=1}^{N}\int_{-\infty}^{\infty}\mathcal{K}_{\alpha}(m_{i},x_{i})\mathcal{K}_{\beta}(x_{i},y_{i})dx_{i}\\}dy_{1}dy_{2}...dy_{n}$
(22)
Now using the result from [2],
$\int_{-\infty}^{\infty}\mathcal{K}_{\alpha}(m,x)K_{\beta}(x,y)du=\mathcal{K}_{\alpha+\beta}(m,y)$
(23)
Using the above property,
$\mathcal{F}^{\alpha}.\mathcal{F}^{\beta}(f)(m_{1},m_{2},...,m_{n})=\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}f(y_{1},y_{2},...,y_{N})\prod_{i=1}^{N}K_{\alpha+\beta}(m_{i},y_{i})dy_{1}dy_{2}...dy_{n}=\mathcal{F}^{\alpha+\beta}$
(24)
Therefore, we conclude that,
$\mathcal{K}_{\alpha}(m_{1},x_{1},m_{2},x_{2}...,m_{n},x_{n})=\prod_{i=1}^{N}\mathcal{K}_{\alpha}(m_{i},x_{i})$
(25)
###### Definition B.10.
Let $f$ and $g$ be square integrable functions, Let, $h(x)=(f*g)(x)$ (where
$*$ denotes the convolution), i.e.,
$h(x)=(f*g)(x)=\int_{-\infty}^{\infty}f(t)g(x-t)\,dt.$ (26)
###### Definition B.11.
For any function $f(x)$, let us define the function
$e_{\alpha}f(x)=e_{\alpha}(x)f(x)$. For any two functions $f$ and $g$, we
define the convolution operation $\ast$ by
$h(x)=(f\ast g)(x)=A_{\alpha}e_{-\alpha}(x)(e_{\alpha}f\ast e_{\alpha}g)(x),$
(27)
where $\ast$ is the convolution operation as defined in Eq. 26.
###### Theorem B.12 (Convolution Theorem for Fractional Fourier Transform
(FrFT)).
Suppose $f$ and $g$ are square-integrable functions. Define
$e_{\alpha}(m)=e^{i\pi\mid m\mid^{2}cot(\alpha)}$ and $h(x)=(f*g)(x)$ where
$*$ denotes the usual convolution integral with $\mathcal{F}^{\alpha}$,
$\mathcal{G}^{\alpha}$, $\mathcal{H}^{\alpha}$ respectively denoting the FrFT
of $f$, $g$ and $h$, respectively. Then,
$\mathcal{H}^{\alpha}(m)=\mathcal{F}^{\alpha}(m)\mathcal{G}^{\alpha}(m)e_{-\alpha}(m).$
(28)
Proof: By FrFT definition from Eq. 2, we know that
$\begin{split}\mathcal{H}^{\alpha}(h)(m)&=A_{\alpha}\int_{-\infty}^{\infty}h(t)e_{\alpha}(t)e_{\alpha}(m,t)e_{\alpha}(m)dt\\\
&=A_{\alpha}^{2}\int_{-\infty}^{\infty}e_{\alpha}(t)e_{\alpha}(m,t)e_{\alpha}(m)e_{-\alpha}(t)dt\int_{-\infty}^{\infty}e_{\alpha}(x)f(x)e_{\alpha}(t-x)g(t-x)dx\\\
&=A_{\alpha}^{2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x)g(t-x)e_{\alpha}(t)e_{\alpha}(m,t)e_{\alpha}(m)e_{-\alpha}(t)e_{\alpha}(x)e_{\alpha}(t-x)dxdt\end{split}$
(29)
By substituting $t-x=v$ in the above eq. , we obtain
$\begin{split}\mathcal{H}^{\alpha}(h)(m)&=A_{\alpha}^{2}e_{\alpha}(m)\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x)g(v)e_{\alpha}(x+v)e_{\alpha}(m,x+v)e_{-\alpha}(x+v)e_{\alpha}(x)e_{\alpha}(v)dxdv\\\
&=A_{\alpha}^{2}e_{-\alpha}(m)\int_{-\infty}^{\infty}f(x)e_{\alpha}(x)e_{\alpha}(m)e_{\alpha}(m,x)dx\int_{-\infty}^{\infty}g(v)e_{\alpha}(v)e_{\alpha}(m)e_{\alpha}(m,v)dv\\\
&=\mathcal{F}^{\alpha}(m)\mathcal{G}^{\alpha}(m)e_{-\alpha}(m)\end{split}$
(30)
Therefore, we have,
$\mathcal{H}^{\alpha}(m)=\mathcal{F}^{\alpha}(m)\mathcal{G}^{\alpha}(m)e_{-\alpha}(m).$
(31)
###### Definition B.13 (Fractional Fourier Projection Operator).
Let’s define the Fractional Fourier wave-number as follows:
$\mathcal{K}_{N}=\\{k\in\mathbb{Z}^{n}:|k|_{\infty}\leq N\\}$ (32)
And define fractional projection operator to truncate high order coefficient
of a function as follows:
$\mathcal{P}_{N}(f)(x)=\mathcal{F}^{-\alpha}(\mathcal{F}^{\alpha}(f)(k).\boldsymbol{1}_{\mathcal{K}_{N}}(k))$
(33)
where $1_{\mathcal{K}_{N}}(k)$ indicates indicator function which takes 1 when
$k\in\mathcal{K}_{N}$ and $0$ otherwise.
###### Theorem B.14 (Convergence of Fractional Fourier Series).
Let $\alpha\in\mathbb{R}$ and $\alpha\neq\pi\mathbb{Z}$, and $f\in
e_{-\alpha}L^{p}(\mathcal{T}_{\alpha}^{n})$ ($1\leq p<\infty$),
$\mathcal{P}_{N}(f)(x)$ denotes as follows:
$\mathcal{P}_{N}(f)(x)=\mathcal{F}^{-\alpha}(\mathcal{F}^{\alpha}(f)(k).\boldsymbol{1}_{\mathcal{K}_{N}}(k))$
(34)
where, $\mathcal{F}^{\alpha}(f)(k)$ denotes the $k^{th}$ fractional
coefficient. Then,
$\lim_{N\rightarrow\infty}\sup_{x}||f(x)-\mathcal{P}_{N}(f)(x)||\rightarrow 0$
(35)
###### Theorem B.15 (Universal Approximation of CoNO).
Let $s,s^{\prime}>0$ and $\alpha\in\mathbb{R}$; Suppose
$\mathcal{G}:H^{s}(\mathcal{T}_{\alpha}^{d};\mathbb{R}^{d_{\text{a}}})\rightarrow
H^{s^{\prime}}(\mathcal{T}_{\alpha}^{d};\mathbb{R}^{d_{\text{u}}})$ represent
a continuous operator between Sobolev spaces where $\mathcal{T}_{\alpha}^{d}$
denotes the fractional order torus and $d_{a},d_{u}\in\mathbb{N}$; and
$K\subset H^{s}(\mathcal{T}_{\alpha}^{d};\mathbb{R}^{d_{\text{in}}})$ is a
compact subset. Then, for any $\varepsilon>0$, there exists CoNO layers
$\mathcal{N}:H^{s}(\mathcal{T}_{\alpha}^{d};\mathbb{R}^{d_{\text{in}}})\rightarrow
H^{s^{\prime}}(\mathcal{T}_{\alpha}^{d};\mathbb{R}^{d_{\text{out}}})$
satisfying:
$\sup_{v\in K}\|\mathcal{G}(v)-\mathcal{N}(v)\|_{L^{2}}\leq\varepsilon$ (36)
Proof: Let $\alpha\in\mathbb{R}$ and define the following fractional
orthogonal projection operator:
$\mathcal{G}_{N}:H^{s}(\mathcal{T}_{\alpha}^{d})\rightarrow
H^{s^{\prime}}(\mathcal{T}_{\alpha}^{d}),\>\>\mathcal{G}_{N}(v)=\mathcal{P}_{N}\mathcal{G}(\mathcal{P}_{N}(v))$
(37)
Using Thm. B.14, we have for $\forall\varepsilon$, there exists $N\geq 0$ such
that:
$\|\mathcal{G}(v)-\mathcal{G}_{N}(v)\|_{L^{2}}\leq\varepsilon,\>\>\forall v\in
K$ (38)
We need to find $\mathcal{N}$ to approximate $\mathcal{G}_{N}$.
Next we define Fractional Conjugate of $\mathcal{G}_{N}$ as follows:
$\mathcal{\hat{G}}_{N}:\mathbb{C}^{\mathcal{K}_{N}}\rightarrow\mathbb{C}^{\mathcal{K}_{N}},\>\>\mathcal{\hat{G}}_{N}(\hat{v})=\mathcal{F}^{\alpha}(\mathcal{G}_{N}(\mathcal{F}^{-\alpha}(\hat{v})))$
(39)
we can show that,
$\mathcal{G}_{N}=\mathcal{F}^{-\alpha}\circ\mathcal{\hat{G}}_{N}\circ(\mathcal{F}^{\alpha}\circ\mathcal{P}_{N})$
(40)
Now with above decomposition, we can construct CoNO by separately
approximating each individual terms $\mathcal{F}^{-\alpha}$,
$\mathcal{\hat{G}}_{N}$ and $\mathcal{F}^{\alpha}\circ\mathcal{P}_{N}$.
Approximating $\mathcal{\hat{G}}_{N}$: For $\forall\varepsilon\geq 0$ and
$\mathcal{\hat{G}}_{N}:\mathbb{C}^{\mathcal{K}_{N}}\rightarrow\mathbb{C}^{\mathcal{K}_{N}}$,
we have $l$ CoNO layers
$\mathcal{L}_{l}\circ\mathcal{L}_{l-1}\circ...\circ\mathcal{L}_{1}$ which
satisfy:
$||\mathcal{\hat{G}}_{N}(\hat{v})(x)-(\mathcal{L}_{l}\circ\mathcal{L}_{l-1}\circ...\circ\mathcal{L}_{1})(\hat{w})(x)||_{L^{2}}\leq\varepsilon,\>\>\forall
v\in K,\forall x\in\mathcal{T}^{d}_{\alpha}$ (41)
where, $\hat{w}:\mathcal{T}^{d}_{\alpha},x\rightarrow\hat{v}$ is constant
function defined on $\mathcal{T}^{d}_{\alpha}$.
From CoNO layer definition, we have,
$\mathcal{L}_{l}(v)(x)=Wv(x)+\mathcal{F}^{-\alpha}(K_{l}(k)\mathcal{F}^{\alpha}(\mathcal{P}_{N}v)(k))$
(42)
To approximate $\mathcal{\hat{G}}_{N}$, we can use Universal Approximation
Theorem for CVNNs [63] by setting $K_{l}(k)$ to be identity. Then,
$\mathcal{F}^{-\alpha}(K_{l}(k)\mathcal{F}^{\alpha}(\mathcal{P}_{N}v)(k))\approx\mathcal{P}_{N}(v),\>\>\forall
l,$ (43)
then we have,
$\mathcal{L}_{l}(v)(x)=Wv(x)+\mathcal{P}_{N}(v)(x)$ (44)
And then using the Universal Approximation Theorem for CVNNs will guarantee
that $\mathcal{L}_{l}\circ\mathcal{L}_{l-1}\circ...\circ\mathcal{L}_{1}$ can
approximate $\mathcal{\hat{G}}_{N}$ to any desired precision.
Approximating $\mathcal{F}^{\alpha}\circ\mathcal{P}_{N}$: For any
$\varepsilon\geq 0$ there exists $l\geq 0$ and
$\mathcal{L}_{l}\circ\mathcal{L}_{l-1}\circ...\circ\mathcal{L}_{1}\circ\mathcal{P}$
that satisfy following:
$||\mathcal{F}^{\alpha}(\mathcal{P}_{N}v)-(\mathcal{L}_{l}\circ\mathcal{L}_{l-1}\circ...\circ\mathcal{L}_{1}\circ\mathcal{P})(v)(x)||_{L^{2}}\leq\varepsilon,\>\>\forall
v\in K,\forall x\in\mathcal{T}^{d}$ (45)
Let’s define $\mathcal{R}(k,x)=e_{\alpha}(k)e_{\alpha}(k,x)$ which constitutes
the orthonormal basis for FrFT, which we call as fractional fourier basis.
Firstly we will show that there exists
$\mathcal{N}=\mathcal{L}_{l}\circ\mathcal{L}_{l-1}\circ...\circ\mathcal{L}_{1}\circ\mathcal{P}$
that satisfies:
$\begin{cases}||\mathcal{N}(v)_{1,k}-\mathcal{P}_{N}v(x).Re(\mathcal{R}(k,x))||_{L_{2}}<\varepsilon\\\
||\mathcal{N}(v)_{1,k}-\mathcal{P}_{N}v(x).Im(\mathcal{R}(k,x))||_{L_{2}}<\varepsilon\end{cases}$
(46)
To construct $\mathcal{N}$, we define the lifting operator $\mathcal{P}$ and
use position embedding as follows:
$\mathcal{P}(v)=R(v(x)=\\{v(x),Re(\mathcal{R}(k,x)),v(x),Im(\mathcal{R}(k,x))\\}_{k\in\mathcal{K}_{N}}$
(47)
where trigonometric polynomial of order $\alpha$ are directly embedded in CoNO
layer. $\mathcal{P}$ lifts the range of function from $\mathbb{R}^{d_{a}}$ to
$\mathbb{R}^{d_{u}}$. Then, by leveraging the Universal Approximation Theorem
for CVNNs where we concatenate the
$\\{v.Re(\mathcal{R}(k,x)),v.Im(\mathcal{R}(k,x))\\}_{k\in\mathcal{K}_{N}}$,
we can have multiple layers satisfies:
$\scriptstyle(\mathcal{L}_{l}\circ...\circ\mathcal{L}_{1})(\\{v(x),Re(\mathcal{R}(k,x)),v(x),Im(\mathcal{R}(k,x))\\}_{k\in\mathcal{K}_{N}})\approx\\{v.Re(\mathcal{R}(k,x)),v.Im(\mathcal{R}(k,x))\\}_{k\in\mathcal{K}_{N}},\>\>\forall
v\in K$ (48)
And can achieve the desired precision by adjusting the width and depth of the
layers. We next note that,
$\mathcal{P}_{N}v(x)=\sum_{k\in\mathcal{K}_{N}}\hat{v}_{k}K_{-\alpha}(k,x)$
(49)
where $\hat{v}_{k}$ denotes the $k^{th}$ coefficient of FrFT. Then by
definition of FrFT in Eq 2:
$\mathcal{F}^{\alpha}(v)(0)=\int_{\mathcal{T}^{d}_{\alpha}}v(x)A_{\alpha}e_{\alpha}(x)dx=Re(\hat{v}_{0})+Im(\hat{v}_{0})$
(50)
$\mathcal{F}^{\alpha}(v.\mathcal{R}(k,x))(0)=\int_{\mathcal{T}^{d}_{\alpha}}v(x)A_{\alpha}e_{\alpha}(x)e_{\alpha}(k)e_{\alpha}(k,x)dx=Re(\hat{v}_{k})+Im(\hat{v}_{k})$
(51)
Thus in layer $\mathcal{L}_{l+1}$:
$\begin{cases}K_{l+1}(k)=0&\text{if}\>\>k=0\\\
K_{l+1}(k)=-Id&\text{if}\>\>k\neq 0\\\ \end{cases}$ (52)
And then we have:
$\scriptstyle\mathcal{F}^{-\alpha}(K_{l+1}(k).\mathcal{F}^{\alpha}(\\{v.Re(\mathcal{R}(k,x)),v.Im(\mathcal{R}(k,x))\\}_{k\in\mathcal{K}_{N}}))=\\{Re(\hat{v}_{k})-v.Re(\mathcal{R}(k,x)),Im(\hat{v}_{k})-v.Im(\mathcal{R}(k,x))\\}_{k\in\mathcal{K}_{N}}$
(53)
Finally, we can set $W_{l+1}$ in layer $\mathcal{L}_{l+1}$ to be identity we
get,
$\mathcal{L}_{l+1}(\\{v.Re(\mathcal{R}(k,x)),v.Im(\mathcal{R}(k,x))\\}_{k\in\mathcal{K}_{N}}))=\\{Re(\hat{v}_{k}),Im(\hat{v}_{k})\\}_{k\in\mathcal{K}_{N}}=\mathcal{F}^{\alpha}(\mathcal{P}_{N}v)$
(54)
Which is the desired approximation.
Approximating $\mathcal{F}^{-\alpha}$: For any $\varepsilon$, we have $l\geq
0$ and
$\mathcal{Q}\circ\mathcal{L}_{l}\circ\mathcal{L}_{l-1}\circ...\circ\mathcal{L}_{1}$
that satisfies:
$||\mathcal{F}^{-\alpha}(\hat{v})-(\mathcal{Q}\circ\mathcal{L}_{l}\circ\mathcal{L}_{l-1}\circ...\circ\mathcal{L}_{1})(v)||_{L^{2}}\leq\varepsilon,\>\>\forall
v\in K$ (55)
where $\hat{v}\in\mathbb{C}^{\mathcal{K}_{N}}$ is the truncated fractional
fourier coefficient.
We can use the previous steps here as well. We can construct
$Re(\hat{v}_{k}).\mathcal{R}(k,x)$ and $Im(\hat{v}_{k}).\mathcal{R}(k,x)$
using $\mathcal{L}_{l}\circ\mathcal{L}_{l-1}\circ...\circ\mathcal{L}_{1}$. We
can construct projection operator $\mathcal{Q}$ to recover the original
function.
$\mathcal{Q}(\\{\hat{v}_{k}.\mathcal{R}(k,x)\\}_{k\in\mathcal{K}_{N}})=\sum_{k\in\mathcal{K}_{N}}Re(\hat{v}_{k}).\mathcal{R}(k,x)-Im(\hat{v}_{k}).\mathcal{R}(k,x)=v(x)$
(56)
Therefore, it completes our proof.
Using the above approximation for $\mathcal{G}_{N}$ we can approximate the
operator to desired precision. Thus establishing the Universal Approximation
Theorem for CoNO.
## Appendix C Details for the Benchmark
### C.1 Details for benchmarks tasks
The following sections comprehensively elucidate details for benchmark tasks.
Figure 8: The diagram presents an overview of the Operator learning task
applied to distinct Partial Differential Equations (PDEs) classified as Fluid
Physics and Solid Physics. (Top) showcases three specific PDEs: Darcy Flow,
Airfoil, and Plasticity. (Bottom), an additional set of three PDEs is
featured: Navier Stokes, Pipe, and Elasticity. Table 4: Detailed benchmark
tasks provide a thorough and systematic exploration of each task’s specific
characteristics used in experimentation.
Dataset | Geometry | Task | Input | Output
---|---|---|---|---
ELASTICITY-P | Point Cloud | Estimate Stress | Material Structure | Inner Stress
ELASTICITY-G | Regular Grid | Estimate Stress | Material Structure | Inner Stress
PLASTICITY | Structured Mesh | Model Deformation | Boundary Condition | Mesh Displacement
NAVIER-STOKES | Regular Grid | Predict Future | Past Velocity | Future Velocity
AIRFOIL | Structured Mesh | Estimate Velocity | Structure | Fluid Velocity
PIPE | Structured Mesh | Estimate Velocity | Structure | Fluid Velocity
DARCY | Regular Grid | Estimate Pressure | Porous Medium | Fluid Pressure
### C.2 Description of Datasets
Table 4 and Fig. 8 comprehensively present the benchmark details. The
categorization of generation details is presented according to the governing
partial differential equations (PDEs) as follows:
Elasticity-P and Elasticity-G Dataset [42]: The dataset is designed to
evaluate internal stress within an incompressible material characterized by an
arbitrary void at its center, subject to external tension. The material’s
structural configuration constitutes the input, and the resulting internal
stress is the output. Notably, two distinct approaches are employed in
modeling the material’s geometry: Elasticity-P utilizes a point cloud
comprising 972 points. Elasticity-G represents the data on a structured grid
with dimensions $41\times 41$, obtained through interpolation from the
Elasticity-P dataset.
Plasticity Dataset [42]: This dataset addresses the plastic forging scenario,
wherein a die with an arbitrary shape impacts a plastic material from above.
The input to the benchmark is characterized by the die’s shape, encoded in a
structured mesh. The benchmark aims to predict the deformation of each mesh
point over the subsequent 20 timesteps. The structured mesh employed has a
resolution of $101\times 31$.
Navier Stokes Dataset [41]: 2D Navier-Stokes equation mathematically describes
the flow of a viscous, incompressible fluid in vorticity form on the unit
torus as follows:
$\begin{split}\partial_{t}w(x,t)+u(x,t)\cdot\nabla w(x,t)&=\nu\Delta
w(x,t)+f(x),\quad x\in(0,1)^{2},\,t\in(0,T]\\\ \nabla\cdot u(x,t)&=0,\quad
x\in(0,1)^{2},\,t\in[0,T]\\\ w(x,0)&=w_{0}(x),\quad x\in(0,1)^{2}\end{split}$
(57)
where, $u$ represents the velocity field, $w=\nabla\times u$ is the vorticity,
$w_{0}$ is the initial vorticity, $\nu$ is the viscosity coefficient, and $f$
is the forcing function. In this dataset, the viscosity ($\nu$) is fixed at
$10^{-5}$, and the 2D field has a resolution of $64\times 64$. Each sample
within the dataset comprises 20 consecutive frames. The objective is to
predict the subsequent ten frames based on the preceding ten.
Pipe Dataset [42]: This dataset focuses on the incompressible flow through a
pipe. The governing equations are Equation 59:
$\nabla\cdot\mathbf{U}=0,$ (58)
$\frac{\partial\mathbf{U}}{\partial
t}+\mathbf{U}\cdot\nabla\mathbf{U}=\mathbf{f}^{-1}\frac{1}{\rho}\nabla
p+\nu\nabla^{2}\mathbf{U}.$ (59)
The dataset is constructed on a geometrically structured mesh with a
$129\times 129$ resolution. We employ the mesh structure as input data for
experimental purposes, with the output being the horizontal fluid velocity
within the pipe.
Airfoil Dataset [42]: The dataset pertains to transonic flow over an airfoil.
Due to the negligible viscosity of air, the viscous term $\nu\nabla^{2}U$ is
omitted from the Navier-Stokes equation. Consequently, the governing equations
for this scenario are expressed as follows:
$\frac{\partial\rho f}{\partial t}+\nabla\cdot(\rho fU)=0$ (60)
$\frac{\partial(\rho fU)}{\partial t}+\nabla\cdot(\rho fUU+pI)=0$ (61)
$\frac{\partial E}{\partial t}+\nabla\cdot((E+p)U)=0,$ (62)
where $\rho f$ represents fluid density, and $E$ denotes total energy. The
data is on a structured mesh with dimensions $200\times 50$, and the mesh
point coordinates are utilized as inputs. The corresponding output is the Mach
number at each mesh point.
Darcy Flow Dataset [41]: It represents the flow through porous media. 2D Darcy
flow over a unit square is given by
$\nabla\cdot(a(x)\nabla u(x))=f(x),\quad x\in(0,1)^{2},$ (63)
$u(x)=0,\quad x\in\partial(0,1)^{2}.$ (64)
where $a(x)$ is the viscosity, $f(x)$ is the forcing term, and $u(x)$ is the
solution. This dataset employs a constant value of forcing term $F(x)=\beta$.
Further, Equation 63 is modified in the form of a temporal evolution as
$\partial_{t}u(x,t)-\nabla\cdot(a(x)\nabla u(x,t))=f(x),\quad x\in(0,1)^{2},$
(65)
In this dataset, the input is represented by the parameter $a$, and the
corresponding output is the solution $u$. The dataset comprises samples
organized on a regular grid with a resolution of $85\times 85$.
## Appendix D Implementation Details
The following section provides a comprehensive overview of the training and
testing samples, including details about the shapes of input and output
tensors.
Table 5: Training details for benchmark datasets. The input-output
resolutions are presented in the shape of (temporal, spatial, variate). The
symbol "/" indicates dimensions excluded.
Dataset | Training Samples | Testing Samples | Input Tensor | Output Tensor
---|---|---|---|---
ELASTICITY-P | 1000 | 200 | (/, 972, 2) | (/, 972, 1)
ELASTICITY-G | 1000 | 200 | (/, 41 $\times$ 41, 1) | (/, 41 $\times$ 41, 1)
PLASTICITY | 900 | 80 | (/, 101 $\times$ 31, 2) | (20, 101 $\times$ 31, 4)
NAVIER-STOKES | 1000 | 200 | (10, 64 $\times$ 64, 1) | (10, 64 $\times$ 64, 1)
AIRFOIL | 1000 | 100 | (/, 200 $\times$ 50, 2) | (/, 200 $\times$ 50, 1)
PIPE | 1000 | 200 | (/, 129 $\times$ 129, 2) | (/, 129 $\times$ 129, 1)
DARCY | 1000 | 200 | (/, 85 $\times$ 85, 1) | (/, 85 $\times$ 85, 1)
### D.1 Training Details
Table 5 presents a comprehensive overview of the experimental setup, including
details regarding the training and testing split and the shapes of the input
and output tensors. This information is crucial for understanding the specific
configurations employed in our experimentation process. In training CoNO, we
need to initialize the fractional orders, which are learned in the same way as
other matrices that are part of our network using optimizers such as Adam.
Notably, fractional orders can vary across axes, and there is no requirement
for uniformity initialization of fractional orders across different axes.
Furthermore, we conducted each experiment five times and observed that the
standard deviation falls within the ranges of 0.0003 for Darcy, Airfoil, and
Pipe datasets, 0.01 for Navier Stokes, 0.002 for Elasticity-P and
Elasticity-G, and 0.0002 for Plasticity.
### D.2 Discrete Implementation of Fractional Fourier Transform
The discrete implementation of the Fractional Fourier Transform (FrFT) is
essential for CoNO. In CoNO, we utilize a matrix multiplication-based discrete
FrFT. This approach leverages the spectral expansion of the fractional
integral kernel using a complete set of eigenfunctions of the FrFT, which are
Hermite-Gaussian functions [9]. We have used the following PyTorch based
discrete implementation of FrFT: Torch FrFT.
### D.3 Complex Neural Networks (CVNNs)
Figure 9: Complex Residual Network used in CoNO for learning the complex part
for CVNNs.
Given that all datasets are inherently real-valued, we designed a neural
network optimized for real-valued operations within the complex domain. Using
Complex-Valued Neural Networks (CVNNs), the network effectively handles
complex data components. It converts real-valued data to complex
representations through a residual block, as shown in Fig. 9. This
architecture allows the neural network to process the real-valued datasets in
the complex domain efficiently. Complex-valued backpropagation was implemented
using the Wirtinger calculus [3], which generalizes the notion of complex
derivatives and trains complex-valued neural networks (generalized complex
chain rule for real-valued loss function) [5, 13]. If $L$ is a real-valued
loss function and $z$ is a complex variable such that $z=x+iy$ where
$x,y\in\mathbb{R}$:
$\nabla_{z}L=\frac{\partial L}{\partial z}=\frac{\partial L}{\partial
x}+i\frac{\partial L}{\partial y}=\frac{\partial
L}{\partial(\text{Re}(z))}+i\frac{\partial
L}{\partial(\text{Im}(z))}=(\nabla_{Re(z)}L+i(\nabla_{Im(z)}L)$ (66)
We have used Pytorch to build blocks for CoNO based on the following GitHub
repositories:
1. 1.
Deep Complex Network
2. 2.
Pytorch Complex
### D.4 Hyperparameters
Table 6: Table detailing the hyperparameters used in CoNO in Darcy Flow
Benchmark.
Parameters | Values
---|---
Learning Rate | 0.001
Batch Size | 20
Latent Dim | 64
Padding | 11
Gamma | 0.5
Step-Size | 100
This section details the hyperparameter values used in CoNO for the Darcy Flow
dataset. We will specify the settings and configurations, such as learning
rates, batch sizes, the number of layers, activation functions, and any other
relevant parameters that were employed to optimize the performance of CoNO on
this dataset as shown in Table 6. Further, as shown in Fig. 2 we initialize
$\alpha=1$ and $\alpha^{\prime}=0.5$. We further used pointwise convolution in
$R^{\alpha^{\prime}}$ and linear transformation in $R^{\alpha}$ by truncating
the higher frequency similar as FNO.
### D.5 Mitigation of Aliasing
In Neural Operator learning, the utilization of non-linear operations, such as
non-linear pointwise activations, can introduce high-frequency components into
the output signal. The manifestation of aliasing induced by nonlinearity can
result in distortion of symmetry inherent in the physical signal, leading to
undesirable effects. Additionally, the pursuit of translational invariance, a
key characteristic in neural operators, becomes susceptible to degradation due
to aliasing. We propose a two-step process to address the challenges of
aliasing errors within the continuous equivariance paradigm. Firstly, before
applying any activation function, we employ an upsampling operation on the
input function, exceeding its frequency bandwidth. Subsequently, a non-linear
operation is applied to the upsampled signal, followed by a sinc-based filter
and downsampling. Incorporating the sinc-based low-pass filter effectively
attenuates higher frequency components in the output signal. This method
enhances the accuracy of neural network predictions by minimizing distortions
caused by aliasing, which is especially critical in applications dealing with
complex signal-processing tasks. Although we experimentally haven’t found much
improvement in performance using alias-free activation.
## Appendix E Prediction Visualization
As illustrated in Fig. 10, the performance exhibited by CoNO in predictive
tasks surpasses that of other benchmark datasets, notably outperforming the
state-of-the-art operator LSM. This superiority is particularly evident in
time-dependent and independent partial differential equations (PDEs)
scenarios. CoNO showcases enhanced predictive accuracy and significantly
reduces artifacts. These compelling findings underscore the efficacy of CoNO
as a robust solution for a diverse range of PDE applications, marking a
significant advancement in scientific machine learning.
Figure 10: (Top) Showcase of Darcy Flow (Left) and Navier Stokes (Right).
(Middle) Showcase of Airfoil (Right) and Pipe (Left). (Bottom) Showcase of
Plasticity (Left) and Elasticity (Right). For Comparison of predicted output,
we have plotted the heatmap of the absolute value of the difference between
Ground Truth and Prediction.
## Appendix F Experimental Results
The following section provides an exhaustive examination of the results
obtained from multiple tasks associated with the operator. These tasks
encompass a broad spectrum, including different resolutions, resilience to
noise, performance in out-of-generalization tasks, data efficiency, and
training stability. The comprehensive analysis presented in this section aims
to offer a detailed insight into the performance and capabilities of the
operator across a range of critical aspects, contributing to a nuanced
understanding of its practical utility and effectiveness in diverse scenarios.
All baseline models are instantiated and implemented utilizing the official
source code.
### F.1 Performance Under Different Resolution
To assess the efficacy of our operator at diverse resolutions, we conducted
experiments on Darcy and Navier-Stokes problems across varying spatial
resolutions. Notably, CoNO consistently outperformed other operators across
these resolutions, demonstrating its superior effectiveness. The comparative
results, as presented in Table 7 and Table 8, unequivocally highlight the
robust performance and versatility of CoNO in addressing challenges associated
with different spatial resolutions in the context of both Darcy and Navier-
Stokes scenarios.
Table 7: Neural Operators performance comparison on Darcy Flow under
Different Resolutions.
Resolution | UNET | FNO | MWT | UNO | FFNO | HTNET | LSM | CoNO
---|---|---|---|---|---|---|---|---
32x32 | 0.0059 | 0.0128 | 0.0083 | 0.0148 | 0.0103 | 0.0058 | 0.0049 | 0.0048
64x64 | 0.0052 | 0.0067 | 0.0078 | 0.0079 | 0.0064 | 0.0046 | 0.0042 | 0.0036
128x128 | 0.0054 | 0.0057 | 0.0064 | 0.0064 | 0.0050 | 0.0040 | 0.0038 | 0.0034
256x256 | 0.0251 | 0.0058 | 0.0057 | 0.0064 | 0.0051 | 0.0044 | 0.0043 | 0.0039
512x512 | 0.0496 | 0.0057 | 0.0066 | 0.0057 | 0.0042 | 0.0063 | 0.0039 | 0.0035
1024x1024 | 0.0754 | 0.0062 | 0.0077 | 0.0058 | 0.0069 | 0.0163 | 0.0050 | 0.0044
Table 8: Neural Operators performance comparison on the Navier-Stokes
Benchmark Under Different Resolutions. "/" indicates poor $l2$ error
performance.
Resolution | UNET | FNO | MWT | UNO | FFNO | HTNET | LSM | CoNO
---|---|---|---|---|---|---|---|---
64×64 | 0.1982 | 0.1556 | 0.1541 | 0.1713 | 0.2322 | 0.1847 | 0.1535 | 0.1287
128×128 | / | 0.1028 | 0.1099 | 0.1068 | 0.1506 | 0.1088 | 0.0961 | 0.0817
### F.2 Robustness to Noise
To evaluate the robustness of our operator in the presence of noisy training
input, we systematically conducted experiments across varying input noise
levels. Our objective was to comprehensively understand the impact of noise on
the performance of the proposed operator. Remarkably, our findings revealed
that CoNO, even when subjected to 0.1% data noise, consistently outperformed
LSM—the best-performing operator trained without exposure to noisy data. This
result is a compelling confirmation of the enhanced robustness exhibited by
our proposed operator under challenging conditions involving noisy training
inputs, as highlighted in Table 9.
We observe relative error increases of 22%, 47%, and 28% for CoNO, LSM, and
FNO, respectively. Here, CoNO outperforms all the models. Furthermore, under
0.5% data noise conditions, our findings indicate relative error increases of
62%, 186%, and 45% for CoNO, LSM, and FNO, respectively. Here, it performs
significantly better than LSM but poorer than FNO. However, it should be noted
that the absolute performance of CoNO in relation to FNO is considerably
better. Thus, it can be argued that CoNO is robust against noise at least as
much as FNO, if not better and better LSM, which is the SOTA model.
Table 9: Neural Operators performance comparison on Darcy Flow under
different noise Levels in the training dataset. "/" indicates poor $\ell_{2}$
error performance.
Noise % | UNET | FNO | MWT | UNO | FFNO | HTNET | LSM | CoNO
---|---|---|---|---|---|---|---|---
0 % | 0.0080 | 0.0108 | 0.0082 | 0.0113 | 0.0083 | 0.0079 | 0.0065 | 0.0050
0.001 % | 0.0094 | 0.0114 | 0.0089 | 0.0115 | 0.0089 | 0.0087 | 0.0085 | 0.0056
0.01 % | 0.0105 | 0.0137 | 0.0097 | 0.0121 | 0.0095 | 0.0097 | 0.0087 | 0.0058
0.1 % | 0.0113 | 0.0139 | 0.0125 | 0.0126 | 0.0106 | 0.0124 | 0.0096 | 0.0061
0.5 % | / | 0.0157 | 0.0135 | 0.0138 | 0.0125 | 0.0148 | 0.0186 | 0.0081
### F.3 Out of Distribution Generalization
In this study, we conducted extensive experiments utilizing the Navier-Stokes
dataset, training our model with a viscosity coefficient set to $10^{-5}$.
Subsequently, we rigorously assessed the out-of-distribution generalization
capabilities by subjecting the trained model to a viscosity coefficient of
$10^{-4}$, as depicted in Table 10. Our empirical observations consistently
demonstrate that the CoNO model exhibits a notably superior generalization
performance, showcasing an impressive increment of $64.3\%$ compared to the
FNO model’s performance. Furthermore, our findings underscore the critical
importance of capturing latent variable information, a task effectively
accomplished by the UNET architecture, particularly exemplified by the Latent
Spectral Operator. Significantly, LSM outperforms all other operators,
including CoNO, emphasizing its role in enhancing generalization capabilities
in fluid dynamics modeling.
Table 10: Neural Operators performance comparison on the Navier-Stokes
Benchmark Under Out of distribution performance. Model is trained on NS
viscosity coefficient $1e^{-5}$ and tested on NS viscosity coefficient
$1e^{-4}$. "/" indicates poor $\ell_{2}$ error performance.
NS Viscosity Coefficient | UNET | FNO | MWT | UNO | FFNO | HTNET | LSM | CoNO
---|---|---|---|---|---|---|---|---
$1e^{-5}$ | 0.1982 | 0.1556 | 0.1541 | 0.1713 | 0.2322 | 0.1847 | 0.1535 | 0.1287
$1e^{-4}$ | / | 0.6621 | 0.5864 | 0.5436 | 0.5606 | 0.4888 | 0.1887 | 0.2321
### F.4 Effect of Number of Layers
We conducted experiments to analyze the relationship between the number of
layers and performance, in comparison to FNO, on the Darcy flow dataset. Our
findings, presented in the table below, reveal that performance initially
improves with increased layers in FNO. However, a significant drop in
performance occurs due to the vanishing gradient problem. Importantly, our
method consistently avoids the vanishing gradient problem even with an
increase in the number of layers as shown in Table 11.
Table 11: Neural Operators performance comparison on Darcy Benchmark under different numbers of layers. Number of Layers | 2 | 4 | 6 | 8
---|---|---|---|---
FNO | 0.0114 | 0.0108 | 0.0087 | 0.0098
CoNO | 0.0066 | 0.0052 | 0.0053 | 0.0052
### F.5 Long Term Prediction
In further substantiating our evidence regarding the enhanced stability of
CoNO in long-horizon predictions, we conducted an additional experiment
utilizing the Navier Stokes model with a viscosity coefficient of $10^{-4}$.
Here, we performed forecasts for the subsequent ten steps based solely on the
preceding ten observations and extrapolated the results over ten more steps.
The outcomes of this experiment are presented in the tables below. Our
findings indicate that although not explicitly trained for predicting the
subsequent 20 timestamps, CoNO consistently outperforms LSM and FNO in
extrapolating beyond the prediction horizon. This observation underscores the
heightened stability and robustness of CoNO in long-horizon prediction tasks
as shown in Table 12.
Table 12: A comparative analysis of the performance of Neural Operators on the Navier-Stokes equations, with a viscosity coefficient of $10^{-4}$, involves training models to predict the subsequent 10 timestamps based on the preceding 10 timestamps. Subsequently, the extrapolated results are utilized to forecast the subsequent 10 timestamps. Neural Operator | $T=5$ | $T=10$ | $T=15$ | $T=20$
---|---|---|---|---
FNO | 0.028 | 0.050 | 0.17 | 0.34
LSM | 0.065 | 0.13 | 0.24 | 0.38
CoNO | 0.020 | 0.045 | 0.14 | 0.31
### F.6 Data Efficiency
As shown in Table 13, the performance of CoNO is superior, showcasing its
competitive capabilities comparable to the second-best operator LSM when
trained on 60% of the available data. Remarkably, across diverse training
dataset ratios, CoNO consistently surpasses all other operators, emphasizing
its remarkable data efficiency compared to state-of-the-art (SOTA) operators.
This observation underscores the efficacy and robustness of CoNO across
varying training scenarios, positioning it as a noteworthy solution in the
realm of operator-based learning. With a data ratio of 0.6, we observed
relative error increases of 34%, 44%, and 35% for CoNO, LSM, and FNO,
respectively. Thus, it is evident that our approach consistently outperforms
the second-best operator, LSM, across various ratio settings. Further, the
absolute performance of CoNO is significantly better than that of FNO.
Table 13: Neural Operators performance comparison on Darcy Benchmark under
different training dataset ratios. "/" indicates poor $\ell_{2}$ error
performance.
Ratio | UNET | FNO | MWT | UNO | FFNO | HTNET | LSM | CoNO
---|---|---|---|---|---|---|---|---
0.2 | / | 0.2678 | 0.2854 | 0.2734 | 0.2573 | 0.2564 | 0.2465 | 0.2234
0.4 | / | 0.0176 | 0.0165 | 0.0183 | 0.0153 | 0.0145 | 0.0138 | 0.0105
0.6 | 0.1234 | 0.0146 | 0.0113 | 0.0153 | 0.0142 | 0.0105 | 0.0094 | 0.0067
0.8 | 0.0107 | 0.0122 | 0.0096 | 0.0134 | 0.0095 | 0.0094 | 0.0082 | 0.0056
1.0 | 0.0080 | 0.0108 | 0.0082 | 0.0113 | 0.0077 | 0.0079 | 0.0065 | 0.0050
### F.7 Inference Time Comparison
The section examines inference time complexity across diverse neural operators
applied to the Darcy Flow benchmark dataset within the Fluid Physics field. It
aims to elucidate these operators’ computational efficiency and efficacy in
modeling fluid flow phenomena, as shown in Table 14.
Table 14: The evaluation of inference time complexity across various neural operators on the Darcy Flow benchmark dataset in the domain of Fluid Physics. Neural Operator | UNET | UFNO | FNO | WMT | LSM | CoNO
---|---|---|---|---|---|---
Inference (s) | 0.035 | 0.042 | 0.135 | 0.045 | 0.020 | 0.055
## Appendix G Limitation and Future Work
To enhance our understanding of CoNO, it is crucial to explore its
mathematical and algorithmic principles thoroughly. We aim to uncover the
latent space’s learning mechanisms and build a solid theoretical foundation
for complex operators. Our research highlights several key challenges that
need careful investigation, including refining initialization procedures for
fractional orders, designing streamlined architectures for complex neural
operators, developing equivariant complex operators, and understanding the
role of the Fractional Fourier Transform (FrFT) in the continuous dynamics of
complex systems. Additionally, our work raises questions about creating
foundational models for Partial Differential Equations (PDEs). These research
directions offer opportunities to further understand CoNO and contribute to
the broader field of Scientific Machine Learning (SciML).
## Appendix H Broader Impact
This paper introduces innovative tools to revolutionize scientific machine
learning (SciML). By integrating Complex Neural Networks (CVNNs) and the
Fractional Fourier Transform (FrFT) into the Neural Operator framework, we
offer a novel approach to addressing the complex challenges of partial
differential equations (PDEs), particularly Fractional PDEs, which lack
explicit differential forms and are common in natural phenomena [1, 38]. Our
research has significant implications for diverse fields such as biology,
physics, and civil engineering, addressing a crucial scientific problem and
paving the way for transformative advancements in interdisciplinary problem-
solving. There is no serious ethical issue of the proposed methodology.
|
The Secret Hyperbolic Life of Positive Scalar Curvature
Joachim Lohkamp
Mathematisches Institut, Universität Münster, Einsteinstrasse 62, Germany
_e-mail<EMAIL_ADDRESS>
Abstract This survey introduces to the _hyperbolic unfolding_ correspondence
that links the geometric analysis of minimal hypersurfaces with that of Gromov
hyperbolic spaces. Problems caused from hypersurface singularities oftentimes
become solvable on associated Gromov hyperbolic spaces. Applied to scalar
curvature geometry this yields _smoothing schemes_ that eliminate such
singularities. We explain the two main lines of such smoothings: a top-down
singular analysis using iterated blow-ups of the given hypersurface and
bottom-up smoothings following the tree of blow-ups backwards.
###### Contents
1. 1 Introduction
2. 2 Singularities as Regular Boundaries
1. 2.1 Hyperbolic Unfoldings
2. 2.2 Reduction to Hyperbolic Geodesics
3. 2.3 Applications in Geometric Analysis
3. 3 Minimal Splitting Factors - Top-Down Analysis
1. 3.1 Minimal Splitting Factors
2. 3.2 Stable Isoperimetry - Hyperbolic Geodesics Again
4. 4 Smoothing Techniques - Bottom-up Constructions
1. 4.1 Surgeries Revisited
2. 4.2 Inductive Removal of Singularities
3. 4.3 From Ordinary to Locally Finite Homology
## 1 Introduction
The story began in the 70s when Hawking [H, Ch. 4] and Schoen and Yau [SY1]
discovered the first cases of a remarkable $scal>0$-heredity one may state, in
dimensions $n\geq 2$, as follows:
_Given a compact manifold $M^{n+1}$ with $scal>0$, any area minimizing
hypersurface $(H^{n},g_{H})\subset(M^{n+1},g_{M})$ also has $scal>0$, after
conformal deformations of $g_{H}$._
We indicate how to derive this powerful yet simple result. The first and the
second variation of $Area(H)$ of a hypersurface $H\subset M$ by a $f\cdot\nu$,
where $\nu$ is the outward normal vector field of $H$, $f\in
C^{\infty}(H,\mathbb{R})$ with $supp\>f\subset reg(H)$ (= the set of regular
points of $H$), are given by
$\displaystyle Area^{\prime}(f)$ $\displaystyle=\int_{H}trA_{H}(z)\cdot
f(z)\>dVol,$ $\displaystyle Area^{\prime\prime}(f)$
$\displaystyle=\int_{H}|\nabla_{H}f|^{2}+\left((trA_{H})^{2}-|A|^{2}-Ric(\nu,\nu)\right)\cdot
f^{2}\>dVol$
where $trA_{H}$ is the mean curvature of $H$, $|A|^{2}$ is the sum of the
squares of principal curvatures of $H$ and $Ric$ the Ricci tensor of $M$.
Since $H$ is supposed to be area minimizing, we have $Area^{\prime}(f)=0$ and
$Area^{\prime\prime}(f)\geq 0$. That is, $trA_{H}=0$ and this gives
(1) $\quad
Area^{\prime\prime}(f)=\int_{H}|\nabla_{H}f|^{2}-\left(|A|^{2}+Ric(\nu,\nu)\right)\cdot
f^{2}\>dVol\geq 0$
Now we recall the Gauß–Codazzi equations for hypersurfaces
$\small|A|^{2}+Ric(\nu,\nu)=1/2\cdot\left(|A|^{2}+scal_{M}-scal_{H}+(trA_{H})^{2}\right)$
where $scal_{H}$ and $scal_{M}$ denote the scalar curvature of $H$ and $M$.
Since $trA_{H}=0$ we may rewrite (1) as follows (here we assume $n\geq 3$, the
computations slightly deviate when $n=2$):
(2) $\int_{H}|\nabla f|^{2}+\frac{n-2}{4(n-1)}\cdot scal_{H}\cdot f^{2}dA\\\
\geq\int_{H}\frac{n}{2(n-1)}\cdot|\nabla
f|^{2}+\frac{n-2}{4(n-1)}\cdot\left(|A|^{2}+scal_{M}\right)\cdot f^{2}dA.$
The left hand side of $(\ref{3})$ is the variational integral for the first
eigenvalue $\lambda_{1}$ of the conformal Laplacian $L_{H}$ on $H$. That is,
when $scal_{M}>0$, inequality $(\ref{3})$ shows that $\lambda_{1}>0$.
Following Kazdan and Warner [KW], the transformation law for scalar curvature
under conformal transformation by the first eigenfunction $f_{1}>0$ can be
used to show that
$scal(f_{1}^{4/(n-2)}\cdot g_{H})\cdot
f_{1}^{\frac{n+2}{n-2}}=L_{H}(f_{1})=\lambda_{1}\cdot f_{1}>0.$
In other words, we observe a reproduction of the scalar curvature constraint
on $M$ on the lower dimensional space $H$, also called a scalar curvature
_splitting factor_ of $M$. This suggests an appealing and versatile strategy
to study scalar curvature from an inductive dimensional splitting until one
reaches a space already understood.
The problem is that only in low dimensions $n\leq 7$ these hypersurfaces are
smooth submanifolds. The ultimate goal of this paper is to explain how to use
hyperbolic geometry to get such smooth splitting factors in arbitrary
dimensions. The basic reference are the papers [L1]–[L5]. The program is
illustrated in the following diagram.
Figure 1: The hyperbolic unfolding gives use means to control the elliptic
analysis on $H$ but it also reveals some new geometric structures like
canonical Semmes families on $H$.
Hyperbolic geometry is extensively used right from the beginning, in chapter
2. The smoothings occupy chapters 3 and 4.
As a general remark, the theory we present naturally extends to considerably
larger classes of almost minimizers. Some of them do not even result from
variational principles. Nevertheless, for the most part we are talking about
area minimizers to simplify the exposition.
## 2 Singularities as Regular Boundaries
The problem we encounter in dimensions $\geq 8$ is how to control $L_{H}$ near
the singular set of $H$. The critical dimension $8$ comes from classical
regularity theory. It says that an area minimizing hypersurface $H^{n}$ that
is smooth outside a potentially complicated singular set $\Sigma_{H}\subset
M^{n+1}$ of codimension $\geq 8$. To overcome these subtleties we reinterpret
$\Sigma_{H}$ as a boundary of $H\setminus\Sigma_{H}$ and show that, relative
to $H\setminus\Sigma_{H}$, the singular set $\Sigma_{H}$ becomes regular, in a
sense we explain below.
We make use of some pieces of geometric measure theory but we neither employ
any structural details of $\Sigma$ nor the full codimension estimate but only
that $\Sigma$ has codimension $\geq 3$.
Motivation To motivate and properly describe the adequate notion of regularity
of $\Sigma$ we consider the case of Euclidean domains. The validity of the
boundary Harnack inequality on a Euclidean domain $G\subset\mathbb{R}^{n}$, in
(3) below, is a litmus test for a reasonable elliptic analysis on $G$. It
asserts that there are constants $A(G)$, $C(G)>1$ such that for any
$p\in\partial G$, small $R>0$ and any two harmonic functions $u$, $v>0$ on
$B_{A\cdot R}(p)\cap G$ which vanish along $B_{A\cdot R}(p)\cap\partial G$,
(3) $u(x)/v(x)\leq C\cdot u(y)/v(y)\mbox{ for all }x,\,y\in B_{R}(p)\cap G.$
Remarkably, there is a purely geometric way to characterize such domains. The
validity of (3) is essentially equivalent [Ai] to the regularity condition
that $G$ is a uniform domain.
###### Definition 2.1
A Euclidean domain $G\subset\mathbb{R}^{n}$ is called uniform if there is some
$c\geq 1$ such that any two points $p,q\in G$ can be joined by a uniform
curve. This is a rectifiable curve $\gamma_{p,q}:[a,b]\rightarrow G$ going
from $p$ to $q$ such that:
* •
_Quasi-geodesic:_ $l(\gamma_{p,q})\leq c\cdot d(p,q)$.
* •
_Twisted double cones:_ For any $z\in\gamma_{p,q}$ let
$l_{min}(\gamma_{p,q}(z))$ be the minimum of the lengths of the two subcurves
of $\gamma_{p,q}$ from $p$ to $z$ and from $z$ to $q$. Then
$l_{min}(\gamma_{p,q}(z))\leq c\cdot dist(z,\partial X).$
One may describe uniformity as a quantitative and scaling invariant form of
connectivity.
$\mathcal{S}$-Structures Turning to $H\setminus\Sigma_{H}$ relative to its
boundary $\Sigma_{H}$, we observe that $G\subset\mathbb{R}^{n}$ is flat until
we reach $\partial G$ whereas $H\setminus\Sigma_{H}$ degenerates towards
$\Sigma_{H}$ with diverging second fundamental form. The isoperimetric
inequality for the area minimizer $H$ allows us to compensate the additional
twist from the divergence of second fundamental form and to establish the even
stronger $\mathcal{S}$-uniformity of $H$. To make this precise we introduce a
measure $\langle A\rangle_{H}$ for this degeneration. An assignment
$H\mapsto\langle A\rangle_{H}$ of a non-negative, measurable function to any
connected area minimizing hypersurface $H$ is called an
$\mathcal{S}$-transform provided
* •
$\langle A\rangle_{H}$ is naturally assigned to $H$, in other words, the
assignment commutes with the convergence of sequences of underlying area
minimizers.
* •
$\langle A\rangle_{H}\geq|A_{H}|$ with $\langle A\rangle_{H}\equiv 0$, if
$H\subset M$ is totally geodesic, otherwise, $\langle A\rangle_{H}$ is
strictly positive.
* •
When $H$ is not totally geodesic, the $\mathcal{S}$-distance $\delta_{\langle
A\rangle}:=1/\langle A\rangle$ is well-defined and it is $L_{\langle
A\rangle}$-Lipschitz regular, for some constant $L_{\langle
A\rangle}=L(\langle A\rangle,n)>0$:
(4) $|\delta_{\langle A\rangle}(p)-\delta_{\langle A\rangle}(q)|\leq
L_{\langle A\rangle}\cdot d(p,q),\mbox{ for }p,q\in H\setminus\Sigma.$
The $\mathcal{S}$-distance is a measure for the distance to singular and
highly curved portions of $H$ that takes also the curvature into account.
Constructions of $\mathcal{S}$-transforms are given in [L1] where we merge
$g_{H}$ and $A_{H}$ in a suitable way. The geometric main application is the
following result.
###### Theorem 2.2
There exists some $c>0$ such that $H\setminus\Sigma$ is an
$\mathcal{S}$-uniform space. This means that any pair $p,q\in
H\setminus\Sigma$ can be joined by an $\mathcal{S}$-uniform curve in
$H\setminus\Sigma$, i.e., a rectifiable curve $\gamma_{p,q}:[a,b]\rightarrow
H\setminus\Sigma$ with $\gamma_{p,q}(a)=p$, $\gamma_{p,q}(b)=q$ and so that:
* •
_Quasi-geodesic:_ $l(\gamma)\leq c\cdot d(p,q).$
* •
_Twisted double $\mathcal{S}$-cones:_ $l_{min}(\gamma_{p,q}(z))\leq
c\cdot\delta_{\langle A\rangle}(z)$ for any $z\in\gamma_{p,q}$.
Figure 2: Schematic view of $H$, where $\Sigma$ is just the bottom line. The
cold colors indicate low and the hot color high curvature of $H$.
$\mathcal{S}$-uniform curves are also sensitive to the underlying curvature.
Some Remarks In this theory the totally geodesic hypersurfaces play the rôle
of the trivial case. They are always smooth submanifolds [L1, Corollary A.6]
and in the non-compact case they are just Euclidean hyperplanes. In this case,
many results in this paper are either obvious or they degenerate to
conventions.
The Lipschitz regular $\mathcal{S}$-distance $\delta_{\langle A\rangle}$
admits a Whitney type $C^{\infty}$-smoothing $\delta_{\langle A\rangle^{*}}$
which satisfies (S1)–(S3) and is quasi-natural in the sense that
$c_{1}\cdot\delta_{\langle A\rangle}(x)\leq\delta_{\langle
A\rangle^{*}}(x)\leq c_{2}\cdot\delta_{\langle A\rangle}(x)$, for some
constants $c_{1}$, $c_{2}>0$, cf. [L1, Proposition B.3].
We note in passing that in older papers, now superseded by [L1]–[L3], we used
the term _skin transform_ for the subclass of $\mathcal{S}$-transforms that
includes a so-called Hardy inequality, now called _Hardy
$\mathcal{S}$-transforms_, cf. [L3]. The renaming had no deeper reasons. The
Hardy inequality is only needed in applications but not in the basic theory.
Secondly, originally the level sets of $\langle A\rangle_{H}$, the _$|A|$
-skins_, served important technical purposes now covered from functional
relations we have for $\langle A\rangle_{H}$.
### 2.1 Hyperbolic Unfoldings
Remarkably, the uniformity of $G$ is also equivalent to the purely geometric
condition that the quasi-hyperbolic metric, which we get from conformally
deforming the Euclidean metric $g_{Eucl}$ to $dist(\cdot,\partial G)^{-2}\cdot
g_{Eucl}$, is Gromov hyperbolic, has bounded geometry and its Euclidean
boundary is homeomorphic to the Gromov boundary, cf. [BHK]. To make this
plausible, we first observe that a (twisted) cone conformally deformed by
$dist(\cdot,\partial G)^{-2}$ roughly looks like a piece of a hyperbolic space
(a generalized Poincaré metric) and the scaling invariance of the uniformity
conditions globalizes this to the whole space.
We will see that an area minimizer $H$ with singular set $\Sigma$ (= boundary
of $H\setminus\Sigma$) has a similar hyperbolic nature. This is more
challenging since $H\setminus\Sigma$ degenerates while we approach $\Sigma$.
In zeroth order this can be compared with the extension of the Riemann
uniformization from complex domains to arbitrary Riemann surfaces.
For starters we recall the notions of Gromov hyperbolicity and of bounded
geometry. It has no local impact but strong consequences for the geometry near
infinity. We mention [BH] as a general reference.
###### Definition 2.3
A locally compact geodesic metric space $X$ is Gromov hyperbolic if its
geodesic triangles are $\mathbf{\delta}$-thin for some $\delta=\delta_{X}>0$.
That is, each point on the edge of any geodesic triangle is within
$\delta$-distance of one of the other two edges.
Two rays in $X$ are _equivalent_ if they have finite Hausdorff distance. The
set $\partial_{G}X$ of equivalence classes $[\gamma]$ of geodesic rays from a
fixed base point $p\in X$ is called the Gromov boundary of $X$. This
definition of $\partial_{G}X$ is independent of $p$. The space
$\overline{X}_{G}=X\cup\partial_{G}X$ admits a natural topology that makes it
a compact metrizable space. It is called the Gromov compactification of $X$.
###### Definition 2.4
A Riemannian manifold $M$ has $(\varrho,\ell)$-bounded geometry if there exist
constants $\varrho={\varrho_{M}}>0$ and $\ell=\ell_{M}\geq 1$ for $M$ such
that for each ball $B_{\varrho}(p)\subset M$ there is a smooth $\ell$-bi-
Lipschitz chart $\phi_{p}$ onto an open set $U_{p}\subset\mathbb{R}^{n}$ with
its Euclidean metric.
The $\mathcal{S}$-metric on $H\setminus\Sigma$ In general the quasi-hyperbolic
metric $dist(\cdot,\Sigma_{H})^{-2}\cdot g_{H}$ is not well-behaved on
$H\setminus\Sigma$. It neither has bounded geometry nor does it have a good
blow-up behavior, that is, the corresponding quasi-hyperbolic metric on
tangent cones of points in $\Sigma_{H}\subset H$ does not approximate that on
$H\setminus\Sigma$. The key point about $\mathcal{S}$-uniformity of $H$ is
that it implies all the desirable properties for the so-called
$\mathcal{S}$-metric $d_{\langle A\rangle}=d_{\langle A\rangle_{H}}$ defined
by
(5) $d_{\langle A\rangle}(x,y):=\inf\Bigl{\\{}\int_{\gamma}\langle
A\rangle\,\,\Big{|}\,\gamma\subset H\setminus\Sigma\mbox{ rectifiable curve
joining }x\mbox{ and }y\Bigr{\\}}$
for $x$, $y\in H\setminus\Sigma$. The metric $d_{\langle A\rangle}$ is even
well-defined for _smooth_ $H$ where $\Sigma=\emptyset$. Alternatively, the
$\mathcal{S}$-metric can be written $\langle A\rangle^{2}\cdot g_{H}$, but
this is not a regular Riemannian metric since $\langle A\rangle$ is merely a
locally Lipschitz function. However, one may use the Whitney type
$C^{\infty}$-smoothing $\delta_{\langle A\rangle^{*}}$ if one needs a smooth
version. Now we can formulate the following hyperbolization result, cf. [L1,
Theorem 1.11, Proposition 3.10 and Theorem 1.13].
###### Theorem 2.5
The $\mathcal{S}$-metric $d_{\langle A\rangle}$ has the following properties:
* •
The metric space $(H\setminus\Sigma,d_{\langle A\rangle})$ and its quasi-
isometric Whitney smoothing, i.e., the smooth Riemannian manifold
$(H\setminus\Sigma,d_{\langle
A\rangle^{*}})=(H\setminus\Sigma,1/\delta_{\langle A\rangle^{*}}^{2}\cdot
g_{H})$, are complete Gromov hyperbolic spaces with bounded geometry.
* •
$d_{\langle A\rangle}$ is natural, that is, the assignment $H\mapsto
d_{\langle A\rangle_{H}}$ commutes with compact convergence of regular domains
of the underlying area minimizers. The typical example is the blow-up around
singular points and the resulting tangent cone approximations.
* •
For any _singular_ $H$ the identity map on $H\setminus\Sigma$ extends to
homeomorphisms
$\displaystyle\widehat{H}$
$\displaystyle\cong\overline{(H\setminus\Sigma,d_{\langle
A\rangle})}_{G}\cong\overline{(H\setminus\Sigma,d_{\langle
A\rangle^{*}})}_{G}\,\mbox{ and }$ $\displaystyle\widehat{\Sigma}$
$\displaystyle\cong\partial_{G}(H\setminus\Sigma,d_{\langle
A\rangle})\cong\partial_{G}(H\setminus\Sigma,d_{\langle A\rangle^{*}}),$
where for $X=(H\setminus\Sigma,d_{\langle A\rangle})$ or
$(H\setminus\Sigma,d_{\langle A\rangle^{*}})$, $\overline{X}_{G}$ and
$\partial_{G}(X)$ denote the Gromov compactification and the Gromov boundary,
respectively.
The spaces $(H\setminus\Sigma,d_{\langle A\rangle})$ and
$(H\setminus\Sigma,d_{\langle A\rangle^{*}})$ are conformally equivalent to
the original space $(H\setminus\Sigma,g_{H})$. We refer to both these spaces
as hyperbolic unfoldings of $(H\setminus\Sigma,g_{H})$.
Here, $\widehat{H}$ and $\widehat{\Sigma}$ denote the one-point
compactifications of $H$ and $\Sigma$ in the non-compact case of Euclidean
hypersurfaces with the extra condition that for $H\subset\mathbb{R}^{n+1}$ we
always add the point $\infty$ to $\Sigma$, even for compact $\Sigma$.
### 2.2 Reduction to Hyperbolic Geodesics
Hyperbolic unfoldings reveal that the potential theory of elliptic operators
$L$, like the conformal Laplacian or the Jacobi field operator, on area
minimizers conforms to the underlying geometry. Intuitively, the evolution of
the minimal Green’s function $\widetilde{G}$ of $\widetilde{L}=\delta_{\langle
A\rangle^{*}}^{2}\cdot L$ on $(H\setminus\Sigma,d_{\langle A\rangle^{*}})$
concentrates along hyperbolic geodesics and this it largely controls the
global analysis of $\widetilde{L}$. (The minimal Green’s function
$\widetilde{G}$ of $\widetilde{L}$, $G(x,y)>0$ is a smallest function on
$M\times M$, singular on $\\{(x,x)\>|\>x\in M\\}$, satisfying the equation
$\widetilde{L}\,G(\cdot,y)=\delta_{y}$, where $\delta_{y}$ is the Dirac
function in $y$.)
This focussing/squeezing effect of hyperbolic geodesics is reflected in so-
called $\boldsymbol{3G}$-inequalities, due to the triple appearance of $G$ in
one inequality, one may interpret as follows:
_Let $x,y,z\in(H\setminus\Sigma,d_{\langle A\rangle^{*}})$ such that $y$ lies
on a hyperbolic geodesic connecting $x$ and $z$. Then, up to universal
constants, there are as many “Brownian particles” travelling directly from $x$
to $z$, measured by $\widetilde{G}(x,z)$, as there are particles travelling
from $x$ to $y$, measured by $\widetilde{G}(x,y)$, and then from $y$ to $z$,
measured by $\widetilde{G}(y,z)$. That is, we have
$\widetilde{G}(x,z)\approx\widetilde{G}(x,y)\cdot\widetilde{G}(y,z)$._
The potential theory on hyperbolic manifolds of bounded geometry naturally
extends to hyperbolic graphs of bounded valence, where the constant $\delta$
measures the deviation of the graph from a tree. In turn, we can approximate
$(H\setminus\Sigma,d_{\langle A\rangle^{*}})$, along with its potential
theory, by such graphs. This formalizes the intuition of a reduction to
hyperbolic geodesics.
The admissible operators on $(H\setminus\Sigma,d_{\langle A\rangle^{*}})$ to
make this work are the adapted weakly coercive operators. These are the linear
second order elliptic operators $L$ which are uniformly elliptic on
$(H\setminus\Sigma,d_{\langle A\rangle^{*}})$ and weakly coercive. That is,
there is a $u>0$ with $L\,u\geq\varepsilon\cdot u$ for some $\varepsilon>0.$
For an exposition of this potential theory on Gromov hyperbolic spaces, which
is largely due to Ancona [A1, A2], see [KL] for an exposition.
An interesting aspect of this theory is that the geometric and the analytic
conditions work hand in hand: the pair of conditions _bounded geometry_
$\leftrightarrow$ _uniformly elliptic_ is employed for most of the basic
estimates and relations and then these results are critically improved from a
tandem use of _Gromov hyperbolicity_ $\leftrightarrow$ _weak coercivity_.
The point about hyperbolic unfoldings is that the transparent potential theory
we have on $(H\setminus\Sigma,d_{\langle A\rangle^{*}})$ transfers one-by-one
to corresponding classes of operators on the original area minimizer. This way
we get the same type of potential theoretic results we have for uniform
Euclidean domains on the $\mathcal{S}$-uniform spaces $H\setminus\Sigma$ with
the difference that we have literally outsourced all the analysis to the
hyperbolic unfolding as our workbench. This is the unfolding corrrespondence
and the relevant operators $L$ on $(H\setminus\Sigma,g_{H})$ are called
_$\mathcal{S}$ -adapted_ provided their counterpart $\delta_{\langle
A\rangle^{*}}^{2}\cdot L$ on $(H\setminus\Sigma,d_{\langle A\rangle^{*}})$ is
adapted weakly coercive. More explicitly, we set
###### Definition 2.6
An elliptic operator $L$ is $\mathcal{S}$-adapted when
$-L(u)=\sum_{i,j}a_{ij}\cdot\frac{\partial^{2}u}{\partial x_{i}\partial
x_{j}}+\sum_{i}b_{i}\cdot\frac{\partial u}{\partial x_{i}}+c\cdot u$ so that
for some $k\geq 1$ and suitable charts
(6)
$k^{-1}\cdot\sum_{i}\xi_{i}^{2}\leq\sum_{i,j}a_{ij}(p)\cdot\xi_{i}\xi_{j}\leq
k\cdot\sum_{i}\xi_{i}^{2},$ (7) $\delta^{\beta}_{\langle
A\rangle}(p)\cdot|a_{ij}|_{C^{\beta}(B_{\Theta(p)}(p))}\leq k,\delta_{\langle
A\rangle}(p)\cdot|b_{i}|_{L^{\infty}(B_{\Theta(p)}(p))}\leq k\mbox{ and }\\\
\delta^{2}_{\langle A\rangle}(p)\cdot|c|_{L^{\infty}(B_{\Theta(p)}(p))}\leq
k,\mbox{ for }\Theta(p)=c/\langle A\rangle(p),\mbox{ for some }c(H)>0.$
and there is a $u>0$ with $L\,u\geq\varepsilon\cdot\langle A\rangle^{2}\cdot
u\mbox{ for some }\varepsilon>0.$
A frequently considered type of elliptic problems is that of eigenvalues,
typically for symmetric operators. For them it is possible and useful to bring
the weak coercivity condition into a variational form. Then the weak
coercivity condition is equivalent to the existence of some positive constant
$\tau>0$ such that the Hardy type inequality
(8) $\int_{H}f\cdot Lf\,dV\,\geq\,\tau\cdot\int_{H}\langle A\rangle^{2}\cdot
f^{2}dV\mbox{ holds for any }f\in C^{\infty}_{0}(H\setminus\Sigma).$
That is, $L$ has an $\langle A\rangle$-weighted positive first eigenvalue
$\lambda^{\langle A\rangle}_{1}(L)>0$, in this singular case commonly called
the principal eigenvalue. We also mention that in the singular case, in sharp
contrast to the smooth case, we have positive $\langle A\rangle$-weighted
eigenfunctions for any $\lambda<\lambda^{\langle A\rangle}_{1}(L)$.
The boundary Harnack inequality we get for $\mathcal{S}$-adapted $L$ along the
boundary $\widehat{\Sigma}$ of $(H\setminus\Sigma,g_{H})$ differs in two ways
from that in the case of uniform domains equipped with the Laplacian.
* •
In place of the balls of (3), we choose hyperbolic halfspaces
$\mathcal{N}^{\delta}_{i}(z)$ in $(H\setminus\Sigma,d_{\langle
A\rangle^{*}})$, with completions $\mathbf{N}^{\delta}_{i}(z)$, uniformly
contracting to $z\in\widehat{\Sigma}$ for $i\rightarrow\infty$. One may
compare this with the Poincaré disc model on $B_{1}(0)\subset\mathbb{R}^{2}$
where the $B_{\rho}(p)\cap B_{1}(0)$, $\rho\in(0,1)$, for flat discs
$B_{\rho}(p)$, $p\in\partial D$, become hyperbolic halfspaces.
* •
Solutions $u>0$ of $L\,f=0$ will usually diverge to infinity when we approach
$\Sigma$. The generalization of the vanishing boundary data is that of
solutions of minimal growth towards $\Sigma_{H}$ when compared to other
solutions $v>0$.
###### Theorem 2.7
There exists a constant $C(H,L)>1$ such that for any $z\in\widehat{\Sigma}$
and any two solutions $u$, $v>0$ of $L\,f=0$ on $H\setminus\Sigma$ with
minimal growth along $\mathbf{N}^{\delta}_{i}(z)\cap\widehat{\Sigma}$, we have
(9) $u(x)/v(x)\leq C\cdot u(y)/v(y)\mbox{ \emph{for all}
}x,\,y\in\mathcal{N}^{\delta}_{i+1}(z).$
Examples To relate the general setup of $\mathcal{S}$-adapted operators to
some geometrically relevant examples, we upgrade the $\mathcal{S}$-transform
$\langle A\rangle_{H}$ to a Hardy $\mathcal{S}$-transform, cf. [L3]. This is
an $\mathcal{S}$-transform that additionally satisfies the following Hardy
type inequality: for any $f\in C^{\infty}(H\setminus\Sigma,\mathbb{R})$
compactly supported in $H\setminus\Sigma$ we have
$\int_{H}|\nabla f|^{2}+|A_{H}|^{2}\cdot f^{2}dA\geq\tau\cdot\int_{H}\langle
A\rangle_{H}^{2}\cdot f^{2}dA,\mbox{ for some }\tau=\tau(\langle
A\rangle,H)\in(0,1).$
Let $H^{n}\subset M^{n+1}$ be a singular area minimizer. Then we have for any
Hardy $\mathcal{S}$-transform:
* •
For $scal_{M}\geq 0$ we have $\lambda^{\langle A\rangle}_{1}(L_{H})>0$ for the
conformal Laplacian $L_{H}$.
* •
We have $\lambda^{\langle A\rangle}_{1}(J_{H})\geq 0$ for the Jacobi field
operator $J_{H}:=-\Delta_{H}-|A|^{2}-Ric_{M}(\nu,\nu)$.
On area minimizers, we can turn any operator that satisfies (6) and (7) into
an $\mathcal{S}$-adapted operator $L$ provided $\lambda^{\langle
A\rangle}_{1}(L)>-\infty$ since $L-\lambda\cdot\langle A\rangle^{2}\cdot Id$
becomes $\mathcal{S}$-adapted for $\lambda<\lambda^{\langle A\rangle}_{1}(L)$.
### 2.3 Applications in Geometric Analysis
A basic application of the boundary Harnack inequality is a transparent Martin
theory.
###### Definition 2.8
Let $X$ be a non-compact Riemannian manifold and $L$ be a linear second order
elliptic operator on $X$ with a minimal Green’s function $G:X\times
X\rightarrow(0,\infty]$. We choose a base point $p$ and consider the space $S$
of sequences $s=\\{p_{n}\\}$ in $X$, $n\geq 1$, such that
* •
$s$ has no accumulation points in $X$.
* •
$K(x,p_{n}):=G(x,p_{n})/G(p,p_{n})\rightarrow K_{s}(x)$ compactly to some
function $K_{s}$ on $X$ as $n\rightarrow\infty$.
The Martin boundary $\partial_{M}(X,L)$ is the quotient of $S$ modulo the
following relation on $S$: $s\sim s^{*}$ if and only if $K_{s}\equiv
K_{s^{*}}$. Moreover, we define the Martin kernel $k(x;y)$ on
$X\times\partial_{M}(X,L)$ by $k(x;y):=K_{s}(x)$, for some sequence $s$
representing $y\in\partial_{M}(X,L)$. As for the Gromov boundary, these
definitions do not depend on the choice of the base point $p$.
The (metrizable) Martin topology on $\overline{X}_{M}:=X\cup\partial_{M}(X,L)$
is defined through the convergence of the function $K(x,p_{n})$.
$\overline{X}_{M}$ and $\partial_{M}(X,L)$ turn out to be compact.
$\overline{X}_{M}$ is called the Martin compactification of $(X,L)$. A point
in $S_{L}(X)$ is called extremal if it cannot be written as a non-trivial
convex combination of other points of $S_{L}(X)$. The subset
$\partial^{0}_{M}(X,L)\subset\partial_{M}(X,L)$ of extremal functions
belonging to $\partial_{M}(X,L)$ is called the minimal Martin boundary.
The point about $\partial^{0}_{M}(X,L)$ is that for any solution $v>0$ of
$L\,f=0$ we have a unique finite Radon measure $\mu=\mu(v)$ on
$\partial^{0}_{M}(X,L)$ so that
(10) $v(x)=\int_{\partial^{0}_{M}(X,L)}k(x;y)\,d\mu(y).$
The problem with this Martin integral is that a proper understanding of
$\partial^{0}_{M}(X,L)$ can be very difficult. Different from classical
contour formulas, $\partial_{M}(X,L)$ and $\partial^{0}_{M}(X,L)$ may strongly
depend on $L$ and they usually differ from intrinsically defined topological
boundaries of $X$. Remarkably, these problems disappear for
$\mathcal{S}$-adapted operators on area minimizers.
###### Theorem 2.9
For any $\mathcal{S}$-adapted operator $L$ on $H\setminus\Sigma$ we have
* •
the identity map on $H\setminus\Sigma$ extends to a homeomorphism between
$\widehat{H}$ and the Martin compactification
$\overline{(H\setminus\Sigma)}_{M}$.
* •
all Martin boundary points are minimal:
$\partial^{0}_{M}(H\setminus\Sigma,L)\equiv\partial_{M}(H\setminus\Sigma,L)$.
Thus, $\widehat{\Sigma}$ and the minimal Martin boundary
$\partial^{0}_{M}(H\setminus\Sigma,L)$ are homeomorphic.
Quantitative Results The unfolding correspondence is not restricted to the
transition between $L$ on $(H\setminus\Sigma,g_{H})$ and $\delta_{\langle
A\rangle^{*}}^{2}\cdot L$ on $(H\setminus\Sigma,d_{\langle A\rangle^{*}})$.
Inspired from the Doob transform ín stochastic analysis we define
(11) $L^{\mathcal{S}}\phi:=\delta^{(n+2)/2}_{\langle A\rangle^{*}}\cdot
L({\delta^{-(n-2)/2}_{\langle A\rangle^{*}}}\cdot\phi)\mbox{ for sufficiently
regular }\phi\mbox{ on }(H\setminus\Sigma,d_{\langle A\rangle^{*}})$
We call $L^{\mathcal{S}}$ the $\mathcal{S}$-Doob Transform of $L$. The minimal
Green’s functions $G$ of an $\mathcal{S}$-adapted $L$ on
$(H\setminus\Sigma,d_{H})$ and $G^{\mathcal{S}}$ of the adapted weakly
coercive $L^{\mathcal{S}}$ on $(H\setminus\Sigma,d_{\langle A\rangle^{*}})$
satisfy
(12) $G^{\mathcal{S}}(x,y)=\delta_{\langle
A\rangle^{*}}^{(n-2)/2}(x)\cdot\delta_{\langle A\rangle^{*}}^{(n-2)/2}(y)\cdot
G(x,y),\mbox{ for }x\neq y\in H\setminus\Sigma.$
There are constants $\beta(H),\alpha(H)>0$ so that (typically applied along
hyperbolic geodesics)
(13) $G^{\mathcal{S}}(x,y)\leq\beta\cdot\exp(-\alpha\cdot d_{\langle
A\rangle^{*}}(x,y)),\mbox{ for }x,y\in H\setminus\Sigma\mbox{ and }\\\
d_{\langle A\rangle^{*}}(x,y)>2\cdot\varrho_{(H\setminus\Sigma,d_{\langle
A\rangle^{*}})}.$
In the case of the conformal Laplacian, the combination of (12) and (13) with
the axioms for $\langle A\rangle$ yields upper radius and distance estimates
after conformal deformations by $G$ and, via boundary Harnack inequality, for
arbitrary solutions of minimal growth.
Minimal Growth under Blow-Ups In the case of a singular minimal cone $C$ the
Martin theory shows that there is exactly one solution $u>0$ with minimal
growth towards $\Sigma_{C}$, up to multiples. This means that $\mu_{u}$ is the
Dirac measure in $\infty\in\widehat{\Sigma}_{C}$. Now we assume that $L$
reproduces under scalings, $L_{C}$ and $J_{C}$ are examples. This implies a
separation of variables: $u=\psi_{C}(\omega)\cdot
r^{\alpha_{C}},\,(\omega,r)\in\partial B_{1}(0)\cap
C\setminus\Sigma_{C}\times\mathbb{R}^{>0}$, for some function $\psi_{C}$ on
$\partial B_{1}(0)\cap C\setminus\Sigma_{C}$ and $\alpha_{C}<0$.
When $u$ solves $L\,v=0$ with minimal growth along $B\cap\widehat{\Sigma}$ for
a ball $B\subset H$ around some $p\in\Sigma$ and we consider any solution $v$
on any tangent cone $C$ of $H$ in $p$ induced by $u$ under blow-up, we find
that $v$ has minimal growth towards all of $\Sigma_{C}$. $u$ asymptotically
looks like $\psi_{C}(\omega)\cdot r^{\alpha_{C}}$. The outcome is a top-down
to bottom-up asymptotic analysis towards $p$.
$\bullet$ Top-Down Given a solution $u>0$ on $H\setminus\Sigma_{H}$ of minimal
growth near some $p\in\Sigma_{H}$, choose some tangent cone $C$ and consider
the (uniquely determined) induced solution of minimal growth towards
$\Sigma_{C}\subset C$ on $C\setminus\Sigma_{C}$. Blow-up in a point of
$\Sigma_{C}\setminus\\{0\\}$ and iterate this process, at most $\dim H-7$
times, until we reach a product cone $\mathbb{R}^{m}\times C^{n-m}$, for some
$C^{n-m}\subset\mathbb{R}^{n-m+1}$ singular only in $0$. We think of this as a
terminal node in a blow-up tree with root $H$. The uniqueness shows that the
induced minimal solutions we end up with are $\mathbb{R}^{n-m+1}$-translation
symmetric and amenable to an explicit description.
$\bullet$ Bottom-Up Starting from terminal nodes $\mathbb{R}^{m}\times
C^{n-m}$, for some $C^{n-m}\subset\mathbb{R}^{n-m+1}$ singular only in $0$, we
transfer our understanding of the induced solutions on the tangent cones
stepwise backwards to $H$ in the blow-up tree. We get an asymptotic portrait
of the solution $u$ on $H\setminus\Sigma$ near $p$ from the family of Martin
theories on $H\setminus\Sigma$ and on its (iterated) tangent cones.
## 3 Minimal Splitting Factors - Top-Down Analysis
For a compact singular area minimizer $H^{n}$ in some $scal>0$-manifold
$M^{n+1}$ we recall that $\lambda^{\langle A\rangle}_{H}>0$. We apply our
theory to get a nicely controllable conformal but still singular
$scal>0$-geometry on $H^{n}$, the minimal splitting factor geometry. One may
think of it as some kind of $\boldsymbol{scal>0}$-twin of $H^{n}$ and we will
explain why.
### 3.1 Minimal Splitting Factors
To this end, we can neither use $L_{H}$ nor $L_{H,\lambda^{\langle
A\rangle}_{H}}=L_{H}-\lambda^{\langle A\rangle}_{H}\cdot\langle
A\rangle^{2}\cdot Id$. When we deform $H$, more precisely $H\setminus\Sigma$,
by solutions of $L_{H}\phi=0$ we get a scalar flat metric. One may feel
tempted to consider solutions of $L_{H,\lambda^{\langle A\rangle}_{H}}\phi=0$,
but this operator is no more $\mathcal{S}$-adapted simply because it obviously
has a vanishing principal eigenvalue and many of the results do not apply.
Instead we choose (super)solutions of
$L_{H,\lambda}\phi=L_{H}\phi-\lambda\cdot\langle A\rangle^{2}\cdot\phi=0$ for
(14) $0<\lambda<\lambda^{\langle A\rangle}_{H}.$
Due to the locally Lipschitz regular coefficients of
$L_{H,\lambda}=L_{H}-\lambda\cdot\langle A\rangle^{2}\cdot Id$, solutions of
$L_{H,\lambda}\,\phi=0$ are $C^{2,\alpha}$-regular, for any $\alpha\in(0,1)$.
This suggests the following regularity assumptions. For
$\lambda<\lambda^{\langle A\rangle}_{H}$, let $\Phi>0$ be a
$C^{2,\alpha}$-supersolution of $L_{H,\lambda}\phi=0$ on
$H\setminus\Sigma_{H}$ so that:
* •
For compact area minimizers $H$: $\Phi$ is a solution in a neighborhood of
$\Sigma$ and it has minimal growth towards $\Sigma$.
* •
For non-compact Euclidean area minimizers $H$: $\Phi$ is a solution on
$H\setminus\Sigma_{H}$ with minimal growth towards $\Sigma$.
###### Theorem 3.1
The metric completion
$(\widehat{H\setminus\Sigma},\widehat{d_{\mathcal{S}}}(\Phi))$ of
$(H\setminus\Sigma,\Phi^{4/(n-2)}\cdot g_{H})$ is homeomorphic to $(H,d_{H})$.
Thus, we can write it as $(H,d_{\mathcal{S}}(\Phi))$. The Hausdorff dimension
of $\Sigma$ relative to $(H,d_{\mathcal{S}}(\Phi))$ is $\leq n-7$.
Here $(H,d_{H})$ is the metric space that results from the embedding
$H^{n}\subset M^{n+1}$. We write simply $d_{\mathcal{S}}$ when the specific
choice of $\Phi$ is not needed or already known from the context.
Some Details To get the homeomorphism, we use the $\mathcal{S}$-Doob transform
(11) and the relation (12) to transfer the basic upper growth estimate (13) to
the minimal Green’s function of $L_{H,\lambda}$ on $(H\setminus\Sigma,g_{H})$.
It also applies to $\Phi$ by the boundary Harnack inequality. This shows that
$(\widehat{H\setminus\Sigma},\widehat{d_{\mathcal{S}}})$ is not larger than
$(H,d_{H})$. This works as soon as $\lambda<\lambda^{\langle A\rangle}_{H}$.
Remarkably, we need $\lambda>0$ to also get lower estimates showing that it is
not smaller and, hence, to establish the homeomorphism from the
Bombieri–Giusti Harnack inequality [BG].
One might expect that the Hausdorff dimension of $\Sigma$ relative to
$(H,d_{\mathcal{S}})$ must be the same as for area minimizers. However, the
Hausdorff dimension is not a topological but only a bi-Lipschitz invariant
while the identity map from $(H,d_{H})$ to $(H,d_{\mathcal{S}})$ is not
Lipschitz continuous. Hölder continuous homeomorphisms can increase the
dimension of a subset of dimension $a\in(0,n)$ to any value $b\in(a,n)$, cf.
[Bi]. We actually need the minimal growth of $\Phi$ and essential parts of the
hyperbolic unfolding theory to control the Hausdorff dimension.
Metric Measure Spaces We augment $(H,d_{\mathcal{S}}(\Phi_{H}))$ to a metric
measure space. To this end we show that there is a canonical extension of
$\Phi^{2\cdot n/(n-2)}\cdot\mu_{H}$ on $H\setminus\Sigma$ to a measure
$\mu_{\mathcal{S}}$ on $H$, where $\mu_{H}$ is the $n$-dimensional Hausdorff
measure on $(H^{n},g_{H})\subset(M^{n+1},g_{M})$. In fact, this extension
$\mu_{\mathcal{S}}$ is a Borel measure on $(H,d_{\mathcal{S}})$, cf. [HKST,
pp. 62–64].
###### Definition 3.2
We call $(H,d_{\mathcal{S}})$ a minimal spitting factor of its ambient space
$M$ and we define the minimal factor measure $\mu_{\mathcal{S}}$ by
$\mu_{\mathcal{S}}(E):=\int_{E\setminus\Sigma_{H}}\Phi^{2\cdot n/(n-2)}\cdot
d\mu_{H},$ for any Borel set $E\subset H$.
The small Hausdorff dimension of $\Sigma\subset(H,d_{\mathcal{S}}(\Phi_{H}))$
ensures that $\mu_{\mathcal{S}}$ is an outer regular measure and it is still
sufficient to define $\mu^{n-1}_{\mathcal{S}}$ for hypersurfaces within
$(H,d_{\mathcal{S}}(\Phi_{H}))$. We get $d\mu^{n-1}_{\mathcal{S}}$ from
extending $\Phi^{2\cdot(n-1)/(n-2)}\cdot d\mu^{n-1}_{H}$ on
$H\setminus\Sigma$, where $d\mu_{H}^{n-1}$ is the hypersurface element on
$(H\setminus\Sigma,g_{H})$.
###### Theorem 3.3
We consider $(H,d_{\mathcal{S}}(\Phi_{H}))$, some $p\in\Sigma_{H}$ and some
tangent cone $C$ in $p$. Then we get the following blow-up invariance:
Any sequence $(H,\tau_{i}\cdot d_{\mathcal{S}}(\Phi_{H}))$, scaled by a
sequence $\tau_{i}\rightarrow\infty$, $i\rightarrow\infty$, around $p$,
subconverges and the limit of any converging subsequence is
$(C,d_{\mathcal{S}}(\Phi_{C}))$ for some tangent cone $C$.
$(C,d_{\mathcal{S}}(\Phi_{C}))$ is invariant under scaling around $0\in C$,
that is, it is again a cone.
This is the geometric counterpart of the top-down blow-up analysis of minimal
growth solutions we discussed in the previous chapter. The result means that
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$ has a simple asymptotic geometry near
$\Sigma$ and it admits inductive tangent cone reductions similar to area
minimizers. Note that the principal eigenvalues of $H$ and $C$ usually differ,
with $\lambda^{\langle A\rangle}_{H}<\lambda^{\langle A\rangle}_{C}$. That is,
we would encounter non-principal eigenvalues in the blow-up analysis even if
we started from $\lambda^{\langle A\rangle}_{H}$.
### 3.2 Stable Isoperimetry - Hyperbolic Geodesics Again
A vital feature of minimal spitting factors is that they still satisfy an
isoperimetric inequality.
###### Theorem 3.4
There are some constants $\gamma(H)>0,$ $\gamma(H^{*})>0$ so that for every
ball $B_{\rho}$ and any open set $U\subset H$ with compact closure and
rectifiable boundary $\partial U$
(15)
$\mu_{\mathcal{S}}(U)^{(n-1)/n}\leq\gamma\cdot\mu^{n-1}_{\mathcal{S}}(\partial
U),$ (16) $\min\\{\mu_{\mathcal{S}}(B_{\rho}\cap
U),\mu_{\mathcal{S}}(B_{\rho}\setminus
U)\\}^{(n-1)/n}\leq\gamma^{*}\cdot\mu^{n-1}_{\mathcal{S}}(B_{\rho}\cap\partial
U).$
The proof has two steps. We first show that
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$ is doubling and has a volume decay
property of order $n$. Both follow from the stronger Ahlfors regularity
estimate that for some $a(H),b(H)>0$,
(17) $a\cdot r^{n}\leq Vol(B_{r}(q),d_{\mathcal{S}})\leq b\cdot r^{n}.$
The trick is to check that $Vol(B_{1}(q),d_{\mathcal{S}})$ for Euclidean
hypersurfaces can be bounded in terms of constants that only depend on the
dimension. This uses the same techniques as the proof of the homeomorphism
theorem above. Scalings to other radii commute with scaling of the original
area minimizing metric and readily give these growth rates.
The main step is the proof of the Poincaré inequality on
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$. It says that there is a constant
$C_{0}=C_{0}(H,\Phi)>0$, so that when $B\subset H$ is an open ball,
$u:B\rightarrow\mathbb{R}$ is an $L^{1}$-function on $H$ that is $C^{1}$ on
$H\setminus\Sigma$. Then we have, setting $|\nabla u|\equiv 0$ on $\Sigma$,
(18) $\fint_{B}|u-u_{B}|\,d\mu_{\mathcal{S}}\leq C_{0}\cdot\fint_{B}|\nabla
u|\,d\mu_{\mathcal{S}},\\\ \mbox{ where
}u_{B}:=\fint_{B}u\,d\mu_{\mathcal{S}}:=\frac{1}{\mu_{\mathcal{S}}(B)}\int_{B}u\,d\mu_{\mathcal{S}}.$
The volume decay property of order $n$ then improves this Poincaré inequality
to the Sobolev inequality with exponent $n$ and, from this, we get the
isoperimetric inequalities for $(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$. A
broad reference is [HKST].
Semmes Families To derive the Poincaré inequality on rather general metric
measure spaces, Semmes has decompiled the classical proof of Poincaré
inequality on $\mathbb{R}^{n}$ [Se, He, HKST]. In an important step, we
encounter uniformly distributed families of curves linking any two given
points. The abstracted concept is that of _thick families of curves_ , also
called _Semmes families_ , satisfying the two conditions (i) and (ii) below.
For reasonable metric spaces, the presence of Semmes families implies the
validity of a Poincaré inequality. On area minimizers, we can use hyperbolic
unfoldings to define canonical Semmes families on $(H,d_{H})$ with the
interesting feature that they are still Semmes families relative to
$(H,d_{\mathcal{S}})$.
###### Proposition 3.5
There is a $C=C(H)>0$ and for any two $p,q\in H$ a family $\Gamma_{p,q}$ of
rectifiable curves $\gamma:I_{\gamma}\rightarrow H$,
$I_{\gamma}\subset\mathbb{R}$, joining $p$ and $q$, so that:
1. (i)
For any $\gamma\in\Gamma_{p,q}$: $l(\gamma|_{[s,t]})<C\cdot
d(\gamma(s),\gamma(t))$, for $s,t\in I_{\gamma}$.
2. (ii)
Each family $\Gamma_{p,q}$ carries a probability measure $\sigma_{p,q}$ so
that for any Borel set $A\subset X$, the assignment $\gamma\mapsto
l(\gamma\cap A)$ is $\sigma$-measurable with
(19) $\int_{\Gamma_{p,q}}l(\gamma\cap A)\,d\sigma(\gamma)\leq
C\cdot\int_{A_{C,p,q}}\left(\frac{d(p,z)}{\mu(B_{d(p,z)}(p))}+\frac{d(q,z)}{\mu(B_{d(q,z)}(q))}\right)d\mu(z)$
for $A_{C,p,q}:=(B_{C\cdot d(p,q)}(p)\cup B_{C\cdot d(p,q)}(q))\cap A$.
The family $\Gamma_{p,q}$ uniformly surrounds a central curve, its core
$\gamma_{p,q}$. It is a hyperbolic geodesic linking $p$ and $q$ in the Gromov
compactification of the hyperbolic unfolding.
The first assertion, for the core, is part of the proof that
$(H\setminus\Sigma,d_{\langle A\rangle^{*}})$ is Gromov hyperbolic in [L1]. It
shows that each segment of $\Gamma_{p,q}$ is an $\mathcal{S}$-uniform curve
relative to $(H,d_{H})$. For the other curves, this follows from the way we
define $\Gamma_{p,q}$.
To define $\Gamma_{p,q}$ and the probability measure $\sigma_{p,q}$, we start
with any two points $x,y\in\mathbb{R}^{n}$ and consider the hyperplane
$L^{n-1}(x,y)$ orthogonal to the line segment $[x,y]\subset\mathbb{R}^{n}$
passing through the midpoint $m(x,y)$ of $[x,y]$. For $\rho=d(x,y)$, we
consider a ball $B_{r}=B_{r}^{n-1}(m(x,y))\subset L^{n-1}(x,y)$ of radius
$r\in(0,\rho]$ which we will choose later. For any $z\in B_{r}$, let
$\gamma_{z}$ be the unit speed curve from $x$ to $y$ we get when we follow the
line segments $[x,z]$ and $[z,y]$. We define
$\Gamma^{\mathbb{R}^{n}}_{x,y}=\Gamma^{\mathbb{R}^{n}}_{x,y}(r):=\\{\gamma_{z}\,|\,z\in
B_{r}\\}$ and the point set $D_{r}(x,y)$ we get as a union of the curves. We
introduce the probability measure $\alpha^{\mathbb{R}^{n}}_{x,y}$ on
$\Gamma^{\mathbb{R}^{n}}_{x,y}$ as follows: if
$W\subset\Gamma^{\mathbb{R}^{n}}_{x,y},$
(20) $\alpha^{\mathbb{R}^{n}}_{x,y}(W):=\mathcal{H}^{n-1}(\\{z\in
B_{r}^{n-1}\,|\,\gamma_{z}\in W\\})/\mathcal{H}^{n-1}(B_{r}^{n-1}).$
For any Borel set $A\subset\mathbb{R}^{n}$, the function
$\gamma\mapsto\ell(\gamma\cap A)$ on $\Gamma^{\mathbb{R}^{n}}_{x,y}$ is
$\alpha^{\mathbb{R}^{n}}_{x,y}$-measurable. Now assume that all points in $A$
are closer to $x$ than to $y$. For the distance between corresponding points
on any two segments, we have
(21) $d(s\cdot z_{1}+(1-s)\cdot x,s\cdot z_{2}+(1-s)\cdot x)\leq 2\cdot r\cdot
s,\mbox{ for }s\in[0,1],z_{1},z_{2}\in B_{r}.$
The coarea formula gives the following inequality for the annuli $A_{j}:=A\cap
B(x,\ 2^{-j}d(x,\ y))\setminus B(x,2^{-j-1}d(x,y))$, $j\in\mathbb{Z}^{\geq
0}$:
$\int_{B_{r}}\ell(\gamma_{z}\cap A)d\mathcal{H}^{n-1}(z)=\sum_{j=0}^{\infty}\
\int_{B_{r}}\ell(\gamma_{z}\cap
A_{j})d\mathcal{H}^{n-1}(z)\leq\sum_{j=0}^{\infty}2^{(j+2)(n-1)}\mu(A_{j})$
$\leq 4^{n-1}\cdot\int_{A\cap
B(x,d(x,y))}\frac{d(x,z)}{\mu(B_{d(x,z)}(x))}d\mu(z)$
Thus we have a Semmes family on $\mathbb{R}^{n}$ and can now identify the line
segment $[x,y]\subset\mathbb{R}^{n}$ with the hyperbolic geodesic
$\gamma_{p,q}$ and use its $\mathcal{S}$-uniformity to transfer the Semmes
family $\Gamma^{\mathbb{R}^{n}}_{x,y}$ to $(H,d_{H})$. In other words, what we
really do is to define a curve fibration of the twisted double
$\mathcal{S}$-cone surrounding $\gamma_{p,q}$ which we already had from the
$\mathcal{S}$-uniformity of $H$. This is the desired canonical Semmes family
on $(H,d_{H})$.
Once again, the estimates for minimal Green’s functions along hyperbolic
geodesics show that, with other constants and probability measures,
$\Gamma_{p,q}$ is still a Semmes family for
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$ and thus we get the desired Poincaré
inequality on $(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$.
The isoperimetric inequalities also control area minimizers within
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$ from the following type of standard
consequences:
###### Corollary 3.6
There is a $\rho_{H}>0$ so that for any $r\in(0,\rho_{H})$ and any area
minimizing boundary $L^{n-1}$ bounding some open set $L^{+}\subset H$ in
$(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$:
(22) $\kappa^{-}_{n}\cdot r^{n}\leq\mu_{\mathcal{S}}(L^{+}\cap
B_{r}(p))\leq\kappa^{+}_{n}\cdot r^{n},$
where $\kappa^{-}_{n},\kappa^{+}_{n}>0$ denote constants depending only on the
dimension $n$.
This volume growth estimate shows that there are no horn-shaped pieces of area
minimizers in $(H,d_{\mathcal{S}},\mu_{\mathcal{S}})$ entering narrow hollow
cylinders. In particular, such an area minimizer cannot _stretch out_ along
$\Sigma$. This is a crucial detail to improve the efficiency of smoothing
techniques we discuss in the next chapter.
## 4 Smoothing Techniques - Bottom-up Constructions
We describe how to deform the still singular minimal splitting factors
$(H,d_{\mathcal{S}})$ to regular $scal>0$-geometries on $H\setminus U$ with
minimal boundary $\partial U$, where is $U$ is a small neighborhood of
$\Sigma_{H}$. This yields schemes of an inductive dimensional descent with a
built-in partial regularization.
The point is that $\partial U$ may also have singularities but of lower
dimension and the minimality of $\partial U$ matches the intended inductive
descent:
Area minimizers in $H\setminus U$ either coincide with $\partial U$ or they
are entirely supported in $H\setminus\overline{U}$. Thus in each loop of the
induction such area minimizers experience a smooth $scal>0$-environment and we
eventually reach exclusively smooth geometries in dimensions $\leq 7$.
### 4.1 Surgeries Revisited
We shed some new light on the well-known $scal>0$-preserving surgeries on a
manifold $M$ one can do along any given submanifold $N\subset M$ of
codimension $\geq 3$, [GL] and [SY2]. We approach these surgeries in a way
that represents the gluing boundaries, after removing tubes around $N$, as
minimal surfaces. One may keep the intermediate bounded manifolds per se or
glue some complementary piece to them to get closed $scal>0$-manifolds with
altered topology.
This illustrates the smoothing results for minimal splitting factors we
discuss below. In a quite similar manner, we remove tubes around the
singularities $\Sigma\subset H$. Again, we will get minimal boundaries and
keep $scal>0$. One may use this manifold with minimal boundary directly or one
can glue a smooth complementary part to it to get a new closed
$scal>0$-manifold.
###### Proposition 4.1
Let $M^{n}$, $n\geq 3$, be a closed manifold and assume that the first
eigenvalue $\lambda_{1}$ of the conformal Laplacian $L_{M}$ is positive. For
any submanifold $N^{k}\subset M^{n}$ of dimension $k\leq n-3$ there are
arbitrarily small neighborhoods $U$ of $N$, such that $M\setminus U$ is
conformal to a $scal>0$-manifold $X=X_{U}$ a (locally) area minimizing
boundary $\partial X$.
We break the construction of the conformal deformation into two steps: In Step
A we choose a canonical conformal deformation to reach a _basic
$scal>0$-metric_ on $M$. In Step B we add a _secondary_ conformal change of
$M\setminus N$ using some positive function diverging to $+\infty$ while we
approach $N$ to achieve the desired _bending_ effect towards $N$. When we
evaluate these conformal deformations, we find two neighborhoods $U\subset V$
of $N$ with $\partial V$ having _positive_ mean curvature and $\partial U$
with _negative_ mean curvature111The sign convention is made so that $\partial
B_{1}(0)\subset\mathbb{R}^{n+1}$, viewed from $0\in\mathbb{R}^{n+1}$, has
positive mean curvature..
We think of $\partial V$ as an outer barrier and $\partial U$ as an inner
barrier. They keep area minimizers from escaping $V\setminus U$. In fact, the
mean curvature constraints show that we may replace any
$Area_{n-1}$-minimizing sequence of boundaries $T_{i}\subset V$ _surrounding_
$U$, that is $T_{i}=\partial W_{i}$, for open $W_{i}$ with $U\subset
W_{i}\subset V$, so that for $i\rightarrow\infty$:
$Area_{n-1}(T_{i})\rightarrow\inf\\{Area_{n-1}(T)\,|\,T=\partial W\mbox{ for
an open }W,U\subset W\subset V\\},$
by another $Area_{n-1}$-minimizing sequence $T^{*}_{i}$ with
$Area_{n-1}(T^{*}_{i})\leq Area_{n-1}(T_{i})$ and support outside a
neighborhood of $\partial V\cup\partial U$. Then standard arguments from basic
geometric measure theory [Gi, 1.20] show that there is an area minimizer
$T^{*}=\partial W^{*}$ for some open $W^{*}$ with $U\subset W^{*}\subset V$.
We set $X:=M\setminus W^{*}$ and have $\partial X=T^{*}$.
Step A We can use the first eigenfunction u=$u_{M}>0$ of $L_{M}$ on $M$, the
assumption $\lambda_{1}>0$ and the transformation law for scalar curvature
under conformal transformation to see:
(23) $scal(u^{4/(n-2)}\cdot g_{M})\cdot
u^{\frac{n+2}{n-2}}=L_{M}(u)=\lambda_{1}\cdot u>0.$
That is $scal(u^{4/(n-2)}\cdot g_{M})>0$. This observation is due to Kazdan
and Warner [KW]. $u^{4/(n-2)}\cdot g_{M}$ is our basic $scal>0$-metric on $M$.
Step B We also choose a function $\psi>0$ on $M\setminus N$ with $L_{M}\psi=0$
and
(24) $\psi=\begin{cases*}r^{2+k-n}+O(r^{3+k-n}),&when $k<n-3$\\\
r^{2+k-n}+O(log(r)^{-1}),&when $k=n-3$,\end{cases*}$
where $r$ is the distance function to $N$. A construction of $\psi$ is given
in [SY2, Appendix]. Roughly speaking, one averages a Green’s function for
$L_{M}$ along $N$.
Now we add $\delta\cdot\psi$ to $u$, for some small $\delta>0$, as a secondary
deformation: we choose $g_{\delta}:=(\delta\cdot\psi+u)^{4/(n-2)}\cdot g_{M}$.
The linearity of $L_{M}$ shows that $scal(g_{\delta})>0$.
We denote the $r$-distance neighborhoods of $N$ relative to $u^{4/(n-2)}\cdot
g_{M}$ by $V_{r}$. Then, for any small $\varepsilon>0$, we have that $\partial
V_{\varepsilon}$ is diffeomorphic to $N^{k}\times S^{n-k-1}$. Thus for small
enough $\delta(\varepsilon)>0$ we readily see that $\partial V_{\varepsilon}$
is essentially determined from $u^{4/(n-2)}\cdot g_{M}$ and it has _positive
mean curvature_ $\approx(n-k-1)/\varepsilon$ largely as in the Euclidean case
of distance tubes around $\mathbb{R}^{k}\subset\mathbb{R}^{n}$. This makes
$\partial V_{\varepsilon}$ an _outer barrier_ for area minimizers in
$V_{\varepsilon}$ homologous to $\partial V_{\varepsilon}$. It keeps them
inside $V_{\varepsilon}$.
In turn, for such a (now fixed) $\delta>0$ and $\rho>0$ small enough we claim
that $\partial V_{\rho}$ has _negative mean curvature_. To see this, we recall
that the second fundamental form $A_{L}(g)$ of a submanifold $L$ with respect
to some metric $g$ on the ambient manifold transforms under conformal
deformations $u^{4/(n-2)}\cdot g$, for smooth $u>0$, according to the formula
[Be, 1.163, p. 60]
(25) $A_{L}(u^{4/(n-2)}\cdot
g)(v,w)=A_{L}(g)(v,w)-\frac{2}{n-2}\cdot{\cal{N}}(\nabla u/u)\cdot g(v,w),$
where ${\cal{N}}(\nabla u/u)$ is the normal component of $\nabla u/u$ with
respect to $L$. In our case, we choose $u=\psi$ and locally consider
$V_{\rho}$ as a tube around $\mathbb{R}^{k}$ in
$\mathbb{R}^{k}\times\mathbb{R}^{n-k}$ where $\mathbb{R}^{k}$ represents $N$.
Thus we have from (24) that $tr\,A_{\partial
V_{\rho}}(g)\approx-(n-k-1)/\rho$, since the $\mathbb{R}^{n-k}$-factor of
$V_{\rho}$ is totally geodesic. The trace of the second summand is $(n-1)\cdot
2/(n-2)\cdot(2+k-n)/\rho$ and we get:
$\displaystyle\qquad\psi^{4/(n-2)}\cdot tr\,A_{\partial
V_{\rho}}(\psi^{4/(n-2)}\cdot g)$ $\displaystyle=-\left((n-k-1)+(n-1)\cdot
2/(n-2)\cdot(2+k-n)\right)\cdot\rho^{-1}$ $\displaystyle=(2-(3+k)\cdot
n+n^{2})/(n-2)\cdot\rho^{-1}$ (26) $\displaystyle\geq
2/(n-2)\cdot\rho^{-1}>0,\mbox{ since }n-3\geq k.$
This makes $\partial V_{\rho}$ an _inner barrier_ for area minimizers in
$M\setminus V_{\rho}$ homologous to $\partial V_{\rho}$, keeping them from
reaching $N$. Thus for any $\varepsilon>0$, there is some
$\delta_{\varepsilon}>0$ so that there is a neighborhood $U_{\varepsilon}$ of
$N$, with $V_{\rho}\subset U_{\varepsilon}\subset V_{\varepsilon}$, such that
$\partial U_{\varepsilon}$ is (locally) area minimizing relative to
$g_{\delta_{\varepsilon}}$.
The resulting $scal>0$-manifold $X=X_{U}$ with its minimal boundary $\partial
X$ can be used in several ways:
Doublings We take a second copy $X^{*}$ of $X$ and glue $X^{*}$ to $X$ along
$\partial X$ to get a doubling $X\cup_{\div}X^{*}$ with the obvious
identification $\div$ along $\partial X$. This can be made to get a smooth
$scal>0$-metric on $X\cup_{\div}X^{*}$ since one can approximate $\partial X$,
slightly (re)enlarging $X$, by a smooth hypersurface $Z$ with positive mean
curvature [G2] and then a standard bending of a collar of $Z$ gives a
$scal>0$-metric with totally geodesic boundary $Z$.
If there is another manifold $Y$ with boundary $\partial Y$ being locally
isometric to $X^{*}$ near $\partial X^{*}$ then we can alternatively glue $Y$
to $X$ along $\partial X$. This is just a variant of the classical surgery in
[GL] and [SY2].
### 4.2 Inductive Removal of Singularities
As already indicated in the codimension $\geq 3$-surgery reconstructed above,
the main result is a partial regularization method where we replace the
singularities of $(H^{n},d_{\mathcal{S}})$ by minimal boundaries:
###### Theorem 4.2
Let $H^{n}$ be a singular area minimizing hypersurface in some compact
$scal\geq 0$-manifold $M^{n+1}$. Then there are arbitrarily small
neighborhoods $U$ of $\Sigma$, such that $H\setminus U$ is conformal to some
$\boldsymbol{scal>0}$-manifold $X_{U}$ with locally area minimizing boundary
$\partial X_{U}$.
This is the generic step in an inductive dimensional descent with a built-in
partial regularization scheme. Starting from a singular area minimizing
hypersurface $H^{n}$ in some smooth $scal\geq 0$ ambient space we descent to a
possibly again singular area minimizing hypersurface $L^{n-1}$ in the smooth
$scal>0$ ambient space $X_{U}$. The process shifts the singular issues with
$H^{n}$ to lower dimensions:
A. Let $L^{n-1}\subset\overline{X_{U}^{n}}$ be an area minimizer. At this
point $\partial X_{U}$ merely constrains $L^{n-1}$ to stay within
$\overline{X_{U}^{n}}$. It does not matter whether $\partial X_{U}$ is smooth
[Gi, Th.1.20, Rm.1.22].
B. The point is that $\partial X_{U}$ is minimal. From this the strict maximum
principle [Si] shows, componentwise, that either $L^{n-1}\equiv\partial
X_{U}^{n-1}$ or $L^{n-1}\cap\partial X_{U}^{n-1}=\emptyset$. This renders
$L^{n-1}$ as an actually unconstraint minimal hypersurface in an extension
$Y_{U}^{n}$ of $\overline{X_{U}^{n}}$ surrounding $L^{n-1}$ as a smooth
$scal>0$-manifold.
C. $L^{n-1}$ may be singular, with a lower dimensional singular set, but we
can reapply Theorem 4.2 to $L^{n-1}$ in dimension $n-1$. This and further
iterations inductively sweep out all singular issues to lower dimensions
before they ultimately disappear in dimension $7$.
To indicate the proof of this theorem, we follow the lines of the case without
singularities as discussed in the previous section.
Step A: Minimal Splitting Factors The counterpart to Step A is the transition
from the compact singular area minimizer $H^{n}$ in some $scal>0$-manifold
$M^{n+1}$ to an associated minimal splitting factor
$(H,d_{\mathcal{S}}(\Phi_{H}))$, where $\Phi_{H}>$ is a positive supersolution
of $L_{H,\lambda}\,\phi=0$ for some $0<\lambda<\lambda^{\langle
A\rangle}_{H}$. We actually choose a rather small $\lambda$ to get uniform
growth estimates for $\Phi$ towards $\Sigma$ which are used to control the
secondary deformations in the following Step B.
Step B: Removal of Singularities We deform $(H,d_{\mathcal{S}})$ in a similar
way as in Step B in the smooth sample case above. We find a small neighborhood
$U$ of $\Sigma$ and an elementary deformation of $g_{\mathcal{S}}$ to a
$scal>0$-metric on $H\setminus U$ so that $\partial U$ becomes minimal.
We construct this elementary deformation of $(H,d_{\mathcal{S}})$ from the
top-down analysis of the previous chapter. We apply tangent cone reductions in
the class of minimal splitting factors. To this end, we cover $\Sigma\subset
H$ with upper bounded intersection numbers by finitely many small balls
$B_{r_{i}}(p_{i})\subset H$ which, after scaling to unit size, are well-
approximated by the ball $B_{1}(0)\cap C_{i}$ in some tangent cone $C_{i}$ in
$p_{i}$ with its minimal splitting factor geometry.
Similarly, we choose a ball cover for $\partial
B_{1}(0)\cap\Sigma_{C_{i}}\subset\partial B_{1}(0)\cap C_{i}$ and finitely
many locally approximating tangent cones. The iteration of this process
defines a blow-up tree $\mathbb{T}$ of cones with root $H$. The branching of
$\mathbb{T}$ ends, after at most $n-7$ blow-ups, with some product cone
$\mathbb{R}^{k}\times C^{n-k}$ singular only along
$\mathbb{R}^{k}\times\\{0\\}$.
In this case, the minimal splitting factor metric has a simple form. For some
function $c:\partial B_{1}(0)\cap C\rightarrow\mathbb{R}^{>0}$ and a
$\gamma\in(-(n-2)/4,0)$, we have
$g_{\mathcal{S}}=(c(\zeta)\cdot\rho^{\gamma})^{4/(n-2)}\cdot g_{C}\mbox{ on
}\mathbb{R}^{k}\times C^{n-k}\setminus\mathbb{R}^{k}\times\\{0\\},$
where $\rho(x)=dist(x,\mathbb{R}^{k}\times\\{0\\})$, $\zeta(x)=\pi_{2}(x/|x|)$
and $\pi_{2}:\mathbb{R}^{k}\times C^{n-k}\rightarrow C^{n-k}$ is the
projection. For this metric, we easily find $\mathbb{R}^{k}$-translation
invariant conformal deformations, supported away from
$\mathbb{R}^{k}\times\\{0\\}$, to another $scal>0$-metric which contains an
elementary inner barrier for area minimizers illustrated in the left and
middle part of Figure 3. An inner barrier is a bumpy deformation of
$g_{\mathcal{S}}$ that locally increases the volume element of
$g_{\mathcal{S}}$ to keep local area minimizers away from $\Sigma$. These
barriers result from a simple cut-off construction of a secondary deformation
added to the minimal factor metric.
Figure 3: The minimal factors are drawn as simple planes. The singular set is
visualized as centered spots and lines. The red rubber bands $L^{n-1}$
illustrate locally area minimizing hypersurfaces kept from contracting to
(parts of) $\Sigma$ by these barriers.
To assemble a global barrier along $\Sigma\subset H$, we localize the barriers
in the terminal nodes of $\mathbb{T}$ keeping $scal>0$: we truncate the bump
deformation to a deformation supported in a compact subset of
$C\setminus\Sigma_{C}$, indicated on the right portion of the picture. This
particular localization is important for the transfer between different nodes
of the blow-up tree. The transfer uses smooth tangent cone approximations,
that is, approximations described as sections of the normal bundle of the
tangent cone in $C^{k}$-norm, for some $k\geq 2$, and we use these sections to
transfer constructions between different spaces via pull-back or push-forward.
But these smooth approximations exist only in positively upper and lower
bounded distance to the singular sets.
These transfer maps between nodes in the blow-up tree allow us to bring the
harvest home: we transport the individual bump deformations backwards to the
next node in $\mathbb{T}$ and inductively assemble barriers until we reach
$H$. This is a simple process but it needs a considerable amount of
bookkeeping using families of well-controlled coverings. The isoperimetry of
$g_{\mathcal{S}}$ is applied when these deformations have been completed. The
isoperimetric inequality appears in the guise of the volume estimates in Cor.
3.6.
Auto-Aligned Coverings: As an alternative to the use of blow-up trees there is
a refined covering argument, in [L5], that guides the placement of the local
barriers.
* •
The local barriers are placed disjointedly to keep control over their curving
effect. The resulting global barrier along $\Sigma$ does not topologically
separate the singular region from the main regular part of $H$. However, using
the isoperimetric inequality of the underlying minimal splitting factor, it
separates geometrically in the sense that it keeps area minimizers from
approaching $\Sigma$. Intuitively, one may compare these topologically
permeable barriers with a Faraday cage.
* •
In turn, these volume estimates also ensure that there is such an area
minimizer since they show that there is a larger neighborhood of the deformed
region that area minimizers do not leave.
Doublings As in the sample case of codimension $\geq 3$ surgeries, we have
arbitrarily small neighborhoods $U$ of $\Sigma$, such that after some
smoothing of $\partial X_{U}$ we can take a second copy $X_{U}^{*}$ of $X_{U}$
and glue $X_{U}^{*}$ to $X_{U}$ so that $X_{U}\cup_{\div}X_{U}^{*}$ is smooth
with $scal>0$.
### 4.3 From Ordinary to Locally Finite Homology
Here we will see how to apply the partial regularization method in scalar
curvature geometry and general relativity to extend results valid in
dimensions $\leq 7$ to higher dimensions. We start with one of the most
prominent examples: the Riemannian positive mass theorem. We refer to [L6] for
the explicit statement and its physical relevance. In geometric terms it can
be formulated as a non-existence of $\boldsymbol{scal>0}$-islands:
###### Theorem 4.3
There exists no complete Riemannian manifold $(M^{n+1},g)$, $n\geq 2$, such
that:
* •
$scal(g)>0$ on a non-empty open set $U\subset M^{n+1}$ with compact closure.
* •
$(M^{n+1}\setminus U,g)$ is isometric to $(\mathbb{R}^{n+1}\setminus
B_{1}(0),g_{Eucl})$.
We indicate the contradiction argument of [L6]. Let us assume we had such a
Riemannian manifold $(M^{n+1},g)$. Then we can place a large cube around
$B_{1}(0)$ and compactify $(M^{n+1},g)$ to a closed $scal>0$-manifold
diffeomorphic to $N^{n+1}\char 35\relax T^{n+1}$, for some closed manifold
$N^{n+1}$. Now we can find a possibly singular area minimizer $H^{n}\subset
N^{n+1}\char 35\relax T^{n+1}$ homologous to the standard $T^{n}\subset
T^{n+1}$ and apply Theorem 4.2 to get the $scal>0$-manifold $X_{U}$ for some
small neighborhood $U$ of $\Sigma_{H}$.
The regularity theory also shows that $H^{n}$ again contains a nearly flat
torus summand, even after the conformal deformation of 4.2, i.e.,
$T^{n}\setminus B\subset H^{n}$ for some small ball $B$. This means that there
is a compact area minimizer $L^{n-1}\subset X_{U}\subset H^{n}$ homologous to
$T^{n-1}$ with $\partial U\cap L^{n-1}=\emptyset$ and, again, $L^{n-1}$
contains a nearly flat torus summand. We iterate this process until we reach a
$scal>0$-surface $F^{2}\char 35\relax T^{2}$ that does not exist, due to the
Gauss–Bonnet theorem. This proves the non-existence of $scal>0$-islands and,
hence, the positive mass theorem.
Locally Finite Homology The argument above exploited the existence of a nearly
flat torus component of $N^{n+1}\char 35\relax T^{n+1}$ to iteratively bypass
the singular sets from a suitable selection of homology classes.
Figure 4: The left hand picture represents the case of a compact
$L^{n-1}\cap\partial U=\emptyset$. In the second picture we have a homology
class that hits $\overline{U}$. Here we can still find an intrinsically
complete local area minimizer $L^{n-1}\cap\partial U=\emptyset$ that
asymptotically approaches $\partial U$.
Theorem 4.2 can also be used in a different way to include more general
homology classes. We start with any non-trivial family of classes
$\alpha[1],\dots,\alpha[k]\in H^{1}(M^{n+1},\mathbb{Z})$. Geometric measure
theory shows that there is an area minimizer $H^{n}\subset M^{n+1}$ that
represents $\alpha[1]\cap[M]$ in the homology of integral currents. Since $M$
is smooth, the integral current homology groups of $M$ are isomorphic to
ordinary homology groups [D], but, in general, a singular $H^{n}$ does _not_
represent the associated class in ordinary homology.
* •
$H^{n}$ might not be a singular $n$-cycle or have a non-finitely generated
homology and $H^{n}\setminus\Sigma$ does not represent a class in ordinary
homology.
This advocates the interpretation of the
$\alpha[1]\cap\cdots\cap\alpha[m]\cap[M]$, $m\leq k$, as classes in integral
current homology. However, Theorem 4.2 and the regularity theory for minimal
hypersurfaces, proving the low dimensionality of the singular set, suggest
still another interpretation. We think of the
$\alpha[1]\cap\cdots\cap\alpha[m]\cap[M]$ as classes in a _locally finite_
homology [HR].
In more detail, for small $U$, the isoperimetric inequality and the fact that
$\partial U$ is locally area minimizing can be used to show that there is an
intrinsically complete local area minimizer $L^{n-1}\subset Y_{U}\subset
H^{n}$ so that, componentwise, either $L^{n-1}$ is compact and
$L^{n-1}\cap\partial X_{U}=\emptyset$ (or $L^{n-1}=\partial X_{U}$), this is
the case we have already discussed above, or
* •
$L^{n-1}$ is non-compact with end. It asymptotically approaches $\partial
X_{U}$, like a geodesic ray on a surface that approaches a closed geodesic in
infinitely many loops. We apply Theorem 4.2 to $\partial X_{U}$ and eventually
transfer its metric to $L^{n-1}$ to get a periodic $scal>0$-end structure.
For the present we assume that $L^{n-1}$ is smooth. Then $L^{n-1}$ represents
a class in the Borel–Moore homology $H_{n-1}^{BM}(X_{U})$ of $X_{U}$.
$L^{n-1}$ can be thought to represent $\alpha[1]\cap\alpha[2]\cap[M]$. To make
this intuition more precise, the local finiteness of chains in this homology
becomes important. $L^{n-1}$ spins around $\partial X_{U}$, but each of the
infinitely many loops can be annihilated by one ordinary homology operation.
That is, $L^{n-1}$ is Borel–Moore homologous to the restriction
$L_{*}^{n-1}\cap X_{U}$ of a hypersurface $L_{*}^{n-1}\subset H$ representing
$\alpha[1]\cap\alpha[2]\cap[M]$ in integral current homology.
In our context, this suffices to view $L^{n-1}$ as a representing cycle of
$\alpha[1]\cap\alpha[2]\cap[M]$ since area minimizers reaching into the
periodic end of $L^{n-1}$ can be made to stay in a compact subset of
$L^{n-1}$. A sample of such a truncation style process is explained in the
proof that topologically large manifolds cannot admit $scal>0$-metrics [L7].
_It would be nice to formalize such ad-hoc truncation arguments in a suitable
locally finite or coarse homology theory where $L^{n-1}$ properly represents
$\alpha[1]\cap\alpha[2]\cap[M]$, cf. [HR], [R] for some background._
To finish the argument, we inductively apply the same strategy to get
potentially singular and non-compact area minimizers with ends that, in the
sense above, represent $\alpha[1]\cap\cdots\cap\alpha[m]\cap[M]$ in smooth
$scal>0$-manifolds of dimension $n-m+1$. Then we regularize them, using
Theorem 4.2, before we start the next loop. This way, we inductively sweep out
all singular issues to lower dimensions before they ultimately disappear in
dimension $7$.
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* [KW] Kazdan, J. and Warner, F.: Existence and Conformal Deformations of Metrics with prescribing Gaussian and scalar curvature. _Ann. of Math_. 101, 317–331 (1975)
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|
# MA-VAE: Multi-head Attention-based Variational Autoencoder Approach for
Anomaly Detection in Multivariate Time-series Applied to Automotive Endurance
Powertrain Testing
Lucas Correia12, Jan-Christoph Goos1, Philipp Klein1, Thomas Bäck2 and Anna V.
Kononova2
1Mercedes-Benz AG, Stuttgart, Germany
2Leiden University, Leiden, The Netherlands
<EMAIL_ADDRESS>
###### Abstract
A clear need for automatic anomaly detection applied to automotive testing has
emerged as more and more attention is paid to the data recorded and manual
evaluation by humans reaches its capacity. Such real-world data is massive,
diverse, multivariate and temporal in nature, therefore requiring modelling of
the testee behaviour. We propose a variational autoencoder with multi-head
attention (MA-VAE), which, when trained on unlabelled data, not only provides
very few false positives but also manages to detect the majority of the
anomalies presented. In addition to that, the approach offers a novel way to
avoid the bypass phenomenon, an undesirable behaviour investigated in
literature. Lastly, the approach also introduces a new method to remap
individual windows to a continuous time series. The results are presented in
the context of a real-world industrial data set and several experiments are
undertaken to further investigate certain aspects of the proposed model. When
configured properly, it is 9% of the time wrong when an anomaly is flagged and
discovers 67% of the anomalies present. Also, MA-VAE has the potential to
perform well with only a fraction of the training and validation subset,
however, to extract it, a more sophisticated threshold estimation method is
required.
## 1 Introduction
Powertrain testing is an integral part of the wider automotive powertrain
development and is undertaken at different stages of development. Each of
these stages is composed of many integration levels. These integration levels
range from powertrain sub-component testing, such as the electric drive unit
(EDU) controller or high-voltage battery (HVB) management system, to whole
vehicle powertrain testing. Each of these has its special type of controlled
environment, called a test bench. The use-case in this paper is on an
endurance powertrain test bench, where the EDU and HVB on their own are tested
under different conditions and loads for longer periods to simulate wear over
time. Given the costly maintenance and upkeep costs of such test benches, it
is desirable to keep downtime at a minimum and to avoid faulty measurements.
Also, it is desirable to detect problems early to prevent damage to the
testee. Given that evaluation is done manually by inspection, it is not
feasible to analyse every single measurement, also evaluation tends to be
delayed, only being undertaken days after the measurement is recorded, hence
there is a clear need for automatic, fast and unsupervised evaluation
methodology which can flag anomalous measurements before the next measurement
is started.
To achieve this, we propose a multi-head attention variational autoencoder
(MA-VAE). MA-VAE consists of a bidirectional long short-term memory (BiLSTM)
variational autoencoder architecture that maps a time-series window into a
temporal latent distribution [Park et al., 2018] [Su et al., 2019]. Also, a
multi-head attention (MA) mechanism is added to further enhance the sampled
latent matrix before it is passed on to the decoder. As shown in the ablation
study, this approach avoids the so-called bypassed phenomenon [Bahuleyan et
al., 2018], which is the first contribution. Furthermore, this paper offers a
unique methodology for the reverse-window process. It is used for remapping
the fixed-length windows the model is trained on to continuous variable-length
sequences.
This paper is structured as follows: First, a short background is provided in
Section 2 on the powertrain testing methodology specific to this use case, as
well as the theory behind VAE and MA mechanisms. Then, related work in
variational autoencoder-based time-series anomaly detection is presented in
Section 3, followed by an in-depth introduction of the real-world data set and
the approach we propose in Section 4. Then, several experiments testing
different aspects of the proposed method are conducted and discussed in
Section 5, along with the final results. Finally, conclusions from this work
are drawn and an outlook into future work is provided in Section 6. The source
code for the data pre-processing, model training as well as evaluation can be
found under https://github.com/lcs-crr/MA-VAE.
## 2 Background
### 2.1 Real-world Application
During endurance testing a portfolio of different driving cycles is run, where
a cycle is a standardised driving pattern, which enables repeatability of
measurements. For this type of testing the portfolio consists exclusively of
proprietary cycles, which differ from the public cycles used, for example, for
vehicle fuel/energy consumption certification like the New European Driving
Cycle (NEDC) or the Worldwide Harmonised Light Vehicles Test Cycle (WLTC). The
reason why proprietary cycles are used for endurance runs is that they allow
for more extensive loading of the powertrain.
Given the presence of a battery in the testee, some time has to be dedicated
to battery soaking (sitting idle) and charging. These procedures are also
standardised using cycles, although, for the intents and purposes of this
paper, they are omitted. What is left are the eight dynamic driving cycles
representing short, long, fast, slow and dynamic trips ranging from $5$ to
$30$ minutes. There are multiple versions of the same cycle, which mostly
differ in starting conditions such as state-of-charge (SoC) and temperature of
the battery.
On powertrain test benches, there are several control methods to ensure the
testee maintains the given driving cycle. In this particular test bench, the
regulation is done by the acceleration pedal and the EDU revolutions-per-
minute (rpm), which is nothing more than a non-SI version of the angular
velocity.
### 2.2 Data Set
This real-world data set consists of 3385 normal measurement files, each of
which contains hundreds of (mostly redundant or empty) channels. A measurement
is considered normal when the testee behaviour conforms to the norm. For this
work, a list of $d_{\textbf{X}}=13$ channels was hand-picked in consultation
with the test bench engineers to choose a reasonable and representative number
of channels. This list includes the vehicle speed, EDU torque, current,
voltage, rotor temperature and stator temperature, left and right wheel shaft
torque, HVB current, voltage, temperature and SoC and inverter temperature.
Given that some channels (such as torque) are sampled much faster than others
(like temperature and SoC), a common sampling rate of $2$Hz is chosen.
Channels sampled slower than $2$Hz are linearly interpolated, which is seen as
permissible due to the lower amplitude resolution of those channels. Channels
sampled faster than $2$Hz are passed through a low-pass filter with a cut-off
frequency of $1$Hz and then resampled to $2$Hz, as is consistent with the
Whittaker–Nyquist–Shannon theorem [Shannon, 1949]. Then the driving cycles are
z-score normalised, i.e. transformed such that the mean for each channel lies
at $0$ and the standard deviation at $1$. Lastly, the driving cycles
(generally referred to as sequences in this paper) are windowed to create a
set of fixed-length sub-sequences, or windows. First, each channel is auto-
correlated to obtain the number of lags of the slowest dynamic process present
in the signal. Then, the window size $W$ is set as the smallest power of two
larger than the longest lag, in this case, $W=256$ time steps or $128$
seconds. Each window overlaps their preceding and succeeding windows by half a
window, i.e. the shift between windows is $W/2=128$ time steps, in order to
reduce computational load compared to a shift of one time step.
Due to the absence of labelled anomalies in the dataset, realistic anomalous
events are intentionally simulated and recorded following the advice of test
bench engineers. To this end, five anomaly types were recorded. In the first
type, the virtual wheel diameter is changed, such that the resulting vehicle
speed deviates from the norm. The wheel diameter is a parameter as resistances
are connected to the shafts rather than actual wheels. The second type of
anomaly involves changing the driving mode from comfort to sport, which leads
to a higher HVB SoC drop over the cycle and a different torque response. In
the third anomaly, the recuperation level is turned from maximum to zero,
hence the minimum EDU torque is always non-negative and the HVB SoC
experiences a higher drop in SoC. In the case of the fourth anomaly, the HVB
is swapped for a battery simulator, where the HVB voltage behaviour deviates
from a real battery. The inverter and EDU share a cooling loop, whose cooling
capacity is reduced at the beginning or middle of the cycle, leading to higher
EDU rotor, EDU stator and inverter temperatures than normal. Every anomaly
type is recorded during every cycle at least once, leading to $60$ anomalous
driving cycles that are all used as the anomalous subset of the test set.
A plot of one normal and one wheel-diameter anomalous cycle is shown in Figure
1. Due to the long channel names, the plot only shows the channel indices, a
table containing the legend is shown in Table 1 for context. Visual inspection
may suggest that the red plot is anomalous, since the EDU and HVB voltage,
temperature and state of charge deviate from the black plot. This deviation is
to be expected because they depend on how charged the battery is and on how
much the battery is used previous to the current cycle. In the case of this
anomaly, the only channel that demonstrates anomalous behaviour is the vehicle
speed, since:
$v_{\text{vehicle}}=r\times\omega$ (1)
where $r$ is the wheel radius and $\omega$ the angular velocity. Evidently,
the anomalous behaviour is most visible at higher speeds.
Figure 1: Features of a normal (black) and an anomalous (red) cycle plotted with respect to time. The anomalous cycle plotted represents a scenario where the wheel diameter has not been set correctly. The amplitude axis is z-score normalised to comply with confidentiality guidelines. Table 1: Legend for the channel names in Figure 1. No. | Name
---|---
1 | Vehicle Speed
2 | EDU Torque
3 | Left Axle Torque
4 | Right Axle Torque
5 | EDU Current
6 | EDU Voltage
7 | HVB Current
8 | HVB Voltage
9 | HVB Temperature
10 | HVB State of Charge
11 | EDU Rotor Temperature
12 | EDU Stator Temperature
13 | Inverter Temperature
In an operative environment, it is desirable to find out whether the
previously recorded sequence had any problems to analyse before the next
measurement is recorded. Also, a model that performs as well as possible with
as little data as possible translates to faster deployment. Good performance
is indicated by a model that can detect as many anomalies as possible and
rarely labels normal measurements wrongly. To investigate the required
training subset size of the model, it is trained with $1$h, $8$h, $64$h, and
$512$h worth of dynamic testing data, which corresponds to the first $6$,
$44$, $348$, and $2785$ driving cycles, respectively. The results are also
presented in Section 5. In each of the above-mentioned cases, the training
subset is further split into a training ($80\%$) and a validation ($20\%$)
subsets. Both the training and validation subsets are batched to sets of 512
windows. Given the anomalous subset size of $60$ driving cycles, $600$ normal
driving cycles recorded after the ones in the training subset are chosen to
make up the normal test subset. This would imply that 9% of measurements at
the test bench are anomalous, however in reality this value is estimated to be
much lower. This amount of anomalous data in relation to normal data is used
as it approximately matches the anomaly ratio in public data sets and because
the data set is not large enough to create a larger normal test subset.
### 2.3 Variational Autoencoders
The variational autoencoder [Kingma and Welling, 2014][Rezende et al., 2014]
is a generative model that structurally resembles an autoencoder, but is
theoretically derived from variational Bayesian statistics. As opposed to the
regular deterministic autoencoder, the VAE uses the evidence lower bound
(ELBO), which is a lower bound approximation of the so-called log evidence
$\log p_{\theta}(\textbf{X})$, as its objective function. The ELBO, Equation
2, can be expressed as the reconstruction log-likelihood and the negative
Kullback-Leibler Divergence ($D_{\text{KL}}$) between the approximate
posterior $q_{\phi}(\textbf{Z}|\textbf{X})$ and the prior
$p_{\theta}(\textbf{Z})$, which is typically assumed to be a Gaussian
distribution [Goodfellow et al., 2016].
$\begin{split}\mathcal{L}_{\theta,\phi}(\textbf{X})&=\mathbb{E}_{\textbf{Z}\sim
q_{\phi}(\textbf{Z}|\textbf{X})}\left[\log
p_{\theta}(\textbf{X}|\textbf{Z})\right]\\\
&-D_{\text{KL}}(q_{\phi}(\textbf{Z}|\textbf{X})||p_{\theta}(\textbf{Z}))\\\
\end{split}$ (2)
where $\textbf{Z}\in\mathbb{R}^{W\times d_{\textbf{Z}}}$ is the sampled latent
matrix and $\textbf{X}\in\mathbb{R}^{W\times d_{\textbf{X}}}$ is the input
window. $W$ refers to the window length, whereas $d_{\textbf{X}}$ and
$d_{\textbf{Z}}$ refer to the input window and latent matrix dimensionality,
respectively. Gradient-based optimisation minimises an objective function and
the goal is the maximisation of the ELBO, hence the final loss function is
defined as the negative of Equation 2, shown in Equation 3.
$\mathcal{L}_{\text{VAE}}=-\mathcal{L}_{\theta,\phi}(\textbf{X})$ (3)
Finally, to enable the backpropagation through the otherwise intractable
gradient of the ELBO, the reparametrisation trick [Kingma and Welling, 2014]
is applied, shown in Equation 4.
$\textbf{Z}=\boldsymbol{\mu}_{\textbf{Z}}+\epsilon\cdot\boldsymbol{\sigma}_{\textbf{Z}}$
(4)
where $\epsilon\sim\mathcal{N}(0,1)$ and
$(\boldsymbol{\mu}_{\textbf{Z}},\log\boldsymbol{\sigma}_{\textbf{Z}}^{2})=q_{\phi}(\textbf{X})$.
### 2.4 Multi-head Attention Mechanism
To simplify the explanation of MA as employed in this work, multi-head self-
attention (MS) will be explained instead with the small difference between MA
and MS being pointed out at the end.
MS consists of two different concepts: self-attention and its multi-head
extension. Self-attention is nothing more than scaled dot-product attention
[Vaswani et al., 2017] where the key, query and value are the same. The scaled
dot-product attention score is the softmax [Bridle, 1990] of the product
between query matrix Q and key matrix K which is scaled by
$\sqrt{d_{\textbf{K}}}$. The product between the attention score and the value
matrix V yields the context matrix C, as shown in Equation 5.
$\textbf{C}=\mathrm{Softmax}\left(\frac{\textbf{Q}\textbf{K}^{T}}{\sqrt{d_{\textbf{K}}}}\right)\textbf{V}$
(5)
Compared to recurrent or convolutional layers, self-attention offers a variety
of benefits, such as the reduction of computational complexity, as well as an
increased amount of operations that can be parallelised. [Vaswani et al.,
2017]. Also, self-attention inherits an advantage over Bahdanau-style
attention [Bahdanau et al., 2015] from the underlying scaled dot-product
attention mechanism: it can run efficiently in matrix multiplication manner
[Vaswani et al., 2017].
Multi-head self-attention then allows the attention model to attend to
different representation subspaces [Vaswani et al., 2017], in addition to
learning useful projections rather than it being a stateless transformation
[Chollet, 2021]. This is achieved using weight matrices $\textbf{W}^{Q}_{i}$,
$\textbf{W}^{K}_{i}$, $\textbf{W}^{V}_{i}$, which contain trainable parameters
and are unique for each head $i$, as shown in Equation6.
$\textbf{Q}_{i}=\textbf{Q}\textbf{W}^{Q}_{i}\quad\textbf{K}_{i}=\textbf{K}\textbf{W}^{K}_{i}\quad\textbf{V}_{i}=\textbf{V}\textbf{W}^{V}_{i}$
(6)
Once the query, key and value matrices are linearly transformed via the weight
matrices, the context matrix $\textbf{C}_{i}$ for each head $i$ is computed
using Equation 7.
$\textbf{C}_{i}=\mathrm{Softmax}\left(\frac{\textbf{Q}_{i}\textbf{K}_{i}^{T}}{\sqrt{d_{\textbf{K}}}}\right)\textbf{V}_{i}$
(7)
Then, for $h$ heads, the different context matrices are concatenated and
linearly transformed again via the weight matrix $\textbf{W}^{O}$, resulting
in the multi-head context matrix $\textbf{C}\in\mathbb{R}^{W\times
d_{\textbf{Z}}}$, Equation 8.
$\textbf{C}=[\textbf{C}_{1},...,\textbf{C}_{h}]\textbf{W}^{O}$ (8)
The underlying mechanism of MA is identical to MS, with the only difference
being that K $=$ Q $\not=$ V. The benefit of this alteration is discussed in
Section 4.
## 3 Related Work
MA-VAE belongs to the so-called generative model class, which encompasses both
variational autoencoders, as well as generative adversarial networks. This
section focuses solely on the work on VAE proposed in the context of time-
series anomaly detection.
In time-series anomaly detection literature, the only other model that uses
the combination of a VAE and an attention mechanism is by [Pereira and
Silveira, 2018]. For the purpose of our paper, it is named VS-VAE. Their
approach consists of a BiLSTM encoder and decoder, where, for an input window
of length W, the $t=W$ encoder hidden states of each direction are passed on
to the variational self-attention (VS) mechanism [Bahuleyan et al., 2018]. The
resulting context vector is then concatenated with the sampled latent vector
and then passed on to the decoder. The author claims that applying VS to the
VAE model solves the bypass phenomenon, however, no evidence for this claim is
provided.
The first published time-series anomaly detection approach based on VAE was
LSTM-VAE [Park et al., 2018]. One of the contributions is its use of a dynamic
prior $\mathcal{N}(\mu_{p},1)$, rather than a static one $\mathcal{N}(0,1)$.
In addition to that, they introduce a state-based threshold estimation method
consisting of a support-vector regressor (SVR), which maps the latent
distribution parameters $(\mu_{\textbf{z}},\sigma_{\textbf{z}})$ to the
resulting anomaly score using the validation data. Hence, the dynamic
threshold can be obtained through Equation 9.
$\eta_{t}=\text{SVR}(\mu_{\textbf{z},t},\sigma_{\textbf{z},t})+c$ (9)
where $c$ is a pre-defined constant to control sensitivity.
OmniAnomaly [Su et al., 2019] attempts to create a temporal connection between
latent distributions by applying a linear Gaussian state space model to them.
For the purpose of this paper, it is called (OmniA). Also, it concatenates the
last gated recurrent unit (GRU) hidden state with the latent vector sampled in
the previous time step. In addition to that, it uses planar normalising flow
[Rezende and Mohamed, 2015] by applying $K$ transformations to the latent
vector in order to approximate a non-Gaussian posterior, as shown in Equation
10.
$f^{k}(\textbf{z}_{t}^{k-1})=\textbf{u}\tanh(\textbf{w}\textbf{z}^{k-1}_{t})+\textbf{b}$
(10)
where u, w and b are trainable parameters.
A simplified VAE architecture [Pereira and Silveira, 2019] based on BiLSTM
layers is also proposed. For the purpose of our paper, it is called W-VAE.
Unlike its predecessor [Pereira and Silveira, 2018], it drops the attention
mechanism but provides contributions elsewhere. It offers two strategies to
detect anomalies based on the VAE outputs. The first involves clustering the
space characterised by the mean parameter of the latent distribution into two
clusters and labelling the larger one as normal. This strategy has a few
weaknesses: it cannot be used in an operative environment as it requires some
sort of history of test windows to form the clusters and it assumes that there
are always anomalous samples present. The second strategy finds the
Wasserstein similarity measure (hence the W in the name) between the latent
mean space mapping of the test window in question and the respective mapping
$i$ resulting from a representative data subset, such as the validation
subset. Equation 11 shows how the Wasserstein similarity measure is computed
$W_{i}(\textbf{z}_{\text{test}},\textbf{z}_{i})=\|\mu_{\textbf{z}_{\text{test}}}-\mu_{\textbf{z}_{i}}\|_{2}^{2}+\|\Sigma_{\textbf{z}_{\text{test}}}^{1/2}-\Sigma_{\textbf{z}_{i}}^{1/2}\|_{F}^{2}$
(11)
where the first term represents the $L2$-Norm between the mean distribution
parameters resulting from the test window and each point of the representative
subset. The second term represents the Frobenius norm between the covariance
matrix resulting from the test window and each point of the representative
subset.
SWCVAE [Chen et al., 2020] is the first that applies convolutional neural
networks (CNN) to VAE for multivariate time-series anomaly detection.
Peculiarly, 2D CNN layers are used with the justification of being able to
process the input both spatially and temporally. We, however, doubt the
ability of the model to properly detect anomalies through spatial processing,
as a kernel moving along the feature axis can only capture features adjacent
to each other. To create a continuous anomaly score from windows they append
the last value of each window to the previous one. For the purpose of this
paper, this process is referred to as last-type reverse-windowing.
SISVAE [Li et al., 2021] tries to improve the modelling robustness by the
addition of a smoothing term in the loss function which contributes to the
reduction of sudden changes in the reconstructed signal, making it less
sensitive to noisy time steps.
As part of the VASP framework [von Schleinitz et al., 2021], a variational
autoencoder architecture is proposed to increase the robustness of time-series
prediction when faced with anomalies. While the main contribution is
attributed to the framework itself, not the VAE, it should be noted that
during inference only the mean parameter of the latent distribution is passed
to the decoder.
## 4 Proposed approach
Figure 2: An illustration of the proposed MA-VAE model. Blue shapes designate
trainable models, orange deterministic tensors and green distribution
parameters. The shape of each tensor is designated below it. During training Z
is used as the value matrix, denoted by the solid arrow, whereas during
inference $\boldsymbol{\mu}_{\textbf{Z}}$ is used as the value matrix, denoted
by the traced arrow.
### 4.1 Overview
To detect anomalies in multivariate time-series data, we propose a variational
autoencoder architecture consisting of BiLSTM layers. The model architecture
is illustrated in Figure 2. During training, the encoder $q_{\phi}$ maps
multivariate input window X to a temporal distribution with parameters
$\boldsymbol{\mu}_{\textbf{Z}}$ and $\log\boldsymbol{\sigma}_{\textbf{Z}}^{2}$
in the forward pass, Equation 12.
$(\boldsymbol{\mu}_{\textbf{Z}},\log\boldsymbol{\sigma}_{\textbf{Z}}^{2})=q_{\phi}(\textbf{X})$
(12)
Given the latent distribution parameters $\boldsymbol{\mu}_{\textbf{Z}}$ and
$\log\boldsymbol{\sigma}_{\textbf{Z}}^{2}$, the latent matrix is sampled from
the resulting distribution, as shown in Equation 13.
$\textbf{Z}\sim\mathcal{N}(\boldsymbol{\mu}_{\textbf{Z}},\log\boldsymbol{\sigma}_{\textbf{Z}}^{2})$
(13)
Then, the input window X is linearly transformed to obtain the query matrices
$\textbf{Q}_{i}$ and key matrices $\textbf{K}_{i}$ for each head $i$ .
Likewise, the sampled latent matrix Z is also transformed to the value matrix
$\textbf{V}_{i}$, as shown in Equation 14.
$\textbf{Q}_{i}=\textbf{X}\textbf{W}^{Q}_{i}\quad\textbf{K}_{i}=\textbf{X}\textbf{W}^{K}_{i}\quad\textbf{V}_{i}=\textbf{Z}\textbf{W}^{V}_{i}$
(14)
To output the context matrix $\textbf{C}_{i}$ for each head $i$, the softmax
of the through $\sqrt{d_{\textbf{K}}}$ normalised query and key product is
multiplied with the value matrix, Equation 15.
$\textbf{C}_{i}=\mathrm{Softmax}\left(\frac{\textbf{Q}_{i}\textbf{K}_{i}^{T}}{\sqrt{d_{\textbf{K}}}}\right)\textbf{V}_{i}$
(15)
The final context matrix C is the result of the linearly-transformed
concatenation of each head-specific context matrix $\textbf{C}_{i}$, as
expressed in Equation 16.
$\textbf{C}=[\textbf{C}_{1},...,\textbf{C}_{h}]\textbf{W}^{O}$ (16)
The decoder $p_{\theta}$ then maps the context matrix C to an output
distribution with parameters $\boldsymbol{\mu}_{\textbf{X}}$ and
$\log\boldsymbol{\sigma}_{\textbf{X}}^{2}$, as shown in Equation 17.
$(\boldsymbol{\mu}_{\textbf{X}},\log\boldsymbol{\sigma}_{\textbf{X}}^{2})=p_{\theta}(\textbf{C})$
(17)
### 4.2 Inference Mode
Despite the generative capabilities of VAE, MA-VAE does not leverage
generation for anomaly detection. Rather than sampling a latent matrix as
shown in Equation 13 during inference, sampling is disabled and only
$\boldsymbol{\mu}_{\textbf{Z}}$ is taken as the input for the multi-head
attention mechanism, like in [von Schleinitz et al., 2021]. Equation 13 in the
forward pass, therefore, is replaced by Equation 18.
$\textbf{Z}=\boldsymbol{\mu}_{\textbf{Z}}$ (18)
This not only accelerates inference by eliminating the sampling process but is
also empirically found to be a good approximation of an averaged latent matrix
if it were sampled several times like in [Pereira and Silveira, 2018]. The MA-
VAE layout during inference is shown in Figure 2, where the traced arrow
designates the information flow from the encoder to the MA mechanism.
### 4.3 Threshold Estimation Method
Anomalies are by definition very rare events, hence an ideal anomaly detector
only flags measurements very rarely but accurately. Test bench engineers
prefer an algorithm that only flags a sequence it is sure is an anomaly, in
other words, an algorithm that outputs very few to no false positives. A high
false positive count would lead to a lot of stoppages and therefore lost
testing time and additional cost. Of course, the vast majority of measurements
evaluated will be normal and hence it is paramount to classify them correctly,
naturally leading to a high precision value. Also, there is no automatic
evaluation methodology currently running at test benches, other than
rudimentary rule-based methods, therefore a solution that plugs into the
existing system that automatically detects some or most anomalies undetectable
by rules is already a gain. To achieve this, the threshold $\tau$ is set as
the maximum log probability observed when the model is fed with validation
data.
### 4.4 Bypass Phenomenon
VAE, when combined with an attention mechanism, can exhibit a behaviour called
the bypass phenomenon [Bahuleyan et al., 2018]. When the bypass phenomenon
happens the latent path between encoder and decoder is ignored and information
flow occurs mostly or exclusively through the attention mechanism, as it has
deterministic access to the encoder hidden states and therefore avoids
regularisation through the $D_{\text{KL}}$ term. In an attempt to avoid this,
[Bahuleyan et al., 2018] propose variational attention, which, like the VAE,
maps the input to a distribution rather than a deterministic vector. Applied
to natural language processing, [Bahuleyan et al., 2018] demonstrate that this
leads to a diversified generated portfolio of sentences, indicating
alleviation of the bypassing phenomenon. As previously mentioned, only
[Pereira and Silveira, 2018] applies this insight in the anomaly detection
domain, however, they do not present any proof that it alleviates the bypass
phenomenon in their work. MA-VAE on the other hand, cannot suffer from the
bypass phenomenon in the sense that information flow ignores the latent
variational path between encoder and decoder since the MA mechanism requires
the value matrix V from the encoder to output the context matrix. Assuming the
bypass phenomenon also applies to a case where information flow ignores the
attention mechanism, one could claim that MA-VAE is not immune. To disprove
this claim, the attention mechanism is removed from the model in an ablation
study to see if anomaly detection performance remains the same. In this case,
V is instead directly input into the decoder. If it drops, it is evidence of
the contribution of the attention mechanism to the model performance and hence
is not bypassed. The results for this ablation study are shown and discussed
in Section 5.
### 4.5 Impact of Seed Choice
Given the stochastic nature of the VAE, the chosen seed can impact the anomaly
detection performance as it can lead to a different local minimum during
training. To investigate the impact the seed choice has on model training, MA-
VAE is trained on three different seeds, the respective results are also shown
in Section 5.
### 4.6 Reverse-window Process
Since the model is trained to reconstruct fixed-length windows, the same
applies during inference. However, to decide whether a given measurement
sequence $\mathcal{S}\in\mathbb{R}^{T\times d_{\textbf{X}}}$ is anomalous, a
continuous reconstruction of the measurement is required. The easiest way to
do so would be to window the input measurement using a shift of $1$, input the
windows into the model and chain the last time step from each output window to
obtain a continuous sequence [Chen et al., 2020]. Considering the BiLSTM
nature of the encoder and decoder, the first and last time steps of a window
can only be computed given the states from one direction, making these values,
in theory, less accurate, however. To overcome this, we propose averaging
matching time steps in overlapping windows, which is called mean-type reverse-
window method. This is done by pre-allocating an array with NaN values,
filling it, and taking the mean for each time step while ignoring the NaN
values. This process and the general anomaly detection process are described
in Algorithm 1. This reverse-window process is done for the mean and variance
parameters of the output distribution, then the variance is converted to
standard deviation since two distributions cannot be combined by averaging the
standard deviations. With a continuous mean and standard deviation, the
continuous negative log probability, i.e. the anomaly score $s$, is computed
for the respective measurement. A comparison between the mean, last and first
reverse-window process is provided in Section 5.
Algorithm 1 Anomaly Detection Process
$\text{Sequence }\mathcal{S}\in\mathbb{R}^{T\times
d_{\textbf{X}}},\quad\text{Threshold }\tau$
$\text{Label }l$
$n_{\text{windows}}\leftarrow T-W+1$
$\boldsymbol{\mu}_{\textbf{X},\text{temp}}\leftarrow\text{zeros}(n_{\text{windows}},T,d_{\textbf{X}})+\text{NaN}$
$\boldsymbol{\sigma}_{\textbf{X},\text{temp}}^{2}\leftarrow\text{zeros}(n_{\text{windows}},T,d_{\textbf{X}})+\text{NaN}$
for $i=1\to n_{\text{windows}}$ do
$\textbf{X}\leftarrow\mathcal{S}[i:W+i]$
$(\boldsymbol{\mu}_{\textbf{Z}},\log\boldsymbol{\sigma}_{\textbf{Z}}^{2})\leftarrow
q_{\phi}(\textbf{X})$
$\textbf{C}\leftarrow\text{MA}(\textbf{X},\textbf{X},\boldsymbol{\mu}_{\textbf{Z}})$
$(\boldsymbol{\mu}_{\textbf{X}},\log\boldsymbol{\sigma}_{\textbf{X}}^{2})\leftarrow
p_{\theta}(\textbf{C})$
$\boldsymbol{\mu}_{\textbf{X},\text{temp}}[i,i:i+W]\leftarrow\boldsymbol{\mu}_{\textbf{X}}$
$\boldsymbol{\sigma}_{\textbf{X},\text{temp}}^{2}[i,i:i+W]\leftarrow\boldsymbol{\sigma}_{\textbf{X}}^{2}$
end for
$\boldsymbol{\mu}_{\textbf{X},\text{seq}}\leftarrow\text{nanmean}(\boldsymbol{\mu}_{\textbf{X},\text{temp}})$
$\boldsymbol{\sigma}_{\textbf{X},\text{seq}}^{2}\leftarrow\text{nanmean}(\boldsymbol{\sigma}_{\textbf{X},\text{temp}}^{2})$
$s\leftarrow-\log
p(\textbf{X}|\boldsymbol{\mu}_{\textbf{X},\text{seq}},\boldsymbol{\sigma}_{\textbf{X},\text{seq}})$
$l\leftarrow\text{max}(s)>\tau$
## 5 Results
### 5.1 Setup
The encoder and decoder both consist of two BiLSTM layers, with the outer ones
having 512 hidden- and cell-state sizes and the inner ones 256. All other
parameters are left as the default in the TensorFlow API.
During training only, input windows are corrupted using Gaussian noise using
$0.01$ standard deviation to increase robustness to noise.
Key factors that are investigated in Section 5 are given a default value which
applies to all experiments unless otherwise specified. These factors are
training/validation subset size, which is set to $512$h, seed choice, which
has been kept at $1$, reverse-window method, where the mean-type is used, the
latent dimension size, which is set to $d_{\textbf{Z}}=16$ and the MA
mechanism, which is set up as proposed in [Vaswani et al., 2017] with a head
count of $h=8$ and a key dimension size $d_{\textbf{K}}=\lfloor
d_{\textbf{X}}/h\rfloor=1$.
The optimiser used is the AMSGrad optimiser with the default parameters in the
TensorFlow API.
Cyclical $D_{\text{KL}}$ annealing [Fu et al., 2019] is applied to the
training of MA-VAE, to avoid the $D_{\text{KL}}$ vanishing problem. The
$D_{\text{KL}}$ vanishing problem occurs when regularisation is too strong at
the beginning of training, i.e. the Kullback-Leibler divergence term has a
larger magnitude in relation to the reconstruction term. Cyclical
$D_{\text{KL}}$ annealing allows the model to weigh the Kullback-Leibler
divergence lower than the reconstruction term in a cyclical manner through a
weight $\beta$. This callback is configured with a grace period of 25 epochs,
where $\beta$ is linearly increased from $0$ to $10^{-8}$. After the grace
period, $\beta$ is set to $10^{-8}$ and is gradually increased linearly to
$10^{-2}$ throughout the following $25$ epochs, representing one loss cycle.
This loss cycle is repeated until the training stops.
All priors in this work are set as standard Gaussian distributions, i.e.
$p=\mathcal{N}(0,1)$.
To prevent overfitting, early stopping is implemented. It works by monitoring
the log probability component of the validation loss during training and
stopping if it does not improve for $250$ epochs. Logically, the model weights
at the lowest log probability validation loss are saved.
Training is done on a workstation configured with an NVIDIA RTX A6000 GPU. The
library used for model training is TensorFlow 2.10.1 on Python 3.10 on Windows
10 Enterprise LTSC version 21H2.
The results provided are given in the form of the calibrated and uncalibrated
anomaly detection performance, i.e. with and without consideration of
threshold $\tau$, respectively. Recall that the threshold used is the absolute
maximum negative log probability obtained from the validation set. Calibrated
metrics are the precision, recall and $F_{1}$ score. Precision $P$ represents
the ratio between correctly identified anomalies (true positives) and all
positives (true and false), shown in Equation 19, recall $R$ represents the
ratio between true positives and all anomalies, shown in Equation 19, and
$F_{1}$ score represents the harmonic mean of the precision and recall, shown
in Equation 20. The underlying metrics used to calculate all of the below are
the true positives ($TP$), false negatives ($FN$) and false positives ($FP$).
$P=\frac{TP}{TP+FP}\quad R=\frac{TP}{TP+FN}$ (19)
$F_{1}=\frac{TP}{TP+0.5*(FP+FN)}=2*\frac{P*R}{P+R}$ (20)
The theoretical maximum $F_{1}$ score, $F_{1,\text{best}}$, is also provided
to aid discussion. This represents the best possible score achievable by the
approach if the ideal threshold were known, i.e. the point on the precision-
recall curve that comes closest to the $P=R=1$ point, though, in reality, this
value is not observable and hence cannot be obtained in an unsupervised
manner.
The uncalibrated anomaly detection performance, i.e. the performance for a
range of thresholds, each $0.1$ apart, is represented by the area under the
continuous precision-recall curve $A_{\text{PRC}}^{\text{cont}}$, Equation 21.
$A_{\text{PRC}}^{\text{cont}}=\int_{0}^{1}P\,dR$ (21)
As the integral cannot be computed for the continuous function, the area under
the discrete precision-recall curve $A_{\text{PRC}}^{\text{disc}}$ is used
which is done using the trapezoidal rule, Equation 22.
$A_{\text{PRC}}^{\text{disc}}=\sum^{N}_{k=1}\frac{f(R_{k-1})+f(R_{k})}{2}\Delta
R_{k}$ (22)
where $N$ is the number of discrete sub-intervals, $k$ the index of sub-
intervals and $\Delta R_{k}$ the sub-intervals length at index $k$. Precision
is a function of recall, i.e. $P=f(R)$.
### 5.2 Ablation Study
MA-VAE is tested without the MA mechanism and with a direct connection from
the encoder to the decoder to observe whether it impacts results.
The anomaly detection performance of MA-VAE and its counterpart without MA,
henceforth referred to as No MA model, are shown in Table 2. While the
precision value of the No MA model is slightly higher than the MA-VAE, the
recall value on the other hand is much lower. Overall, MA-VAE has a higher
$F_{1}$ score, as well as a higher theoretical maximum $F_{1}$ score, although
both values are so close enough to each other that one could claim the
threshold is near ideal. The uncalibrated performance is also higher in the
case of the MA-VAE, as evident in the precision-recall plot in Figure 3.
Interestingly, MA-VAE may feature a lower precision value for the chosen
unsupervised threshold but has the potential to have a higher maximum recall
at $P=1$.
The results hence point towards an improvement brought about by the addition
of the MA mechanism and therefore the bypass phenomenon can be ruled out.
Table 2: Precision $P$, recall $R$, $F_{1}$ score, theoretical best $F_{1}$ score $F_{1,\text{best}}$ and area under the precision-recall curve $A_{PRC}$ results for the model variant without the MA mechanism and MA-VAE. The best values for each metric are given in bold. Model | $P$ | $R$ | $F_{1}$ | $F_{1,\text{best}}$ | $A_{\text{PRC}}$
---|---|---|---|---|---
No MA | 1.00 | 0.35 | 0.52 | 0.54 | 0.52
MA-VAE | 0.92 | 0.55 | 0.69 | 0.70 | 0.66
Figure 3: Precision-recall curves for the model variation without MA and MA-
VAE.
### 5.3 Data Set Size Requirements
To evaluate how much data is required to train MA-VAE to a point of adequate
anomaly detection performance, it has been trained with $1$h, $8$h, $64$h, and
$512$h worth of dynamic testing data.
The results for this experiment are presented in Table 3. On the one hand, as
the training/validation subset increases in size, the precision value
improves, with the largest jump occurring when the dynamic testing time goes
from $1$h to $8$h. The recall value on the other hand decreases as the subset
grows. This can be attributed to the fact that smaller subset sizes lead to a
small validation set and therefore less data to obtain a threshold from. With
a limited amount of data to obtain a threshold from, it is more difficult to
get a representative error distribution, leading to a threshold that is very
small and hence marks most anomalies correctly but also leads to a lot of
false positives. $F_{1}$ score reaches a point of diminishing returns with the
$8$h subset onwards, this can also be observed in the case of the theoretical
maximum $F_{1}$ score, $F_{1,\text{best}}$, as well as in the $A_{\text{PRC}}$
value, further supported by the precision-recall plot in Figure 4. Lastly, the
$F_{1}$ score seems to approach the $F_{1,\text{best}}$ score as the subset
grows, also backing the fact that with a small subset size, a good threshold
cannot easily be obtained.
Table 3: Precision $P$, recall $R$, $F_{1}$ score, threoretical best $F_{1}$ score $F_{1,\text{best}}$ and area under the precision-recall curve $A_{PRC}$ results for the different training/validation subset sizes. The best values for each metric are given in bold. Size | $P$ | $R$ | $F_{1}$ | $F_{1,\text{best}}$ | $A_{\text{PRC}}$
---|---|---|---|---|---
$1$h | 0.09 | 0.88 | 0.17 | 0.55 | 0.49
$8$h | 0.66 | 0.63 | 0.64 | 0.72 | 0.69
$64$h | 0.71 | 0.57 | 0.63 | 0.69 | 0.68
$512$h | 0.92 | 0.55 | 0.69 | 0.70 | 0.66
Figure 4: Precision-recall curves for the model trained on different
training/validation subset sizes.
Therefore, for application at the test bench, the largest subset size is
desirable due to the higher precision value and a closer-to-ideal threshold
value.
### 5.4 Impact of Seed Choice
To illustrate the impact it has on the performance metrics, they are presented
using three different seeds.
Table 4 shows that while the precision values are roughly in the same range
for all seeds, the recall values vary more significantly, which also reflects
on the $F_{1}$ score. However, by inspecting the $F_{1,\text{best}}$ and
$A_{\text{PRC}}$ values it becomes clear that the seeds are not as far apart
as the recall value suggests and that the issue may lie with the threshold
choice. Figure 5 further supports this, as all lines have roughly the same
path, with the exception of seed $3$ at very high precision values. The plot
clearly shows that a more suitable (lower) threshold would lead to seed $3$
having a comparable recall value to the other seeds while maintaining high
precision.
Some differences can be observed between the seeds, especially in the recall
values, however, this can be attributed to the unsupervised threshold choice.
Table 4: Precision $P$, recall $R$, $F_{1}$ score, threoretical best $F_{1}$ score $F_{1,\text{best}}$ and area under the precision-recall curve $A_{PRC}$ results for the different seeds. The best values for each metric are given in bold. Seed | $P$ | $R$ | $F_{1}$ | $F_{1,\text{best}}$ | $A_{\text{PRC}}$
---|---|---|---|---|---
1 | 0.92 | 0.55 | 0.69 | 0.70 | 0.66
2 | 0.90 | 0.60 | 0.72 | 0.73 | 0.67
3 | 0.96 | 0.40 | 0.56 | 0.70 | 0.64
Figure 5: Precision-recall curves for the model trained on different seeds.
### 5.5 Reverse-window Process
To investigate the effect of the mean-type reverse-window method, it is
compared with the first-type and last-type methods where the first and last
values of each window are carried over, respectively.
The results in this subsection, Table 5 and Figure 6, tell a similar story to
the previous subsection. The metrics independent of the chosen threshold are
very similar regardless of the reverse-window method, implying that they are
comparable and that any differences in the calibrated metrics can be
attributed to the chosen threshold. The mean-type reverse-window method
results in a higher computational load, though negligible. For a rather long
sequence of $4000$ time steps, i.e. around $33$ minutes long, the mean-type
method only takes around $2$ seconds longer. One source of delay that can
appear, however, is during online anomaly detection. An online anomaly
detection algorithm is defined as an algorithm which evaluates the sequence as
it is being recorded. To obtain time step $t$ using the mean-type (or the
first-type) you have to wait for time step $t+W$ while $t<W$. This translates
to a delay of around $2$ minutes in the real world, given the chosen window
size. If the evaluation is done offline, i.e. when $t=W$, then this delay is
eliminated since the last value does not have other overlapping values to
compute the mean.
Table 5: Precision $P$, recall $R$, $F_{1}$ score, threoretical best $F_{1}$ score $F_{1,\text{best}}$ and area under the precision-recall curve $A_{PRC}$ results for the different reverse-window types. The best values for each metric are given in bold. Type | $P$ | $R$ | $F_{1}$ | $F_{1,\text{best}}$ | $A_{\text{PRC}}$
---|---|---|---|---|---
first | 0.97 | 0.48 | 0.64 | 0.69 | 0.64
last | 0.88 | 0.58 | 0.71 | 0.71 | 0.67
mean | 0.92 | 0.55 | 0.69 | 0.70 | 0.66
Figure 6: Precision-recall curves for different reverse-window methods.
### 5.6 Hyperparameter Optimisation
As part of the hyperparameter optimisation of MA-VAE, a list of latent
dimension sizes $d_{\textbf{Z}}$ in combination with a list of key dimension
sizes $d_{\textbf{K}}$ is tested. Despite the larger learning capacity
associated with a higher $d_{\textbf{K}}$, the concatenation is always
transformed to a matrix of size $d_{\textbf{O}}=d_{\textbf{Z}}$. For the two
variables, values of 1, 4, 16, and 64 are tested.
The best result is achieved with $d_{\textbf{Z}}=d_{\textbf{K}}=64$. Given
that they are the respective highest values of $d_{\textbf{Z}}$ and
$d_{\textbf{K}}$, even higher values should be experimented with in the
future, though they will lead to higher model complexity and
training/inference time. The attention head count $h$ was also experimented
with using the same range of values as for $d_{\textbf{Z}}$ and
$d_{\textbf{K}}$, however, none performed better than the $h=8$ configuration.
The results are presented in Table 6, the corresponding precision-recall plot
is shown in Figure 7. $91\%$ of the sequences marked as normal were actually
normal and $67\%$ of the total number of anomalous sequences in the test set
were detected. One example of the anomalous cycles and the respective
reconstructions is plotted in Figure 8.
Table 6: Precision $P$, recall $R$, $F_{1}$ score, threoretical best $F_{1}$ score $F_{1,\text{best}}$ and area under the precision-recall curve $A_{PRC}$ result for the best $d_{\textbf{Z}}$, $d_{\textbf{K}}$ and $h$ values. $d_{\textbf{Z}}$ | $d_{\textbf{K}}$ | $h$ | $P$ | $R$ | $F_{1}$ | $F_{1,\text{best}}$ | $A_{\text{PRC}}$
---|---|---|---|---|---|---|---
64 | 64 | 8 | 0.91 | 0.67 | 0.77 | 0.79 | 0.74
Figure 7: Precision-recall curve for the final MA-VAE. Figure 8: Wheel
diameter anomaly plotted in black and the output distribution in red, as well
as anomaly score plotted in blue and the threshold as a straight line in
orange.
### 5.7 Benchmarking
Of course, MA-VAE is not the first model proposed for time-series anomaly
detection. To underline its anomaly detection performance, it is compared with
a series of other models based on variational autoencoders. The chosen subset
of models is based on the work discussed in Section 3 which either linked
source code or contained enough information for implementation. The models are
implemented using hyperparameters specified in their respective publications.
To even the playing field, the models are trained on the $512$h subset with
early stopping, which is parametrised equally across all models. The anomaly
detection process specified in Algorithm 1 is also applied to all models,
along with the threshold estimation method. The results can be seen in Table
7.
Table 7: Precision $P$, recall $R$, $F_{1}$ score, threoretical best $F_{1}$ score $F_{1,\text{best}}$ and area under the precision-recall curve $A_{PRC}$ results for competing models and MA-VAE (Ours). The best values for each metric are given in bold. Model | $P$ | $R$ | $F_{1}$ | $F_{1,\text{best}}$ | $A_{\text{PRC}}$
---|---|---|---|---|---
VS-VAE | 1.00 | 0.33 | 0.50 | 0.56 | 0.51
W-VAE | 1.00 | 0.30 | 0.46 | 0.46 | 0.41
OmniA | 0.96 | 0.37 | 0.53 | 0.58 | 0.53
SISVAE | 1.00 | 0.30 | 0.46 | 0.50 | 0.51
MA-VAE | 0.91 | 0.67 | 0.77 | 0.79 | 0.74
As is evident, MA-VAE outperforms all other models in every metric, except for
precision. As stated in Section 4 a high precision figure is important in this
type of powertrain testing, however, the reduced precision is still considered
tolerable. Also, it comes at the benefit of a much higher recall figure, which
is reflected in the superior $F_{1}$ figure. Furthermore, the
$F_{1,\text{best}}$ figure, which is obtained at $P=0.98$ and $R=0.67$,
suggests that MA-VAE has the potential to achieve even higher precision
without sacrificing recall if the threshold were optimised. The higher
$A_{PRC}$ also shows that MA-VAE has a higher range of thresholds at which it
performs well.
## 6 Conclusion and Outlook
In this paper, a multi-head attention variational autoencoder (MA-VAE) for
anomaly detection in automotive testing is proposed. It not only features an
attention configuration that avoids the bypass phenomenon but also introduces
a novel method of remapping windows to whole sequences. A number of
experiments are conducted to demonstrate the anomaly detection performance of
the model, as well as to underline the benefits of key aspects introduced with
the model.
From the results obtained, MA-VAE clearly benefits from the MA mechanism,
indicating the avoidance of the bypass phenomenon. Moreover, the proposed
approach only requires a small training/validation subset size but fails to
obtain a suitable threshold, as with increasing subset size only the
calibrated anomaly detection performance increases. Training with different
seeds also is shown to have little impact on the anomaly detection metrics,
provided the threshold is chosen suitably, further underlining the previous
point. Moreover, mean-type reverse windowing fails to significantly outperform
its first-type and last-type counterparts, while introducing additional lag if
it is applied to online anomaly detection. Lastly, the hyperparameter
optimisation revealed that the MA-VAE variant with the largest latent
dimension and attention key dimension resulted in the best anomaly detection
performance. It is only $9\%$ of the time wrong when an anomaly is flagged and
manages to discover $67\%$ of the anomalies present in the test data set.
Also, it outperforms all other competing models it is compared with.
In the future, a method of threshold choice involving active learning will be
investigated, which can use user feedback to hone in on a better threshold.
Also, MA-VAE is set to be tested in the context of online anomaly detection,
i.e. during the driving cycle measurement.
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|
Universal relation for operator complexity
Zhong-Ying Fan1†
1† Department of Astrophysics, School of Physics and Material Science,
Guangzhou University, Guangzhou 510006, China
ABSTRACT
We study Krylov complexity $C_{K}$ and operator entropy $S_{K}$ in operator
growth. We find that for a variety of systems, including chaotic ones and
integrable theories, the two quantities always enjoy a logarithmic relation
$S_{K}\sim\mathrm{ln}{C_{K}}$ at long times, where dissipative behavior
emerges in unitary evolution. Otherwise, the relation does not hold any
longer. Universality of the relation is deeply connected to irreversibility of
operator growth.
Email<EMAIL_ADDRESS>,
###### Contents
1. 1 Introduction
2. 2 Preliminaries: recursion method and irreversibility of operator dynamics
1. 2.1 The recursion method
2. 2.2 Irreversibility and ergodicity
3. 3 Operator growth at initial times
1. 3.1 Initial growth
2. 3.2 Testing the initial growth
4. 4 Operator growth at long times
1. 4.1 Chaotic systems
1. 4.1.1 With logarithmic correction
2. 4.2 Integrable theories
1. 4.2.1 With logarithmic correction
3. 4.3 With bounded support and beyond
5. 5 Finite chains
6. 6 Conclusion and discussion
## 1 Introduction
Time evolution in quantum mechanical systems is generally local and unitary.
However, it is also known that many quantum systems have effectively
irreversible hydrodynamic descriptions, for example the transport properties.
It is a central goal to understand emergence of this thermal behavior in
theoretical research on quantum many body systems. Operator growth and the
relation to thermalization is an exciting topic is in this direction, for
example see [1, 2, 3, 4, 5].
Consider an initial operator $\mathcal{O}_{0}$ and suppose it can be written
as a sum of a few basis vectors in any local basis. Heisenberg evolution of
the space of operators, $\mathcal{O}(t)=e^{iHt}\mathcal{O}_{0}e^{-iHt}$, is
described by a set of nested commutators of $\mathcal{O}_{0}$ with the
Hamiltonian $H$. Evaluation of all these commutators is equivalent to solving
dynamics of the operator completely. However, for complex systems, the number
of these commutators (or the nonzero coefficients of $\mathcal{O}(t)$ in any
local basis) increases monotonically in the evolution and can blow up
exponentially: the initially simple operator $\mathcal{O}_{0}$ grows
irreversibly into a complex one. Because of exponential size of the problem, a
statistical description should emerge for the process. This implies that
operator growth should have some form of universality, similar to statistical
mechanics. It is of great interests to search universal features of operator
dynamics.
In this paper, we would like to study information quantities, the Krylov
complexity ( or K-complexity) [6] and the operator entropy (or K-entropy) [7]
in operator growth. Previously, the two quantities were examined separately in
literature [8, 9, 10, 11, 12, 13]. Here we are interested in examining their
functional relation in the time evolution. We will show that the two enjoys a
logarithmic relation at long times111In this paper, the base of the logarithm
is the constant $e$.
$S_{K}(t)=\tilde{\eta}\,\mathrm{ln}{C_{K}(t)}+\cdots\,,$ (1)
where dissipative behavior emerges. Here $\tilde{\eta}$ is a positive
constant, depending on the choice of dynamics. We propose that it is up
bounded as $\tilde{\eta}\leq 1$. The relation is universal to all irreversible
growth of operators, not necessarily exponential growth, as far as we can
check.
The paper is organized as follows. In section 2, we briefly review the
recursion method and a quantitative measure for irreversibility of operator
growth. In section 3, we study the functional relation between $S_{K}$ and
$C_{K}$ at initial times. We show that they enjoy a product logarithmic
relation $S_{K}\sim-C_{K}\log{C_{K}}$ to leading order. In section 4, we study
the long time behavior of $S_{K}$ and $C_{K}$ for a variety of systems with
emergence of dissipative behaviors222In this paper, “dissipative behavior” was
referred to the decaying behavior of operator wave functions at long times.,
including chaotic ones and integrable theories. We find that they always enjoy
a logarithmic relation $S_{K}\sim\log{C_{K}}$ at leading order, irrespective
of dynamical details. In section 5, as a comparison, we examine simple
systems, which have no emergence of dissipative behaivors and show that in
this case, the logarithmic relation does not hold any longer. We conclude in
section 6.
## 2 Preliminaries: recursion method and irreversibility of operator dynamics
For a lattice system with Hamiltonian $H$, the initial operator
$\mathcal{O}_{0}$ evolves in time according to the Heisenberg equation
$\partial_{t}\mathcal{O}(t)=i[H\,,\mathcal{O}(t)]$ or explicitly
$\mathcal{O}(t)=e^{iHt}\mathcal{O}_{0}e^{-iHt}=\sum_{n=0}{\frac{(it)^{n}}{n!}}\tilde{\mathcal{O}}_{n}\,,$
(2)
where $\tilde{\mathcal{O}}_{n}$ stands for nested commutators:
$\tilde{\mathcal{O}}_{1}=[H,\mathcal{O}_{0}]\,,\tilde{\mathcal{O}}_{2}=[H,\tilde{\mathcal{O}}_{1}]\,,\cdots\,,\tilde{\mathcal{O}}_{n}=[H,\tilde{\mathcal{O}}_{n-1}]\,.$
However, evaluation of these commutators is of great difficult for complex
systems. Sometimes it is helpful to thick of the problem as solving a
Schrödinger equation by taking the operator as a wave function. Define the
Liouvillian $\mathcal{L}\equiv[H\,,\cdot]$, the operator wave function evolves
as
$|\mathcal{O}(t)\rangle=e^{i\mathcal{L}t}|\mathcal{O}_{0}\rangle=\sum_{n=0}{\frac{(it)^{n}}{n!}}|\tilde{\mathcal{O}}_{n}\rangle\,,$
(3)
where
$|\tilde{\mathcal{O}}_{n}\rangle=\mathcal{L}^{n}|\mathcal{O}_{0}\rangle.$ In
this language, one generally needs to introduce a proper inner product in the
operator Hilbert space. For example, in the infinite temperature limit, one
usually has $\langle A|B\rangle=\mathrm{Tr}\big{(}A^{{\dagger}}B\big{)}$. We
refer the readers to [14] for more details. The physical information about
operator growth is essentially encoded in the so-called auto-correlation
function, which is defined as
$C(t):=\langle\mathcal{O}_{0}|\mathcal{O}(t)\rangle=\langle\mathcal{O}_{0}|e^{i\mathcal{L}t}|\mathcal{O}_{0}\rangle\,.$
(4)
The same information can also be extracted from the moments $\mu_{2n}$
$\mu_{2n}:=\langle\mathcal{O}_{0}|\mathcal{L}^{2n}|\mathcal{O}_{0}\rangle={\frac{d^{2n}}{dt^{2n}}}C(-it)\Big{|}_{t=0}\,.$
(5)
Note
$C(t)=\sum_{n=0}{\frac{\mu_{2n}(it)^{2n}}{(2n)!}}=1-{\frac{1}{2}}\mu_{2}t^{2}+\cdots$,
implying that the initial growth of operators is determined by the lowest
moment $\mu_{2}$. We will come to this point in the next section.
Another two quantities encoding the same information about the dynamics of
operators are the relaxation function $\phi_{0}(z)$ and the spectral density
$\Phi(\omega)$. They are defined as the Laplace transform and the Fourier
transform of $C(t)$, respectively
$\displaystyle\phi_{0}(z):=\int_{0}^{+\infty}dt\,e^{-zt}C(t)\,,$
$\displaystyle\Phi(\omega):=\int_{-\infty}^{+\infty}dt\,e^{-i\omega t}C(t)\,.$
(6)
Note that they (and the moments ) are linearly related to the auto-correlation
function. In particular, one has [14]
$\Phi(\omega)=2\lim_{\varepsilon\rightarrow
0}\mathrm{Re}\big{[}\phi_{0}(\varepsilon-i\omega)\big{]}\,.$ (7)
In the remaining of this paper, we will frequenctly switch between these
quantities and use the one that is most convenient to discuss the physics we
are interested in. The readers should not be confused.
### 2.1 The recursion method
In general, the original operator basis
$\\{|\tilde{\mathcal{O}}_{n}\rangle\\}$ is not orthogonal. Just like in
ordinary quantum mechanics, one can study dynamics of the operator wave
function using different basises, of which a particularly interesting one is
orthonormal. This leads to a different approach to operator growth. Let us
briefly review it in the following.
Using Gram-Schmidt scheme, one starts with a normalized vector
$|\mathcal{O}_{0}\rangle$. Then the first vector is given by
$|\mathcal{O}_{1}\rangle:=b_{1}^{-1}\mathcal{L}|\mathcal{O}_{0}\rangle$, where
$b_{1}:=\langle\mathcal{O}_{0}\mathcal{L}|\mathcal{L}\mathcal{O}_{0}\rangle^{1/2}$.
For the $n-$th vector, one inductively defines
$\displaystyle|A_{n}\rangle:=\mathcal{L}|\mathcal{O}_{n-1}\rangle-
b_{n-1}|\mathcal{O}_{n-2}\rangle\,,$
$\displaystyle|\mathcal{O}_{n}\rangle:=b_{n}^{-1}|A_{n}\rangle\,,\quad
b_{n}:=\langle A_{n}|A_{n}\rangle^{1/2}\,.$ (8)
The output of the above procedure is a set of orthomornal vectors
$\\{|\mathcal{O}_{n}\rangle\\}$, called Krylov basis and a sequence of
positive numbers $\\{b_{n}\\}$, referred to as Lanczos coefficients, which
have units of energy (in this paper, time is measured in units of the inverse
of Lanczos coefficients, for example $1/b_{1}$). The physical information
encoded in the auto-correlation function or moments can be equivalently
extracted from the Lanczos coefficients. However, they are related via
nonlinear transformations. In Krylov basis, the Liouvillian is tridiagonal
$L_{nm}:=\langle
O_{n}|\mathcal{L}|O_{m}\rangle=\left(\begin{array}[]{cccccccc}0&b_{1}&0&0&\ldots\\\
b_{1}&0&b_{2}&0&\ldots\\\ 0&b_{2}&0&b_{3}&\ldots\\\ 0&0&b_{3}&0&\ddots\\\
\vdots&\vdots&\vdots&\ddots&\ddots\\\ \end{array}\right)\,,$ (9)
whereas the moments are given by
$\mu_{2n}=\langle\mathcal{O}_{0}|\mathcal{L}^{2n}|\mathcal{O}_{0}\rangle=(L^{2n})_{00}\,.$
(10)
Some low lying examples are
$\mu_{2}=b_{1}^{2}\,,\quad\mu_{4}=b_{1}^{4}+b_{1}^{2}b_{2}^{2}\,,\quad\cdots\,.$
(11)
General transformations between these two sets of constants can be found in
[14] (or Appendix A of [6]). The nonlinear information coding of Lanczos
coefficients plays an indispensable role in demonstrating universality of
operator growth [6].
In Krylov basis, evolution of the operator $\mathcal{O}(t)$ can be formally
written as
$|\mathcal{O}(t)\rangle:=\sum_{n=0}i^{n}\varphi_{n}(t)|\mathcal{O}_{n}\rangle\,,$
(12)
where $\varphi_{n}(t):=i^{-n}\langle\mathcal{O}_{n}|\mathcal{O}(t)\rangle$ is
a discrete set of wave functions. The Heisenberg evolution of $\mathcal{O}(t)$
gives rise to a discrete set of equations
$\partial_{t}\varphi_{n}=b_{n}\varphi_{n-1}-b_{n+1}\varphi_{n+1}\,,$ (13)
subject to the boundary condition $\varphi_{n}(0)=\delta_{n0}$ and
$b_{0}=0=\varphi_{-1}(t)$ by convention. This defines a one-dimensional
quantum mechanical problem on a half chain. This uniquely determines the wave
functions $\varphi_{n}(t)$ for a given set of Lanczos coefficients. Since the
auto-correlation function is simply $C(t)=\varphi_{0}(t)$, it is completely
equivalent to the Lanczos coefficients.
Taking the Laplace transform of (13), one finds
$z\phi_{n}(z)-\delta_{n0}=b_{n}\phi_{n-1}(z)-b_{n+1}\phi_{n+1}(z)\,,$ (14)
where $\phi_{n}(z)$ is the Laplace transform of $\varphi_{n}(t)$. The relation
gives the continued fraction representation of the relaxation function
$\phi_{0}(z)=\dfrac{1}{z+\dfrac{b_{1}^{2}}{z+\dfrac{b_{2}^{2}}{z+\ddots}}}\,.$
(15)
For finite chains, the recursion method naturally terminates at some finite
order and the relaxation function will have a compact expression. This already
solves the operator dynamics for simple systems completely, see sec. 5 for
more details. For (sufficiently) complex systems, the one dimensional chain
will be semi-infinite and different approaches are needed.
### 2.2 Irreversibility and ergodicity
It turns out that the condition for the emergence of ergodic behavior for
complex systems can be formulated in terms of operator growth directly. Define
a quantity $W$ as [15]
$W={\frac{b_{2}^{2}\,b_{4}^{2}\,b_{6}^{2}\cdots}{b_{1}^{2}\,b_{3}^{2}\,b_{5}^{2}\cdots}}\,,$
(16)
which is referred to as the canonical form of $W$. Equivalently, it can be
evaluated using either the relaxation function
$W=\phi_{0}(0)\,,$ (17)
or the auto-correlation function directly as
$W=\int_{0}^{+\infty}dt\,\varphi_{0}(t)\,.$ (18)
For simple systems, since the recursion method terminates at some finite order
so that $b_{2k}=0$ or $b_{2k+1}=0$, one will have $W=0$ or $W=\infty$. The
process will be reversible. Otherwise, a finite $W$ (except zero) describes an
irreversible process. It was shown [15] that this is equivalent to the Kubo’s
condition, which is formulated in terms of time averages of correlation
functions. However, evaluation of $W$ provides a simpler way to test
ergodicity of the theory.
There are also other ways that were believed to probe irreversibility of
operator growth. For example, at long times the auto-correlation function
should approach to zero. However, as pointed out in [15], this is only a
necessary condition. In fact, it is hard to search an alternate condition
equivalent to a finite $W$. Our new observation is information quantities: the
relation between K-complexity and K-entropy may be such a candidate.
## 3 Operator growth at initial times
According to (12), the wave function $\varphi_{n}(t)$ on the one dimensional
chain could be interpreted as probabilities, as in ordinary quantum mechanics.
Normalization of the operator wave function $|\mathcal{O}(t)\rangle$ implies
$\sum_{n=0}|\varphi_{n}(t)|^{2}=1$. The K-complexity was defined as the
average positions of the chain [6]
$C_{K}:=\langle\mathcal{O}(t)|n|\mathcal{O}(t)\rangle=\sum_{n=0}n|\varphi_{n}|^{2}\,.$
(19)
Clearly, it depends on the Krylov basis, the initial operator and the dynamics
under considerations. Yet physical meaning of $C_{K}$ is far from transparent.
It was established [6] that K-complexity provides an upper bound for other
notions of complexity, such as OTOCs. Application of the theorem to chaotic
systems leads to an upper bound on Lyapunov exponent $\lambda_{L}\leq
2\alpha$, where $\alpha$ is the asymptotic growth rate of Lanczos coefficient
$b_{n}\rightarrow\alpha n+\gamma$ as $n\rightarrow+\infty$. Moreover,
extension of the analysis to finite temperatures, it was argued that the bound
is tighter than the MSS bound [16] $\lambda_{L}\leq 2\alpha\leq 2\pi T$.
Though these results are interesting, the K-complexity has not been connected
to irreversibility of operator growth, as far as we know.
On the other hand, the operator entropy (or $K$-entropy) was defined as [7]
$S_{K}(t):=-\sum_{n=0}|\varphi_{n}(t)|^{2}\,\mathrm{ln}{|\varphi_{n}(t)|^{2}}\,.$
(20)
Similar to $C_{K}$, $S_{K}$ depends on the Krylov basis, the initial operator
and the choice of dynamics. Its physical meaning is not clear as well.
Properties of $S_{K}$ were previously examined in the scrambling regime and
the post-scrambling regime for chaotic systems [7].
In this paper, we will investigate $C_{K}$ and $S_{K}$ for a variety of
systems with dissipative behavior emerging at long times. We will show that
they always enjoy a logarithmic relation $S_{K}\sim\log{C_{K}}$ in this case.
### 3.1 Initial growth
Before studying the long time dynamics, let us first address operator growth
at initial times.
According to the discrete Schrödinger equation (13) and the boundary
conditions, one easily finds
$\varphi_{n}(0)=\delta_{n0}\,,\quad\dot{\varphi}_{n}(0)=b_{1}\delta_{n1}\,,\quad\ddot{\varphi}_{n}(0)=-b_{1}^{2}\delta_{n0}+b_{1}b_{2}\delta_{n2}\,,$
(21)
where a dot denotes a derivative with respect to $t$. Using these results, we
deduce
$C_{K}(0)=0\,,\quad\dot{C}_{K}(0)=0\,,\quad\ddot{C}_{K}(0)=2\mu_{2}\,.$ (22)
It implies that initially, the Krylov complexity grows quadratically
$C_{K}(t)=\mu_{2}t^{2}+\cdots\,.$ (23)
Extension of the analysis to the operator entropy yields
$S_{K}(0)=0\,,\quad\dot{S}_{K}(0)=0\,,\quad\ddot{S}_{K}(0)=-2\mu_{2}\,\mathrm{ln}{(\mu_{2}t^{2})}-4\mu_{2}\,.$
(24)
To leading order, the operator entropy behaves as
$S_{K}(t)=-\mu_{2}t^{2}\,\mathrm{ln}{(\mu_{2}t^{2})}+O(\mu_{2}t^{2})\,.$ (25)
The results imply a product-logarithmic relation between $S_{K}$ and $C_{K}$
at initial times
$S_{K}(t)=-C_{K}(t)\,\mathrm{ln}{C_{K}(t)}+O\big{(}C_{K}(t)\big{)}\,.$ (26)
Notice that the relation is universal to both reversible and irreversible
process. It does not capture any information about long time dynamics.
Comparing it to the result at long times will help us to understand the long
time dynamics better.
### 3.2 Testing the initial growth
Let us test the relation (26) using several exact examples.
The first is $b_{n}=\alpha\sqrt{n(n-1+\eta)}$, which naturally arises in SYK
model. The wave function on the semi-infinite chain can be solved as [6]
$\varphi_{n}(t)=\sqrt{{\textstyle{\frac{\scriptstyle\Gamma(n+\eta)}{\scriptstyle
n!\Gamma(\eta)}}}}\,{\frac{\tanh^{n}(\alpha t)}{\cosh^{\eta}{(\alpha t)}}}\,.$
(27)
Evaluating K-complexity yields
$C_{K}=\sum_{n}n|\varphi_{n}|^{2}=\eta\sinh^{2}{(\alpha t)}$. Clearly, at
initial times, $C_{K}\simeq\eta\alpha^{2}t^{2}=\mu_{2}t^{2}$, where
$\mu_{2}=b_{1}^{2}=\eta\alpha^{2}$. On the other hand,
$\log{|\varphi_{n}|^{2}}\simeq n\,\mathrm{ln}{(\alpha^{2}t^{2})}\simeq
n\mathrm{ln}{C_{K}}$, leading to
$S_{K}\simeq-\sum_{n}n|\varphi_{n}|^{2}\,\mathrm{ln}{C_{K}}=-C_{K}\,\mathrm{ln}{C_{K}}$.
This coincides with the relation (26).
The second example is $b_{n}=\alpha\sqrt{n}$, which appears in several
integrable models [6]. The operator state $|\mathcal{O}(t)\rangle$ can be
viewed as the Glauber coherent state in the operator Hilbert space [8]. This
gives
$\varphi_{n}(t)=e^{-\alpha^{2}t^{2}/2}{\frac{\alpha^{n}t^{n}}{\sqrt{n!}}}\,.$
(28)
The Krylov complexity can be evaluated as
$C_{K}=\alpha^{2}t^{2}=\mu_{2}t^{2}$. The initially quadratic growth persists
in the full time evolution. On the other hand, evaluation of the operator
entropy yields
$S_{K}=-C_{K}\,\mathrm{ln}{C_{K}}+C_{K}+\sum_{n}|\varphi_{n}|^{2}\log{n!}\,.$
(29)
Clearly, at initial times, the first term on the r.h.s is dominant. This again
coincides with the relation (26).
The third example is $b_{n}=\alpha\sqrt{n(2j-n+1)}$, where $j$ is an integer
or half integer [8]. The operator Hilbert space has a finite dimension with
$0\leq n\leq 2j$. It describes a reversible process since $W=\infty$ ( or
$W=0$) for an integer (or half integer) $j$. One has
$\varphi_{n}(t)=\sqrt{{\textstyle{\frac{\scriptstyle\Gamma(2j+1)}{\scriptstyle
n!\Gamma(2j-n+1)}}}}\,{\frac{\tan^{n}(\alpha t)}{\cos^{-2j}{(\alpha t)}}}\,.$
(30)
Evaluation of K-complexity and K-entropy yields $C_{K}=2j\sin^{2}{(\alpha t)}$
and
$S_{K}=-C_{K}\,\mathrm{ln}{\tan^{2}{(\alpha
t)}}-\,\mathrm{ln}{\Big{(}\Gamma(2j+1)\cos^{4j}(\alpha
t)\Big{)}}+\sum_{n}|\varphi_{n}|^{2}\,\mathrm{ln}{\Big{(}n!\,\Gamma(2j-n+1)\Big{)}}\,.$
(31)
Again at initial times $C_{K}\simeq\mu_{2}t^{2}$ and $S_{K}\simeq-
C_{K}\,\mathrm{ln}{C_{K}}$, consistent with the relation (26).
In fact, for reversible process, the operator wave functions can always be
solved exactly and hence there are a lot of examples that can test the
relation (26), for example see (5).
## 4 Operator growth at long times
We are moving to study the functional relation between K-complexity and
K-entropy for a variety of dissipative systems, including chaotic ones,
integrable theories and many others.
We will adopt the continuum limit as well as numerical approach. For semi-
infinite chains, the continuum limit is good at capture long time behaviors of
K-complexity and K-entropy using coarse grained wave functions. We shall
briefly review it by following [7].
Introducing a lattice cutoff $\epsilon$ and defining a coordinate $x=\epsilon
n$ and velocity $v(x)=2\epsilon b_{n}$. The interpolating wave function is
defined as $\varphi(x\,,t)=\varphi_{n}(t)$. Continuum version of the discrete
equation (13) is given by
$\partial_{t}\varphi(x\,,t)={\frac{1}{2\epsilon}}\Big{[}v(x)\varphi(x-\epsilon)-v(x+\epsilon)\varphi(x+\epsilon)\Big{]}\,.$
(32)
Expansion in powers of $\epsilon$, one finds a chiral wave equation to leading
order
$\partial_{t}\varphi=-v(x)\partial_{x}\varphi-{\frac{1}{2}}\partial_{x}v(x)\varphi+O(\epsilon)\,,$
(33)
with a position-dependent velocity $v(x)$ and mass
${\frac{1}{2}}\partial_{x}v(x)$. Introducing a new coordinate $y$ as
$v(x)\partial_{x}=\partial_{y}$ and a rescaled wave function
$\psi(y\,,t)=\sqrt{v(y)}\,\varphi(y\,,t)\,,$ (34)
the equation simplifies to
$(\partial_{t}+\partial_{y})\psi(y\,,t)=0+O(\epsilon)\,.$ (35)
The general solution is given by
$\psi(y\,,t)=\psi_{i}(y-t)\,,$ (36)
where $\psi_{i}(y)=\psi(y\,,0)$ stands for the initial amplitude. It implies
that at leading order the coarse grained wave function moves ballistically. It
turns out that this leading order approximation always derives the growth of
K-complexity correctly. However, for K-entropy, some higher order corrections
should be included for certain cases. In fact, the method just captures the
leading long time dependence for both K-complexity and K-entropy
qualitatively. Hence, we will also adopt numerical approach as a supplement.
Normalization condition in both $x$-frame and $y$-frame reads
$1=\sum_{n}|\varphi_{n}(t)|^{2}={\frac{1}{\epsilon}}\int\mathrm{d}x\,\varphi^{2}(x\,,t)={\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi^{2}(y\,,t)\,.$
(37)
The K-complexity can be evaluated as
$\displaystyle C_{K}(t)$ $\displaystyle=$
$\displaystyle\sum_{n}n|\varphi_{n}(t)|^{2}={\frac{1}{\epsilon}}\int\mathrm{d}x\,{\frac{x}{\epsilon}}\,\varphi^{2}(x\,,t)$
(38) $\displaystyle=$
$\displaystyle{\frac{1}{\epsilon}}\int\mathrm{d}y\,{\frac{x(y)}{\epsilon}}\,\psi_{i}^{2}(y-t)$
$\displaystyle=$
$\displaystyle{\frac{1}{\epsilon}}\int\mathrm{d}y\,{\frac{x(y+t)}{\epsilon}}\,\,\psi_{i}^{2}(y)\,,$
and the K-entropy reads
$\displaystyle S_{K}(t)$ $\displaystyle=$
$\displaystyle-\sum_{n}|\varphi_{n}(t)|^{2}\,\mathrm{ln}{|\varphi_{n}(t)|^{2}}$
(39) $\displaystyle=$
$\displaystyle-{\frac{1}{\epsilon}}\int\mathrm{d}x\,\varphi^{2}(x\,,t)\,\mathrm{ln}{\varphi^{2}(x\,,t)}$
$\displaystyle=$
$\displaystyle-{\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi_{i}^{2}(y-t)\,\mathrm{ln}{\Big{[}{\frac{\psi_{i}^{2}(y-t)}{v(y)}}\Big{]}}$
$\displaystyle=$
$\displaystyle-{\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi_{i}^{2}(y)\,\mathrm{ln}{\psi_{i}^{2}(y)}+{\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi_{i}^{2}(y)\,\mathrm{ln}{v(y+t)}\,,$
where the time dependence of $S_{K}$ is only contained in the second term on
the r.h.s. Once we know the transformation between the two frames, we are able
to extract the leading time dependence of $C_{K}$ and $S_{K}$ at long times
immediately.
### 4.1 Chaotic systems
It was first conjectured in [6] that for chaotic systems, the Lanczos
coefficient grows asymptotically linearly $b_{n}\rightarrow\alpha n+\gamma$ as
$n\rightarrow+\infty$ (This is valid to the infinite temperature limit. At
finite temperatures, it was shown in [11] that the asymptotically linear
behavior of $b_{n}$ can also be obtained for free quantum field theories,
which however do not probe chaos ). This gives rise to $v(x)=2\alpha
x+2\epsilon\gamma$, where the subleading order correction $\gamma$ plays only
a short time in the scrambling regime and hence is negligible in the continuum
limit. One finds
$y={\frac{1}{2\alpha}}\mathrm{ln}{\big{(}{\frac{x}{\epsilon}}\big{)}}\qquad\mathrm{or}\qquad
x=\epsilon\,e^{2\alpha y}\,.$ (40)
Evaluation of K-complexity yields
$C_{K}(t)={\frac{1}{\epsilon}}\int\mathrm{d}y\,e^{2\alpha(y+t)}\,\psi_{i}^{2}(y)=C_{K}(0)\,e^{2\alpha
t}\,.$ (41)
It grows exponentially with correct exponent. This coincides with the SYK
model. The operator entropy is given by
$\displaystyle S_{K}(t)$ $\displaystyle=$
$\displaystyle{\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi_{i}^{2}(y)\,\mathrm{ln}{v(y+t)}+\cdots$
(42) $\displaystyle=$
$\displaystyle{\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi_{i}^{2}(y)\,2\alpha(y+t)+\cdots$
$\displaystyle=$ $\displaystyle 2\alpha t+\cdots\,,$
which grows linearly with time. These results have already been derived in
[7]. Our new contribution here is we realize that at long times333Asymptotic
growth of $C_{K}$ implies that the time scale at which the logarithmic
relation emerges may be defined as $C_{K}\sim O(1)$. This coincides with the
scrambling time $\sim\log S$ for chaotic systems, where $S$ stands for the
number of degrees of freedoms. they imply a logarithmic relation
$S_{K}(t)=\mathrm{ln}{C_{K}(t)}+\cdots\,,$ (43)
which is very similar to the celebrated Boltzmann relation in statistical
mechanics. Indeed, operator randomization is very efficient for fast
scramblers [7] so that a statistical description should emerge in the
scrambling regime (and beyond). This inspires us that the above relation may
signal irreversibility of operator growth for general cases, irrespective of
choice of dynamics. This motivates us to study the relation between $S_{K}$
and $C_{K}$ for many other systems.
However, it should be emphasized that in the relation (43) the proportional
coefficient is undetermined since continuum limit just captures the leading
time dependence of $C_{K}$ and $S_{K}$ qualitatively. In general, we may take
the relation in the form of Eq.(1). However, it turns out that the relation
(43) is correct for chaotic systems. For example, consider the SYK model with
$\eta=1$, the K-complexity and K-entropy can be evaluated exactly as
$\displaystyle C_{K}=\sinh^{2}{(\alpha t)}\,,$ $\displaystyle
S_{K}=\cosh^{2}{(\alpha t)}\,\mathrm{ln}{\cosh^{2}{(\alpha
t)}}-\sinh^{2}{(\alpha t)}\,\mathrm{ln}{\sinh^{2}{(\alpha t)}}\,.$ (44)
In the long time limit, $C_{K}\rightarrow e^{2\alpha t}/4$ and
$S_{K}\rightarrow 2\alpha t$, which leads to the relation (43), with the
proportional coefficient exactly equals to unity. For generic $\eta$, we find
that this is always true. From a statistical point of view, the Boltzmann-like
relation (43) emerges from a uniform distribution. This inspires us that the
relation may hold for general chaotic systems, in which operator randomization
is most efficient. We check this idea for a variety of chaotic models and find
that it is indeed true. It also implies that for chaotic systems, the operator
wave functions at long times are effectively described by a uniform
distribution at leading order.
As an example, consider a class of model spectral density [14]
$\Phi(\omega)={\frac{\pi}{\omega_{0}\Gamma(\nu+1)}}\Big{|}{\frac{\omega}{\omega_{0}}}\Big{|}^{\nu}\mathrm{exp}\Big{(}-\big{|}{\frac{\omega}{\omega_{0}}}\big{|}\Big{)}\,,$
(45)
where $\omega_{0}=2\alpha/\pi$. Notice that exponential decay of
$\Phi(\omega)$ at large frequency is equivalent to asymptotically linear
growth of Lanczos coefficient [14]. Hence, the model spectral density
describes certain chaotic systems. It turns out that in this case the
frequency moments can be written in closed form as
$\mu_{2n}=\omega_{0}^{2n}\,\Gamma(1+\nu+2n)/\Gamma(1+\nu)\,.$ (46)
The Lanczos coefficients can be computed using recurrence relations [14]. The
auto-correlation function turns out to be
$C(t)=\varphi_{0}(t)={\frac{1}{2\big{(}1-i\omega_{0}t\big{)}^{1+\nu}}}+{\frac{1}{2\big{(}1+i\omega_{0}t\big{)}^{1+\nu}}}\,.$
(47)
Figure 1: The functional relation $S_{K}\sim\mathrm{ln}C_{K}$ is shown for the
model spectral density (45), where $\nu=0\,,1\,,2$ from left to right. At long
times, the linear relation Eq.(1) holds, with (shown in dashed lines)
$S_{K}=0.976348\,\mathrm{ln}{C_{K}}+1.06661$ ($\nu=0$),
$S_{K}=0.978129\,\mathrm{ln}{C_{K}}+0.934119$ ($\nu=1$) and
$S_{K}=1.01102\,\mathrm{ln}{C_{K}}+0.63022$ ($\nu=2$). Within our numerical
accuracy $\tilde{\eta}\simeq 1$ for all these cases. To have a nice
presentation, we have moved the results along the horizontal axis properly.
The remaining wave functions $\varphi_{n}(t)$’s can be deduced using the
discrete equation (13). To simplify matter, we set $\omega_{0}=2/\pi$ so that
$\alpha=1$. The functional relation $S_{K}(C_{K})$ for several $\nu$’s value
was shown numerically in Fig. 1. It is easily seen that at long times the
logarithmic relation (1) indeed holds, with the proportional coefficient
$\tilde{\eta}$ close to unity444Because of exponential growth of $C_{K}$,
computational cost increases exponentially for chaotic models. This limits our
numerical accuracy.
$\displaystyle\nu=0\,,\qquad\tilde{\eta}=0.976348\,,$
$\displaystyle\nu=1\,,\qquad\tilde{\eta}=0.978129\,,$
$\displaystyle\nu=2\,,\qquad\tilde{\eta}=1.011020\,.$ (48)
This supports our intuitive idea that the Boltzmann relation (43) holds for
general chaotic systems.
To end this subsection, let us comment on the case beyond the scrambling
regime. It was argued [7] that in this case, the Lanczos coefficient will
approach to a constant $b_{n}\rightarrow b=\alpha S$, for systems with $S$
extensive degrees of freedoms. Using continuum limit, it was shown that
K-complexity grows linearly
$C_{K}(t)_{\mathrm{post-scrambling}}\sim 2b(t-t_{*})+\cdots\,,$ (49)
until arriving at its maximum value of order $\sim e^{O(S)}$, where $t_{*}$
stands for the scrambling time. On the other hand, a careful examination of
the long time tails of wave functions leads to
$S_{K}(t)\sim\mathrm{ln}{\big{(}2bt\big{)}}+\cdots\,.$ (50)
The operator entropy continues to grow until arriving at the maximum value
$S_{K}\sim O(S)$. This again leads to the logarithmic relation (1), though the
constant $\tilde{\eta}$ is undetermined. However, we argue that in this case
$\tilde{\eta}$ should be still equal to unity since in the post-scrambling
regime, the operator wave functions should still obey a uniform distribution
to leading order.
The above result supports our intuitive idea that the logarithmic relation (1)
simply signals irreversibility, irrespective of dynamical details of the
process. We would like to extend the analysis to general systems with
emergence of dissipative behaviors, where the proportional constant
$\tilde{\eta}$ is undetermined. Here it is worth emphasizing that the
K-entropy approaches to its maximum value for a uniform distribution because
of maximal entropy principle. Hence, we propose that the constant
$\tilde{\eta}$ is bounded as
$0<\tilde{\eta}\leq 1\,,$ (51)
where the up bound is saturated for fast scramblers, including $\mathrm{1d}$
chaotic systems. We find that this is true as far as we can check.
#### 4.1.1 With logarithmic correction
It was argued in [6] that for $\mathrm{1d}$ chaotic systems, the asymptotic
behavior of Lanczos coefficient acquires a logarithmic correction:
$b_{n}=\alpha n/\mathrm{ln}{n}+\cdots$. However, for our purpose, we may take
it as a nearly chaotic model in diverse dimensions: the Lanczos coefficient
grows faster than integrable theories (which have a power law $b_{n}\sim
n^{\delta}\,,0<\delta<1$) but still slower than fast scramblers. A more
general case may be taken as $b_{n}=\alpha n/(\mathrm{ln}{n})^{\sigma}$, where
$\sigma>0$, corresponding to a velocity $v(x)=2\alpha
x/(\mathrm{ln}{{\textstyle{\frac{\scriptstyle
x}{\scriptstyle\epsilon}}}})^{\sigma}$. In this case, the two frames are
connected by
Figure 2: The functional relation $S_{K}(C_{K})$ for nearly chaotic models.
$b_{n}={\textstyle{\frac{\scriptstyle\alpha
n}{\scriptstyle\mathrm{ln}{(n+1)}}}}$ for the left panel and
$b_{n}={\textstyle{\frac{\scriptstyle\alpha
n}{\scriptstyle\mathrm{ln}^{2}{(n+1)}}}}$ for the right panel. We set
$\alpha=1$ in numerical calculations. At long times, the functional relation
is very well fitted by
$S_{K}=0.994787\,\mathrm{ln}{C_{K}}-0.958367\,\mathrm{ln}{\mathrm{ln}{C_{K}}}+1.04432$
(left) and
$S_{K}=0.634549\,\mathrm{ln}{C_{K}}+0.593231\,\mathrm{ln}{\mathrm{ln}{C_{K}}}+0.59138$
(right), matching the continuum limit.
$x=\epsilon\,\mathrm{exp}\Big{[}\big{(}2\tilde{\alpha}y\big{)}^{{\frac{1}{1+\sigma}}}\Big{]}\,,$
(52)
where $\tilde{\alpha}=\alpha(1+\sigma)$. It is straightforward to deduce
$C_{K}(t)={\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi_{i}^{2}(y)\,\mathrm{exp}\Big{[}\big{(}2\tilde{\alpha}(y+t)\big{)}^{{\frac{1}{1+\sigma}}}\Big{]}\sim\mathrm{exp}\Big{[}\big{(}2\tilde{\alpha}t\big{)}^{{\frac{1}{1+\sigma}}}\Big{]}\,,$
(53)
and
$\displaystyle S_{K}(t)$ $\displaystyle=$
$\displaystyle{\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi_{i}^{2}(y)\,\Big{[}\big{(}2\tilde{\alpha}(y+t)\big{)}^{{\frac{1}{1+\sigma}}}-\mathrm{ln}{\big{(}2\tilde{\alpha}(y+t)\big{)}^{{\frac{1}{1+\sigma}}}}\Big{]}+\cdots$
(54) $\displaystyle\sim$
$\displaystyle\big{(}2\tilde{\alpha}t\big{)}^{{\frac{1}{1+\sigma}}}-\mathrm{ln}{\big{(}2\tilde{\alpha}t\big{)}^{{\frac{1}{1+\sigma}}}}+\cdots\,,$
where a tilde means taking the long time limit and we have ignored the
constant coefficient in each term. Combining these results again leads to the
logarithmic relation (1) except that the subleading order correction is of
order $O(\mathrm{ln}\mathrm{ln}{C_{K}})$. However, from the method itself, we
do not expect it can extract the subleading order term of $S_{K}$ (or $C_{K}$)
correctly since the leading term is only determined qualitatively.
Nevertheless, the analysis is consistent with our numerical calculations for
several examples, see Fig. 2. However, numerically it is also hard to ensure
the form of the subleading order corrections. For numerical data established
in the figure, including $\mathrm{ln}\mathrm{ln}{C_{K}}$ term just fits the
numerical data slightly better than that without it but it should not be
considered conclusive. However, an exception occurs for $\mathrm{1d}$ chaotic
systems. We find that in this case $\tilde{\eta}$ is equal to unity if and
only if the $\mathrm{ln}\mathrm{ln}{C_{K}}$ term is included!
### 4.2 Integrable theories
Theories with asymptotic growth of Lanczos cofficients $b_{n}\rightarrow\alpha
n^{\delta}$, where $0<\delta<1$ were referred to as integrable ones [6]. In
this case, $v(x)=2\alpha\epsilon^{1-\delta}x^{\delta}$ and the two frames are
connected by
$y={\frac{1}{2\tilde{\alpha}}}\Big{(}{\frac{x}{\epsilon}}\Big{)}^{1-\delta}\qquad\mathrm{or}\qquad
x=\epsilon\,\big{(}2\tilde{\alpha}y\big{)}^{{\frac{1}{1-\delta}}}\,,$ (55)
where $\tilde{\alpha}=\alpha(1-\delta)$. Evaluation of K-complexity yields
$C_{K}(t)={\frac{1}{\epsilon}}\int\mathrm{d}y\,\big{[}2\tilde{\alpha}(y+t)\big{]}^{{\frac{1}{1-\delta}}}\,\psi_{i}^{2}(y)\sim\big{(}2\tilde{\alpha}t\big{)}^{{\frac{1}{1-\delta}}}+\cdots\,.$
(56)
It grows in a power law at sufficiently long time. On the other hand, the
operator entropy is given by
$\displaystyle S_{K}(t)$ $\displaystyle=$
$\displaystyle{\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi_{i}^{2}(y)\,\mathrm{ln}{\big{[}2\tilde{\alpha}(y+t)\big{]}^{\frac{\delta}{1-\delta}}}+\cdots$
(57) $\displaystyle\sim$
$\displaystyle{\frac{\delta}{1-\delta}}\,\mathrm{ln}{\big{(}2\tilde{\alpha}t\big{)}}+\cdots$
$\displaystyle\sim$ $\displaystyle\delta\,\mathrm{ln}{C_{K}(t)}+\cdots\,.$
Again this leads to the logarithmic relation (1). However, unlike previous
cases, the proportional constant $\tilde{\eta}=\delta$ is exact for
$\delta\geq 1/2$ (from the method itself, this should be simply a
coincidence). We check it using a lot of numerical examples and find that it
is always true. For example, consider integrable models with $b_{n}=\alpha
n^{\delta}$. Numerically evaluation of the operator entropy (29) yields at
long times
$\displaystyle\delta=1/2\,,\qquad
S_{K}(t)=0.500917\,\mathrm{ln}{C_{K}(t)}+1.41373\,,$
$\displaystyle\delta=2/3\,,\qquad
S_{K}(t)=0.669477\,\mathrm{ln}{C_{K}(t)}+1.43501\,,$
$\displaystyle\delta=3/4\,,\qquad
S_{K}(t)=0.752683\,\mathrm{ln}{C_{K}(t)}+1.44995\,.$ (58)
In all these cases, the proportional constant $\tilde{\eta}$ is equal to
$\delta$, within our numerical accuracy. It is a pity that we do not have a
physical interpretation for this.
#### 4.2.1 With logarithmic correction
To test whether the above result can be extended to slightly different cases,
let us consider logarithmic corrections to integrable models. For instance,
the Lanczos coefficient behaves asymptotically as $b_{n}=\alpha
n^{\delta}/\mathrm{ln}{n}$, giving rise to
$v(x)=2\alpha\epsilon\big{(}{\textstyle{\frac{\scriptstyle
x}{\scriptstyle\epsilon}}}\big{)}^{\delta}/\mathrm{ln}{\big{(}{\textstyle{\frac{\scriptstyle
x}{\scriptstyle\epsilon}}}\big{)}}$. The two frames are connected by
$y={\frac{1}{2\bar{\alpha}}}\big{(}X\,\mathrm{ln}{X}-X\big{)}\,,\quad
X=\big{(}{\textstyle{\frac{\scriptstyle
x}{\scriptstyle\epsilon}}}\big{)}^{1-\delta}\,,$ (59)
where $\bar{\alpha}=\alpha(1-\delta)^{2}$. Notice at very large $y$,
$X\sim(2\bar{\alpha}y)/\mathrm{ln}{(2\bar{\alpha}y)}$ to leading order. Hence,
at long times the K-complexity and K-entropy behave as
$\displaystyle C_{K}(t)\sim
x(t)/\epsilon\sim(2\bar{\alpha}t)^{\gamma}/\mathrm{ln}^{\gamma}{(2\bar{\alpha}t)}\,,$
$\displaystyle
S_{K}(t)\sim\mathrm{ln}{v(t)}\sim\delta\,\mathrm{ln}\big{(}{\textstyle{\frac{\scriptstyle
x(t)}{\scriptstyle\epsilon}}}\big{)}\sim\delta\,\mathrm{ln}{C_{K}(t)}\,,$ (60)
where $\gamma=1/(1-\delta)$. It turns out that the K-complexity grows a bit
slower than the integrable models. However, while the logarithmic relation (1)
still holds, the proportional coefficient $\tilde{\eta}=\delta$ is falsified
by our numerical results.
One may also consider positively corrections to Lanzos coefficient as
$b_{n}=\alpha n^{\delta}\,\mathrm{ln}{n}$, corresponding to
$v(x)=2\alpha\epsilon\big{(}{\textstyle{\frac{\scriptstyle
x}{\scriptstyle\epsilon}}}\big{)}^{\delta}\,\mathrm{ln}{\big{(}{\textstyle{\frac{\scriptstyle
x}{\scriptstyle\epsilon}}}\big{)}}$. The two frames are connected as
$y={\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2\alpha}}}\mathrm{li}(X)\,,$
(61)
where $\mathrm{li}(x)$ stands for the logarithmic integral function. At large
$y$, one has $X\sim 2\alpha y\,\mathrm{ln}{(2\alpha y)}$, implying that the
K-complexity grows a bit faster than integrable models
$C_{K}\sim(2\alpha t)^{\gamma}\,\mathrm{ln}^{\gamma}{(2\alpha t)}\,.$ (62)
Again the K-entropy turns out to be
$S_{K}(t)\sim\mathrm{ln}{v(t)}\sim\delta\,\mathrm{ln}{C_{K}(t)}$, but the
constant coefficient $\tilde{\eta}=\delta$ is falsified by our numerical
results. It seems that the result $\tilde{\eta}=\delta$ is only valid to
integrable theories with $\delta\geq 1/2$.
### 4.3 With bounded support and beyond
Consider $\delta\rightarrow 0$ limit of integrable theories. The Lanczos
coefficient approaches to a constant $b_{n}\rightarrow b$ asymptotically,
corresponding to a constant velocity $v(x)=v=2\epsilon b$. This case is very
similar to the post-scrambling regimes of chaotic systems [7], except the
initial amplitude. Careful analysis using the continuum limit (with higher
order corrections) implies that the K-entropy grows logarithmically at long
times
$S_{K}(t)\sim\mathrm{ln}{(2bt)}\,,$ (63)
whereas the K-complexity increases linearly $C_{K}(t)\sim 2bt$. This again
leads to the logarithmic relation (1).
To test the result, consider two simple models555These two models arise from a
chain of classical harmonic oscillator of $2N$ atoms with periodic boundary
conditions [17]. . The first is $b_{n}=\omega_{0}/2$. The wave function is
solved in terms of Bessel functions
$\varphi_{n}(t)=J_{n}(\omega_{0}t)+J_{n+2}(\omega_{0}t)\,.$ (64)
Numerically evaluation of K-complexity and K-entropy yields at long times
$S_{K}(t)=0.729302\,\mathrm{ln}{C_{K}(t)}+0.353124\,.$ (65)
The second example is $b_{1}=\omega_{0}/\sqrt{2}\,,b_{n}=\omega_{0}/2$ for
$n>1$. The wave function is given by
$\varphi_{n}(t)=c_{n}J_{n}(\omega_{0}t)\,,$ (66)
where $c_{-1}=0\,,c_{0}=1$ and $c_{n}=\sqrt{2}$ for $n>1$. We find at long
times
$S_{K}(t)=0.870024\,\mathrm{ln}{C_{K}(t)}+0.603974\,.$ (67)
These results support the continuum limit analysis.
To proceed, consider a critical case where the Lanczos coefficient grows
asymptotically faster than the bounded case but still slower than a power law.
For example: $b_{n}=\alpha\,\mathrm{ln}{n}$. It corresponds to a velocity
$v(x)=2\epsilon\alpha\,\mathrm{ln}{({\textstyle{\frac{\scriptstyle
x}{\scriptstyle\epsilon}}})}$ and the two frames are transformed as
$y(x)={\textstyle{\frac{\scriptstyle 1}{\scriptstyle
2\alpha}}}\mathrm{li}\big{(}{\textstyle{\frac{\scriptstyle
x}{\scriptstyle\epsilon}}}\big{)}\,.$ (68)
Since at large $y$, $x/\epsilon\sim 2\alpha y\,\mathrm{ln}{(2\alpha y)}$, one
has
$C_{K}(t)\sim x(t)/\epsilon\sim 2\alpha t\,\mathrm{ln}{(2\alpha t)}\,,$ (69)
which is a product logarithmic law. The K-complexity grows faster than the
bounded case (which is linear) but still slower than the integrable theories.
This is in accordance with our expectations.
Figure 3: Left panel: $b_{n}=\alpha\,\mathrm{ln}{(n+1)}$. Right panel:
$b_{n}=\alpha\,\mathrm{ln}{n}+\gamma$. We set $\alpha=\gamma=1$. In both
cases, the logarithmic relation (1) already appears at the time scale where
$C_{K}(t_{c})\sim O(1)$ (see the orange lines) but the coefficient
$\tilde{\eta}$ is changed at later times (see the red lines). Around $t\sim
t_{c}$, we have $S_{K}=0.613581\,\mathrm{ln}{C_{K}}+1.35088$ (left) and
$S_{K}=0.619693\,\mathrm{ln}{C_{K}}+1.36223$ (right) whereas at long times
$S_{K}=0.166437\,\mathrm{ln}{C_{K}}+2.59585$ (left) and
$S_{K}=0.170329\,\mathrm{ln}{C_{K}}+2.40017$ (right). Within our numerical
accuracy, both cases have the same coefficient $\tilde{\eta}$ at the same time
regimes.
However, at this order, the K-entropy turns out to be
$S_{K}\sim\mathrm{ln}v(t)\sim\mathrm{ln}\mathrm{ln}{C_{K}}$, violating the
logarithmic relation (1) apparently. To resolve the issue, one may include
higher order corrections in the wave equation as the bounded case.
Unfortunately, we do not find a definite answer using this approach.
Nevertheless, our numerical results still suggest that the logarithmic
relation (1) holds in this case, see Fig. 3.
To end this section, let us discuss the relation $S_{K}(C_{K})$ in the full
time evolution. We have seen that for systems described by semi-infinite
chains $S_{K}\sim-C_{K}\,\mathrm{ln}{C_{K}}$ at initial times and
$S_{K}\sim\mathrm{ln}{C_{K}}$ at long times where dissipative behavior
emerges. These two regimes are universal, irrespective of the choice of
dynamics. The former is determined by the lowest moment $\mu_{2}$ whereas the
latter probably signals irreversibility of the process. There is a smooth
crossover between them, which however depends on the dynamical details. Since
the functional function $S_{K}(C_{K})$ is well defined in the full time
evolution, it makes sense to study it in operator dynamics and connect it to
emergence of ergodic behavior of the theories.
## 5 Finite chains
In this section, we would like to study K-complexity and K-entropy for finite
chains and compare the results with previous cases. In this case, the
recursion method naturally comes to a stop at some order $K$ so that
$b_{K+1}=0$. As a consequence, the continued-fraction representation of the
relaxation function terminates at the $K$-th level:
$\phi_{0}(z)=\dfrac{1}{z+\dfrac{b_{1}^{2}}{z+\dfrac{b_{2}^{2}}{\begin{array}[]{cc}z+\cdots&\\\
&\cdots+\dfrac{b_{K}^{2}}{z}\end{array}}}}\,.$ (70)
This is a rational function $p_{K}(z)/q_{K+1}(z)$. Here there are two cases,
depending on $K$ is odd or even, corresponding to $W=0$ or $W=\infty$. For odd
$K$,
$\displaystyle p_{K}(z)=z^{K}+z^{K-2}+\cdots+z\,,$ $\displaystyle
q_{K+1}(z)=z^{K+1}+z^{K-1}+\cdots+c\,,$ (71)
where we have omitted the constant coefficient for each term in the
polynomials and $c\neq 0$. This leads to $W=p_{K}(0)/q_{K+1}(0)=0$. On the
other hand, for even $K$, the results (5) still hold but with the last term
interchanged between $p_{K}(z)$ and $q_{K+1}(z)$, giving rise to $W=\infty$.
Hence, for both cases, the dynamics of operator is nonergodic. We will show
that as a consequence, the functional relation $S_{K}\sim\log{C_{K}}$ at long
times does not hold any longer.
To proceed, consider odd $K$ at first. The relaxation function contains $K+1$
poles, which are all located on the imaginary axis. The spectral function,
inferred via
$\Phi(\omega)=\lim_{\varepsilon\rightarrow
0}2\mathrm{Re}\big{[}\phi_{0}(\varepsilon-i\omega)\big{]}\,,$ (72)
consists of $L=(K+1)/2$ pairs of $\delta$-functions
$\Phi(\omega)=\pi\sum_{\ell=1}^{L}a_{\ell}\big{[}\delta(\omega-\omega_{\ell})+\delta(\omega+\omega_{\ell})\big{]}\,,$
(73)
where all the frequencies $\omega_{\ell}$ are generally nonzero. The auto-
correlation function turns out to be
$\varphi_{0}(t)=\sum_{\ell=1}^{L}a_{\ell}\cos{(\omega_{\ell}t)}\,.$ (74)
However, if one of the $L$ frequencies $\omega_{\ell}$ happens to be zero, the
total number of Lanczos coefficients will be reduced by one so that $K=2L-2$.
This is exactly the even $K$ case. Nevertheless, the result (74) is valid to
both cases.
Given the above auto-correlation function, it turns out that all the remaining
(nonzero) wave functions $\varphi_{n}$’s will be a sum of sine or cosine
functions. It is immediately seen that in this case both K-complexity and
K-entropy will no longer grow monotonically in the time evolution. Of course,
the functional relation $S_{K}\sim\mathrm{ln}{C_{K}}$ will not hold at long
times any longer.
Figure 4: Trajectories on $C_{K}$-$S_{K}$ plane for finite chains. Left panel:
$K=1$. Middle panel: $K=3$ and $\omega_{2}=2\omega_{1}$. In both cases, the
particle moves periodically between $O$ and $F$. In the first half period of
$C_{K}$, $t\in[0\,,T_{C}/2]$, it moves to the right, from $O$ to $F$ whereas
in the next half period $t\in[T_{c}/2\,,T_{C}]$, it moves in the opposite
direction, similar to a harmonic oscillator. Right panel: $K=3$ and
$\omega_{2}=\sqrt{3}\omega_{1}$. The motion of the particle is not periodic
and the trajectory becomes more and more complex as time increases. Here we
have chosen $t\in[0\,,20]$.
Let us consider several examples. The first is $K=1$: $b_{1}=\omega$ and
$b_{n}=0$ otherwise. The auto-correlation function is simply a cosine function
$\varphi_{0}(t)=\cos{(\omega t)}$ and $\varphi_{1}(t)=\sin{(\omega t)}$. One
easily finds
$\displaystyle C_{K}(t)=\sin^{2}(\omega t)\,,$ $\displaystyle
S_{K}(t)=-\cos^{2}(\omega t)\,\mathrm{ln}{\cos^{2}(\omega t)}-\sin^{2}(\omega
t)\,\mathrm{ln}{\sin^{2}(\omega t)}\,.$ (75)
It is clear that the both are periodic in times. The minimal period is
$T=\pi/\omega$ for $C_{K}$ and $T=\pi/2\omega$ for $S_{K}$. The functional
relation $S_{K}(C_{K})$ is shown in the left panel of Fig. 4. We may think of
the time evolution as a particle moving on the $C_{K}$-$S_{K}$ plane. In the
first half period of $C_{K}$, $t\in[0\,,T_{C}/2]$, the particle moves to the
right, starting at $O$ and stopping at $F$, giving rise to a finite path. In
the next half period $t\in[T_{c}/2\,,T_{C}]$, the particle still moves along
the same trajectory but in the opposite direction exactly: it goes from $F$ to
$O$. This is very similar to a harmonic oscillator. The same behavior will be
repeated as time increases.
The second example is $K=3$:
$b_{1}^{2}={\frac{\omega_{1}^{2}+\omega_{2}^{2}}{2}}\,,\quad
b_{2}^{2}={\frac{(\omega_{1}^{2}-\omega_{2}^{2})^{2}}{2(\omega_{1}^{2}+\omega_{2}^{2})}}\,,\quad
b_{3}^{2}={\frac{2\omega_{1}^{2}\omega_{2}^{2}}{\omega_{1}^{2}+\omega_{2}^{2}}}\,.$
(76)
The auto-correlation function is given by
$\varphi_{0}(t)=\big{(}\cos(\omega_{1}t)+\cos(\omega_{2}t)\big{)}/2$. If the
ratio $\omega_{2}/\omega_{1}$ is rational, $\varphi_{0}(t)$ will be periodic
as well as the K-complexity and K-entropy. The situation is quite similar to
the previous case except some slight differences, as shown in the middle panel
of Fig. 4. However, when $\omega_{2}/\omega_{1}$ is irrational, the auto-
correlation function will not be periodic. The particle does not move on a
fixed path and the trajectory on the plane will become more and more complex
in the time evolution, see the right panel of Fig. 4. These are general
features for odd $K$.
For even $K$, as previously emphasized, the results can be obtained from
$(K+1)$-case by setting one of the $L$ frequencies $\omega_{\ell}$ equal to
zero. For example, for $K=2$ case, the auto-correlation function is given by
$\varphi_{0}(t)=a_{0}+a_{1}\cos{(\omega t)}$, where $a_{0}+a_{1}=1$. Of
course, the time evolution is periodic except that $W=\infty$ because of the
constant term $a_{0}\neq 0$. For $K=4$ case,
$\varphi_{0}(t)=a_{0}+a_{1}\cos{(\omega_{1}t)}+a_{2}\cos{(\omega_{2}t)}$,
where $a_{0}+a_{1}+a_{2}=1$. Whether the time evolution is periodic or not
depends on the ratio $\omega_{2}/\omega_{1}$ is rational or irrational. In any
case, the trajectory on the $C_{K}$-$S_{K}$ plane shows the same features as
the odd $K$ case qualitatively. Therefore, we safely conclude that for
reversible process characterized by $W=0$ or $W=\infty$, the functional
relation $S_{K}\sim\log{C_{K}}$ at long times does not hold any longer.
## 6 Conclusion and discussion
In this paper, we study quantum information quantities: K-complexity
$C_{K}(t)$ and K-entropy $S_{K}(t)$ in operator growth. We are trying to
search their interesting features which may diagnose irreversibility of the
process. We have studied a variety of systems with emergence of dissipative
behaviors, including chaotic ones and integrable theories. Our main result is
for irreversible process, the two quantities enjoy a logarithmic relation (1)
to leading order at sufficiently long times, where the proportional constant
is up bounded as $\tilde{\eta}\leq 1$. It should be emphasized that in
practical calculations we choose a certain inner product in the operator space
and take the thermodynamic limit. However, the logarithmic relation only
depends on the asymptotic behavior of Lanczos coefficients $b_{n}$. Hence
choosing different inner product or considering finite temperatures will not
change the relation if the asymptotic behavior of $b_{n}$ takes the same
asymptotic form as those studied in the paper. Inspired by similarity of the
relation to the Boltzmann formula in statistical mechanics, we propose that
the relation is a sufficient condition for irreversibility of operator growth,
irrespective of the choice of dynamics. This is true as far as we can check
although we cannot prove it analytically. Physical consequences of the
relation deserves further investigations.
model | chaotic | $1d$-chaotic | integrable | unknown | unknown | bounded
---|---|---|---|---|---|---
$b_{n}$ | $\alpha n$ | $\alpha n/\mathrm{ln}{n}$ | $\alpha n^{\delta}$ | $\alpha n^{\delta}(\mathrm{ln}{n})^{\pm}$ | $\alpha\,\mathrm{ln}{n}$ | $b$
$C_{K}$ | $e^{2\alpha t}$ | $e^{\sqrt{4\alpha t}}$ | $(2\alpha t)^{\gamma}$ | $(2\alpha t)^{\gamma}\,\mathrm{ln}^{\pm\gamma}{(2\alpha t)}$ | $2\alpha t\,\mathrm{ln}{(2\alpha t)}$ | $2bt$
$S_{K}$ | $2\alpha t$ | $\sqrt{4\alpha t}$ | $\mathrm{ln}{(2\alpha t)}$ | $\mathrm{ln}{(2\alpha t)}$ | $\mathrm{ln}{(2\alpha t)}$ | $\mathrm{ln}{(2bt)}$
Table 1: Asymptotic growth of Lanczos coefficient $b_{n}$ and the leading long
time dependence of K-complexity and K-entropy. The constants $\alpha$ and $b$
have dimension of energy and $\gamma={\frac{1}{1-\delta}}$, $0<\delta<1$. The
last column corresponds to the case with bounded support, of which a
particularly interesting example is chaotic systems in the post-scrambling
regimes. The fifth/sixth columns can be viewed as integrable models/bounded
case with logarithmic corrections.
In addition to the logarithmic relation, behavior of K-complexity itself is
also closely related to irreversibility. In table 1, we summarize asymptotic
growth of Lanczos coefficient $b_{n}$ and the leading long time dependence of
$C_{K}$ (and $S_{K}$). It is intriguing to observe that the former is somehow
positively related to the latter: when $b_{n}$ grows faster asymptotically,
$C_{K}$ grows faster in the time evolution (in a one-to-one mapping) as well.
If this is correct, from the long time behavior of $C_{K}$, one can read off
the asymptotic behavior of the Lanczos coefficients. The underlying relation
between $C_{K}$ and emergence of dissipative behavior certainly deserves
further investigations.
## Acknowledgments
Z.Y. Fan was supported in part by the National Natural Science Foundations of
China with Grant No. 11805041 and No. 11873025.
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|
\left.-\hat{u}(\tilde{F}-F(0)+w_{A})-\hat{\tilde{v}}(F-F(0)+w_{B})\right)\right]\end{aligned}\end{aligned}$
$\displaystyle\quad-\frac{\mu}{R_{u}}Q_{f^{\prime}}\bar{f}(0)(G_{-}(0)-w_{D_{-}})-\frac{\mu}{R_{v}}\bar{Q}_{\bar{f}^{\prime}}f(0)(G_{+}(0)-w_{D_{+}})\
.$ (A.23)
The first line vanishes identically, using (2.25). The terms proportional to
$\Theta$ functions add up to
$\displaystyle\begin{aligned} \frac{\mu}{2}\int d\sigma
d{\tilde{\sigma}}&\left[f^{\prime}\frac{\mathcal{H}+\mathcal{P}}{R_{u}}\frac{\tilde{\bar{f}}^{\prime}}{R_{v}}\left(-\tilde{v}^{\prime}\tilde{G}_{-}+\mu^{2}(\tilde{\mathcal{H}}-\tilde{\mathcal{P}})\tilde{F}\right)\Theta(\sigma-{\tilde{\sigma}})\right.\\\
&~{}\left.+\frac{f^{\prime}}{R_{u}}\tilde{\bar{f}}^{\prime}\frac{\tilde{\mathcal{H}}-\tilde{\mathcal{P}}}{R_{v}}\left(-u^{\prime}G_{+}+\mu^{2}(\mathcal{H}+\mathcal{P})F\right)\Theta({\tilde{\sigma}}-\sigma)\right]\end{aligned}$
$\displaystyle=-\frac{\mu}{2}\int d\sigma
d{\tilde{\sigma}}f^{\prime}\tilde{\bar{f}}^{\prime}\frac{\mathcal{H}+\mathcal{P}}{R_{u}}\frac{\tilde{\mathcal{H}}+\tilde{\mathcal{P}}}{R_{v}}[\Theta(\sigma-{\tilde{\sigma}})+\Theta({\tilde{\sigma}}-\sigma)]=-\frac{2\mu}{R_{u}R_{v}}Q_{f^{\prime}}\bar{Q}_{\bar{f}^{\prime}}\
.$ (A.24)
Similarly, the terms proportional to $\hat{u}$ and $\hat{\tilde{v}}$ give
$\displaystyle\frac{\mu}{R_{u}^{2}}Q_{uf^{\prime}}\int
d{\tilde{\sigma}}\frac{\tilde{\bar{f}}^{\prime}}{R_{v}}\left[\tilde{v}^{\prime}\tilde{G}_{-}-\mu^{2}(\tilde{\mathcal{H}}-\tilde{\mathcal{P}})(\tilde{F}-F(0)+w_{A})\right]$
$\displaystyle+\frac{\mu}{R_{v}^{2}}\bar{Q}_{v\bar{f}^{\prime}}\int
d\sigma\frac{f^{\prime}}{R_{u}}\left[u^{\prime}G_{+}-\mu^{2}(\mathcal{H}+\mathcal{P})(F-F(0)+w_{B})\right]$
$\displaystyle=\frac{2\mu}{R_{u}R_{v}}\left[\frac{1}{R_{u}}Q_{uf^{\prime}}\bar{Q}_{\bar{f}^{\prime}}(1+\mu^{2}[F(0)-w_{A}])+\frac{1}{R_{v}}Q_{f^{\prime}}\bar{Q}_{v\bar{f}^{\prime}}(1+\mu^{2}[F(0)-w_{B}])\right]\
.$ (A.25)
Using these equations to work out the Poisson bracket of the charges, we
finally find
$\displaystyle\\{Q_{f},\bar{Q}_{\bar{f}}\\}$ (A.26)
$\displaystyle=\frac{\mu}{R_{u}}Q_{f^{\prime}}\left(\bar{f}(0)(w_{D_{-}}-G_{-}(0))+\int
d{\tilde{\sigma}}\frac{\tilde{\bar{f}}^{\prime}}{R_{v}}\left[(\tilde{\mathcal{H}}-\tilde{\mathcal{P}})\left(-\frac{1}{2}+\hat{\tilde{v}}(1+\mu^{2}[F(0)-w_{B}])+\mu^{2}\tilde{B}\right)-\tilde{v}^{\prime}\tilde{D}_{-}\right]\right)$
$\displaystyle\,+\frac{\mu}{R_{v}}\bar{Q}_{\bar{f}^{\prime}}\left(f(0)(w_{D_{+}}-G_{+}(0))+\int
d\sigma\frac{f^{\prime}}{R_{u}}\left[(\mathcal{H}+\mathcal{P})\left(-\frac{1}{2}+\hat{u}(1+\mu^{2}[F(0)-w_{A}])+\mu^{2}A\right)-u^{\prime}D_{+}\right]\right)$
The terms proportional to $u,v$ in the integrand, which represent the
contribution of the state-dependent radii to the Poisson brackets, are
dangerous, as they would produce terms proportional to $Q_{uf^{\prime}}$ on
the right-hand side, which are not conserved due to the non-periodicity of the
associated function. A very similar derivation for the left-left and right-
right components of the Poisson bracket algebra gives
$\displaystyle\\{Q_{f},Q_{g}\\}$
$\displaystyle=\frac{1}{R_{u}}Q_{fg^{\prime}-f^{\prime}g}$
$\displaystyle\quad-\frac{\mu^{2}}{R_{u}}Q_{f^{\prime}}\left(g(0)(F(0)-w_{B})+\int\frac{d{\tilde{\sigma}}}{R_{u}}\tilde{g}^{\prime}\left[\tilde{u}^{\prime}\tilde{B}-(\tilde{\mathcal{H}}+\tilde{\mathcal{P}})\left(\tilde{D}_{-}+\hat{\tilde{u}}[G_{-}(0)-w_{D_{-}}]\right)\right]\right)$
$\displaystyle\quad+\frac{\mu^{2}}{R_{u}}Q_{g^{\prime}}\left(f(0)(F(0)-w_{B})+\int\frac{d\sigma}{R_{u}}f^{\prime}\left[u^{\prime}B-(\mathcal{H}+\mathcal{P})\left(D_{-}+\hat{u}[G_{-}(0)-w_{D_{-}}]\right)\right]\right)\
.$ (A.27) $\displaystyle\\{\bar{Q}_{\bar{f}},\bar{Q}_{\bar{g}}\\}$
$\displaystyle=\frac{1}{R_{v}}\bar{Q}_{\bar{f}^{\prime}\bar{g}-\bar{f}\bar{g}^{\prime}}$
$\displaystyle\quad-\frac{\mu^{2}}{R_{v}}\bar{Q}_{\bar{f}^{\prime}}\left(\bar{g}(0)(F(0)-w_{A})+\int\frac{d{\tilde{\sigma}}}{R_{v}}\tilde{\bar{g}}^{\prime}\left[\tilde{v}^{\prime}\tilde{A}-(\tilde{\mathcal{H}}-\tilde{\mathcal{P}})\left(\tilde{D}_{+}+\hat{\tilde{v}}[G_{+}(0)-w_{D_{+}}]\right)\right]\right)$
$\displaystyle\quad+\frac{\mu^{2}}{R_{v}}\bar{Q}_{\bar{g}^{\prime}}\left(\bar{f}(0)(F(0)-w_{A})+\int\frac{d\sigma}{R_{v}}\bar{f}^{\prime}\left[v^{\prime}A-(\mathcal{H}-\mathcal{P})\left(D_{+}+\hat{v}[G_{+}(0)-w_{D_{+}}]\right)\right]\right)\
.$ (A.28)
With these results, we can deduce how to choose the functions $A$, $B$ and
$D_{\pm}$ in order to guarantee, for consistency, that the right-hand sides
are conserved. In the following, we will analyse this requirement step by
step.
Concretely, we need to cancel the radius contributions proportional with
$Q_{uf^{\prime}}$ and $\bar{Q}_{v\bar{f}^{\prime}}$, which are not conserved
due to the lack of periodicity of the functions that label them. Let us first
consider the result (A.3), and understand the possible choices for the winding
of $D_{-}$ so as to cancel this term. There are three types of winding one can
consider: proportional to $\hat{u}(\sigma)$, proportional to $\Theta(\sigma)$,
or proportional to $\hat{v}(\sigma)$. In the case of $D_{-}$, the first type
of winding does not help cancel the non-conserved terms proportional to
$Q_{uf^{\prime}}$ in the total Poisson bracket. Moreover, it introduces $u$ \-
type winding in (A.26), which cannot be cancelled by $u$ \- type winding in
$B$ (or the other functions) without introducing additional $u$ \- type
winding in (A.3). Consequently, the $u$ \- type winding in $B$ and $D_{-}$
must be zero. A similar argument sets the $v$ \- type winding in $A$ and
$D_{+}$ to zero.
For whatever the other type of winding we may have, the vanishing of the terms
proportional to $Q_{uf^{\prime}}$ in (A.3) and $\bar{Q}_{v\bar{f}^{\prime}}$
in (A.3) sets $w_{D_{\pm}}=G_{\pm}(0)$. The conservation equation (A.4) then
requires that $w_{A}=w_{B}=F(0)$, so the charge algebra becomes
$\displaystyle\\{Q_{f},\bar{Q}_{\bar{f}}\\}$ $\displaystyle=\frac{\mu
Q_{f^{\prime}}}{R_{u}R_{v}}\int
d\sigma\bar{f}^{\prime}\left[\left(\hat{v}-\frac{1}{2}+\mu^{2}B\right)(\mathcal{H}-\mathcal{P})-v^{\prime}D_{-}\right]$
$\displaystyle+\frac{\mu\bar{Q}_{\bar{f}^{\prime}}}{R_{u}R_{v}}\int d\sigma
f^{\prime}\left[\left(\hat{u}-\frac{1}{2}+\mu^{2}A\right)(\mathcal{H}+\mathcal{P})-u^{\prime}D_{+}\right]$
(A.29) $\displaystyle\\{Q_{f},Q_{g}\\}$
$\displaystyle=\frac{1}{R_{u}}Q_{fg^{\prime}-f^{\prime}g}-\frac{\mu^{2}}{R_{u}^{2}}Q_{f^{\prime}}\int
d\sigma
g^{\prime}\,(u^{\prime}B-(\mathcal{H}+\mathcal{P})D_{-})+\frac{\mu^{2}}{R_{u}^{2}}Q_{g^{\prime}}\int
d\sigma f^{\prime}\,(u^{\prime}B-(\mathcal{H}+\mathcal{P})D_{-})$
and a similar equation for $\\{\bar{Q}_{\bar{f}},\bar{Q}_{\bar{g}}\\}$. The
first term in the first equation can be cancelled if we choose a $v$ \- type
of winding $D_{-}$ and/or $B$; however, if we do not want it to introduce non-
conserved terms in the second equation, we need to choose
$u^{\prime}B^{(v)}-(\mathcal{H}+\mathcal{P})D_{-}^{(v)}=0\;,\;\;\;\;\;\;v^{\prime}D_{-}^{(v)}-\mu^{2}(\mathcal{H}-\mathcal{P})B^{(v)}=\mathcal{H}-\mathcal{P}$
(A.30)
where the superscript $(v)$ indicates the coefficient of $\hat{v}$ in the
corresponding function. These equations determine $D_{-}^{(v)}=G_{-}$ and
$B^{(v)}=F$. Since the total winding of the functions $D_{-}$ and $B$ is
$G_{-}(0)$ and respectively $F(0)$, we see that the $v$-type winding accounts
for all of it, leaving no room for a $\Theta$ \- type of winding. An identical
argument sets
$v^{\prime}A^{(u)}-(\mathcal{H}-\mathcal{P})D_{+}^{(u)}=0\;,\;\;\;\;\;\;\;u^{\prime}D_{+}^{(u)}-\mu^{2}(\mathcal{H}+\mathcal{P})A^{(u)}=\mathcal{H}+\mathcal{P}$
(A.31)
Therefore, we find that the winding part of the functions $A,B,D_{\pm}$ is
entirely fixed by charge conservation and the non-appearance of non-conserved
terms in the charge algebra.
This however is not sufficient to fix the charge algebra, as the periodic
parts of these functions have not yet been fixed. The integrals we are left
with are only conserved if they are of the form $\mathcal{H}+\mathcal{P}$
times a periodic function of $\hat{u}$ or $\mathcal{H}-\mathcal{P}$ times a
function of $\hat{v}$. Combining the first line of section A.3 with the third,
we find the requirement
$\displaystyle(\tilde{\mathcal{H}}-\tilde{\mathcal{P}})\left(-\frac{1}{2}+\mu^{2}\tilde{B}^{(p)}\right)-\tilde{v}^{\prime}\tilde{D}_{-}$
$\displaystyle=-(\tilde{\mathcal{H}}-\tilde{\mathcal{P}})\bar{X}_{p}(\hat{v})\
,$
$\displaystyle\tilde{u}^{\prime}\tilde{B}^{(p)}-(\tilde{\mathcal{H}}+\tilde{\mathcal{P}})\tilde{D}_{-}^{(p)}$
$\displaystyle=-(\tilde{\mathcal{H}}+\tilde{\mathcal{P}})X_{p}(\hat{u})\ ,$
(A.32)
where the superscript $(p)$ indicates the periodic part and where
$X_{p}(\hat{u})$ and $\bar{X}_{p}(\hat{v})$ are periodic functions that are
arbitrary for now. Comparing this with (A.13) however, we find that $X_{p}=X$
and $\bar{X}_{p}=\bar{X}-\hat{v}$.
Similarly, the other contributions to (A.26)–(A.3) are conserved when (2.35)
holds with $Y_{p}\equiv Y-\hat{u}$ and $\bar{Y}_{p}\equiv\bar{Y}$ periodic.
The resulting algebra is
$\displaystyle\\{Q_{f},\bar{Q}_{\bar{f}}\\}$
$\displaystyle=-\frac{\mu}{R_{u}R_{v}}\left(Q_{f^{\prime}}Q_{\bar{X}_{p}\bar{f}^{\prime}}+\bar{Q}_{\bar{f}^{\prime}}Q_{Y_{p}f^{\prime}}\right)\
,$ $\displaystyle\\{Q_{f},Q_{g}\\}$
$\displaystyle=\frac{1}{R_{u}}Q_{fg^{\prime}-f^{\prime}g}+\frac{\mu^{2}}{R_{u}^{2}}\left(Q_{f^{\prime}}Q_{X_{p}g^{\prime}}-Q_{g^{\prime}}Q_{X_{p}f^{\prime}}\right)\
,$ $\displaystyle\\{\bar{Q}_{\bar{f}},\bar{Q}_{\bar{g}}\\}$
$\displaystyle=\frac{1}{R_{v}}\bar{Q}_{\bar{f}^{\prime}\bar{g}-\bar{f}\bar{g}^{\prime}}+\frac{\mu^{2}}{R_{v}^{2}}\left(\bar{Q}_{\bar{f}^{\prime}}\bar{Q}_{\bar{Y}_{p}\bar{g}^{\prime}}-\bar{Q}_{\bar{g}^{\prime}}\bar{Q}_{\bar{Y}_{p}\bar{f}^{\prime}}\right)\
.$ (A.33)
We can finally rewrite $\partial_{u}f=f^{\prime}/R_{u}$ and
$\partial_{v}\bar{f}=\bar{f}^{\prime}/R_{v}$ to recover section 2.2.
The undetermined functions $X_{p},\bar{X}_{p},Y_{p},\bar{Y}_{p}$ could more
generally be constrained by the Jacobi identities that the Poisson bracket
algebra must satisfy. These involve product of $\delta$ function
distributions, so their full analysis is rather involved. We will not pursue
it here.161616Note added in v2: to simplify the analysis, it is possible to
study the Jacobi identities involving a “smeared” version of the field
dependent coordinates $u_{0}\equiv\int_{0}^{R}d\sigma\,u(\sigma)$,
$v_{0}\equiv\int_{0}^{R}d\sigma\,v(\sigma)$ and the currents, as was done in
[73] for $J\bar{T}$. The simplest Jacobi identity is
$\\{\\{P,\tilde{P}\\},u_{0}\\}+\text{cyclic}=0$. Plugging in the minimal
solution, we find that this Jacobi identity _is not_ satisfied. More
generally, it seems to be very difficult to find a solution to the Jacobi
identities that has the non-trivial winding required by the charge
conservation equation (A.4). If no such solution exists, this means that the
functions $u,v$ as defined in this note are not good functions on phase space
when the $T\bar{T}$ \- deformed theory lives on compact space. This does not,
however, preclude the existence of related field-dependent coordinates whose
action phase space is well-defined, as was the case for $J\bar{T}$.
## Appendix B The $J\bar{T}$ charge algebra
In this appendix, we present some details of the calculation of Poisson
brackets of the conserved charges in $J\bar{T}$ \- deformed CFTs. As explained
in the main text, the undeformed commutators (3.29) and the general formula
(3.25) allow us to compute the commutator of the deformed right-moving
Hamiltonian with various quantities of interest. These read
$\\{\mathcal{H}_{R}(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}=\frac{-\mathcal{H}_{R}^{(0)}(\sigma)-\mathcal{H}_{R}^{(0)}(\tilde{\sigma})+\frac{\lambda^{2}}{2}\mathcal{H}_{R}(\sigma)\mathcal{H}_{R}(\tilde{\sigma})}{\sqrt{\left(1-\lambda\mathcal{J}_{+}(\sigma)\right)^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\sigma)}\sqrt{\left(1-\lambda\mathcal{J}_{+}(\tilde{\sigma})\right)^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\tilde{\sigma})}}\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(B.1)
$\\{\mathcal{P}(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}=\frac{\mathcal{H}_{R}^{(0)}(\sigma)+\mathcal{H}_{R}^{(0)}(\tilde{\sigma})+\lambda\mathcal{J}_{+}(\sigma)\mathcal{H}_{R}(\tilde{\sigma})}{\sqrt{(1-\lambda\mathcal{J}_{+}(\tilde{\sigma}))^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\tilde{\sigma})}}\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(B.2)
$\\{\mathcal{H}_{R}(\sigma),\mathcal{J}_{+}(\tilde{\sigma})\\}=\frac{\lambda\mathcal{H}_{R}(\sigma)}{2\sqrt{\left(1-\lambda\mathcal{J}_{+}(\sigma)\right)^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}}}\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(B.3)
$\\{\mathcal{H}_{R}(\sigma),\mathcal{J}_{-}(\tilde{\sigma})\\}=-\frac{\mathcal{J}_{-}(\sigma)}{\sqrt{\left(1-\lambda\mathcal{J}_{+}(\sigma)\right)^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\sigma)}}\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(B.4)
Using the fact that $\mathcal{J}_{+}-\mathcal{J}_{-}=\phi^{\prime}_{1}$ and
that the last two commutators are total $\tilde{\sigma}$ derivatives, we can
deduce the commutator of $\mathcal{H}_{R}$ with $\phi$, which will be used to
compute the Poisson bracket of the energy currents with the field-dependent
coordinate $v=V-\lambda\phi_{1}$. We find
$\\{\mathcal{H}_{R}(\sigma),\phi_{1}(\tilde{\sigma})\\}=\frac{-\mathcal{J}_{-}(\sigma)-\lambda\mathcal{H}_{R}(\sigma)/2}{\sqrt{(1-\lambda\mathcal{J}_{+}(\sigma))^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}(\sigma)}}\delta(\sigma-\tilde{\sigma})$
(B.5)
Note that, in principle, we could have added an arbitrary integration
function, $A(\sigma)$, to the right-hand-side of this commutator. However, the
fact that the commutator is local with respect to $\sigma-{\tilde{\sigma}}$
compels us to set this potential integration function to zero.
Using the above commutators, we can easily compute those of
$\mathcal{H}_{L}=\mathcal{H}_{R}+\mathcal{P}$. As noted in the main text, the
$\\{\mathcal{H}_{L},\mathcal{H}_{L}\\}$ commutator can be shown to be
equivalent (using the criteria in appendix C) to
$\\{\mathcal{H}_{L}(\sigma),\mathcal{H}_{L}(\tilde{\sigma})\\}=(\mathcal{H}_{L}(\sigma)+\mathcal{H}_{L}(\tilde{\sigma}))\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(B.6)
The $\\{\mathcal{H}_{L}(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}$ commutator
is somewhat involved; however, in the calculations below, in which we will
integrate over ${\tilde{\sigma}}$, it suffices to know that it is proportional
to $\partial_{\sigma}\delta(\sigma-{\tilde{\sigma}})$ which implies, using
(C.1), that we only need to know its value at ${\tilde{\sigma}}=\sigma$, as
well as that of its first ${\tilde{\sigma}}$ derivative evaluated at
${\tilde{\sigma}}=\sigma$,
$\left.\\{\mathcal{H}_{L}(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}\right|_{\tilde{\sigma}=\sigma}=\mathcal{H}_{R}\left(1-\frac{1}{\sqrt{\left(1-\lambda\mathcal{J}_{+}\right)^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}}}\right)\;,\;\;\;\;\;\;\partial_{\tilde{\sigma}}\left.\\{\mathcal{H}_{L}(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}\right|_{\tilde{\sigma}=\sigma}=0$
(B.7)
The $\\{\mathcal{H}_{L},\phi\\}$ commutator can be inferred from (B.5).
We can now compute the algebra of the symmetry generators (3.12). It is
trivial to show that the left-movers satisfy a Virasoro algebra, since
$\\{Q_{f},Q_{g}\\}=\int d\sigma
d\tilde{\sigma}f(U)g(\tilde{U})\\{\mathcal{H}_{L}(\sigma),\mathcal{H}_{L}(\tilde{\sigma})\\}=\int
d\sigma\mathcal{H}_{L}(\sigma)(fg^{\prime}-f^{\prime}g)=Q_{fg^{\prime}-f^{\prime}g}$
(B.8)
The commutator of the left and the right generators can be shown to vanish
$\displaystyle\\{Q_{f},\bar{Q}_{\bar{f}}\\}$ $\displaystyle=$
$\displaystyle\int d\sigma
d\tilde{\sigma}f(U)\left[\\{\mathcal{H}_{L}(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}\bar{f}(\tilde{v})+\bar{f}^{\prime}(\tilde{v})\\{\mathcal{H}_{L}(\sigma),\tilde{v}\\}\mathcal{H}_{R}(\tilde{\sigma})\right]$
(B.9) $\displaystyle=\;\int d\sigma
f(U)\left[\left.\\{\mathcal{H}_{L}(\sigma),\mathcal{H}_{R}(\tilde{\sigma})\\}\right|_{\tilde{\sigma}=\sigma}\partial_{\sigma}\bar{f}(v)-\lambda\bar{f}^{\prime}(v)\left.\\{\mathcal{H}_{L}(\sigma),\phi(\tilde{\sigma})\\}\right|_{\tilde{\sigma}=\sigma}\mathcal{H}_{R}(\sigma)\right]=0$
where we used the fact that the field-dependent coordinate
$v=V-\lambda\phi_{1}$ (and so
$\partial_{\sigma}\bar{f}=\bar{f}^{\prime}(1-\lambda\phi^{\prime}_{1})$), as
well as the commutators (B.7) and we used the formula (C.5) for the integral
of a derivative of a delta function.
Finally, the right-moving generators have commutation relations
$\displaystyle\\{\bar{Q}_{\bar{f}},\bar{Q}_{\bar{g}}\\}$ $\displaystyle=$
$\displaystyle\int d\sigma
d\tilde{\sigma}\left[\bar{f}(v)\bar{g}(\tilde{v})\\{\mathcal{H}_{R},\tilde{\mathcal{H}}_{R}\\}+\bar{f}^{\prime}(v)\\{v,\mathcal{H}_{R}(\tilde{\sigma})\\}\mathcal{H}_{R}\bar{g}(\tilde{v})+\bar{f}(v)\bar{g}^{\prime}(\tilde{v})\\{\mathcal{H}_{R},\tilde{v}\\}\tilde{\mathcal{H}}_{R}\right]$
(B.10) $\displaystyle=$ $\displaystyle\int
d\sigma\left[\bar{f}(v)\left.\partial_{\tilde{\sigma}}\left(\bar{g}(\tilde{v})\\{\mathcal{H}_{R},\tilde{\mathcal{H}}_{R}\\}\right)\right|_{\tilde{\sigma}=\sigma}+\lambda(\bar{f}^{\prime}\bar{g}-\bar{g}^{\prime}\bar{f})\mathcal{H}_{R}\left.\\{\mathcal{H}_{R}(\sigma),\phi(\tilde{\sigma})\\}\right|_{\tilde{\sigma}=\sigma}\right]$
Note there is no $\\{v,\tilde{v}\\}$ commutator, as $v=V-\lambda\phi_{1}$ only
involves $\phi$. Using the fact that
$\left.\partial_{\tilde{\sigma}}\left(\\{\mathcal{H}_{R},\tilde{\mathcal{H}}_{R}\\}\right)\right|_{\tilde{\sigma}=\sigma}=\frac{2}{\lambda^{2}}\partial_{\sigma}\left(1-\frac{1-\lambda\mathcal{J}_{+}}{\sqrt{\left(1-\lambda\mathcal{J}_{+}\right)^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}}}\right)=-\partial_{\sigma}\frac{\mathcal{H}_{R}}{\sqrt{\left(1-\lambda\mathcal{J}_{+}\right)^{2}-\lambda^{2}\mathcal{H}_{R}^{(0)}}}$
(B.11)
Integrating by parts and using the specific values of the commutators, we find
$\\{\bar{Q}_{\bar{f}},\bar{Q}_{\bar{g}}\\}=\int
d\sigma(\bar{f}^{\prime}\bar{g}-\bar{g}^{\prime}\bar{f})\mathcal{H}_{R}=\bar{Q}_{\bar{f}^{\prime}\bar{g}-\bar{f}\bar{g}^{\prime}}$
(B.12)
Let us now compute the Poisson brackets with the currents. It is easy to show,
using the explicit expression (3.16) for $K_{U}$, that
$\\{Q_{f},P_{\chi}\\}=\int d\sigma
d\tilde{\sigma}f(U)\chi(\tilde{U})\\{\mathcal{H}_{L}(\sigma),K_{U}(\tilde{\sigma})\\}=\int
d\sigma f(U)\chi^{\prime}(U)K_{U}(\sigma)=P_{f\chi^{\prime}}$ (B.13)
The commutator of two $U(1)$ charges gives the usual central extension,
$\\{P_{\chi},P_{\eta}\\}=1/2\int d\sigma\chi\eta^{\prime}$ and the commutator
with the right-moving Virasoro generators vanishes. As for the right-moving
$U(1)$
$\bar{P}_{\bar{\chi}}=\int d\sigma\frac{\pi-\phi^{\prime}}{2}\bar{\chi}(v)$
(B.14)
it can be easily shown that they commute with the left-moving $U(1)$ and the
left-moving Virasoro. The algebra of these $U(1)$ currents is however not Kac-
Moody
$\\{\bar{P}_{\bar{\chi}},\bar{P}_{\bar{\eta}}\\}=\frac{1}{2}\int
d\sigma\left(-\bar{\chi}\bar{\eta}^{\prime}+\lambda\bar{\chi}\bar{\eta}^{\prime}\frac{\pi+\phi^{\prime}}{2}-\lambda\bar{\chi}^{\prime}\bar{\eta}\frac{\pi-\phi^{\prime}}{2}\right)$
(B.15)
$\\{\bar{Q}_{\bar{f}},\bar{P}_{\bar{\eta}}\\}=-\int
d\sigma\left(\bar{f}\bar{\eta}^{\prime}\mathcal{J}_{-}+\frac{\lambda}{2}\bar{f}^{\prime}\bar{\eta}\,\mathcal{H}_{R}\right)$
(B.16)
As explained in the main text, we can obtain an operator that has the standard
commutator with the right-moving Virasoro by adding a multiple of
$T_{\alpha\bar{z}}$ to the current. It turns out that the combination
$\bar{P}_{\bar{\eta}}^{KM}=\int
d\sigma\left(\frac{\pi-\phi^{\prime}}{2}+\frac{\lambda}{2}\mathcal{H}_{R}\right)\bar{\eta}$
(B.17)
does the job. The commutator of these new charges is
$\\{\bar{P}_{\bar{\chi}},\bar{P}_{\bar{\eta}}\\}=-\frac{1}{2}\int
d\sigma\bar{\chi}\bar{\eta}^{\prime}(1-\lambda\phi^{\prime})=-\frac{1}{2}\int
d\sigma\bar{\chi}\partial_{\sigma}\bar{\eta}$ (B.18)
Thus, we find two copies of the Virasoro - Kac-Moody algebra. As explained in
the main text, on a compact space we need to divide the coordinates by the
associated radii, but this does not affect the charge algebra, because the
zero modes of $\mathcal{J}_{\pm}$ that enter $R_{v}$ commute with all the
currents.
## Appendix C Equivalence of distributions
The derivative of a Dirac-$\delta$ distribution acts on test functions as
$\int
d\tilde{\sigma}f(\sigma)g(\tilde{\sigma})\partial_{\sigma}\delta(\sigma-\tilde{\sigma})=f(\sigma)g^{\prime}(\sigma)$
(C.1)
It is useful to know when two distributions of the form
$F(\sigma,{\tilde{\sigma}})\partial_{\sigma}\delta(\sigma-{\tilde{\sigma}})$
are equivalent. The following criterion is necessary and sufficient:
$\displaystyle F_{1}\partial_{\sigma}\delta(\sigma-{\tilde{\sigma}})\cong
F_{2}\partial_{\sigma}\delta(\sigma-{\tilde{\sigma}})$
$\displaystyle\Leftrightarrow\begin{cases}F_{1}(\sigma,\sigma)&=F_{2}(\sigma,\sigma)\\\
\partial_{\sigma}F_{1}(\sigma,{\tilde{\sigma}})|_{{\tilde{\sigma}}=\sigma}&=\partial_{\sigma}F_{2}(\sigma,{\tilde{\sigma}})|_{{\tilde{\sigma}}=\sigma}\\\
\partial_{\tilde{\sigma}}F_{1}(\sigma,{\tilde{\sigma}})|_{{\tilde{\sigma}}=\sigma}&=\partial_{\tilde{\sigma}}F_{2}(\sigma,{\tilde{\sigma}})|_{{\tilde{\sigma}}=\sigma}\\\
\end{cases}\ .$ (C.2)
An example of equivalent distributions is given by
$F_{1}=\sqrt{f(\sigma)f({\tilde{\sigma}})}$ and
$F_{2}=[f(\sigma)+f({\tilde{\sigma}})]/2$, or
$[f(\sigma)g(\tilde{\sigma})+g(\sigma)f(\tilde{\sigma})]\partial_{\sigma}\delta(\sigma-\tilde{\sigma})=[f(\sigma)g(\sigma)+f(\tilde{\sigma})g(\tilde{\sigma})]\partial_{\sigma}\delta(\sigma-\tilde{\sigma})$
(C.3)
which indeed satisfies eq. C.2.
The remainder of this subsection is dedicated to showing eq. C.2. To this end,
note that two distributions $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are
equivalent iff they integrate to the same result against arbitrary test
functions, i.e.
$\displaystyle\int_{\sigma_{1}}^{\sigma_{2}}d\sigma\,\mathcal{F}_{1}$
$\displaystyle=f(\sigma_{1},\sigma_{2},{\tilde{\sigma}})=\int_{\sigma_{1}}^{\sigma_{2}}d\sigma\,\mathcal{F}_{2}\
,$ (C.4a)
$\displaystyle\int_{{\tilde{\sigma}}_{1}}^{{\tilde{\sigma}}_{2}}d{\tilde{\sigma}}\,\mathcal{F}_{1}$
$\displaystyle=g({\tilde{\sigma}}_{1},{\tilde{\sigma}}_{2},\sigma)=\int_{{\tilde{\sigma}}_{1}}^{{\tilde{\sigma}}_{2}}d{\tilde{\sigma}}\,\mathcal{F}_{2}\
,$ (C.4b)
$\displaystyle\int_{\sigma_{1}}^{\sigma_{2}}d\sigma\int_{{\tilde{\sigma}}_{1}}^{{\tilde{\sigma}}_{2}}d{\tilde{\sigma}}\,\mathcal{F}_{1}$
$\displaystyle=h(\sigma_{1},\sigma_{2},{\tilde{\sigma}}_{1},{\tilde{\sigma}}_{2})=\int_{\sigma_{1}}^{\sigma_{2}}d\sigma\int_{{\tilde{\sigma}}_{1}}^{{\tilde{\sigma}}_{2}}d{\tilde{\sigma}}\,\mathcal{F}_{2}$
(C.4c)
Obviously, if $f$ or $g$ are normal functions (i.e. not distributions), the
requirement eq. C.4c follows from either eq. C.4a or eq. C.4b. For
distributions proportional to $\delta^{\prime}$, the left-hand side of the
requirement eq. C.4a gives
$\displaystyle\int_{\sigma_{1}}^{\sigma_{2}}d\sigma\,F(\sigma,{\tilde{\sigma}})\partial_{\sigma}\delta(\sigma-{\tilde{\sigma}})$
(C.5)
$\displaystyle=F(\sigma_{2},\sigma_{2})\delta({\tilde{\sigma}}-\sigma_{2})-F(\sigma_{1},\sigma_{1})\delta({\tilde{\sigma}}-\sigma_{1})-\partial_{\sigma}F(\sigma,{\tilde{\sigma}})|_{\sigma={\tilde{\sigma}}}\Theta(\sigma_{1}<{\tilde{\sigma}}<\sigma_{2})\
.$
The result is not a simple function, but it is a distribution with only
$\delta$ functions for which we know the equivalence relations: the
coefficients of the delta functions have to agree, as well as the remainder.
This leads to the first two equations on the right-hand side of eq. C.2. The
third equation follows from eq. C.4b. Equation C.4c does not provide
additional constraints.
The fact that the result of eq. C.5 contains $\delta$ functions, suggests a
generalization to distributions of the form
$[F(\sigma,{\tilde{\sigma}})\partial_{\sigma}+G(\sigma,{\tilde{\sigma}})]\delta(\sigma-{\tilde{\sigma}})$.
Two such distributions are equivalent iff
$\displaystyle F_{1}(\sigma,\sigma)$ $\displaystyle=F_{2}(\sigma,\sigma)\ ,$
(C.6a)
$\displaystyle\partial_{\sigma}F_{1}(\sigma,{\tilde{\sigma}})-G_{1}(\sigma,{\tilde{\sigma}})|_{\sigma={\tilde{\sigma}}}$
$\displaystyle=\partial_{\sigma}F_{2}-G_{2}(\sigma,{\tilde{\sigma}})|_{\sigma={\tilde{\sigma}}}\
,$ (C.6b)
$\displaystyle\partial_{\tilde{\sigma}}F_{1}(\sigma,{\tilde{\sigma}})+G_{1}(\sigma,{\tilde{\sigma}})|_{\sigma={\tilde{\sigma}}}$
$\displaystyle=\partial_{\tilde{\sigma}}F_{2}(\sigma,{\tilde{\sigma}})+G_{2}(\sigma,{\tilde{\sigma}})|_{\sigma={\tilde{\sigma}}}\
.$ (C.6c)
The derivation is identical.
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|
#
Chenyuan Feng, Howard H. Yang, , Deshun Hu, Zhiwei Zhao, , Tony Q. S. Quek, ,
and Geyong Min C. Feng and T. Q. S. Quek are with Information Systems
Technology and Design, Singapore University of Technology and Design, 487372,
Singapore (email<EMAIL_ADDRESS><EMAIL_ADDRESS>.H.
H. Yang is with the Zhejiang University/University of Illinois at Urbana-
Champaign Institute, Zhejiang University, Haining 314400, China, (email:
[email protected]).D. Hu is with the Department of Communication
Engineering, Harbin Institute of Technology, Harbin 150001, China
(email:[email protected]).Z. Zhao is with the School of Computer Science
and Engineering, University of Electronic Science and Technology of China,
Chengdu 610051, China (e-mail: [email protected]).G. Min is with the Department
of Computer Science, College of Engineering Mathematics and Physical Sciences,
The University of Exeter, Exeter EX4 4QF, U.K. (e-mail:
[email protected]).The corresponding author of this paper is H. H. Yang.
# Mobility-Aware Cluster Federated Learning in Hierarchical Wireless Networks
Chenyuan Feng, Howard H. Yang, , Deshun Hu, Zhiwei Zhao, , Tony Q. S. Quek, ,
and Geyong Min C. Feng and T. Q. S. Quek are with Information Systems
Technology and Design, Singapore University of Technology and Design, 487372,
Singapore (email<EMAIL_ADDRESS><EMAIL_ADDRESS>.H.
H. Yang is with the Zhejiang University/University of Illinois at Urbana-
Champaign Institute, Zhejiang University, Haining 314400, China, (email:
[email protected]).D. Hu is with the Department of Communication
Engineering, Harbin Institute of Technology, Harbin 150001, China
(email:[email protected]).Z. Zhao is with the School of Computer Science
and Engineering, University of Electronic Science and Technology of China,
Chengdu 610051, China (e-mail: [email protected]).G. Min is with the Department
of Computer Science, College of Engineering Mathematics and Physical Sciences,
The University of Exeter, Exeter EX4 4QF, U.K. (e-mail:
[email protected]).The corresponding author of this paper is H. H. Yang.
###### Abstract
Implementing federated learning (FL) algorithms in wireless networks has
garnered a wide range of attention. However, few works have considered the
impact of user mobility on the learning performance. To fill this research
gap, firstly, we develop a theoretical model to characterize the hierarchical
federated learning (HFL) algorithm in wireless networks where the mobile users
may roam across multiple edge access points (APs), leading to incompletion of
inconsistent FL training. Secondly, we provide the convergence analysis of HFL
with user mobility. Our analysis proves that the learning performance of HFL
deteriorates drastically with highly-mobile users. And this decline in the
learning performance will be exacerbated with small number of participants and
large data distribution divergences among user’s local data. To circumvent
these issues, we propose a mobility-aware cluster federated learning (MACFL)
algorithm by redesigning the access mechanism, local update rule and model
aggregation scheme. Finally, we provide experiments to evaluate the learning
performance of HFL and our MACFL. The results show that our MACFL can enhance
the learning performance, especially for three different cases: the case of
users with non-independent and identical distribution (non-IID) data, the case
of users with high mobility, and the cases with a small number of users.
###### Index Terms:
Hierarchical federated learning, user mobility, data heterogeneity,
convergence analysis
## I Introduction
To support emerging intelligent services and applications, federated learning
(FL) has been proposed in [1] as a promising approach to generate high quality
models without collecting the distributed data. It provides great convenience
for parallel processing and can reduce the costs of message exchanging
significantly [2, 3]. As for the learning efficiency, the convergence analysis
of the FL in wireless networks has been studied in [4, 5], which can
characterize the impact of unreliable communication links. To accelerate the
convergence of the FL, sophisticated user scheduling schemes in wireless
networks have been designed in [6], and a theoretical bound has been analyzed
in [7]. To fully exploit the processing power of cloud and edge computing
servers, a client-edge-cloud hierarchical federated learning (HFL) paradigm
has been proposed in [8], and the resource optimization has been designed in
[9].
Albeit the popularity, it has been demonstrated that FL yields suboptimal
results if the local clients’ data distributions diverge, which causes
significant decline of model accuracy [10]. Due to the heterogeneous
environment, a shared global model trained by conventional FL might not
generalize well for each user. To tackle this problem, a new version named
cluster multi-task FL is proposed in [11] and [12]. Cluster multi-task FL can
improve the FL framework by dividing clients into different clusters, and
different learning tasks and shared models are learned by different clusters.
In [13], the authors have proved that cluster FL will converge faster than
single cell FL. Therefore, cluster FL is more suitable for the large-scale
hierarchical wireless networks.
Aside from dividing users into different clusters, another technique named
personalized FL is proposed to cope with the data heterogeneity among users.
In [14], the authors propose the Per-FedAvg algorithm based on a model-
agnostic meta-learning formulation. In [15], the pFedMe algorithm is proposed
by adding regularization terms to local loss functions in the version of
Moreau envelopes. The authors of [16] propose the FedAMP algorithm to
facilitate the collaboration between similar clients based on attention
mechanism. However, most of the works related to personalized FL are
formulated for single-cell networks rather than hierarchical wireless
networks.
In addition to data heterogeneity, user mobility is another important factor
in hierarchical wireless networks. However, most of the previous works only
consider a static network topology while the impact of user mobility has been
ignored. Indeed, in many practical scenarios, one mobile user may download the
latest global model from a certain access point (AP) and keep moving and might
leave the AP’s coverage area during the local training procedure. As far as we
know, the state-of-the-art works that considered user mobility in the design
of the FL paradigms are [17] and [18]. However, in [17] and [18], the APs do
not select the users with high mobility that might leave their coverage area,
which can lead to a decrease in the number of participants, hence results in
missing valid training data and under-fitting global model.
Although cluster FL with mobile users seems promising and more practical,
several essential challenges still exist, which can be listed as follows:
Firstly, it is difficult to evaluate the impact of user mobility on the
convergence rate and accuracy of the HFL. Besides single-cell networks, the
convergence rate of multi-layer FL is analyzed in [19, 20, 21]. However, there
is little work considering the convergence performance of HFL with mobile
users. Secondly, it is difficult to efficiently assign mobile users with non-
IID dataset into different clusters with limited communication and computation
resources. It is neither efficient for clustering schemes with additional
communication and computation costs, nor practical due to the neglect of
physical accessibility [22, 23, 24, 25]. Finally, the existing aggregation
schemes are based on the conventional equal-weighted averaging scheme [1],
which become the performance bottleneck due to the distribution divergences of
non-IID data and user mobility.
Motivated by these critical issues, we analyze the the performance of HFL
algorithm with mobile users and propose a mobility-aware cluster federated
learning (MACFL) algorithm. Our main contributions are summarized as follows:
* •
We develop a theoretical framework for the analysis of HFL with mobile users,
whereas convergence rates are derived by accounting for the impacts of user
mobility as well as the heterogeneity from both dataset and network
architecture.
* •
Building upon the analytical results, we redesign the access mechanism to
allow mobile users to effectively participate in collaboration. We also
introduce personalized FL and attentive averaging schemes to boost up the
convergence rate and enhance model accuracy.
* •
We conduct extensive experiments to evaluate the performance of our proposed
scheme, and also study the impact of network parameters on the learning
performance. The experiment results show that our proposed schemes can
significantly improve the convergence and accuracy performance of HFL with
mobile users.
## II System Model
In this section, we introduce the architecture of the hierarchical wireless
network, the mobility model of users, as well as the implementation of FL in
such a system.
Figure 1: An illustration of the system model. Fig.1: The hierarchical
wireless network consists of one cloud server, multiple edge APs, and all
users will communicate with their nearest edge AP and might lose connection
due to mobility. Fig.1 shows an example of the procedure of HFL with
$\kappa_{1}=3$ and $\kappa_{2}=2$.
### II-A Network Setup
We consider a statistical learning task conducted in a hierarchical wireless
network. The network consists of $M$ mobile users, $N$ edge APs, and one cloud
server, as depicted in Fig.1. We denote the set of users by
$\\{u_{m}\\}_{m=1}^{M}$, where a generic user $u_{m}$ holds a local dataset
$\mathcal{D}_{u_{m}}$. The size of dataset $\mathcal{D}_{u_{m}}$ is denoted as
$|\mathcal{D}_{u_{m}}|$, where $|\cdot|$ represents the cardinality of a set.
We further denote the set of edge APs by $\\{c_{n}\\}_{n=1}^{N}$, and assume
that each edge AP is equipped with an edge server, we interchangeably use the
term edge AP and edge server in the rest of paper. Without loss of generality,
we consider $M\gg N$ because an AP is usually connected by multiple users in
the real world. In this work, we consider a simple cluster scheme based on
physical accessibility. Specifically, each mobile user will connect to the
nearest edge AP based on real time locations without additional communication
and computation overhead. As such, we denote $\mathcal{C}_{n}$ as the set of
mobile users connected to the edge AP $c_{n}$.
The objective of all the entities in this network is to jointly minimize the
global loss functions, $F(\mathbf{w},\mathcal{D})$, given as follows
$\displaystyle F(\mathbf{w},\mathcal{D})$
$\displaystyle=\frac{1}{|\mathcal{D}|}\sum_{(\mathbf{x}_{i},y_{i})\in\mathcal{D}}\ell(\mathbf{w};\mathbf{x}_{i},y_{i})=\sum_{m=1}^{M}\alpha_{u_{m}}F_{u_{m}}(\mathbf{w},\mathcal{D}_{u_{m}})$
(1)
in which $\mathcal{D}=\cup_{m=1}^{M}\mathcal{D}_{u_{m}}$ is the dataset
aggregated from all the mobile users and
$\ell(\mathbf{w};\mathbf{x}_{i},y_{i})$ is the loss function assigned on the
$i$-th data sample pair $(\mathbf{x}_{i},y_{i})$. Particularly,
$\mathbf{x}_{i}\in\mathbb{R}^{d}$ denotes the $i$-th input sample and $y_{i}$
the corresponding label, and $\mathbf{w}\in\mathbb{R}^{d}$ is the minimizer,
which can fully parametrize the machine learning model, and $\alpha_{u_{m}}$
denotes the linear combination weight of user $u_{m}$ and satisfies
$\sum_{m=1}^{M}\alpha_{u_{m}}=1$ with $\alpha_{u_{m}}\in[0,1]$. Moreover, the
function $F_{u_{m}}(\mathbf{w},\mathcal{D}_{u_{m}})$ represents the empirical
loss function comprised by the local dataset of the mobile user $u_{m}$, given
as
$F_{u_{m}}(\mathbf{w},\mathcal{D}_{m})=\frac{1}{|\mathcal{D}_{u_{m}}|}\sum_{(\mathbf{x}_{i},y_{i})\in\mathcal{D}_{u_{m}}}\ell(\mathbf{w};\mathbf{x}_{i},y_{i}).$
(2)
Due to privacy concerns of the users, the local dataset is not accessible by
neither the APs nor the cloud server. Therefore, the global loss function
$F(\mathbf{w},\mathcal{D})$ cannot be directly evaluated and its minimization
needs to be carried out by means of federated computing. Fig. 1 illustrates a
typical procedure of HFL. Particularly, inside the coverage – also referred to
as cluster – of each edge AP, model training is conducted between the AP and
the mobile users connected to it. As such, the objective function of this
cluster is given by
$\displaystyle
f_{c_{i}}(\mathbf{w}_{c_{i}},\mathcal{D}_{c_{i}})=\sum_{u_{m}\in\mathcal{C}_{i}}\alpha_{u_{m}}^{c}F_{u_{m}}(\mathbf{w},\mathcal{D}_{u_{m}})$
(3)
where
$\mathcal{D}_{c_{i}}=\bigcup_{u_{m}\in\mathcal{C}_{i}}\mathcal{D}_{u_{m}}$
denotes the the training dataset owned by all users that connects to edge AP
$c_{i}$ and $\alpha_{u_{m}}^{c}\in[0,1]$ denotes the weight of $u_{m}$ at edge
aggregation in its assigned cluster, whereas
$\sum_{m=1}^{M}\alpha_{u_{m}}^{c}=1$. Similarly, the edge APs will further
aggregate their parameters at the cloud server to minimize the following
function
$\displaystyle
f(\mathbf{w},\mathcal{D})=\sum_{i=1}^{N}\alpha_{c_{i}}f_{c_{i}}(\mathbf{w},\mathcal{D}_{c_{i}})$
(4)
where $\alpha_{c_{i}}\in[0,1]$ is the weight of $c_{i}$ and we have
$\sum_{i=1}^{N}\alpha_{c_{i}}=1$. Recalling cluster and global functions, we
have $\alpha_{u_{m}}=\alpha_{u_{m}}^{c}\alpha_{c_{u_{m}}}$.
In this work, we consider the communications between users and edge APs happen
every $\kappa_{1}$ times of local updates. And the communication between the
cloud and edge servers happens every $\kappa_{2}$ times of edge aggregations.
### II-B User Mobility Model
In the presence of user mobility, the set of mobile users associated with each
edge AP varies over time. In order to characterize this feature, we assume all
the users are uniformly distributed over the entire network at the beginning
of time, and then each user will stay or move to a neighboring cluster
according to a certain probability during the local training.
We use a Markov chain to model the dynamics of mobile users. Specifically, we
use a vector $\mathbf{\pi}_{u_{m}}^{t}\in\mathbb{R}^{N}$ to capture the state
of connection for $u_{m}$ at the $t$-th iteration, where the entries are
defined as follows
$\pi^{t}_{u_{m}}[i]=\left\\{\begin{matrix}1,&\text{if}~{}u_{m}\in\mathcal{C}_{i}^{t}\\\
0,&\text{otherwise}\end{matrix}\right.$ (5)
in which $\mathcal{C}_{i}^{t}$ denotes the set of mobile users connected to
$c_{n}$ at the $t$-th iteration. Moreover, we use an adjacency matrix
$\mathbf{A}\in\mathbb{R}^{N\times N}$ to characterize the spatial topology
between the edge APs. The elements of this matrix are defined
$\mathbf{A}_{ij}=\left\\{\begin{matrix}1,&~{}\text{if}~{}~{}c_{j}\in\mathcal{N}(c_{i})\\\
0,&\\!\\!\\!\\!\\!\\!\\!\\!\text{otherwise}\end{matrix}\right.$ where
$\mathcal{N}(c_{n})$ denotes the set of neighboring APs of $c_{n}$, including
itself, and the size of $\mathcal{N}(c_{n})$ is given by
$|\mathcal{N}(c_{i})|=\sum_{j=1}^{N}\mathbf{A}_{ij}$.
Figure 2: An example of a linear graph and the transmission probability of
mobile users between different clusters.
We assume all the user are uniformly and randomly distributed over the entire
network at the beginning of time, and the probability of any specific user
moving to different neighboring APs are equal. Therefore, the element of
transition probability matrix can be written as
$\mathbf{P}^{t}_{i,j}=\left\\{\begin{matrix}p_{s,c_{i}}^{t},&\text{if}~{}i=j\\\
\frac{1-p_{s,c_{i}}^{t}}{\left|\mathcal{N}(c_{n})\right|-1},&\text{if}~{}~{}c_{j}\in\mathcal{N}(c_{i})\setminus
c_{i}\\\ 0,&\text{otherswise}\end{matrix}\right.$ (6)
If $p_{s,c_{i}}^{t}=1,\forall i,\forall t$, the scenario boils down to the
conventional HFL with static users.
In this paper, we assume all the mobile users have the same transition
probability matrix. Then, at the $t$-th local computation step, let
$\mathbf{P}^{t}\in\mathbb{R}^{N\times N}$ denote the transition probability
matrix, which can be used to characterize the trajectory of mobile user at
$t$-th iteration. As shown in Fig. 2, the entry at ($i$, $j$), i.e.,
$\mathbf{P}^{t}_{ij}$, denotes the probability that users staying in AP
$c_{i}$ are moving to $c_{j}$ at $t$-th local computation time. In our work,
we assume the transition probability matrix are the same for all users and
each user will stay at the same cluster at $t$-th local computation at the
probability of $p_{s,c_{i}}^{t}$, and leave for different neighbors at the
probability of $1-p_{s,c_{i}}^{t}$. Given current observation vector and
transmission probability matrix, the future state vector can be estimated by
$\pi^{t+1}_{u_{m}}=\pi^{t}_{u_{m}}\mathbf{P}^{t}=\pi^{0}_{u_{m}}\prod_{\tau=0}^{t}\mathbf{P}^{\tau}$
(7)
Following the above, we can calculate the size $S_{i}^{t}$ of the set
$\mathcal{C}_{i}^{t}$ as follows
$S_{i}^{t}=\sum_{m=1}^{M}\pi^{t}_{u_{m}}[i]=\sum_{m=1}^{M}\pi^{0}_{u_{m}}\left(\prod_{\tau=0}^{t}P^{\tau}_{u_{m}}\right)_{[:,i]}$
(8)
where $(\cdot)_{[:,i]}$ represents the $i$-th column of a matrix. The
evolution of $S_{i}^{t}$ is a complicated quasi-birth and death process, which
depends on the initial conditions, Markov process intensity matrix, and the
number of iterations [26]. For the sake of tractability, we consider an
equilibrium condition with steady-state probability distribution as follows:
$S_{i}^{t}(1-p_{s,c_{i}}^{t})=\sum_{c_{j}\in\mathcal{N}(c_{i})\setminus
c_{i}}S_{j}^{t}p_{c_{j},c_{i}}^{t},\forall i,\forall t.$ (9)
This equation indicates the number of incoming and departing users in each
cluster is balanced, then we have $S_{i}^{t}=S_{i},\forall t,\forall i$. It is
noteworthy that the indexes of user in $\mathcal{C}_{i}^{t}$ are time varying
albeit the size keeps the same.
## III HFL with Mobile Users
In this section, we extend the conventional HFL algorithm [8] to account for
cases with mobile users. The general scheme is given in Algorithm 1, whereas
the key steps, as well as the convergence analysis, are elaborated in the
sequel.
### III-A Algorithm Description
Algorithm 1 Conventional HFL with Mobile Users
1:Initialize $\\{\mathcal{D}_{u_{m}}\\}_{m=1}^{M}$,
$\\{\pi_{u_{m}}^{0}\\}_{m=1}^{M}$, $\mathbf{P}$,
$\mathcal{C}^{0}=\\{\mathcal{C}_{1}^{0},...,\mathcal{C}_{N}^{0}\\}$,
${\\{\mathbf{w}_{c_{i}}^{(0)}\\}}_{i=1}^{N}$.
2:for each edge communication round $b=0,1,...,B-1$ do
3: for each user $m=1,2,...,M$ in parallel do
4: $t=b\kappa_{1}$
5: Local update based on (10) and (11).
6: for each edge $n=1,....,N$ in parallel do
7: Edge update based on (12).
8: if $i\mod\kappa_{2}=0$ then
9: Cloud update based on (13).
10: for each edge $n=1,....,N$ in parallel do
11: $\mathbf{w}_{c_{n}}^{t+1}\leftarrow\mathbf{w}_{g}^{t+1}$
12:return Global model $\mathbf{w}_{g}^{T}$
The key steps of the HFL algorithm are presented in Algorithm 1. During the
local training procedure, each mobile user will download the cluster model
from its closest AP and then update local model after the local training. Only
mobile users who stay in the same coverage area at the local training
procedure can upload successfully to the original edge AP, and the cloud
server will make an aggregation of all cluster models from all the edge APs’
models after every $\kappa_{2}$ edge model aggregations. The procedures of
user update, edge update, and cloud update are detailed in below,
respectively.
#### III-A1 Local Update
Let $t=b\kappa_{1}+j$ denote the index of local update iteration, with $i$
denoting the index of edge communication round and $j$ denoting the index of
epoch during the local training. $\mathbf{w}^{t}_{u_{m}}$ denotes the local
model parameter of $u_{m}$ at the $t$-th local update iteration, when
$t\mod\kappa_{1}=0$, each user downloads the latest cluster model from its
tagged edge AP, i.e.,
$\mathbf{w}^{t}_{u_{m}}=\mathbf{w}^{t}_{c_{n}},\text{ if
}\pi_{u_{m}}^{t}[n]=1,$ (10)
and then performs $\kappa_{1}\geq 1$ steps of local updates via SGD methods,
which can be expressed as
$\displaystyle\mathbf{w}^{t+1}_{u_{m}}\leftarrow\mathbf{w}^{t}_{u_{m}}-\eta
g(\mathbf{w}^{t}_{u_{m}},\xi^{t+1}_{u_{m}}),$ (11)
where $\eta$ is the learning rate, $\xi^{t}_{u_{m}}\subset\mathcal{D}_{u_{m}}$
is a mini-batch independently and identically sampled from the local dataset
at the $t$-th local update iteration, and
$g(\mathbf{w}^{t}_{u_{m}},\xi^{t}_{u_{m}})$ denotes the mini batch gradient of
local loss function, and it satisfies
$\mathbb{E}_{\xi}\\{g(\mathbf{w}^{t}_{u_{m}},\xi^{t}_{u_{m}})\\}=\nabla
F(\mathbf{w}^{t}_{u_{m}},\mathcal{D}_{u_{m}})$. For notation simplicity, we
use $g(\mathbf{w}^{t}_{u_{m}})$ and $\nabla F(\mathbf{w}^{t}_{u_{m}})$ for
short in the rest of this paper.
#### III-A2 Edge update
The communications between the users and their associated edge APs take place
in every $\kappa_{1}$ local update iterations. Due to mobility, the set of
mobile users in cluster $c_{n}$ at $b$-th edge communication round might be
different from that at the $(b+1)$-th edge communication round. As such, the
edge AP only collects the updated gradients from the users that are connect to
$c_{n}$ at the $(b+1)$-th edge aggregation can upload successfully. Let a
Boolean variable $\mathbb{I}_{u_{m}}^{b\kappa_{1}}$ represent the uploading
state, i.e., $\mathbb{I}_{u_{m}}^{b\kappa_{1}}=1$ if
$\pi_{u_{m}}^{(b+1)\kappa_{1}}[i]=\pi_{u_{m}}^{b\kappa_{1}}[i],\forall i$,
otherwise $\mathbb{I}_{u_{m}}^{b\kappa_{1}}=0$. Then the cluster model is
updated as follows:
$\mathbf{w}_{c_{i}}^{(b+1)\kappa_{1}}=\mathbf{w}_{c_{i}}^{b\kappa_{1}}-\eta\sum_{u_{m}\in\mathcal{C}_{i}^{t}}\sum_{\tau=1}^{\kappa_{1}}\alpha_{u_{m}}^{c}g(\mathbf{w}^{b\kappa_{1}+\tau}_{u_{m}})\mathbb{I}_{u_{m}}^{b\kappa_{1}}$
(12)
where $\mathbf{w}_{c_{i}}^{b\kappa_{1}}$ and
$\mathbf{w}_{c_{i}}^{(b+1)\kappa_{1}}$ are the cluster models of the edge AP
$c_{i}$ after the $b$-th and $(b+1)$-th edge communication round,
respectively. If $b\mod\kappa_{2}\neq 0$, the edge AP will broadcast the
latest cluster model to its connected users. And after every $\kappa_{2}$ edge
aggregations, the edge APs will upload the cluster models to the cloud server,
and then update the cluster model after receiving the global model from the
cloud, and finally broadcast to the connected users.
#### III-A3 Cloud Update
The global aggregation happens in every $\kappa_{2}$ edge aggregations, which
means the global model is updated as follows when $b\mod\kappa_{2}=0$:
$\mathbf{w}_{g}^{b\kappa_{1}}=\sum_{n=1}^{N}\alpha_{c_{n}}\mathbf{w}_{c_{n}}^{b\kappa_{1}}$
(13)
where $\alpha_{c_{n}}$ denotes the weight of updated cluster models from
$c_{n}$. Since the global aggregation happens every $\kappa_{1}\kappa_{2}$
local update iterations, we also have
$\mathbf{w}_{g}^{(b+\kappa_{2})\kappa_{1}}=\mathbf{w}_{g}^{b\kappa_{1}}-\eta\sum_{m=1}^{M}\sum_{q=0}^{\kappa_{2}-1}\sum_{\tau=1}^{\kappa_{1}}\alpha_{u_{m}}g(\mathbf{\tilde{w}}^{(b+q)\kappa_{1}+\tau}_{u_{m}})\mathbb{I}_{u_{m}}^{(b+q)\kappa_{1}}$
(14)
Compared to the time spent in local training, the time duration of parameter
uploading, model aggregation, and downloading is relatively short. Therefore,
we assume the connection states of the mobile users will only change during
the local training procedures and remain the same during communication and
model aggregation procedures.
### III-B Convergence Analysis
Owing to mobility, users may drop off from the associated edge APs, which
leads to a decrease in the number of participants in the FL. In this section,
we analyze the impact of user mobility on the convergence rate of the HFL.
Similar to previous works, we make the following assumptions:
* •
The loss functions are lower-bounded: $f(\mathbf{w})\geq
f_{\text{inf}},\forall\mathbf{w}$.
* •
The gradients of loss functions are bounded: $\|\nabla
F(\mathbf{w})\|^{2}\leq G^{2},\forall\mathbf{w}$.
* •
The loss functions are $L$-smooth: $\|\nabla F(\mathbf{w}_{1})-\nabla
F(\mathbf{w}_{2})\|\leq
L\|\mathbf{w}_{1}-\mathbf{w}_{2}\|,\forall\mathbf{w}_{1},\mathbf{w}_{2}$.
* •
The mini-batch gradients are unbiased with bounded variance:
$\mathbb{E}_{\xi|\mathbf{w}}\\{g(\mathbf{w})\\}=\nabla F(\mathbf{w})$,
${\|g(\mathbf{w})-\nabla
F(\mathbf{w})\|}^{2}\leq\sigma^{2},\forall\mathbf{w},\xi$.
* •
The divergences of the local, cluster and global loss functions are bounded,
for all $u_{m},c_{i},\mathbf{w}$, we have $\frac{1}{N}\sum_{i=1}^{N}{\|\nabla
f_{c_{i}}(\mathbf{w})-\nabla f(\mathbf{w})\|}^{2}\leq\epsilon_{g}^{2}$,
$\frac{1}{S_{i}}\sum_{u_{m}\in\mathcal{C}_{i}}{\|\nabla
F_{u_{m}}(\mathbf{w})-\nabla
f_{c_{i}}(\mathbf{w})\|}^{2}\leq\epsilon_{c}^{2}.$
Although the averaged global model is not observable when
$t\mod\kappa_{1}\kappa_{2}\neq 0$ in this system, here we use
$\mathbf{\bar{w}}_{g}^{t}$ for analysis similar to related work [20, 21].
Because the objective function $F(\mathbf{w},\mathcal{D})$ can be non-convex,
the learning algorithm via SGD methods may converge to a local minimum or
saddle point. Similar to [20, 21] and [28, 29], we leverage the expected
gradient norm as an indicator of convergence, namely the training algorithm
achieves an $\theta$-suboptimal solution if
$\mathbb{E}\left[\frac{1}{T}\sum_{t=1}^{T}{\|\nabla
f(\mathbf{\bar{w}}_{g}^{t})\|}^{2}\right]\leq\theta$, where $\theta>0$. This
condition can guarantee the convergence of an algorithm to a stationary point.
###### $\mathbf{Lemma}$ 1
For the employed FL system, when the learning rate satisfies
$\eta<\frac{1}{L}$, after $T$ rounds of global aggregations, we have
$\displaystyle\theta_{HFL}\leq$ $\displaystyle\frac{2}{\eta
p_{s}T}\big{[}\,\mathbb{E}f(\mathbf{\bar{w}}^{0}_{g})-f_{\inf})\,\big{]}+\eta
L\sigma^{2}\sum_{m=1}^{M}(\alpha_{u_{m}})^{2}+4\epsilon_{g}^{2}+4\epsilon_{c}^{2}$
$\displaystyle+\frac{4L^{2}}{T}\sum_{t=0}^{T-1}\bigg{(}\sum_{i=1}^{N}\alpha_{c_{n}}\mathbb{E}\Big{[}\|\mathbf{\bar{w}}^{t}_{g}-\mathbf{\bar{w}}_{c_{i}}^{t}\|^{2}\Big{]}+\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\Big{[}\|\mathbf{\bar{w}}_{c_{u_{m}}}^{t}-\mathbf{w}_{u_{m}}^{t}\|^{2}\Big{]}\bigg{)}.$
(15)
###### Proof:
Please refer to Appendix -A. ∎
Notably, the first term on the right hand side of (1) is similar to the
setting of centralized learning [29] when $p_{s}=1$, and the second term is
introduced by the randomness arises from the mini-batch gradients, the third
and fourth terms correspond to the divergences among the local, cluster, and
global loss functions, the fifth and sixth terms are introduced by the
divergences among local, cluster, and global model parameters. When $p_{s}=0$,
the right hand of the inequality of (1) is unbounded, which means the training
algorithm does not converge if no mobile user is participating in the model
aggregation.
We further bound the mean square error (MSE) of the local, as well as cluster,
model parameters in below.
###### $\mathbf{Lemma}$ 2
For the employed FL system, if $\eta<\frac{1}{\sqrt{12}L\kappa_{1}}$, the MSE
of the local model parameters can be bounded as follows:
$\displaystyle\frac{1}{T}\sum_{t=0}^{T-1}\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\|\mathbf{w}^{t}_{u_{m}}-\mathbf{\bar{w}}_{c_{u_{m}}}^{t}\|^{2}$
$\displaystyle\leq\frac{1}{1-12\eta^{2}L^{2}\kappa_{1}^{2}p_{s}^{2}}\bigg{(}6\eta^{2}p_{s}^{2}\kappa_{1}^{2}\epsilon_{c}^{2}+2\eta^{2}\kappa_{1}p_{s}\left(\sigma^{2}+(1-p_{s})G^{2}\right)\sum_{m=1}^{M}\alpha_{u_{m}}(1-\alpha_{u_{m}}^{c})\bigg{)}.$
(16)
###### Proof:
Please refer to Appendix -B. ∎
###### $\mathbf{Lemma}$ 3
For the employed FL system, if
$\eta<\frac{1}{\sqrt{12}L\kappa_{1}\kappa_{2}}$, the MSE of the cluster model
parameters can be bounded as follows:
$\displaystyle\frac{1}{T}\sum_{t=0}^{T-1}\sum_{i=1}^{N}\alpha_{c_{n}}\mathbb{E}\|\mathbf{\bar{w}}^{t}_{g}-\mathbf{\bar{w}}_{c_{i}}^{t}\|^{2}$
$\displaystyle\leq\frac{1}{1-12\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}p_{s}^{2}}\bigg{\\{}6\eta^{2}p_{s}^{2}\kappa_{1}^{2}\kappa_{2}^{2}\epsilon_{g}^{2}+4\eta^{2}\kappa_{1}\kappa_{2}p_{s}(\sigma^{2}+(1-p_{s})G^{2})\sum_{m=1}^{M}\alpha_{u_{m}}(\alpha_{u_{m}}^{c}-\alpha_{u_{m}})$
$\displaystyle+\frac{4\eta^{2}p_{s}^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}}{1-12\eta^{2}L^{2}\kappa_{1}^{2}p_{s}^{2}}\bigg{[}2\eta^{2}\kappa_{1}p_{s}(\sigma^{2}+(1-p_{s})G^{2})\sum_{m=1}^{M}\alpha_{u_{m}}(1-\alpha_{u_{m}}^{c})+6\eta^{2}p_{s}^{2}\kappa_{1}^{2}\epsilon_{c}^{2}\bigg{]}\bigg{\\}}$
(17)
###### Proof:
Please refer to Appendix -C. ∎
We are now in position to obtain the upper bound of $\theta_{HFL}$ by applying
Lemma 2 and Lemma 3 back into Lemma 1.
###### $\mathbf{Theorem}$ 1
Under the employed FL system, when the learning rate is chosen as
$\eta<\frac{1}{\sqrt{12}L\kappa_{1}\kappa_{2}}$, then after $T$ rounds of
global aggregations, Algorithm 1 achieves an $\theta_{HFL}$-suboptimal
solution, where $\theta_{HFL}$ is bounded as
$\displaystyle\theta_{HFL}$ $\displaystyle\leq\frac{2}{\eta
p_{s}T}\big{[}\mathbb{E}f(\mathbf{\bar{w}}^{0}_{g})-f_{\inf}\big{]}+\eta
L\sigma^{2}\sum_{m=1}^{M}\alpha_{u_{m}}^{2}+\frac{4(1-6L^{2}\eta^{2}p_{s}^{2}\kappa_{1}^{2}\kappa_{2}^{2})}{1-12\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}p_{s}^{2}}\epsilon_{g}^{2}$
$\displaystyle+4\bigg{[}1+\frac{(1-8\eta^{2}p_{s}^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2})6\eta^{2}L^{2}p_{s}^{2}\kappa_{1}^{2}}{(1-12\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}p_{s}^{2})(1-12\eta^{2}L^{2}\kappa_{1}^{2}p_{s}^{2})}\bigg{]}\epsilon_{c}^{2}+\frac{8L^{2}\eta^{2}\kappa_{1}p_{s}(\sigma^{2}+(1-p_{s})G^{2})}{1-12\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}p_{s}^{2}}$
$\displaystyle\times\sum_{m=1}^{M}\alpha_{u_{m}}\bigg{[}2\kappa_{2}(\alpha_{u_{m}}^{c}-\alpha_{u_{m}})+\frac{1-8\eta^{2}p_{s}^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}}{1-12\eta^{2}L^{2}\kappa_{1}^{2}p_{s}^{2}}(1-\alpha_{u_{m}}^{c})\bigg{]}.$
(18)
From the right hand side of (1), we can see that the convergence error floor
depends on the initial states, divergences between local, cluster, and global
loss functions, SGD variance, and rate of descent of loss functions.
Remark 1: By taking the partial derivatives with respect to $\kappa_{1}$ and
$\kappa_{2}$, respectively, we find that $\theta_{HFL}$ monotonously goes down
with a decreasing value of $\kappa_{1}$ or $\kappa_{2}$, which implies the
more often the users and edge APs exchange information, the faster the
training algorithm converges. Additionally, if the product of
$\kappa_{1}\kappa_{2}$ is fixed, $\theta_{HFL}$ will decline with a decrease
in $\kappa_{1}$ or an increase in $\kappa_{2}$. This observation suggests
exchanging local models frequently helps more than exchanging global models,
when the duration of global aggregation is fixed.
Remark 2: By taking a partial derivative with respect to $p_{s}$, we can see
that $\theta_{HFL}$ is not a monotonic function of $p_{s}$ which depends on
the properties of training data and loss functions employed by local training.
Particularly, when $p_{s}=0$, the algorithm will never converge since there is
no mobile user participating in the model aggregation, which means a new model
aggregation scheme needs to be devised for users with high mobility.
In addition, by recalling the definition of $\epsilon_{g}^{2}$ and
$\epsilon_{c}^{2}$, it is clear that the divergences of local and cluster loss
functions with IID data is smaller than those with non-IID data.
###### $\mathbf{Corollary}$ 1
Under the employed FL system, if the weights for parameter aggregations are
chosen as $\alpha_{c_{n}}=\frac{1}{N}$, $\alpha_{u_{m}}=\frac{1}{M}$, and
$\alpha_{u_{m}}^{c}=\frac{N}{M}$, and the learning rate is set as
$\eta<\frac{1}{\sqrt{12}L\kappa_{1}\kappa_{2}}$, then after $T$ rounds of
global aggregations, Algorithm 1 can achieve an $\theta_{HFL}$-suboptimal
solution, where $\theta_{HFL}$ is bounded as
$\displaystyle\theta_{HFL}$
$\displaystyle\leq\frac{2(\mathbb{E}f(\mathbf{\bar{w}}^{0}_{g})-f_{\inf})}{\eta
p_{s}T}+\frac{\eta L}{M}\sigma^{2}$
$\displaystyle+\frac{4(1-6L^{2}\eta^{2}p_{s}^{2}\kappa_{1}^{2}\kappa_{2}^{2})}{1-12\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}p_{s}^{2}}\epsilon_{g}^{2}+4\bigg{[}1+\frac{(1-8\eta^{2}p_{s}^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2})6\eta^{2}L^{2}p_{s}^{2}\kappa_{1}^{2}}{(1-12\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}p_{s}^{2})(1-12\eta^{2}L^{2}\kappa_{1}^{2}p_{s}^{2})}\bigg{]}\epsilon_{c}^{2}$
$\displaystyle+\frac{8L^{2}\eta^{2}\kappa_{1}p_{s}(\sigma^{2}+(1-p_{s})G^{2})}{1-12\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}p_{s}^{2}}\bigg{[}2\kappa_{2}\frac{N-1}{M}+\frac{1-8\eta^{2}p_{s}^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}}{1-12\eta^{2}L^{2}\kappa_{1}^{2}p_{s}^{2}}\frac{M-N}{M}\bigg{]}.$
(19)
Following Corollary 1, we can see that if $M=N$, there is $\epsilon_{c}^{2}=0$
and $\alpha_{u_{m}}^{c}=1$, if $N=1$, we have $\epsilon_{g}^{2}=0$,
$\kappa_{2}=1$ and $\alpha_{u_{m}}^{c}=\alpha_{u_{m}}$, the case turns into
single-cell FL. If $M=N=1$, we have $\theta_{HFL}\leq\frac{2}{\eta
p_{s}T}(\mathbb{E}f(\mathbf{\bar{w}}^{0}_{g})-f_{\inf})+\eta L\sigma^{2}$,
which is in accord with centralized learning with SGD method [29] with
$p_{s}=1$.
## IV Mobility-Aware Cluster federated learning
In this section, we propose the MACFL algorithm by redesigning the access
mechanism, personalized local model update rule, and weighted average schemes.
Specifically, our proposed MACFL algorithm allows a mobile user to download a
cluster model from one edge AP and, if it roams to another cell, upload the
parameter to the corresponding AP, which leads to more mobile users to
successfully participate in model aggregations. Based on the analysis in
Theorem 1, the shared cluster models learned by conventional HFL can not
generalize well due to the user mobility and data heterogeneity. Therefore, we
also introduce personalized FL and attentive weighted averaging schemes to
reduce the performance degradation caused by SGD variance and the divergences
among local, cluster, and global loss functions. In this sections, we apply
these state-of-the-art schemes to HFL with mobile users and analyze its
convergence rate.
### IV-A Learning Task
In our proposed MACFL algorithm, the newly-arrived users can update their
local models based on the cluster model downloaded by one edge AP, and then
upload the updated results to a different edge AP. This mechanism leads to an
increasing number of participants. However, considering user mobility, the
shared cluster model aims to learn a common model for the time-varing user
set, and does to adapt to each user. Inspired by the personalized FL in
single-cell networks such as Per-FedAvg [14], we propose a novel personalized
cluster FL for hierarchical wireless networks to learn shared cluster and
global models for each server and personalized local models for each mobile
user at the same time. Moreover, motivated by attention-based FL in single-
cell networks such as FedAMP [16], we introduce an attentive collaboration
into HFL to boost the collaboration effectiveness between users without
leakage of private data, which can also mitigate the impact of user mobility
and data heterogeneity.
In our work, we aim to learn personalized local and cluster models and shared
gloabl model at the same time, similar to [14], we first rewrite the global
loss function in the following way
$\displaystyle\tilde{f}(\\{\mathbf{w}_{u_{m}}\\}_{m=1}^{M},\mathbf{w}_{g},\mathcal{D})=\sum_{m=1}^{M}\beta_{u_{m}}\tilde{F}_{u_{m}}(\mathbf{w}_{u_{m}},\mathbf{w}_{g},\mathcal{D}_{u_{m}}),$
$\displaystyle\tilde{F}_{u_{m}}(\mathbf{w}_{u_{m}},\mathbf{w}_{g},\mathcal{D}_{u_{m}})=F_{u_{m}}((\mathbf{w}_{g}-\rho\nabla
F_{u_{m}}(\mathbf{w}_{u_{m}},\mathcal{D}_{u_{m}})),\mathcal{D}_{u_{m}})$ (20)
where
$\tilde{f}(\\{\mathbf{w}_{u_{m}}\\}_{m=1}^{M},\mathbf{w}_{g},\mathcal{D})$ and
$\tilde{F}_{u_{m}}(\mathbf{w}_{u_{m}},\mathbf{w}_{g},\mathcal{D}_{u_{m}})$
denote the global and local loss function in MACFL algorithm with $F_{u_{m}}$
defined as (2), and $\beta_{u_{m}}$ is the weight assigned on the local models
learned by $u_{m}$ at global aggregation. And the global model
$\mathbf{w}_{g}$ is obtained by collaborative learning of all users, and the
personalized local model is learned by each user based on the shared model and
one more step of gradient decent, which can be computed as
$\mathbf{w}_{g}-\rho\nabla F_{u_{m}}(\mathbf{w}_{u_{m}},\mathcal{D}_{u_{m}})$
with $\rho$ denoting the stepsize of local training.
### IV-B Algorithm Description
#### IV-B1 Local Update
With a loss function given as (IV-A), at the $b$-th edge communication round,
each user will at first download the latest personalized model from nearest
edge AP, i.e.,
$\mathbf{w}^{b\kappa_{1}}_{u_{m}}=\mathbf{w}^{b\kappa_{1}}_{c_{n}}$ if
$u_{m}\in\mathcal{C}_{n}^{b\kappa_{1}}$, and then perform $\kappa_{1}\geq 1$
steps of local updates via SGD. Let $t=b\kappa_{1}+j,j=0,...,\kappa_{1}-1$,
the evolution of each iteration can be expressed as
$\displaystyle\mathbf{w}^{t+1}_{u_{m}}=\mathbf{w}^{t}_{u_{m}}-\eta\tilde{g}(\mathbf{w}^{t}_{u_{m}}),~{}\text{with
}\tilde{g}(\mathbf{w}^{t}_{u_{m}})=(\mathbf{I}-\rho\nabla
g(\mathbf{w}^{t}_{u_{m}}))g(\mathbf{w}^{t}_{u_{m}}-\rho
g(\mathbf{w}^{t}_{u_{m}}))$ (21)
where $\tilde{g}(\mathbf{w}^{t}_{u_{m}})$ denotes the mini-batch gradient of
local loss function $\tilde{F}_{u_{m}}(\mathbf{w}^{t}_{u_{m}})$ given in
(IV-A) at $i$-th local update after $t$-th edge comunication round,
$\mathbf{I}$ denotes the identity matrix, and $\nabla
g(\mathbf{w}^{t}_{u_{m}})$ denotes the gradient of mini-batch gradient
$g(\mathbf{w}^{t}_{u_{m}})$ given in (11). As states in [14], the gradient
estimates can be approximated by first-order approximations with a slight loss
of accuracy in order to avoid the high computation complexity of computing
hessian term, namely, $\tilde{g}(\mathbf{w}^{t}_{u_{m}})\approx
g(\mathbf{w}^{t}_{u_{m}}-\rho g(\mathbf{w}^{t}_{u_{m}}))$.
#### IV-B2 Edge Update
In conventional FL, the aggregation strategy usually weights the importance of
each local model equally or proportional to the size of the local data.
However, this averaging scheme might not be the true weight of local data of
each user in mixture data distribution of cluster and global model. Moreover,
the mobile user sets in each cluster changes every communication round, the
new users who might have different cluster model at last iteration, therefore
a new edge update method needs to be designed. As shown in [10], the
performance of model aggregation depends on a reasonable design of weight
coefficients. Motivated by the attention scheme employed in FedAMP[16], we
redesign the edge update rule for $(b+1)$-th edge communication round as
follows
$\mathbf{w}_{c_{n}}^{(b+1)\kappa_{1}}=\sum_{u_{m}\in\mathcal{C}_{n}^{(b+1)\kappa_{1}}}\beta_{u_{m}}^{c}\mathbf{w}^{(b+1)\kappa_{1}}_{u_{m}},~{}\text{with
}\beta_{u_{m}}^{c}=\frac{e^{-\sigma_{1}cos(\mathbf{w}_{u_{m}}^{(b+1)\kappa_{1}},\mathbf{w}_{c_{n}}^{b\kappa_{1}})}}{\sum_{u_{m}\in\mathcal{C}_{n}^{(b+1)\kappa_{1}}}e^{-\sigma_{1}cos(\mathbf{w}_{u_{m}}^{(b+1)\kappa_{1}},\mathbf{w}_{c_{n}}^{b\kappa_{1}})}},$
(22)
where $\beta_{u_{m}}^{c}$ is the attention coefficient, $\sigma_{1}$ is a
scalar hyper-parameter, the attention coefficient $\beta_{u_{m}}^{c}$ denotes
the linear combination weight of model parameter aggregating from user $u_{m}$
to AP $c_{n}$, and
$\cos(\mathbf{w}_{u_{m}}^{(b+1)\kappa_{1}},\mathbf{w}_{c_{n}}^{b\kappa_{1}})$
is the cosine similarity between $\mathbf{w}_{u_{m}}^{(b+1)\kappa_{1}}$ and
$\mathbf{w}_{c_{n}}^{b\kappa_{1}}$ which is defined as
$\cos(\mathbf{x},\mathbf{y})=\frac{<\mathbf{x},\mathbf{y}>}{\|\mathbf{x}\|^{2}\|\mathbf{y}\|^{2}}$.
In this work, we employ a simple but effective attention mechanism with only
one scalar hyper-parameter. Compared with other clustered FL with
deterministic weight, the weight coefficients are learnable parameters in our
work. There are other complex methods to learn attention coefficients, such as
neural networks [30], which will result in a larger computation overhead.
Future work will focus on sophisticated attention scheme with limited
computation resources. After edge update, the edge server will distribute the
cluster model to connecting users, and when it is time for cloud update, the
cluster model will be sent to the cloud server.
#### IV-B3 Cloud Update
When $b\mod\kappa_{2}=0$, the cloud server will collect the cluster models
from all the edge APs and update the global parameter as follows
$\displaystyle{\mathbf{w}}_{g}^{b\kappa_{1}}=\sum_{n=1}^{N}\beta_{c_{n}}{\mathbf{w}}^{b\kappa_{1}}_{c_{n}},~{}\text{with
}\beta_{c_{n}}=\frac{e^{-\sigma_{2}cos(\mathbf{w}_{c_{n}}^{b\kappa_{1}},\mathbf{w}_{g}^{(b-\kappa_{2})\kappa_{1}})}}{\sum_{n=1}^{N}e^{-\sigma_{2}cos(\mathbf{w}_{c_{n}}^{b\kappa_{1}},\mathbf{w}_{g}^{(b-\kappa_{2})\kappa_{1}})}}$
(23)
where $\beta_{c_{n}}$ is the attention coefficient, $\sigma_{2}$ is a scalar
hyper-parameter.
After cloud update, the cloud server will send latest global model to all edge
APs, and all edge servers will distribute the latest shared model to its
connecting users for a new round of local computing.
### IV-C Convergence Analysis
By invoking similar approaches in Section III, we can analyze the convergence
rate of the proposed training scheme.
###### $\mathbf{Theorem}$ 2
For the employed FL system, if the learning rate is chosen as
$\eta\leq\frac{1}{\sqrt{12}L\kappa_{1}\kappa_{2}}$, then after $T$ rounds of
global aggregations, Algorithm 2 can achieve an $\epsilon_{MACFL}$-suboptimal
solution, where $\epsilon_{MACFL}$ is bounded as follows
$\displaystyle\theta_{MACFL}$ $\displaystyle\leq\frac{2}{\eta
T}\big{[}\mathbb{E}\tilde{f}(\mathbf{\bar{w}}^{0}_{g})-\tilde{f}_{\inf}\big{]}+\eta
L\sigma_{M}^{2}\sum_{m=1}^{M}\beta_{u_{m}}^{2}$
$\displaystyle+\frac{12L^{2}\eta^{2}\kappa_{1}^{2}\kappa_{2}^{2}}{1-12\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}}\epsilon_{M,g}^{2}+\frac{12L^{2}\eta^{2}\kappa_{1}^{2}(1-8\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2})}{(1-12\eta^{2}L^{2}\kappa_{1}^{2})(1-12\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2})}\epsilon_{M,c}^{2}$
$\displaystyle+\frac{4L^{2}\eta^{2}\kappa_{1}}{1-12\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}}\sigma_{M}^{2}\sum_{m=1}^{M}\beta_{u_{m}}\bigg{(}\frac{1-8\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}}{1-12\eta^{2}L^{2}\kappa_{1}^{2}}(1-\beta_{u_{m}}^{c})+2\kappa_{2}(\beta_{u_{m}}^{c}-\beta_{u_{m}})\bigg{)}.$
(24)
###### Proof:
Please refer to Appendix -D. ∎
## V Performance Evaluation
In this section, we conduct experimental evaluation of the cluster FL
algorithm on heterogeneous models and dataset configurations to verify the
efficacy of our analysis and proposed scheme.
### V-A Experimental Settings
In our experiments, we consider a FL system consisting of 50 users and 5
clusters formed by the APs and one cloud server, and the 5 clusters constitute
a linear graph as shown in Fig. 2. All users are uniformly and randomly
assigned to 5 clusters at the initial time. We consider an image
classification learning task based on the standard MNIST [31] dataset, which
consists of 60,000 images for training and 10,000 images for testing and
contains 10 different hand-written digits. At the user side, we consider the
Convolutional Neural Network (CNN) models with cross-entropy loss functions
for local training. The mini-batch size at each SGD step is set as 10, and the
learning rate is set as 0.001. We set the total iterations to be 2000.
Because data heterogeneity is critical in FL, we employ two configurations of
data distributions: IID and non-IID dataset cases. The size of local dataset
owned by each user is set as $|\mathcal{D}_{u_{m}}|=600$ for both IID and non-
IID dataset cases. As for the IID dataset case, we assume the training dataset
of each user is an IID sample of the global training dataset. As for the non-
IID dataset case, we consider the pathological non-IID setting in [1], in
which each user only has at most two categories of digits.
### V-B Impact of User Mobility on Conventional HFL
We first carry out experiments to assess the convergence rate of the HFL
algorithm with mobile users to illustrate the impact of user mobility with
different system parameters. Prior to giving out the results, we would like to
note that on the one hand, user mobility leads to a decreasing number of
participants at each model aggregation round, which may result in missing
training data and under-fitting training model. On the other hand, mobility of
the users also has the potential to shuffle the data, which helps reducing the
divergences among clusters and construct a representative sample at each SGD
step of global model.
#### V-B1 Impact of the Frequency of Aggregation
Figure 3: Evaluation of test accuracy performance with different
$\kappa_{1},\kappa_{2}$, when $p_{s}=0.5,M=50$ and $N=5$.
Fig. 3 demonstrates the impact of parameter aggregation intervals,
$\kappa_{1}$ and $\kappa_{2}$, on the accuracy of the trained model. It is
worthwhile to note that $\kappa_{1}$ and $\kappa_{2}$ characterize the
frequencies of edge aggregation and global aggregation, respectively, and
there are $\kappa_{2}$ times of communication rounds between users and edge
servers and one communication round between cloud and edge servers during one
cloud update procedure. From Fig. 3 we can see that the convergence rates
under IID and non-IID dataset can be boosted up by reducing $\kappa_{1}$ or
$\kappa_{2}$, i.e., increasing the frequency of communications between either
the users and edge APs or the edge APs and cloud server. Additionally, if we
fix the product of $\kappa_{1}$ and $\kappa_{2}$, the convergence rate is
accelerated with decreasing $\kappa_{1}$, which suggests that allowing more
frequent edge aggregations helps to enhance learning performance. These
observations verify the convergence analysis given in Theorem 1 and accord
with related multi-layer FL settings [20] and [21]. This result poses a
dissent on the conclusions drawn from single-cell FL [1], where the test
accuracy can be improved by increasing $\kappa_{1}$. The main reason
attributes to the fact that increasing $\kappa_{1}$ will enlarge the
divergences among local models and increasing $\kappa_{2}$ will also aggravate
the weight difference among cluster models. Similar to [21], our results show
that high frequency of local average, instead of large number of local update
iterations, is beneficial to the learning performance in HFL. Moreover, it can
be seen that the learning performance of IID dataset case is better than that
of non-IID dataset case, which also coincides with the conclusions indicated
by Theorem 1. This is because the value of the divergences of local and
cluster loss functions and the variance of SGD of IID dataset case is smaller
than those of non-IID dataset case.
#### V-B2 Impact of Staying Probability
Figure 4: Evaluation of test accuracy performance with different $p_{s}$ when
$\kappa_{1}=20,\kappa_{2}=1,M=50$ and $N=5$.
In Fig. 4, we illustrate the impact of user staying probability $p_{s}$ on the
convergence rate of HFL algorithm. From this figure, we first notice that the
convergence rate changes slightly for IID dataset case with different
$p_{s}>0$, which implies that the user mobility has a mild effect on the
learning performance as long as a portion of the mobile users can participate
in the model aggregation. This is because the test accuracy is insensitive to
the number of participants for the ideal FL with IID dataset in both single-
cell [1] and multi-cell [19] networks as long as the mini-batch sample is a
representative IID sample of the entire dataset. Therefore, the test accuracy
will not deteriorate severely by reducing $p_{s}$ as long as $p_{s}\neq 0$.
For non-IID dataset case, we observe that increasing $p_{s}$ can bring along a
marked gain in the learning performance. This comes from the fact that when
dataset is non-IID, it is more likely that a large number of participants
contribute a representative sample at each model aggregation step [1, 19].
Therefore, increasing $p_{s}$ retains more effective participants in each
cluster, which can enhance the learning performance with non-IID datasets.
Moreover, this staying probability also has a relation to the impact of
$\kappa_{1}$ in Fig. 3 in real life. Particularly, a small $\kappa_{1}$ means
a shorter duration of local update, and thus the staying probability during
local update becomes larger, which helps to speed up HFL for non-IID dataset
case. Additionally, increasing $p_{s}>0$ also helps to reduce the vibrations
of the convergence curves, especially for non-IID dataset case. Two special
scenarios are also noteworthy: When $p_{s}=0$, there are no updated results
collected successfully by edge servers due to high mobility, and the test
accuracy merely depends on the initialization model for both IID and non-IID
dataset cases. When $p_{s}=1$, it turns to the conventional HFL with static
users. And the test accuracy with $p_{s}=1$ for IID dataset case is higher
comparing with non-IID dataset case, which is reasonable.
#### V-B3 Impact of the Number of Users
Figure 5: Evaluation of test accuracy performance with different $M$ when
$p_{s}=0.5,\kappa_{1}=20,\kappa_{2}=1$ and $N=5$.
Fig. 5 depicts the convergence rate of HFL under different user number $M$. It
can be seen that the convergence rate increases with respect to $M$ for both
IID and non-IID dataset cases, which is in accord with convergence analysis in
conventional FL [20] and [29]. Additionally, increasing $M$ also helps to
reduce the vibrations of the convergence curves, especially when the number of
total global aggregations is small. It suggests that increasing the number of
participating users benefits the learning performance when $T$ is small for
IID dataset. This result confirms our analysis given in Theorem 1, namely,
updates via SGD from a larger number of users will reduce the variance of SGD
at each iteration, which is also in line with [27].
### V-C Comparison between HFL and Our Proposed MACFL Algorithm
We now turn our attention to the performance of the proposed MACFL. In this
part, we perform several comparisons between our proposed MACFL and the
conventional HFL algorithm. For a fair comparison, we adopt the same system
and common hyperparameters, namely $\eta$, $\kappa_{1}$, $\kappa_{2}$, and $M$
for both algorithms. We also use the same initial state for both two
algorithms. For the hyperparameters only used in MACFL algorithm, namely,
$\sigma_{1}$, $\sigma_{2}$, and $\rho$, we conduct a grid search to figure out
the combination of fine-tuned parameters via learning from the experiences of
Per-FedAvg [14] and FedAMP [16]. And in our simulation we set
$\sigma_{1}=\sigma_{2}=25$ and $\rho=0.001$ for both IID and non-IID datasets.
Figure 6: Comparison of the evaluation of test accuracy performance between
HFL and our proposed MACFL algorithm with different data distributions and
system parameters $\kappa_{1},\kappa_{2}$ when $p_{s}=0.5,M=50$ and $N=5$.
Fig. 6 plots the convergence rates of MACFL and HFL algorithm under different
values of $\kappa_{1}$ and $\kappa_{2}$. Firstly, the convergence rate of our
proposed MACFL algorithm can be enhanced with increasing $\kappa_{1}$ and
$\kappa_{2}$ for both IID and non-IID data distributions, which are in
consistence with Theorem 2. Secondly, our proposed MACFL algorithm attains
faster convergence rate than the HFL algorithm in all cases. Particularly, for
the IID dataset case, the test accuracy can be improved from $87.75\%$ to
$89.10\%$ with $\kappa_{1}=20,\kappa_{2}=2$ by using our proposed scheme, and
from $92.96\%$ to $94.01\%$ with $\kappa_{1}=20,\kappa_{2}=1$, and from
$93.32\%$ to $94.57\%$ with $\kappa_{1}=10,\kappa_{2}=1$. As for the non-IID
datasetcase, the performance gains are more pronounced, whereas the test
accuracy can be improved from $58.92\%$ to $66.07\%$ with
$\kappa_{1}=20,\kappa_{2}=2$ with MACFL algorithm, and from $77.95\%$ to
$86.01\%$ with $\kappa_{1}=20,\kappa_{2}=1$, and from $85.35\%$ to $91.97\%$
with $\kappa_{1}=10,\kappa_{2}=1$. Moreover, the fluctuations in the
convergence curve can also be largely reduced, especially for non-IID dataset
case. Such an improvement in the learning performance is ascribed to the
sophisticated weighted average scheme and more effective model update and
aggregation rules employed in our proposed MACFL algorithm.
Figure 7: Comparison of the evaluation of test accuracy performance between
HFL and our proposed MACFL algorithm with different data distributions and
system parameters $p_{s}$ when $\kappa_{1}=20,\kappa_{2}=1,M=50$ and $N=5$ .
In Fig. 7, we compare the performance of our proposed MACFL and conventional
HFL algorithm under various staying probability $p_{s}$ of mobile users. The
first noteworthy observation is that different from HFL algorithm, the test
accuracy of our proposed MACFL algorithm is not sensitive to the variants of
$p_{s}$ for both IID and non-IID data distributions. This phenomenon can be
explained by our proposed model aggregation rule, in which the stay
probability $p_{s}$ does not affect the number of effective participants.
Secondly, it shows that our proposed algorithm outperforms the conventional
HFL algorithm with mobile users for different $p_{s}$ settings, especially for
$p_{s}=0$, where the test accuracy can be improved from $11.37\%$ to $93.86\%$
for IID dataset case by using our proposed scheme, and from $11.37\%$ to
$80.85\%$ for non-IID dataset case. Unlike HFL algorithm with $p_{s}=0$, in
which the global model never get updated due to the extremely high mobility of
users, our proposed model aggregation rule allows all the mobile user to
upload their computed results at each communication round even though no user
is connected with the same tagged edge AP during local training phase.
Moreover, our algorithm also helps to reduce the vibrations of convergence
curves, especially for non-IID dataset case, owing to attentive weighted
scheme in the proposed algorithm.
Figure 8: Comparison of the evaluation of test accuracy performance between
HFL and our proposed MACFL algorithm with different data distributions and
system parameters $M$ when $p_{s}=0.5,\kappa_{1}=20,\kappa_{2}=1$ and $N=5$.
In Fig. 8, the test accuracy performance of our proposed MACFL algorithm under
different $M$ is evaluated, and conventional HFL with mobile users is chosen
as a benchmark. Firstly, similar to HFL algorithm, we note that the test
accuracy of our proposed MACFL algorithm can be improved by increasing $M$ for
both IID and non-IID data distributions. This can be explained by the same
reasons for HFL algorithm, and the results prove the correctness of our
convergence analysis given in Theorem 2. Secondly, it shows that our proposed
algorithm outperforms the conventional HFL algorithm with mobile users,
especially for small value of $M$. In Fig. 8, it shows that for the IID
dataset case, the test accuracy can be improved from $65.83\%$ to $71.31\%$
with $M=5$ with our proposed scheme, and from $93.06\%$ to $94.01\%$ with
$M=50$, and from $95.47\%$ to $95.93\%$ with $M=100$. As for non-IID dataset
case, the improvements are more pronounced. In particular, as Fig. 8 shows,
the test accuracy can be improved from $45.89\%$ to $56.85\%$ with $M=5$ by
using our proposed averaging scheme, and from$77.95\%$ to $86.01\%$ with
$M=50$, and from $85.32\%$ to $93.60\%$ with $M=100$. Such enhancements mainly
owes to our proposed model update and aggregation rules, which can help each
user to learn a more accurate personalized local models and help both edge
servers and cloud server to construct more precise shared models, and it also
unveils the importance of a well designed averaging scheme in FL system.
## VI Conclusion
In this paper, we studied the performance of FL in the context of a
hierarchical wireless network consisted of one cloud server, multiple edge
APs, and a large number of mobile users. We derived analytical expressions for
the convergence rate of FL in the conventional setup, by accounting for
several key factors of the system, including the heterogeneity arises from
both the dataset and network architecture, as well as the user mobility. The
analysis revealed that increasing the communication frequency amongst the
users and their connected APs can accelerate the convergence rate. In
contrast, an increase in user mobility leads to a dropout of participants and
decreases the convergence rate. Furthermore, by exploiting the correlations
amongst the parameters of mobile users and edge APs, we proposed a mobility-
aware scheme to aggregate the users’ parameters. The efficacy of the proposed
approach is corroborated via extensive experiments, whereas the gain over the
conventional training method is particularly pronounced when mobile users
possess non-IID datasets. Future extensions of this work can focus on the more
delicate design of HFL with mobile users, such as personalized local model
updates, data-driven model aggregation schemes, etc.
### -A Proof of Lemma 1
In this section, we use $g(\mathbf{w}_{u_{m}}^{t})$,
$F(\mathbf{w}_{u_{m}}^{t})$, $f_{c_{i}}(\mathbf{w}_{c_{i}}^{t})$ and
$f(\mathbf{w}_{g}^{t})$ as the short version of mini-batch gradients, local,
cluster and global loss functions in this section. Similar to [20, 21], we use
averaged global model parameters to analysis even though the they are not
observed for each iteration, and the virtual global model parameters can be
written as
$\mathbf{\bar{w}}_{g}^{t+1}=\mathbf{\bar{w}}_{g}^{t}-\eta\sum_{m=1}^{M}\alpha_{u_{m}}g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}$
to analysis, then we have
$\displaystyle\mathbb{E}f(\mathbf{\bar{w}}^{t+1}_{g})$
$\displaystyle=\mathbb{E}f\left(\mathbf{\bar{w}}^{t}_{g}-\sum_{m=1}^{M}\alpha_{u_{m}}g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}\right)$
$\displaystyle\overset{a}{\leq}\mathbb{E}f(\mathbf{\bar{w}}^{t}_{g})+\frac{\eta^{2}L}{2}\mathbb{E}\|\sum_{m=1}^{M}\alpha_{u_{m}}g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}\|^{2}-\eta\mathbb{E}<\nabla
f(\mathbf{\bar{w}}^{t}),\sum_{m=1}^{M}g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}>$
(25)
where step (a) holds because of smoothness of the loss functions. For the
second term of in (-A), we have
$\displaystyle\mathbb{E}\|\sum_{m=1}^{M}\alpha_{u_{m}}g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}\|^{2}\overset{a}{=}$
$\displaystyle\mathbb{E}\|\sum_{m=1}^{M}\alpha_{u_{m}}\left(g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}-p_{s}\nabla
F(\mathbf{w}_{u_{m}}^{t})\right)\|^{2}+\mathbb{E}\|p_{s}\sum_{m=1}^{M}\
\alpha_{u_{m}}\nabla F(\mathbf{w}_{u_{m}}^{t})\|^{2}$
$\displaystyle\overset{b}{\leq}$
$\displaystyle\mathbb{E}\|\sum_{m=1}^{M}\alpha_{u_{m}}\left(g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}-p_{s}\nabla
F(\mathbf{w}_{u_{m}}^{t})\right)\|^{2}+p_{s}^{2}\sum_{m=1}^{M}\
\alpha_{u_{m}}\mathbb{E}\|\nabla F(\mathbf{w}_{u_{m}}^{t})\|^{2}$ (26)
where the step (a) in (-A) holds because
$\mathbb{E}\|\mathbf{x}\|^{2}=\mathbb{E}\|\mathbf{x}-\mathbb{E}\mathbf{x}\|^{2}-(\mathbb{E}\mathbf{x})^{2}$,
step (b) in (-A) holds because
$\|\sum_{i=1}^{N}a_{i}x_{i}\|^{2}\leq\sum_{i=1}^{N}a_{i}\|x_{i}\|^{2}$, with
$0\leq a_{i}\leq 1,\sum_{i=1}^{N}a_{i}=1$. For the first term in inequality
(-A), we have
$\displaystyle\mathbb{E}\|\sum_{m=1}^{M}\alpha_{u_{m}}\left(g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}-p_{s}\nabla
F(\mathbf{w}_{u_{m}}^{t})\right)\|^{2}$ $\displaystyle=$
$\displaystyle\mathbb{E}\bigg{\|}\sum_{m=1}^{M}\alpha_{u_{m}}\bigg{(}g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}-\nabla
F(\mathbf{w}_{u_{m}}^{t})+(1-p_{s})\nabla
F(\mathbf{w}_{u_{m}}^{t})\bigg{)}\bigg{\|}^{2}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}^{2}\mathbb{E}\bigg{\|}g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}-\nabla
F(\mathbf{w}_{u_{m}}^{t})+(1-p_{s})\nabla
F(\mathbf{w}_{u_{m}}^{t})\bigg{\|}^{2}+\sum_{m=1}^{M}\sum_{j=1,j\neq
m}^{M}\alpha_{u_{m}}\alpha_{u_{j}}\mathbb{E}\bigg{\\{}<g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}$
$\displaystyle-\nabla F(\mathbf{w}_{u_{m}}^{t})+(1-p_{s})\nabla
F(\mathbf{w}_{u_{m}}^{t}),g(\mathbf{w}_{u_{j}}^{t})\mathbb{I}_{u_{j}}^{t}-\nabla
F(\mathbf{w}_{u_{j}}^{t})+(1-p_{s})\nabla
F(\mathbf{w}_{u_{j}}^{t})>\bigg{\\}}$ (27)
For the first term in inequality (-A), we have
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}^{2}\mathbb{E}\bigg{\|}g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}-\nabla
F(\mathbf{w}_{u_{m}}^{t})+(1-p_{s})\nabla
F(\mathbf{w}_{u_{m}}^{t})\bigg{\|}^{2}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}^{2}\bigg{(}\mathbb{E}\|g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}-\nabla
F(\mathbf{w}_{u_{m}}^{t})\|^{2}+(1-p_{s})^{2}\mathbb{E}\|\nabla
F(\mathbf{w}_{u_{m}}^{t})\|^{2}+\mathbb{E}<(1-p_{s})\nabla
F(\mathbf{w}_{u_{m}}^{t}),$ $\displaystyle
g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}-\nabla
F(\mathbf{w}_{u_{m}}^{t})>+\mathbb{E}<g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}-\nabla
F(\mathbf{w}_{u_{m}}^{t}),(1-p_{s})\nabla F(\mathbf{w}_{u_{m}}^{t})>\bigg{)}$
$\displaystyle\overset{a}{=}$
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}^{2}\mathbb{E}\|g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}-\nabla
F(\mathbf{w}_{u_{m}}^{t})\|^{2}-\sum_{m=1}^{M}\alpha_{u_{m}}^{2}(1-p_{s})^{2}\mathbb{E}\|\nabla
F(\mathbf{w}_{u_{m}}^{t})\|^{2}$ $\displaystyle\leq$
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}^{2}\bigg{(}p_{s}\sigma^{2}+p_{s}(1-p_{s})\mathbb{E}\|\nabla
F(\mathbf{w}_{u_{m}}^{t})\|^{2}\bigg{)},$ (28)
where step (a) holds because the mini-batch gradients are assumed to be
unbiased with bounded variance, and the independence between the mobility
variable and the mini-batch sampling, we have
$\mathbb{E}\\{g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}\\}=p_{s}\nabla
F(\mathbf{w}_{u_{m}}^{t})$. For the second term in inequality (-A), we have
$\sum_{m=1}^{M}\sum_{j=1,j\neq
m}^{M}\alpha_{u_{m}}\alpha_{u_{j}}\mathbb{E}\big{\\{}<g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}-\nabla
F(\mathbf{w}_{u_{m}}^{t})+(1-p_{s})\nabla
F(\mathbf{w}_{u_{m}}^{t}),g(\mathbf{w}_{u_{j}}^{t})\mathbb{I}_{u_{j}}^{t}-\nabla
F(\mathbf{w}_{u_{j}}^{t})$ $+(1-p_{s})\nabla
F(\mathbf{w}_{u_{j}}^{t})>\big{\\}}=0$ because of the unbiasedness of mini-
batch gradients and the independence between user mobility and random
sampling.
Plugging $\eqref{eq:ft31}$ back into inequality (-A), we have
$\displaystyle\mathbb{E}\|\sum_{m=1}^{M}\alpha_{u_{m}}g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}\|^{2}\leq
p_{s}\sigma^{2}\sum_{m=1}^{M}\alpha_{u_{m}}^{2}+\sum_{m=1}^{M}\alpha_{u_{m}}p_{s}\left(p_{s}+\alpha_{u_{m}}(1-p_{s})\right)\mathbb{E}\|\nabla
F(\mathbf{w}_{u_{m}}^{t})\|^{2}$ (29)
For the third term in (-A), we have
$\displaystyle-\eta\mathbb{E}<\nabla
f(\mathbf{\bar{w}}_{g}^{t}),\sum_{m=1}^{M}\alpha_{u_{m}}g_{u_{m}}(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}>$
$\displaystyle\overset{a}{=}$ $\displaystyle-\eta
p_{s}\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}<\nabla
f(\mathbf{\bar{w}}_{g}^{t}),\nabla F(\mathbf{w}_{u_{m}}^{t})>$
$\displaystyle\overset{b}{=}$ $\displaystyle-\frac{\eta
p_{s}}{2}\sum_{m=1}^{M}\alpha_{u_{m}}\bigg{(}\mathbb{E}\|\nabla
f(\mathbf{\bar{w}}^{t}_{g})\|^{2}+\mathbb{E}\|\nabla
F(\mathbf{w}_{u_{m}}^{t})\|^{2}-\mathbb{E}\|\nabla
f(\mathbf{\bar{w}}^{t}_{g})-\nabla F(\mathbf{w}_{u_{m}}^{t})\|^{2}\bigg{)}$
(30)
where a conditional expectation is made in step (a), and step (b) holds
because
$<\mathbf{x},\mathbf{y}>=\frac{1}{2}(\|\mathbf{x}-\mathbf{y}\|^{2}-\|\mathbf{x}\|^{2}-\|\mathbf{y}\|^{2})$.
For the last term in (-A), we have
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\|\nabla
f(\mathbf{\bar{w}}^{t}_{g})-\nabla F(\mathbf{w}_{u_{m}}^{t})\|^{2}$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\bigg{\|}\nabla
f(\mathbf{\bar{w}}^{t}_{g})\pm\nabla
f(\mathbf{\bar{w}}^{t}_{c_{u_{m}}})\pm\nabla
f_{c_{u_{m}}}(\mathbf{\bar{w}}^{t}_{c_{u_{m}}})\pm\nabla
F(\mathbf{\bar{w}}^{t}_{c_{u_{m}}})-\nabla
F(\mathbf{w}_{u_{m}}^{t})\bigg{\|}^{2}$ $\displaystyle\overset{a}{\leq}$
$\displaystyle
4L^{2}\sum_{i=1}^{N}\alpha_{c_{i}}\mathbb{E}\|\mathbf{\bar{w}}^{t}_{g}-\mathbf{\bar{w}}_{c_{i}}^{t}\|^{2}+4\epsilon_{g}^{2}+4L^{2}\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\|\mathbf{\bar{w}}_{c_{u_{m}}}^{t}-\mathbf{w}_{u_{m}}^{t}\|^{2}+4\epsilon_{c}^{2}$
(31)
where $\mathbf{\bar{w}}_{c_{u_{m}}}^{t}$ denotes the averaged cluster model
owned by the edge AP which the $u_{m}$ located in at $t$-th iteration, step
(a) holds because of the assumptions of smoothness and bounded divergences
among the local, cluster and global loss functions.
Plugging (29), (-A) and (-A) back into inequality (-A), and rearranging the
order, dividing both side by $\frac{\eta p_{s}}{2}$ and taking the average
over time, we have
$\displaystyle\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}\|\nabla
f(\mathbf{\bar{w}}^{t}_{g})\|^{2}$ $\displaystyle\leq\frac{2}{\eta
p_{s}T}(\mathbb{E}f(\mathbf{\bar{w}}^{0}_{g})-f_{\inf})+\eta
L\sigma^{2}\sum_{m=1}^{M}\alpha_{u_{m}}^{2}+4\epsilon_{g}^{2}+4\epsilon_{c}^{2}$
$\displaystyle+\frac{4L^{2}}{T}\sum_{t=0}^{T-1}\sum_{i=1}^{N}\alpha_{c_{i}}\mathbb{E}\|\mathbf{\bar{w}}^{t}_{g}-\mathbf{\bar{w}}_{c_{i}}^{t}\|^{2}+\frac{4L^{2}}{T}\sum_{t=0}^{T-1}\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\|\mathbf{\bar{w}}_{c_{u_{m}}}^{t}-\mathbf{w}_{u_{m}}^{t}\|^{2}$
$\displaystyle+\frac{1}{T}\sum_{t=0}^{T-1}\sum_{m=1}^{M}\alpha_{u_{m}}\left(\eta
L(p_{s}+\alpha_{u_{m}}(1-p_{s}))-1\right)\mathbb{E}\|\nabla
F(\mathbf{w}_{u_{m}}^{t})\|^{2}$ (32)
The last term in (-A) characterizes the efficiency of gradient descent with
specific loss functions, if the learning rate is small enough, the last term
is less than zero. Specifically, if $\eta\leq\frac{1}{L}$, we have $\eta
L(p_{s}+\alpha_{u_{m}}(1-p_{s}))\leq 1$ since $0\leq p_{s}\leq 1$ and
$0\leq\alpha_{u_{m}}\leq 1$. By far, we conclude the proof of Lemma 1.
### -B Proof of Lemma 2
For the MSE of the local and cluster model parameters, if $t\mod\kappa_{1}=0$,
we have $\mathbf{w}_{u_{m}}^{t}=\mathbf{w}_{c_{u_{m}}}^{t}$ , otherwise, let
$b=\lfloor\frac{t}{\kappa_{1}}\rfloor$, we have
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\|\mathbf{w}^{t}_{u_{m}}-\mathbf{\bar{w}}_{c_{u_{m}}}^{t}\|^{2}$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\bigg{\|}(\mathbf{w}^{b\kappa_{1}}_{u_{m}}-\eta\sum_{\tau=b\kappa_{1}}^{t-1}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-(\mathbf{\bar{w}}_{c_{u_{m}}}^{b\kappa_{1}}-\eta\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\sum_{\tau=b\kappa_{1}}^{t-1}\alpha_{u_{m}}^{c}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}})\bigg{\|}^{2}$
$\displaystyle=$
$\displaystyle\eta^{2}\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\bigg{\|}\sum_{\tau=b\kappa_{1}}^{t-1}\bigg{(}\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}\bigg{)}\bigg{\|}^{2}$
$\displaystyle=$
$\displaystyle\eta^{2}\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\bigg{\|}\sum_{\tau=b\kappa_{1}}^{t-1}\bigg{(}\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}\pm
p_{s}\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\pm p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})-g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}\bigg{)}\bigg{\|}^{2}$
$\displaystyle\leq$ $\displaystyle
2\eta^{2}\sum_{m=1}^{M}\alpha_{u_{m}}\bigg{\\{}\mathbb{E}\bigg{\|}\sum_{\tau=b\kappa_{1}}^{t-1}\bigg{[}\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}\bigg{(}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\bigg{)}-\bigg{(}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\bigg{)}\bigg{]}\bigg{\|}^{2}$
$\displaystyle+p_{s}^{2}\mathbb{E}\bigg{\|}\sum_{\tau=b\kappa_{1}}^{t-1}\bigg{(}\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})-\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\bigg{)}\bigg{\|}^{2}\bigg{\\}}$ (33)
For the first term in (-B), we have
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}\bigg{\\{}\mathbb{E}\bigg{\|}\sum_{\tau=b\kappa_{1}}^{t-1}\bigg{[}\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}\bigg{(}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\bigg{)}-\bigg{(}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\bigg{)}\bigg{]}\bigg{\|}^{2}$
$\displaystyle\overset{a}{=}$
$\displaystyle\sum_{n=1}^{N}\sum_{u_{m}\in\mathcal{C}_{n}}\alpha_{c_{n}}\alpha_{u_{m}}^{c}\bigg{[}\mathbb{E}\bigg{\|}\sum_{\tau=b\kappa_{1}}^{t-1}\bigg{(}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})-\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}(g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}}))\bigg{)}\bigg{\|}^{2}\bigg{]}$
$\displaystyle\overset{b}{=}$
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\big{\|}\sum_{\tau=b\kappa_{1}}^{t-1}\big{(}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\big{)}\big{\|}^{2}-\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\big{\|}\sum_{\tau=b\kappa_{1}}^{t-1}\sum_{u_{m}\in\mathcal{C}_{n}}\alpha_{u_{m}}^{c}\big{(}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\big{)}\big{\|}^{2}$ $\displaystyle\overset{c}{=}$
$\displaystyle\sum_{m=1}^{M}\sum_{\tau=b\kappa_{1}}^{t-1}\alpha_{u_{m}}\bigg{(}\mathbb{E}\bigg{\|}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\bigg{\|}^{2}-\alpha_{u_{m}}^{c}\mathbb{E}\bigg{\|}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\bigg{\|}^{2}\bigg{)}$ $\displaystyle\leq$
$\displaystyle\kappa_{1}\sum_{m=1}^{M}\alpha_{u_{m}}(1-\alpha_{u_{m}}^{c})\mathbb{E}\bigg{\|}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\bigg{\|}^{2}$ (34)
where step (a) holds because
$\alpha_{u_{m}}=\alpha_{u_{m}}^{c}\alpha_{c_{u_{m}}}$, step (b) holds because
$\sum_{i=1}^{N}a_{i}\|x_{i}-\sum_{i=1}^{N}a_{i}x_{i}\|^{2}=\sum_{i=1}^{N}a_{i}\|x_{i}\|^{2}-(\sum_{i=1}^{N}a_{i}x_{i})^{2}$
with $\sum_{i=1}^{N}a_{i}=1,0\leq a_{i}\leq 1$, step (c) holds because
$\|\sum_{i=1}^{N}x_{i}-\mathbb{E}\\{x_{i}\\}\|^{2}=\sum_{i=1}^{N}\|x_{i}-\mathbb{E}\\{x_{i}\\}\|^{2}$
with the independence among different $x_{i}$. And then we have
$4\kappa_{1}\sum_{m=1}^{M}\alpha_{u_{m}}(1-\alpha_{u_{m}}^{c})\mathbb{E}\bigg{\|}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\bigg{\|}^{2}\leq\kappa_{1}p_{s}(\sigma^{2}+(1-p_{s})G^{2})\sum_{m=1}^{M}\alpha_{u_{m}}(1-\alpha_{u_{m}}^{c})$
because
$\mathbb{E}\|x_{i}-\mathbb{E}\\{x_{i}\\}\|^{2}=\mathbb{E}\|x_{i}\|^{2}-(\mathbb{E}\\{x_{i}\\})^{2}$.
For the second term in (-B), we have
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\|\sum_{\tau=b\kappa_{1}}^{t-1}(\nabla
F(\mathbf{w}^{\tau}_{u_{m}})-\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}\nabla
F(\mathbf{w}^{\tau}_{u_{m}}))\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{M}\alpha_{u_{m}}\mathbb{E}\bigg{\|}\sum_{\tau=b\kappa_{1}}^{t-1}\bigg{(}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\pm\nabla
F(\mathbf{\bar{w}}^{\tau}_{c_{u_{m}}})\pm\nabla
f_{c_{u_{m}}}(\mathbf{\bar{w}}^{\tau}_{c_{u_{m}}})-\sum_{u_{j}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}\nabla
F(\mathbf{w}^{\tau}_{u_{j}})\bigg{)}\bigg{\|}^{2}$ $\displaystyle\leq$
$\displaystyle
3\sum_{m=1}^{M}\alpha_{u_{m}}\bigg{(}\mathbb{E}\bigg{\|}\sum_{\tau=b\kappa_{1}}^{t-1}\bigg{(}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})-\nabla
F(\mathbf{\bar{w}}^{\tau}_{c_{u_{m}}})\bigg{)}\bigg{\|}^{2}+\mathbb{E}\bigg{\|}\sum_{\tau=b\kappa_{1}}^{t-1}\bigg{(}\nabla
F(\mathbf{\bar{w}}^{\tau}_{c_{u_{m}}})-\nabla
f_{c_{u_{m}}}(\mathbf{\bar{w}}^{\tau}_{c_{u_{m}}})\bigg{)}\bigg{\|}^{2}$
$\displaystyle+\mathbb{E}\bigg{\|}\sum_{\tau=b\kappa_{1}}^{t-1}(\nabla
f_{c_{u_{m}}}(\mathbf{\bar{w}}^{\tau}_{c_{u_{m}}})-\sum_{u_{j}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}\nabla
F(\mathbf{w}^{\tau}_{u_{j}}))\bigg{\|}^{2}\bigg{)}$ $\displaystyle\leq$
$\displaystyle
6L^{2}\kappa_{1}\sum_{m=1}^{M}\sum_{\tau=b\kappa_{1}}^{t-1}\alpha_{u_{m}}\mathbb{E}\|\mathbf{w}^{\tau}_{u_{m}}-\mathbf{\bar{w}}^{\tau}_{c_{u_{m}}}\|^{2}+3\kappa_{1}^{2}\epsilon_{c}^{2}$
(35)
Plugging (-B) and (-B) back into (-B), and taking average of time, if
$\eta<\frac{1}{\sqrt{12}L\kappa_{1}}$ , we can conclude the proof of Lemma 2.
### -C Proof of Lemma 3
Similar to [20] [21], we use averaged cluster model parameters to analysis
even though the they are not observed for each iteration, and the virtual
cluster model parameters can be written as
$\mathbf{\bar{w}}_{c_{n}}^{t+1}=\mathbf{\bar{w}}_{c_{n}}^{t}-\eta\sum_{u_{m}\in\mathcal{C}_{n}^{t}}\alpha_{u_{m}}^{c}g(\mathbf{w}_{u_{m}}^{t})\mathbb{I}_{u_{m}}^{t}$
to analysis. We use $c(u_{m})$ to indicate the index of edge AP that is
connected by mobile user $u_{m}$ in the rest of this paper. For the MSE of the
cluster and global model parameters, if $t\mod\kappa_{1}\kappa_{2}=0$, we have
$\mathbf{w}_{c_{n}}^{t}=\mathbf{w}_{g}^{t}$ if $t\mod\kappa_{1}\kappa_{2}=0$,
otherwise, let $b_{1}=\lfloor\frac{t}{\kappa_{1}\kappa_{2}}\rfloor$. Similar
to Lemma 2, we have
$\displaystyle\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\|\mathbf{\bar{w}}^{t}_{g}-\mathbf{\bar{w}}_{c_{n}}^{t}\|^{2}=\eta^{2}\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\sum_{m=1}^{M}\alpha_{u_{m}}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}\pm
p_{s}\nabla f_{c_{u_{m}}}(\mathbf{\bar{w}}^{\tau}_{c(u_{m})})$
$\displaystyle\pm p_{s}\sum_{j=1}^{N}\alpha_{c_{j}}\nabla
f_{c_{j}}(\mathbf{\bar{w}}^{\tau}_{c_{j}})-\sum_{u_{m}\in\mathcal{C}_{u_{m}}^{\tau}}\alpha_{u_{m}}^{c}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}\bigg{)}\bigg{\|}^{2}$
$\displaystyle\leq$ $\displaystyle
2\eta^{2}\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
f_{c_{u_{m}}}(\mathbf{\bar{w}}^{\tau}_{c(u_{m})})-(\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}$
$\displaystyle-p_{s}\sum_{j=1}^{N}\alpha_{c_{j}}\nabla
f_{c_{j}}(\mathbf{\bar{w}}^{\tau}_{c_{j}}))\bigg{)}\bigg{\|}^{2}+2\eta^{2}p_{s}^{2}\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\nabla
f_{c_{u_{m}}}(\mathbf{\bar{w}}^{\tau}_{c(u_{m})})-\sum_{j=1}^{N}\alpha_{c_{j}}\nabla
f_{c_{j}}(\mathbf{\bar{w}}^{\tau}_{c_{j}})\bigg{)}\bigg{\|}^{2}$
$\displaystyle\leq$ $\displaystyle
4\eta^{2}\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\sum_{u_{m}\in\mathcal{C}_{u_{m}}^{\tau}}\alpha_{u_{m}}^{c}(g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}}))-\sum_{m=1}^{M}\alpha_{u_{m}}(g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}$
$\displaystyle-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}}))\bigg{)}\bigg{\|}^{2}+4\eta^{2}p_{s}^{2}\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\sum_{u_{m}\in\mathcal{C}_{u_{m}}^{\tau}}\alpha_{u_{m}}^{c}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})-\nabla
f_{c_{u_{m}}}(\mathbf{\bar{w}}^{\tau}_{c_{u_{m}}})\bigg{)}\bigg{\|}^{2}$
$\displaystyle+2\eta^{2}p_{s}^{2}\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\nabla
f_{c_{u_{m}}}(\mathbf{\bar{w}}^{\tau}_{c(u_{m})})-\sum_{j=1}^{N}\alpha_{c_{j}}\nabla
f_{c_{j}}(\mathbf{\bar{w}}^{\tau}_{c_{j}})\bigg{)}\bigg{\|}^{2}$ (36)
For the first term in (-C), we have
$\displaystyle\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\sum_{u_{m}\in\mathcal{C}_{u_{m}}^{\tau}}\alpha_{u_{m}}^{c}(g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-\nabla
F(\mathbf{w}^{\tau}_{u_{m}}))-p_{s}\sum_{m=1}^{M}\alpha_{u_{m}}(g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}}))\bigg{)}\bigg{\|}^{2}$
$\displaystyle\overset{a}{=}$
$\displaystyle\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\sum_{u_{m}\in\mathcal{C}_{u_{m}}}\alpha_{u_{m}}^{c}(g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}}))\bigg{)}\bigg{\|}^{2}-\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\sum_{m=1}^{M}\alpha_{u_{m}}(g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}$
$\displaystyle-p_{s}\nabla F(\mathbf{w}^{\tau}_{u_{m}}))\bigg{)}\bigg{\|}^{2}$
$\displaystyle=$
$\displaystyle\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\sum_{m=1}^{M}\alpha_{u_{m}}\bigg{(}\alpha_{u_{m}}^{c}\mathbb{E}\|g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\|^{2}-\alpha_{u_{m}}\mathbb{E}\|g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-p_{s}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\|^{2}\bigg{)}$ $\displaystyle\overset{b}{=}$
$\displaystyle\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\sum_{m=1}^{M}\alpha_{u_{m}}(\alpha_{u_{m}}^{c}-\alpha_{u_{m}})\bigg{(}\mathbb{E}\bigg{\|}g(\mathbf{w}^{\tau}_{u_{m}})\mathbb{I}^{\tau}_{u_{m}}-\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\bigg{\|}^{2}-\mathbb{E}\bigg{\|}(p_{s}-1)\nabla
F(\mathbf{w}^{\tau}_{u_{m}})\bigg{\|}^{2}\bigg{)}$ $\displaystyle\leq$
$\displaystyle\kappa_{1}\kappa_{2}p_{s}(\sigma^{2}+(1-p_{s})G^{2})\sum_{m=1}^{M}\alpha_{u_{m}}(\alpha_{u_{m}}^{c}-\alpha_{u_{m}})$
(37)
where step (a) holds because
$\sum_{i=1}^{N}a_{i}\|x_{i}-\sum_{i=1}^{N}a_{i}x_{i}\|^{2}=\sum_{i=1}^{N}a_{i}\|x_{i}\|^{2}-(\sum_{i=1}^{N}a_{i}x_{i})^{2}$
with $\sum_{i=1}^{N}a_{i}=1,0\leq a_{i}\leq 1$, step (b) holds because
$\mathbb{E}\|x_{i}-\mathbb{E}\\{x_{i}\\}\|^{2}=\mathbb{E}\|x_{i}\|^{2}-(\mathbb{E}\\{x_{i}\\})^{2}$.
For the second term in (-C), we have
$\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\sum_{u_{m}\in\mathcal{C}_{n}}\alpha_{u_{m}}^{c}\nabla
F(\mathbf{w}^{\tau}_{u_{m}})-\nabla
f_{c_{n}}(\mathbf{\bar{w}}^{\tau}_{c_{n}})\bigg{)}\bigg{\|}^{2}\\\ \leq
4\eta^{2}p_{s}^{2}\kappa_{1}\kappa_{2}L^{2}\sum_{m=1}^{M}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\alpha_{u_{m}}\mathbb{E}\bigg{\|}\mathbf{w}^{\tau}_{u_{m}}-\mathbf{\bar{w}}^{\tau}_{c_{u_{m}}}\bigg{\|}^{2}$
because $\|\sum_{i=1}^{N}x_{i}\|^{2}\leq N\sum_{i=1}^{N}\|x_{i}\|^{2}$, and
$\|\sum_{i=1}^{N}a_{i}x_{i}\|^{2}\leq\sum_{i=1}^{N}a_{i}\|x_{i}\|^{2}$ with
$\sum_{i=1}^{N}a_{i}=1,0\leq a_{i}\leq 1$. For the third term in (-C), we have
$\displaystyle
2\eta^{2}p_{s}^{2}\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\nabla
f_{c_{u_{m}}}(\mathbf{\bar{w}}^{\tau}_{c(u_{m})})-\sum_{j=1}^{N}\alpha_{c_{j}}\nabla
f_{c_{j}}(\mathbf{\bar{w}}^{\tau}_{c_{j}})\bigg{)}\bigg{\|}^{2}$
$\displaystyle=$ $\displaystyle
2\eta^{2}p_{s}^{2}\sum_{n=1}^{N}\alpha_{c_{n}}\mathbb{E}\bigg{\|}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\bigg{(}\nabla
f_{c_{n}}(\mathbf{\bar{w}}^{\tau}_{c_{n}})\pm\nabla
f(\mathbf{\bar{w}}^{\tau}_{c_{n}})\pm\nabla
f(\mathbf{\bar{w}}^{\tau}_{g})-\sum_{j=1}^{N}\alpha_{c_{j}}\nabla
f_{c_{j}}(\mathbf{\bar{w}}^{\tau}_{c_{j}})\bigg{)}\bigg{\|}^{2}$
$\displaystyle\overset{a}{\leq}$ $\displaystyle
12\eta^{2}p_{s}^{2}\kappa_{1}\kappa_{2}L^{2}\sum_{n=1}^{N}\sum_{\tau=b_{1}\kappa_{1}\kappa_{2}}^{t-1}\alpha_{c_{n}}\mathbb{E}\|\mathbf{\bar{w}}^{\tau}_{c_{n}}-\mathbf{\bar{w}}^{\tau}_{g}\|^{2}+6p_{s}^{2}\eta^{2}\kappa_{1}^{2}\kappa_{2}^{2}\epsilon_{g}^{2}$
(38)
where step (a) holds because of assumptions of bounded variance of mini-batch
gradients and bounded divergences among loss functions and
$\|\sum_{i=1}^{N}x_{i}\|^{2}\leq N\sum_{i=1}^{N}\|x_{i}\|^{2}$.
Plugging (-C) and (-C) back into (-C), and taking average of time, by the help
of Lemma 2, if $\eta<\frac{1}{\sqrt{12}L\kappa_{1}\kappa_{2}}$, we can
conclude the proof of Lemma 3.
### -D Proof of Theorem 2
Similar to HFL, the mini-batch gradients, local, cluster and global function
in our proposed MACFL algorithm are denoted as
$\tilde{g}(\mathbf{w}_{u_{m}}^{t})$, $\tilde{F}(\mathbf{w}_{u_{m}}^{t})$,
$\tilde{f}_{c_{i}}(\mathbf{w}_{c_{i}}^{t})$ and
$\tilde{f}(\mathbf{w}_{g}^{t})$, the averaged cluster and global model can be
rewritten as
$\mathbf{\bar{w}}_{c_{n}}^{t+1}=\mathbf{\bar{w}}_{c_{n}}^{t}-\sum_{u_{m}\in\mathcal{C}_{i}^{t}}\beta_{u_{m}}^{c}(\mathbf{\bar{w}}_{c_{n}}^{t}-\mathbf{w}_{u_{m}}^{t+1})$,
and
$\mathbf{\bar{w}}_{g}^{t+1}=\mathbf{\bar{w}}_{g}^{t}-\sum_{i=1}^{N}\beta_{c_{i}}(\mathbf{\bar{w}}_{g}^{t}-\mathbf{\bar{w}}_{c_{n}}^{t+1})$.
In our proposed algorithm, all mobile users can participate in edge model
aggregation at each iteration, similar to the proof of Lemma 1, we have
$\mathbb{E}\tilde{f}(\mathbf{\bar{w}}^{t+1}_{g})\leq\mathbb{E}\tilde{f}(\mathbf{\bar{w}}^{t}_{g})-\eta\mathbb{E}<\nabla\tilde{f}(\mathbf{\bar{w}}_{g}^{t}),\sum_{m=1}^{M}\beta_{u_{m}}\tilde{g}(\mathbf{w}_{u_{m}}^{t})>+\frac{\eta^{2}L}{2}\mathbb{E}\|\sum_{m=1}^{M}\beta_{u_{m}}\tilde{g}(\mathbf{w}_{u_{m}}^{t})\|^{2}$
(39)
For the second term, we have
$\displaystyle-\mathbb{E}<\nabla\tilde{f}(\mathbf{\bar{w}}_{g}^{t}),\sum_{m=1}^{M}\beta_{u_{m}}\tilde{g}(\mathbf{w}_{u_{m}}^{t})>$
$\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}\mathbb{E}\|\nabla\tilde{f}(\mathbf{\bar{w}}^{t}_{g})-\sum_{m=1}^{M}\beta_{u_{m}}\nabla\tilde{F}(\mathbf{w}_{u_{m}}^{t})\|^{2}-\mathbb{E}\|\nabla\tilde{f}(\mathbf{\bar{w}}^{t}_{g})\|^{2}-\mathbb{E}\|\sum_{m=1}^{M}\beta_{u_{m}}\nabla\tilde{F}(\mathbf{w}_{u_{m}}^{t})\|^{2}\bigg{)}$
$\displaystyle\leq$ $\displaystyle
L^{2}\big{(}\sum_{i=1}^{N}\beta_{c_{i}}\mathbb{E}\|\mathbf{\bar{w}}^{t}_{g}-\mathbf{\bar{w}}_{c_{i}}^{t}\|^{2}+\sum_{m=1}^{M}\beta_{u_{m}}\mathbb{E}\|\mathbf{\bar{w}}_{c_{u_{m}}}^{t}-\mathbf{w}_{u_{m}}^{t}\|^{2}\big{)}-\frac{1}{2}\big{(}\mathbb{E}\|\nabla\tilde{f}(\mathbf{\bar{w}}^{t}_{g})\|^{2}+\mathbb{E}\|\sum_{m=1}^{M}\beta_{u_{m}}\nabla\tilde{F}(\mathbf{w}_{u_{m}}^{t})\|^{2}\big{)}$
(40)
For the third term, we have
$\displaystyle\mathbb{E}\|\sum_{m=1}^{M}\beta_{u_{m}}\tilde{g}(\mathbf{w}_{u_{m}}^{t})\|^{2}=$
$\displaystyle\bigg{(}\mathbb{E}\|\sum_{m=1}^{M}\beta_{u_{m}}(\tilde{g}(\mathbf{w}_{u_{m}}^{t})-\nabla\tilde{F}(\mathbf{w}_{u_{m}}^{t}))\|^{2}+\mathbb{E}\|\sum_{m=1}^{M}\beta_{u_{m}}\nabla\tilde{F}(\mathbf{w}_{u_{m}}^{t})\|^{2}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\sigma_{M}^{2}\sum_{m=1}^{M}\beta_{u_{m}}^{2}+\mathbb{E}\|\sum_{m=1}^{M}\beta_{u_{m}}\nabla\tilde{F}(\mathbf{w}_{u_{m}}^{t})\|^{2}$
(41)
where $\theta_{M}^{2}$ is the bound of variance of
$\tilde{g}(\mathbf{w}_{u_{m}}^{t})$.
Rearranging the order, dividing both side by $\frac{\eta}{2}$ and taking the
average over time, if $\eta\leq\frac{1}{L}$, we have
$\displaystyle\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}\|\nabla\tilde{f}(\mathbf{\bar{w}}^{t}_{g})\|^{2}\leq$
$\displaystyle\frac{2}{\eta
T}(\mathbb{E}\tilde{f}(\mathbf{\bar{w}}^{0}_{g})-\tilde{f}_{\inf})+\eta
L\sigma_{M}^{2}\sum_{m=1}^{M}\beta_{u_{m}}^{2}$
$\displaystyle+\frac{2L^{2}}{T}\sum_{t=0}^{T-1}\bigg{(}\sum_{i=1}^{N}\beta_{c_{i}}\mathbb{E}\|\mathbf{\bar{w}}^{t}_{g}-\mathbf{\bar{w}}_{c_{i}}^{t}\|^{2}+\sum_{m=1}^{M}\beta_{u_{m}}\mathbb{E}\|\mathbf{\bar{w}}_{c_{u_{m}}}^{t}-\mathbf{w}_{u_{m}}^{t}\|^{2}\bigg{)}$
(42)
Similar to Lemma 2, if $\eta<\frac{1}{\sqrt{12}L\kappa_{1}}$ , we can obtain
the upper bound of the MSE of the local and cluster model parameters as
follows:
$\frac{1}{T}\sum_{t=0}^{T-1}\sum_{m=1}^{M}\beta_{u_{m}}\mathbb{E}\|\mathbf{w}^{t}_{u_{m}}-\mathbf{\bar{w}}_{c_{u_{m}}}^{t}\|^{2}\leq\frac{2\eta^{2}\kappa_{1}}{1-12\eta^{2}L^{2}\kappa_{1}^{2}}\bigg{(}3\kappa_{1}\epsilon_{M,c}^{2}+\sigma_{M}^{2}\sum_{m=1}^{M}\beta_{u_{m}}(1-\beta_{u_{m}}^{c})\bigg{)}$
(43)
Similar to Lemma 3, if $\eta<\frac{1}{\sqrt{12}L\kappa_{1}\kappa_{2}}$, we can
obtain the upper bound of local model parameters MSE as follows:
$\displaystyle\frac{1}{T}\sum_{t=0}^{T-1}\sum_{i=1}^{N}\beta_{c_{i}}\mathbb{E}\|\mathbf{\bar{w}}^{t}_{g}-\mathbf{\bar{w}}_{c_{i}}^{t}\|^{2}$
$\displaystyle\leq\frac{2\eta^{2}\kappa_{1}\kappa_{2}}{1-12\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}^{2}}\bigg{[}3\kappa_{1}\kappa_{2}\epsilon_{M,g}^{2}+\frac{12\eta^{2}L^{2}\kappa_{1}^{3}\kappa_{2}}{1-12\eta^{2}L^{2}\kappa_{1}^{2}}\epsilon_{M,c}^{2}$
$\displaystyle+2\sigma_{M}^{2}\sum_{m=1}^{M}\beta_{u_{m}}\bigg{(}(\beta_{u_{m}}^{c}-\beta_{u_{m}})+\frac{2\eta^{2}L^{2}\kappa_{1}^{2}\kappa_{2}}{1-12\eta^{2}L^{2}\kappa_{1}^{2}}(1-\beta_{u_{m}}^{c})\bigg{)}\bigg{]}$
(44)
Plugging (43) and (-D) back into (-D), if all models are initialized at a same
point, and if $\eta<\frac{1}{\sqrt{12}L\kappa_{1}\kappa_{2}}$, we can conclude
the proof of Theorem 2.
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|
# Recurrent neural network based parameter estimation of Hawkes model on high-
frequency financial data
Kyungsub Lee111Department of Statistics, Yeungnam University, Gyeongsan,
Gyeongbuk 38541, Korea
###### Abstract
This study examines the use of a recurrent neural network for estimating the
parameters of a Hawkes model based on high-frequency financial data, and
subsequently, for computing volatility. Neural networks have shown promising
results in various fields, and interest in finance is also growing. Our
approach demonstrates significantly faster computational performance compared
to traditional maximum likelihood estimation methods while yielding comparable
accuracy in both simulation and empirical studies. Furthermore, we demonstrate
the application of this method for real-time volatility measurement, enabling
the continuous estimation of financial volatility as new price data keeps
coming from the market.
## 1 Introduction
We propose a real-time estimation and volatility measurement scheme.
Specifically, we continuously estimate the financial model parameters and
volatility in real-time as new price process become available during the
market operation. Our method uses a recurrent neural network to estimate the
parameters of a Hawkes model based on high-frequency financial data, and these
estimates are subsequently used to compute volatility. This approach exhibits
significantly faster computational performance compared to the traditional
maximum likelihood estimation (MLE) method.
The Hawkes process introduced by Hawkes (1971) is a type of point process used
to model the occurrence of events over time. It is frequently utilized in
finance to model the arrival of trades or other financial events in the
market; here, we use it to describe the fluctuations in the price in the tick
structure. As a two dimensional Hawkes process with symmetric kernel in our
study, it is characterized by two key features: self-excitation and mutual-
excitation (Bacry et al., 2013). Self-excitation means that the occurrence of
an event increases the likelihood of the same types of events occurring, while
mutual-excitation implies that the occurrence of one event can increase the
likelihood occurrence of other types of events.
For the estimation, we use a long short-term memory (LSTM) network introduced
by Hochreiter and Schmidhuber (1997). This is a type of recurrent neural
network that is capable of learning long-term dependencies in data. LSTMs can
be used for a variety of tasks that involve sequential data, including
language modeling, machine translation, speech recognition, time series based
on historical data, and financial analysis (Zhang et al., 2021; Ghosh et al.,
2022). They are particularly useful for tasks where capturing long-term
dependencies is important, as the gating mechanism allows the network to
retain important earlier information in the sequence while discarding
irrelevant or outdated information.
Thus, our method uses a neural network for parameter estimation. Similar
attempts have been made in recent years in various fields (Wlas et al., 2008;
Wang et al., 2022; Wei and Jiang, 2022), but there is still limited research
on using neural networks for estimation in financial time series.
Specifically, we use a direct parameter estimation approach in which the
neural network is trained to directly predict the model parameters based on
the observed data. This method is an example of a small aspect of network-
based parameter estimation and expected to be applied to more complex models.
## 2 Method
### 2.1 Price model
First, we explain the stochastic model to describe high-frequency stock price
movements. We model the up and down movements of the tick-level price process
as a marked Hawkes process, which captures both the timing and size of the
movements. This process is defined by the random measures,
$\bm{M}(\mathrm{d}u\times\mathrm{d}z)=\begin{bmatrix}M_{1}(\mathrm{d}u\times\mathrm{d}z_{1})\\\
M_{2}(\mathrm{d}u\times\mathrm{d}z_{2})\end{bmatrix}$ (1)
in the product space of time and jump size, $\mathbb{R}\times E_{i}$, for
$i=1,2$, where $E_{i}=\mathbb{N}\times\\{i\\}$ denotes the space of mark
(jump) sizes for up and down price movements, respectively. Each measure in
Eq. (1) is associated with a sequence of $E_{i}$-valued random variables
$\\{Z_{i,n}\\}$ in addition to the sequence of random times $\\{\tau_{i,n}\\}$
for each $i$. That is,
$M_{i}(\mathrm{d}u\times\mathrm{d}z_{i})=\sum_{n}\delta_{\tau_{i,n},Z_{i,n}}(\mathrm{d}u\times\mathrm{d}z_{i})$
with the Dirac measure $\delta$, which is defined as follows: for any time
interval $I$ and $A_{i}\subset E_{i}$
$\delta_{\tau_{i,n},Z_{i,n}}(I\times
A_{i})=\left\\{\begin{array}[]{lr}1,\text{ if }\tau_{i,n}\in I\text{ and
}Z_{i,n}\in A_{i},\\\ 0,\text{ otherwise.}\end{array}\right.$
A vector of càdlàg counting processes is defined by
$\bm{N}_{t}=\begin{bmatrix}N_{1}(t)\\\ N_{2}(t)\end{bmatrix}=\int_{(0,t]\times
E}\mathrm{Dg}(z)\bm{M}(\mathrm{d}u\times\mathrm{d}z),\quad\mathrm{Dg}(z)=\begin{bmatrix}z_{1}&0\\\
0&z_{2}\end{bmatrix},\quad E=E_{1}\cup E_{2}$
which counts the number of events weighted by their size, that is,
$N_{i}(t)=N_{i}((0,t])=\sum_{n}Z_{i,n}\mathbbm{1}_{\\{0<\tau_{i,n}\leq
t\\}}=\\#\textrm{ of }\tau_{i,n}\in(0,t],\quad\textrm{for }i=1,2.$
###### Assumption 1.
The stochastic intensity $\lambda_{i}$ for $N_{i}$ is represented by the
following:
$\bm{\lambda}_{t}=\begin{bmatrix}\lambda_{1}(t)\\\
\lambda_{2}(t)\end{bmatrix}=\bm{\mu}+\int_{(-\infty,t]\times
E}\bm{\alpha}\circ\bm{b}(t-u)\bm{M}(\mathrm{d}u\times\mathrm{d}z)$ (2)
where $\bm{\mu}$ is a $2\times 1$ positive constant base intensity vector,
$\bm{\alpha}$ is a positive $2\times 2$ constant matrix. $\bm{h}$ is a decay
function matrix and $\circ$ denotes the element-wise product. From the
definition of Eq. (2), for simplicity, we assume that the future impact of an
event on intensities is independent of jump size, as the integrand of the
equation does not contain the jump variable $z$. In addition, for further
parsimony, we assume that:
$\bm{\mu}=\begin{bmatrix}\mu\\\
\mu\end{bmatrix},\quad\bm{\alpha}=\begin{bmatrix}\alpha_{1}&\alpha_{2}\\\
\alpha_{2}&\alpha_{1}\end{bmatrix},\quad\bm{b}(t)=\begin{bmatrix}\mathrm{e}^{-\beta
t}&\mathrm{e}^{-\beta t}\\\ \mathrm{e}^{-\beta t}&\mathrm{e}^{-\beta
t}\end{bmatrix}.$ (3)
Hence, the set of parameters to be estimated is
$\\{\mu,\alpha_{1},\alpha_{2},\beta\\}$. We also assume that the mark $Z_{i}$
at time $t$ is independent from the $\sigma$-algebra generated by
$(N_{j}(s),\lambda_{j}(s))_{s<t}$ for $j=1,2$. The intensity process is
assumed to be stationary and that the spectral radius of
$|\int_{0}^{\infty}\bm{\alpha}\circ\bm{b}(t)\mathrm{d}t|$ is less than 1.
Under this assumption, the Hawkes volatility of price movements – the standard
deviation of total up and down net movements – is represented by
$\mathrm{SD}(N_{1}(t)-N_{2}(t))=\sqrt{\bm{\mathrm{u}}^{\top}\left[\mathcal{T}\left\\{\overline{\bm{\mathrm{Z}}}\circ\bm{\mathrm{B}}\right\\}+\overline{\bm{\mathrm{Z}}}^{(2)}\circ\mathrm{Dg}(\mathbb{E}[\bm{\lambda}_{t}])\right]\bm{\mathrm{u}}t},\quad\bm{\mathrm{u}}=\begin{bmatrix}1\\\
-1\end{bmatrix}$ (4)
where $\mathcal{T}$ is an operator such that
$\mathcal{T}(\bm{\mathrm{M}})=\bm{\mathrm{M}}+\bm{\mathrm{M}}^{\top}$ for a
square matrix $\bm{\mathrm{M}}$ and $\mathrm{Dg}(\cdot)$ denotes a diagonal
matrix whose diagonal entry is composed of the argument. Furthermore,
$\mathbb{E}[\bm{\lambda}_{t}]=(\bm{\beta}-\bm{\alpha})^{-1}\bm{\beta}\bm{\mu},\quad\bm{\beta}=\begin{bmatrix}\beta&0\\\
0&\beta\end{bmatrix}$ (5)
and
$\mathbb{E}[\bm{\lambda}_{t}\bm{\lambda}_{t}^{\top}]=(\bm{\beta}-\bm{\alpha})^{-1}\left(\frac{1}{2}\bm{\alpha}\mathrm{Dg}(\mathbb{E}[\bm{\lambda}_{t}])\bm{\alpha}+\bm{\beta}\bm{\mu}\mathbb{E}[\bm{\lambda}_{t}^{\top}]\right)$
(6)
and
$\bm{\mathrm{B}}=\left\\{\overline{\bm{\mathrm{Z}}}^{\top}\circ\mathbb{E}[\bm{\lambda}_{t}\bm{\lambda}_{t}^{\top}]+\mathrm{Dg}(\mathbb{E}[\bm{\lambda}_{t}])\left(\bm{\alpha}\circ\overline{\bm{\mathrm{Z}}}\right)^{\top}-\mathrm{Dg}(\overline{\bm{\mathrm{Z}}})\mathbb{E}[\bm{\lambda}_{t}]\mathbb{E}[\bm{\lambda}_{t}]^{\top}\right\\}(\bm{\beta}-\bm{\alpha})^{-1}$
(7)
and by the mark independent assumption,
$\overline{\bm{\mathrm{Z}}}=\begin{bmatrix}\mathbb{E}[Z_{1}]&\mathbb{E}[Z_{2}]\\\
\mathbb{E}[Z_{1}]&\mathbb{E}[Z_{2}]\end{bmatrix},\quad\overline{\bm{\mathrm{Z}}}=\begin{bmatrix}\mathbb{E}[Z_{1}^{2}]&\mathbb{E}[Z_{2}^{2}]\\\
\mathbb{E}[Z_{1}^{2}]&\mathbb{E}[Z_{2}^{2}]\end{bmatrix}.$
To calculate the volatility of price changes, rather than the number of
movements, we multiply the minimum tick size to Eq. (4). Further details can
be found in Lee and Seo (2017) and Lee (2022).
### 2.2 Network model
Next, we construct a recurrent neural network for parameter estimation. The
traditional method of estimating the parameters of a Hawkes process is MLE,
which involves maximizing the log-likelihood function of the model:
$L(T,\bm{\theta})=\sum_{i=1}^{2}\left(\int_{(0,T]\times
E}\log\lambda_{i}(u)M_{i}(\mathrm{d}u\times\mathrm{d}z_{i})-\int_{0}^{T}\lambda_{i}(u)\mathrm{d}u\right)$
to estimate the parameters most likely to generate the observed data.
In contrast, neural network based parameter estimation involves training a
neural network to predict the parameters of a Hawkes process based on input
data. To do this, numerous sample paths of inter-arrival times and movement
types (up or down) are generated, where the parameters of each path are
determined randomly. These sample paths are then used as feature variables,
and the associated true parameter values are used as the target variables.
The neural network is trained on these data. Once trained, it can be used to
predict the parameter values of a new sample path of Hawkes process data.
These predicted parameter values can then be used to compute further
complicated formulae, such as the Hawkes volatility in Eq. (4), which is a
measure of the variability of the process over time.
Here, we use an LSTM model with three layers as the neural network. The LSTM
network is known for its capability of retaining information over a long
duration, making it appropriate for tasks that require context comprehension
or state preservation. The gates mechanism in the LSTM architecture that
regulates the flow of information between memory cells and the network enables
the network to choose which information should be preserved or discarded. We
also tested gated recurrent unit networks (Cho et al., 2014); however, the
LSTM performed slightly better in our problem. A thorough account of the
network’s implementation is presented in the following section.
## 3 Simulation result
In this simulation study, we generate a set of paths of Hawkes processes to
create a training dataset for the neural network. The dataset comprises a
sufficient quantity of synthetic data, which are utilized to train the network
to predict the real parameters of the Hawkes process. The real parameters for
each Hawkes process are randomly selected to cover the entire range of
possible values, with each path having distinct parameters. For the ranges of
the parameters, we use the ranges of the estimates obtained by fitting the
past intraday price process of various stocks to the Hawkes model.
Approximately 30 symbols of stocks, including AAPL, AMZN, C, FB, GOOG, IBM,
MCD, MSFT, NVDA, and XOM from 2018 to 2019 are used.
For the estimation, we focus on high-frequency rather than ultra-high-
frequency. More precisely, the raw data are filtered as follows. We observe
the mid-price at intervals of $\Delta t=0.1$ seconds, noting any changes from
the previously observed price. If a change is detected, we record the exact
time of the change and the new price. If the price remains the same, we move
on to the next interval of 0.1 seconds and repeat the process. This method
allows us to filter out unnecessary movements, commonly referred to as
microstructure noise, observed at ultra-high frequencies.
Once the set of estimates is obtained, we generate Hawkes process paths of a
2,000-time step. These are then used for neural network training together with
estimates as target variables. This method yields tens of thousands of
datasets, which are sufficient to construct a comprehensive training set.
The implementation of the LSTM network model is as follows. The first layer
consists of 12 units, which manage sequential data. These data are a two-
dimensional input of time series data that comprise inter-arrivals and types
of movements (up or down). Up and down movements are encoded as 1 and 2,
respectively. The output of this layer is a sequence of 12 length of vectors,
where each vector is the output of the first layer at a given time step. Thus,
if the original time series has a 2,000-time step, then the output of the
first layer is a $2,000\times 12$ matrix, which is the time step $\times$ the
number of units.
The second layer has 12 units and produces a single (not a sequence of) vector
of length 12. The final layer is a dense (fully connected) layer with four
units, which produces output representing the parameters in the Hawkes model,
$\mu,\alpha_{1},\alpha_{2}$ and $\beta$. If we extend the model for more
complexity, the number of units in the last dense layer will be adjusted
accordingly. This is because each unit in the last layer represents each
parameter.
As pointed out by Mei and Eisner (2017), a natural extension of LSTM is deep
LSTM, especially for complex structured data such as high-frequency financial
data, where the effectiveness of multi-layering seems promising. However, we
utilized a relatively parsimonious Hawkes model and achieved sufficient
performance without employing a large number of layers, so we used the LSTM
model proposed above, which is relatively faster to train. If the data
structure and model become more complex, an extension to deep LSTM would be
helpful.
For training, we use 75,000 sample data points; hence, the dataset for the
neural network’s input has a shape of $75,000\times 1,000\times 2$. The Adam
(Kingma and Ba, 2014) optimizer is used for training. Generally, more than 300
epochs are used. For testing, we use 15,000 data points that were not used for
training. This is done to evaluate the model’s ability to make predictions on
these unseen data based on mean squared error (MSE). We then compare the
results with a traditional MLE. The computation times were measured using a
typical commercial PC. The result shows that the MLE has slightly better
performance in terms of MSE; however, the neural network also shows a
reasonable result. Meanwhile, the general numerical method of MLE requires
many iterations and is time consuming. However, a well-trained neural network
computes an estimate very quickly, which is less than a hundredth of the time
required by MLE.
| Neural network | MLE
---|---|---
MSE | 0.0513 | 0.0417
Time (sec) | 0.0120 | 1.763
To understand the basic properties of the neural network estimator, we
investigate its sampling distribution. To examine the sampling distribution,
we generate 100,000 paths of length 2,000 using the fixed parameter values of
$\mu=0.3$, $\alpha_{1}=0.4$, $\alpha_{2}=0.7$, $\beta=1.5$. We then compare
the obtained sampling distributions using the neural network and MLE methods.
Overall, MLE outperforms the neural network slightly. Even so, the general
performance of neural networks is also quite good.
Parameter | True | Neural network | MLE
---|---|---|---
| | Mean | S.D. | Mean | S.D.
$\mu$ | 0.3000 | 0.3151 | 0.0358 | 0.3036 | 0.0314
$\alpha_{1}$ | 0.4000 | 0.4429 | 0.0661 | 0.3988 | 0.0500
$\alpha_{2}$ | 0.7000 | 0.5733 | 0.0697 | 0.7024 | 0.0608
$\beta$ | 1.5000 | 1.5736 | 0.1364 | 1.5078 | 0.1145
Specifically, the aforementioned example compares the performance of the
numerical optimizer (used in MLE) and neural network. Owing to the nature of
simulation studies, the numerical optimizer has several advantages. The
outcome of a numerical optimizer is often influenced by its initial value. In
the example provided above, because the true value of the parameter is known,
it was directly used as the initial value. This may have improved the
performance compared to the result from a random initial value. As finding a
good initial value is sometimes challenging, a numerical optimizer may perform
worse in real-world problems.
In addition, if the numerical optimizer exhibits unexpected behavior, such as
failure to converge appropriately, human engagement may be necessary, such as
adjustments to the initial value or retrying the procedure. As the complexity
of the model and level of noise present in the empirical data increase, the
advantages of the numerical optimizer may decrease. In such cases, further
research may be required to determine whether the numerical optimizer still
outperforms the neural network.
## 4 Empirical result
The approach in the previous section can be directly applied to empirical
data. However, we need to consider whether robust estimation can be made in
situations where the empirical data do not completely follow the Hawkes
process. For example, in filtered high-frequency price process data, a subdue
effect (where an event reduces intensity) can sometimes occur. This can result
in negative estimates for the parameter $\alpha$, which violates the
definition of the Hawkes model. To address this, a more complex model should
be used; however, as this falls outside the scope of this study, an
alternative method is to use a softplus activation function $\log(1+\exp(x))$
for the last layer in the neural network. This approach is similar to
constraint optimization by ensuring positive estimates. Furthermore, instead
of predicting $\beta$ directly, we trained and predicted
$\beta-\alpha_{1}-\alpha_{2}$. By using the softplus function, this method
ensures that the branching ratio condition of the Hawkes model is met.
To further increase robustness, a combination of empirical data and its
maximum likelihood estimates as training data can be used, rather than relying
solely on simulation data. This approach accounts for the possibility of model
mis-specification; for instance, the observed data may not perfectly align
with the Hawkes process. By incorporating the MLE into the training data, the
neural network can better mimic the MLE of the Hawkes model. Thus, if the goal
is to construct a neural network that closely approximates the MLE, even under
the possibility of model mis-specification, this method can be effective.
The following section explains the step-by-step procedure. We select segments
of observed intraday data of inter-arrivals and movement types. Each segment
consists of a 2,000-time step. These selected paths are used to fit the Hawkes
model using MLE. The resulting dataset is then used to train the neural
network, where the inter-arrivals and types of real data serve as feature
variables and the maximum likelihood estimates are the target variables.
Figure 1 illustrates the intraday dynamics of estimates of the Hawkes model on
a specific date. The data used for this illustration are out-of-sample that
are not used for training. Specifically, it was estimated using segments of
data corresponding to every 2,000-time step. This corresponds to a time
horizon of approximately 10-20 minutes. To create a more continuous graph, the
time windows for estimation were moved forward slowly with sufficient overlap.
The neural network shows very consistent results with MLE.
(a) $\mu$
(b) $\beta$
(c) $\alpha_{1}$
(d) $\alpha_{2}$
Figure 1: Intraday estimates for the NBBO of AAPL using MLE and neural network
Figure 2 presents the instantaneous intraday annualized Hawkes volatility
calculated using MLE, a neural network, and nonparametric realized volatility
as a benchmark by Andersen et al. (2003). The realized volatility is
calculated using the observed values at 1-second intervals for the price
process of the period. All three measures have a similar trend throughout the
day. Although it is not possible to present all the results examined, in some
cases, MLE showed unstable dynamics of volatility. This is likely due to the
fact that the 2,000-time length used for estimation is a relatively small
sample size to estimate the parameters of the Hawkes model.
Figure 2: Intraday annualized volatilities for the NBBO of AAPL using MLE and
neural network
## 5 Conclusion
This study shows that a neural network can accurately estimate time series
parameters, with an accuracy similar to MLE and much faster computation. While
the example used here is for calculating Hawkes volatility, but our proposed
method can be applied to various fields. It can be particularly useful in
cases where the model is complex and traditional estimation procedures are
challenging, such as modeling entire limit order book. Further research in
this area is expected to be ongoing and diverse.
## Acknowledgements
This work has supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIT)(No. NRF-2021R1C1C1007692).
## References
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* Cho et al. (2014) Cho, K., B. Van Merriënboer, D. Bahdanau, and Y. Bengio (2014). On the properties of neural machine translation: Encoder-decoder approaches. arXiv preprint arXiv:1409.1259.
* Ghosh et al. (2022) Ghosh, P., A. Neufeld, and J. K. Sahoo (2022). Forecasting directional movements of stock prices for intraday trading using lstm and random forests. Finance Research Letters 46, 102280.
* Hawkes (1971) Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. Journal of the Royal Statistical Society. Series B (Methodological) 33, 438–443.
* Hochreiter and Schmidhuber (1997) Hochreiter, S. and J. Schmidhuber (1997). Long short-term memory. Neural computation 9, 1735–1780.
* Kingma and Ba (2014) Kingma, D. P. and J. Ba (2014). Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980.
* Lee (2022) Lee, K. (2022). Application of hawkes volatility in the observation of filtered high-frequency price process in tick structures. arXiv 2207.05939.
* Lee and Seo (2017) Lee, K. and B. K. Seo (2017). Marked Hawkes process modeling of price dynamics and volatility estimation. Journal of Empirical Finance 40, 174–200.
* Mei and Eisner (2017) Mei, H. and J. M. Eisner (2017). The neural Hawkes process: A neurally self-modulating multivariate point process. Advances in Neural Information Processing Systems 30.
* Wang et al. (2022) Wang, X., J. Feng, Q. Liu, Y. Li, and Y. Xu (2022). Neural network-based parameter estimation of stochastic differential equations driven by lévy noise. Physica A: Statistical Mechanics and its Applications 606, 128146.
* Wei and Jiang (2022) Wei, Y. M. and Z. Jiang (2022). Estimating parameters of structural models using neural networks. USC Marshall School of Business Research Paper.
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|
supSupplementary References figurec
# LiteTransformerSearch: Training-free Neural Architecture Search for
Efficient Language Models
Mojan Javaheripi1, Gustavo H. de Rosa2, Subhabrata Mukherjee2,
Shital Shah2, Tomasz L. Religa3, Caio C.T. Mendes2,
Sebastien Bubeck2, Farinaz Koushanfar1, Debadeepta Dey2
1University of California San Diego, 2Microsoft Research, 3Microsoft
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
The Transformer architecture is ubiquitously used as the building block of
large-scale autoregressive language models. However, finding architectures
with the optimal trade-off between task performance (perplexity) and hardware
constraints like peak memory utilization and latency is non-trivial. This is
exacerbated by the proliferation of various hardware. We leverage the somewhat
surprising empirical observation that the number of decoder parameters in
autoregressive Transformers has a high rank correlation with task performance,
irrespective of the architecture topology. This observation organically
induces a simple Neural Architecture Search (NAS) algorithm that uses decoder
parameters as a proxy for perplexity without need for any model training. The
search phase of our training-free algorithm, dubbed Lightweight Transformer
Search (LTS)111code available at
https://github.com/microsoft/archai/tree/neurips_lts/archai/nlp, can be run
directly on target devices since it does not require GPUs. Using on-target-
device measurements, LTS extracts the Pareto-frontier of perplexity versus any
hardware performance cost. We evaluate LTS on diverse devices from ARM CPUs to
NVIDIA GPUs and two popular autoregressive Transformer backbones: GPT-2 and
Transformer-XL. Results show that the perplexity of $16$-layer GPT-2 and
Transformer-XL can be achieved with up to $1.5\times,2.5\times$ faster runtime
and $1.2\times,2.0\times$ lower peak memory utilization. When evaluated in
zero and one-shot settings, LTS Pareto-frontier models achieve higher average
accuracy compared to the $350$M parameter OPT across $14$ tasks, with up to
$1.6\times$ lower latency. LTS extracts the Pareto-frontier in under $3$ hours
while running on a commodity laptop. We effectively remove the carbon
footprint of hundreds of GPU hours of training during search, offering a
strong simple baseline for future NAS methods in autoregressive language
modeling.
## 1 Introduction
The Transformer architecture [42] has been used as the de-facto building block
of most pre-trained language models like GPT [5]. A common problem arises when
one tries to create smaller versions of Transformer models for edge or real-
time applications (e.g. text prediction) with strict memory and latency
constraints: it is not clear what the architectural hyperparameters should be,
e.g., number of attention heads, number of layers, embedding dimension, and
the inner dimension of the feed forward network, etc. This problem is
exacerbated if each Transformer layer is allowed the freedom to have different
values for these settings. This results in a combinatorial explosion of
architectural hyperparameter choices and a large heterogeneous search space.
For instance, the search space considered in this paper consists of over
$10^{54}$ possible architectures.
Neural Architecture Search (NAS) is an organic solution due to its ability to
automatically search through candidate models with multiple conflicting
objectives like latency vs. task performance. The central challenge in NAS is
the prohibitively expensive function evaluation, i.e., evaluating each
architecture requires training it on the dataset at hand. Thus it is often
infeasible to evaluate more than a handful of architectures during the search
phase. Supernets [31] have emerged as a dominant paradigm in NAS which combine
all possible architectures into a single graph and jointly train them using
weight-sharing. Nevertheless, supernet training imposes constraints on the
expressiveness of the search space [29] and is often memory-hungry [52, 6, 51]
as it creates large networks during search. Additionally, training supernets
is non-trivial as children architectures may interfere with each other and the
ranking between sub-architectures based on task performance is not preserved
[29]222See [29] for a comprehensive treatment of the difficulties of training
supernets..
We consider a different approach by proposing a training-free proxy that
provides a highly accurate ranking of candidate architectures during NAS
without need for costly function evaluation or supernets. Our scope is NAS for
efficient autoregressive Transformers used in language modeling. We design a
lightweight search method that is target hardware-aware and outputs a gallery
of models on the Pareto-frontier of perplexity versus hardware metrics. We
term this method Lightweight Transformer Search (LTS). LTS relies on our
somewhat surprising observation: the decoder parameter count has a high rank
correlation with the perplexity of fully trained autoregressive Transformers.
Given a set of autoregressive Transformers, one can accurately rank them using
decoder parameter count as the proxy for perplexity. Our observations are also
well-aligned with the power laws in [22], shown for homogeneous autoregressive
Transformers, i.e., when all decoder layers have the same configuration. We
provide extensive experiments that establish a high rank correlation between
perplexity and decoder parameter count for _both_ homogeneous and
heterogeneous search spaces.
Figure 1: High-level overview of LTS. We propose a training-free zero-cost
proxy for evaluating the validation perplexity of candidate architectures.
Pareto-frontier search is powered by evolutionary algorithms which use the
proposed proxy along with real latency and memory measurements on the target
hardware to evaluate sampled architectures.
The above phenomenon coupled with the fact that a candidate architecture’s
hardware performance can be measured on the target device leads to a training-
free search procedure: pick one’s favorite discrete search algorithm (e.g.
evolutionary search), sample candidate architectures from the search space;
count their decoder parameters as a proxy for task performance (i.e.,
perplexity); measure their hardware performance (e.g., latency and memory)
directly on the target device; and progressively create a Pareto-frontier
estimate. While we have chosen a reasonable search algorithm in this work, one
can plug and play any Pareto-frontier search method such as those in [20].
Building upon these insights, Figure 1 shows a high-level overview of LTS. We
design the first training-free Transformer search that is performed entirely
on the target (constrained) platform. As such, LTS easily performs a multi-
objective NAS where several underlying hardware performance metrics, e.g.,
latency and peak memory utilization, are simultaneously optimized. Using our
training-free proxy, we extract the $3$-dimensional Pareto-frontier of
perplexity versus latency and memory in a record-breaking time of $<3$ hours
on a commodity Intel Core i7 CPU. Notably, LTS eliminates the carbon footprint
from hundreds of GPU hours of training associated with legacy NAS methods.
To corroborate the effectiveness of our proxy, we train over $2900$
Transformers on three large language modeling benchmark datasets, namely,
WikiText-103 [27], One Billion Word [7], and Pile [17]. We use LTS to search
for Pareto-optimal architectural hyperparameters in two popularly used
autoregressive Transformer backbones, namely, Transformer-XL [10] and GPT-2
[32]. We believe decoder parameter count should be regarded as a competitive
baseline for evaluating Transformer NAS, both in terms of ranking capabilities
and easy computation. We open-source our code along with tabular information
of our trained models to foster future NAS research on Transformers.
## 2 Related Work
Here, we discuss literature on automated search for Transformer architectures
in the language domain. We refer to extensive surveys on NAS [14, 49] for a
broader overview of the field.
Decoder-only Architectures. So et al. [37] search over TensorFlow programs
that implement an autoregressive language model via evolutionary search. Since
most random sequences of programs either have errors or underperform, the
search has to be seeded with the regular Transformer architecture, termed
“Primer”. As opposed to “Primer” which uses large computation to search a
general space, we aim to efficiently search the “backbone” of traditional
decoder-only Transformers. Additionally, the objective in “Primer” is to find
models that train faster. Our objective for NAS, however, is to deliver
Pareto-frontiers for inference, with respect to perplexity and hardware
constraints.
Encoder-only Architectures. Relative to decoder-only autoregressive language
models, encoder-only architectures like BERT [11] have received much more
recent attention from the NAS community. NAS-BERT [50] trains a supernet to
efficiently search for masked language models (MLMs) which are compressed
versions of the standard BERT, Such models can then be used in downstream
tasks as is standard practice. Similar to NAS-BERT, Xu et al. [51] train a
supernet to conduct architecture search with the aim of finding more efficient
BERT variants. They find interesting empirical insights into supernet training
issues like differing gradients at the same node from different child
architectures and different tensors as input and output at every node in the
supernet. The authors propose fixes that significantly improve supernet
training. Tsai et al. [41], Yin et al. [53], Gao et al. [16] also conduct
variants of supernet training with the aim of finding more efficient BERT
models.
Encoder-Decoder Related: Applying the well-known DARTS [24] approach to
Transformer search spaces leads to memory-hungry supernets. To mitigate this
issue, Zhao et al. [61] propose a multi-split reversible network and a memory-
efficient backpropagation algorithm. One of the earliest papers that applied
discrete NAS to Transformer search spaces was [36], which uses a modified form
of evolutionary search. Due to the expense of directly performing discrete
search on the search space, this work incurs extremely large computation
overhead. Follow-up work by [46] uses the Once-For-All [6] approach to train a
supernet for encoder-decoder architectures used in machine translation. Search
is performed on subsamples of the supernet that inherit weights to estimate
task accuracy. For each target device, the authors train a small neural
network regressor on thousands of architectures to estimate latency. As
opposed to using a latency estimator, LTS evaluates the latency of each
candidate architecture on the target hardware. Notably, by performing the
search directly on the target platform, LTS can easily incorporate various
hardware performance metrics, e.g., peak memory utilization, for which
accurate estimators may not exist. To the best of our knowledge, such holistic
integration of multiple hardware metrics in Transformer NAS has not been
explored previously.
## 3 Lightweight Transformer Search
We perform an evolutionary search over candidate architectures to extract
models that lie on the Pareto-frontier. In contrast to the vast majority of
prior methods that deliver a single architecture from the search space, our
search is performed over the entire Pareto, generating architectures with a
wide range of latency, peak memory utilization, and perplexity with one round
of search. This alleviates the need to repeat the NAS algorithm for each
hardware performance constraint.
To evaluate candidate models during the search, LTS uses a training-free proxy
for the validation perplexity. By incorporating training-free evaluation
metrics, LTS, for the first time, performs the entire search directly on the
target (constrained) hardware. Therefore, we can use real measurements of
hardware performance during the search. Algorithm 1 outlines the iterative
process
Input: Search space $\mathcal{D}$, $n_{iter}$
Output: Perplexity-latency-memory Pareto-frontier $\mathbb{F}$
1
$\mathcal{L},\mathcal{M},\mathcal{P},\mathbb{F}\leftarrow\emptyset,\emptyset,\emptyset,\emptyset$
2 while _$N\leq n_{iter}$_ do
3 $\mathbb{F}^{\prime}\leftarrow$ Subsample($\mathbb{F}$)
4 $\mathbb{S}_{N}\leftarrow EA(\mathbb{F}^{\prime},\mathcal{D})$
// hardware profiling
5 $\mathcal{L}\leftarrow\mathcal{L}\bigcup$ Latency($\mathbb{S}_{N}$)
6 $\mathcal{M}\leftarrow\mathcal{M}\bigcup$ Memory($\mathbb{S}_{N}$)
// estimate perplexity
7 $\mathcal{P}\leftarrow\mathcal{P}\bigcup$ Proxy($\mathbb{S}_{N}$)
// update the Pareto-frontier
8 $\mathbb{F}\leftarrow$
LowerConvexHull($\mathcal{P},\mathcal{L},\mathcal{M}$)
Algorithm 1 LTS’s training-free NAS
performed in LTS 333The Pareto-frontier search method in Algorithm 1 is
inspired by [13] and [21]. Other possibilities include variations proposed in
[20], evaluation of which is orthogonal to our contributions in this work. for
finding candidate architectures in the search space ($\mathcal{D}$), that lie
on the $3$-dimensional Pareto-frontier ($\mathbb{F}$) of perplexity versus
latency and memory. At each iteration, a set of points ($\mathbb{F^{\prime}}$)
are subsampled from the current Pareto-frontier. A new batch of architectures
($\mathbb{S}_{N}$) are then sampled from $\mathbb{F^{\prime}}$ using
evolutionary algorithms ($EA(.)$). The new samples are evaluated in terms of
latency ($\mathcal{L}$), peak memory utilization ($\mathcal{M}$), and
validation perplexity ($\mathcal{P}$). Latency and memory are measured
directly on the target hardware while the perplexity is indirectly estimated
using our accurate and training-free proxy methods. Finally, the Pareto-
frontier is recalibrated using the lower convex hull of all sampled
architectures. In the context of multi-objective NAS, Pareto-frontier points
are those where no single metric (e.g., perplexity, latency, and memory) can
be improved without degrading at least one other metric [20]. To satisfy
application-specific needs, optional upper bounds can be placed on the latency
and/or memory of sampled architectures during search.
Search Space. Figure 2 shows all elastic parameters in LTS search space,
namely, number of layers (nlayer), number of attention heads (nhead), decoder
output dimension (dmodel), inner dimension of the feed forward network
(dinner), embedding dimension (dembed), and the division factor ($k$) of
adaptive embedding [3]. These architectural parameters are compatible with
popularly used autoregressive
Figure 2: Elastic parameters in LTS search space.
Transformer backbones, e.g., GPT. For preliminaries on autoregressive
Transformers, please see Appendix A. We adopt a heterogeneous search space
where the backbone parameters are decided on a per-layer basis. This is in
contrast to the homogeneous structure commonly used in Transformers [10, 5],
which reuses the same configuration for all layers. Compared to homogeneous
models, the flexibility associated with heterogeneous architectures enables
them to obtain much better hardware performance under the same perplexity
budget (see Section 4.4).
Heterogeneous search space was previously explored in [46]. However, due to
the underlying supernet structure, not all design parameters can change
freely. As an example, the dimensionality of the Q, K, V vectors inside the
encoder and decoder layers is fixed to a large value of $512$ to accommodate
inheritance from the supernet. Our search space, however, allows exploration
of all internal dimensions without constraints. By not relying on the supernet
structure, our search space easily encapsulates various Transformer backbones
with different configurations of the input/output embedding layers and elastic
internal dimensions.
LTS searches over the following values for the architectural parameters in our
backbones: nlayer$\in\\{2,\dots,16|1\\}$444We use the notation {vmin,…,
vmax—step size} to show the valid range of values.,
dmodel$\in\\{128,\dots,1024|64\\}$, dinner$\in\\{256,\dots,4096|64\\}$, and
nhead$\in\\{2,4,8\\}$. Additionally we explore adaptive input embedding [3]
with dembed$\in\\{128,256,512\\}$ and factor $k\in\\{1,2,4\\}$. Once a dmodel
is sampled, we adjust the lower bound of the above range for dinner to
$2\times$dmodel. Encoding this heuristic inside the search ensures that the
acquired models will not suffer from training collapse. Our heterogeneous
search space encapsulates more than $10^{54}$ different architectures. Such
high dimensionality further validates the critical need for training-free NAS.
### 3.1 Training-free Architecture Ranking
$\blacktriangleright$ Low-cost Ranking Proxies. Recently, Abdelfattah et al.
[1] utilize the summation of pruning scores over all model weights as the
ranking proxy for Convolutional Neural Networks (CNNs), where a higher score
corresponds to higher architecture rank in the search space. White et al. [48]
analyze these and more recent proxies and find that no particular proxy
performs consistently well over various tasks and baselines, while parameter
and floating point operations (FLOPS) count proxies are quite competitive.
However, they did not include Transformer-based search spaces in their
analysis. To the best of our knowledge, low-cost (pruning-based) proxies have
not been evaluated on Transformer search spaces in the language domain. Note
that one cannot naively apply these proxies to language models. Specifically,
since the embedding layer in Transformers is equivalent to a lookup operation,
special care must be taken to omit this layer from the proxy computation.
Using this insight, we perform the first systematic study of low-cost proxies
for NAS on autoregressive Transformers for text prediction.
We leverage various pruning metrics, namely, grad_norm , snip [23], grasp
[45], fisher [40], and synflow [38]. We also study jacob_cov [26] and
relu_log_det [25] which are low-cost scoring mechanisms proposed for NAS on
CNNs in vision tasks. While these low-cost techniques do not perform model
training, they require forward and backward passes over the architecture to
compute the proxy, which can be time-consuming on low-end hardware.
Additionally, the aforesaid pruning techniques, by definition, incorporate the
final softmax projection layer in their score assessment. Such an approach
seems reasonable for CNNs dealing with a few classification labels, however,
it can skew the evaluation for autoregressive Transformers dealing with a
large output vocabulary space. To overcome these shortcomings, we introduce a
zero-cost architecture ranking strategy in the next section that outperforms
the proposed low-cost proxies in terms of ranking precision, is data free, and
does not perform any forward/backward propagation.
Figure 3: Our training-free zero-cost proxy based on decoder parameter count
is highly correlated with the (ground truth) validation perplexity after full
training. Each plot contains $200$ architectures sampled randomly from the
search space of Transformer-XL or GPT-2 backbone.
$\blacktriangleright$ Decoder Parameter Count as a Proxy. We empirically
establish a strong correlation between the parameter count of decoder layers
and final model performance in terms of validation perplexity. We evaluate
$200$ architectures sampled uniformly at random from the search space of two
autoregressive Transformer backbones, namely, Transformer-XL and GPT-2. These
architectures are trained fully on WikiText-103 and One Billion Word (LM1B)
datasets, which consumes over $\mathbf{25000}$ GPU-hours on NVIDIA A100 and
V100 nodes. We compare the ranking obtained using decoder parameter count
proxy and the ground truth ranking after full training in Figure 3. On
WikiText-103, zero-cost ranking using the decoder parameter count obtains a
Spearman’s Rank Correlation (SRC) of $0.97$ with full training. SRC further
increases to $0.99$ for the more complex LM1B benchmark on both backbones.
This validates that the decoder parameter count is strongly correlated with
final model performance, thereby providing a reliable training-free proxy for
NAS.
## 4 Experiments
We conduct experiments to seek answers to the following critical questions:
1
How well can training-free proxies perform compared to training-based methods
for estimating the performance of Transformer models?
2
How does model topology affect the performance of the proposed decoder
parameter proxy?
3
Can our training-free decoder parameter count proxy be integrated inside a
search algorithm to estimate the Pareto-frontier? How accurate is such an
estimation of the Pareto?
4
Which models are on the Pareto-frontier of perplexity, latency, and memory for
different hardware?
5
How well do LTS models perform in zero and one-shot settings compared to hand-
designed variants when evaluated on downstream tasks?
We empirically answer questions 1, 2, 4, and 5 in Sections 4.2, 4.3, 4.4, and
4.5 respectively. We further address question 3 in Appendix C where we show
the Pareto-frontier models extracted by the decoder parameter count proxy are
very close to the ground truth Pareto-frontier with an average of $0.6\%$
perplexity difference. Additionally, we show the efficacy of the decoder
parameter count proxy when performing search on different ranges of model
sizes in Appendix C, Figure 12.
### 4.1 Experimental Setup
Please refer to Appendix B for information about the benchmarked datasets,
along with details of our training and evaluation setup, hyperparameter
optimization, and evolutionary search algorithm.
Backbones. We apply our search on two widely used autoregressive Transformer
backbones, namely, Transformer-XL [10] and GPT-2 [32] that are trained from
scratch with varying architectural hyperparameters. The internal structure of
these backbones are quite similar, containing decoder blocks with attention
and feed-forward layers. The difference between the backbones lies mainly in
their dataflow structure; the Transformer-XL backbone adopts a recurrence
methodology over past states coupled with relative positional encoding which
enables modeling longer term dependencies.
Performance Criteria. To evaluate the ranking performance of various proxies,
we first establish a ground truth ranking of candidate architectures by
training them until full convergence. This ground truth ranking is then
utilized to compute two performance criteria as follows:
$\blacktriangleright$ Common Ratio (CR): We define CR as the percentage
overlap between the ground truth ranking of architectures versus the ranking
obtained from the proxy. CR quantifies the ability of the proxy ranking to
identify the top$k\%$ architectures based on their validation perplexity after
full training.
$\blacktriangleright$ Spearman’s Rank Correlation (SRC): We use this metric to
measure the correlation between the proxy ranking and the ground truth.
Ideally, the proxy ranking should have high correlation with the ground truth
over the entire search space as well as high-performing candidate models.
### 4.2 How do training-free proxies perform compared to training-based
methods?
In this section, we benchmark several proxy methods for estimating the rank of
candidate architectures. Specifically, we investigate three different ranking
techniques, namely, partial training, low-cost methods, and number of decoder
parameters.
$\blacktriangleright$ Partial Training. We first analyze the relationship
between validation perplexity after a shortened training period versus that of
full training for ranking candidate models. We stop the training after
$\tau\in[1.25\%,87.5\%]$ of the total training iterations needed for model
convergence. Figure 4 demonstrates the SRC and CR of partial training with
various $\tau$s, evaluated on $100$ randomly selected models from the
Transformer-XL backbone, trained on WikiText-103. The horizontal axis denotes
the average time required for $\tau$ iterations of training across all sampled
models. Intuitively, a higher number of training iterations results in a more
accurate estimate of the final perplexity. Nevertheless, the increased wall-
clock time prohibits training during search and also imposes the need for
GPUs. Interestingly, very few training iterations, i.e., $1.25\%$, provide a
very good proxy for final performance with an SRC of $>0.9$ on the entire
population. Our training-free proxy, i.e., decoder parameter count, also shows
competitive SRC compared to partial training.
Figure 4: Comparison between partial training and our zero-cost proxy, i.e.,
decoder parameter count, in terms of ranking performance and timing overhead.
Each subplot corresponds to a top$k\%$ of the randomly sampled models, based
on their validation perplexity after full training.
$\blacktriangleright$ Low-cost Proxies. We benchmark various low-cost methods
introduced in Section 3.1 on $200$ randomly sampled architectures from the
Transformer-XL backbone, trained on WikiText-103. Figure 5 shows the SRC
between low-cost proxies and the ground truth ranking after full training.
Figure 5: SRC between low-cost proxies and the ground truth ranking after full
training of 200 randomly sampled Transformers. The decoder parameter count
obtains the best SRC with zero cost.
We measure the cost of each proxy in terms of FLOPs. As seen, the evaluated
low-cost proxies have a strong correlation with the ground truth ranking (even
the lowest performing relu_log_det has $>0.8$ SRC), validating the
effectiveness of training-free NAS on autoregressive Transformers. The lower
performance of relu_log_det can be attributed to the much higher frequency of
ReLU activations in CNNs, for which the method was originally developed,
compared to Transformer-based architectures. Our analysis of randomly selected
models with homogeneous structures also shows a strong correlation between the
low-cost proxies and validation perplexity, with decoder parameter count
outperforming other proxies (see Appendix D).
$\blacktriangleright$ Parameter Count. Figure 6a demonstrates the final
validation perplexity versus the total number of model parameters for $200$
randomly sampled architectures from GPT-2 and Transformer-XL backbones. This
figure contains two important observations: (1) the validation perplexity has
a downward trend as the number of parameters increases, (2) The discontinuity
is caused by the dominance of embedding parameters when moving to the small
Transformer regime. We highlight several example points in Figure 6a where the
architectures are nearly identical but the adaptive input embedding factor $k$
is changed. Changing $k\in\\{1,2,4\\}$ (shown with different colors in Figure
6a) varies the total parameter count without much influence on the validation
perplexity.
Figure 6: (a) Validation perplexity after full training versus total
parameters for $200$ randomly sampled architectures trained on WikiText-103.
The clear downward trend suggests a strong correlation between parameter count
and perplexity. (b), (c) Performance of parameter count proxies for ranking
the randomly sampled architectures with Transformer-XL and GPT-2 backbones.
The above observations motivate us to evaluate two proxies, i.e., total number
of parameters and decoder parameter count. Figures 6b and 6c demonstrate the
CR and SRC metrics evaluated on the $200$ randomly sampled models divided into
top$k\%$ bins based on their validation perplexity. As shown, the total number
of parameters generally has a lower SRC with the validation perplexity,
compared to decoder parameter count. This is due to the masking effect of
embedding parameters, particularly in the Transformer-XL backbone. The total
number of decoder parameters, however, provides a highly accurate, zero-cost
proxy with an SRC of $0.97$ with the perplexity over all models, after full
training. We further show the high correlation between decoder parameter count
and validation perplexity for Transformer architectures with homogeneous
decoder blocks in the supplementary material, Appendix D. While our main focus
is on autoregressive, decoder-only, Transformers, we provide preliminary
results on the ranking performance of parameter count proxies for encoder-only
and encoder-decoder Transformers in Appendix J.
### 4.3 How does variation in model topology affect decoder parameter count
as a proxy?
The low-cost proxies introduced in Section 3.1, rely on forward and backward
passes through the network. As such, they automatically capture the topology
of the underlying architecture via the dataflow. The decoder parameter count
proxy, however, is topology-agnostic. In this section, we investigate the
effect of topology on the performance of decoder parameter count proxy.
Specifically, we seek to answer whether for a given decoder parameter count
budget, the aspect ratio of the architecture, i.e., trading off the width
versus the depth, can affect the final validation perplexity.
We define the aspect ratio of the architecture as dmodel (=width), divided by
nlayer (=depth). This metric provides a sense of how skewed the topology is
and has been used in prior works which study scaling laws for language models
[22]. For a given decoder parameter count budget, we generate several random
architectures from the GPT-2 backbone with a wide range of the width-to-depth
aspect ratios555We control the aspect ratio by changing the width, i.e.,
dmodel while keeping dinner=$2\times$dmodel and nhead=$8$. The number of
layers is then derived such that the total parameter count remains the same..
The generated models span wide, shallow topologies (e.g., dmodel=$1256$,
nlayer=$2$) to narrow, deep topologies (e.g., dmodel=$112$, nlayer=$100$).
Figure 7(a) shows the validation perplexity of said architectures after full
training on WikiText-103 versus their aspect ratio. The maximum deviation
(from the median) of the validation perplexity is $<12.8\%$ for a given
decoder parameter count, across a wide range of aspect ratios $\in[1,630]$.
Our findings on the heterogeneous search space complement the empirical
results by [22] where decoder parameter count largely determines perplexity
for homogeneous Transformer architectures, irrespective of shape (see Figure 5
in [22]).
We observe stable training when scaling models from the GPT-2 backbone up to
$100$ layers, with the perplexity increasing only when the aspect ratio nears
$1$. Nevertheless, such deep models are not part of our search space as they
have high latency and are unsuitable for lightweight inference. For the
purposes of hardware-aware and efficient Transformer NAS, decoder parameter
count proxy holds a very high correlation with validation perplexity,
regardless of the architecture topology as shown in Figure 7(a). We further
validate the effect of topology on decoder parameter count proxy for the
Transformer-XL backbone in Figure 14 of Appendix E. Our results demonstrate
less than $7\%$ deviation (from the median) in validation perplexity for
different aspect ratios $\in[8,323]$.
Note that while models with the same parameter count have very similar
validation perplexities, the topology in fact affects their hardware
performance, i.e., latency (up to $2.8\times$) and peak memory utilization (up
to $5.5\times$), as shown in Figures 7(b) and 7(c). This motivates the need
for incorporating hardware metrics in NAS to find the best topology.
(a)
(b)
(c)
Figure 7: Validation perplexity after full training versus the (a) width-to-
depth aspect ratio, (b) latency, and (c) peak memory utilization. Models are
randomly generated from the GPT-2 backbone and trained on WikiText-103. For a
given decoder parameter count, we observe low variation in perplexity across
different models, regardless of their topology. The topology, however,
significantly affects the latency (up to $2.8\times$) and peak memory
utilization (up to $5.5\times$) for models with the same perplexity.
### 4.4 Pareto-frontier models for various hardware platforms
We run LTS on different target hardware and obtain a range of Pareto-optimal
architectures with various latency/memory/perplexity characteristics. During
search, we fix the adaptive input embedding factor to $k=4$ to search models
that are lightweight while ensuring nearly on-par validation perplexity with
non-adaptive input embedding. As the baseline Pareto, we benchmark the
Transformer-XL (base) and GPT-2 (small) models with homogeneous layers
$\in[1,16]$. This is because the straightforward way to produce architectures
of different latency/memory is varying the number of layers (layer-scaling)
[42, 10]. We compare our NAS-generated architectures with layer-scaled
backbones and achieve better validation perplexity and/or lower latency and
peak memory utilization. All baseline666The best reported result in the
literature for GPT-2 or Transformer-XL might be different based on the
specific training hyperparameters, which is orthogonal to our investigation.
and NAS-generated models are trained using the same setup enclosed in Table 2
of Appendix B.
Figure 8 shows the Pareto-frontier architectures found by LTS versus the
layer-scaled baseline. Here, all models are trained on the LM1B dataset (See
Figure 16 in Appendix G for results on WikiText-103). Note that the Pareto-
frontier search is performed in a $3$-dimensional space, an example of which
is enclosed in Appendix F, Figure 15. For better visualization, in Figure 8 we
plot $2$-dimensional slices of the Pareto-frontier with validation perplexity
on the y-axis and one hardware performance metric (either latency or memory)
on the x-axis.
As seen, in the low-latency regime, LTS consistently finds models that have
significantly lower perplexity compared to naive scaling of the baseline
Transformer-XL or GPT-2. On the Transformer-XL backbone, LTS finds
architectures with an average of $19.8\%$ and $28.8\%$ lower latency and
memory, while achieving similar perplexity compared to the baseline on ARM
CPU. Specifically, the perplexity of the $16$-layer Transformer-XL base can be
replicated on the ARM device with a lightweight model that is $1.6\times$
faster and utilizes $1.9\times$ less memory during execution. On the Corei7
CPU, the Pareto-frontier models found by LTS are on average $25.8\%$ faster
and consume $30.0\%$ less memory under the same validation perplexity
constraint. In this setting, LTS finds a model that replicates the perplexity
of the $16$-layer Transformer-XL base while achieving $1.7\times$ faster
runtime and $1.9\times$ less peak memory utilization. The savings are even
higher on the GPU device, where the NAS-generated models achieve the same
perplexity as the baseline with average $30.5\%$ lower latency and $27.0\%$
less memory. Specifically, an LTS model with the same perplexity as the
$16$-layer Transformer-XL base has $2.5\times$ lower latency and consumes
$2.0\times$ less peak memory on TITAN Xp.
On the GPT-2 backbone, NAS-generated models consume on average $11.8\%$ less
memory while achieving the same validation perplexity and latency on an ARM
CPU. The benefits are larger on Corei7 and TitanXP where the latency savings
are $13.8\%$ and $11.9\%$, respectively. The peak memory utilization also
decreases by $9.7\%$ and $12.9\%$, on average, compared to the baseline GPT-2s
on Corei7 and TITAN Xp. Notably, NAS finds new architectures with the same
perplexity as the $16$-layer GPT-2 with $1.3\times$, $1.5\times$ faster
runtime and $1.2\times$ lower memory utilization on Corei7 and TITAN Xp.
Our heterogeneous search space allows us to find a better parameter
distribution among decoder layers. Therefore, LTS delivers architectures with
better performance in terms of perplexity, while reducing both latency and
memory when compared to the homogeneous baselines. We provide the architecture
of all baseline and LTS models shown in Figure 8 in Tables 4-7 of Appendix I.
Figure 8: 2D visualization of the perplexity versus latency and memory Pareto-
frontier found by LTS, versus the scaled backbone models with varying number
of layers, trained on the LM1B dataset. Architectural parameters for models
shown here are detailed in Appendix I.
$\blacktriangleright$ Search Efficiency. The main component in LTS search time
is the latency/peak memory utilization measurement for candidate architectures
since evaluating the model perplexity is instant using the decoder parameter
count. Therefore, our search finishes in a few hours on commodity hardware,
e.g., taking only $0.9$, $2.6$, and $17.2$ hours on a TITAN Xp GPU, Corei7
CPU, and an ARM core, respectively. To provide more context into the timing
analysis, full training of even one $16$-layer Transformer-XL base model on
LM1B using a machine with $8\times$ NVIDIA V100 GPUs takes $15.8$ hours. Once
the Pareto-frontier models are identified, the user can pick a model based on
their desired hardware constraints and fully train it on the target dataset.
LTS is an alternate paradigm to that of training large supernets; our search
can run directly on the target device and GPUs are only needed for training
the final chosen Pareto-frontier model after search.
In Table 1 we study the ranking performance of partial training ($500$ steps)
versus the decoder parameter count proxy for evaluating $1200$ architectures
from the Transformer-XL backbone during LTS search. Astonishingly the decoder
parameter count proxy gets higher SRC compared to partial training, while
effectively removing training from the inner loop of search for NAS.
| | Train
---
Iter
| GPU
---
Hours
| $CO_{2}$e
---
(lbs)
SRC
Full Training | 40,000 | 19,024 | 5433 | 1.0
Partial Training | 500 | 231 | 66 | 0.92
5,000 | 2690 | 768 | 0.96
# Decoder Params | 0 | 0 | $\sim$0 | 0.98
Table 1: Ranking abilities of full and partial training versus our proxy for
$1200$ models sampled during LTS search. Training time is reported for
WikiText-103 and NVIDIA V100 GPU. Decoder parameter count proxy obtains an SRC
of $0.98$ using zero compute. Figure 9: Average zero and one-shot accuracy
obtained by LTS models (dots) and the baseline OPT-$350$M (triangle) across
$14$ NLP tasks. Latency is measured on an A6000 NVIDIA GPU. Architectural
parameters for all models shown here are detailed in Appendix I.
### 4.5 Zero and one-shot performance comparison of LTS models with OPT
Zhang et al. [58] open-source a set of pre-trained decoder-only language
models, called OPT, which can be used for zero or few-shot inference on
various NLP tasks. Below, we compare the performance of LTS Pareto-frontier
models with the hand-designed OPT architecture in zero and one-shot settings.
We use LTS to search for language models with a GPT-2 backbone which have
$300$M to $500$M total parameters to compare with the $350$M parameter OPT.
Our search space is detailed in Appendix H. The search is conducted with
latency as the target hardware metric and decoder parameter count as a proxy
for perplexity. Once the search concludes, We train $20$ models from the
Pareto-frontier along with OPT-$350$M on $28$B tokens from the Pile [17]. The
pretrained models are then evaluated on $14$ downstream NLP tasks, namely,
HellaSwag [57], PIQA [4], ARC (easy and challenge) [9], OpenBookQA [28],
WinoGrande [34], and SuperGLUE [44] benchmarks BoolQ, CB, COPA, WIC, WSC,
MultiRC, RTE, and ReCoRD. The training hyperparameters and the evaluation
setup are outlined in Appendix B. Figure 9 shows the overall average accuracy
obtained across all $14$ tasks versus the inference latency for LTS models and
the baseline OPT. As shown, NAS-generated models achieve a higher average
accuracy with lower latency compared to the hand-designed OPT-$350$M model. We
provide a per-task breakdown of zero and one-shot accuracy in Appendix H,
Figure 17.
$\blacktriangleright$ Zero-shot Performance. Figure 17(a) demonstrates the
zero-shot accuracy obtained by LTS and OPT-$350$M on the benchmarked tasks.
Compared to the OPT-$350$M architecture, LTS finds models that achieve higher
accuracy and lower latency in the zero-shot setting on all evaluated
downstream tasks. Specifically, the maximum achievable accuracy of our NAS-
generated models is $0.2-8.6\%$ higher than OPT-$350$M with an average speedup
of $1.2\times$. If latency is prioritized, LTS delivers models which are, on
average, $1.5\times$ faster and up to $4.6\%$ more accurate than OPT-$350$M .
$\blacktriangleright$ One-shot Performance. Similar trends can be observed for
one-shot evaluation as shown for different tasks in Figure 17(b). LTS Pareto-
frontier models improve the per-task accuracy of OPT-$350$M on $12$ out of
$14$ tasks by $0.1-8.0\%$, while achieving an average speedup of $1.2\times$.
On the same tasks, LTS Pareto-frontier includes models that enjoy up to
$1.6\times$ speedup over OPT-$350$M with an average $1.5\%$ higher accuracy.
On the RTE task, the best LTS model has $0.4\%$ lower accuracy but $1.6\times$
faster runtime. On the WSC task, the best performing LTS model obtains a
similar one-shot accuracy as OPT-$350$M, but with $1.5\times$ faster runtime.
## 5 Limitations and Future Work
Decoder parameter count provides a simple yet accurate proxy for ranking
autoregressive Transformers. This should serve as a strong baseline for future
works on Transformer NAS. Our focus is mainly on autoregressive, decoder-only
transformers. We therefore, study perplexity as the commonly used metric for
language modeling tasks. Nevertheless, recent literature on scaling laws for
Transformers suggest a similar correlation between parameter count and task
metrics may exist for encoder only (BERT-style) Transformers or encoder-
decoder models used in neural machine translation (NMT) [19]. Additionally,
recent findings [39] show specific scaling laws exist between model size and
downstream task metrics, e.g., GLUE [43]. Inspired by these observations, we
provide preliminary studies that suggest parameter count proxies may be
applicable to Transformers in other domains. Detailed investigations of such
zero-cost proxies for NAS on heterogeneous BERT-style or NMT models with new
performance metrics is an important future avenue of research.
## 6 Acknowledgements
Professor Farinaz Koushanfar’s effort has been in parts supported by the NSF
TILOS AI institute, award number CCF-2112665.
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## Checklist
1. 1.
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1. (a)
Do the main claims made in the abstract and introduction accurately reflect
the paper’s contributions and scope? [Yes]
2. (b)
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Appendix E
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## Appendix A Preliminaries on Autoregressive Transformers
Perplexity. Perplexity is a widely used metric for evaluating the performance
of autoregressive language models. This metric encapsulates how well the model
can predict a word. Formally, perplexity of a language model $M$ is derived
using the entropy formula as:
$Perplexity(M)=2^{H(L,M)}=2^{-\sum_{x}L(x).log(M(x)))}$ (1)
where $L$ represents the ground truth words. As seen, the perplexity is
closely tied with the cross-entropy loss of the model, i.e., $H(L,M)$.
Parameter count. Contemporary autoregressive Transformers consist of three
main components, namely, the input embedding layer, hidden layers, and the
final (softmax) projection layer. The embedding layer often comprises look-up
table-based modules that map the input language tokens to vectors. These
vectors then enter a stack of multiple hidden layers a.k.a, the decoder
blocks. Each decoder block is made up of an attention layer and a feed-forward
network. Once the features are extracted by the stack of decoder blocks, the
final prediction is generated by passing through the softmax projection layer.
When counting the number of parameters in an autoregressive Transformer, the
total parameters enclosed in the hidden layers is dubbed the decoder parameter
count or equivalently, the non-embedding parameter count. These parameters are
architecture-dependent and do not change based on the underlying tokenization
or the vocabulary size. The embedding parameter count, however, accounts for
the parameters enclosed in the input embedding layer as well as the final
softmax projection layer as they are both closely tied to the word embedding
and vocabulary size. We visualize an autoregressive Transformer in Figure 10,
where the orange blocks contain the decoder parameters and grey blocks hold
the embedding parameters.
Figure 10: High-level visualization of different components in autoregressive
Transformers. Here, the parameters enclosed in the orange blocks are counted
as decoder parameters, while the parameters contained in the gray boxes are
included in the embedding parameter count.
## Appendix B Experimental Setup
Datasets. We conduct experiments on three datasets, namely, WikiText-103,
LM1B, and the Pile. The datasets are tokenized using word-level and byte-pair
encoding for models with Transformer-XL and GPT-2 backbones, respectively.
Training and Evaluation. We adopt the open-source code by [15] and [30] to
implement the GPT-2 and Transformer-XL backbones, respectively. We further use
the source code provided in [2] to implement the baseline OPT-$350$M and LTS
models used in zero and one-shot evaluations. Table 2 encloses the
hyperparameters used for training. In this paper, each model is trained
separately from scratch. In many scenarios, the user only needs to train one
model from the Pareto-frontier, which is selected based on their needs for
perplexity, latency, and memory. However, if the users are interested in
multiple models, they can either train all models separately or fuse them and
train them simultaneously using weight sharing as in [46, 55]. As an example,
if the user is interested in two particular models from the Pareto-frontier
which have $3$ and $5$ layers, the user can fuse them into a single $5$-layer
(super)net and train both models at the same time using weight sharing. The
cost of training this supernet is roughly the same as training a $5$-layer
model. Therefore, this simple trick can amortize the training cost for Pareto-
frontier models.
Throughout the paper, validation perplexity is measured over a sequence length
of $192$ and $32$ tokens for WikiText-103 and LM1B datasets, respectively. For
our zero and one-shot evaluations, we adopt the open-source code by Gao et al.
[18]. Inference latency and peak memory utilization are measured on the target
hardware for a sequence length of $192$, averaged over $10$ measurements. The
sequence length is increased to $2048$ for latency comparison with the OPT
baseline. We utilize PyTorch’s native benchmarking interface for measuring the
latency and memory utilization of candidate architectures.
Choice of Training Hyperparameters. For each backbone, dataset, and task, we
use the same training setup for all models generated by NAS. This is the
common setting used in the vast majority of NAS papers, including popular
benchmarks [12, 35, 56], due to the extremely high cost of NAS combined with
hyperparameter optimization (HPO). The setup for our training hyperparameters
is based on the evidence provided in prior art in Transformer design [22, 33,
19] and NAS [46, 62, 8]. Specifically, for the range of model sizes studied in
this paper, prior work adopts the same batch size (see Table 2.1 in GPT-3
[5]), which suggests there is no significant benefit in optimizing the batch
size per architecture. The original GPT-3 paper [5] also adopts the same
learning rate scheduler for all models, regardless of their size. Similarly,
authors of [22] show that the choice of learning rate scheduler does not have
a significant effect on final model performance, which further validates that
exploration of the scheduler will not alter the empirical findings in this
paper.
Authors of [22] further provide a rule-of-thumb for setting the optimal
learning rate (see Equation D.1 of [22]). This rule shows that changes in the
optimal learning rate are negligible for the range of model sizes in our
search space. We validate this by conducting an experiment that aims to find
the optimal learning rate per architecture. We sweep the learning rate
$\in[0.0001,0.001,0.01,0.1]$ for $100$ randomly sampled models from the GPT-2
backbone and train them on WikiText-103. The studied models span a wide range
of configurations with $2-16$ layers and $2-65$M total parameters. We then
pick the optimal learning rate for each architecture, i.e., the one which
results in the lowest perplexity. We remeasure the correlation between newly
obtained perplexities and the decoder parameter county proxy. Our learning
rate optimization experiment results in two important observations: 1) for the
vast majority of the architectures ($98\%$), the optimal learning rate is
equal to $0.01$, i.e., the value used in all experiments (see Table 2), and 2)
the ranking of architectures after convergence remains largely unchanged,
leading to a correlation of $0.93$ with decoder parameter count, compared to
$0.96$ when using the same learning rate for all models. The above evidence
suggests that the same training setup can be used for all architectures in the
search space, without affecting the results.
Table 2: LTS training hyperparameters for different backbones. Here, DO
represents dropout layers.
Backbone | Dataset | Tokenizer | # Vocab | Optim. | # Steps | Batch size | LR | Scheduler | Warmup | DO | Attn DO
---|---|---|---|---|---|---|---|---|---|---|---
Transformer-XL | WT103 | Word | 267735 | LAMB [54] | 4e4 | 256 | 1e-2 | Cosine | 1e3 | 0.1 | 0.0
LM1B | Word | 267735 | Adam | 1e5 | 224 | 2.5e-4 | Cosine | 2e4 | 0.0 | 0.0
GPT-2 | WT103 | BPE | 50264 | LAMB [54] | 4e4 | 256 | 1e-2 | Cosine | 1e3 | 0.1 | 0.1
LM1B | BPE | 50264 | LAMB [54] | 1e5 | 224 | 2.5e-4 | Cosine | 2e4 | 0.1 | 0.1
Pile | BPE | 50272 | Adam | 5.48e4 | 256 | 3e-5 | Linear | 715 | 0.1 | 0.0
Search Setup. Evolutionary search is performed for $30$ iterations with a
population size of $100$; the parent population accounts for $20$ samples out
of the total $100$; $40$ mutated samples are generated per iteration from a
mutation probability of $0.3$, and $40$ samples are created using crossover.
## Appendix C How Good is the Decoder Parameters Proxy for Pareto-frontier
Search?
In this Section, we validate whether the decoder parameter count proxy
actually helps find Pareto-frontier models which are close to the ground truth
Pareto front. We first fully train all $1200$ architectures sampled from the
Transformer-XL backbone during the evolutionary search (1). Using the
validation perplexity obtained after full training, we rank all sampled
architectures and extract the ground truth Pareto-frontier of perplexity
versus latency. We train the models on the WikiText-103 dataset and benchmark
Intel Xeon E5-2690 CPU as our target hardware platform for latency measurement
in this experiment.
Figure 11 represents a scatter plot of the validation perplexity (after full
training) versus latency for all sampled architectures during the search. The
ground truth Pareto-frontier, by definition, is the lower convex hull of the
dark navy dots, corresponding to models with the lowest validation perplexity
for any given latency constraint. We mark the Pareto-frontier points found by
the training-free proxy with orange color. As shown, the architectures that
were selected as the Pareto-frontier by the proxy method are either on or very
close to the ground truth Pareto-frontier.
Figure 11: Perplexity versus latency Pareto obtained from full training of
$1200$ architectures sampled during NAS on Transformer-XL backbone. Orange
points are the Pareto-frontier extracted using decoder parameter count proxy,
which lies close to the actual Pareto-frontier. Decoder parameter count holds
an SRC of $0.98$ with the ground truth perplexity after full training.
We define the mean average perplexity difference as a metric to evaluate the
distance ($d_{avg}$) between the proxy and ground truth Pareto-frontier:
$d_{avg}=\frac{1}{N}\sum_{i=1}^{N}\frac{|p_{i}-p_{gt,i}|}{p_{gt,i}}$ (2)
Here, $p_{i}$ denotes the $i$-th point on the proxy Pareto front and
$p_{gt,i}$ is the closest point, in terms of latency, to $p_{i}$ on the ground
truth Pareto front. The mean average perplexity difference for Figure 11 is
$d_{avg}=0.6\%$. This small difference validates the effectiveness of our
zero-cost proxy in correctly ranking the sampled architectures and estimating
the true Pareto-frontier. In addition to the small distance between the prxoy-
estimated Pareto-frontier and the ground truth, our zero-cost proxy holds a
high SRC of $0.98$ over the entire Pareto, i.e., all $1200$ sampled
architectures.
We further study the decoder parameter proxy in scenarios where the range of
model sizes provided for search is limited. We categorize the total $1200$
sampled architectures into different bins based on the decoder parameters.
Figure 12 demonstrates the SRC between the decoder parameter count proxy and
the validation perplexity after full training for different model sizes. The
proposed proxy provides a highly accurate ranking of candidate architectures
even when exploring a small range of model sizes.
Figure 12: SRC between the decoder parameter count proxy and validation
perplexity. Results are gathered on $1200$ models grouped into four bins based
on their decoder parameter count. Our proxy performs well even when exploring
within a small range of model sizes.
## Appendix D Analysis on Homogeneous Models
In this section, we evaluate the efficacy of the proposed proxies on the
homogeneous search space, i.e., when all decoder layers have the same
parameter configuration. In this scenario, the parameters are sampled from the
valid ranges in Section 3 to construct one decoder block. This block is then
replicated based on the selected nlayer to create the Transformer
architecture. In what follows, we provide experimental results gathered on
$100$ randomly sampled Transformer models from the Transformer-XL backbone
with homogeneous decoder blocks, trained on WikiText-103.
$\blacktriangleright$ Low-cost Proxies. Figure 13(a) demonstrates the SRC
between various low-cost methods and the validation perplexity after full
training. On the horizontal axis, we report the total computation required for
each proxy in terms of FLOPs. Commensurate with the findings on the
heterogeneous models, we observe a strong correlation between the low-cost
proxies and validation perplexity, with the decoder parameter count
outperforming other proxies. Note that we omit the relu_log_det method from
Figure 13(a) as it provides a low SRC of $0.42$ due to heavy reliance on ReLU
activations.
(a)
(b)
Figure 13: Experiments conducted on $100$ randomly sampled Transformers with
homogeneous decoder blocks, trained on WikiText-103. (a) SRC between the
ranking obtained from low-cost proxies and the ground truth ranking after full
training. The decoder parameter count obtains the best SRC with zero cost. (b)
Performance of parameter count proxies. The decoder parameter count provides a
very accurate ranking proxy with an SRC of $0.95$ over all models.
$\blacktriangleright$ Parameter Count. As seen in Figure 13(b), the total
parameter count has a low SRC with the validation perplexity while the decoder
parameter count provides an accurate proxy with an SRC of $0.95$ over all
architectures. These findings on the homogeneous search space are well-aligned
with the observations in the heterogeneous space.
## Appendix E How Does Model Topology Affect the Training-free Proxies?
Figure 14(a) shows the validation perplexity versus the aspect ratio of random
architectures sampled from the Transformer-XL backbone and trained on
WikiText-103. Here, the models span wide, shallow topologies (e.g.,
dmodel=$1024$, nlayer=$3$) to narrow, deep topologies (e.g., dmodel=$128$,
nlayer=$35$). The maximum change in the validation perplexity for a given
decoder parameter count is $<7\%$ for a wide range of aspect ratios
$\in[8,323]$. Nevertheless, for the same decoder parameter count budget, the
latency and peak memory utilization vary by $1.3\times$ and $2.0\times$ as
shown in Figures 14(b) and 14(c).
For deeper architectures (more than $40$ layers) with the Transformer-XL
backbone, we observe an increase in the validation perplexity, which results
in a deviation from the pattern in Figure 14a. This observation is associated
with the inherent difficulty in training deeper architectures, which can be
mitigated with the proposed techniques in the literature [47]. Nevertheless,
such deep models have a high latency, which makes them unsuitable for
lightweight inference. For hardware-aware and efficient Transformer NAS, our
search space contains architectures with less than $16$ layers. In this
scenario, the decoder parameter count proxy holds a very high correlation with
validation perplexity, regardless of the architecture topology as shown in
Figure 14(a).
(a)
(b)
(c)
Figure 14: Validation perplexity after full training versus (a) the width-to-
depth aspect ratio, (b) latency, and (c) peak memory utilization. Models are
randomly generated from the Transformer-XL backbone and trained on
WikiText-103. For a given decoder parameter count, we observe low variation in
perplexity across different models, regardless of their topology. The
topology, however, significantly affects the latency and peak memory
utilization for models with the same perplexity.
## Appendix F 3D Pareto Visualization
Figure 15 visualizes the $3$-dimensional Pareto obtained during search on the
GPT-2 backbone. Here, the black and blue points denote regular and Pareto-
frontier architectures, respectively. The pair of red dots are architectures
which match in both memory and decoder parameter count ($\sim$ perplexity).
However, as shown, their latency differs by $2\times$. The pair of green
points correspond to models with the same decoder parameter count ($\sim$
perplexity) and latency, while the memory still differs by $30$MB, which is
non-negligible for memory-constrained application. In a $2$-objective Pareto-
frontier search of perplexity versus memory (or latency), each pair of red (or
green) dots will result in similar evaluations. While in reality, they have
very different characteristics in terms of the overlooked metric. This
experiment validates the need for multi-objective Pareto-frontier search,
which simultaneously takes into account multiple hardware performance metrics.
Figure 15: 3D visualization of our multi-objective NAS for the GPT-2 backbone
on TITAN Xp GPU. Architectures with similar memory and decoder parameter count
can result in drastically different runtimes (up to $2\times$ difference).
Similarly, architectures with similar decoder parameter count and latency may
have different peak memory utilization. Therefore, it is important to perform
multi-objective NAS where several hardware characteristics are simultaneously
taken into account when extracting the Pareto-frontier. Figure 16: 2D
visualization of the perplexity versus latency and memory Pareto-frontier
found by LTS and scaled backbone models with varying number of layers. All
models are trained on the WikiText-103 dataset. The architectural parameters
for all models are enclosed in Appendix I.
## Appendix G LTS Pareto-frontier on WikiText-103
We compare the Pareto-frontier architectures found by LTS with the baseline
after full training on the WikiText-103 dataset in Figure 16. Commensurate
with the findings on the LM1B dataset, the NAS-generated models outperform the
baselines in at least one of the three metrics, i.e., perplexity, latency, and
peak memory utilization. We note that the gap between the baseline models and
those obtained from NAS is larger when training on the LM1B dataset. This is
due to the challenging nature of LM1B, which exceeds the WikiText-103 dataset
size by $\sim 10\times$. Thus, it is harder for hand-crafted baseline models
to compete with the optimized LTS architectures on LM1B.
On the Transformer-XL backbone, the models on LTS Pareto-frontier for the ARM
CPU have, on average, $3.8\%$ faster runtime and $20.7\%$ less memory under
the same validation perplexity budget. On the Corei7, the runtime and memory
savings increase to $13.2\%$ and $19.6\%$, respectively, while matching the
baseline perplexity. We achieve our highest benefits on TITAN Xp GPU where LTS
Pareto-frontier models have, on average, $31.8\%$ lower latency and $21.5\%$
lower memory utilization. Notably, the validation perplexity of the baseline
$16$-layer Transformer-XL base can be achieved with a lightweight model with
$2.1\times$ lower latency while consuming $1.6\times$ less memory at runtime.
On the GPT-2 backbone, LTS achieves $6.3-11.2$ lower perplexity in the low-
latency-and-memory regime. As we transition to larger models and higher
latency, our results show that the GPT-2 architecture is nearly optimal on
WikiText-103 when performing inference on a CPU. The benefits are more
significant when targeting a GPU; For any given perplexity achieved by the
baseline, LTS Pareto-frontier on TITAN Xp delivers, on average, $9.0\%$ lower
latency and $4.5\%$ lower memory. Therefore, the perplexity and memory of the
baseline $16$-layer GPT-2 can be achieved by a new model that runs $1.4\times$
faster and consumes $1.2\times$ less memory on TITAN Xp.
## Appendix H Zero and One-Shot Evaluation of LTS Models
(a) zero-shot
(b) one-shot
Figure 17: LTS Pareto-frontier models (dots) achieve a higher zero and one-
shot accuracy with lower latency compared to the hand-designed OPT-$350$M
model (triangle). Latency is measured on an A6000 NVIDIA GPU. Architectural
parameters for all models shown here are detailed in Appendix I.
For this experiment, we design our search space to cover models with a similar
parameter count budget as the OPT-$350$M model. To this end, we search over
the following values for the architectural parameters:
nlayer$\in\\{3,\dots,29|1\\}$, dmodel$\in\\{512,\dots,1472|64\\}$,
dinner$\in\\{512,\dots,6080|64\\}$, and nhead$\in\\{2,4,8,16\\}$. To directly
compare with OPT, we use a generic, non-adaptive embedding layer for our
models. Therefore, the search space does not include the $k$ factor and
dembed=dmodel.
Figures 17(a) and 17(b) show the per-task zero and one-shot performance of LTS
models and OPT-$350$M. Please refer to Section 4.5 of the main paper for a
summarization of the results in these figures.
## Appendix I Architecture Details
Tables 4, 5, 6, 7 enclose the architecture parameters for the baseline and
NAS-generated models in Figures 8 and 16 for Transformer-XL and GPT-2
backbones. Table 3 further holds the architecture details of models used in
our zero and one-shot evaluations of Figures 9 and 17. For each target
hardware, the rows of the table are ordered based on increasing decoder
parameter count (decreasing validation perplexity). For all models,
dhead=dmodel/nhead and dembed=dmodel. For models in Tables 4, 5, 6, 7, the
adaptive input embedding factor is set to $k=4$. The models in Table 3,
however, use the generic, non-adaptive, input embedding ($k=1$) following the
original OPT architecture [58].
## Appendix J Transformers in other Domains
In what follows, we perform preliminary experiments on Transformers used on
other domains to investigate the applicability of parameter-based proxies for
ranking.
Encoder-only Transformers. BERT [11] is a widely popular Transformer composed
of encoder blocks, which is used in a variety of tasks, e.g., question
answering and language inference. The main difference between BERT and the
Transformers studied in this paper is the usage of bidirectional versus causal
attention. Specifically, the encoder blocks in BERT are trained to compute
attention between each input token and all surrounding tokens. In
autoregressive models, however, attention is only computed for tokens
appearing prior to the current token. BERT is trained with a mixture of masked
language modeling and next sentence prediction objectives to ensure
applicability to language modeling as well as downstream language
understanding tasks. We use the architectural parameters described in Section
3 to construct the search space and randomly sample $300$ models from the BERT
backbone. We then train all models on WikiText-103 for 40K steps following the
training setup provided in the original BERT paper [11] for the batch size,
sequence length, optimizer, learning rate, vocabulary size, and tokenizer.
Figure 18 demonstrates the CR and SRC of encoder parameter count and test
perplexity measured on various top$k\%$ performing BERT models. As seen, both
the encoder and total parameter count provide a highly accurate proxy for test
perplexity of BERT, achieving an SRC of $0.96$ and $0.98$, respectively. This
trend suggests that parameter-based proxies for NAS can be applicable to
encoder-only search spaces as well.
Figure 18: Performance of parameter count proxies on $300$ randomly sampled
models from the BERT backbone, trained on WikiText-103. Both encoder and total
parameter counts provide a very accurate ranking proxy with an SRC of 0.96 and
0.98 over all models, respectively.
Encoder-Decoder Transformers. Transformers in this domain comprise both
encoder and decoder layers with bidirectional and causal attention
computation. This unique structure makes these models suitable for sequence-
to-sequence tasks such as Neural Machine Translation (NMT). Recent work [19]
shows that the performance of encoder-decoder Transformers also follows a
scaling law with model size. This power-law behavior between model size and
performance can be leveraged to develop training-free proxies for ranking
these architectures during search. We test our hypothesis by performing
experiments on the open-source NMT benchmark by [60, 59] which consists of
$2000$ Transformers trained on various language pairs. The pre-trained
Transformers in this benchmark have homogeneous layers, i.e., the
architectural parameters are the same for all layers and identical for the
encoder and the decoder. In addition to architectural parameters, the search
space for this benchmark also includes various BPE tokenization and learning
rates. We, therefore, pre-process the benchmark by gathering all instances of
Transformers for a fixed BPE. Then for each given architecture, we keep the
results corresponding to the best-performing learning rate.
Figure 19 shows a heatmap of the SRC between parameter count proxies and
perplexity as well as the BLEU score. As seen, the ranking performance of
total parameter count versus non-embedding parameter count, i.e., parameters
enclosed in the encoder and decoder blocks, is largely similar. On certain
tasks, e.g., ‘ja-en’, ‘so-en’, and ‘sw-en’ the parameter count proxies perform
quite well, achieving a high SRC with both the BLEU score and perplexity.
Interestingly, on ‘so-en’ and ‘sw-en’, the parameter count and performance are
inversely correlated, which may be due to the limited training data for these
language pairs which gives smaller models a leading advantage over larger
architectures. While these preliminary results show promise for parameter-
based proxies in NAS for NMT, several aspects require further investigation,
e.g., the effect of architectural heterogeneity and dataset size on the
performance of these proxies. Studying these aspects may perhaps lead to a new
formulation of training-free proxies for NMT and are out of scope for this
paper.
Figure 19: SRC between parameter count proxies and performance metrics, i.e.,
BLEU score and perplexity (PPL) for translation between various language
pairs. The “NonEmbParams” label denotes the parameters enclosed in the encoder
and decoder blocks while the “TotalParams” label corresponds to the total
parameter count including those in the embedding layers. Here, darker versus
lighter colors show a high positive and negative correlation, respectively.
## Appendix K Ethics Statement and Broader Impact
We provide an extremely lightweight method for NAS on autoregressive
Transformers. Our work is likely to increase the adoption of NAS in the NLP
domain, providing several prevalent benefits:
Firstly, more widespread adoption of automated techniques, e.g., NAS
eliminates the need for laborious trials and error for manual design of
Transformer architectures, freeing up hundreds of hours of man-power as well
as computational resources. Secondly, automating architecture design can
trigger the generation of new models with superior performance, which benefits
the ever-growing applications of NLP in the everyday life. Finally, by making
the search algorithm efficient, we ensure it can be accessible to the general
scientific public without need for any expensive mode training, thereby
minimizing the unwanted byproducts of the Deep Learning era such as the carbon
footprint, and power consumption. While the benefits of automation in NLP are
plenty, it can lead to potential side effects that have not been yet fully
unveiled. Since our work advances the use of NAS in the NLP design pipeline,
there is need for scrutiny of the models which have been automatically
designed with respect to aspects such as bias, misinformation, and nefarious
activity, to name a few.
Table 3: Detailed architectural parameters for all models in Figure 9 with
GPT-2 backbone.
| nlayer | dmodel | nhead | dinner | DecoderParams (M)
---|---|---|---|---|---
baseline (OPT-$350$M) | 24 | 1024 | 16 | 4096 | 304.4
M1 | 26 | 1024 | 16 | 2816 | 261.4
M2 | 15 | 1280 | 16 | 4480 | 273.2
M3 | 24 | 1280 | 8 | 1856 | 274.3
M4 | 16 | 1344 | 8 | 3840 | 283.8
M5 | 14 | 1344 | 8 | 4800 | 284.8
M6 | 20 | 1216 | 4 | 3456 | 289.2
M7 | 16 | 1344 | 16 | 4096 | 294.8
M8 | 28 | 1344 | 8 | 1344 | 306.6
M9 | 28 | 1088 | 8 | 2816 | 306.7
M10 | 26 | 1152 | 16 | 2816 | 309.4
M11 | 25 | 832 | 2 | 5760 | 310.9
M12 | 20 | 1280 | 16 | 3456 | 310.9
M13 | 19 | 1280 | 8 | 3840 | 314.2
M14 | 26 | 1152 | 4 | 3008 | 320.9
M15 | 19 | 1472 | 8 | 2816 | 325.5
M16 | 13 | 1472 | 4 | 5568 | 329.0
M17 | 14 | 1480 | 2 | 5824 | 367.3
M18 | 20 | 1152 | 8 | 5760 | 374.3
M19 | 26 | 1024 | 4 | 5696 | 414.8
M20 | 25 | 1408 | 8 | 3136 | 422.3
Table 4: Detailed architectural parameters for all models in Figure 8 with
Transformer-XL backbone.
| nlayer | dmodel | nhead | dinner | DecoderParams (M)
---|---|---|---|---|---
baseline | $\in$[1,16] | 512 | 8 | 2048 | -
ARM | M1 | 2 | 512 | [2, 2] | [1216, 1280] | 5.2
M2 | 3 | 320 | [2, 4, 2] | [1472, 2368, 3392] | 6.2
M3 | 2 | 512 | [2, 2] | [2560, 2176] | 7.5
M4 | 2 | 512 | [2, 2] | [3904, 1792] | 8.5
M5 | 2 | 640 | [2, 2] | [3520, 3456] | 13.0
M6 | 2 | 704 | [8, 2] | [3904, 3968] | 16.1
M7 | 2 | 832 | [2, 2] | [3264, 3968] | 19.0
M8 | 2 | 960 | [2, 2] | [3648, 3968] | 23.9
M9 | 2 | 960 | [2, 2] | [3904, 3968] | 24.4
M10 | 3 | 960 | [2, 2, 2] | [1856, 2368, 3392] | 28.5
M11 | 3 | 832 | [2, 2, 2] | [3904, 3968, 3008] | 28.5
M12 | 3 | 960 | [2, 4, 2] | [3328, 2368, 3200] | 30.9
M13 | 3 | 960 | [4, 2, 2] | [3648, 3584, 3584] | 34.6
M14 | 3 | 960 | [2, 2, 2] | [3904, 3584, 3456] | 34.9
M15 | 3 | 960 | [2, 2, 8] | [4032, 3968, 3904] | 36.7
M16 | 4 | 896 | [4, 2, 8, 2] | [3904, 3008, 3520, 3584] | 41.2
M17 | 4 | 960 | [8, 8, 8, 4] | [4032, 3968, 2880, 3200] | 45.5
M18 | 4 | 960 | [2, 2, 2, 2] | [3840, 3904, 3520, 3072] | 46.0
M19 | 4 | 960 | [2, 2, 2, 2] | [4032, 3648, 3136, 4032] | 47.0
M20 | 4 | 960 | [8, 2, 4, 8] | [4032, 3584, 3840, 3584] | 47.4
M21 | 4 | 960 | [2, 2, 4, 2] | [3904, 3968, 3840, 3584] | 47.8
M22 | 5 | 960 | [2, 2, 2, 2, 2] | [3904, 3968, 3264, 3456, 3200] | 57.3
M23 | 5 | 960 | [2, 2, 2, 8, 2] | [3904, 3648, 3136, 3648, 3840] | 58.0
M24 | 6 | 960 | [2, 2, 2, 2, 2, 8] | [3328, 2624, 3392, 2944, 3008, 3904] | 64.6
M25 | 6 | 960 | [2, 2, 4, 2, 8, 8] | [3584, 2624, 3392, 3968, 3008, 3328] | 65.9
M26 | 6 | 960 | [2, 4, 2, 2, 2, 2] | [2112, 3840, 3328, 3264, 3968, 3648] | 66.4
M27 | 6 | 960 | [2, 4, 2, 2, 8, 2] | [3904, 3008, 3392, 3648, 3392, 3584] | 67.9
M28 | 6 | 960 | [2, 2, 2, 2, 2, 4] | [3968, 3968, 3456, 3456, 3776, 2432] | 68.1
M29 | 6 | 960 | [2, 4, 8, 4, 2, 8] | [3904, 3008, 3392, 3200, 3968, 3904] | 68.8
M30 | 6 | 960 | [8, 8, 2, 4, 2, 4] | [3904, 3648, 3136, 3648, 3200, 3840] | 68.8
M31 | 6 | 960 | [8, 4, 8, 4, 2, 8] | [3904, 3648, 3392, 3200, 3968, 3840] | 69.9
M32 | 8 | 896 | [4, 2, 2, 4, 4, 2, 4, 8] | [3584, 3968, 3392, 3904, 2240, 1856, 2560, 3264] | 76.6
M33 | 8 | 896 | [4, 2, 2, 2, 4, 4, 4, 2] | [3584, 3584, 3520, 2368, 2752, 4032, 3520, 3264] | 79.9
M34 | 9 | 896 | [4, 2, 4, 4, 8, 2, 8, 8, 2] | [3840, 3136, 3520, 2880, 3200, 3008, 3328, 2560, 3136] | 87.5
M35 | 8 | 960 | [2, 4, 4, 4, 4, 8, 2, 2] | [3968, 3584, 3520, 3072, 3968, 4032, 1856, 3712] | 90.2
M36 | 12 | 832 | [2, 4, 4, 2, 2, 8, 8, 8, 4, 4, 2, 8] | [3136, 2112, 2112, 2368, 2752, 2432, 2432, 2176, 3456, 3712, 2880, 3712] | 97.0
M37 | 9 | 960 | [4, 4, 8, 2, 2, 2, 8, 8, 2] | [2112, 3008, 3520, 3648, 3968, 4032, 1984, 3200, 3520] | 97.2
M38 | 9 | 960 | [8, 2, 4, 2, 8, 8, 8, 2, 2] | [3968, 3008, 3520, 3200, 3200, 4032, 1984, 2816, 3520] | 97.7
M39 | 12 | 832 | [4, 4, 4, 2, 2, 8, 4, 8, 2, 8, 2, 8] | [3136, 3968, 2112, 2368, 3072, 2240, 2624, 2112, 3456, 3072, 2880, 3264] | 98.7
Corei7 | M1 | 2 | 384 | [2, 2] | [896, 2816] | 4.3
M2 | 2 | 576 | [2, 2] | [1792, 2816] | 8.6
M3 | 2 | 576 | [2, 2] | [1408, 3776] | 9.3
M4 | 2 | 832 | [2, 2] | [1728, 1536] | 12.4
M5 | 2 | 768 | [2, 2] | [3776, 1920] | 14.7
M6 | 2 | 768 | [2, 2] | [2112, 3584] | 14.7
M7 | 2 | 832 | [2, 2] | [3776, 3392] | 18.9
M8 | 2 | 832 | [2, 2] | [3968, 3584] | 19.5
M9 | 2 | 960 | [2, 4] | [1984, 3840] | 20.4
M10 | 2 | 960 | [8, 8] | [3968, 3584] | 23.7
M11 | 2 | 960 | [2, 2] | [3904, 3904] | 24.2
M12 | 3 | 896 | [2, 2, 2] | [2304, 3904, 3904] | 30.2
M13 | 3 | 960 | [2, 2, 4] | [2176, 3840, 2880] | 30.9
M14 | 3 | 960 | [2, 2, 4] | [3776, 2880, 3904] | 34.1
M15 | 3 | 960 | [2, 8, 2] | [3840, 3840, 3904] | 36.1
M16 | 3 | 960 | [2, 8, 8] | [3904, 3840, 3904] | 36.2
M17 | 3 | 960 | [2, 2, 8] | [3968, 3904, 3904] | 36.5
M18 | 4 | 960 | [2, 4, 2, 2] | [3904, 2112, 4032, 3584] | 44.6
M19 | 4 | 960 | [2, 2, 2, 4] | [2112, 3840, 3904, 3904] | 44.9
M20 | 4 | 960 | [2, 4, 8, 4] | [3776, 3392, 3520, 3904] | 46.5
M21 | 4 | 960 | [2, 2, 2, 4] | [3904, 3776, 3904, 3904] | 48.2
M22 | 5 | 960 | [2, 2, 2, 2, 2] | [3776, 1984, 3904, 3904, 3456] | 55.8
M23 | 5 | 960 | [2, 4, 2, 4, 2] | [3968, 3584, 3520, 3904, 3200] | 58.0
M24 | 5 | 960 | [2, 4, 4, 4, 2] | [3776, 3840, 3904, 3904, 3968] | 60.3
M25 | 6 | 960 | [2, 4, 4, 2, 2, 4] | [3776, 3840, 3904, 3904, 3008, 2304] | 67.5
M26 | 6 | 960 | [2, 4, 2, 4, 2, 4] | [3776, 2112, 4032, 3584, 3200, 4032] | 67.5
M27 | 6 | 960 | [2, 4, 2, 4, 4, 4] | [3776, 3840, 3904, 4032, 3648, 2432] | 69.2
M28 | 6 | 960 | [4, 2, 8, 4, 2, 2] | [3840, 3712, 3520, 4032, 3200, 4032] | 70.6
M29 | 7 | 960 | [2, 2, 8, 4, 2, 2, 4] | [3776, 3840, 3904, 1856, 3072, 3648, 4032] | 78.7
M30 | 8 | 960 | [2, 2, 2, 4, 2, 4, 8, 2] | [3392, 1792, 3904, 3904, 3200, 2432, 1792, 2496] | 80.9
M31 | 8 | 960 | [2, 2, 4, 4, 2, 4, 8, 2] | [3776, 3008, 4032, 3904, 3520, 3136, 1984, 3648] | 88.8
M32 | 8 | 960 | [8, 2, 2, 4, 8, 4, 4, 8] | [3776, 3008, 3904, 3904, 2176, 4032, 4032, 3648] | 91.6
M33 | 13 | 768 | [2, 8, 2, 4, 2, 2, 4, 2, 2, 8, 8, 8, 4] | [3776, 2112, 1600, 3904, 3840, 2880, 2304, 3200, 2048, 2944, 2816, 3328, 3968] | 97.9
M34 | 9 | 960 | [4, 2, 4, 4, 4, 4, 8, 8, 2] | [3840, 3136, 3520, 4032, 3200, 4032, 3648, 2112, 2368] | 98.9
M35 | 9 | 960 | [8, 2, 8, 8, 2, 4, 8, 2, 2] | [3520, 3008, 2880, 4032, 3200, 2432, 4032, 3904, 3136] | 99.4
Table 5: Detailed architectural parameters for all models in Figure 8 with
Transformer-XL backbone.
| nlayer | dmodel | nhead | dinner | DecoderParams (M)
---|---|---|---|---|---
baseline | $\in$[1,16] | 512 | 8 | 2048 | -
TITAN Xp | M1 | 2 | 384 | [2, 2] | [1152, 2432] | 4.2
M2 | 2 | 448 | [8, 2] | [2944, 3008] | 7.4
M3 | 2 | 576 | [2, 2] | [2048, 1728] | 7.7
M4 | 2 | 512 | [2, 2] | [2368, 3072] | 8.2
M5 | 2 | 832 | [8, 2] | [3264, 3072] | 17.5
M6 | 2 | 768 | [2, 2] | [3968, 4032] | 18.2
M7 | 2 | 896 | [8, 4] | [4032, 2880] | 20.4
M8 | 2 | 960 | [4, 8] | [3968, 3008] | 22.6
M9 | 2 | 960 | [4, 8] | [3968, 3648] | 23.9
M10 | 2 | 960 | [2, 2] | [3840, 3968] | 24.2
M11 | 3 | 896 | [8, 4, 8] | [4032, 2112, 3392] | 29.2
M12 | 3 | 896 | [2, 2, 2] | [3840, 2880, 3840] | 31.0
M13 | 3 | 960 | [2, 2, 2] | [3584, 3072, 2624] | 31.7
M14 | 3 | 960 | [4, 2, 2] | [3840, 3008, 3840] | 34.4
M15 | 3 | 960 | [8, 2, 8] | [4032, 4032, 3520] | 36.1
M16 | 3 | 960 | [2, 2, 8] | [3584, 4032, 4032] | 36.2
M17 | 3 | 960 | [2, 2, 8] | [4032, 4032, 3840] | 36.7
M18 | 3 | 960 | [8, 4, 8] | [4032, 4032, 4032] | 37.1
M19 | 4 | 896 | [4, 4, 8, 8] | [4032, 3456, 3328, 3392] | 41.6
M20 | 4 | 960 | [4, 2, 8, 8] | [3840, 3008, 3328, 3584] | 44.9
M21 | 4 | 960 | [2, 2, 8, 8] | [4032, 3968, 3904, 3840] | 48.7
M22 | 4 | 960 | [4, 2, 4, 4] | [3840, 4032, 3904, 4032] | 48.8
M23 | 5 | 960 | [4, 2, 4, 4, 8] | [3840, 3008, 3392, 2496, 4032] | 55.3
M24 | 5 | 960 | [4, 2, 8, 8, 8] | [3840, 3008, 3840, 3328, 3968] | 57.6
M25 | 5 | 960 | [2, 2, 4, 4, 4] | [3968, 4032, 3328, 4032, 2752] | 57.9
M26 | 6 | 896 | [8, 4, 8, 4, 8, 8] | [3328, 2112, 3392, 3904, 3328, 3264] | 58.8
M27 | 5 | 960 | [8, 2, 8, 8, 4] | [4032, 3008, 3840, 3904, 3968] | 59.1
M28 | 5 | 960 | [2, 4, 2, 2, 8] | [3968, 3968, 3840, 4032, 3904] | 60.9
M29 | 6 | 896 | [2, 2, 4, 4, 2, 2] | [3840, 3968, 3840, 3328, 3904, 3904] | 65.0
M30 | 6 | 960 | [4, 8, 8, 4, 8, 4] | [3072, 3584, 3392, 3840, 3328, 3712] | 67.9
M31 | 6 | 960 | [4, 8, 8, 8, 4, 4] | [3840, 3584, 3392, 3328, 3968, 3776] | 69.7
M32 | 6 | 960 | [4, 8, 8, 8, 8, 2] | [3840, 3840, 3392, 3840, 3328, 3712] | 69.9
M33 | 6 | 960 | [4, 2, 2, 4, 2, 8] | [3840, 3008, 3840, 3904, 4032, 3392] | 70.0
M34 | 6 | 960 | [2, 4, 8, 8, 4, 2] | [3840, 3968, 3840, 3328, 4032, 3776] | 71.5
M35 | 7 | 960 | [4, 8, 8, 8, 8, 2, 8] | [3840, 3968, 3840, 3328, 3968, 3328, 4032] | 82.8
M36 | 8 | 960 | [4, 2, 8, 8, 8, 4, 8, 8] | [3840, 3968, 3840, 3328, 3072, 3328, 4032, 3072] | 91.6
M37 | 10 | 896 | [8, 4, 8, 8, 8, 2, 8, 2, 4, 8] | [4032, 3008, 3840, 2560, 3904, 3904, 3072, 3264, 2368, 2496] | 98.4
M38 | 12 | 832 | [2, 4, 8, 8, 8, 8, 8, 8, 8, 8, 4, 2] | [3840, 2816, 2112, 3584, 3648, 2432, 2304, 3008, 2880, 1664, 2432, 3776] | 99.0
M39 | 9 | 960 | [8, 8, 8, 4, 4, 8, 8, 4, 2] | [2752, 3456, 2880, 3904, 2752, 3904, 4032, 3264, 3136] | 99.3
M40 | 10 | 896 | [8, 8, 8, 2, 8, 2, 2, 2, 8, 2] | [3840, 3072, 3840, 2560, 3648, 3328, 3840, 3008, 2880, 3328] | 100.0
Table 6: Detailed architectural parameters for all models in Figure 8 with
GPT-2 backbone.
| nlayer | dmodel | nhead | dinner | DecoderParams (M)
---|---|---|---|---|---
baseline | $\in$[1,16] | 1024 | 12 | 3072 | -
TITAN Xp | M1 | 3 | 256 | [2, 2, 2] | [3072, 3776, 3904] | 6.3
M2 | 2 | 448 | [2, 2] | [3456, 3776] | 8.1
M3 | 2 | 448 | [2, 4] | [4032, 3904] | 8.7
M4 | 3 | 384 | [2, 2, 2] | [3072, 2176, 4032] | 8.9
M5 | 2 | 576 | [2, 2] | [3456, 3584] | 10.8
M6 | 4 | 448 | [2, 2, 2, 2] | [4032, 3904, 1920, 3072] | 14.8
M7 | 4 | 512 | [2, 2, 4, 2] | [3904, 3136, 1280, 2624] | 15.4
M8 | 2 | 832 | [8, 2] | [3456, 3584] | 17.3
M9 | 2 | 960 | [2, 8] | [3456, 3648] | 21.0
M10 | 2 | 960 | [2, 2] | [3968, 3584] | 21.9
M11 | 5 | 640 | [2, 2, 2, 2, 2] | [4032, 2560, 2176, 2304, 3136] | 26.4
M12 | 3 | 832 | [2, 8, 4] | [3840, 3840, 3776] | 27.4
M13 | 5 | 704 | [2, 2, 2, 4, 4] | [2368, 3648, 1856, 3712, 3200] | 30.8
M14 | 3 | 960 | [2, 2, 2] | [3584, 3648, 4032] | 32.7
M15 | 3 | 960 | [2, 2, 2] | [3904, 3520, 4032] | 33.1
M16 | 6 | 640 | [2, 2, 2, 2, 2, 2] | [2624, 2560, 2880, 3776, 3648, 3840] | 34.6
M17 | 4 | 896 | [2, 2, 4, 2] | [4032, 3712, 3328, 3072] | 38.2
M18 | 5 | 832 | [2, 2, 2, 4, 4] | [3392, 3648, 2880, 3712, 3200] | 41.9
M19 | 4 | 960 | [2, 2, 4, 2] | [3904, 3136, 3328, 3776] | 42.0
M20 | 4 | 960 | [8, 8, 2, 4] | [3904, 3712, 4032, 3776] | 44.4
M21 | 6 | 832 | [2, 2, 4, 2, 2, 2] | [3904, 3456, 4032, 1792, 3072, 2496] | 47.9
M22 | 5 | 896 | [4, 2, 2, 2, 4] | [3968, 3200, 3840, 3328, 3648] | 48.3
M23 | 5 | 960 | [2, 2, 2, 2, 2] | [3904, 3264, 3328, 3776, 3392] | 52.4
M24 | 5 | 960 | [2, 2, 4, 2, 2] | [3584, 3456, 3776, 2944, 4032] | 52.7
M25 | 5 | 960 | [2, 8, 2, 4, 2] | [3904, 3648, 4032, 3776, 3968] | 55.6
M26 | 6 | 960 | [8, 8, 2, 2, 2, 2] | [3904, 2560, 2880, 3776, 2240, 3840] | 59.1
M27 | 6 | 960 | [2, 2, 2, 4, 2, 2] | [2496, 3456, 3328, 3904, 3968, 2944] | 60.8
M28 | 6 | 960 | [4, 2, 4, 4, 2, 8] | [4032, 3456, 3328, 3776, 4032, 2752] | 63.2
M29 | 6 | 960 | [2, 2, 2, 4, 4, 4] | [3968, 3648, 3840, 3776, 3584, 2624] | 63.4
M30 | 7 | 960 | [2, 2, 2, 4, 2, 4, 2] | [3904, 2368, 4032, 3008, 3520, 2944, 2496] | 68.7
M31 | 7 | 960 | [2, 2, 4, 2, 2, 2, 4] | [3072, 3648, 3520, 3584, 3136, 1984, 3584] | 69.1
M32 | 7 | 960 | [4, 2, 2, 2, 8, 2, 2] | [3712, 3648, 3584, 3520, 2752, 3008, 3392] | 71.2
M33 | 8 | 960 | [2, 4, 4, 2, 2, 2, 2, 2] | [3904, 2816, 3072, 1920, 3328, 3456, 2304, 2368] | 74.1
M34 | 8 | 960 | [2, 2, 2, 4, 2, 2, 8, 2] | [3520, 2368, 4032, 1792, 3200, 3776, 3200, 3648] | 78.6
M35 | 8 | 960 | [4, 2, 4, 4, 8, 8, 4, 2] | [3520, 3712, 3328, 3776, 3200, 2752, 3200, 2112] | 78.7
M36 | 8 | 960 | [8, 4, 2, 8, 2, 2, 2, 2] | [3520, 3840, 3328, 3776, 3200, 3776, 3968, 3648] | 85.4
M37 | 10 | 960 | [2, 8, 2, 4, 2, 2, 4, 2, 8, 8] | [3648, 2560, 3776, 1792, 3968, 2752, 3200, 2368, 4032, 2368] | 95.5
M38 | 10 | 960 | [2, 4, 2, 2, 4, 2, 4, 2, 4, 8] | [3840, 2240, 3328, 3776, 3648, 3200, 2944, 2368, 3968, 2880] | 98.8
M39 | 10 | 960 | [2, 4, 2, 2, 2, 2, 4, 2, 4, 8] | [3840, 2240, 3328, 3776, 3200, 3200, 3968, 2368, 3968, 2816] | 99.8
Table 7: Detailed architectural parameters for all models in Figure 8 with
GPT-2 backbone.
| nlayer | dmodel | nhead | dinner | DecoderParams (M)
---|---|---|---|---|---
baseline | $\in$[1,16] | 1024 | 12 | 3072 | -
ARM | M1 | 2 | 512 | [2, 2] | [1920, 1920] | 6.0
M2 | 3 | 320 | [8, 2, 4] | [1920, 1920, 3712] | 6.1
M3 | 2 | 576 | [2, 2] | [1344, 3200] | 7.9
M4 | 3 | 384 | [2, 8, 2] | [3840, 2368, 3328] | 9.1
M5 | 5 | 384 | [4, 4, 2, 4, 4] | [2880, 1920, 960, 2496, 1280] | 10.3
M6 | 2 | 768 | [2, 2] | [1600, 2240] | 10.6
M7 | 5 | 320 | [4, 2, 2, 4, 2] | [1344, 2240, 3776, 3008, 3648] | 11.0
M8 | 3 | 768 | [2, 2, 4] | [1856, 1792, 1920] | 15.7
M9 | 3 | 704 | [2, 2, 2] | [3136, 2112, 1920] | 16.1
M10 | 2 | 960 | [4, 2] | [3584, 2304] | 18.7
M11 | 6 | 448 | [4, 4, 2, 2, 4, 2] | [3072, 2112, 4032, 2688, 1600, 3072] | 19.7
M12 | 3 | 960 | [4, 4, 2] | [2368, 2560, 2048] | 24.5
M13 | 4 | 704 | [4, 8, 4, 2] | [3008, 3776, 2560, 3648] | 26.3
M14 | 5 | 704 | [4, 2, 4, 2, 8] | [3584, 3136, 3776, 3072, 1856] | 31.7
M15 | 3 | 960 | [2, 2, 2] | [3392, 3648, 3840] | 32.0
M16 | 4 | 960 | [4, 2, 8, 2] | [2048, 3328, 1984, 1856] | 32.5
M17 | 7 | 704 | [2, 4, 4, 4, 8, 2, 2] | [3008, 2560, 1920, 1856, 2112, 1728, 3136] | 36.9
M18 | 4 | 960 | [2, 2, 4, 8] | [3392, 3456, 2432, 2304] | 37.0
M19 | 5 | 832 | [4, 4, 4, 4, 4] | [3840, 1920, 4032, 3072, 3968] | 41.9
M20 | 5 | 960 | [8, 4, 2, 2, 4] | [2560, 2048, 3648, 1728, 2304] | 42.1
M21 | 5 | 960 | [4, 4, 2, 2, 2] | [3072, 2240, 1984, 2176, 3520] | 43.4
M22 | 5 | 960 | [2, 4, 4, 4, 2] | [2496, 3648, 3328, 3392, 2112] | 47.2
M23 | 6 | 832 | [4, 2, 4, 4, 2, 4] | [2496, 3200, 1664, 3904, 3520, 3840] | 47.7
M24 | 6 | 960 | [8, 2, 2, 2, 8, 4] | [2304, 3328, 3456, 1856, 1792, 2112] | 50.7
M25 | 5 | 960 | [4, 8, 2, 4, 4] | [3264, 2688, 4032, 3968, 3712] | 52.4
M26 | 6 | 960 | [2, 4, 4, 2, 2, 2] | [3008, 2624, 4032, 2688, 3520, 2624] | 57.7
M27 | 6 | 960 | [2, 4, 4, 2, 8, 2] | [2304, 3648, 3328, 3648, 3904, 1728] | 57.8
M28 | 6 | 960 | [4, 4, 2, 4, 2, 2] | [3072, 2368, 4032, 4032, 3776, 3264] | 61.6
M29 | 7 | 960 | [2, 2, 2, 8, 4, 8, 4] | [3008, 2304, 1920, 1984, 3520, 2816, 3712] | 62.9
M30 | 7 | 960 | [2, 4, 4, 4, 4, 2, 2] | [3200, 4032, 2048, 2624, 2112, 2752, 2880] | 63.6
M31 | 7 | 960 | [2, 4, 4, 4, 4, 2, 4] | [3584, 3648, 3328, 3392, 3200, 1984, 3200] | 68.8
M32 | 7 | 960 | [2, 4, 8, 8, 2, 2, 8] | [3008, 3648, 3584, 3648, 3008, 1728, 3712] | 68.8
M33 | 7 | 960 | [4, 4, 2, 4, 4, 8, 4] | [3584, 3840, 3328, 3392, 3136, 2944, 2496] | 69.5
M34 | 8 | 960 | [8, 2, 2, 8, 2, 2, 8, 2] | [3008, 3648, 1792, 1984, 3008, 2816, 3712, 3520] | 74.7
M35 | 8 | 960 | [2, 2, 2, 2, 8, 4, 4, 2] | [3008, 2304, 1792, 3008, 3520, 2880, 3712, 3456] | 75.1
M36 | 8 | 960 | [2, 2, 2, 2, 2, 2, 4, 8] | [3008, 1792, 3840, 3392, 3520, 3136, 3712, 3520] | 79.4
| M37 | 9 | 960 | [2, 2, 4, 4, 8, 8, 4, 2, 4] | [1664, 1792, 2240, 3904, 3648, 3264, 2176, 3712, 1856] | 79.9
| M38 | 11 | 832 | [8, 4, 2, 4, 4, 2, 8, 4, 4, 8, 8] | [3072, 2368, 4032, 3968, 1664, 3968, 2176, 2624, 3840, 2176, 2112] | 83.8
| M39 | 9 | 960 | [4, 2, 4, 8, 2, 2, 4, 2, 4] | [2496, 3648, 3328, 3392, 3648, 1728, 2880, 3520, 2368] | 85.1
| M40 | 9 | 960 | [4, 2, 4, 8, 4, 2, 4, 2, 4] | [3072, 2816, 4032, 2560, 3648, 1728, 3840, 3264, 3456] | 87.8
| M41 | 10 | 960 | [8, 2, 4, 4, 2, 2, 4, 8, 2, 4] | [3648, 1792, 2432, 1856, 3392, 2304, 3776, 2944, 3136, 3904] | 93.0
| M42 | 10 | 960 | [8, 2, 2, 4, 2, 2, 2, 4, 2, 2] | [3264, 2048, 3520, 3904, 3840, 3840, 2624, 3072, 3776, 2304] | 98.8
| M43 | 12 | 896 | [4, 4, 4, 2, 4, 2, 4, 8, 8, 2, 4, 2] | [2048, 3136, 4032, 1792, 3584, 1728, 3136, 3008, 2560, 3200, 3648, 1728] | 98.9
| M44 | 10 | 960 | [4, 2, 8, 4, 2, 8, 4, 4, 4, 2] | [3584, 3968, 3328, 3904, 2368, 2112, 3904, 3520, 3328, 2688] | 99.8
| M45 | 10 | 960 | [8, 2, 4, 4, 4, 4, 4, 2, 2, 8] | [2688, 3200, 3840, 3392, 3520, 3136, 3392, 3520, 2880, 3200] | 99.9
Corei7 | M1 | 2 | 384 | [2, 2] | [3840, 2432] | 6.0
M2 | 3 | 320 | [2, 2, 2] | [2176, 3072, 2496] | 6.2
M3 | 2 | 512 | [2, 2] | [1408, 2624] | 6.2
M4 | 3 | 384 | [2, 2, 2] | [3264, 3456, 3584] | 9.7
M5 | 2 | 576 | [2, 2] | [3136, 3648] | 10.5
M6 | 3 | 448 | [2, 2, 2] | [4032, 3648, 4032] | 12.9
M7 | 4 | 448 | [2, 2, 4, 4] | [3072, 3648, 4032, 1792] | 14.5
M8 | 2 | 768 | [2, 2] | [3968, 3328] | 15.9
M9 | 4 | 576 | [2, 2, 2, 2] | [3072, 2752, 3456, 3136] | 19.6
M10 | 2 | 960 | [2, 2] | [3840, 3264] | 21.0
M11 | 4 | 640 | [2, 2, 2, 2] | [2176, 3648, 3584, 1920] | 21.1
M12 | 3 | 960 | [2, 2, 2] | [2176, 3264, 2432] | 26.2
M13 | 4 | 768 | [2, 2, 2, 2] | [3584, 2112, 3392, 1920] | 26.4
M14 | 4 | 768 | [2, 2, 2, 2] | [3584, 2560, 3776, 1536] | 27.1
M15 | 4 | 832 | [2, 2, 2, 2] | [3904, 1984, 3392, 3136] | 31.8
M16 | 3 | 960 | [2, 2, 2] | [3968, 4032, 2880] | 32.0
M17 | 5 | 768 | [2, 2, 4, 2, 2] | [3648, 3072, 3392, 1984, 2944] | 34.9
M18 | 4 | 960 | [2, 2, 2, 2] | [3136, 1984, 3392, 2944] | 36.8
M19 | 4 | 960 | [2, 2, 2, 4] | [3968, 3456, 3584, 3136] | 42.0
M20 | 6 | 768 | [4, 2, 2, 4, 2, 4] | [3584, 2112, 3456, 3136, 3840, 2560] | 42.9
M21 | 7 | 768 | [2, 4, 2, 4, 4, 4, 2] | [2624, 1984, 2496, 3968, 2880, 2112, 4032] | 47.5
M22 | 5 | 960 | [2, 2, 4, 2, 4] | [2176, 3264, 3392, 3008, 3328] | 47.6
M23 | 6 | 960 | [4, 4, 2, 4, 2, 2] | [2048, 2624, 3520, 1984, 2880, 2624] | 52.3
M24 | 6 | 960 | [2, 4, 4, 4, 2, 2] | [1792, 3456, 2752, 2240, 1664, 3840] | 52.4
M25 | 6 | 960 | [4, 2, 2, 2, 4, 4] | [2176, 1664, 3648, 3136, 3968, 3904] | 57.7
M26 | 7 | 960 | [2, 2, 4, 4, 2, 2, 8] | [2816, 1792, 3968, 1728, 1664, 3328, 2944] | 60.9
M27 | 7 | 896 | [2, 2, 4, 2, 2, 2, 2] | [3904, 3264, 3328, 3968, 1728, 2624, 4032] | 63.5
M28 | 7 | 960 | [4, 2, 4, 2, 2, 2, 2] | [3584, 2560, 1792, 1920, 3968, 2112, 3968] | 64.1
M29 | 8 | 960 | [2, 2, 2, 4, 2, 2, 2, 4] | [3328, 2432, 2624, 2752, 1664, 2240, 2304, 2816] | 68.3
M30 | 7 | 960 | [4, 2, 4, 2, 2, 2, 2] | [3904, 2304, 2368, 3584, 3264, 2880, 3904] | 68.5
M31 | 8 | 960 | [4, 2, 4, 2, 2, 4, 2, 4] | [2560, 3648, 2624, 2112, 3328, 2112, 1792, 3328] | 70.9
| M32 | 8 | 960 | [4, 4, 4, 2, 2, 4, 2, 4] | [2560, 2304, 2624, 4032, 2688, 2624, 3840, 2816] | 74.7
| M33 | 9 | 960 | [2, 4, 2, 4, 2, 4, 2, 2, 4] | [3072, 3264, 2944, 1984, 2880, 3520, 2112, 2624, 1728] | 79.6
| M34 | 10 | 896 | [2, 2, 4, 2, 2, 2, 2, 2, 4, 2] | [2816, 3264, 3584, 1792, 3136, 3584, 2240, 2240, 1920, 2752] | 81.2
| M35 | 9 | 960 | [8, 2, 2, 2, 4, 4, 2, 4, 4] | [3904, 3648, 2432, 3136, 3264, 2816, 2240, 3072, 3840] | 87.7
| M36 | 10 | 960 | [4, 4, 2, 2, 4, 4, 2, 4, 4, 2] | [2176, 3264, 2752, 3136, 3968, 3520, 3776, 3328, 1728, 2496] | 94.9
| M37 | 10 | 960 | [4, 2, 4, 2, 2, 2, 2, 4, 2, 2] | [3904, 2112, 2496, 3968, 3968, 2624, 3904, 2304, 3200, 3840] | 99.0
| M38 | 11 | 960 | [4, 2, 2, 4, 2, 4, 2, 2, 4, 4, 4] | [2176, 4032, 3264, 3840, 2688, 1984, 1728, 2944, 1920, 2368, 3840] | 99.8
|
# Predictable Bandwidth Slicing with Open vSwitch
Jesse Chen and Behnam Dezfouli {jschen<EMAIL_ADDRESS>Internet of Things
Research Lab, Department of Computer Science and Engineering, Santa Clara
University, USA
###### Abstract
Software switching, a.k.a virtual switching, plays a vital role in network
virtualization and network function virtualization, enhances configurability,
and reduces deployment and operational costs. Software switching also
facilitates the development of edge and fog computing networks by allowing the
use of commodity hardware for both data processing and packet switching.
Despite these benefits, characterizing and ensuring deterministic performance
with software switches is harder, compared to physical switching appliances.
In particular, achieving deterministic performance is essential to adopt
software switching in mission-critical applications, especially those deployed
in edge and fog computing architectures. In this paper, we study the impact of
switch configurations on bandwidth slicing and predictable packet latency. We
demonstrate that latency and predictability are dependent on the
implementation of the bandwidth slicing mechanism and that the packet
schedulers used in OVS Kernel-Path and OVS-DPDK each focus on different
aspects of switching performance.
###### Index Terms:
Software Switching, Deterministic Performance, Latency Prediction, Edge
Computing, Fog Computing
## I Introduction
With the arrival of new paradigms such as edge and fog computing, the
necessity for comprehensive understanding of network behavior becomes
increasingly important as tasks often have a multitude of requirements that
depend on network performance guarantees, such as minimum flow bandwidth or
packet latency constraints. Some requirements are easy to fulfill. For
example, it is straightforward to track the available processing and memory
resources of edge and fog nodes. However, the prediction of network parameters
such as end-to-end packet latency is dependent on a variety of factors and
requires a comprehensive understanding of network configuration and topology
[1, 2]. An essential step towards this understanding is the characterization
of packet switching behaviors.
In this paper, we focus on software switching considering its high
applicability in edge and fog computing scenarios [3]. For example, software
switches are utilized to build multi-function nodes in edge and fog systems,
where each node performs both networking and computation tasks. Specifically,
with commodity hardware that is capable of computation, software switches are
used to add switching capability to a network, resulting in lower costs for
deployment, maintenance, and upgrades. Software switches also offer greater
configuration flexibility, making them more suitable for edge and fog networks
which need to handle highly-dynamic workloads. In contrast, traditional
hardware switches, even when enabled with Software Defined Networking (SDN)
protocols such as OpenFlow and NETCONF, are limited in their ability to
accomplish effective bandwidth slicing because of queue limitations. While
hardware switches are usually limited to eight queues per port, software
switches do not impose this limitation [4]. Furthermore, the size of software
switches’ flow tables is flexible and can be extended in ways that hardware
switches’ can not. Software switches open up the possibility of efficient
bandwidth isolation for each task’s data flows and simplify the process of
developing and applying new network policies [3, 5].
Although one of the main benefits of software switches when compared to
traditional hardware switches is the high degree of flexibility offered, there
remains areas of study that have largely been neglected in existing analyses.
Specifically, existing studies overwhelmingly focus on the switches’ maximum
throughput capabilities [6, 7, 8] or latency measurements in best-case, non-
realistic scenarios [9, 1, 10, 11, 12, 2, 13]. These studies are important to
understand the performance limitations of software switches, but they provide
very little to characterize performance in real-world scenarios. In
particular, these studies fail to provide relevant analysis of packet latency
in edge and fog networking scenarios where the bandwidth is sliced to provide
queue rate guarantees.
In this work, we fill the gap in existing literature by studying how the
various aspects of bandwidth slicing such as packet scheduling and queue rate
affect latency. We study and evaluate bandwidth slicing using OVS-Kernel Path
(OVS-KP) and OVS-DPDK and identify their strengths and weaknesses in terms of
latency and resource efficiency. In addition to characterizing the latency
patterns in bandwidth slicing scenarios, we also identify and analyze the
underlying causes of these latency patterns. We observe that although the
packet latency of OVS-DPDK is considerably lower than that of OVS-KP, the
latency of OVS-KP is stable and predictable using M/M/1 queuing. This is
because OVS-KP is able to efficiently utilize the available queue buffers.
OVS-DPDK achieves its lower latency by minimizing the time spent by the
packets in the packet scheduler queue; however, this comes at the cost of
inefficient resource utilization. To keep the queue length short, it drops all
packets that are in excess of the allocated queue rate, which results in high
TCP retransmsission rates and the need for excess ingress bandwidth in order
to maintain the target throughput. The observations of this paper can be
leveraged to employ software switching in various scenarios, such as for
building edge and fog computing systems that need to handle the diverse
latency and throughput requirements of IoT applications.
The rest of this paper is organized as follows. We present the related work in
Section II. In Section III, we overview the two software switches used in this
work. In Section IV, we discuss the importance and extraction of effective
queue rate. In Section V, we show that the latency of OVS-KP is predictable
using the M/M/1 queueing model. We discuss the latency of software switching
with a user-space data plane in Section VI. In Section VII, we discuss the
resource efficiency tradeoffs between different variants of the Open vSwitch.
In Section VIII, we present discussions on the current and future
applicability of this work, highlight its significance, and conclude the
paper.
## II Related Work
Existing works on the performance evaluation of software switches are either
limited in scope and ignore latency as a performance parameter, or present an
evaluation of oversimplified use-cases that are not representative of real-
world network configurations.
Fang et al. [6] evaluate a broad spectrum of the available software switching
solutions and present a direct comparison of the maximum throughput values of
each of the evaluated switches. Their evaluation of software switches remains
surface-level as they focus on the intricacies of inter-switch comparability,
leaving much to be desired in terms of performance analysis. McGuinness et al.
[7] focus on performance evaluations of the BESS software switch in the
context of high throughput datacenter use-cases. Although they evaluate the
throughput accuracy of the rate limiter, its effect on latency has been
neglected. Furthermore, datacenters cannot be compared to edge and fog
networking scenarios, as the two types of networks have different hardware and
applications. Meyer et al. [8] present a model for software switch
performance, but limit the scope of their model to only include throughput
measurements. Overall, these studies evaluate the performance of software
switches primarily in terms of throughput and neglect to include any
measurements of latency.
In [9], Zhang et al. perform evaluations across various state-of-the-art
software switching solutions. They analyze performance of the OVS-DPDK, snabb,
BESS, FastClick, VPP, and netmap VALE software switches and present
comparisons of their maximum throughput and packet latency. Despite the
breadth of comparisons, their performance analysis is narrow and only
encompasses the most basic of configurations and measurements. Emmerich et al.
[1] present an in-depth performance evaluation of Open vSwitch (OVS). However,
their work primarily focuses on throughput, and the analysis of latency is for
very simple scenarios that are insufficient to model edge and fog networking
use-cases. They evaluate latency only as a function of flow throughput and
ignore the performance impacts of bandwidth slicing in multi-queue scenarios.
He et al. [14] evaluate a software switch bandwidth slicing mechanism;
however, their tests were performed in simple scenarios and their results are
presented without a thorough analysis of the latency values. The same
shortcoming is exhibited in [10, 11, 12, 13] in terms of latency evaluation:
their models of packet latency are for simple, synthetic testing scenarios.
The latency of a single flow has been evaluated in [10, 11, 12], while [13]
only evaluates the latency of a single packet. These scenarios are rudimentary
and cannot be used to accurately generate models of bandwidth-sliced software
switch behaviors.
## III Methodology and Background
Figure 1: The testbed used for the studies of this paper. The two software
switches used are OVS-KP and OVS-DPDK.
### III-A Testbed Setup
Figure 1 presents the testbed architecture. To measure the latency of OVS
packet switching, we ran experiments on a testbed consisting of three
machines: a traffic source, a software switch, and a traffic destination. We
used Intel 82580 1GbE and Intel X550T 10GbE NICs. The traffic source sends UDP
and TCP traffic to the traffic destination through the OVS. For the UDP flows
with fixed bandwidth, we used iPerf, which supports specification of UDP flow
bandwidth. In the tests, we modified the number of queues, the allocated
throughput of each queue, and the type of data flows in each queue. We will
further discuss the details of each test in their relevant sections.
Packets are captured using tshark at the egress and ingress ports of the
traffic source and traffic destination, respectively. To ensure the
synchronization of timestamp values, the traffic source and traffic
destination are two VMs running on a single machine, and the clocks of these
two VMs are synchronized with that of the hypervisor. This configuration
allows us to accurately measure and analyze latency values.
### III-B Open vSwitch
Open vSwitch (OVS) [15, 16, 17] is an open source, production quality software
switch that is compatible with various hypervisors and container systems. OVS
is highly programmable and is configured using the OpenFlow and OVSDB
protocols [18]. We consider the two main variants of OVS: (i) OVS Kernel-Path
(OVS-KP), which implements its data path via a kernel module, and (ii) OVS
with the Data Plane Development Kit (OVS-DPDK), which implements its data path
through Poll Mode Drivers (PMDs) in the user-space. We use OVS 2.15.0 and DPDK
20.11.1.
We perform bandwidth slicing on the switches by using their packet schedulers.
The packet schedulers that we utilize are configured to shape flows via
minimum guaranteed rate and/or maximum limited rate parameters. When combined
with flow rules that direct packets to the queues, data flows are shaped to
specific minimum and maximum rates.
For OVS-KP, we use the Hierarchical Token Bucket (HTB) packet scheduler since
it is widely used and available in the Linux traffic control module. HTB is a
classful queuing discipline that supports hierarchical traffic shaping. Its
rate control mechanisms are implemented with the token bucket filter
algorithm, and its hierarchical token borrowing system allows parent classes
to share tokens with their child classes. This token sharing system allows
each child class to enforce a guaranteed minimum rate, while also sharing
excess available bandwidth with their sibling classes.
OVS-DPDK uses a different packet scheduler based on the Two-Rate, Three Color
Marker (TRTCM) algorithm. Similar to HTB, TRTCM also uses a token bucket for
rate control and provides traffic shaping abilities such as guaranteed minimum
and maximum queue rates.
Although HTB and TRTCM are very similar on the surface, their rate control
mechanisms are significantly different, which results in different latency and
throughput behaviors for queues with the same allocated rate. HTB is a
hierarchical implementation of the token bucket filter algorithm, meaning that
when a packet arrives at the head of the queue and there are no tokens
available, the packet waits in the queue until tokens become available, at
which time the packet is dequeued and sent to the NIC. On the other hand, for
OVS-DPDK, when a packet arrives at the head of the queue, the TRTCM token
buckets are checked for tokens, and if there are not enough tokens for the
packet, the packet is dropped. Most significantly, this difference in behavior
results in different dequeue rates from the queues; as we will discuss in
Sections IV and VI, the dequeue rate impacts flow latency and throughput.
## IV Extracting Effective Queue Rate
In a switching system, a packet experiences three types of latency:
transmission latency, processing latency, and queueing latency. Transmission
latency is directly related to NIC’s transmission rate and can be easily
calculated. Processing latency is caused by various factors, including copying
a packet to and from different queues, looking up a packet’s forwarding
decision in the tables, or waiting for hardware I/O operations. Since hardware
switches are solely dedicated to one function, packet processing delay in a
hardware switch is consistently low. On the other hand, the processing delay
of software switches is usually higher and with greater statistical variation.
Choice of packet scheduler also affects processing delay. Software switches
sacrifice processing latency in exchange for greater flexibility. Last, and
the focus of this section, is queueing latency ($L_{q}$), which is defined as
the time spent by a packet in the switch’s queue, waiting to be transmitted.
This latency depends on the queue rate, flow rate, and packet scheduler.
In every software switch, each packet must traverse three queues: the ingress
NIC queue, the egress NIC queue, and the packet scheduler’s queue. In this
system, throughput is limited by the lowest-rate queue, which is usually the
user-allocated packet scheduler queue. As a result, a packet spends the most
time waiting in the packet scheduler queue because queueing latency
experienced by a packet increases exponentially as the queue input rate
approaches the queue service rate. We show that this queueing latency follows
an M/M/1 queue trend and that with enough knowledge of the system, the latency
can be predicted when using OVS-KP.
### IV-A Observing Packet Scheduler Behavior
An important factor in predicting packet latency through any queueing system
is the queue’s dequeue behavior. In our case, we need to understand how the
packet schedulers dequeue traffic from their rate-limited queues. Although
rate-limited queues are allocated with bits/s or bytes/s values, the packet
scheduler is not actually dequeueing in such small increments. The dequeue
behavior varies, depending on how the packet scheduler is implemented, and
even different packet schedulers with similar queue parameters behave
differently. This often-overlooked variable causes packets to experience
different queueing latency values, even if the allocated queue rates are
identical.
(a) Burst Latency for OVS-KP
(b) Per Packet Latency for OVS-KP
(c) Burst Latency for OVS-DPDK
(d) Per Packet Latency for OVS-DPDK
Figure 2: Packets are dequeued from HTB and TRTCM queues in different sized
groupings. HTB dequeues in groupings of 16 packets while TRTCM deqeuues in
groupings of 8 packets.
In these experiments, we generate UDP flows with various packet burst sizes,
then we track the individual packet latencies and the latency of each burst as
a whole. Figure 2 presents the results. As Figures 2(a) and 2(c) demonstrate,
when we increase the number of packets in each burst, the latency of the whole
burst increases in a stepped pattern. This indicates that both HTB and TRTCM
dequeue packets from their queues in bursts of packets, instead of one at a
time. To further support this, the analysis of individual packet latencies in
Figures 2(b) and 2(d) show that the latencies of each packet in the bursts are
grouped in segments of about 16 and 8 packets each, although there exist small
variation in grouping sizes for each dequeue segment. Figure 2(b) and 2(d)
also highlight that the grouping pattern holds true for bursts of all sizes:
no matter how big the burst, packets are always dequeued in fixed size groups.
Using this information, we extrapolate that the effective rate of HTB queues
must be calculated in units of 16 packets, and the effective rate of TRTCM
queues must be calculated in units of 8 packets.
Figure 3: A scenario where an IoT edge device needs to send data and a command
to a server. Only case (c) allows faster transmission of data to the server.
Once the server receives the data, it can start processing, and when the
command is received, the action can be performed immediately.
These observations can be leveraged to enhance communication determinism and
performance in various contexts. For example, this knowledge of burst latency
behaviors provides devices with the ability to send messages that take
advantage of the fact that the latency of a 5-packet message is equal to that
of a 15-packet message. For example, assume an IoT edge device needs to send
data and command to a server. In the first scenario, data and command are sent
as a single message, which is segmented into 20 packets, as Figure 3a shows.
This results in the delivery of data and command at once. In the second
scenario, two messages are generated for data and one message for command.
Assume the first message is segmented into 7 packets, the second into 8
packets, and the third into 5 packets. Once these messages are received by the
transport layer of the device, the packets are sent in an interleaved manner.
As it can be observed in Figure 3b, the command message is delivered first,
which cannot be used because the data messages have not been received yet. In
the third scenario we rely on the behavior of software switches and generate
two messages: one for data, which is transmitted first, and one for command,
transmitted second. As Figure 3c shows, once the data is received by the
server, it can start processing the data, and when the command arrives, the
server can perform the action immediately. Therefore, the third solution
provides the minimum latency and better utilization of resources. A similar
method can be used regarding controller-switch communication in SDNs. To
ensure timely delivery and execution of commands, the controller can manage
the ordering of sent packets based on the command type and size.
## V Delay Prediction of OVS-KP Bandwidth Slicing
We confirm that the queuing latency of OVS-KP switching follows an M/M/1
trend. We set up an experiment to measure the relationship between the latency
of packets in a TCP flow and the queue rate. We generate and route a TCP flow
through a rate-limited queue in the switch. We do not set any flow rate at the
traffic source because the TCP flow rate naturally increases until it detects
packet loss caused by the rate-limiter in the software switch.
(a) 1GbE NIC
(b) 10GbE NIC
Figure 4: The queueing latency of TCP packets when switched by OVS-KP. The
delay is dependent on the allocated queue rate, and with the knowledge of
queue ingress rate and packet scheduler implementation, the expected latency
can be predicted.
We present the results in Figure 4. This figure demonstrates that (i) queue
rates are the determining factor for rate-limited TCP packet latencies, and
(ii) the pattern of observed latencies align with the latency values that one
would expect when modeling each queue as an M/M/1 queue. The results for 1GbE
(Figure 4(a)) and 10GbE (Figure 4(b)) NICs are presented side-by-side to show
that NIC line-rate is not a significant factor in this experiment.
It is important to note that when calculating the expected latency, the queue
rate must be represented via the amount of data that is dequeued in one
instance, i.e., the effective queue rate. In Section IV-A, we showed that HTB
dequeues 16 packets at a time. Thus, instead of calculating expected latency
using the queue’s bit-rate value, we calculate the expected latency using
queue rate in units of 16 packets. This is where knowledge of average packet
size is important, as we now combine average packet size, queue bit rate, and
packet scheduler dequeue behavior to generate effective queue rate values. We
calculate the expected latency via: $L_{q}=\frac{1}{\mu(1-\rho)}$ (1) where
$\mu$ is the effective queue rate, and $\rho$ is the queue utilization ratio
[19]. We include the latency predictions in Figure 4 for comparison against
the observed values. The relationship between queue rate and observed latency
is what is expected from an M/M/1 queueing system. In Figure 4, we validate
that predictions based off of observed throughput ($\rho$) and packet
scheduler knowledge ($\mu$) are accurate. We generate values for $\rho$ by
comparing the observed throughput at the traffic source’s egress port and the
allocated queue rates. The observed throughput is extracted from the wireshark
captures at the traffic source’s egress port. $\mu$ is calculated by
converting the units of each queue rate from bits per second to packet
groupings per second. Given that small variations as low as 0.5% in queue
utilization ratio significantly affect latency prediction, the results confirm
that this latency prediction methodology is valid and accurate.
We showed that for bandwidth-sliced flows, queueing latency is the most
significant portion of end-to-end latency and that it overshadows transmission
latencies; transmission latency on a 1GbE link for a single packet is on the
order of microseconds, while the observed switching latencies are up to four
orders of magnitude greater. Current task allocation schemes either ignore
latency as a task request parameter, or assume that network latencies consist
only of trivially calculated transmission latencies. In contrast, our method
allows the prediction of communication latency, which can be leveraged to
address the requirements of various tasks in edge and fog computing systems.
As another example of leveraging this method, a SDN controller can accurately
enforce bandwidth slicing schemes that satisfy the expected communication
latency between edge devices and switches. Also, to configure switches with
latency bounds, the controller can enforce bandwidth slicing methods along all
the controller-switch paths.
## VI OVS-DPDK Bandwidth Slicing
In this section, we focus on OVS-DPDK and the effect of the TRTCM packet
scheduler on packet latency, in comparison to OVS-KP’s HTB.
(a) 1GbE NIC
(b) 10GbE NIC
Figure 5: For OVS-DPDK, the end-to-end packet latency is not as predictable as
that of OVS-KP due to the differences between HTB and TRTCM.
For a direct comparison between the two variants of OVS, this time we use OVS-
DPDK and run an experiment similar to that of Section V. We present the
results in Figure 5. The results show that the latency behaviors are not
similar at all to that of Figure 4. OVS-DPDK queues that are rate-limited with
TRTCM cannot be modeled as an M/M/1 queueing system because the queues are not
being dequeued at the allocated queue rate. Although the rate of data sent to
the egress NIC matches the allocated rate, the rate at which packets are
removed from the queue depends on the CPU frequency and OVS-DPDK’s tick rate.
Unlike HTB, which uses the availability of tokens to limit the rate at which
packets are removed from the queue, TRTCM uses the availability of tokens to
decide which actions to take. If there are tokens available in the bucket when
a packet is dequeued, the packet is passed on to the NIC. If there are not
enough tokens for the packet, the packet is dropped. The token bucket is
refilled at the allocated queue rate, hence, the amount of data sent to the
NIC is limited by that value. This approach results in a very high dequeue
rate for all TRTCM queues, and the effective dequeue rate is on the order of
several Gbps. For OVS-KP, the value of $\rho$ in Equation (4) is close to 1
because the flow rate is approaching the effective queue rate, whereas for
OVS-DPDK, that value is now much closer to 0 because the effective queue rate
is much higher than the flow rate. This results in packets spending
significantly less time waiting in the packet scheduler’s queues. As we can
see from a direct comparison of Figures 4(a) and 5(a), for TCP flows that are
rate limited to 500 Mbps, the latency is reduced from 19.22 ms with HTB to
0.27 ms with TRTCM, a 70x reduction. Although the magnitude of latency
reduction varies depending on the allocated queue rate and NIC line rate, this
shows that a significant portion of the latency experienced by the packets
that traverse OVS-KP’s rate-limited queues is the time spent waiting in the
packet scheduler’s queue. OVS-DPDK’s rate control mechanism is able to avoid
these long queueing latencies, while still being able to accurately rate-limit
the traffic to the egress NIC.
## VII Resource Efficiency Comparisons
On the surface, OVS-KP and OVS-DPDK’s rate control mechanisms accomplish the
same goal: limit the rate of traffic that is sent to the egress NIC. Although
differences in implementation have significant implications for latency,
another implication that is just as important is the effect of the packet
scheduler on resource utilization efficiency. One of the main goals in
edge/fog task allocation is to utilize resources effectively and efficiently,
which, for network resources, is usually accomplished through bandwidth
slicing and rate control mechanisms. We have observed that the HTB and TRTCM
packet schedulers are capable of accurately rate-limiting a queue; however,
our observations also show that OVS-DPDK’s choice to drop all packets that are
in excess of the allocated queue rate is a tradeoff between latency and
effective bandwidth utilization.
Figure 6: Comparing the performance of switching a TCP flow through OVS-KP and
OVS-DPDK using 1GbE ((a) and (b)) and 10GbE ((c) and (d)) NICs. The queue rate
control method of OVS-DPDK is considerably less efficient than that of OVS-KP.
We run experiments with the same setup as Figure 4 and 5, and we use iperf3 to
capture data for TCP retransmission rate and TCP congestion window size.
Figures 6a and 6c present the results for TCP retransmission rate. We observe
that rate-limiting with TRTCM causes significantly more TCP retransmissions
compared to HTB. Rate-limiting to 500 Mbps with TRTCM results in 4233 and 4652
TCP retransmissions per second for 1GbE and 10GbE NICs, respectively. This
indicates that for TCP traffic, maintaining an egress throughput of 500 Mbps
out of the switch requires an additional 50.8 Mbps and 55.8 Mbps of
retransmission traffic, due to the large number of packets that TRTCM drops.
More importantly, although OVS-DPDK is able to switch individual packets with
lower latency than OVS-KP, the high rate of packet drops/retransmissions has
an adverse effect on application message latency. The application layer is not
dependent on individual packet latency, rather, is dependent on the latency of
messages which can be composed of multiple packets. In a situation with a 10%
retransmission rate, large application layer messages are very likely to
experience retransmissions and slowdowns due to the inefficiencies of TRTCM.
Figures 6b and 6d show that rate-limited flows with TRTCM have much smaller
congestion window sizes than flows that are rate-limited with HTB. Once again,
this is related to TRTCM—dropping all packets that are over the allocated
queue rate is extremely limiting for TCP congestion window size. Each time a
packet is dropped and retransmitted, the congestion window of that TCP
connection is halved. For TCP flows with high retransmission rates like those
we observed with TRTCM, the congestion windows are severely limited and are
unable to grow due to the constant packet drops and subsequent window size
adjustments. The frequent congestion window size adjustments also results in
spikes and dips in flow throughput, which have a detrimental effect on latency
predictability. As such, application layer messages that are sent through OVS-
DPDK always have an element of unpredictability due to high retransmission
rates while messages sent through OVS-KP do not.
Since OVS-DPDK operates completely in user-space, it achieves its high
performance by constantly consuming 100% of at least one processor core. For
high performance use-cases, a separate core is used for each port, resulting
in several processor cores being dedicated entirely to running the DPDK user-
space data path. In low-cost and low-energy edge and fog computing scenarios,
this is not desirable, especially when compared to OVS-KP, which consumes less
than 5% of a single processor core with HTB while switching 10Gbps traffic
with hundreds of flow rules and queues.
## VIII Conclusion and Future Work
In this paper, we studied how packet schedulers affect switching latency and
resource efficiency. We first developed models to predict the latency of the
M/M/1 queueing system that can be found in the HTB packet scheduler.
Specifically, we analyzed the behavior of the packet scheduler, then used
these observations to predict packet latency of TCP flows. We then discussed
the design differences between OVS-KP and OVS-DPDK packet schedulers and
showed how each achieves either low latency or resource utilization efficiency
at the cost of the other. The results presented in this work provide a
foundation from which we can begin to build deterministic software switching
systems that can be specifically used to build low-cost processing and packet
switching systems using commodity hardware.
The design decisions that allow OVS-DPDK’s TRTCM to achieve low latency in
comparison to OVS-KP’s HTB come at the cost of inefficient bandwidth usage,
throughput instability, and reduced latency predictability. This information
is especially important to design networks with specific performance metrics
in mind. Besides, this information can be leveraged to design packet
schedulers that combine the desired properties of HTB and TRTCM. For example,
a new packet scheduler that seeks to enforce latency bounds while also
achieving flow reliability could dequeue packets according to the queue rate
similar to HTB, and also dynamically adjust queue length so that packets that
do not meet the packet latency requirements are dropped, similar to TRTCM.
This way, the benefits of using the queue buffers can be realized, while also
keeping queueing latency within established bounds.
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|
# Roller-Coaster in a Flatland: Magnetoresistivity in Eu-intercalated Graphite
A. L. Chernyshev Department of Physics and Astronomy, University of
California, Irvine, California 92697, USA O. A. Starykh Department of
Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA
###### Abstract
Novel phenomena in magnetically-intercalated graphite has been a subject of
much research, pioneered and promoted by M. S. and G. Dresselhaus and many
others in the 1980s. Among the most enigmatic findings of that era was a
dramatic, roller-coaster-like behavior of the magnetoresistivity in EuC6
compound, in which magnetic Eu2+ ions form a triangular lattice that is
commensurate to graphite honeycomb planes. In this study, we provide a long-
awaited microscopic explanation of this behavior, demonstrating that the
resistivity of EuC6 is dominated by spin excitations in Eu-planes and their
highly nontrivial evolution with the magnetic field. Together with our
theoretical analysis, the present study showcases the power of the synthetic
2D materials as a source of potentially significant insights into the nature
of exotic spin excitations.
###### Contents
1. I Introduction
2. II Phenomenology and modeling
1. II.1 Electronic structure of EuC6
2. II.2 Spin model and parameters
1. II.2.1 Classical ground states
2. II.2.2 Tilt angles and critical fields
3. II.2.3 Parameters
3. III Spin excitations
1. III.1 General case of a coplanar state
2. III.2 $1/S$-expansion
3. III.3 LSWT Hamiltonian
4. III.4 Diagonalization
5. III.5 Magnon eigenenergies
4. IV Kondo coupling and resistivity
1. IV.1 Kondo coupling
2. IV.2 Resistivity
1. IV.2.1 Large-${\bf q}$ insights
2. IV.2.2 Estimate of $J_{K}$
5. V Results
1. V.1 $T$-dependence of resistivity
2. V.2 Magnetoresistivity vs biquadratic-exchange
3. V.3 Magnetoresistivity, role of $k_{F}$
4. V.4 Outlook
6. VI Summary
7. A First order transitions
1. A.1 Y-UUD transition
2. A.2 V-FM transition
8. B Particular cases
1. B.1 Polarized state
2. B.2 120° state
3. B.3 Plateau state
9. C Transport formalism, $1/\tau$ approximation
1. C.1 Basics and conventions
2. C.2 Phonon-like spin-conserving scattering
1. C.2.1 Ansatz and solution
2. C.2.2 2D case
3. C.3 Spin-flip scattering
4. C.4 Quasiparticle $1/\tau_{\rm qp}$, angular dependence
1. C.4.1 Quasiparticle vs transport $1/\tau$
2. C.4.2 Angular dependence of $1/\tau_{\rm qp}$
## I Introduction
The two-dimensional (2D) world of Flatland, a mathematical abstraction and a
subject of popular culture [1], has, arguably, received its ultimate physical
realization in the form of graphene [2, 3], whose unique properties [4] have
ushered in a new era of making artificial heterostructures via a Lego-like [5]
assembly of layered materials. Perhaps with an exception of research in the
twisted bi- and $n$-layer graphene [6], the fledging field of van der Waals
magnets holds most promise in opening new horizons for the fundamental studies
and applications along the path of using this technology [7, 8, 9, 10, 11].
Historically, a more traditional, if not ancient [12], way of achieving
similar goals of synthesizing materials with novel properties from a stack of
carbon layers and various elements and compounds has relied on the process of
intercalation, also known in popular culture [13]. The research in graphite
intercalation compounds (GICs) has attracted significant attention in the
past, with the evolution of such studies and understanding of these materials
outlined in several books and, specifically, in the reviews by M. S. and G.
Dresselhaus, whose efforts contributed to much of the progress in this area,
see Refs. [14, 15, 16, 17].
Figure 1: (a) The schematics of EuC6. Small and large dots are C and Eu atoms,
respectively. (b) The EuC6 in-plane resistivity data vs magnetic field,
$\rho(H)$, for various temperatures, see Ref. [18]. Arrows are the fields of
the anomalies in $\rho(H)$ that correspond to transitions between magnetic
states, extrapolated to $T\\!=\\!0$. [Reprinted with permission from S. T.
Chen, M. S. Dresselhaus, G. Dresselhaus, H. Suematsu, H. Minemoto, K. Ohmatsu,
and Y. Yosida, Phys. Rev. B 34, 423 (1986), Ref. [18]. Copyright (1986) by the
American Physical Society.] (c) Our representative results for the transport
scattering rate.
Of the fundamental footprint of this research, it is the magnetically-
intercalated compounds that have produced the most intriguing phenomena [16].
The case of EuC6, made of alternating honeycomb layers of carbon and
triangular-lattice layers of Eu ions, shown schematically in Fig. 1(a),
particularly stands out. A highly dramatic, roller-coaster-like dependence of
the in-plane resistivity on a magnetic field, reproduced from Ref. [18] in
Fig. 1(b), is clearly indicative of an intricately intertwined magnetic and
electronic degrees of freedom of this material. Incidentally, EuC6 is also the
first magnet to exhibit the fabled $1/3$-magnetization plateau [19] and was
inspirational for an understanding of this state [20].
The pioneering studies of EuC6 [21, 22, 23, 18] have analyzed and successfully
identified key exchange terms of the triangular-lattice spin-$7/2$ model
Hamiltonian of the localized $4f$ orbitals of Eu2+ ions that are necessary to
understand the field-induced phases and the concomitant magnetization data
[19]. However, while yielding a reasonable estimate of the Kondo coupling, the
sole attempt to explain magnetoresistivity itself [24] has provided a largely
unsuccessful modeling of it via a crude consideration of the spin-scattering
of electrons and suggested a rather relic backflow mechanism to explain the
$T$-dependence of the resistivity.
Thus, it is fair to say that, by and large and to the best of our knowledge,
there exists no proper explanation of the key resistivity results observed in
EuC6, shown in Fig. 1(b). Furthermore, the refocusing of the research of the
1980s and 1990s on correlated systems and high-temperature superconductivity
has left these striking results in their enigmatic state.
In this work, we provide a microscopic theory of the magnetoresistivity of
EuC6 and demonstrate that its highly nontrivial evolution with the magnetic
field must be dominated by the scattering off the spin excitations in Eu-
planes. Figure 1(c) demonstrates representative results of our theory, which
capture most of the qualitative and quantitative features of the experimental
data in Fig. 1(b), with the details of the theory provided below. In an
anticipation of the upcoming revival of the research in the graphite
intercalation compounds driven by an interest in the novel magnetic systems,
our effort will help bringing together different approaches to the search of
new phenomena [8, 17] in the graphite-derived artificial magnetic materials.
More broadly, we would also like to highlight that there is a number of
conducting magnetic materials that exhibit a highly non-monotonic
magnetoresistivity [25, 26, 27, 28], showing that such measurements can serve
as a very sensitive probe of the field-induced phase transitions. However,
most theoretical explanations, if any, are limited to an associative
construction of phenomenological spin models to match the number of phase
transitions and broad trends in magnetization [28], without any attempt to
explicate scattering mechanisms and calculate resistivity. In that respect,
our present study is also the one that accomplishes precisely this goal: a
fully microscopic calculation of the resistivity throughout all the phases in
the phase diagram of the underlying spin model. We anticipate our results not
only be inspirational for the broader research in metallic magnets, but also
to provide the technical guidance for similar studies.
We outline, in broad strokes, our approach and results. We build on the
achievements of the prior work on EuC6 [21, 22, 23, 18] and reanalyze
phenomenological constraints on the triangular-lattice spin-$7/2$ model of
Eu2+ layers. In this analysis, we also use more recent experimental insights
into the magnetic ground state of EuC6 [29] and density-functional theory of
its electronic structure [30].
Thus, we establish bounds on the exchange parameters as related to the
phenomenology of different magnetic phases of EuC6, examine ranges of
parameters that make transitions between the phases first-order, and formulate
a minimal model to describe EuC6. We proceed by constructing the spin-wave
theory for all the field-induced phases of that model. Although a numerical
procedure is generally needed to obtain magnon eigenenergies, the approach
leading to it, as well as the results for some of the phases, are fully
analytical.
While the Kondo coupling between conducting electronic states and Eu2+ spins
is fully local, the matrix elements of electron scattering on magnons have a
nontrivial form, owing to the internal structure of quasiparticle eigenstates
in different phases. This structure leads, among other things, to the non-
spin-flip scattering processes in the non-collinear phases. We articulate that
these matrix elements are essential for a consistent calculation of the
transport scattering rate. The expression for the latter, given in a concise
form, is derived using Boltzmann formalism, which we revisit for both spin-
flip and non-spin-flip channels, providing a thorough derivation of the
relaxation-time approximation in the process.
The temperature-dependence of the resistivity anticipated from our theory is
discussed for all field-induced phases. Significantly, the zero-field results
of our theory demonstrate an analogue of the phonon-dominated resistivity
behavior, but due to scattering off the acoustic magnons of the 120$\degree$
state, with a 2D equivalent of the Bloch-Grüneisen low-temperature asymptote
of $\rho\\!\propto\\!T^{4}$ and the high-temperature Ohm’s law,
$\rho\\!\propto\\!T$. Given the extent of the magnon bandwidth, the nearly
linear trend of $\rho(T)$ observed in Ref. [24] above 8 K is shown to be well
within the onset of the Ohm’s regime.
The resistivity calculations are performed at experimentally relevant
temperatures for various parameters of the minimal model to demonstrate
qualitative trends and for a specific set of parameters that best describes
EuC6. We also investigate the dependence of our results on the filling
fraction of electronic bands, encoded in the Fermi momentum $k_{F}$, and
conclude that the relatively smaller values of $k_{F}\\!\lesssim\\!\pi/3$
provide a better correspondence to the EuC6 phenomenology, inviting more
research into a verification of its electronic properties. Other intriguing
features of the resistivity for the larger values of $k_{F}$, potentially
controllable by doping, are also discussed.
Altogether, the results of our model for the transport relaxation rate,
offered in Fig. 1(c) for a representative $k_{F}\\!=\\!\pi/3$, show a striking
similarity to the experimental data in Fig. 1(b), with the possible origin of
the discrepancies at higher field discussed below. Our theory implicitly
contains the field-dependence via that of the magnon spectra and scattering
matrix elements, which, in turn, depend on the spin arrangement in each of the
field-induced phases. It also properly accounts for the effect of the thermal
population of magnetic scatterers on the resistivity. One of the qualitative
messages of our study is the importance of the non-spin-flip channel of the
scattering, which is present in the phases with the non-collinear spin
configurations, but is absent for the collinear ones. This effect explains the
weaker scattering and lower resistivity in the $1/3$-magnetization plateau and
fully polarized phases.
The general picture that emerges from our analysis is that of the resistivity
as a very informative probe of not only field-induced phase transitions, but
also of the elementary spin excitations in these phases. The provided thorough
theoretical analysis of the iconic two-dimensional triangular-lattice
antiferromagnet coupled to conduction electrons showcases a largely untapped
power of the synthetic 2D materials as a source of potentially significant
insights into the nature of exotic spin excitations. Our approach and findings
can be applied, for example, to the electron scattering by the fractionalized
spinons of the Kitaev spin liquid [31, 32] and to the other magnetically-
intercalated systems such as chalcogenides [33, 34].
The paper is organized as follows. Section II.1 discusses electronic structure
of EuC6 and the approximate values of the Fermi momenta. Section II.2 gives an
overview of the phenomenologically-motivated spin model of EuC6, its classical
ground states and critical fields, and parameters of the minimal model.
Details on the first-order transitions are delegated to Appendix A. Spin
excitations of the model for all field-induced phases are discussed in Section
III, which provides details of the spin-wave formalism and results for
representative magnon eigenenergiesa and the eigenfunctions. The fully
analytical results for the polarized, 120$\degree$, and plateau phases are
given in Appendix B.
The Kondo coupling and its estimate, as well as resistivity and some
qualitative insight into it, are discussed in Section IV. This consideration
relies, not in a small way, on a detailed derivation of the relaxation rates
for the spin-flip and non-spin-flip channels from the Boltzmann formalism,
provided in Appendix C, which also discusses possible limitations of this
approach and potential new phenomena at large values of $2k_{F}$. The
temperature-dependence of magnetoresistivity, results for various values of
the key model parameters and Fermi momentum, and an outlook on the possible
future extensions are given in Section V. We provide a summary in Section VI.
Figure 2: (a) Energy bands of graphene in its full BZ (dashed lines), and
folded onto Eu-lattice BZ to represent rigid-band structure of Eu-intercalated
graphite (solid lines) along the paths M${}_{\rm gr}\Gamma$M(K)KgrMgr shown in
(b); high-symmetry points are highlighted. Two Dirac bands are color-coded to
indicate their origin before folding. Energies are in units of the graphene
tight-binding hopping parameter $t\\!=\\!3.16$ eV [4]. Horizontal line is the
Fermi energy $E_{F}\\!=\\!0.83t\\!\approx\\!2.62$ eV. It matches the Fermi
momenta $k_{F,{\rm min}}\\!\approx\\!0.48\pi/a$ and $k_{F,{\rm
min}}\\!\approx\\!0.7\pi/a$ from Ref. [30] of the trigonally warped Fermi
surfaces shown in (b) that approximately correspond to the $0.5e$ filling of
the bands. Insert: crystal structure of Eu-GIC from Fig. 1 with primitive
translational vectors of Eu and C lattices. (b) Brillouin zones of the
graphene and of the Eu-based triangular lattice. Fermi surfaces at $E_{F}$ are
color-coded according to the bands in (a). Two representative circular Fermi
surfaces, with $k_{F}\\!=\\!0.4\pi/a$ and $0.7\pi/a$, (dashed lines). High-
symmetry paths are indicated by the arrows.
## II Phenomenology and modeling
### II.1 Electronic structure of EuC6
The electronic structure of Eu-intercalated graphite EuC6 has been
investigated experimentally and theoretically in the mid-90s of the last
century [30], with the summary of these efforts given in Ref. [17].
Structurally, EuC6 is the so-called stage-I intercalated compound, meaning
that the Eu layers alternate with that of carbon. Viewed from a graphite
layer, the rare-earth atoms are located on top of the centers of the graphite
hexagons and form a $\sqrt{3}\times\\!\sqrt{3}$ superstructure as is
illustrated in Fig. 1(a). The material is characterized by the so-called
$A\alpha A\beta$ stacking (space group $P6_{3}/mmc$), in which Eu atoms form a
hexagonal closed packed structure with alternating positions $\alpha$ and
$\beta$ between consecutive layers, while carbons follow the $AA$ stacking
[35, 36, 29]. This arrangement of carbon sheets is different from the $AB$, or
Bernal, stacking of the graphite.
As a result, the principal unit associated with the Eu-based triangular
lattice can be seen as containing one Eu and six carbon atoms, while the
structural unit cell contains two Eu atoms and twelve carbons. As is shown in
Fig. 2(b), the two-dimensional (2D) Brillouin zone (BZ) of the triangular Eu
lattice is three times smaller than that of the graphene. The lattice
constants of the triangular Eu lattice and that of the honeycomb graphene
lattice are related as $a\\!=\\!\sqrt{3}a_{\rm gr}$, see Fig. 2.
The key features of the electronic band structure of EuC6 can be understood
within the “rigid-band” approximation, see Ch. 5 of Ref. [17]. One assumes
that the band structure of the graphene layer is not changed by the Eu
intercalation, with the latter resulting only in a partial filling of the
graphene bands up to a Fermi energy $E_{F}$, illustrated in Fig. 2(a) by a
horizontal line.
Upon folding onto the Eu-based Brillouin zone, the Dirac bands are mapped from
the proximities of the Kgr and K${}^{\prime}_{\rm gr}$ points of the graphene
Brillouin zone onto the neighborhood of the $\Gamma$ point, see Fig. 2. These
bands are equivalent up to a $\pi/3$ rotation, with a representative constant-
energy cut demonstrating characteristic “flower-petal” Fermi surfaces
originating from the trigonal $C_{3}$ symmetry of the graphene lattice, see
Fig. 2(b). These two bands from the two valleys at Kgr and K${}^{\prime}_{\rm
gr}$ are the ones being filled away from the charge-neutrality point by the
doping provided by the intercalated Eu.
To estimate the size of the Fermi surfaces produced by doping, one can
approximate them as circles with a radius $k_{F}$, neglecting their trigonal
warping. Naturally, the Fermi momentum $k_{F}$ is determined by the 2D density
of electrons donated to a graphene sheet by the Eu layer. Taking into account
band (valley) and spin degeneracy factors yields
$n_{e}^{2D}\\!=\\!k_{F}^{2}/\pi$ [17]. The nominal valence state of Eu is
Eu2+. Assuming that all 2$e$/Eu go into the conduction bands and using the 2D
volume of the Eu unit cell $V_{c}\\!=\\!a^{2}\sqrt{3}/2$, one obtains
$k_{F,2e}\\!=\\!(4\pi/\sqrt{3}a^{2})^{1/2}\\!\approx\\!0.86\pi/a$. The same
result can be obtained by matching the area (2D volume) of the fully occupied,
doubly-degenerate triangular-lattice Brillouin zone of the Eu lattice,
$V_{BZ}^{\triangle}\\!=\\!8\pi^{2}/\sqrt{3}a^{2}$, with the four-fold
degenerate (valley$\times$spin) Fermi circle of radius $k_{F,2e}$. Altogether,
the Fermi surface in EuC6, estimated within this approach, is expected to be
large.
The detailed calculations of electronic structure of EuC6 in Ref. [30] feature
the band-structure that is not unlike the rigid-band picture in Fig. 2(a),
with the bands that are crossing the Fermi level clearly reminiscent of the
folded graphene bands. However, two key differences are a significantly lower
doping of the carbon $\pi$-orbitals, which accounts for about $0.5e$ per Eu2+,
and the rest of electrons filling up the Eu-derived $spd$-hybrid band, with
the latter absent in the rigid-band description [30, 17]. These findings are
also supported by the angle-resolved photoemission studies of stage-I EuC6 and
stage-II EuC12 materials, reported in Ref. [30].
The most direct implication of the first result for our analysis of the Fermi
surfaces is the four times smaller density of donated electrons, which
straightforwardly translates into the two times smaller Fermi momentum in the
graphene conduction bands, $k_{F,e/2}\\!\approx\\!0.43\pi/a$. We also estimate
the Fermi momenta of the “true,” trigonally warped Fermi surfaces from the
band structure in Ref. [30] as $k_{F,{\rm max}}\\!\approx\\!0.7\pi/a$ and
$k_{F,{\rm min}}\\!\approx\\!0.45\pi/a$, in a qualitative agreement with the
estimate of $k_{F,e/2}$ above. Our choice of the representative
$E_{F}\\!=\\!0.83t$ (in units of $t\\!=\\!3.16$ eV [4]) in Fig. 2(a) and of
the resultant Fermi surfaces in Fig. 2(b) is made to match the Fermi momenta
from Ref. [30], which, in turn, should approximately correspond to the $0.5e$
filling of the bands.
The other shortcoming of the rigid-band approximation is the omission of the
Eu-derived partially filled $sd$-hybrid band [30, 17]. It was also argued that
the hybridization of the Eu $sd$-orbitals and graphene $\pi$-orbitals is
responsible for the mediation of the strong Kondo interaction between the
localized $4f$-orbital spins of Eu and conduction $\pi$-orbital electrons of
graphite, estimated at $J_{K}\\!\approx\\!0.15$ eV [24]. This key element of
our study is described in Sec. IV.1.
In our analytical treatment of the scattering rate in Sec. IV.2 and Appendix
C, we are motivated by the analysis and discussion provided in this Section
and approximate the relevant electronic degrees of freedom of EuC6 by the two
degenerate bands with the circular Fermi surfaces of radius $k_{F}$ centered
around $\Gamma$ point. We treat $k_{F}$ as a parameter and show how the key
features of the calculated magnetoresistivity evolve with it, see Sec. V. We
expect the renewed interest in the problem to result in a convergence of the
band-structure calculations with the experimental data regarding the relevant
electronic structure and parameters of EuC6 and other GICs.
### II.2 Spin model and parameters
It has been proposed in Refs. [18, 21, 22, 23] that the minimal model that
describes phenomenology of the magnetism in EuC6 is the triangular-lattice
$S\\!=\\!7/2$ model
$\displaystyle\mathcal{H}=$ $\displaystyle\sum_{\langle
ij\rangle_{n}}J_{n}\,\mathbf{S}_{i}\cdot\mathbf{S}_{j}-B\sum_{\langle
ij\rangle_{1}}\left(\mathbf{S}_{i}\cdot\mathbf{S}_{j}\right)^{2}-\mathbf{h}\cdot\sum_{i}\mathbf{S}_{i},$
(1)
where $\langle ij\rangle_{1(2)}$ denote the (next-)nearest-neighbor bonds with
the corresponding exchanges $J_{1(2)}$ and ${\bf h}\\!=\\!g\mu_{B}\mathbf{H}$
in the Zeeman term. A crucial ingredient of this model is the biquadratic
term. While $B$ may be small compared to the exchanges, it is important
because of the $S^{2}$ amplification factor.
It was argued in Refs. [18, 21, 22, 23] that this minimal model would not be
complete without the ring-exchange term, which is discussed below in some more
detail. While the biquadratic and ring-exchange terms play similar role in
stabilizing the up-up-down (UUD or plateau) state in a wide range of fields,
our analysis of the EuC6 phenomenology provided below points to the values of
the ring exchange that are secondary to $B$, differing from the values
advocated in Refs. [18, 21]. However, given a close similarity of their
effects, this variation is likely inconsequential and amounts to a different
parametrization of such effects within an effective model. In the spin-wave
consideration that follows, we will ignore the ring-exchange term entirely,
citing cumbersomeness of its treatment.
Another difference of our model from the consideration of Refs. [18, 21, 22,
23] is that the exchange terms in (1) are taken as Heisenberg, not $XY$. This
makes no difference for the classical phase diagram in the in-plane field,
which was simulated using classical Monte-Carlo in Ref. [18] in the $XY$
limit. However, the actual anisotropy in EuC6 is unlikely to exceed 10%, as is
evidenced by the very similar saturation fields in the in-plane and out-of-
plane magnetization and by the nearly isotropic $g$-factors [18], justifying
our choice of the isotropic limit of the model.
Figure 3: Triangular lattice, its elementary translation vectors
${\bm{\delta}}_{\alpha}$, primitive unit cell for the three-sublattice spin
structures (shaded) with its basis vectors ${\bf a}_{\alpha}$, and an example
of such a structure with $\\{{\rm A},{\rm B},{\rm C}\\}$ sublattices.
Laboratory reference frame $\\{x_{0},y_{0},z_{0}\\}$ and the field direction
are indicated.
#### II.2.1 Classical ground states
In this work, we focus exclusively on the field orientation that is in the
plane of Eu2+ ions, see Fig. 3. While for the isotropic approximation that we
choose in model (1) the direction of the field is irrelevant, the
phenomenology that follows identifies with that of the in-plane field data for
EuC6, which exhibits a weak easy-plane ($XXZ$) anisotropy [21]. The out-of-
plane field direction in this latter case yields a different, and much
simpler, magnetization and ground state evolution [37].
Triangular-lattice antiferromagnets host a rather astonishing variety of the
unconventional field-induced phases, see Refs. [38, 39, 40]. As we have noted
above, EuC6 was the first material in which the best-known of such
unconventional phases, the UUD magnetization plateau state, has been
identified [19].
Figure 4: (a) The schematics of the evolution of magnetic order with the field
from 120${\degree}$ state at $H\\!=\\!0$ to the Y phase, with the transition
to the UUD plateau phase at $H_{c1}$, from the UUD phase to V phase at
$H_{c2}$, followed by a transition to the saturated FM phase at the saturation
field $H_{s}$. The representative three-sublattice spin structures are shown.
(b) Angles of spins with the laboratory $z_{0}$ axis (field direction) for an
arbitrary coplanar three-sublattice structure. Angles
$\widetilde{\alpha}\\!=\\!\\{\beta,\alpha_{1},\alpha_{2}\\}$ correspond to the
$\alpha\\!=\\!\\{{\rm A},{\rm B},{\rm C}\\}$ sublattices. (c) Sets of
$\widetilde{\alpha}$ for all phases.
For the model (1), the field-evolution of the classical ground states is known
from the earlier works [20, 39, 21] with the schematics of the evolution of
magnetic order with the field shown in Fig. 4(a). At $H\\!=\\!0$, spins assume
a 120${\degree}$ configuration that was confirmed for EuC6 by the muon-spin
spectroscopy [29]. A finite field continuously deforms it into the so-called
Y-structure followed by a transition to the UUD (plateau) state at $H_{c1}$.
The spin angles and the field direction are shown in Fig. 4(b) and (c). The
higher field induces a transition from the UUD phase to the V phase at
$H_{c2}$ and to the fully polarized FM phase at the saturation field $H_{s}$.
It is worth noting that in all ordered phases, spin configurations are
coplanar and belong to the three-sublattice structure with the same unit cell,
see Fig. 3 and Fig. 4.
In the earlier studies of EuC6 [18, 21, 22, 23], the minimal model (1) was
also augmented by the ring-exchange term
$\displaystyle\mathcal{H}_{K}=K\sum_{\langle ijk\ell\rangle}P_{ijk\ell},$ (2)
$\displaystyle
P_{ijk\ell}=\left(\mathbf{S}_{i}\mathbf{S}_{j}\right)\left(\mathbf{S}_{k}\mathbf{S}_{\ell}\right)+\left(\mathbf{S}_{i}\mathbf{S}_{\ell}\right)\left(\mathbf{S}_{j}\mathbf{S}_{k}\right)-\left(\mathbf{S}_{i}\mathbf{S}_{k}\right)\left(\mathbf{S}_{j}\mathbf{S}_{\ell}\right),$
where spins belong to the elementary nearest-neighbor four-site plaquettes
$\langle ijk\ell\rangle$.
After some deliberation, one can write the classical energy of the model (1)
with the ring-exchange term (2) for an arbitrary coplanar three-sublattice
structure as
$\displaystyle\frac{E_{cl}}{NS^{2}J_{1}}=(1-k)\big{(}\cos\widetilde{\alpha}_{AB}+\cos\widetilde{\alpha}_{AC}+\cos\widetilde{\alpha}_{BC}\big{)}$
$\displaystyle\quad\quad\ \
+3j_{2}-b\big{(}\cos^{2}\widetilde{\alpha}_{AB}+\cos^{2}\widetilde{\alpha}_{AC}+\cos^{2}\widetilde{\alpha}_{BC}\big{)}$
(3)
$\displaystyle-h\big{(}\cos\widetilde{\alpha}_{A}+\cos\widetilde{\alpha}_{B}+\cos\widetilde{\alpha}_{C}\big{)}+2k\big{(}\cos\widetilde{\alpha}_{AB}\cos\widetilde{\alpha}_{AC}$
$\displaystyle\phantom{-h\big{(}\cos\widetilde{\alpha}_{A}+}\ \ \ \
+\cos\widetilde{\alpha}_{AB}\cos\widetilde{\alpha}_{BC}+\cos\widetilde{\alpha}_{AC}\cos\widetilde{\alpha}_{BC}\big{)},$
where $N$ is the number of sites in the triangular lattice, the dimensionless
field and exchange parameters are in units of the nearest-neighbor exchange
$J_{1}$, $h\\!=\\!g\mu_{B}H/3J_{1}S$, $j_{2}\\!=\\!J_{2}/J_{1}$,
$b\\!=\\!BS^{2}/J_{1}$, and $k\\!=\\!KS^{2}/J_{1}$, spin angles with the field
direction
$\widetilde{\alpha}_{\alpha}\\!=\\!\\{\beta,\alpha_{1},\alpha_{2}\\}$
correspond to the $\alpha\\!=\\!\\{{\rm A},{\rm B},{\rm C}\\}$ sublattices
according to Fig. 4(b) and (c), and mutual angles of spins are
$\widetilde{\alpha}_{AB}\\!=\\!\beta-\alpha_{1}$,
$\widetilde{\alpha}_{AC}\\!=\\!\beta-\alpha_{2}$, and
$\widetilde{\alpha}_{BC}\\!=\\!\alpha_{1}-\alpha_{2}$.
#### II.2.2 Tilt angles and critical fields
Energy minimization in (3) at a fixed field with respect to spin angles should
produce both the equilibrium spin configurations and critical fields for the
transitions between phases. Figure 4 shows that in Y and V phases spin angles
depend on the field continuously, while spins are (anti)collinear with the
field for the full extent of the UUD and FM phases. In the Y and V phases, the
general form of the classical energy in (3) simplifies, with the energy of the
Y phase controlled by one independent angle and for the V phase by two angles.
For the Y phase, a straightforward algebra gives an equation for the angle
$\alpha_{1}$
$\frac{\partial E^{\rm
Y}_{cl}}{\partial\alpha_{1}}=0=(1+b)\,x-a_{0}-6k\,x^{2}-4b\,x^{3}\,,$ (4)
where $x\\!=\\!\cos\alpha_{1}$ and $a_{0}\\!=\\!(1+h-3k)/2$. Since cubic
equation allows for analytical solutions [41], the angles of the spin
configuration within the Y phase are fully determined by such a solution of
(4). In the spin-wave treatment of the model (1) presented below, the
equilibrium spin configuration in the Y phase is obtained from a $k\\!=\\!0$
version of (4).
As was first noted in Ref. [21], the evolution of $\alpha_{1}$ with $H$
becomes discontinuous and transition to the UUD phase turns first-order at
larger values of $B\\!>\\!0$ and $K\\!>\\!0$. However, leaving this detail
aside for a moment, one can always find a solution for a transition field
between the Y and UUD phases by assuming it to be continuous and putting
$\cos\alpha_{1}\\!=\\!1$ in (4), which yields
$h_{c1}=1-6b-9k\,,$ (5)
in agreement with Ref. [21].
The meaning of this critical field is twofold. It is the true critical field
for a phase transition at the smaller values of $B$ and $K$ where it is
continuous. In what follows, we focus on $K\\!=\\!0$, “$B$-only” model (1),
for which continuous transition can be shown to exists up to
$b_{c}\\!=\\!1/11$. For the values of $b\\!>\\!b_{c}$, the Y phase is stable
up to the _higher_ critical field
$\widetilde{h}_{c1}=\sqrt{\frac{4(1+b)^{3}}{27b}}-1\,$ (6)
at which the angle changes discontinuously. However, the critical field in (5)
continues to define the region of $\widetilde{h}_{c1}\\!>\\!h\\!>\\!h_{c1}$
where the plateau phase is (meta)stable, meaning that the spin excitations
defined within the UUD phase are stable down to $h_{c1}$ in (5). A detailed
consideration of the critical fields associated with the first-order
transitions is provided in Appendix A.
Somewhat fortuitously, our choice of parameters for EuC6 discussed below
corresponds to $b$ only very slightly larger than $b_{c}$, so the transitions
that we find are very marginally first-order. Experiments in EuC6 [18] have
also indicated small hysteresis effects in magnetoresistance [18], suggesting
a correspondence between the two.
For the V phase, energy minimization in (3) yields the following equations in
the angles, $\sin\beta\\!=\\!2\sin\alpha_{1}$ and
$h\,\sin\beta=2\sin\gamma\,\big{(}1+k+2(k-b)\cos\gamma\big{)}\,,$ (7)
where $\gamma\\!=\\!\alpha_{1}+\beta$. For the “$B$-only” model (1) that we
focus on below, one can simplify (7) to the equation for $\beta$ in the form
$h\\!=\\!F(\cos\beta,b)$, with
$F(x,b)=\big{(}x+\sqrt{x^{2}+3}\big{)}\big{(}1-b\big{(}x\sqrt{x^{2}+3}-1+x^{2}\big{)}\big{)}\,,$
(8)
which can be solved numerically to find $\alpha_{1}$ and $\beta$ angles of the
equilibrium spin configuration in the V phase.
An approach to the transitions from the UUD to V and from V to the FM phases
by assuming their continuity and (anti)collinearity of the spins in (7) yields
$h_{c2}=1+2b-k\,,\ \ \ h_{s}=3\big{(}1-2b+3k\big{)}\,,$ (9)
also in agreement with Ref. [21]. While a transition at $h_{c2}$ remains
continuous for a wide range of parameters, transition to the saturated phase
for the “$B$-only” model (1) turns first-order at the same $b_{c}\\!=\\!1/11$
as the Y-to-UUD transition at $h_{c1}$ discussed above, showing a similar
phenomenology. Given that the range of parameters discussed below is only
weakly affected by the associated discontinuities, we will carry on referring
to $h_{c1}$, $h_{c2}$, and $h_{s}$ in Eq. (5) and Eq. (9) as to the “true”
critical fields, see Appendix A for more detail.
#### II.2.3 Parameters
It is useful to consider pure Heisenberg limit of the model (1) as a
reference. In that case, the dimensionless critical fields
$h^{0}_{c1}\\!=\\!h^{0}_{c2}\\!=\\!1$ and $h^{0}_{s}\\!=\\!3$, all in units of
$3J_{1}S/g\mu_{B}$. Thus, as one can see from (5) and (9), for $B,K\\!>\\!0$
the biquadratic and ring-exchange terms necessarily open up a finite range of
fields for the plateau phase. However, while both terms drive down $h_{c1}$
from its $h^{0}_{c1}$ value, their effects on $h_{c2}$ and $h_{s}$ are
opposite to each other. Most importantly, if the additional terms are
dominated by the biquadratic one, the critical fields $h_{c1}$ and $h_{c2}$
split away from their Heisenberg value in the opposite directions, with
$h_{c1}$ below and $h_{c2}$ above $h^{0}_{c1}$. If, however, the ring-exchange
term is the leading one, both $h_{c1}$ and $h_{c2}$ shift down from
$h^{0}_{c1}$.
This observation has a direct impact on the analysis of the phenomenology of
EuC6 and parameters of the model that follow from it. A summary of the
experimental data that is relevant to such an analysis can be found in Ref.
[18]. Eu2+ spins order antiferromagnetically at $T_{N}\\!\approx\\!40$ K, with
the 120${\degree}$ structure of their zero-field ground state confirmed more
recently [29]. The critical fields of all the transitions discussed above can
be inferred directly from the $T\\!\rightarrow\\!0$ extrapolations of the
associated anomalies in the resistivity data in Fig. 1(b), which is reproduced
from Ref. [18]. Thus, the experimental value of the saturation field is
$H_{s}^{\rm exp}\\!\approx\\!21.5$ T, while the Y-to-UUD and UUD-to-V
transitions are at $H_{c1}^{\rm exp}\\!\approx\\!1.6$ T and $H_{c2}^{\rm
exp}\\!\approx\\!9.0$ T, respectively, see also Table 1.
Given Eq. (5) and Eq. (9), the experimental values of the three critical
fields are sufficient to uniquely determine three parameters of the model,
$J_{1}$, $B$, and $K$. In broad strokes, an overall energy scale dictated by
$J_{1}$ sets an extent of the ordered phases that is determined from the
saturation field $H_{s}$, while the width of the plateau between $H_{c1}$ and
$H_{c2}$ and their relation to $H_{s}/3$ fixes $B$ and $K$. The results are
listed in the first line of Table 1 where we have also used the Lande g-factor
g=1.94 [18].
In agreement with the prior estimates [18] and general expectations, the
biquadratic and ring-exchange terms are much smaller than the leading
exchanges, yet they are essential for the existence of the unconventional UUD
phase. Importantly, the ring-exchange is subleading to the biquadratic term
with the ratio $B/K\\!\approx\\!3$. Provided our discussion above, the
dominance of $B$ over $K$ is clear already from the fact that the UUD-to-V
critical field $H_{c2}^{\rm exp}$ is substantially larger than $H_{s}^{\rm
exp}/3$.
It is, therefore, rather puzzling to find almost exactly opposite hierarchy of
$B$ and $K$ in Refs. [18, 21, 22, 23], based on the same data for EuC6. The
reason for the difference is the following. With the rest of the
phenomenological constraints being the same, the UUD-to-V critical field in
Refs. [18, 21] is chosen as $\widetilde{H}_{c2}^{\rm exp}\\!\approx\\!6.4$ T,
which is less than $H_{s}^{\rm exp}/3$, hence implying the dominance of $K$
over $B$. The smaller critical field is inferred from a rather broad
magnetization data, which, given the second-order nature of the UUD-to-V
transition, is strongly affected by the finite-temperature effects, see also
Ref. [42] on a different material highlighting the same effect. It is
difficult for us to understand why the lower $\widetilde{H}_{c2}^{\rm exp}$
was insisted upon in the prior works, except for the premeditated importance
of the ring-exchange terms.
| $J_{1}$ | $J_{2}$ | $BS^{2}$ | $KS^{2}$ | $H_{c1}$ | $H_{c2}$ | $H_{s}$
---|---|---|---|---|---|---|---
Exp. | 0.974 | -0.783 | 0.086 | 0.029 | 1.6 | 9.0 | 21.5
Model | 1.085 | -0.728 | 0.1 | 0 | 3.91 | 10.35 | 21.39
Table 1: Exchange parameters (K) and critical fields (T), $S\\!=\\!7/2$. Exp.:
experimental values of the fields define parameters of the model as described
in text. Model: chosen parameters of the model (1) with the resultant critical
fields.
The remaining parameter of the model (1) is the second-neighbor exchange
$J_{2}$, which is necessary to reconcile the value of the ordering
temperature, $T_{N}$, with that of the saturation field, as the two are not
fully compatible for the model that contains only the nearest-neighbor
exchanges. Since the leading mechanism that provides spin couplings in EuC6 is
believed to be of the RKKY-type [18], the $J_{2}$-term with $J_{2}\\!<\\!0$ is
seen as natural.
Another element of the consideration that is easy to justify is the use of the
mean-field approximation for the ordering temperature despite the quasi-2D
character of EuC6 and continuous symmetries of the model (1). The large spin
value $S\\!=\\!7/2$, aforementioned $XXZ$ anisotropy, and the presence of
small interplane couplings [18] that are ignored in our model, all give strong
ground for the use of the mean-field approach [43]
$T_{N}^{\rm MF}=-\frac{S(S+1)}{3k_{B}}\ \lambda_{\rm min}(\mathbf{Q}),$ (10)
where $\lambda_{\rm min}(\mathbf{Q})$ is the lowest eigenvalue of the Fourier
transform of the exchange matrix in (1) at the ordering vector $\mathbf{Q}$.
For the three-sublattice orders, $\mathbf{Q}\\!=\\!(\pm 4\pi/3,0)$ and
$\lambda_{\rm min}(\mathbf{Q})$ can also be inferred from the classical energy
in (3) as $\lambda_{\rm min}(\mathbf{Q})\\!=\\!2E_{cl}^{120\degree}/NS^{2}$ to
yield
$T_{N}^{\rm MF}\approx S(S+1)\big{(}J_{1}-2J_{2}\big{)},$ (11)
where contributions of small interplane couplings are ignored and we have also
dropped even smaller and nearly canceling contributions from the $B$ and $K$
terms. Since $J_{1}$ is already determined from the critical fields,
$T_{N}^{exp}\\!=\\!T_{N}^{\rm MF}$ in Eq. (11) gives $J_{2}$ in the first line
of Table 1.
Having established the secondary role of the ring-exchange term in EuC6
phenomenology, we are going to completely ignore such a term in the model
consideration of the scattering of electrons by spin excitations presented
next. This step is motivated by both strong similarity of the effects provided
by the ring-exchange to that of the biquadratic terms and a considerable
cumbersomness of the spin-wave treatment of the ring-exchange in the
triangular lattice, see Refs. [44, 45].
With the number of model parameters reduced, there are more phenomenological
constraints than there parameters. Fixing one of $H_{c2}$ or $H_{c1}$ to their
experimental value either narrows or widens the extent of the plateau by about
$4$ T compared to the data, with $BS^{2}$ being 0.06K and 0.17K, respectively.
Instead, we fix $BS^{2}$ to an intermediate value of 0.1K, which leads to only
a slightly narrower plateau and somewhat higher critical fields than in
experiment, $H^{th}_{c1}\\!\approx\\!3.9$ T and $H^{th}_{c2}\\!\approx\\!10.4$
T, see Table 1 for a full set of the model parameters. This is the set of
parameters that will be used henceforth in all calculations of the
magnetoresistivity. It corresponds to the dimensionless parameters
$j_{2}\\!=\\!J_{2}/J_{1}\\!=\\!-0.671$ and
$b\\!=\\!BS^{2}/J_{1}\\!=\\!0.0922$. For the representative pictures of the
spin-wave spectra shown in Sec. III.5 below we choose a close set of
$j_{2}\\!=\\!-0.8$ and $b\\!=\\!0.1$.
## III Spin excitations
In this Section, a general spin-wave approach is formulated for all coplanar
three-sublattice states in Fig. 4. In Appendix B, we provide a consideration
of the FM, 120${\degree}$, and UUD states for which a simplified approach is
possible, allowing to obtain fully analytical results.
We would like to note that the biquadratic exchange has been widely employed
to emulate quantum effects in a variety of spin models, including Heisenberg
and $XXZ$ triangular-lattice models to stabilize their plateau state. However,
we are not aware of the spin-wave theory consideration of the model (1) in the
literature, with an exception of the early work [20], which provided a
consideration of the zone-center, ${\bf k}\\!=\\!0$, modes. The possibility of
a consistent spin-wave expansion for an arbitrary coplanar three-sublattice
structure presented next was motivated, in part, by a general formalism in
Ref. [46].
### III.1 General case of a coplanar state
For a spin-wave expansion, the laboratory reference frame
$\\{x_{0},y_{0},z_{0}\\}$ in Fig. 3 and Fig. 4 needs to be rotated to the
local reference frame $\\{x,y,z\\}$ on each site, so that the $z$ axis is
along the direction dictated by a classical spin configuration obtained in
Sec. II.2.1. For the coplanar states in Fig. 4, such a transformation is a
simple rotation in the $x_{0}$–$z_{0}$ plane, such that
$S_{\alpha}^{y_{0}}\\!=\\!S_{\alpha}^{y}$ and
$\displaystyle
S^{x_{0}}_{\alpha}=S^{x}_{\alpha}\cos\widetilde{\alpha}-S^{z}_{\alpha}\sin\widetilde{\alpha},$
(12) $\displaystyle
S^{z_{0}}_{\alpha}=S^{z}_{\alpha}\cos\widetilde{\alpha}+S^{x}_{\alpha}\sin\widetilde{\alpha},$
where $\alpha$ and $\widetilde{\alpha}$ are, respectively, the sublattices and
corresponding spin angles in Fig. 4(b).
### III.2 $1/S$-expansion
Consider the $1/S$-expansion of each individual term in the model (1)
separately. For the nearest-neighbor $J_{1}$ term, it is convenient to rewrite
it first as
$\displaystyle\mathcal{H}_{J_{1}}=J_{1}\sum_{\langle
ij\rangle_{1}}\left(\hat{h}_{ij}^{(e)}+\hat{h}_{ij}^{(o)}\right),$ (13)
where the “even” (e) and “odd” (o) parts
$\displaystyle\hat{h}_{ij}^{(e)}$
$\displaystyle=S_{i}^{y}S_{j}^{y}+\cos\widetilde{\alpha}_{ij}\left(S_{i}^{x}S_{j}^{x}+S_{i}^{z}S_{j}^{z}\right),$
(14) $\displaystyle\hat{h}_{ij}^{(o)}$
$\displaystyle=\sin\widetilde{\alpha}_{ij}\left(S_{i}^{z}S_{j}^{x}-S_{i}^{x}S_{j}^{z}\right),$
are separated to distinguish their subsequent contribution of the even and odd
powers of the bosonic operators to the $1/S$-expansion; here
$\widetilde{\alpha}_{ij}\\!=\\!\widetilde{\alpha}_{i}-\widetilde{\alpha}_{j}$
are the angles between neighboring spins.
In the lowest orders, the even part yields a contribution to the classical
energy and to the harmonic, ${\cal O}(S)$, linear spin-wave theory (LSWT)
order of the expansion
$\displaystyle\hat{h}_{ij}^{(e)}$ $\displaystyle\Rightarrow
S^{2}\cos\widetilde{\alpha}_{ij}+\hat{h}_{ij,LSWT}^{(e)}\,,$ (15)
while the odd part in (14) gives the linear order, ${\cal O}(S^{3/2})$, which
must vanish upon a summation in (13) for the classical energy minimum,
followed by the higher-order, ${\cal O}(S^{1/2})$, anharmonic interactions
that can be neglected for the large spin values.
However, for the biquadratic term of the model (1)
$\mathcal{H}_{B}=-B\sum_{\langle
ij\rangle_{1}}\big{(}\mathbf{S}_{i}\cdot\mathbf{S}_{j}\big{)}^{2}=-B\sum_{\langle
ij\rangle_{1}}\left(\hat{h}_{ij}^{(e)}+\hat{h}_{ij}^{(o)}\right)^{2},$ (16)
both even and odd parts play a role in its LSWT order
$\left(\mathbf{S}_{i}\cdot\mathbf{S}_{j}\right)^{2}\Rightarrow
2S^{2}\cos\widetilde{\alpha}_{ij}\,\hat{h}_{ij,LSWT}^{(e)}+\Big{(}\hat{h}_{ij}^{(o)}\Big{)}^{2}_{LSWT}\,,\
\ $ (17)
with their contributions, obtained from the standard Holstein-Primakoff
bozonization of spins in the rotated reference frame,
$S_{i}^{z}\\!=\\!S-a^{\dagger}_{i}a^{\phantom{{\dagger}}}_{i}$ and
$S_{i}^{-}\\!=\\!a_{i}^{\dagger}\sqrt{2S}$, are
$\displaystyle\hat{h}_{ij,LSWT}^{(e)}=-\frac{S}{2}\Big{[}\big{(}a_{i}^{\dagger}-a_{i}\big{)}\big{(}a_{j}^{\dagger}-a_{j}\big{)}$
(18)
$\displaystyle\quad\quad+\cos\widetilde{\alpha}_{ij}\left(2\big{(}a_{i}^{\dagger}a_{i}+a^{\dagger}_{j}a_{j}\big{)}-\big{(}a_{i}^{\dagger}+a_{i}\big{)}\big{(}a_{j}^{\dagger}+a_{j}\big{)}\right)\Big{]},$
$\displaystyle\Big{(}\hat{h}_{ij}^{(o)}\Big{)}^{2}_{LSWT}=\frac{S^{3}}{2}\sin^{2}\widetilde{\alpha}_{ij}\left(a_{i}^{\dagger}+a_{i}-a_{j}^{\dagger}-a_{j}\right)^{2}.$
Of the remaining terms in model (1), the Zeeman term is particularly simple,
and so is the next-nearest-neighbor $J_{2}$ term, as the former involves only
local energy of bosons while the latter connects spins that belong to the same
sublattices, giving, in the LSWT order,
$\displaystyle\mathcal{H}_{H}=$ $\displaystyle
g\mu_{B}H\sum_{i}\cos\widetilde{\alpha}_{i}\,a_{i}^{\dagger}a_{i}\,,$ (19)
$\displaystyle\mathcal{H}_{J_{2}}=$ $\displaystyle-J_{2}S\sum_{\langle
ij\rangle_{2}}\left(a_{i}^{\dagger}a_{i}+a^{\dagger}_{j}a_{j}-\big{(}a_{i}^{\dagger}a_{j}+a^{\dagger}_{j}a_{i}\big{)}\right).$
(20)
We point out, as a side remark, that it is relatively straightforward to
modify model (1) and the resultant LSWT Hamiltonian to include the effects of
the easy-plane anisotropy that is present in EuC6. However, we did not find
significant qualitative changes in the results for some of the key phases
studied in this work. Given the extra cumbersomeness this anisotropy would
introduce in the LSWT matrix below, we leave a detailed study of such an
extension to a future work.
### III.3 LSWT Hamiltonian
The LSWT order of the model (1), explicated in Eqs. (13)–(20), is obtained for
a general coplanar state. To make further progress, one needs to specify spin
arrangement for the classical ground state. In our case, all such states of
interest can be represented as the three-sublattice states, highlighted in
Fig. 4. Thus, a general approach to all of them can be pursued [46].
The first step is to switch from the site notation $i$ to the one of the unit
cells of the three-sublattice structure $\ell$ and sublattice index $\alpha$:
$i\\!\rightarrow\\!\\{\alpha,\ell\\}$. As a result, the Holstein-Primakoff
boson operators are split into three species
$a^{({\dagger})}_{\alpha,\ell}\\!=\\!\\{a_{\ell}^{({\dagger})},b_{\ell}^{({\dagger})},c_{\ell}^{({\dagger})}\\}$
corresponding to $\alpha\\!=\\!\\{{\rm A},{\rm B},{\rm C}\\}$ sublattices. The
Fourier transformation for them is
$a^{\phantom{{\dagger}}}_{\alpha,\ell}=\frac{1}{\sqrt{N_{c}}}\sum_{\bf
q}\,a^{\phantom{{\dagger}}}_{\alpha,\bf q}\,e^{-i{\bf q}{\bf
r}_{\alpha,\ell}},$ (21)
where ${\bf r}_{\alpha,\ell}\\!=\\!{\bf R}_{\ell}\\!+\\!\bm{\rho}_{\alpha}$
and $N_{c}\\!=\\!N/3$ is the number of unit cells. The sublattice coordinates
within the unit cell can be chosen as $\bm{\rho}_{A}\\!=\\!0$,
$\bm{\rho}_{B}\\!=\\!-\bm{\delta}_{2}$, and
$\bm{\rho}_{C}\\!=\\!\bm{\delta}_{3}$, see Fig. 3.
After some algebra, using these boson species and their Fourier transforms,
the LSWT Hamiltonian for an arbitrary coplanar three-sublattice state reads as
$\displaystyle\hat{\cal H}^{(2)}=\frac{3J_{1}S}{2}\sum_{\bf q}\hat{\bf x}_{\bf
q}^{\dagger}\hat{\bf H}_{\bf q}\hat{\bf x}_{\bf q}^{\phantom{\dagger}}\,,$
(22)
where $\hat{\bf x}^{\dagger}_{\bf q}\\!=\\!\left(a^{\dagger}_{\bf
q},b^{\dagger}_{\bf q},c^{\dagger}_{\bf q},a^{\phantom{{\dagger}}}_{-\bf
q},b^{\phantom{{\dagger}}}_{-\bf q},c^{\phantom{{\dagger}}}_{-\bf q}\right)$
and $\hat{\bf H}_{\bf q}$ is a matrix
$\displaystyle\hat{\bf H}_{\bf q}=\left(\begin{array}[]{cc}\hat{\bf
A}^{\phantom{\dagger}}_{\bf q}&\hat{\bf B}^{\phantom{\dagger}}_{\bf
q}\\\\[2.15277pt] \hat{\bf B}^{\dagger}_{\bf q}&\hat{\bf A}^{*}_{-\bf
q}\end{array}\right),$ (25)
with the $3\times 3$ matrices $\hat{\bf A}^{\phantom{{\dagger}}}_{\bf q}$ and
$\hat{\bf B}^{\phantom{{\dagger}}}_{\bf q}$
$\displaystyle\hat{\bf A}^{\phantom{{\dagger}}}_{\bf
q}=\left(\begin{array}[]{ccc}A_{\bf q}&D_{\bf q}&E^{*}_{\bf q}\\\ D^{*}_{\bf
q}&B_{\bf q}&F_{\bf q}\\\ E_{\bf q}&F^{*}_{\bf q}&C_{\bf
q}\end{array}\right),\ \ \ \hat{\bf B}^{\phantom{{\dagger}}}_{\bf
q}=\left(\begin{array}[]{ccc}G&J_{\bf q}&K^{*}_{\bf q}\\\ J^{*}_{\bf
q}&H&L_{\bf q}\\\ K_{\bf q}&L^{*}_{\bf q}&I\end{array}\right).\ \ \ \ \ \ $
(32)
The elements of the $\hat{\bf A}^{\phantom{{\dagger}}}_{\bf q}$ matrix are
given by
$\displaystyle A_{\bf q}=$
$\displaystyle\,h\cos\beta-\cos\widetilde{\alpha}_{AB}-\cos\widetilde{\alpha}_{AC}-2j_{2}\big{(}1-\gamma^{(2)}_{\bf
q}\big{)}$ $\displaystyle+b\big{(}1+3\left(\cos 2\widetilde{\alpha}_{AB}+\cos
2\widetilde{\alpha}_{AC}\right)/2\big{)},$ $\displaystyle B_{\bf q}=$
$\displaystyle\,h\cos\alpha_{1}-\cos\widetilde{\alpha}_{AB}-\cos\widetilde{\alpha}_{BC}-2j_{2}\big{(}1-\gamma^{(2)}_{\bf
q}\big{)}$ $\displaystyle+b\big{(}1+3\left(\cos 2\widetilde{\alpha}_{AB}+\cos
2\widetilde{\alpha}_{BC}\right)/2\big{)},$ $\displaystyle C_{\bf q}=$
$\displaystyle\,h\cos\alpha_{2}-\cos\widetilde{\alpha}_{BC}-\cos\widetilde{\alpha}_{AC}-2j_{2}\big{(}1-\gamma^{(2)}_{\bf
q}\big{)}$ $\displaystyle+b\big{(}1+3\left(\cos 2\widetilde{\alpha}_{BC}+\cos
2\widetilde{\alpha}_{AC}\right)/2\big{)},$ (33) $\displaystyle D_{\bf q}=$
$\displaystyle\,\gamma_{\bf
q}\big{(}1+2b(1-2\cos\widetilde{\alpha}_{AB})\big{)}\cos^{2}(\widetilde{\alpha}_{AB}/2),$
$\displaystyle E_{\bf q}=$ $\displaystyle\,\gamma_{\bf
q}\big{(}1+2b(1-2\cos\widetilde{\alpha}_{AC})\big{)}\cos^{2}(\widetilde{\alpha}_{AC}/2),$
$\displaystyle F_{\bf q}=$ $\displaystyle\,\gamma_{\bf
q}\big{(}1+2b(1-2\cos\widetilde{\alpha}_{BC})\big{)}\cos^{2}(\widetilde{\alpha}_{BC}/2),$
and of the $\hat{\bf B}^{\phantom{{\dagger}}}_{\bf q}$ matrix, respectively,
$\displaystyle G=$
$\displaystyle-b\big{(}\sin^{2}\widetilde{\alpha}_{AB}+\sin^{2}\widetilde{\alpha}_{AC}\big{)},$
$\displaystyle H=$
$\displaystyle-b\big{(}\sin^{2}\widetilde{\alpha}_{AB}+\sin^{2}\widetilde{\alpha}_{BC}\big{)},$
$\displaystyle I=$
$\displaystyle-b\big{(}\sin^{2}\widetilde{\alpha}_{BC}+\sin^{2}\widetilde{\alpha}_{AC}\big{)},$
(34) $\displaystyle J_{\bf q}=$ $\displaystyle-\gamma_{\bf
q}\big{(}1-2b(1+2\cos\widetilde{\alpha}_{AB})\big{)}\sin^{2}(\widetilde{\alpha}_{AB}/2),$
$\displaystyle K_{\bf q}=$ $\displaystyle-\gamma_{\bf
q}\big{(}1-2b(1+2\cos\widetilde{\alpha}_{AC})\big{)}\sin^{2}(\widetilde{\alpha}_{AC}/2),$
$\displaystyle L_{\bf q}=$ $\displaystyle-\gamma_{\bf
q}\big{(}1-2b(1+2\cos\widetilde{\alpha}_{BC})\big{)}\sin^{2}(\widetilde{\alpha}_{BC}/2),$
where $h\\!=\\!g\mu_{B}H/3J_{1}S$, $j_{2}\\!=\\!J_{2}/J_{1}$, and
$b\\!=\\!BS^{2}/J_{1}$ as before,
$\widetilde{\alpha}_{AB}\\!=\\!\beta-\alpha_{1}$,
$\widetilde{\alpha}_{AC}\\!=\\!\beta-\alpha_{2}$,
$\widetilde{\alpha}_{BC}\\!=\\!\alpha_{1}-\alpha_{2}$, and
$\displaystyle\gamma_{\bf q}=\frac{1}{3}\sum_{\alpha}e^{i{\bf
q}\bm{\delta}_{\alpha}},\ \ \ \gamma^{(2)}_{\bf
q}=\frac{1}{3}\sum_{\alpha}\cos{\bf q}\bm{a}_{\alpha}\,,$ (35)
with the first- and second-neighbor translation vectors
$\bm{\delta}_{1}\\!=\\!(1,0)a$, $\bm{\delta}_{2}\\!=\\!(-1,\sqrt{3})a/2$,
$\bm{\delta}_{3}\\!=\\!-(1,\sqrt{3})a/2$, and
$\bm{a}_{1}\\!=\\!(3,-\sqrt{3})a/2$, $\bm{a}_{3}\\!=\\!(0,\sqrt{3})a$,
$\bm{a}_{2}\\!=\\!-(3,\sqrt{3})a/2$, respectively, see Fig. 3; $a$ is the
lattice constant.
### III.4 Diagonalization
The eigenvalues of $\hat{\bf g}\hat{\bf H}_{\bf q}$ in (25), give magnon
eigenenergies $\\{\omega_{1{\bf q}},\omega_{2{\bf q}},\omega_{3{\bf
q}},-\omega_{1-{\bf q}},-\omega_{2-{\bf q}},-\omega_{3-{\bf q}}\\}$ (in units
of $3J_{1}S$). Here $\hat{\bf g}$ is a diagonal matrix $[1,1,1,-1,-1,-1]$, see
Ref. [47]. While magnon energies are crucial for our consideration of the
spin-wave scattering of electrons that follows, an essential role is also
played by the matrix elements, which are related to the $U$ and $V$ parameters
of the generalized Bogolyubov transformation from the Holstein-Primakoff
bosons to the ones of the quasiparticle eigenmodes
$\displaystyle a^{\phantom{{\dagger}}}_{\alpha,{\bf
q}}=\sum_{\gamma}\left(U_{\alpha,{\bf
q}}^{({\gamma})}\,A^{\phantom{{\dagger}}}_{\gamma,{\bf q}}+V_{\alpha,{\bf
q}}^{({\gamma})}\,A^{{\dagger}}_{\gamma,-{\bf q}}\right),$ (36)
with the quasiparticle operators $A_{\gamma,{\bf q}}\\!=\\!\left\\{A_{\bf
q},B_{\bf q},C_{\bf q}\right\\}$ and
$\displaystyle\sum_{\gamma}\left(\big{|}U_{\alpha,{\bf
q}}^{({\gamma})}\big{|}^{2}-\big{|}V_{\alpha,{\bf
q}}^{({\gamma})}\big{|}^{2}\right)=1\,.$ (37)
The transformation (36) can be written in a matrix form
$\displaystyle\hat{\bf x}_{\bf q}\\!=\\!\left(\\!\begin{array}[]{c}\hat{\bf
a}^{\phantom{\dagger}}_{\bf q}\\\\[2.15277pt] \hat{\bf a}^{\dagger}_{-\bf
q}\end{array}\\!\right)\\!=\\!\left(\\!\begin{array}[]{cc}\hat{\bf U}_{\bf
q}&\hat{\bf V}_{\bf q}\\\\[2.15277pt] \hat{\bf V}^{*}_{-\bf q}&\hat{\bf
U}^{*}_{-\bf q}\end{array}\\!\right)\\!\left(\\!\begin{array}[]{c}\hat{\bf\cal
A}^{\phantom{\dagger}}_{\bf q}\\\\[2.15277pt] \hat{\bf\cal A}^{\dagger}_{-\bf
q}\end{array}\\!\right)\\!=\hat{\bf S}_{\bf q}\cdot\hat{\bf z}_{\bf q},\ \ \ $
(44)
where vectors $\hat{\bf a}_{\bf q}\\!=\\!\big{[}a_{\bf q},b_{\bf q},c_{\bf
q}\big{]}^{T}$, $\hat{\bf a}^{\dagger}_{-\bf q}\\!=\\!\big{[}a^{\dagger}_{-\bf
q},b^{\dagger}_{-\bf q},c^{\dagger}_{-\bf q}\big{]}^{T}$ and $\hat{\bf\cal
A}_{\bf q}\\!=\\!\big{[}A_{\bf q},B_{\bf q},C_{\bf q}\big{]}^{T}$,
$\hat{\bf\cal A}^{\dagger}_{-\bf q}\\!=\\!\big{[}A^{\dagger}_{-\bf
q},B^{\dagger}_{-\bf q},C^{\dagger}_{-\bf q}\big{]}^{T}$ are introduced. It
follows that the transformation matrix $\hat{\bf S}_{\bf q}$ diagonalizes
$\hat{\bf g}\hat{\bf H}_{\bf q}$ in Eq. (25), see Refs. [47, 48]. Thus, the
$U_{\alpha}^{({\gamma})}$ and $V_{\alpha}^{({\gamma})}$ parameters can be
extracted as the elements of the properly normalized eigenvectors of $\hat{\bf
g}\hat{\bf H}_{\bf q}$ from a diagonalization procedure.
With all components of the $\hat{\bf A}_{\bf q}$ and $\hat{\bf B}_{\bf q}$
matrices (32) given explicitly in (33) and (34), the $6\\!\times\\!6$ LSWT
Hamiltonian (25) has to be diagonalized numerically. We have implemented such
a procedure using MATHEMATICA. In Sec. III.5, we provide plots of magnon
energies throughout the Brillouin zone (BZ) in Fig. 5 for the representative
field values from all the phases in Fig. 4(a).
Figure 5: Full BZ of the triangular lattice (outer hexagon) and magnetic BZ of
the three-sublattice structures (inner hexagon), high-symmetry points in units
of inverse lattice spacing $1/a$, and the direction of a representative
K$\Gamma$MK cut.
We also point out that although the approach to the multi-flavor boson problem
discussed here is very general, there are significant simplifications in our
case owing to the high symmetry of the model (1) and the lattice.
Specifically, $\hat{\bf A}^{*}_{-\bf q}\\!=\\!\hat{\bf A}_{\bf q}$ and
$\hat{\bf B}^{\dagger}_{\bf q}\\!=\\!\hat{\bf B}_{\bf q}$ in (25) as their
off-diagonal matrix elements (33), (34) are simply proportional to the complex
hopping amplitude $\gamma^{*}_{-\bf q}\\!=\\!\gamma_{\bf q}$ (35). As a
result, all eigenenergies of $\hat{\bf g}\hat{\bf H}_{\bf q}$ are reciprocal,
$\omega_{\gamma-{\bf q}}\\!=\\!\omega_{\gamma{\bf q}}$, and $\\{\hat{\bf
V}^{*}_{-\bf q},\hat{\bf U}^{*}_{-\bf q}\\}\\!=\\!\\{\hat{\bf V}_{\bf
q},\hat{\bf U}_{\bf q}\\}$ in Eq. (44).
### III.5 Magnon eigenenergies
Figure 6: Magnon energies in units of $3SJ_{1}$ for several representative
field values from the phases sketched in Fig. 4,
$j_{2}\\!=\\!J_{2}/J_{1}\\!=\\!-0.8$ and $b\\!=\\!BS^{2}/J_{1}\\!=\\!0.1$. (a)
120$\degree$-phase, $h\\!=\\!0$, (b) Y phase, $h\\!=\\!0.3$, (c) UUD phase,
$h\\!=\\!h_{c1}\\!=\\!0.4$, $h\\!=\\!0.8$, and $h\\!=\\!h_{c2}\\!=\\!1.2$, (d)
V phase, $h\\!=\\!2.0$, (e) FM phase $h\\!=\\!h_{s}\\!=\\!2.4$,
$h\\!=\\!h_{s}+1$, and $h\\!=\\!h_{s}+2$. Transitions are at
$h_{c1}\\!=\\!1-6b$, $h_{c2}\\!=\\!1+2b$, and $h_{s}\\!=\\!3(1-2b)$. The 1D
phase diagram vs field with representative field values is sketched in (f).
In Fig. 6 we provide plots of magnon eigenenergies for several representative
field values from all of the phases sketched in Fig. 4(a) and for the
parameters in model (1) $j_{2}\\!=\\!J_{2}/J_{1}\\!=\\!-0.8$ and
$b\\!=\\!BS^{2}/J_{1}\\!=\\!0.1$. Energies are in units of $3SJ_{1}$ and
dimensionless field is $h\\!=\\!g\mu_{B}H/3J_{1}S$. All plots are along a
representative cut K$\Gamma$MK through the full Brillouin zone shown in Fig.
5.
In zero field, $h\\!=\\!0$, magnetic order is the canonical 120$\degree$ phase
with the $O(3)$ symmetry that is spontaneously broken. Since it is broken
fully by the choice of the ordering plane and by the spin arrangement within
the plane, there are three Goldstone modes that one can observe in Fig. 6(a).
As we discuss in some more detail in Appendix B for the 120$\degree$ case, the
three magnon branches that are defined within the magnetic BZ can be related
to a single branch defined within the full BZ using “rotated” reference frame
for the spin quantization axes. This allows to represent the full spectrum as
the “original” branch, labeled by $\omega_{\bf k}$ in Fig. 6(a), and two modes
that are “shifted” by the ordering vector $\pm{\bf Q}\\!=\\!(4\pi/3,0)$.
For a finite in-plane field, the symmetry of the model (1) is lowered to
$U(1)$ by the field. Spontaneous breaking of the $U(1)$ symmetry within the Y
phase in Fig. 4(a) at $h\\!<\\!h_{c1}$ results in a single magnon branch with
a Goldstone mode and two gapped branches, as is shown in Fig. 6(b) for
$h\\!=\\!0.3$. A characteristic feature of the gapless branch is an upward
curvature of the dispersion in the long-wavelength limit, $\omega_{\bf
k}\\!\approx\\!c|{\bf k}|+r|{\bf k}|^{3}$ with $r\\!>\\!0$.
The UUD phase in Fig. 4(a) is sandwiched between two critical fields,
$h_{c1}\\!=\\!1-6b\\!=\\!0.4$ and $h_{c2}\\!=\\!1+2b\\!=\\!1.2$. Since the
$U(1)$ symmetry is preserved throughout this phase, the spectrum is,
generally, gapped except at the transition points, see Fig. 6(c) that shows
magnon spectra at both $h_{c1}$ and $h_{c2}$ and at intermediate
$h\\!=\\!0.8$. The partially polarized, $U(1)$-preserving UUD state is, in a
way, similar to the fully polarized FM phase, with the spectra for the latter
for the fields at and above the saturation field $h_{s}\\!=\\!3(1-2b)$ shown
in Fig. 6(e). Because of the continuous $U(1)$ symmetry of the model (1),
magnetic field couples to a conserved total magnetization in both UUD and FM
cases, which leads to the linear dependence of magnon energies in Fig. 6(c)
and (e) on the field. This also makes the transitions at $h_{s}$ and
$h_{c1(2)}$ analogous to the Bose-Einstein condensation (BEC) [49, 50]. We
note that the absolute minima of $\omega_{\bf k}$ and the corresponding BEC
condensation points in the FM case are at the ordering vectors of the three-
sublattice order, $\pm{\bf Q}\\!=\\!\pm$K, not at the $\Gamma$ point. To
emphasize this feature of the FM phase, the magnon energies in Fig. 6(e) are
shown without folding on the magnetic BZ, see also Appendix B.1.
The last phase of the model (1) with a spontaneously broken $U(1)$ symmetry
that is realized at $h_{c2}\\!<\\!h\\!<\\!h_{s}$ is the V phase, see Fig.
4(a). Its spectrum is similar to that of the Y phase, having one concave
Goldstone and two gapped modes, see Fig. 6(d). It can be seen as interpolating
between the spectra of the UUD phase at $h_{c2}$ and that of the FM phase at
the saturation field.
## IV Kondo coupling and resistivity
In this Section we derive the electron-magnon interaction Hamiltonian,
originating from the Kondo coupling, for a general case of a coplanar spin
arrangement and present the expression for the electronic transport relaxation
rate due to such a scattering mechanism.
### IV.1 Kondo coupling
The most reasonable minimal model for the interaction of conduction electrons
with the local spins in the magnetic layers of EuC6 is the Kondo coupling
$\displaystyle{\cal H}_{int}=J_{K}\sum_{i}{\bf s}_{i}\cdot{\bf S}_{i},$ (45)
where electron spin operators are
$s^{a}_{i}\\!=\\!\frac{1}{2}f^{\dagger}_{i,\alpha}\hat{\sigma}^{a}_{\alpha\beta}f^{\phantom{{\dagger}}}_{i,\beta}$
with $\hat{\bm{\sigma}}_{i}$ being Pauli matrices. With the external field
providing spin quantization axis, it is natural to split (45) into spin-flip
and non-spin-flip parts, ${\cal H}_{int}\\!=\\!{\cal H}_{int}^{+-}+{\cal
H}_{int}^{zz}$,
$\displaystyle{\cal H}_{int}^{+-}=$
$\displaystyle\,\frac{J_{K}}{2}\sum_{i}\big{(}f^{\dagger}_{i,\uparrow}f^{\phantom{{\dagger}}}_{i,\downarrow}S_{i}^{-_{0}}+f^{\dagger}_{i,\downarrow}f^{\phantom{{\dagger}}}_{i,\uparrow}S_{i}^{+_{0}}\big{)},$
(46) $\displaystyle{\cal H}_{int}^{zz}=$
$\displaystyle\,\frac{J_{K}}{2}\sum_{i}\big{(}f^{\dagger}_{i,\uparrow}f^{\phantom{{\dagger}}}_{i,\uparrow}-f^{\dagger}_{i,\downarrow}f^{\phantom{{\dagger}}}_{i,\downarrow}\big{)}S_{i}^{z_{0}},$
where $f^{({\dagger})}_{i,\uparrow(\downarrow)}$ are operators of the
conduction electrons and $\\{-_{0},+_{0},z_{0}\\}$ indices in the operators of
local spins refer to the “laboratory” reference frame
$\\{x_{0},y_{0},z_{0}\\}$ associated with the field direction, see Fig. 3 and
Fig. 4.
For a general coplanar spin configuration in Fig. 4, the local axes are
rotated from the laboratory ones (12) to introduce quantized spin excitations
for a given spin arrangement. Consider the non-spin-flip part. Here, according
to (12),
$S^{z_{0}}_{i}\\!=\\!S^{z}_{i}\cos\widetilde{\alpha}+S^{x}_{i}\sin\widetilde{\alpha}$,
with $\widetilde{\alpha}$ being the angle of spin’s $z$-axis on a site $i$
with $z_{0}$. Upon quantization, $S^{z}_{i}$ converts into two-magnon term,
while $S^{x}_{i}$ yields one-magnon emission/absorption. Similarly to the
problem of electron-phonon scattering, it is the lowest-order coupling that
needs to be considered, unless it is forbidden for a symmetry reason or its
scattering kinematics is suppressed. In our case, there are no such
constraints and the $S^{z}$ part is also of the higher order in $1/S$ sense.
Therefore, we approximate the local spin operators in (46) by their single-
magnon components
$\displaystyle S_{i}^{+_{0}}\approx$
$\displaystyle\,2\,\sqrt{\frac{S}{2}}\,\Big{(}\cos^{2}\left(\widetilde{\alpha}/2\right)\,a^{\phantom{{\dagger}}}_{i}-\sin^{2}\left(\widetilde{\alpha}/2\right)\,a^{{\dagger}}_{i}\Big{)},$
$\displaystyle S_{i}^{z_{0}}\approx$
$\displaystyle\,\sqrt{\frac{S}{2}}\,\sin\widetilde{\alpha}\,\Big{(}a^{\phantom{{\dagger}}}_{i}+a^{{\dagger}}_{i}\Big{)},$
(47)
where the angle $\widetilde{\alpha}$ depends on the sublattice.
Using Fourier transform (21) in (46) together with (47), one arrives at
$\displaystyle{\cal H}_{int}^{+-}=$
$\displaystyle\frac{2\widetilde{J}_{K}}{\sqrt{3N}}\sum_{{\bf k},{\bf
q}}\Big{[}f^{\dagger}_{{\bf k}-{\bf q}\uparrow}f^{\phantom{{\dagger}}}_{{\bf
k}\downarrow}\sum_{\alpha}\Big{(}\cos^{2}\left(\widetilde{\alpha}/2\right)a^{{\dagger}}_{\alpha,{\bf
q}}$ $\displaystyle\phantom{\frac{2\widetilde{J}_{K}}{\sqrt{3N}}\sum_{{\bf
k},{\bf q}}\Big{[}f^{\dagger}_{{\bf k}-{\bf
q}\uparrow}f}-\sin^{2}\left(\widetilde{\alpha}/2\right)a^{\phantom{{\dagger}}}_{\alpha,-{\bf
q}}\Big{)}+{\rm H.c.}\Big{]},$ $\displaystyle{\cal H}_{int}^{zz}=$
$\displaystyle\,\frac{\widetilde{J}_{K}}{\sqrt{3N}}\sum_{{\bf k},{\bf
q}}\Big{[}f^{\dagger}_{{\bf k}-{\bf q}\uparrow}f^{\phantom{{\dagger}}}_{{\bf
k}\uparrow}-f^{\dagger}_{{\bf k}-{\bf
q}\downarrow}f^{\phantom{{\dagger}}}_{{\bf k}\downarrow}\Big{]}$ (48)
$\displaystyle\phantom{\frac{\widetilde{J}_{K}}{\sqrt{3N}}\sum_{{\bf k},{\bf
q}}\Big{[}f^{\dagger}_{{\bf k}-{\bf
q}\uparrow}}\times\sum_{\alpha}\sin\widetilde{\alpha}\,\Big{(}a^{{\dagger}}_{\alpha,{\bf
q}}+a^{\phantom{{\dagger}}}_{\alpha,-{\bf q}}\Big{)},$
where $\widetilde{J}_{K}\\!=\\!\frac{1}{2}J_{K}\sqrt{S/2}$, $N$ is the total
number of sites and summation in ${\bf k}$ and ${\bf q}$ is over the full
Brillouin zone of the triangular lattice, Fig. 5. We note that the single-
magnon non-spin-flip terms are nonzero in the 120$\degree$, Y, and V phases,
in which the angles $\widetilde{\alpha}\\!\neq\\!\\{0,\pi\\}$, because in
these states the symmetry of the Hamiltonian (1) is broken completely and a
spin-flip does not correspond to a particular spin value.
The last transformation is to the quasiparticle operators given by Eq. (36),
which yields
$\displaystyle{\cal H}_{int}^{+-}=$
$\displaystyle\frac{\widetilde{J}_{K}}{\sqrt{3N}}\sum_{{\bf k},{\bf
q}}\Big{[}f^{\dagger}_{{\bf k}-{\bf q}\uparrow}f^{\phantom{{\dagger}}}_{{\bf
k}\downarrow}\sum_{\gamma}\Big{(}M^{+-}_{\gamma,{\bf
q}}A^{{\dagger}}_{\gamma,{\bf q}}$
$\displaystyle\phantom{\frac{\widetilde{J}_{K}}{\sqrt{3N}}\sum_{{\bf k},{\bf
q}}\Big{[}f^{\dagger}_{{\bf k}-{\bf q}\uparrow}f^{\phantom{{\dagger}}}_{{\bf
k}\downarrow}}+N^{+-}_{\gamma,{\bf q}}A^{\phantom{{\dagger}}}_{\gamma,-{\bf
q}}\Big{)}+{\rm H.c.}\Big{]},$ $\displaystyle{\cal H}_{int}^{zz}=$
$\displaystyle\,\frac{\widetilde{J}_{K}}{\sqrt{3N}}\sum_{{\bf k},{\bf
q}}\Big{[}f^{\dagger}_{{\bf k}-{\bf q}\uparrow}f^{\phantom{{\dagger}}}_{{\bf
k}\uparrow}-f^{\dagger}_{{\bf k}-{\bf
q}\downarrow}f^{\phantom{{\dagger}}}_{{\bf k}\downarrow}\Big{]}$ (49)
$\displaystyle\phantom{\frac{\widetilde{J}_{K}}{\sqrt{3N}}\sum_{{\bf k},{\bf
q}}\Big{[}f^{\dagger}_{{\bf k}-{\bf
q}\uparrow}}\times\sum_{\gamma}M^{zz}_{\gamma,{\bf
q}}\Big{(}A^{{\dagger}}_{\gamma,{\bf q}}+A^{\phantom{{\dagger}}}_{\gamma,-{\bf
q}}\Big{)},$
with the matrix elements
$\displaystyle M^{+-}_{\gamma,{\bf q}}=$
$\displaystyle\,2\sum_{\alpha}\Big{(}\cos^{2}\left(\widetilde{\alpha}/2\right)U_{\alpha,-{\bf
q}}^{({\gamma})}-\sin^{2}\left(\widetilde{\alpha}/2\right)V_{\alpha,-{\bf
q}}^{({\gamma})}\Big{)},$ $\displaystyle N^{+-}_{\gamma,{\bf q}}=$
$\displaystyle\,2\sum_{\alpha}\Big{(}\cos^{2}\left(\widetilde{\alpha}/2\right)V_{\alpha,-{\bf
q}}^{({\gamma})}-\sin^{2}\left(\widetilde{\alpha}/2\right)U_{\alpha,-{\bf
q}}^{({\gamma})}\Big{)},$ $\displaystyle M^{zz}_{\gamma,{\bf q}}=$
$\displaystyle\,\sum_{\alpha}\sin\widetilde{\alpha}\,\Big{(}U_{\alpha,-{\bf
q}}^{({\gamma})}+V_{\alpha,-{\bf q}}^{({\gamma})}\Big{)}.$ (50)
One can see that while the structure of the non-spin-flip term in (49) is
similar to that of the electron-phonon scattering, the spin-flip part is
different as the amplitudes of magnon emission and absorption by the same
$\uparrow(\downarrow)$ electron are, generally, different. This is, of course,
most obvious in the polarized FM state, in which magnons do have a definite
spin, and, therefore, can be emitted only by electrons with the spin
$\downarrow$ and absorbed only by electrons with the spin $\uparrow$.
With the electron-magnon couplings explicated in Eqs. (49) and (50), one has a
clear path toward a calculation of the electron’s relaxation rate and,
therefore, resistivity as a function of the field and temperature.
The derivation of the electron-magnon interaction above and the calculation of
the relaxation rate that follows can be repeated for individual particular
cases of the 120$\degree$, FM, and UUD phases with an alternative spin-wave
formulation considered in Appendix B. Each of these considerations follows the
same structure with a varying degree of simplification compared to the general
case described above. While we do not expose these alternative solutions here
as they lead to identical outcomes, they do offer an important verification
and an analytical insight into the makeup of our solution.
Figure 7: Combinations $\Phi_{\gamma,{\bf k}}$ (53) of the matrix elements in
(50) for several representative field values along a representative cut
K$\Gamma$MK through the full BZ, see Fig. 5: Y phase, $h\\!=\\!0.3$, UUD
phase, $h\\!=\\!0.8$, and V phase, $h\\!=\\!2.0$.
We have repeatedly emphasized the importance of the field-induced changes in
magnon energies and in electron-magnon matrix elements for our key results
that follow next. With the representative magnon energies shown in Fig. 6, we
complement them with the similar representative plots of the combinations of
matrix elements given in (53), which enter the integral expression for the
resistivity, see Eq. (52) below. In Fig. 7 we show combinations
$\Phi_{\gamma,{\bf k}}$ from (53) for the three representative field values
from different phases, Y phase at $h\\!=\\!0.3$, UUD phase at $h\\!=\\!0.8$,
and V phase at $h\\!=\\!2.0$. The plot is along a representative cut
K$\Gamma$MK through the full BZ as in Fig. 6 and for the same parameter
choices in model (1) as above, $j_{2}\\!=\\!J_{2}/J_{1}\\!=\\!-0.8$ and
$b\\!=\\!BS^{2}/J_{1}\\!=\\!0.1$.
Since the numeration of magnon modes in Fig. 6(b)-(d) is from the lowest to
highest in energy, the solutions for the matrix elements in Fig. 7 switch
between branches whenever branches cross. Some of the crossings are at the
high symmetry points and some are not. A general trend that can be observed in
Fig. 7 is that some of the matrix elements are strongly suppressed around the
$\Gamma$ point and are either maximal or singular at the $K$ point.
An interesting feature of the matrix elements in the UUD phase can be noted.
There is no dependence of $\Phi_{\gamma,{\bf k}}$ on the field, only switching
between branches according to their numeration. That is, while there is a
definite reshuffling of the magnon modes vs field that can be seen in Fig.
6(c), the same combination of matrix elements as depicted in the middle panel
of Fig. 7 corresponds to any other point on the magnetization plateau (UUD)
phase. This observation provides an interesting connection between the
structure of the quasiparticle states, encoded in their wave-functions, and
conserved magnetization.
### IV.2 Resistivity
Similarly to the theory of electron-phonon scattering in the resistivity of
metals, Fermi energy $E_{F}$ is by far the most dominant energy scale of the
problem, perhaps even more so in our case, as the magnon bandwidth, field
strengths, and temperature range of interest are all $\lesssim\\!50$ K while
$E_{F}\\!\sim\\!1$–3 eV [30]. Because of that, magnon-induced scattering of
electrons is happening within a thin energy shell around the Fermi surface.
For the effectively 2D magnetic excitations, transport scattering rate can be
shown to reduce to a 1D integral that is limited by that shell.
With the technical details of the Boltzmann equation approach to the electron-
magnon scatterings in Eq. (49) delegated to Appendix C and a mild assumption
of the circular 2D Fermi surface, we obtain the transport relaxation rate for
electrons with both spin projections
$\displaystyle\frac{\hbar}{\tau_{F}}=\frac{\sqrt{3}}{\pi}\,\frac{|\widetilde{J}_{K}|^{2}}{E_{F}}\,(k_{F}a)^{2}\,I_{k_{F}}(T,H)\,,$
(51)
where $\widetilde{J}_{K}\\!=\\!\frac{1}{2}J_{K}\sqrt{S/2}$,
$E_{F}\\!=\\!\hbar^{2}k_{F}^{2}/2m$ with $m$ being effective electron mass,
$k_{F}$ is a Fermi momentum, and the 1D integral given by
$\displaystyle I_{k_{F}}(T,H)=$
$\displaystyle\,\int_{0}^{1}\frac{z^{2}\,dz}{\sqrt{1-z^{2}}}$ (52)
$\displaystyle\,\phantom{\int_{0}^{1}}\times\frac{1}{3}\sum_{\gamma}\Phi_{\gamma,{\bf
q}}\,{\rm n}^{0}_{\gamma,{\bf q}}\big{(}{\rm n}^{0}_{\gamma,{\bf
q}}+1\big{)}\,\frac{\omega_{\gamma,{\bf q}}}{T}\,,$
with the 2D momentum parametrization along the 1D contour ${\bf
q}\\!=\\!2k_{F}(z^{2},z\sqrt{1-z^{2}})$, Bose distribution function ${\rm
n}^{0}_{\gamma,{\bf q}}$ for a magnon with the energy $\omega_{\gamma,{\bf
q}}$, $\gamma$ numerating magnon branches, and $\Phi_{\gamma,{\bf q}}$
abbreviating matrix element contribution
$\displaystyle\Phi_{\gamma,{\bf q}}=2\big{|}M^{zz}_{\gamma,{\bf
q}}\big{|}^{2}+\big{|}M^{+-}_{\gamma,{\bf
q}}\big{|}^{2}+\big{|}N^{+-}_{\gamma,{\bf q}}\big{|}^{2}.$ (53)
This result in (51)-(53) combines the effort of the entire work in a concise
form. It accumulates the solution of the transport theory that proves the
validity of the $1/\tau$-approximation in our case, implicitly contains the
field-dependence of the magnon spectra and matrix elements (50) via the spin
angles $\widetilde{\alpha}$ and parameters of the generalized Bogolyubov
transformation (36), incorporates field-induced transitions between different
phases, and includes the effect of thermal population of magnetic scatterers
on the resistivity. As is discussed in the Section V, contributions of the
thermal distribution of magnons and matrix element component (53) are both
essential for the resistivity results. While the general expressions (51) and
(52) may not be too intuitive, the following consideration will provide
essential ingredients for such an intuition.
Figure 8: The kernel ${\cal K}_{2k_{F}}$, Eq. (54), vs $H$ for
$k_{F}\\!=\\!\pi/4$ and for (a) $b\\!=\\!0$, and (b) $b\\!=\\!1/11$.
Contributions of the spin-flip and non-spin-flip channels are represented by
shadings, Y, UUD, V, and FM phases and transitions between them are indicated.
#### IV.2.1 Large-${\bf q}$ insights
The key elements of the physics packed in Eq. (51) can be extracted from the
kernel of the integral in (52). We begin by noting that the $z^{2}$ factor in
(52) originates from a suppression of the small-angle scattering processes of
electrons in the transport relaxation rate. Thus, the integral is dominated by
the large-${\bf q}$ scattering events that correspond to $z\\!\rightarrow\\!1$
and $q\\!\rightarrow\\!2k_{F}$. The $1/\sqrt{1-z^{2}}$ factor is due to
angular integration in 2D and also contributes to an enhancement of the
large-${\bf q}$ contributions.
Further intuition, which also lays out expectations for the results presented
in the next Section, is provided by the remainder of the kernel in the second
line of (52) taken at a “typical” momentum ${\bf q}^{*}\\!=\\!(2k_{F},0)$ and
in the high-temperature limit, approximating Bose-factors as ${\rm
n}^{0}_{\gamma,{\bf q}}\\!\approx\\!T/{\omega_{\gamma,{\bf q}}}$ and omitting
an overall prefactor $T/3$,
${\cal K}_{2k_{F}}=\sum_{\gamma}\frac{\Phi_{\gamma,{\bf
q}^{*}}}{\omega_{\gamma,{\bf q}^{*}}}.$ (54)
Referred to as “the kernel” below, ${\cal K}_{2k_{F}}$ is a sum over the
branch index $\gamma$ of the ratios of the matrix elements (53) and magnon
energies, taken at ${\bf q}^{*}$. We also note that the high-temperature
approximation is closely relevant to EuC6 phenomenology discussed in Sec. V.
The kernel ${\cal K}_{2k_{F}}$ allows one to analyze contributions of
different magnon modes to the resistivity and also to compare the relative
importance of the spin-flip and non-spin-flip channels in the scattering.
While the latter is absent in the collinear UUD and FM phases, it is present
in the noncollinear Y and V phases, where $\sin\widetilde{\alpha}\\!\neq\\!0$,
see (50). This partitioning of (54) into the channels is done by separating
the non-spin-flip matrix element contribution $|M^{zz}_{\gamma,{\bf
q}^{*}}|^{2}$ in $\Phi_{\gamma,{\bf q}^{*}}$ from the rest, see (53).
Figure 8 shows ${\cal K}_{2k_{F}}$ (54) vs magnetic field for a representative
$k_{F}\\!=\\!\pi/4$ and for two values of the biquadratic parameter $b$ in
model (1): (a) Heisenberg, $b\\!=\\!0$, and (b) $b\\!=\\!b_{c}\\!=\\!1/11$.
The rest of the parameters are from Table 1, see Sec. II.2.3. In the
Heisenberg limit, the UUD phase reduces to a critical point separating Y and V
phases. Contributions of the spin-flip and non-spin-flip scattering channels
are shaded by different colors.
The key lesson of Figure 8 is that the non-spin-flip scattering channel,
although secondary to the spin-flip one, is responsible for an enhancement of
${\cal K}_{2k_{F}}$ in the noncollinear Y and V phases relative to the
collinear UUD and FM phases where it is not available. Accordingly, one should
expect the higher transport relaxation rates (51) and higher resistivity in
these noncollinear phases. Fig. 8 also shows that the biquadratic interaction
enhances non-spin-flip scattering and causes stronger variations of the kernel
near Y-UUD and V-FM transitions. One can anticipate all these trends to
persist in the results for the resistivity discussed in Sec. V.
#### IV.2.2 Estimate of $J_{K}$
Using considerations of the electronic structure of EuC6 provided in Sec. II.1
and assuming two doubly-degenerate bands with cylindrical Fermi surfaces to
describe it, the 3D electronic concentration $n$ is related to the value of
the Fermi momentum $k_{F}$ via
$n=\frac{k_{F}^{2}}{\pi c}\,,$ (55)
where $c$ is the interplane distance between Eu layers. With that and some
rearranging, the expression for the resistivity can be cast in the following
form
$\rho=\frac{m}{ne^{2}\tau_{F}}=\frac{c}{4}\,R_{K}\cdot\frac{\hbar}{E_{F}\tau_{F}},$
(56)
in which the von Klitzing constant $R_{K}\\!=\\!h/e^{2}\\!\approx\\!25.8$
k$\Omega$ and interplane distance $c$ set the proper units and the relaxation
rate of (51) is made dimensionless by a normalization to the Fermi energy.
One can use the expression for $\rho$ in Eq. (56) with $\hbar/\tau_{F}$ from
(51) to estimate the Kondo coupling constant $J_{K}$ in (45) that is needed to
reproduce experimental values of $\rho$ in EuC6. By taking $\rho_{H=0}(24{\rm
K})\\!\approx\\!12.5$ $\mu\Omega$cm from Fig. 1(b) and $c\\!=\\!4.87$ Å[21],
one obtains $\hbar/E_{F}\tau_{F}\\!\approx\\!0.091$, which, if matched to the
theory results for $k_{F}a\\!=\\!\pi/3$ in Fig. 1(c), yields
$J_{K}/E_{F}\\!\approx\\!0.275$. By scaling of the value of the Fermi energy
in Fig. 2(a) to what it would be for $k_{F}a\\!=\\!\pi/3$, one has
$E_{F}\\!\approx\\!1.17$ eV and $J_{K}\\!\approx\\!0.32$ eV. This estimate is
of the same order, albeit somewhat larger, than the value 0.15 eV quoted in
the early literature [24]. However, it seems that most of the discrepancy
could be in the factor of two difference in the definition of electronic spin
in the Kondo coupling (45), making the remaining difference rather academic.
## V Results
With all the elements of our approach and qualitative and quantitative
considerations and estimates provided above, we can now offer a detailed
overview of the results that follow from our theory. A comparison of the
experimental data for the magnetoresistivity in EuC6 vs field with our
calculations for the model parameters from Table 1 and for a representative
value of $k_{F}\\!=\\!\pi/3$ is given in Figs. 1(b) and (c). Given the
simplicity of our model and potential additional unaccounted effects discussed
in more detail in Sec. V.4, the similarity between experiment and theory is
rather astounding.
This similarity includes high resistivity in the Y phase and its quick roll-
down near the Y-UUD transition, a gentle downward slope of $\rho$ vs $H$ in
the UUD phase, followed by a smooth rise in the V phase. The temperature
evolution of the $\rho(H)$ curves is also consistent with the data, perhaps
with an exception of the lowest temperatures. A discrepancy can also be seen
in the larger values of $\rho$ in the FM phase and a strong rise toward it
near the V-FM transition in the theory results. This is likely due to a
proximity to the $H$–$T$ phase transition boundary, where interactions between
magnetic excitations, neglected in our consideration, become important.
This successful comparison strongly suggests the correctness of the advocated
mechanism of the magnetoresistance in magnetically intercalated graphite as
dominated by electron scattering on magnetic excitations, which, in turn,
allows insights into the nature of such excitations.
In the following, we present further evidence of the success of our theory
together with a detailed analysis of the dependence of our results on the key
model parameters, such as biquadratic interaction of spins $b$ and electron
Fermi momentum $k_{F}$, summarized in Figures 9–11. This analysis provides
implications for the microscopic parameters that should describe EuC6 and also
offers a glimpse of the prospective new phenomena that can be induced in
intercalated magnetic materials and similar systems by means of the chemical-,
pressure-, or gate-doping.
Figure 9: The $T$-dependence of $\rho_{\rm mag}$ due to magnetic scattering
(51) on the (a) linear and (b) log-log scale in the 120$\degree$ state,
$H\\!=\\!0$, and in the middle of the UUD phase, $H\\!=\\!(H_{c2}+H_{c1})/2$,
for the parameters from Table 1 and $k_{F}\\!=\\!\pi/3$. Dashed lines are
Bloch-Grüneisen’s, $\sim\\!T^{4}$ in 2D, Ohm’s, $\sim\\!T$, and exponential
asymptotics, see text.
### V.1 $T$-dependence of resistivity
We complement our results for the field-dependence of $1/\tau_{F}$ in Fig.
1(c) by the temperature-dependence of $\rho(H,T)$ at fixed $H$. Our Figure
9(a) shows the results for two field values: $H\\!=\\!0$, 120$\degree$ spin
state, and for the middle of the UUD phase, $H\\!=\\!(H_{c2}+H_{c1})/2$. The
results are for the same optimal choice of parameters to describe EuC6 from
Table 1 as in Fig. 1(c), and for $k_{F}\\!=\\!\pi/3$.
Our Fig. 9(b) shows the same data on the log-log scale in order to emphasize
two distinct temperature regimes, the “low-$T$” and the “high-$T$.” The
overall energy scale for scattering is set by the magnon bandwidth, which
plays the role analogous to that of the Debye energy in the electron-phonon
resistivity [51]. Drawing from this analogy, a transition between the low- and
high-$T$ regimes can be expected at a fraction of the magnetic Debye energy
[51, 52, 53], which can be estimated from the magnon spectra in Fig. 6 as
$W_{\rm m}\\!\approx\\!10J_{1}S$ with some variation between phases. Using
$J_{1}\\!\approx\\!1.1$ K, see Table 1, and $S\\!=\\!7/2$ yields $W_{\rm
m}\\!\approx\\!40$ K. Indeed, the transition between the two regimes can be
observed in Fig. 9 at $W_{\rm m}/5\\!\approx\\!8$ K.
This consideration implies that the majority of experimental results on EuC6
in Refs. [21, 22, 23], in our Fig. 1(b), which is reproduced from Ref. [18],
and in all our theoretical plots are in that “high-$T$” regime,
$T\\!\gtrsim\\!8$ K. The nature of this regime, where resistivity crosses over
to a linear-$T$ dependence as is indicated by the asymptotes in Fig. 9, is
simply an equivalent of the Ohm’s law. Approximating Bose-factors in (52) by
their high-temperature limit, ${\rm n}^{0}_{\gamma,{\bf
q}}\\!\approx\\!T/{\omega_{\gamma,{\bf q}}}$, naturally yields
$1/\tau_{F}\\!\propto\\!T$. Parenthetically, this also motivates our high-$T$
approximation used in the consideration of the kernel in Sec. IV.2.1.
Figure 10: $1/\tau_{F}$ vs $H$ from (51) for representative
$k_{F}\\!=\\!\pi/3$ and temperatures, exchange parameters are from Table 1.
(a) Results for the biquadratic exchange $b\\!=\\!0$, 0.03, 0.06, and 0.0922
are displaced down for clarity by the increments of 0.05 in the given units.
Circles/squares mark transitions between magnetic phases. (b) $b\\!=\\!0.13$,
$j_{2}\\!=\\!0.08$, dotted lines indicate discontinuities.
Thus, the nearly-linear $T$-dependence of $\rho(T)$ in EuC6 observed in Ref.
[24] above 8 K is simply within the onset of the Ohm’s regime, with no need
for an artificial backflow scenario proposed in that work. Needless to say,
details of the spectra do not matter at high temperatures and the same
$\rho(T)\\!\propto\\!T$ dependence should hold for all field-induced phases,
as is shown by a comparison of the UUD and 120$\degree$ states in Fig. 9.
Naturally, in very high fields, the field-induced gaps
$g\mu_{B}(H\\!-\\!H_{s})\\!\gg\\!k_{B}T$ will lead to a freeze-out of the
magnon scattering.
The low-$T$ regime is a bit more subtle and depends on the magnetic phase. The
case of the $120^{\circ}$ state and, by proxy, of the Y and V phases with the
Goldstone modes that are linear at low energies, $\omega_{\bf
q}\\!\propto\\!q$, see Fig. 6(a), (b) and (d), is very much similar to the
textbook case of acoustic phonons. For the $120^{\circ}$ state, one can show
from Appendix B.2 that the matrix element contribution (53) associated with
the coupling to such a mode is also linear in $q$ in that limit,
$\Phi_{\gamma,{\bf q}}\\!\propto\\!q$. Then, a simple power-counting in (52)
for $T\\!\gtrsim\\!\omega_{\bf q}$ using $z\\!\propto\\!q\\!\propto\\!T$,
yields a 2D analogue of the Bloch-Grüneisen asymptotic regime
$1/\tau_{F}\\!\propto\\!T^{4}$ shown in Fig. 9.
As opposed to the gapless phases, the UUD and FM phases are gapped away from
the transition points, see Fig. 6(c) and (e). Thus, one can expect to see an
activated behavior of the resistivity, $\rho\\!\propto\\!e^{-\Delta/T}$, at
sufficiently low temperatures. While this regime can be visible in Fig. 9(b),
in practice its detection requires reaching temperatures $T\ll\Delta$. There
is also an additional smallness due to a $\propto\\!T^{7/2}$ prefactor of the
exponent associated with a suppressed coupling to the lowest mode,
$\Phi_{\gamma,{\bf q}}\\!\propto\\!q^{4}$. An estimate for the gap in the
middle of the UUD phase for EuC6 gives
$\Delta\\!\approx\\!1.1SJ_{1}\\!\approx\\!4.2$ K, providing a guidance for the
future observations.
The locus of magnon momenta ${\bf q}$ that are involved in the scattering
depends on the value of $k_{F}$ as we discuss below. However, in the field-
polarized FM case, it is, generally, away from the energy minimum in Fig.
6(e), leading to a larger gap in the exponent, which is further increased by
the Zeeman energy $g\mu_{B}(H-H_{s})$ away from the saturation field,
accompanied by a more favorable prefactor $\propto\\!T^{1/2}$, so the
freezing-out should be readily observable in the FM phase at higher
temperatures.
Lastly, one can naively expect a power-law that is different from
$\propto\\!T^{4}$ near the Y-UUD and UUD-V transition points as both are
affiliated with the BEC-like transitions, in which magnon energy is quadratic,
$\omega_{\bf q}\\!\propto\\!q^{2}$, see Fig. 6(c). Although we refrain from
discussing it in any significant detail, the situation is more complicated as
the coupling to these BEC modes is different at $H_{c1}$ and $H_{c2}$. In the
first case the coupling vanishes, maintaining an exponential trend due to
higher energy modes, while in the second case it indeed leads to a different
power-law $\propto\\!T^{7/2}$ due to a suppressed coupling, $\Phi_{\gamma,{\bf
q}}\\!\propto\\!q^{4}$.
Altogether, the consideration given above presents further evidence of the
validity of our theoretical approach, providing a physically transparent
description of the temperature-dependence of the resistivity of EuC6 in the
previously accessed temperature regime. It also invites further such studies
in finite fields and especially at lower temperatures, where resistivity
should be a sensitive probe of the spin excitation spectra.
### V.2 Magnetoresistivity vs biquadratic-exchange
The discussion provided below serves two goals. First is to investigate how
prevalent are the strong anomalies in the magnetoresistivity, $\rho$ vs $H$,
in the model of the conduction electrons coupled via a coupling (45) to the
spins that are described by the Heisenberg-biquadratic model (1). Second is to
demonstrate that the magnetoresistivity of EuC6 is consistent with the
substantial biquadratic-exchange parameter $b$ in such a model.
In Figure 10, we present the transport relaxation rate vs field obtained from
(51) for several representative temperatures, Fermi momentum
$k_{F}\\!=\\!\pi/3$, and exchange parameters from Table 1, except that now we
vary the key biquadratic-exchange parameter $b\\!=\\!BS^{2}/J_{1}$. The
corresponding magnetoresistivity is related to these results by a dimensional
constant factor, see Eq. (56).
Figure 10(a) shows two sequences of curves, offset for clarity, with the
biqudratic exchange increasing in nearly equal steps from the Heisenberg
limit, $b\\!=\\!0$, to the value $b\\!=\\!0.0922$ that we use as an optimal
choice for EuC6, see Sec. II.2.3. Figure 10(b) shows results for the
biquadratic exchange $b\\!=\\!0.13$ that substantially exceeds the “critical”
value $b_{c}\\!=\\!1/11$, which corresponds to a change of the Y-UUD and V-FM
transitions to the first-order type as discussed in detail in Sec. II.2.2 and
Appendix A.
The evolution of $1/\tau_{F}$ with $b$ in Fig. 10(a) features already
anticipated trends. First, the opening of the $1/3$-magnetization plateau
(UUD) phase away from the Heisenberg limit, see Sec. II.2, is clearly visible.
Second, the results in Fig. 10(a) are in a close accord with with the behavior
of the kernel, discussed in Sec. IV.2.1 and illustrated in Fig. 8, providing
an explicit confirmation that the transport relaxation rate and
magnetoresistivity are dominated by the $2k_{F}$ scattering processes.
The key observation from the results in Fig. 10(a) is that strong roller-
coaster like variations in magnetoresistivity, such as the ones observed in
EuC6, must be associated with the nearly critical values of the biquadratic
exchange within our model. Although some aspects resembling strong variations
and indicating clear differences of $\rho$ vs $H$ dependence between different
phases can already be observed in the pure Heisenberg model, see, for example,
a kink-like feature at the Y-V boundary in the upper curves in Fig. 10(a),
others are much less pronounced, see a rather small change of slope at the the
V-FM transition at $H_{s}$ in the same results.
Our Fig. 10(a) demonstrates that the role of the biquadratic term in model (1)
goes far beyond just establishing the UUD phase boundaries, which are clearly
marked by the kinks in $1/\tau_{F}$. With increasing $b$, the Y-UUD transition
becomes steeper upon shifting to the lower fields, showing a divergent
derivative for $b\\!=\\!0.0922$ that is related to a similar behavior of spin
angles in Fig. 12. Still, the biggest change takes place at the V-FM
transition, which too becomes weakly first-order, as is elaborated on in
Appendix A. Here, the $1/\tau_{F}$ field-dependence evolves from a nearly
featureless one in the Heisenberg limit to a “shock-wave”-like shape for
$b\\!=\\!0.0922$. In contrast, the UUD-V transition remains continuous
throughout these transformations, although the slope of $1/\tau_{F}$ at
$H_{c2}$ also changes visibly.
Increasing $b$ beyond the critical $b_{c}$ should lead to a hysteresis in the
magnetoresistance. Figure 10(b) illustrates the case of
$b\\!=\\!0.13\\!>\\!b_{c}$. These results are obtained by using the local
stability of the solutions for the magnetic configurations, of their
corresponding spin-wave energies, and of electron-magnon matrix elements
within the overlap regions of the coexisting phases. For example, from within
the Y phase, the Y magnetic configuration persist up to $\widetilde{h}_{c1}$,
as is described in Appendix A.1. From within the UUD phase, the same field
region can be accessed starting from $h_{c1}\\!<\\!\widetilde{h}_{c1}$.
Therefore, in the overlap region, $h_{c1}\\!<\\!h\\!<\\!\widetilde{h}_{c1}$,
the relaxation rate $1/\tau_{F}$ can be calculated in two different ways,
resulting in the sizable discontinuities in Fig. 10(b), indicated by vertical
dotted lines marking the overlap intervals of
$h_{c1}\\!<\\!h\\!<\\!\widetilde{h}_{c1}$ for the Y-UUD and
$h_{s}\\!<\\!h\\!<\\!\widetilde{h}_{s}$ for the V-FM transition.
We note that the transition regions and discontinuities in Fig. 10(b) are only
illustrative. As is discussed in Sec. A.1, the transition between the two
overlapping phases should take place at $h_{c}^{*}$, at which the energies of
the two phases become equal. At a finite temperature, a proper consideration
of the first-order transition should include entropic contribution to the free
energy of the competing phases. In addition, one can expect the co-existence
region to be affected by secondary anisotropies that are neglected in our
minimal model. Nonetheless, we believe that Fig. 10(b) faithfully represents a
qualitative effect of a strong biquadratic interaction on the
magnetoresistance across the first-order transition.
Altogether, the results presented in this section provide an important
overview of the characteristic evolution of the magnetotransport within the
model of electrons coupled to the spin subsystem, which is described by the
Heisenberg-biquadratic model.
As is discussed in Sec. II.2.3, the microscopic parameters of the spin model
(1) describing EuC6 are determined entirely from the thermodynamic quantities,
such as critical fields and transition temperature. Therefore, for claiming a
success of a theoretical description it is crucial that the resulting set of
microscopic parameters yields distinctive features that are in accord with a
wider phenomenology of the material, especially the one that involves less
trivial quantities such as dynamical response and transport. We can claim such
a success here, as the parameters chosen to describe EuC6 in Table 1 are also
the ones that produce sharp, nearly singular features in magnetoresistivity
results that follow from our theory and also match closely the observed ones.
Figure 11: $1/\tau_{F}$ from (51) vs $H$ for (a) $k_{F}\\!=\\!\pi/4$, (b)
$k_{F}\\!=\\!0.4\pi$, (c) $k_{F}\\!=\\!0.5\pi$, and (d) $k_{F}\\!=\\!0.6\pi$.
Parameters are from Table 1.
### V.3 Magnetoresistivity, role of $k_{F}$
Two more aspects of our study merit further discussion. First, as is mentioned
in Sec. II, the electronic band filling fraction in EuC6 and the Fermi
momentum $k_{F}$ parametrizing it are not well-determined. While the nominal
Eu2+ valence naively implies a large Fermi surface, the electronic structure
and angle-resolved photoemission study [30] suggested a substantially smaller
electron fraction in the relevant carbon orbitals and a smaller $k_{F}$. We
would like to weigh in on this subject, with the magnetoresistivity in our
model arguing for a still somewhat smaller Fermi surface, with
$k_{F}\\!\lesssim\\!\pi/3$.
Second, much of the interest in the synthetic materials in general and in the
graphite-derived systems in particular is due to a significant flexibility
regarding electronic density manipulation. Then, in addition to varying
parameters of the spin model, it is also important to explore the outcomes of
our theory in a wider range of electronic parameters in order to anticipate
potential new effects that can be accessible due to such a flexibility. To
that end, we discuss some of the larger-$k_{F}$ results.
Our Figure 11 shows the constant-$T$ curves of the transport relaxation rate
$1/\tau_{F}$ vs $H$ calculated using (51) as in Fig. 1(c) and Fig. 10 for a
set of representative temperatures from $T\\!=\\!24$ K down to $2$ K in 2 K
steps. Results are for the model parameters from Table 1 which describe EuC6,
and for the four different values of the Fermi momentum, $k_{F}\\!=\\!\pi/4$,
$0.4\pi$, $0.5\pi$, and $0.6\pi$. In this case, the field-independent constant
factor that relates $1/\tau_{F}$ to magnetoresistivity $\rho(H,T)$, Eq. (56),
is different for the four sets as they correspond to different electronic
concentrations $n$ via (55).
Consider $k_{F}\\!=\\!\pi/4$ results in Fig. 11(a) first. All of the features
in the data are the same as in Fig. 1(c) and as discussed in Sec. V.2 for Fig.
10(a), including the steep Y-UUD transition, a “shock-wave” feature at the
V-FM boundary, and a decline in the FM phase due to the Zeeman-induced gap
that is depleting magnon population. In agreement with the analysis of $\rho$
vs $T$ in Sec. V.1, the temperature-induced offset of the curves is nearly
linear in $T$ except for the lowest sets.
On a closer and more quantitative side, one can argue that in terms of the
overall trends in magnetoresistivity curves, the $k_{F}\\!=\\!\pi/3$ results
in Fig. 1(c) provide a somewhat better fit to the EuC6 data in Fig. 1(b) than
the $k_{F}\\!=\\!\pi/4$ ones. Moreover, to match experimental data, the
decrease of $k_{F}$ requires a nearly proportional increase of the Kondo
coupling constant (45) relative to the Fermi energy, $J_{K}/E_{F}$, thus
restricting $k_{F}$ from being too small.
A surprising trend starts being revealed by the results for the larger
$k_{F}\\!=\\!0.4\pi$ in Fig. 11(b). Although the features in the constant-$T$
curves are qualitatively similar to the $k_{F}\\!=\\!\pi/4$ case, changes at
the transitions are less steep and less like the ones in the experimental data
in Fig. 1(b). They are nearly gone for the Y-UUD boundary in the
$k_{F}\\!=\\!0.5\pi$ results in Fig. 11(c) and the V-FM transition for this
$k_{F}$ is also marked by the spike-like structures, certainly unlike anything
observed in EuC6. The $k_{F}\\!=\\!0.6\pi$ results in Fig. 11(d) complete this
unexpected trend, with all the transitions, including the formerly rather
featureless UUD-V one, showing spikes.
These qualitative transformations signify a change in the dominant scattering
that contributes to the resistivity. Regardless of its nature, which we
discuss below, an immediate outcome of this analysis is in the
phenomenological restriction on the size of the Fermi surface in EuC6. As was
described in Sec. II.1, the trigonally warped Fermi surfaces from the band
structure in Ref. [30] have the extent from $k_{F,{\rm
min}}\\!\approx\\!0.45\pi$ to $k_{F,{\rm max}}\\!\approx\\!0.7\pi$, in a
qualitative agreement with a rigid-band estimate assuming circular Fermi
surface and $e/2$ per Eu2+ doping of the carbon bands that gives
$k_{F,e/2}\\!\approx\\!0.43\pi$, see also Fig. 2. However, the
magnetoresistivity of EuC6 within our theory suggests a still smaller Fermi
surface with an optimal $k_{F}$ near $\pi/3$. These results invite more
research into the band structure and direct measurements of the Fermi surface
of EuC6.
To understand the transformation of the relaxation rates with $k_{F}$ in Fig.
11, we need to return to the analysis of $1/\tau_{F}$ in Eq. (51) and in Sec.
IV.2.1. Because of the hierarchy $\omega_{\bf q},T\\!\ll\\!E_{F}$, electrons
participating in a conduction process scatter between momenta that are in a
close vicinity of the Fermi surface. With an assumption of the circular 2D
Fermi surface, the magnon momenta that are involved in such a scattering also
form a circular locus of points in the ${\bf q}$ space, see Fig. 13(b) and
Fig. 16 in Appendix C. These momenta extend from $|{\bf q}|\\!=\\!0$ to the
maximum of $|{\bf q}|\\!=\\!2k_{F}$, with the small-momentum contribution to
the transport scattering rate in (51) suppressed and large-momentum
contribution enhanced, as is discussed in Sec. IV.2.1.
Then it follows for the $k_{F}\\!=\\!\pi/3$ case that the typical large-
momentum “$2k_{F}$”-magnons, responsible for most of the scattering, are from
the set of $|{\bf q}|$ near $2\pi/3$. Referring to the Brillouin zones in Fig.
5, this value corresponds to the proximity of the $\widetilde{M}$ point of the
magnetic Brillouin zone and to the high-energy magnons near the maxima of
$\omega_{\gamma,{\bf q}}$, see Fig. 6.
However, further increase of $k_{F}$ drives the extent of the ${\bf
q}$-contour outside of the first magnetic BZ and also brings the
$2k_{F}$-magnon energy down. Then, the truly “dangerous” value of the Fermi
momentum of the circular Fermi surface is $k_{F}\\!=\\!2\pi/3$, as it allows
magnon momenta to reach the corners of the full Brillouin zone, $K$ and
$K^{\prime}$, which correspond to the ordering vector of all ordered phases,
${\bf Q}\\!=\\!\pm(4\pi/3,0)$, with gapless or nearly gapless modes. Thus, it
is the approach of $k_{F}\\!\to\\!2\pi/3$, or, rather, $2k_{F}\\!\to\\!|{\bf
Q}|$, which is responsible for the dramatic changes in Fig. 11 from (a) to
(d).
This analysis also shows that at a given $T$, the population of the relevant
scatterers for $k_{F}\\!=\\!\pi/3$ is lower than that for the larger $k_{F}$
values, which explains an order-of-magnitude enhancement of $1/\tau_{F}$ from
$k_{F}\\!=\\!\pi/3$ in Fig. 1(c) to $k_{F}\\!=\\!0.6\pi$ in Fig. 11(d) that is
only partially accounted for by the $(k_{F}a)^{2}$ factors in (51).
Since the arguments above relies only on $2k_{F}\\!\rightarrow\\!|{\bf Q}|$,
they suggests a degree of universality. Specifically, we argue that
$1/\tau_{F}$ in all gapless phases should diverge in this limit as
$\propto\\!|2k_{F}-Q|^{-1}$, with a field-dependent prefactor, leading to an
overall increase of the relaxation rates observed in Fig. 11(d). In addition,
the $2k_{F}\\!\to\\!|{\bf Q}|$ behavior of $1/\tau_{F}$ should apply equally
to the pure Heisenberg case, which offers an opportunity for a quantitative
analytical insight. Using expressions for the FM and 120$\degree$ phases from
Appendix B and high-$T$ limit for the Bose factors in (52), neglecting $b$ and
$j_{2}$, and expanding ${\bf q}$ near ${\bf Q}$, after some algebra, indeed
yields $I_{k_{F}}(T,H_{s})\\!\approx\\!(8\pi
T/3J_{1}S)(2k_{F}|2k_{F}-Q|)^{-1}$ for the FM phase at the gapless $H_{s}$
point. The result is the same for the 120$\degree$ phase at $H\\!=\\!0$, but
smaller by a factor 3/4.
Since $H_{s}$ is the transition point that exhibits spike-like features in
Fig. 11, while the 120$\degree$ state is away from the transition, this result
confirms our hypothesis that the entire set of $1/\tau_{F}$ is divergent, or
nearly divergent in the weakly gapped UUD phase, with the spikes being a
quantitative effect that is associated with $\propto\\!q^{2}$ Goldstone modes
at the transitions compared to $\propto\\!q$ modes inside the gapless Y and V
phases. We can also verify that the factor 3/4 between the $H_{s}$ and
$H\\!=\\!0$ (120$\degree$ phase) points is indeed in a reasonable accord with
the results in Fig. 11(d). The $\propto\\!|2k_{F}-Q|^{-1}$ divergence is also
consistent with the difference between $k_{F}\\!=\\!0.5\pi$ and
$k_{F}\\!=\\!0.6\pi$ results at $H_{s}$ in Figs. 11(c) and (d).
This study of the divergence brings in one more important aspect of the
problem that has been neglected so far. We use a fairly reasonable and
certainly simplifying assumption of the cylindrical Fermi surface. However, by
itself this assumption does not automatically make $1/\tau_{F}$ independent of
the direction of the electron momentum ${\bf k}$, with the angular-dependence
originating from the discrete lattice symmetry that is still encoded in the
spin excitations and electron-magnon matrix elements.
As may be clear intuitively, the reason for this issue to be important in the
context of the $\propto\\!|2k_{F}-Q|^{-1}$ divergence is that the “dangerous”
${\bf Q}$-vectors correspond to the discrete points (BZ corners) in the
momentum space, see Fig. 5, leading to the truly divergent $1/\tau_{F}$ only
for these directions. This subject is considered in Appendix C.4.2 for a
closely related quasiparticle relaxation rate $1/\tau_{\rm qp}$, for which the
effect of angular dependence can be taken into account without any additional
approximations.
In Appendix C.4.2, we demonstrate that the effect of the angle-dependence in
$1/\tau_{\rm qp}$ is really negligible up to $k_{F}\\!\approx\\!0.5\pi$ and
even for $k_{F}\\!\gtrsim\\!0.6\pi$ it is still very modest, confirming the
accuracy of our results presented in this work and justifying our initial
approximation that omitted this effect.
Given the limitations of the cylindrical Fermi-surface approximation, which
should become problematic for larger $k_{F}$, and possible effects of the
Fermi-surface reconstruction at the magnetic zone boundaries, it is not
entirely clear whether the true divergences will survive, but they may still
have strong effects even if avoided. This points to an interesting venue of
potential studies of the magnetic scattering effects in the large-Fermi-
surface EuC6, induced by the chemical-, pressure-, or gate-doping.
Some of the considered phenomenology is reminiscent of the “hot spots”
phenomena, much discussed in the theory of cuprates [54], where certain parts
of the Fermi surface are suggested to experience strong scattering due to the
low-energy magnetic excitations with a particular ${\bf Q}$-vector. It is not
unthinkable that the suggested further studies of the large-Fermi-surface EuC6
may also be able to shed a new light on this important problem.
### V.4 Outlook
We would like to reiterate that the present study has provided a thorough
consideration of one of the iconic models in frustrated magnetism, the
triangular-lattice Heisenberg model, enriched by the biquadratic exchange and
coupled to conduction electrons, with the goal of understanding
magnetoresistivity throughout its phase diagram in an external magnetic field.
The use of this model as a microscopic description of EuC6, with additional
approximations for electron bands and parameters estimated from experimental
critical fields and temperature, is clearly a simplification. Yet, the
evidence of the success of such a description is undeniable, with many, if not
most, features of the magnetoresistivity reproduced, also leading to
constraints on the model parameters for both localized spins and electron
densities discussed above.
However, the presented description is not complete. Below, we briefly discuss
other possible terms in the model that might be missing, their expected
effects, possible sources of the remaining discrepancies of our theory with
the experimental data, and desirable future studies.
First additional term in the spin model (1) is the ring-exchange term (2),
inspired by the early works [55, 56], see a discussion of it and its secondary
role for EuC6 in Sec. II.2. According to our estimates in Table 1, the ring-
exchange is about three times smaller than the biquadratic term. Since the
symmetry of the model is unaltered by it and the field-induced spin-angle
dependencies on the parameters that make transitions first-order are very
similar to the biquadratic-only case [21], it was reasonable to neglect it.
The only unexplored outcome of the ring-exchange term is a possible
stabilization of an additional magnetic state between V and FM phases, with
some evidence of such a phase in EuC6 suggested by the data, see Fig. 1(b) and
Ref. [18].
Next is the $XXZ$ anisotropy in the $J_{1}$ and $J_{2}$ terms, which is
necessary to explain very different magnetization behavior for the in- and
out-of-plane field direction [18]. Given close values of the saturation fields
for these directions and nearly isotropic g-factor, it is expected to be
relatively weak, at the level of 10% [21], see also Sec. II.2. However, unlike
the ring-exchange, it lowers the symmetry of the model. Therefore, for the in-
plane field, none of the considered phases will have the true Goldstone modes,
and gapless excitations will exist only at the field-induced transitions. This
is going to alter the low-$T$ behavior of the resistivity discussed in Sec
V.1, and also change the dynamical critical exponent at all critical fields
from the BEC-like ($z\\!=\\!2$, $\omega_{\bf q}\\!\propto\\!q^{2}$) to the
relativistic-like ($z\\!=\\!1$, $\omega_{\bf q}\\!\propto\\!q$). However, as
most of experimentally-relevant theory results is pertinent to the “high-$T$”
regime, see Sec V.1, the effect is expected to be secondary on them. We have
performed a limited study of anisotropic $XXZ$ model for some of the phases,
and found no qualitatively significant differences from the Heisenberg limit
results in that regime even for strong anisotropy.
Since Eu2+ spins are large, the dipole-dipole interactions are not necessarily
negligible. However, using the analysis of Ref. [57] and the size of the unit
cell of EuC6, we estimate this term to be at least an order of magnitude
smaller than the values of the exchanges. Since the dipolar terms also break
spin rotational symmetries, one can expect effects similar to that of the
$XXZ$ anisotropy.
The spin and electronic degrees of freedom are not purely 2D in EuC6, with the
3D interplane spin couplings estimated to be of order $0.1J_{1}$ in Ref. [18].
While nominally essential for the finite Néel temperature, the effect on the
spin excitations can be expected to be minimal. However, even if they are
small, both electron and spin 3D dispersions can be crucial for the softening
of the $2k_{F}$ singularities discussed in Sec. V.3.
Perhaps the most significant difference of the outcome of our theory from the
magnetoresistance data in EuC6 in Figs. 1(b) and (c) is the larger values of
$\rho$ in the FM phase and a strong rise toward it near the V-FM transition.
An obvious reason for the discrepancy is the neglect of the temperature
dependence of the phase transition boundaries in our approach. While it can be
dismissed for the Y-UUD and UUD-V boundaries, for which the results are in a
close accord with the data, transition at the V-FM boundary has a substantial
downward suppression with $T$. One of the possible approaches, which we leave
for future studies, is to include temperature dependence of $H_{s}$ by
accounting for the interactions between magnetic excitations that are ignored
in our consideration.
A less straightforward suggestion is related to the form of the Kondo coupling
(45). We have two Fermi surfaces, originated from the downfolding of two Dirac
graphene bands onto the Brillouin zone of the Eu-lattice. The coupling to
local spins is treated as fully diagonal in the band index in (45). This may,
or may not be the full story, with an intriguing prospect that the spin
arrangement can permit or forbid intrerband scattering. This is again an
interesting subject for a further investigation.
Another notable difference of our results in Fig. 1(c) from the data in Fig.
1(b) is at the lowest temperatures that are accessed experimentally. While
scattering on magnetic excitations dies out in our theory, magnetoresistivity
data retains sizable differences between magnetic phases. One of the scenarios
is due to a feedback of the spin orders on the electronic density of states,
producing a different imprint onto the resistivity in different field-induced
phases. Further studies, with a help of electronic structure calculations, can
be envisioned here.
A different scenario for the same effect involves a compelling picture
associated with the impurity-induced spin-textures [58, 59], which generally
arise in frustrated spin systems due to magnetic couplings that are modified
in the presence of impurities. While the impact of such textures on the
dynamical properties has been recently investigated [60, 61], their effect on
the resistivity in the field-induced phases is simply unknown. However, one
can expect the spin-textures to exist readily in the noncolinear and be
suppressed in the collinear phases [58, 60], suggesting a profile that is
similar to the one observed in the magnetoresistivity. Needless to say, this
is a yet another direction for future investigations.
We conclude this Section by suggesting several extensions of experimental work
in EuC6. As is discussed in Sec. V.1 and above, precise low-temperature
measurements would provide a significant source of information on the field-
induced transitions to and from the UUD phase that would allow to study
intriguing critical behaviors and help determine the effective model more
precisely. As we have expounded on in Sec. V.3, the tunable-$k_{F}$
experiments can allow to study singular behavior in resistivity that may have
significant implications to other systems, and as is briefly mentioned in
Appendix C.4.2, a significant violation of the Wiedemann-Franz law can be
expected at the field-induced transitions in EuC6.
## VI Summary
The main goal of the present study has been to develop a microscopic
theoretical description of the highly dramatic evolution of the resistivity in
EuC6 with the magnetic field. The results and discussions presented in the
prior sections provide a strong affirmation that we have succeeded in that
goal, with our results capturing most of the qualitative and quantitative
features of the experimental data. This success is based on a physical picture
of the scattering of electrons from the graphene-derived bands of the carbon
layers by spin excitations from the triangular-lattice Eu-planes.
In the course of this work, we have provided a thorough theoretical
investigation of the ground states, field-induced phase transitions, spin
excitations, and their couplings to the conduction electrons in the
paradigmatic two-dimensional triangular-lattice antiferromagnet with a
biquadratic exchange, throughout its phase diagram. Our effort highlights the
virtues of the full-scale microscopic approach to the problem, not only to the
spin model, but also to the transport formalism for the spin-flip and non-
spin-flip channels, allowing rigorously obtained numerical results to receive
comprehensive analytical and physical insights and interpretations.
The research advanced in the present study yields predictions of new field-
induced and doping-induced phenomena in magnetically-intercalated graphite and
related systems, also offering an inspiration for bringing together different
approaches in the search of new effects in the graphite-derived artificial
magnetic materials. We anticipate our effort to be relevant to a broader
research in metallic magnets and to provide significant technical guidance for
the similar theoretical studies. Presently, our study invites more research
into EuC6 electronic, thermodynamic, transport, and magnetic properties.
Lastly, our work has advocated resistivity measurements, combined with a
detailed theoretical analysis, as a very informative probe of not only field-
induced phase transitions, but also of the unconventional spin excitations in
magnetic materials. We believe that synthetic 2D materials may become a
significant source of potentially novel insights into the nature of exotic
spin excitations such as, for example, fractionalized spinons in a quantum
spin liquid.
###### Acknowledgements.
We are indebted to Roser Valentí and Vladislav Borisov for a fruitful
discussion regarding electronic structure of intercalated graphite and for
sharing their unpublished data that provided an important first-principles and
moral support to our Fermi-surface consideration. We would like to
wholeheartedly thank KITP for the semi-virtual hospitality during the workshop
of the pandemic-impacted Fall of 2020 when the bulk of this work was
completed. An unexpected and enlightening encounter by one of the authors (A.
L. C.) with Urobatis Halleri during the partial in-person attendance have
undoubtedly impacted the reported results. O. A. S. thanks Hassan Allami and
Dima Pesin for discussions of experiments on EuC6 and initial attempts at the
theoretical formulation of the problem. A. L. C. is grateful to Pavel Maksimov
for a substantial Mathematica help and to Ilya Krivorotov for an illuminating
discussion regarding possible values of the dipole-dipole terms. The work of
A. L. C. was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences under Award No. DE-SC0021221. The work of O. A. S. was
supported by the National Science Foundation CMMT program under Grant No.
DMR-1928919. KITP is supported by the National Science Foundation under Grant
No. NSF PHY-1748958.
## Appendix A First order transitions
Here we describe analysis of the first order Y-UUD and V-FM transitions. We
focus on the Heisenberg-biquadratic model (1) and provide some technical
details on the classical ground states introduced in Sec. II.2.1.
### A.1 Y-UUD transition
Without the ring-exchange term, $k\\!=\\!0$, equation on
$x\\!=\\!\cos\alpha_{1}$ in the Y phase, Eq. (4), reduces to
$x^{3}-\frac{1+b}{4b}x=-\frac{1+h}{8b}.$ (57)
A substitution $x\\!=\\!q\,\sqrt{(1+b)/3b}$ gives
$q^{3}-\frac{3}{4}q=-(1+h)\sqrt{\frac{27b}{(1+b)^{3}}}.$ (58)
Comparison with the trigonometric identity
$\sin^{3}\phi-\frac{3}{4}\sin\phi=-\frac{1}{4}\sin 3\phi,$ (59)
leads to the solution for $x\\!=\\!\cos\alpha_{1}$
$x\\!=\\!\sqrt{\frac{1+b}{3b}}\sin\\!\left(\frac{1}{3}\arcsin\left[(1+h)\sqrt{\frac{27b}{4(1+b)^{3}}}\right]\right).$
(60)
It is now easy to check that the Y-UUD transition for $b\leq b_{c}\\!=\\!1/11$
is continuous and takes place at $h_{c1}\\!=\\!1-6b$, see Eq. (5), at which
$\cos\alpha_{1}\\!=\\!1$. For $b\\!>\\!b_{c}$, the Y phase remains locally
stable up to a critical field
$\widetilde{h}_{c1}\\!=\\!\sqrt{4(1+b)^{3}/27b}-1$, see Eq. (6), which is
found from the condition that the argument of $\arcsin$ in (60) reaches the
maximal value of $1$.
Given that the UUD phase remains locally stable for all $h\\!\geq\\!h_{c1}$,
we observe that field interval $h_{c1}\\!\leq\\!h\\!\leq\\!\widetilde{h}_{c1}$
determines the overlap region of the Y and UUD phases.
Within our approach, however, the actual transition between the two phases
takes place when the classical energies of the two phases become equal. This
defines another field, $h_{c1}^{*}$, which can be found as follows. First,
with the help of (57), the per-site energy of the Y phase
$\widetilde{E}_{Y}\\!=\\!E_{Y}/NS^{2}J_{1}-3j_{2}$ can be simplified to a
quadratic form of $x\\!=\\!\cos\alpha_{1}$
$\widetilde{E}_{Y}=(1+b)x^{2}-1.5(1+h)x+h-1-b,$ (61)
which, upon equating with the energy of the UUD phase $\widetilde{E}_{\rm
UUD}\\!=\\!-1-h-3b$, yields
$x^{*}=\frac{3(1+h)-\sqrt{9(1+h)^{2}-32(1+b)(h+b)}}{4(1+b)}.$ (62)
For a given $b\\!>\\!b_{c}$, the right-hand sides of (62) and (60) determine
the transition field $h_{c1}^{*}$. The solution is easily obtained numerically
using MATHEMATICA.
For our choice of $b\\!=\\!0.0922$, see Table 1, which is only slightly larger
than $b_{c}\\!=\\!1/11$, the resultant critical fields are nearly
indistinguishable from each other:
$\\{h_{c1},h_{c1}^{*},\widetilde{h}_{c1}\\}\\!=\\!\\{0.4468,0.446869,0.446891\\}$.
For the larger values of $b$, the three fields become sufficiently different
and allow one to study resistivity hysteresis. For example, for $b\\!=\\!0.13$
considered in Sec. V.2 we have
$\\{h_{c1},h_{c1}^{*},\widetilde{h}_{c1}\\}\\!=\\!\\{0.22,0.25556,0.282313\\}$.
### A.2 V-FM transition
Figure 12: (a) The cosines and (b) the sines of angles $\alpha_{1}$ and
$\beta$ vs $h$ throughout the Y-UUD-V-FM sequence of the phases in Fig. 4,
$h\\!=\\!g\mu_{B}H/3J_{1}S$. Dashed lines are for the pure Heisenberg model,
$b\\!=\\!0$, solid lines are for $b\\!=\\!BS^{2}/J_{1}\\!=\\!0.0922$ from
Table 1. Small discontinuities due to very weakly first-order transitions can
be seen at the Y-UUD and V-FM transitions, as
$b\\!=\\!0.0922\\!>\\!b_{c}\\!=\\!1/11\\!\approx\\!0.0909$.
In the V phase, denoting $y\\!=\\!\cos\beta$, introducing
$t\\!=\\!y+\sqrt{3+y^{2}}$, and rewriting Eq. (8) as a cubic equation for the
variable $t$ defined in the interval $1\\!\leq\\!t\\!\leq\\!3$ yields
$t^{3}-\frac{2+5b}{b}t=-\frac{2h}{b}.$ (63)
This equation is solved by mapping to the same identity (59) as above, with
the result given by
$t=2\sqrt{\frac{2+5b}{3b}}\sin\left(\frac{1}{3}\arcsin\left[h\sqrt{\frac{27b}{(2+5b)^{3}}}\right]\right),$
(64)
from which the angles are obtained as
$\cos\beta=\frac{t^{2}-3}{2t},\ \ \ \cos\alpha_{1}=\frac{t^{2}+3}{4t}.$ (65)
It is easy to check that the V-FM transition is continuous for $b\leq b_{c}$,
with the same $b_{c}\\!=\\!1/11$ as above, and the critical field of the
transition is given by $h_{s}\\!=\\!3-6b$, see Eq. (9). For $b\\!>\\!b_{c}$, V
phase remains locally stable up to a larger field
$\widetilde{h}_{s}\\!=\\!\sqrt{(2+5b)^{3}/27b}$, which is found from the
argument of $\arcsin$ in (64) being equal to $1$.
Similarly to the case of the discontinuous Y-UUD transition described above,
the actual V-FM transition field $h_{s}^{*}$ is found by equating energies of
the V and FM phases. Some tedious algebra gives the energy of the V phase
$\widetilde{E}_{V}=-\frac{b}{8}t^{4}+\frac{2+5b}{4}t^{2}-ht-\frac{33b}{8}-\frac{3}{2},$
(66)
while $\widetilde{E}_{\rm FM}\\!=\\!3(1-h-b)$. Solving
$\widetilde{E}_{V}\\!=\\!\widetilde{E}_{\rm FM}$ numerically, with $t$ given
by (64), we find that the actual transition field $h_{s}^{*}$ satisfies
$h_{s}\\!<\\!h_{s}^{*}\\!<\\!\widetilde{h}_{s}$.
For $b\\!=\\!0.0922$ used in our work, we find
$\\{h_{s},h_{s}^{*},\widetilde{h}_{s}\\}\\!=\\!\\{2.4468,2.44689,2.44692\\}$,
which are, again, essentially identical. For $b\\!=\\!0.13$ considered in Sec.
V.2, these fields become
$\\{h_{s},h_{s}^{*},\widetilde{h}_{s}\\}\\!=\\!\\{2.22,2.28231,2.30258\\}$.
The evolution of the cosines and sines of spin angles $\alpha_{1}$ and $\beta$
with $h$ throughout the Y-UUD-V-FM sequence of the phases in Fig. 4 is shown
in Fig. 12 for two representative values of $b$.
## Appendix B Particular cases
With the general spin-wave approach for the coplanar three-sublattice states
outlined in Sec. III.1, it is still immensely useful to have a fully
analytical approach developed for some of the states. This is for the sake of
both explicit analytical results and for an independent verification of the
partially numerical approach of Sec. III. For the fully polarized FM and
120${\degree}$ states, a single-sublattice formulation of the SWT is possible.
For the UUD state, the Hamiltonian matrix in (25) can be reduced to
$3\\!\times\\!3$ matrix and solved in a compact form.
### B.1 Polarized state
In the fully polarized FM state, see Fig. 4(a), all angles are the same,
$\widetilde{\alpha}_{\alpha}\\!=\\!0$. In the absence of the easy-plane
anisotropy, the off-diagonal $aa$ ($a^{\dagger}a^{\dagger}$) terms cancel out
and the LSWT Hamiltonians in Eqs. (13)–(20) reduce to a tight-binding form
similar to that of (20). Since there is no distinction between the sublattices
in this case, a Fourier transform of the Holstein-Primakoff bosons
$a^{\phantom{{\dagger}}}_{i}=\frac{1}{\sqrt{N}}\sum_{\bf
q}\,\widetilde{a}^{\phantom{{\dagger}}}_{\bf q}\,e^{-i{\bf q}{\bf r}_{i}},$
(67)
where $N$ is the total number of sites and ${\bf q}$ belongs to the full
Brillouin zone of the triangular lattice, is sufficient to diagonalize the
LSWT model. The magnon energy is
$\omega_{\bf q}=3J_{1}S\Big{(}h-2(1-2b)\big{(}1-\overline{\gamma}_{\bf
q}\big{)}-2j_{2}\big{(}1-\gamma^{(2)}_{\bf q}\big{)}\Big{)}\,,$ (68)
where $h\\!=\\!g\mu_{B}H/3J_{1}S$, $j_{2}\\!=\\!J_{2}/J_{1}$,
$b\\!=\\!BS^{2}/J_{1}$ as before, $\gamma^{(2)}_{\bf q}$ is given in (35), and
$\displaystyle\overline{\gamma}_{\bf q}=\frac{1}{3}\sum_{\alpha}\cos{\bf
q}\bm{\delta}_{\alpha}\,.$ (69)
Here, consideration of the Kondo coupling (45) simplifies substantially as the
single-magnon spin-conserving terms in the electron-magnon interaction in (46)
are not present, laboratory and local spin axes are the same, and all
sublattices are equivalent. Using Fourier transform (67) in (46) with (47) and
$\widetilde{\alpha}_{\alpha}\\!=\\!0$ yields
$\displaystyle{\cal
H}_{int}^{+-}=\frac{2\widetilde{J}_{K}}{\sqrt{N}}\sum_{{\bf k},{\bf
q}}\Big{[}f^{\dagger}_{{\bf k}-{\bf q}\uparrow}f^{\phantom{{\dagger}}}_{{\bf
k}\downarrow}\widetilde{a}^{{\dagger}}_{{\bf q}}+{\rm H.c.}\Big{]},$ (70)
where $\widetilde{J}_{K}\\!=\\!\frac{1}{2}J_{K}\sqrt{S/2}$ as before. The
spin-flip scattering term is simple, with a matrix element containing no
momentum-dependence.
According to Appendix C.3, the spin-flip scattering straightforwardly leads to
the relaxation rate in the form of Eq. (51) with the 1D integral in (52)
taking the form
$\displaystyle I_{k_{F}}(T,H)=$
$\displaystyle\,4\int_{0}^{1}\frac{z^{2}\,dz}{\sqrt{1-z^{2}}}\,{\rm
n}^{0}_{{\bf q}}({\rm n}^{0}_{{\bf q}}+1)\,\frac{\omega_{{\bf q}}}{T}\,,$ (71)
with the same momentum parametrization along the 1D contour ${\bf
q}\\!=\\!2k_{F}(z^{2},z\sqrt{1-z^{2}})$ and Bose distribution function ${\rm
n}^{0}_{{\bf q}}$ with the magnon energy $\omega_{{\bf q}}$ from Eq. (68).
Given the simplicity of the polarized FM state, it may be instructive to
demonstrate a relation of the single-sublattice formalism to the general
three-sublattice one described in Secs. III.3 and III.4, as the latter is
supposed to give an identical description.
For all spins polarized, $\widetilde{\alpha}_{\alpha}\\!=\\!0$, the off-
diagonal term in the Hamiltonian matrix (25) $\hat{\bf
B}^{\phantom{\dagger}}_{\bf q}\\!\equiv\\!0$ and
$\displaystyle\hat{\bf A}^{\phantom{\dagger}}_{\bf q}$ $\displaystyle=C_{\bf
q}\hat{\bf I}+(1-2b)\hat{\bm{\Lambda}}^{\phantom{\dagger}}_{\bf q}\,,$ (72)
$\displaystyle C_{\bf q}=h-2(1-2b)-2j_{2}\big{(}1-\gamma^{(2)}_{\bf
q}\big{)},$
where $\hat{\bf I}$ is a $3\times 3$ identity matrix and
$\displaystyle\hat{\bm{\Lambda}}^{\phantom{\dagger}}_{\bf
q}=\left(\begin{array}[]{ccc}0&\gamma_{\bf q}&\gamma^{*}_{\bf q}\\\
\gamma^{*}_{\bf q}&0&\gamma_{\bf q}\\\ \gamma_{\bf q}&\gamma^{*}_{\bf
q}&0\end{array}\right),$ (76)
with $\gamma_{\bf q}$ from (35). Since
$[\hat{\bm{\Lambda}}^{\phantom{\dagger}}_{\bf q},\hat{\bf I}]\\!=\\!0$, the
three magnon branches have the energies
$\omega_{\gamma\bf q}=3J_{1}S\big{(}C_{\bf q}+(1-2b)\lambda_{\gamma\bf
q}\big{)}\,,$ (77)
where $\lambda_{\gamma\bf q}$ are the eigenvalues of
$\hat{\bm{\Lambda}}^{\phantom{\dagger}}_{\bf q}$. A straightforward algebra
with (76) gives $\lambda_{1\bf q}\\!=\\!2\overline{\gamma}_{\bf q}$ and
$\lambda_{2(3)\bf q}\\!=\\!2\overline{\gamma}_{{\bf q}\pm{\bf Q}}$, with
$\overline{\gamma}_{\bf q}\\!=\\!(\gamma_{\bf q}+\gamma^{*}_{\bf q})/2$ from
(69) and ${\bf Q}\\!=\\!(4\pi/3,0)$. Thus, the three magnon branches are the
“original” single-sublattice result in (68), $\omega_{1\bf
q}\\!=\\!\omega_{\bf q}$, and the other two are “shifted” by the ordering
vectors, $\omega_{2(3)\bf q}\\!=\\!\omega_{{\bf q}\pm{\bf Q}}$.
In the single-sublattice treatment shown above, the one-magnon coupling
involves $\widetilde{a}^{\phantom{{\dagger}}}_{\bf q}$
($\widetilde{a}^{{\dagger}}_{\bf q}$) operators from (67) with a constant
matrix element, see Eq. (70). In the three-sublattice approach, matrix
elements (50) of the general form of electron-magnon coupling in Eq. (49)
require the knowledge of the Hamiltonian eigenfunctions. In the polarized FM
case, anomalous terms in the transformation to quasiparticles (44) are absent,
$\hat{\bf V}_{\bf q}\\!=\\!0$, and all angles are
$\widetilde{\alpha}_{\alpha}\\!=\\!0$, immediately simplifying (50) to just
${\cal H}_{int}^{+-}=\frac{\widetilde{J}_{K}}{\sqrt{3N}}\sum_{{\bf k},{\bf
q},\gamma}\Big{[}M^{+-}_{\gamma,{\bf q}}f^{\dagger}_{{\bf k}-{\bf
q}\uparrow}f^{\phantom{{\dagger}}}_{{\bf
k}\downarrow}a^{{\dagger}}_{\gamma,{\bf q}}\ +{\rm H.c.}\Big{]},$ (78)
with the matrix elements $M^{+-}_{\gamma,{\bf
q}}\\!=\\!2\sum_{\alpha}U_{\alpha,-{\bf q}}^{({\gamma})}$, where the matrix of
vectors $\hat{\bf U}_{\bf q}$ should diagonalize
$\hat{\bm{\Lambda}}^{\phantom{\dagger}}_{\bf q}$ in (76). A simple algebra
yields
$\displaystyle{\bf U}^{(1)}\\!=\\!\frac{1}{\sqrt{3}}(1,1,1)^{T},\ \ {\bf
U}^{(2,3)}\\!=\\!\frac{1}{\sqrt{3}}(1,e^{\pm i\theta},e^{\mp i\theta})^{T},\ \
\ \ \ \ $ (79)
for the “original” $\omega_{1\bf q}$ and “shifted” $\omega_{2(3)\bf q}$,
respectively; here $\theta\\!=\\!4\pi/3$.
Because of the local nature of the Kondo interaction, the matrix element of
the coupling to an eigenmode $\gamma$ in (78) is simply proportional to the
sum of the components of a corresponding vector,
$\sum_{\alpha}U_{\alpha}^{(\gamma)}$. Thus, as it trivially follows from (79),
the resultant matrix elements of the coupling to the “shifted” ${\bf q}\pm{\bf
Q}$ modes are identically zero and only the “original” $\omega_{1\bf q}$-mode
contributes to (78) with $M^{+-}_{\gamma,{\bf q}}\\!=\\!2\sqrt{3}$.
Needless to say, this renders the coupling Hamiltonians in the single-
sublattice and three-sublattice treatment, (70) and (78), equivalent. Their
resultant scattering rate is given in (51) and (71).
A closely associated problem is the relation between the single-sublattice
operators $\widetilde{a}^{\phantom{{\dagger}}}_{\bf q}$
($\widetilde{a}^{{\dagger}}_{\bf q}$) to the three-flavor operators
$a^{\phantom{{\dagger}}}_{\alpha\bf q}$ ($a^{{\dagger}}_{\alpha\bf q}$). A
simple algebra gives
$\displaystyle a^{\phantom{{\dagger}}}_{\bf q}$
|
# Scalable Federated Unlearning via Isolated and Coded Sharding
Anonymous submission Yijing Lin1,2 Zhipeng Gao1∗ Hongyang Du4 Dusit Niyato4
Gui Gui5
Shuguang Cui3,2 Jinke Ren2,3 1 State Key Laboratory of Networking and
Switching Technology, Beijing University of Posts and Telecommunications
2 The Future Network of Intelligence Institute, The Chinese University of Hong
Kong (Shenzhen)
3 School of Science and Engineering, The Chinese University of Hong Kong
(Shenzhen)
4 School of Computer Science and Engineering, Nanyang Technological University
5 School of Automation, Central South University
<EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding authors: Jinke Ren
and Zhipeng Gao.
###### Abstract
Federated unlearning has emerged as a promising paradigm to erase the client-
level data effect without affecting the performance of collaborative learning
models. However, the federated unlearning process often introduces extensive
storage overhead and consumes substantial computational resources, thus
hindering its implementation in practice. To address this issue, this paper
proposes a scalable federated unlearning framework based on isolated sharding
and coded computing. We first divide distributed clients into multiple
isolated shards across stages to reduce the number of clients being affected.
Then, to reduce the storage overhead of the central server, we develop a coded
computing mechanism by compressing the model parameters across different
shards. In addition, we provide the theoretical analysis of time efficiency
and storage effectiveness for the isolated and coded sharding. Finally,
extensive experiments on two typical learning tasks, i.e., classification and
generation, demonstrate that our proposed framework can achieve better
performance than three state-of-the-art frameworks in terms of accuracy,
retraining time, storage overhead, and F1 scores for resisting membership
inference attacks.
## 1 Introduction
With the increasing awareness and stringent regulations of data protection,
there is a growing demand for new machine learning technologies that can reap
the full benefit of rich data while preserving data privacy. Federated
learning (FL) McMahan et al. (2017) has become a promising paradigm to
collaboratively train a learning model across multiple clients without sharing
their local data. As shown in Figure 1(a), each client independently trains a
local model using its local dataset and uploads the model parameters to a
central server. The server collects the model parameters from clients and
broadcasts the aggregated model to all clients. This process is iterated until
model convergence.
Figure 1: Federated Learning vs. Federated Unlearning. (a) In federated
learning, the clients and the server collaboratively train a global model by
exchanging model parameters. (b) In federated unlearning, upon receiving an
unlearning request from a specific client $C^{\prime}$, the well-trained
global model will be unlearned by using the local models of other clients for
calibration, thus removing the corresponding data effect.
Despite the great potential of FL, it encounters challenges in data privacy
since the shared model parameters still contain some data information. On the
other hand, the European Union’s General Data Protection Regulation (GDPR) and
California Consumer Privacy Act (CCPA) in the United States have released
critical requirements for the “right to be forgotten”, which allows clients to
erase the data effects from the model parameters trained on their local
datasets Chen and Yang (2023); Liu et al. (2022); Su and Li (2023), thus
motivates a new computing paradigm called federated unlearning Liu et al.
(2021, 2022); Su and Li (2023). As shown in Figure 1(b), each client performs
local model calibration and sends the calibrated model parameters to the
central server, which removes the data effect from the well-trained FL model
via calibration and aggregation. Since the unlearned model does not need to be
trained from scratch, the computational overhead can be significantly reduced.
Recent studies on federated unlearning have identified two distinct
strategies, including provable and unprovable guarantees. Provable guarantees
Bourtoule et al. (2021) ensure the complete removal of data effects, while
unprovable guarantees indicate that the model has almost forgotten the data
effects of specific clients. To reduce retraining time, a new framework called
FedEraser is proposed in Liu et al. (2021), which removes the data effects of
specific clients by storing intermediate model parameters on the central
server. Although FedEraser achieves provable guarantees and satisfactory
unlearning performance, it is short of scalability due to the extensive
storage overhead. On the other hand, several orthogonal works Liu et al.
(2022); Wu et al. (2022); Wang et al. (2022) employ various unprovable
guarantee-based techniques to enhance the unlearning effectiveness. However,
these works mainly optimize loss functions and prune network layers to bound
the removal level of data effect, which cannot provide stringent provable
guarantees.
To overcome the aforementioned challenges, this paper proposes a scalable
federated unlearning framework to efficiently remove the data effects of
specific clients while maintaining the model accuracy. Specifically, the
entire learning and unlearning process is divided into multiple stages, in
which we construct several isolated shards and perform efficient retraining to
reduce the storage overhead of the central server. Based on this, we carry out
coded computing on different shards to further improve the storage efficiency
and reduce the training time. To demonstrate the effectiveness of the proposed
framework, we conduct extensive experiments on two typical learning tasks,
i.e., classification and generation, and compare our proposed framework with
three state-of-the-art frameworks, i.e., FedEraser Liu et al. (2021),
RapidRetrain Liu et al. (2022), and FedRetrain Wang et al. (2022); Su and Li
(2023). Our main contributions can be summarized as follows:
* •
We for the first time introduce a stage-based isolated sharding mechanism to
reduce the number of affected clients for federated unlearning. Theoretical
analysis proves that the expected time cost for processing unlearning requests
can be significantly reduced in both sequential and concurrent cases.
* •
We design a coded computing-based sharding mechanism to improve the system
scalability. Specifically, it can improve the storage efficiency by
$(1-2\mu)C$ and the throughput by $S/O(C^{2}\log^{2}C\log\log C)$, where $C$
represents the number of clients, $\mu$ is the proportion of erroneous
results, and $S$ is the number of shards.
* •
Experiment results show that the proposed framework can reduce at least 65%
retraining time and 98% storage overhead while achieving comparable unlearning
effectiveness to FedEraser, RapidRetrain, and FedRetrain.
## 2 Related Work
To mitigate the data effects of specific clients, machine unlearning is
proposed by partitioning training data into isolated slices and performing a
joint sharded, isolated, sliced, and aggregated (SISA) method Bourtoule et al.
(2021); Xu et al. (2023). Specifically, upon receiving an unlearning request,
these slices completely remove the data effects and retrain the learning model
to save computational resources and achieve unlearning with provable
guarantees. Besides SISA, some previous works also adopt unprovable guarantee-
based methods, such as gradient ascent Chen and Yang (2023), adding unlearning
layers via a selective teacher-student formulation Jang et al. (2022), and
transforming the unlearning process into a single-class classification task
Yan et al. (2022). Although these methods have achieved good unlearning
performance, they need direct access to client data, thus cannot be applied in
FL due to data privacy.
Recent studies have paid attention to removing the data effects of specific
clients from well-trained FL models. Specifically, the authors in Liu et al.
(2021) initially propose the concept of federated unlearning and develop a new
framework called FedEraser. By using the storage resources of the central
server, it can retain intermediate model parameters to achieve unlearning with
provable guarantees. To reduce computational overhead while maintaining model
accuracy, many unprovable guarantee-based frameworks have also been developed,
such as RapidRetrain Liu et al. (2022), TF-IDF Wang et al. (2022); Salton and
Buckley (1988), FedRecovery Zhang et al. (2023), KNOT Su and Li (2023), and
BFU Wang et al. (2023). Specifically, RapidRetrain modifies the loss function
to accelerate the retraining process and remove the data effects of all
clients. In classification tasks, TF-IDF is utilized to unlearn the
contributions of specific classes by pruning the most relevant class-
discriminative channels Wang et al. (2022). FedRecovery considers differential
privacy in the federated unlearning process, where intermediate model
parameters from clients are utilized to retrain the global model. KNOT
proposes a clustered aggregation-based asynchronous unlearning mechanism to
reduce the retraining cost. BFU develops a parameter self-sharing approach to
maintain model accuracy while erasing the data effects of clients. While these
works have achieved impressive unlearning performance via experiments, they
mainly consider unprovable guarantees such that the unlearning requirements
may not be fulfilled Bourtoule et al. (2021).
## 3 Methodology
In this section, we first describe the federated unlearning framework and then
introduce two mechanisms based on isolated sharding and coded computing.
Figure 2: Scalable Federated Unlearning Framework. For the unlearning requests
initiated at different stages, only affected shards perform calibrations to
remove the data effects of specific clients. To reduce storage overhead and
improve scalability, intermediate model parameters are encoded/decoded as
slices for efficient communication between clients and servers.
### 3.1 Scalable Federated Unlearning Framework
We consider an FL system consisting of $S$ central servers and $C$ distributed
clients, denoted by a set $\mathcal{C}=\\{C_{1},\ldots,C_{C}\\}$. Each client
collects a fraction of data and constitutes its local dataset. A shared
learning model $\mathbf{w}$ needs to be collaboratively trained across all
clients and servers. In the training process, the intermediate model
parameters are stored on the servers for subsequent unlearning purposes. To
remove the data effects of specific clients, we define a subset of clients,
denoted by $\mathcal{C}^{\prime}\subseteq\mathcal{C}$, in which each client
needs to be unlearned based on the unlearning requests. Meanwhile, we define
$\mathcal{D}^{\prime}\subseteq\mathcal{D}$ as the overall dataset associated
with the unlearned clients. Let $\mathbf{w}^{\prime}$ denote the unlearned
global model. Then, according to Kurmanji et al. (2023), the unlearned global
model should satisfy
$\left\\{\begin{split}&I\left(\mathbf{w}(\mathcal{D}^{\prime});\mathbf{w}^{\prime}(\mathcal{D}^{\prime})\right)=0,\\\
&I\left(\mathbf{w}(\mathcal{D}-\mathcal{D}^{\prime});\mathbf{w}^{\prime}(\mathcal{D}-\mathcal{D}^{\prime})\right)=1,\end{split}\right.$
(1)
where $I(\cdot)$ represents the mutual information.
To achieve (1), most existing solutions utilize the intermediate model
parameters stored on the servers to unlearn the well-trained global model,
which inevitably increases the storage overhead of the servers. To solve this
problem, we present a new federated unlearning framework, which considers both
uncoded and coded sharding to meet adaptive and even unlearning requests. As
shown in Figure 2, uncoded sharding utilizes a stage-based isolated sharding
mechanism to erase the data effect in the global model, which will be
illustrated in Section 3.2. Coded sharding compresses intermediate model
parameters into slices, which is able to maintain the storage overhead as the
number of clients increases and will be detailed in Section 3.3.
### 3.2 Stage-based Isolated Sharding Mechanism
Initialization. Since each client may join or leave the system at any time, we
divide the entire learning and unlearning process into multiple stages, where
the learning and unlearning operations are performed within each stage.
Specifically, at each stage, the clients are distributed over multiple shards,
denoted by a set $\mathcal{S}$. In each shard, there is a particular server
for model aggregation. Therefore, the number of shards is equal to the number
of servers, i.e., $S$. The clients in the shard $s$ are defined by a set
$\mathcal{C}_{s}$. In addition, the dataset and the server associated with the
shard $s$ are denoted by $\mathcal{D}_{s}$ and $J_{s}$, respectively.
At each stage, the clients in the same shard perform the FedAvg algorithm
McMahan et al. (2017) to train a global model. Specifically, each client first
trains its local model by $L$ epochs and then transmits the model parameters
to the corresponding server for aggregation. Thereafter, the aggregated model
parameters are broadcast to the clients for local model update. These steps
are iterated for $G$ rounds to obtain a well-trained FL model, which serves as
the foundational model for the subsequent unlearning process. Moreover, the
server stores the intermediate model parameters of clients in the same shard,
which will also be used in the subsequent unlearning process.
Preparation. At each stage, we assume that there are $K$ unlearning requests
across the impacted shards, denoted by a set $\mathcal{S}^{\prime}$. In each
impacted shard $s_{i}\in\mathcal{S}^{\prime}$, the clients and unlearning
clients are denoted by $\mathcal{C}_{s_{i}}$ and
$\mathcal{C}_{s_{i}}^{\prime}$, respectively. Let $J_{s_{i}}$ denote the
server associated with the impacted shard $s_{i}$, which collects intermediate
model parameters $\mathbf{w}_{\mathcal{C}_{s_{i}}}^{g},\forall
g\in\mathcal{G}$ from the clients in the same shard for storage, where
$\mathcal{G}=\\{1,\cdots,G\\}$ is the set of the global learning round. Next,
the server $J_{s_{i}}$ removes the intermediate model parameters associated
with the unlearning clients $\mathcal{C}_{s_{i}}^{\prime}$ from the whole
parameter set, as given by
$\mathbf{w}_{s_{i}}^{g}=\mathbf{w}_{\mathcal{C}_{s_{i}}}^{g}-\mathbf{w}_{\mathcal{C}_{s_{i}}^{\prime}}^{g},\forall
g\in\mathcal{G}$. Then, the server aggregates the model parameters of the
retained clients to obtain the initial global unlearned model, as given by
$\mathbf{w}_{s_{i}}^{g^{\prime}=0}=\frac{1}{M}\sum_{m=1}^{M}\mathbf{w}_{s_{i},m}^{g},$
(2)
where $\mathbf{w}_{s_{i},m}^{g}$ is the local model of the retained client $m$
in the shard $s_{i}$, $M$ is the number of retained clients, $g^{\prime}$
denotes the unlearning round, and $\mathcal{G}^{\prime}=\\{1,\cdots,G\\}$
represents the set of all unlearning rounds. Finally,
$\mathbf{w}_{s_{i}}^{g^{\prime}=0}$ is sent to the retained clients in the
same shard, i.e., $\mathcal{C}_{s_{i}}-\mathcal{C}_{s_{i}}^{\prime}$ for
subsequent retraining.
Retraining. In the unlearning round $g^{\prime}$, instead of retraining from
scratch, each retained client in shard $s_{i}$ receives the global unlearned
model $\mathbf{w}_{s_{i}}^{g^{\prime}}$ and utilizes its local dataset to
update $\mathbf{w}_{s_{i}}^{g^{\prime}}$ by $\dfrac{L}{r}$ local epochs, where
$r$ is a ratio for reducing the number of local retraining rounds Liu et al.
(2021). Then, the updated model parameters are uploaded to the server, which
calibrates and updates the global unlearned model by
$\mathbf{w}_{s_{i}}^{{g^{\prime}}+1}=\mathbf{w}_{s_{i}}^{g^{\prime}}+\frac{1}{M}\sum_{m=1}^{M}\frac{\|\mathbf{w}_{\mathcal{C}_{s_{i},m}}^{g}\|}{\|\mathbf{w}_{\mathcal{C}_{s_{i},m}^{\prime}}^{g^{\prime}}\|}\mathbf{w}_{\mathcal{C}_{s_{i},m}^{\prime}}^{g^{\prime}}.$
(3)
Here, $g=g^{\prime}$ indicates that
$\mathbf{w}_{\mathcal{C}_{s_{i},m}^{\prime}}^{g^{\prime}}$ is used to
calibrate the local model in the same global learning round. Finally,
$\mathbf{w}_{s_{i}}^{g^{\prime}+1}$ is distributed to the retained clients in
the shard. This process will be iterated for $G$ rounds.
According to $\eqref{eq:requirement}$, the provable guarantees of the
unlearning process is achieved if the global unlearned model satisfies
$\left\\{\begin{split}&I\left(\mathbf{w}_{s_{i}}^{g}(\mathcal{D}_{s_{i}}^{\prime});\mathbf{w}_{s_{i}}^{{g}^{\prime}}(\mathcal{D}_{s_{i}}^{\prime})\right)=0,\\\
&I\left(\mathbf{w}_{s_{i}}^{g}(\mathcal{D}_{s_{i}}-\mathcal{D}_{s_{i}}^{\prime});\mathbf{w}_{s_{i}}^{{g}^{\prime}}(\mathcal{D}_{s_{i}}-\mathcal{D}_{s_{i}}^{\prime})\right)=1.\end{split}\right.$
(4)
To achieve this, it is important to maintain the isolation of the shard in the
entire unlearning process, i.e., avoiding cross-shard interactions at each
stage. We shall note that there are scenarios where cross-shard interactions
are necessary at different stages. In such cases, unprovable guarantee-based
methods can be employed. However, it is not the main focus of this paper and
will not be discussed herein.
### 3.3 Coded Computing-based Sharding Mechanism
As stated in Bourtoule et al. (2021), an effective federated unlearning
framework should be lightweight and scalable, which is able to balance the
retraining time, storage overhead, and computational cost for unlearning. To
achieve this goal, we develop a new coded computing-based sharding mechanism,
which is composed of two parts, including coded computation and coded
reconstruction.
Coded Computation. In the isolated sharding mechanism, the server stores the
uncoded intermediate model parameters
$\mathbf{w}_{\mathcal{C}_{s}}^{g},\forall g\in\mathcal{G}$ of the clients in
the same shard, which are used to remove the data effects from the global
model. Conversely, in the coded computing-based sharding mechanism, the server
collects the coded intermediate model parameters, denoted by
$\mathbf{\widetilde{w}}_{i},\forall i\in\\{1,\cdots,C\\}$, which are generated
from $\mathbf{w}_{\mathcal{C}_{s}}^{g}$ and distributed in clients. Without
loss of generality, we utilize the Lagrange interpolation polynomial method
Roth (2006) to compute $\mathbf{\widetilde{w}}_{i}$. Specifically, we first
select $S$ different real numbers $\\{\omega_{1},\ldots,\omega_{S}\\}$, where
$\omega_{s}$ is associated with the shard $s$. Then, the Lagrange polynomial
with variable $\alpha$ is given by
$u(\alpha)=\sum_{s=1}^{S}\mathbf{w}_{\mathcal{C}_{s}}^{g}\prod_{j\neq
s}\frac{\alpha-\omega_{j}}{\omega_{s}-\omega_{j}}.$ (5)
To distribute the intermediate model parameters over all clients, we further
select $C$ different real numbers $\\{\alpha_{1},\ldots,\alpha_{C}\\}$, where
$\alpha_{i}$ is associated with client $i$ and $C$ is the total number of
clients. Based on this, the coded intermediate model parameters stored in
client $i$ can be generated at point $\alpha_{i}$, as
$\displaystyle\mathbf{\widetilde{w}}_{i}$
$\displaystyle=u(\alpha_{i})=\sum_{s=1}^{S}\mathbf{w}_{\mathcal{C}_{s}}^{g}\prod_{j\neq
s}\frac{\alpha_{i}-\omega_{j}}{\omega_{s}-\omega_{j}}.$ (6)
In the coded computing-based sharding mechanism, each client not only stores a
mix of coded intermediate model parameters across different shards but also
generates a series of keys. These keys are shared by the clients and the
server in each shard, which will be used in the following coded
reconstruction. In particular, each client distributes the coded intermediate
model parameters to other clients within or outside the shard.
We shall note that coded computation is utilized in the initialization phase
described in Section 3.2. Instead of directly storing the intermediate model
parameters, the server only collects coded intermediate model parameters from
clients for unlearning. Since the size of the coded model parameters is much
smaller than that of the uncoded ones, the storage overhead of the server can
be significantly reduced.
Coded Reconstruction. In the preparation phase introduced in Section 3.2,
instead of directly utilizing the intermediate model parameters, the server in
each impacted shard utilizes the keys (generated in coded computation) to
retrieve and reconstruct these model parameters from clients. This process is
akin to decoding a Reed-Solomon code Roth (2006). Given the evaluations at $C$
distinct points, the server reconstructs the model parameters by decoding a
Reed-Solomon code with dimension $S$ and length $C$. The detailed decoding
process can be described as follows: Step 1: Let
$\\{\mathbf{\widetilde{w}}_{1},\ldots,\mathbf{\widetilde{w}}_{C}\\}$ denote
the corresponding coded slices of the model parameters among clients, which
can be obtained according to (6); Step 2: The server uses the keys shared by
the clients in the same shard to access these slices; Step 3: The server
reconstructs the original model parameters by solving the following linear
equation derived from the Reed-Solomon decoding process, as
$\mathbf{\widetilde{w}}_{\mathcal{C}_{s}}^{g}=\left[\begin{array}[]{cccc}1&\alpha_{1}&\cdots&\alpha_{1}^{S-1}\\\
1&\alpha_{2}&\cdots&\alpha_{2}^{S-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
1&\alpha_{C}&\cdots&\alpha_{C}^{S-1}\end{array}\right]^{-1}\cdot\left[\begin{array}[]{c}\mathbf{\widetilde{w}}_{1}\\\
\mathbf{\widetilde{w}}_{2}\\\ \vdots\\\
\mathbf{\widetilde{w}}_{C}\end{array}\right],$ (7)
where the matrix formed by $\alpha_{i}$ is a Vandermonde matrix. It is
invertible as long as all elements are distinct. According to (7), the server
can accurately reconstruct the intermediate model parameters from the coded
slices distributed among all clients. Moreover, since each client only
receives a single slice of the coded model parameters, it is difficult for
other clients to reconstruct all coded model parameters, thus preserving the
data privacy.
After finishing the decoding process, the server in each impacted shard
removes the intermediate model parameters associated with the clients
requesting unlearning, initiates the retraining phase, and retrains the global
model. The overall federated unlearning process is summarized in Algorithm 1.
Algorithm 1 Federated Unlearning with Isolated Sharding and Coded Computing
1:for each shard $s\in\mathcal{S}$ do
2: Initialization: // Run on clients
3: Perform coded computation via (6).
4: Distribute $\mathbf{\widetilde{w}}_{i}$ to the clients within or outside
shard.
5: if $s\in\mathcal{S}$ then
6: Preparation: // Run on the server
7: Perform coded reconstruction via (7).
8: Remove the intermediate model parameters of the unlearning clients.
9: Obtain the initial unlearned global model via (2).
10: Retrain: // Run on clients and server
11: Retrain the global model for unlearning via (3).
12: end if
13:end for
## 4 Theoretical Analysis
### 4.1 Time Efficiency for Isolated Sharding
Sequential Unlearning Requests. In the sequential setting, each unlearning
request is addressed individually. Given $S$ shards and $K$ unlearning
requests, the probability that a shard $s$ is selected for retraining $j$
times across $i-1$ unlearning requests is given by
$P_{s}={i-1\choose
j}\left(\frac{1}{S}\right)^{j}\left(1-\frac{1}{S}\right)^{i-1-j},$ (8)
where $\dfrac{1}{S}$ is the probability that affects any given shard. Let
$\overline{C}_{t}$ denote the average time cost associated with all shards.
Then, the expected time cost for processing all unlearning requests can be
estimated by
$\displaystyle T_{s}$
$\displaystyle=\sum_{i=1}^{K}\sum_{j=0}^{i-1}\left(\frac{1}{S}\right)^{j}\left(1-\frac{1}{S}\right)^{i-1-j}\overline{C}_{t}\overset{(a)}{=}K\overline{C}_{t},$
(9)
where $(a)$ is obtained by the binomial theorem Bourtoule et al. (2021).
Concurrent Unlearning Requests. In the concurrent setting, the unlearning
requests are aggregated in a batch for joint processing. Let
$\\{b_{1},\ldots,b_{S}\\}$ denote a set of Bernoulli random variables, where
$b_{s}$ represents whether shard $s$ is affected in the unlearning process.
Therefore, $P(b_{s}=1)=\dfrac{1}{S}$. Accordingly, the expected time cost for
processing all unlearning requests can be estimated by
$\displaystyle T_{c}$
$\displaystyle=\mathbb{E}\left(\sum_{s=1}^{S}b_{s}\overline{C}_{t}\right)\
\overset{(b)}{=}S\overline{C}_{t}\left(1-\left(1-\frac{1}{S}\right)^{K}\right),$
(10)
where $(b)$ is derived from the expected value of the Bernoulli random
variable.
### 4.2 Storage Effectiveness for Coded Sharding
In this work, we use two typical metrics to evaluate the effectiveness of the
coded sharding mechanism: 1) Storage efficiency, denoted by $\gamma$, which is
defined as the ratio of the size of intermediate model parameters to that of
the data stored in the server; 2) Throughput, denoted by $\lambda$, which
characterizes the capacity for processing unlearning requests. The throughput
is mainly determined by two factors, i.e., the number of unlearning requests
being processed and the associated computational cost. We shall note that the
storage efficiency and throughput are the same for the sequential and
concurrent settings since only the size of the model parameters is different.
We take the full storage mechanism Liu et al. (2021) as the benchmark, where
all the intermediate model parameters are stored in the servers. Therefore,
its storage efficiency is set as $\gamma_{f}=1$. Moreover, since the servers
handle all unlearning requests, the throughput is $\lambda_{f}=1$. For the
proposed uncoded sharding mechanism, all clients are divided into $S$ shards.
Therefore, its storage efficiency and throughput are given by $\gamma_{s}=S$
and $\lambda_{s}=S$, respectively. On the other hand, the proposed coded
sharding mechanism is resistant to $\mu C$ erroneous results, where $\mu$ is
the proportion of the erroneous results to the coded slices Roth (2006). Thus,
it follows
$2\mu C\leq C-S,$ (11)
which implies that the upper bound of $S$ is $(1-2\mu)C$. Accordingly, we have
$S\leq\gamma_{c}\leq(1-2\mu)C.$ (12)
Then, given $O(C^{2}\log^{2}C\log\log C)$ as the additional computational
overhead for coded computing Li et al. (2020), the throughput of the coded
sharding mechanism can be expressed as
$\lambda_{c}=S/O(C^{2}\log^{2}C\log\log C).$ (13)
## 5 Experiments
### 5.1 Experimental Settings
Federated Learning and Unlearning. We consider a total of 100 clients in our
experiments to demonstrate the effectiveness of the proposed framework. In the
learning process, only 20 clients are randomly selected in each training
round. These clients are divided into 4 shards such that each shard has 5
clients. The numbers of local epochs and training rounds are set as 10 and 30,
respectively. In the unlearning process, we consider two types of unlearning
requests: 1) Even, where all requests are evenly distributed across shards; 2)
Adapt, where all requests are adaptively initiated in one shard Bourtoule et
al. (2021).
Figure 3: Performance with a single unlearning request. | MNIST | FMNIST | CIFAR-10 | Shakespeare
---|---|---|---|---
F1 Score ($\downarrow$) | IID | Non-IID | IID | Non-IID | IID | Non-IID | IID | Non-IID
FR | 0.5455 | 0.5354 | 0.5423 | 0.5434 | 0.5074 | 0.4742 | 0.5426 | 0.5995
FE | 0.5450 | 0.1953 | 0.4659 | 0.2803 | 0.1649 | 0.6536 | - | -
RR | 0.5496 | 0.2289 | 0.5203 | 0.5553 | 0.5264 | 0.6684 | - | -
SE | 0.5486 | 0.5959 | 0.5447 | 0.5560 | 0.4812 | 0.5141 | 0.6240 | 0.0392
Retraining Time ($\downarrow$) | IID | Non-IID | IID | Non-IID | IID | Non-IID | IID | Non-IID
FR | 565.68 | 563.66 | 556.57 | 558.25 | 573.78 | 571.92 | 2406.96 | 2343.51
FE | 293.84 | 287.40 | 290.90 | 291.94 | 301.57 | 304.22 | 1196.03 | 1213.69
RR | 391.71 | 396.40 | 376.49 | 390.88 | 386.00 | 392.61 | - | -
SE | 96.79 | 96.51 | 96.50 | 98.12 | 105.05 | 103.96 | 148.06 | 136.95
Gain | $\downarrow$ 67.06% | $\downarrow$ 66.41% | $\downarrow$ 66.82% | $\downarrow$ 66.39% | $\downarrow$ 65.16% | $\downarrow$ 65.82% | $\downarrow$ 87.62% | $\downarrow$ 88.71%
Table 1: F1 score and retraining time in IID and non-IID scenarios.
Datasets and Models. We use four commonly-adopted datasets including MNIST
LeCun et al. (1998), Fashion-MNIST Xiao et al. (2017), CIFAR-10 Krizhevsky et
al. (2009), and Tiny Shakespeare McMahan et al. (2017) for experiments, which
are applicable to a diverse range of tasks. On the other hand, we consider two
typical learning tasks, i.e., image classification and language generation. To
accomplish the two tasks, we use two learning models: 1) Convolutional neural
network, which is composed of 2 convolutional, 2 pooling, and 2 fully
connected layers. It will be trained on the MINST, Fashion-MINST, and CIFAR-10
datasets, respectively; 2) NanoGPT111https://github.com/karpathy/nanoGPT
Radford et al. (2019), which consists of a 4-layer transformer with 4
attention heads Vaswani et al. (2017), an embedding layer with dimension = 16,
a block layer, and a vocabulary with size = 109. In particular, NanoGPT is
trained on the Tiny Shakespeare dataset. For data distribution, we consider
two data-partition approaches: 1) Independent and identically distributed
(IID), where all data samples are randomly divided into 100 equal parts, and
each client is assigned with one part; 2) Non-IID. Specifically, for the image
classification task, 80% data samples of each client belong to one primary
class, while the remaining data samples belong to other classes Wang et al.
(2020). For the language generation task, the entire dataset is divided into
several unbalanced buckets, and each client is assigned with two buckets to
ensure non-IID data distribution across different clients.
Baseline Frameworks. We adopt three state-of-the-art baseline frameworks: 1)
FedRetrain (FR) Liu et al. (2021), which retrains the model from scratch to
achieve unlearning purposes; 2) FedEraser (FE) Liu et al. (2021), which
calibrates model parameters using the historical updates from retained
clients; 3) RapidRetrain (RR) Liu et al. (2022), which employs a diagonal
empirical Fisher information matrix to expedite retraining. We name our
proposed framework as SE, which is the abbreviation of Sharding Eraser enabled
by isolated and coded sharding.
Performance Metrics. Without loss of generality, the unlearning effectiveness
can be evaluated by four metrics, including accuracy, retraining time, storage
overhead, and F1 score for resisting membership inference attacks (MIAs)
Shokri et al. (2017). Note that F1 score is utilized to verify whether data is
successfully unlearned from the model Bourtoule et al. (2021); Liu et al.
(2021). These four metrics align with the fundamental principle outlined in
Bourtoule et al. (2021), which focuses on maintaining model accuracy, reducing
unlearning time, as well as providing security guarantees.
### 5.2 Experimental Results
Figure 4: Performance with concurrent unlearning requests. Figure 5:
Communication time and storage overhead of different frameworks with
concurrent adaptive unlearning requests.
Performance with Single Unlearning Request. We first consider a simple
scenario in which each retained client has a single unlearning request. In
this scenario, we test the unlearning performance of the four frameworks, and
the results are shown in Figure 3. As can be seen from Figures 3(a)-(c), the
proposed framework SE can achieve comparable accuracy to the baseline
framework FR in both IID and non-IID cases. Moreover, SE always outperforms
the other two baseline frameworks, i.e., FE and RR. The results on the Tiny
Shakespear dataset are very similar, except that the rapid retraining
framework RR cannot converge Su and Li (2023). On the other hand, since FR
retrains the model from scratch and entirely removes data effects of specific
clients, its F1 score and accuracy should be jointly considered for
performance comparison. As shown in Table 1 and Figure 3, the F1 score of FE
is lower than those of FR, RR, and SE. However, this result is attributed to
the reduced accuracy of FE, which cannot be considered as the improved
unlearning performance. Instead, our proposed framework SE can achieve
comparable F1 scores with FR while maintaining model accuracy. Besides, for
the retraining time, our framework SE always surpasses the other three
frameworks. For example, in the non-IID case, SE can achieve about 66.41%,
66.39%, 65.16%, and 87.62% time reduction on the MNIST, Fashion-MNIST,
CIFAR-10, and Tiny Shakespeare dataset. Therefore, our proposed framework can
significantly reduce the retraining time without sacrificing accuracy.
Performance with Concurrent Unlearning Requests. To demonstrate the
scalability of the proposed framework, we further consider a scenario in which
each retained client has multiple concurrent unlearning requests. In
particular, each request can be evenly or adaptively initiated across
different shards. The experimental results are illustrated in Figure 4.
Specifically, on the CIFAR-10 dataset, the proposed framework SE can
outperform FE and RR, and can offer a comparable F1 score to all baseline
frameworks. Moreover, it reduces the retraining time by 70% in the even
scenario. Similar trends can be observed in the Tiny Shakespear dataset. These
gains are attributed to the fact that SE can significantly reduce the number
of affected clients in the unlearning process. As for the adaptive scenario,
the variance in data quality across different shards may affect the loss of SE
on the Tiny Shakespeare dataset. Nevertheless, its retraining time is still
the shortest among all frameworks, demonstrating the effectiveness of the
proposed framework in terms of time efficiency.
Storage Overhead with Concurrent Unlearning Requests. Since FR and RR do not
utilize storage resources to improve unlearning performance, we only compare
the proposed framework SE with the baseline framework FE. We take the
concurrent adaptive scenario for experiments. Therein, the communication time
for transmitting model parameters includes two parts: 1) Base network delay,
which is set as 0.1 second; 2) Model transmission time, which is computed as
the ratio of the model size (in bits) and the network data rate (in bit/s).
Moreover, to demonstrate the benefit of the coded computing mechanism, we
define Coded SE as the framework with isolated and coded sharding, and denote
Uncoded SE as the framework with only isolated sharding. In both frameworks,
the storage overhead includes the model parameters stored in one shard.
Due to that the storage overhead is independent of data distribution, we take
the IID scenario as an example. Figure 5(a) and Figure 5(b) show the
communication time and storage overhead of FE, Uncoded SE, and Coded SE with
concurrent adaptive unlearning requests. We can observe that Coded SE achieves
the smallest storage overhead and the minimum communication time among all
frameworks. For example, the storage overhead can be reduced by almost 98%. On
the other hand, we perform experiments on the Shakespeare dataset to
investigate the impact of the total number of clients and global rounds on the
storage overhead. The results are depicted in Figures 5(c)-(d). From these
figures, we can see that Coded SE shows a sharp reduction in storage overhead
with a slight increase in communication time as the number of clients
increases. This result is attributed to the increased division of model
parameters into more shards for storage when the number of clients increases.
Also, there exists a marginal increase in the distribution and retrieval time
when the server utilizes the model parameters for unlearning.
## 6 Conclusion
In this paper, we have developed a scalable federated unlearning framework by
introducing two mechanisms based on isolated sharding and coded computing. The
isolated sharding mechanism divides distributed clients into multiple shards
and removes the data effects according to sequential or concurrent unlearning
requests. The coded computing mechanism extends the scalability of the
framework by encoding, decoding, distributing, and retrieving intermediate
model parameters. By doing so, the storage overhead of the central server can
be significantly reduced. Theoretical analysis and experimental results
demonstrate the effectiveness of the proposed framework as compared with three
baseline frameworks. Future works may consider integrating unlearning layers
into model architectures to achieve cross-shard unlearning.
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|
# Optimal charging of open spin-chain quantum batteries via homodyne-based
feedback control
Y. Yao Center for Quantum Sciences and School of Physics, Northeast Normal
University, Changchun 130024, China Center for Advanced Optoelectronic
Functional Materials Research, and Key Laboratory for UV Light-Emitting
Materials and Technology of Ministry of Education, Northeast Normal
University, Changchun 130024, China X. Q. Shao<EMAIL_ADDRESS>Center
for Quantum Sciences and School of Physics, Northeast Normal University,
Changchun 130024, China Center for Advanced Optoelectronic Functional
Materials Research, and Key Laboratory for UV Light-Emitting Materials and
Technology of Ministry of Education, Northeast Normal University, Changchun
130024, China
###### Abstract
We study the problem of charging a dissipative one-dimensional $XXX$ spin-
chain quantum battery using local magnetic fields in the presence of spin
decay. The introduction of quantum feedback control based on homodyne
measurement contributes to improve various performance of the quantum battery,
such as energy storage, ergotropy, and effective space utilization rate. For
the zero temperature environment, there is a set of optimal parameters to
ensure that the spin-chain quantum battery can be fully charged and the energy
stored in the battery can be fully extracted under the perfect measurement
condition, which is found through the analytical calculation of a simple two-
site spin-chain quantum battery and further verified by numerical simulation
of a four-site spin-chain counterpart. For completeness, the adverse effects
of imperfect measurement, anisotropic parameter, and finite temperature on the
charging process of the quantum battery are also considered.
## I Introduction
The quantum batteries have become an important topic concerned by researchers
due to the requirement for the efficient and miniaturized energy storage
devices. Initially, the concept of quantum batteries proposed by Alicke and
Fannes Alicki and Fannes (2013) is defined as a small quantum mechanical
system for temporarily storing energy. Subsequently, a series of quantum
battery models have been proposed in closed systems. Simply put, the charging
process can be achieved by coupling the battery to the charger which can be
regarded as both energy provider and energy mediator. For simple models, such
as a two-level system quantum battery or a quantum harmonic oscillator quantum
battery Andolina _et al._ (2018); Chen and Hasegawa ; Zhang _et al._ (2019);
Delmonte _et al._ (2021); Crescente _et al._ (2020a); Seah _et al._ (2021),
the batteries of these can achieve complete charging directly, but the stored
energy has oscillation behavior because the charging process is unitary. Once
the optimal charging time is missed, the stored energy of the battery is
reduced obviously. To solve this problem, some researchers proposed adiabatic
charging schemes to stabilize the energy storage of battery Santos _et al._
(2019); Dou _et al._ (2020); Santos _et al._ (2020). In addition to the
single-body quantum battery model, there are also many-body quantum battery
models Qi and Jing (2021); Andolina _et al._ (2019a, b); Rossini _et al._
(2019); Julià-Farré _et al._ (2020); Huangfu and Jing (2021); Liu _et al._
(2021); Rossini _et al._ (2020); Lu _et al._ (2021); Crescente _et al._
(2020b); Zhang and blaauboer ; Dou _et al._ (2022); Caravelli _et al._
(2020); Le _et al._ (2018); Ghosh _et al._ (2020); Kamin _et al._ (2020a);
Zhao _et al._ (2022); Barra _et al._ (2022); Arjmandi _et al._ (2022);
Mondal and Bhattacharjee , including Sachdev-Ye-Kitaev batteries Rossini _et
al._ (2020), Tavis-Cummings quantum battery Lu _et al._ (2021), Dicke quantum
battery Crescente _et al._ (2020b); Zhang and blaauboer ; Dou _et al._
(2022), Random quantum batteries Caravelli _et al._ (2020), and spin-chain
quantum battery Le _et al._ (2018); Ghosh _et al._ (2020); Kamin _et al._
(2020a); Zhao _et al._ (2022); Barra _et al._ (2022); Arjmandi _et al._
(2022); Mondal and Bhattacharjee ; Ghosh and Sen (De) and so on. These many-
body quantum batteries exhibit remarkable charging speed compared with single-
body quantum battery Binder _et al._ (2015); Ferraro _et al._ (2018); Gyhm
_et al._ (2022).
As a matter of fact, the quantum batteries cannot be completely isolated from
the environment. Therefore, it is necessary to investigate how to stabilize
the stored energy and resist energy leakage caused by decoherence when the
quantum battery is immersed in the environment Farina _et al._ (2019); Barra
(2019); Pirmoradian and Mølmer (2019); Tacchino _et al._ (2020); Hovhannisyan
_et al._ (2020); Kamin _et al._ (2020b); García-Pintos _et al._ (2020); Ito
and Watanabe ; Gherardini _et al._ (2020); Bai and An (2020); Quach and Munro
(2020); Tabesh _et al._ (2020); Zhao _et al._ (2021); Ghosh _et al._
(2021); Peng _et al._ (2021); Santos (2021); Xu _et al._ (2021); Yao and
Shao (2021); Centrone _et al._ ; Mitchison _et al._ (2021); Carrasco _et
al._ (2022). In 2020, Gherardini et al. Gherardini _et al._ (2020) put
forward a stable charging scheme based on continuous measurement to compensate
the entropy increase and keep the open quantum battery in the highest entropy
state. Later, Quach and Munro Quach and Munro (2020) introduced an open
quantum battery model by using dark states to achieve superextensive capacity
and power density. In general, quantum batteries can hardly be fully charged
in the presence of environmental noise. Therefore how to improve the energy
storage of quantum batteries in specific systems is still a challenging
problem.
At the same time, we should not only pay attention to the stable and effective
charging process of the battery, but also consider the maximum capacity of the
battery itself, so as to provide sufficient energy for other equipment.
Recently, the model of $N$-spin chain quantum battery interacting with
environmental noise is constructed in Ref. Zhao _et al._ (2021), and the
maximum energy of a quantum battery driven by a coherent cavity driving field
or a heat reservoir is investigated. The results show that the nearest-
neighbor hopping interaction enhances the energy storage and ergotropy of
quantum batteries. Then, Ghosh et al. advanced an open spin-chain battery
charging scheme Ghosh _et al._ (2021), where they considered each spin was
connected to two local bosonic reservoirs. During the charging process, the
energy dissipation channel is closed and the energy absorption channel is
opened. Their research shows that when the quantum battery is affected by
Markovian noise, the storage speed of the quantum battery in the transient
region is faster than that without noise, and the maximum recoverable work
quantified by ergotropy is also higher. It is worth noting that the above
results only hold when the ratio of dephasing noise rate to energy absorption
rate is within a certain range, and the advantage of noise disappears when the
system reaches the steady state.
In order to charge the quantum battery stably and effectively, we propose a
dissipative protocol based on homodyne feedback, which is composed of $XXX$
spin chain with open boundary condition. The energy injection into the battery
is performed by applying a local external magnetic field to each spin. During
the charging process, photons of spin radiation induced by the environment are
collected, so that the homodyne current is generated after detection, and then
the strength of local charging field becomes related to the photocurrent
signal. By adjusting the feedback strength and the feedback direction, the
$XXX$ spin-chain quantum battery can be fully charged and stabilized.
Therefore, the energy stored by the battery in steady state is more favorable
than the Markovian scheme of Ref. Ghosh _et al._ (2021). The advantage of the
homodyne-based feedback control mechanism is its simplicity in practical
applications because it avoids the challenge of real-time state estimation
required by Bayesian or state-based feedback Doherty and Jacobs (1999);
Carvalho _et al._ (2008). The relevant feedback theory has been widely used
in various fields Wang _et al._ (2005); Liu _et al._ (2010); Bushev _et
al._ (2006); Wang and Wiseman (2001); Campagne-Ibarcq _et al._ (2016); Genoni
_et al._ (2013) and guided by it some experimental results have been obtained
Smith _et al._ (2002); Armen _et al._ (2002); Geremia _et al._ (2004).
The remainder of the paper is organized as follows. In Sec. II, we first
analyze the stored energy and the maximum energy extracted from the spin-chain
quantum battery (ergotropy) under feedback control, and then introduce the
concept of effective space utilization rate. In Sec. III, we successively
obtain the analytical solution of the charging process for $XXX$ spin-chain
quantum battery consisting of two spins, and numerically verify the case of
multiple spins. In addition, the stochastic dynamics of the battery charging
process, the effects of spin-spin interaction, the imperfect measurement,
anisotropic parameter, and finite temperature on the battery charging process
are also discussed in detail. Finally, we give a summary of the present
protocol in Sec. IV.
## II DISSIPATIVE Spin-chain QUANTUM BATTERY
Figure 1: The spin-chain quantum battery. Each spin is coupled with an
independent reservoir (light grey ellipse) and has a spontaneous emission rate
$\Gamma$. Initially, the battery is prepared in its ground state and each spin
is subjected to a homodyne measurement. After the detection, a corresponding
homodyne photocurrent is produced. Then the feedback control is activated by
applying local magnetic fields depending on the measured results.
### II.1 Master equation of the system
We model the quantum battery as a one-dimensional Heisenberg spin chain
consisting of $N$ spins on the lattice, each spin interacts with a local zero-
temperature reservoir, as shown in Fig. 1. In the absence of charging, the
static Hamiltonian of the battery can be described as ($\hbar=1$)
$\displaystyle H_{B}$ $\displaystyle=$
$\displaystyle\frac{h}{2}\sum\limits_{j=1}^{N}\sigma_{j}^{z}+\sum\limits_{j=1}^{N-1}J[(1+\gamma)\sigma_{j}^{x}\sigma_{j+1}^{x}+(1-\gamma)\sigma_{j}^{y}\sigma_{j+1}^{y}$
(1) $\displaystyle+\Delta\sigma_{j}^{z}\sigma_{j+1}^{z}],$
where $h$ represents the strength of external magnetic field breaking the
degeneracy between $\mid\downarrow\rangle$ and $\mid\uparrow\rangle$, $J$
refers to the nearest-neighbor coupling strength between spins. $\sigma^{k}$
$(k=x,y,z)$ is the corresponding Pauli spin matrix, and $\gamma$ and $\Delta$
are the anisotropic parameters. For convenience, we assume that all the
parameters here are positive real numbers.
To inject energy into battery, the local external magnetic field needs to be
applied to each spin. The form of charging Hamiltonian is chosen as:
$H_{\rm
charging}=\sum_{j=1}^{N}\\{\Omega_{j}[\sigma_{j}^{x}\sin(\alpha)+\sigma_{j}^{y}\cos(\alpha)]\\},$
(2)
where $\alpha\in(-\pi,\pi]$ is viewed as a regulator of changing the direction
of magnetic field, and $\Omega_{j}$ is the corresponding strength applied to
the $j$th spin. The evolution of the system can be characterised by the
following Lindblad master equation
$\mathop{\dot{\rho}}=-i[H_{B}+H_{\rm
charging},\rho]+\Gamma\sum_{j=1}^{N}\mathcal{D}[\sigma_{j}^{-}]\rho,$ (3)
where $\mathcal{D}[o]\bullet=o\bullet
o^{{\dagger}}-({o^{{\dagger}}o\bullet+\bullet o^{{\dagger}}o})/2$ and $\Gamma$
is the spontaneous emission rate of each spin constituting the quantum
battery.
If the above model is considered in a closed system, the energy stored in the
battery usually oscillates over time, and the protocol is only effective for
the instantaneous charging of the quantum battery, and in actual environment,
the non-zero value of $\Gamma$ further reduces the charging performance of the
battery. Therefore, designing an efficient charging scheme in open quantum
systems is a problem worthy of consideration. From this perspective, we
introduce the quantum feedback mechanism based on homodyne measurement into
the charging protocol of quantum battery.
In the process of feedback control, the homodyne measurement of each spin is
performed, and the resulting photocurrent $J^{\rm hom}_{j}$ can be
approximated as signal plus Gaussian white noise Wiseman and Milburn (1993);
Wiseman (1994); Gardiner (1985); Wiseman and Milburn (2009), i.e., (please see
appendix. A)
$J^{\rm
hom}_{j}(t)=\langle\sigma_{j}^{x}\rangle+\frac{\xi(t)}{\sqrt{\eta\Gamma}},$
(4)
where $\eta\leq 1$ is the total measurement efficiency of the detection, and
$\xi(t)=dw(t)/dt$ represents Gaussian white noise with a complex Wiener
increment $dw(t)$ satisfying $[dw(t)]^{2}=dt$ and an average over the noise
$E[dw(t)]=0$ Gardiner (1985). The local magnetic field thus becomes time-
dependent and its strength depending on the measurement results can be
expressed as
$\Omega_{j}(t)=\Omega_{0j}+fJ^{\rm hom}_{j}(t-\tau),$ (5)
where $f$ is the feedback strength, and $\tau$ represents a small time delay
in the feedback loop. For simplicity, the constant amplitude $\Omega_{0j}$ can
be set to 0, then the corresponding feedback Hamiltonian reads
$H_{\rm fb}=\sum_{j=1}^{N}J^{\rm hom}_{j}(t-\tau)F_{j},$ (6)
where $F_{j}=f[\sigma^{x}_{j}\sin(\alpha)+\sigma^{y}_{j}\cos(\alpha)]$. Now
the total conditioned equation including feedback obeys the following rule
Wiseman and Milburn (2009)
$\rho_{J}(t+dt)=\sum_{j=1}^{N}e^{\mathcal{K}_{j}J^{\rm
hom}_{j}(t-\tau)dt}(\rho_{J}(t)+\mathcal{A}[\rho_{J}(t)]),$ (7)
where $\mathcal{K}_{j}$ is a Liouville superoperator satisfying
$\mathcal{K}_{j}\rho=-i[F_{j},\rho]$.
$\mathcal{A}[\rho_{J}(t)]=\Gamma\mathcal{D}[\sigma^{-}_{j}]\rho_{J}(t)dt+\mathcal{H}[-iH_{B}]\rho_{J}(t)dt+\sqrt{\eta\Gamma}d\omega(t)\mathcal{H}[\sigma^{-}_{j}]\rho_{J}(t)$
with $\mathcal{H}$ being a nonlinear superoperator defined as
$\mathcal{H}[o]\bullet=o\bullet+\bullet o^{{\dagger}}-{\rm
Tr}\\{\bullet(o+o^{{\dagger}})\\}\bullet$. Under the Markovian limit of
$\tau\rightarrow 0$, Eq. (7) can be expanded to second order in
$\mathcal{K}_{j}$ while retaining the first order of $dt$. Then, by applying
the rules of Ito calculus (i.e., $[dw(t)]^{2}=dt$ ), the stochastic master
equation describing the charging process of the spin-chain quantum battery is
$\displaystyle\mathop{\dot{\rho}_{J}}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\\{\mathcal{K}_{j}(\sigma^{-}_{j}\rho_{J}+\rho_{J}\sigma^{+}_{j})+\frac{1}{2\eta\Gamma}\mathcal{K}_{j}^{2}\rho_{J}+\sqrt{\eta\Gamma}\xi(t)$
(8)
$\displaystyle\times[\mathcal{H}[\sigma^{-}_{j}]+(\eta\Gamma)^{-1}\mathcal{K}_{j}]\rho_{J}\\}+\mathcal{L}\rho_{J},$
where
$\mathcal{L}\rho_{J}=-i[H_{B},\rho_{J}]+\Gamma\sum_{j=1}^{N}\mathcal{D}[\sigma^{-}_{j}]\rho_{J}$.
Taking the ensemble average of Eq. (8), we have
$\displaystyle\mathop{\dot{\rho}}$ $\displaystyle=$
$\displaystyle-i[H_{B},\rho]+\Gamma\sum_{j=1}^{N}\mathcal{D}[\sigma^{-}_{j}]\rho$
(9)
$\displaystyle-i\sum_{j=1}^{N}\\{[F_{j},\sigma_{j}^{-}\rho+\rho\sigma_{j}^{+}]+\frac{1}{2\eta\Gamma}[F_{j},-i[F_{j},\rho]]\\},$
where the first two terms describe the dynamical evolution caused by the
internal Hamiltonian of battery and the spontaneous decay, respectively. The
last two terms describe the charging coherence introduced by the feedback
operation and the measurement noise fed back into the system. The central idea
is to use continuous measurement records to control the dynamics of system.
### II.2 The relevant performance parameters of the battery
The stored energy of the quantum battery at an arbitrary time $t$ is defined
as
$\Delta E(t)={\rm Tr}[H_{B}\rho_{B}(t)]-{\rm Tr}[H_{B}\rho_{B}(0)],$ (10)
where $\rho_{B}(t)$ is the density matrix of the battery and $\rho_{B}(0)$ is
the initial state. In order to describe the maximum energy that can be
extracted from the battery, we need to calculate the corresponding ergotropy
Allahverdyan _et al._ (2004)
$\mathcal{E}(t)={\rm Tr}[H_{B}\rho_{B}(t)]-\mathop{\rm min}\limits_{U}{\rm
Tr}[U\rho_{B}(t)U^{{\dagger}}H_{B}],$ (11)
the second term in the r.h.s. of Eq. (11) indicates the minimum energy value
which cannot be extracted from the battery by executing all unitaries $U$ on
the system.
If Hamiltonian $H$ and density matrix $\rho$ of a system are written orderly
as $H=\sum_{i}\varepsilon_{i}|\varepsilon_{i}\rangle\langle\varepsilon_{i}|$
($\varepsilon_{1}<\varepsilon_{2}\cdots<\varepsilon_{N}$), and
$\rho=\sum_{k}r_{k}|r_{k}\rangle\langle r_{k}|$ ($r_{1}>r_{2}\cdots>r_{N}$),
respectively, the corresponding passive state Allahverdyan _et al._ (2004);
Lenard (1978) of the system can be expressed as
$\sigma=\sum_{j}r_{j}|\varepsilon_{j}\rangle\langle\varepsilon_{j}|.$ (12)
Thus the ergotropy can be explicitly rewritten as
$\displaystyle\mathcal{E}(t)$ $\displaystyle=$ $\displaystyle{\rm
Tr}[H_{B}\rho_{B}(t)]-{\rm Tr}[H_{B}\sigma]$ (13) $\displaystyle=$
$\displaystyle{\rm Tr}[H_{B}\rho_{B}(t)]-\sum_{j}r_{j}\varepsilon_{j}.$
It is well known that the energy spectrum structure of a spin chain is closely
related to the nearest-neighbor coupling strength between spins, thus it is
not comprehensive to measure the performance of the battery only by the stored
energy of the battery and the ergotropy. Here, we introduce the effective
space utilization rate of the battery, i.e. the ratio of the total battery
space occupied by the stored energy of the battery, which is also an important
parameter for benchmarking the performance of quantum battery with internal
interaction,
$R(t)=\frac{\Delta E(t)}{C_{\rm max}},$ (14)
where $C_{\rm max}=E_{\rm max}-E_{\rm min}$ is the maximum capacity of the
battery, and $E_{\max}$ and $E_{\min}$ are the highest energy and the lowest
energy in the energy spectrum of $H_{B}$, respectively. Undoubtedly, $R=1$
indicates that the battery can be fully charged.
## III the OPTIMAL FEEDBACK CONDITION
### III.1 The charging of two-spin quantum battery under homodyne-based
feedback control
For convenience, we mainly take the $XXX$ spin-chain quantum battery
($\Delta=1$ and $\gamma=0$) consisting of two spins as an toy model to study
the charging process. This simple model can not only reflect the
characteristics obviously different from the single-body quantum battery, but
also help us determine the optimal parameter range when extending to a many-
body spin quantum battery. According to Eq. (1), we can get the following
eigenstates
$\displaystyle|E_{a}\rangle$ $\displaystyle=\mid\downarrow\downarrow\rangle,$
(15a) $\displaystyle|E_{b}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(\mid\uparrow\downarrow\rangle-\mid\downarrow\uparrow\rangle),$
(15b) $\displaystyle|E_{c}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(\mid\uparrow\downarrow\rangle+\mid\downarrow\uparrow\rangle),$
(15c) $\displaystyle|E_{d}\rangle$
$\displaystyle=\mid\uparrow\uparrow\rangle,$ (15d)
with the corresponding eigenvalues $E_{a}=-h+J$, $E_{b}=-3J$, $E_{c}=J$, and
$E_{d}=h+J$, respectively. It is easy to check that the order of eigenvalues
is $E_{a}<E_{b}<E_{c}<E_{d}$ when $J<h/4$, and the order of eigenvalues
becomes $E_{b}<E_{a}<E_{c}<E_{d}$ once $J>h/4$. The above results show that
the ground state of the system is related to the coupling strength $J$ between
spins. But the highest energy state of the battery is always
$|E_{d}\rangle=\mid\uparrow\uparrow\rangle$ regardless of $J$.
The steady-state solution of Eq. (9) (see appendix. B) is
$\displaystyle\rho_{11}(\infty)$
$\displaystyle=\frac{\chi^{4}}{(2\chi^{2}+2\chi\eta\cos\alpha+\eta)^{2}},$
(16a) $\displaystyle\rho_{22}(\infty)$
$\displaystyle=\frac{\chi^{2}(\chi^{2}+2\chi\eta\cos{\alpha}+\eta)}{(2\chi^{2}+2\chi\eta\cos{\alpha}+\eta)^{2}},$
(16b) $\displaystyle\rho_{33}(\infty)$ $\displaystyle=\rho_{22}(\infty),$
(16c) $\displaystyle\rho_{44}(\infty)$
$\displaystyle=\frac{(\chi^{2}+2\chi\eta\cos{\alpha}+\eta)^{2}}{(2\chi^{2}+2\chi\eta\cos{\alpha}+\eta)^{2}},$
(16d)
where we have assumed $\chi=f/\Gamma$ for simplicity. The optimal feedback
condition can be determined through
$\partial\rho_{11}(\infty)/\partial\chi=2(\chi^{2}\eta\cos\alpha+\chi\eta)/(2\chi^{2}+2\chi\eta\cos\alpha+\eta)^{2}=0$.
By selecting $\chi=-1/\cos\alpha$, the maximum value of $\rho_{11}(\infty)$ is
obtained as
$\rho_{11}^{\rm max}(\infty)=\frac{1}{(2-\eta\cos^{2}{\alpha})^{2}}.$ (17)
Under the perfect measurement condition $\eta=1$, the optimal value of
$\rho_{11}^{\rm max}(\infty)$ is 1 when $\alpha=0$ or $\pi$, which means the
battery can be fully charged [$R(\infty)$=1] from its ground state by applying
$y$-direction feedback to the battery, even in the presence of dissipation.
Thus, $\alpha=0$ $(\pi)$ and $\chi=\mp 1$ are viewed as the optimal feedback
parameters of the $XXX$ spin-chain quantum battery. In what follows, we
stipulate that the $y$-direction feedback refers to the case of $\alpha=\pi$,
unless otherwise specified. According to Eqs. (10) and (13), the corresponding
ratio of ergotropy to the stored energy of the battery in steady state is
computed to be $\mathcal{E}(\infty)/\Delta E(\infty)=1$, whenever the ground
state of the $XXX$ model is $\mid\downarrow\downarrow\rangle$ ($J<h/4$) or
$(\mid\uparrow\downarrow\rangle-\mid\downarrow\uparrow\rangle)/\sqrt{2}$
($J>h/4$), which means the stored energy in the battery can be fully
extracted.
### III.2 The charging process of the battery made up with four spins
In the above study, we mainly take the spin-chain quantum battery composed of
two spins as an example to analyze the charging performance of the battery. In
order to verify the similar conclusion is also applicable to the battery with
more spins, we now numerically discuss the case of spin-chain quantum battery
composed of four spins. Considering more spins will undoubtedly increase our
computational time, but will not change the conclusion. In the numerical
simulation, we set the ground state of the battery as the initial state, and
the measurement efficiency $\eta$ is 1.
Figure 2: (a) The stored energy $\Delta E(\infty)/h$ of the battery at steady
state is a function of $\alpha$ and $\chi$. (b) The maximum energy
$\mathcal{E}(\infty)/h$ that can be extracted (ergotropy) from the battery,
other parameters are $\eta=1$ and $J=h$.
In order to determine the optimal feedback parameters, the stored energy and
the ergotropy of the battery in steady state are displayed as functions of
$\alpha$ and $\chi$ in Fig. 2, governed by Eq. (9), and other parameters are
$\eta=1$ and $J=h$. It clearly shows that there are three equivalent optimal
parameters ($\alpha$, $\chi$) in the corresponding parameter range, which is
consistent with the case of two spins. (Please see appendix. C for more
spins). For convenience, we choose the optimal feedback condition
($\alpha=\pi$, $\chi=1$) for the following discussion.
Figure 3: The $y$-direction feedback control is applied to the system, i.e.,
$\alpha=\pi$. (a) The variation of optimal stored energy $\Delta E(\infty)/h$
of the battery at steady state with $J/h$. The inset describes the optimal
feedback parameter $\chi$ at different $J$. (b) The variations of $R(\infty)$
and $\mathcal{E}(\infty)/\Delta E(\infty)$ of the battery in the steady state
with the coupling strength $J$. The measurement efficiency is given as
$\eta=1$.
Fig. 3(a) shows that the optimal stored energy of the battery in the steady
state increases with the increase of coupling strength $J/h$ between spins,
which means the interaction between spins expands the capacity of the battery
itself. From the inset, we see the feedback strength required for the battery
to achieve the optimal energy storage does not change with $J$. In order to
measure the battery performance under feedback mechanism more comprehensively,
we characterize the variation of the space utilization rate [$R(\infty)$] and
the ratio of the ergotropy to the stored energy [$\mathcal{E}(\infty)/\Delta
E(\infty)$] of the battery with the coupling strength between spins in Fig.
3(b). It indicates that the coupling strength between spins is not the factor
affecting the ability of energy storage and energy extraction, because the
highest energy state of the $XXX$ spin-chain battery is always the state all
spins pointing upward, irrespective to the coupling strength between the
spins. After the above analysis of four spins, we qualitatively obtain the
conclusion similar to the case of two spins. It clearly proves that our scheme
is still effective for quantum battery composed of multiple spins.
### III.3 Random dynamics, imperfect measurement, and different initial state
Figure 4: Stochastic charging dynamics of the battery under the optimal
feedback condition. In the inset, the measuring efficiency is set to
$\eta=0.8$. The shallow grey lines represent the random 200 trajectories of
battery storage energy obtained by Eq. (8), the red solid line represents the
average value of 200 energy trajectories, and green dashed line represents the
ensemble average energy obtained by Eq. (9).
In order to show the stochastic nature of the measurement and feedback
processes, in Fig. 4, we numerically simulate 200 random trajectories of the
energy stored in the $XXX$ spin-chain battery varying with $\Gamma t$ based on
the random master equation Eq. (8). Each trajectory represents the result of a
single operation (see the light grey line). Obviously, the energy of different
trajectories has dispersion behavior in the charging process. But, when the
system reaches the steady state, the feedback has a strong inhibitory effect
on the fluctuation of battery energy. In addition, the average of 200 energy
trajectories (the red solid line) is coincide with the energy of the ensemble
average obtained through the master equation Eq. (9) (the green dashed line).
The above results show that the battery can be fully charged not only in the
overall average sense, but also in the single-trajectory. And the inset shows
that the battery can be stably and effectively charged even if the measurement
efficiency is not perfect.
Figure 5: (a) The stored energy $\Delta E(\infty)/h$ (solid line) and the
ergotropy (dash-dotted line) of the spin-chain quantum battery with different
coupling strengths $J/h$. The inset shows the optimal feedback parameter
$\chi$ required by the stored energy (solid line) and ergotropy (dash-dotted
line). (b) The effective utilization rate $R(\infty)$ of the battery as a
function of $J/h$ under the optimal feedback strength. The measurement
efficiency is $\eta=0.8$.
Fig. 5(a) describes the optimal stored energy (solid line) and ergotropy
(dash-dotted line) of the battery in the steady state with imperfect
measurement efficiency $\eta=0.8$. The corresponding optimal feedback
strengths with different $J$ values are shown in the inset. We can clearly see
that the maximum energy extracted from the battery is lower than the optimal
stored energy of the battery. When the spin-spin interaction strength
increases to a certain value $J_{c}$ ($J_{c}=2h$ in terms of energy storage
and $J_{c}=0.8763h$ in terms of energy extraction, please see appendix D for
details), the required optimal feedback intensity should be zero, which means
that the feedback control is invalid in this case. Fig. 5(b) describes the
variation of the effective space utilization rate $R(\infty)$ of the battery
with the $J$ at the optimal feedback strength (the inset of Fig. 5(a)). The
results reflect that the imperfect measurement weakens the role of feedback
control, and reduces the performance of the battery.
Figure 6: The stored energy $\Delta E(\infty)/h$ of the battery in steady
state as a function of $J/h$ under different measurement efficiency $\eta$.
The initial state of the battery is spin-down state and the feedback parameter
$\chi$ at different $J$ values is optimal.
So far, we have studied the charging performance of spin-chain quantum battery
using the ground state as the initial state. Next, we mainly study the
charging process of the battery when all the spin are initialized downward.
Fig. 6 reveals that the stored energy of the battery is independent of $J$
under the condition of perfect measurement (red solid line), because the
stored energy of the battery during the whole charging process is $\Delta
E(\infty)=(3J+2h)-(-2h+3J)=4h$. Nevertheless, when the measurement efficiency
is imperfect ($\eta=0.8$), the stored energy of the battery at steady state is
greatly affected (blue dash line). We can see again that the nearest-neighbor
hopping between spins has a critical value that renders the feedback control
ineffective, as shown in the inset on the left. To better understand this
phenomenon, we use $J/h=3$ to give the energy storage curve of the battery
with the change of feedback strength in the right inset. The result manifests
that the stored energy of the battery becomes negative when the feedback
control is switched on, indicating that the battery has evolved into a lower
energy state than the initial state, where the battery is in the discharging
process.
### III.4 Performance of $XY$ spin-chain quantum battery and $XYZ$ spin-chain
quantum battery
The effect of anisotropic parameter ($\gamma$) on the charging process of the
battery can be characterized by analyzing the related properties of the $XY$
spin-chain quantum battery ($\Delta=0$, $0<\gamma<1$) and $XYZ$ spin-chain
quantum battery ($\Delta=1$, $0<\gamma<1$). In this part, we assume the
measurement efficiency is $\eta=1$, and the feedback parameter $\chi$ takes
the optimal value under their respective $\gamma$.
Figure 7: The effective space utilization rate $R(\infty)$ in steady state of
the $XY$ spin-chain quantum battery ($\Delta=0$) with the spin number of $N=2$
(a) and $N=4$ (b) as a function of the anisotropic parameter $\gamma$ under
different coupling strengths $J$. The insets in (a) and (b) describe the
variations of the stored energy $\Delta E(\infty)/h$ of the battery with
$\gamma$ in steady state. The feedback strengths at different $\gamma$ values
are optimal. Other parameters are $\eta=1$, $\alpha=\pi$ and $\Gamma=h$. (c)
and (d) show the variations of the effective space utilization with $\gamma$
of the battery with spin number of $N=2$ and $N=4$ under different spontaneous
emission rates $\Gamma$ with $J=h$, respectively.
Figs. 7(a) and 7(b) respectively describe the influence of the anisotropic
parameter $\gamma$ on the effective space utilization rate $R(\infty)$ and the
optimal stored energy $\Delta E(\infty)/h$ of the $XY$ spin-chain battery when
the spin numbers of $N=2$ and $N=4$. Different curves correspond to different
spin-spin coupling strengths, where the spontaneous emission rate is set to
$\Gamma=h$. The results in Fig. 7(a) and the inset show that for a weak
$J=0.1h$ the anisotropic parameter $\gamma$ almost does not affect the
effective space utilization rate $R(\infty)$ and the optimal stored energy
$\Delta E(\infty)$. However, for a relatively strong $J=3h$, $R(\infty)$ and
$\Delta E(\infty)$ increase with $\gamma$ significantly. When the number of
particles increases to $N=4$, we can obtain a conclusion similar to that of
$N=2$, as shown in Fig. 7(b). The above results indicate that under the strong
interaction between spins, the increase of anisotropic parameter $\gamma$
plays a positive role in improving the performance of the $XY$ spin-chain
quantum battery. Figs. 7(c) and 7(d) respectively describe that the effective
space utilization rates of the battery with $N=2$ and $N=4$ decrease with the
increase of dissipation rate under a given spin-spin coupling strength
($J=h$), which is completely different from the $XXX$ model mentioned above.
Nevertheless, a relatively weak dissipation rate ranging from $0.1h$ to $h$,
can still be regarded as an independent variable. It is worth noting that the
specific values of $\gamma=0$ and $\gamma=1$ correspond to the typical
transverse $XX$ model and transverse Ising model respectively.
Figure 8: (a) The effective space utilization rate $R(\infty)$ of the $XYZ$
spin-chain quantum battery ($\Delta=1$) in steady state as a function of the
anisotropic parameter $\gamma$ for $N=2$. The inset describes the change of
the effective space utilization rate $R$ of the battery with $\gamma$ in
steady state. (b) The same as (a) but the spin number is $N=4$. Other
conditions are the same with the Fig. 7. (c) and (d) show the variations of
the effective space utilization with $\gamma$ of the battery with spin number
of $N=2$ and $N=4$ under different spontaneous emission rates $\Gamma$ with
$J=h$, respectively.
The influence of anisotropic parameter $\gamma$ on the effective space
utilization rate and optimal energy storage of $XYZ$ spin-chain quantum
batteries with $N=2$ and $N=4$ is illustrated in Figs. 8(a) and 8(b) with
$\Gamma=h$ respectively. It indicates that no matter $N=2$ or $N=4$, when the
spin coupling strength is weak ($J=0.1h$), the anisotropic parameters still
have no significant effect on the effective space utilization rate and optimal
energy storage. However, when the coupling strength between spins increases to
$J=3h$, with the increase of anisotropic parameters, the batteries with $N=2$
and $N=4$ exhibit different behaviors in the terms of optimal energy storage.
Meanwhile, we see the effective space utilization of the batteries with spins
of $N=2$ and $N=4$ decreases with the increase of $\gamma$, which means that
the increase in anisotropy is detrimental to the battery chargeability. Figs.
8(c) and 8(d) show the effects of different spontaneous emission rates
$\Gamma$ on the effective space utilization rate of the $XYZ$ spin-chain
quantum batteries with the spin number of $N=2$ and $N=4$, respectively. The
influence of spontaneous emission rate on the current model is obviously
different from that of $XY$ spin-chain model, especially shown in Figs. 8(d),
the effective space utilization increases with the increase of dissipation
rate. Similarly, the specific value of $\gamma=0$ corresponds to the $XXX$
model in Sec. III.2, and we have shown in the previous discussion that its
performance is not affected by spin-spin interaction strength and the
spontaneous emission rate.
### III.5 The effect of finite temperature
Figure 9: (a)-(b) describe the optimal energy storage and ergotropy of the
battery at steady state when the spin numbers are $N=2$ and $N=4$,
respectively. And the corresponding optimal feedback parameter $\chi$ is shown
in the inset. (c) describes the variation of effective space utilization rate
of the battery with the temperature. (d) shows the evolution of energy stored
in the battery ($N=4$) with time $\Gamma t$ at different temperatures, where
the feedback strengths are adjusted to the optimal values and the initial
state of the battery is set to the corresponding ground state. Other
parameters are $\eta_{d}=\eta_{c}=0.8$ and $J=h$.
When there is a finite temperature $T$ in the environment, the uncollected
photons will enter the thermal radiation reservoir with an average occupancy
of $n_{\rm T}=[\exp(\hbar\omega/k_{B}T)-1]^{-1}$. Therefore, we assume that
the photons radiated by spin enter two channels, one is the thermal reservoir
with finite temperature for the uncollected photons, the other is the
effective zero temperature for the collected photons. Then the conditional
state of the system influenced by homodyne measurement can be written as
$\displaystyle\dot{\rho}_{J}$ $\displaystyle=$
$\displaystyle-i[H_{B},\rho_{J}]+\sum_{j=1}^{N}\\{\eta_{c}\Gamma\mathcal{D}[\sigma^{-}_{j}]\rho_{J}+\sqrt{\eta\Gamma}\xi(t)\mathcal{H}[\sigma^{-}_{j}]\rho_{J}\\}$
(18) $\displaystyle+(1-\eta_{c})\sum_{j=1}^{N}\mathcal{L}_{j\rm T}\rho_{J},$
where $\mathcal{L}_{j\rm T}\rho_{J}=\Gamma\\{(1+n_{\rm
T})\mathcal{D}[\sigma^{-}_{j}]\rho_{J}+n_{\rm
T}\mathcal{D}[\sigma^{+}_{j}]\rho_{J}\\}$ is the standard Lindblad dissipator
of the thermal radiation reservoir Mitchison _et al._ (2021); Wiseman and
Milburn (2009). $\eta$ is the total measurement efficiency of the detection
system satisfying $\eta=\eta_{c}\eta_{d}$, which incorporates the collected
photon efficiency $\eta_{c}$ and the detector efficiency $\eta_{d}$
simultaneously.
The master equation based on the homodyne feedback control should be rewritten
as
$\displaystyle\mathop{\dot{\rho}}$ $\displaystyle=$
$\displaystyle-i[H_{B},\rho]+\eta_{c}\Gamma\sum_{j=1}^{N}\mathcal{D}[\sigma^{-}_{j}]\rho+(1-\eta_{c})\sum_{j=1}^{N}\mathcal{L}_{j\rm
T}\rho$ (19)
$\displaystyle-i\sum_{j=1}^{N}\\{[F_{j},\sigma_{j}^{-}\rho+\rho\sigma_{j}^{+}]+\frac{1}{2\eta\Gamma}[F_{j},-i[F_{j},\rho]]\\}.$
By solving the Eq. (19) with $N=2$, we find that the terms in the steady state
are given by
$\displaystyle\rho_{11}(\infty)$ $\displaystyle=\frac{(f^{2}+n_{\rm
T}\Gamma^{2}\eta-n_{\rm
T}\Gamma^{2}\eta\eta_{c})^{2}}{[2f^{2}-2f\Gamma\eta+(1+2n_{\rm
T})\Gamma^{2}\eta-2n_{\rm T}\Gamma^{2}\eta\eta_{c}]^{2}},$ (20a)
$\displaystyle\rho_{22}(\infty)$
$\displaystyle=-\frac{\Gamma^{2}(-2f+\Gamma)^{2}\eta^{2}}{4[2f^{2}-2f\Gamma\eta+(1+2n_{\rm
T})\Gamma^{2}\eta-2n_{\rm T}\Gamma^{2}\eta\eta_{c}]^{2}}$
$\displaystyle\quad+\frac{1}{4},$ (20b) $\displaystyle\rho_{33}(\infty)$
$\displaystyle=\rho_{22}(\infty),$ (20c) $\displaystyle\rho_{44}(\infty)$
$\displaystyle=\frac{[f^{2}-2f\Gamma\eta+(1+n_{\rm T})\Gamma^{2}\eta-n_{\rm
T}\Gamma^{2}\eta\eta_{c}]^{2}}{[2f^{2}-2f\Gamma\eta+(1+2n_{\rm
T})\Gamma^{2}\eta-2n_{\rm T}\Gamma^{2}\eta\eta_{c}]^{2}}.$ (20d)
For a given temperature and collection efficiency, the optimal feedback
strength $f$ for the stored energy of the battery can be determined as
$f/\Gamma=\frac{1}{2}[1+\sqrt{1+4n_{\rm T}\eta(1-\eta_{c})}],$ (21)
and the corresponding maximum value of the $\rho_{11}(\infty)$ is obtained as
follows
$\rho_{11}^{\rm
max}(\infty)=\frac{1}{4}[1+\frac{\eta\sqrt{1+\mu}}{1+\mu+(1-\eta)\sqrt{1+\mu}}]^{2},$
(22)
where $\mu=4n_{\rm T}\eta(1-\eta_{c})$. The existence of finite temperature
increases the required optimal feedback strength $f$ and reduces the
population of the system in $\rho_{11}(\infty)$, and cause the battery to not
be fully charged.
In Fig. 9, we set the coupling strength between spins as $J=h$, and the total
feedback efficiency is $\eta=0.64$. Whether the spin chain quantum battery is
composed of two spins (Fig. 9 (a)) or four spins (Fig. 9 (b)), it can be
clearly seen that the ergotropy and stored energy of battery in the steady
state decrease monotonously with the increase of temperature. Moreover, the
finite temperature increases the required optimal feedback strength as shown
by the corresponding inset, we once again observe that the feedback control
will fails in term of the ergotropy when the temperature is relatively low,
which is similar to the behavior in Fig. 5 (a). Fig. 9 (c) depicts the change
of the effective space utilization rate of the battery with the temperature,
and the results show that $R(\infty)$ decreases with the increase of
temperature. Finally, the energy stored of the battery at different
temperatures is plotted as a function of $\Gamma t$ in the Fig. 9 (d), where
the feedback intensity is set to the corresponding optimal value. It indicates
that the time to reach the steady state of the system is shortened with the
increase of temperature. Although the finite temperature is not conducive to
the battery energy storage, it can still be maintained at a high value when
the temperature is relatively low ($n_{\rm T}\in[0,0.2])$.
## IV Summary
In summary, we have constructed a dissipative $XXX$ spin-chain quantum battery
model based on homodyne measurement. Its core idea is to use continuous
measurement records to control system dynamics. The analytical form of $XXX$
spin chain model of two spins show that the battery can be charged stably and
fully under the optimal feedback condition. We also extend the model to the
case of four spins, and qualitatively get conclusions similar to that of two-
spin model. The interaction between spins expands the capacity of the $XXX$
spin-chain battery itself, but does not affect the degree of storage and
extraction of battery energy. The stochastic dynamics of the battery charging
process reflects that the present feedback mechanism is not only effective in
the overall average sense, but also effective in a single trajectory. The
imperfect measurement and the finite temperature weaken the influence of
feedback control on the system, which are not conducive to the storage and
extraction of battery energy. In addition, under the imperfect measurement
condition, there is a critical value $J_{c}$ of the coupling strength, and
beyond this critical value, the feedback control will fail. For the $XY$ spin-
chain quantum battery, the increase of anisotropy is beneficial to improve the
effective space utilization rate. Meanwhile, the lower dissipation rate is
more favorable for battery energy storage. For the $XYZ$ spin-chain quantum
battery, the relatively large spontaneous emission will help to improve the
effective space utilization rate. We hope that our work will provide a
valuable reference for future research on open quantum batteries.
## Acknowledgements
X. Q. Shao would like to express his thanks to Dr. Lingzhen Guo for his
valuable discussion on the numerical simulation of homodyne-based feedback
control, and Dr. Gangcheng Wang for his helpful comment on the feature of
$XXX$ spin-chain model. The anonymous reviewers are also thanked for
constructive comments that helped in improving the quality of this paper. This
work is supported by the National Natural Science Foundation of China (NSFC)
under Grants No. 11774047 and No. 12174048.
## Appendix A The relevant derivation of Eq. (4)
In the feedback control process, the photon emitted by each spin is collected
and becomes a beam with annihilation operator $\sqrt{\Gamma}\sigma^{-}_{j}$.
Then the beam enters one port of a 50 : 50 beam splitter, and a strong local
oscillator $\beta$ enters another port Wiseman and Milburn (1993); Wiseman
(1994); Gardiner (1985); Wiseman and Milburn (2009). After output by beam
splitter, two field operators $b_{1j}$ and $b_{2j}$ are obtained in the form
of
$b_{kj}=[\sqrt{\Gamma}\sigma^{-}_{j}-(-1)^{k}\beta]/\sqrt{2},$ (23)
when both fields are detected, the corresponding forms of two photocurrent are
$\overline{J_{kj}}=\langle|\beta|^{2}-(-1)^{k}(\sqrt{\Gamma}\beta\sigma^{+}_{j}+\sqrt{\Gamma}\beta^{\ast}\sigma^{-}_{j})+\Gamma\sigma^{+}_{j}\sigma^{-}_{j}\rangle/2,$
(24)
where
$\sqrt{\Gamma}\beta\sigma^{+}_{j}+\sqrt{\Gamma}\beta^{\ast}\sigma^{-}_{j}$
represents the interference of the system and the local oscillator. The above
equation refers to the average photocurrent. However, instantaneous
photocurrent rather than average photocurrent is recorded in practice. Due to
the randomness in the quantum measurement process, the photocurrent will
change randomly. In the ideal case of homodyne detection, namely
$|\beta|^{2}\gg\Gamma$, the rate of photodetections tends to be infinite, and
the corresponding form of photocounts can be approximated as signal plus
Gaussian white noise.
$J^{\rm{hom}}_{j}(t)=\frac{J_{1j}(t)-J_{2j}(t)}{|\beta|}=\sqrt{\Gamma}\langle
e^{-i\Phi}\sigma^{+}_{j}+e^{i\Phi}\sigma^{-}_{j}\rangle+\xi(t).$ (25)
where $\Phi={\rm arg}\beta$ is the phase of the local oscillator, and
$\xi(t)=dw(t)/dt$ represents Gaussian white noise satisfying normal
distribution with $dw(t)$ a complex Wiener increment. When the detector with
efficiency $\eta$ is considered in the whole process, the above expression can
be modified as
$J^{\rm{hom}}_{j}(t)=\langle\sigma^{x}_{j}\rangle+\frac{\xi(t)}{\sqrt{\eta\Gamma}},$
(26)
where we have set $\Phi=0$ for simplicity.
## Appendix B Steady-state solution of two-body $XXX$ spin-chain quantum
battery
The Hilbert space of considered $XXX$ spin-chain quantum battery consists of
the following four bases $|1\rangle=\mid\uparrow\uparrow\rangle$,
$|2\rangle=\mid\uparrow\downarrow\rangle$,
$|3\rangle=\mid\downarrow\uparrow\rangle$, and
$|4\rangle=\mid\downarrow\downarrow\rangle$. The density matrix $\rho$ of the
system at an arbitrary time $t$ can be written as
$\rho(t)=\left(\begin{array}[]{cccc}\rho_{11}(t)&\rho_{12}(t)&\rho_{13}(t)&\rho_{14}(t)\\\
\rho_{21}(t)&\rho_{22}(t)&\rho_{23}(t)&\rho_{24}(t)\\\
\rho_{31}(t)&\rho_{32}(t)&\rho_{33}(t)&\rho_{34}(t)\\\
\rho_{41}(t)&\rho_{42}(t)&\rho_{43}(t)&\rho_{44}(t)\end{array}\right).$ (27)
By solving Eq. (9), we obtain following coupled differential equation of
diagonal elements
$\dot{\rho}_{11}(t)=-2\Gamma\rho_{11}(t)-4f\cos\alpha\rho_{11}(t)+\frac{f^{2}}{\Gamma\eta}[\rho_{22}(t)+\rho_{33}(t)-2\rho_{11}(t)],$
(28)
$\dot{\rho}_{22}(t)=2iJ[\rho_{23}(t)-\rho_{32}(t)]+(\Gamma+2f\cos\alpha)[\rho_{11}(t)-\rho_{22}(t)]+\frac{f^{2}}{\Gamma\eta}[\rho_{11}(t)-2\rho_{22}(t)+\rho_{44}(t)],$
(29)
$\dot{\rho}_{33}(t)=-2iJ[\rho_{23}(t)-\rho_{32}(t)]+(\Gamma+2f\cos\alpha)[\rho_{11}(t)-\rho_{33}(t)]+\frac{f^{2}}{\Gamma\eta}[\rho_{11}(t)-2\rho_{33}(t)+\rho_{44}(t)],$
(30)
and
$\dot{\rho}_{44}(t)=-\dot{\rho}_{11}(t)-\dot{\rho}_{22}(t)-\dot{\rho}_{33}(t)$.
The corresponding differential equations of off-diagonal elements are
$\displaystyle\dot{\rho}_{12}(t)$ $\displaystyle=$
$\displaystyle\dot{\rho}_{21}^{\ast}(t)=-i\\{h\rho_{12}(t)+2J[\rho_{12}(t)-\rho_{13}(t)]\\}-f\\{[3\rho_{12}(t)+\rho_{21}(t)]\cos\alpha+i[\rho_{12}(t)+\rho_{21}(t)]\sin\alpha\\}$
(31)
$\displaystyle-\frac{f^{2}}{\Gamma\eta}[2\rho_{12}(t)+e^{2i\alpha}\rho_{21}(t)-\rho_{34}(t)]-\frac{3\Gamma\rho_{12}(t)}{2},$
$\displaystyle\dot{\rho}_{13}(t)$ $\displaystyle=$
$\displaystyle\dot{\rho}_{31}^{\ast}(t)=-i\\{h\rho_{13}(t)+2J[\rho_{13}(t)-\rho_{12}(t)]\\}-f\\{[3\rho_{13}(t)+\rho_{31}(t)]\cos\alpha+i[\rho_{13}(t)+\rho_{31}(t)]\sin\alpha\\}$
(32)
$\displaystyle-\frac{f^{2}}{\Gamma\eta}[2\rho_{13}(t)+e^{2i\alpha}\rho_{31}(t)-\rho_{24}(t)]-\frac{3\Gamma\rho_{13}(t)}{2},$
$\dot{\rho}_{14}(t)=\dot{\rho}_{41}^{\ast}(t)=-e^{i\alpha}f[2\rho_{14}(t)+\rho_{23}(t)+\rho_{32}(t)]-2ih\rho_{14}(t)-\Gamma\rho_{14}(t)-\frac{f^{2}}{\Gamma\eta}\\{2\rho_{14}(t)+e^{2i\alpha}[\rho_{23}(t)+\rho_{32}(t)]\\},$
(33) $\displaystyle\dot{\rho}_{23}(t)$ $\displaystyle=$
$\displaystyle\dot{\rho}_{32}^{\ast}(t)=2iJ[\rho_{22}(t)-\rho_{33}(t)]-\Gamma\rho_{23}(t)+\frac{f^{2}}{\Gamma\eta}\\{-2\rho_{23}(t)-e^{-2i\alpha}[\rho_{14}(t)+e^{4i\alpha}\rho_{41}(t)]\\}$
(34)
$\displaystyle-f\cos\alpha[\rho_{14}(t)+2\rho_{23}(t)+\rho_{41}(t)]+if\sin\alpha[\rho_{14}(t)-\rho_{41}(t)],$
$\displaystyle\dot{\rho}_{24}(t)$ $\displaystyle=$
$\displaystyle\dot{\rho}_{42}^{\ast}(t)=[\rho_{13}(t)-\frac{\rho_{24}(t)}{2}]\Gamma-i\\{h\rho_{24}(t)+2J[\rho_{34}(t)-\rho_{24}(t)]\\}+\frac{f^{2}}{\Gamma\eta}[\rho_{13}(t)-2\rho_{24}(t)-e^{2i\alpha}\rho_{42}(t)]$
(35)
$\displaystyle+f\\{[2\rho_{13}(t)-\rho_{24}(t)-\rho_{42}(t)]\cos\alpha-i[\rho_{24}(t)+\rho_{42}(t)]\sin\alpha\\},$
$\displaystyle\dot{\rho}_{34}(t)$ $\displaystyle=$
$\displaystyle\dot{\rho}_{43}^{\ast}(t)=[\rho_{12}(t)-\frac{\rho_{34}(t)}{2}]\Gamma-i\\{h\rho_{34}(t)+2J[\rho_{24}(t)-\rho_{34}(t)]\\}+\frac{f^{2}}{\Gamma\eta}[\rho_{12}(t)-2\rho_{34}(t)-e^{2i\alpha}\rho_{43}(t)]$
(36)
$\displaystyle+f\\{[2\rho_{12}(t)-\rho_{34}(t)-\rho_{43}(t)]\cos\alpha-i[\rho_{34}(t)+\rho_{43}(t)]\sin\alpha\\}.$
Then by imposing the condition $\dot{\rho}(t)=0$, we get the following steady-
state solution
$\rho_{11}(\infty)=\frac{f^{4}}{(2f^{2}+2f\Gamma\eta\cos{\alpha}+\Gamma^{2}\eta)^{2}},\qquad\rho_{22}(\infty)=\frac{f^{2}(f^{2}+2f\Gamma\eta\cos{\alpha}+\Gamma^{2}\eta)}{(2f^{2}+2f\Gamma\eta\cos{\alpha}+\Gamma^{2}\eta)^{2}},$
(37)
and $\rho_{33}(\infty)=\rho_{22}(\infty)$,
$\rho_{44}(\infty)=1-\rho_{11}(\infty)-\rho_{22}(\infty)-\rho_{33}(\infty)$.
According to Eq. (10), the energy stored by the quantum battery is
$\displaystyle\Delta E_{XXX}(\infty)$ $\displaystyle=$
$\displaystyle(J+h)\rho_{11}(\infty)-J\rho_{22}(\infty)+2J\rho_{32}(\infty)+2J\rho_{23}(\infty)-J\rho_{33}(\infty)+(J-h)\rho_{44}(\infty)$
(38)
$\displaystyle-[(J+h)\rho_{11}(0)-J\rho_{22}(0)+2J\rho_{32}(0)+2J\rho_{23}(0)-J\rho_{33}(0)+(J-h)\rho_{44}(0)].$
## Appendix C The highest energy state of a $N$-site $XXX$ spin chain
For a general $N$-site $XXX$ spin chain ($\gamma=0$, $\Delta=1$), the
corresponding Hamiltonian can be rewritten as
$H_{B}=h\sum\limits_{j=1}^{N}S_{j}^{z}+\sum\limits_{j=1}^{N-1}2J[(\bm{S}_{j}+\bm{S}_{j+1})^{2}-\bm{S}_{j}^{2}-\bm{S}_{j+1}^{2}],$
(39)
where
$\bm{S}_{j}\equiv(S^{x}_{j},S^{y}_{j},S^{z}_{j})=(\sigma^{x}_{j},\sigma^{y}_{j},\sigma^{z}_{j})/2$.
Since the two parts of above Hamiltonian commute, it’s easy to check that
highest-energy eigenstate of the $XXX$ spin chain is
$\mid\uparrow\uparrow\cdots\uparrow\rangle$ for all positive real parameters,
which is independent of the nearest-neighbor coupling strength between spins,
and the corresponding eigenenergy is $E_{\rm max}=Nh/2+(N-1)J$.
In the Sec. III.2, we have observed that the optimal feedback conditions of
the quantum battery composed of four spins are the same as that of the two
spins. Now we present the numerical results of the energy storage and
ergotropy of the battery composed of six spins varying with parameters
$\alpha$ and $\chi$ in Fig. 10. For convenience, we also reproduce the
numerical simulation results of the quantum battery composed of two spins.
Figure 10: (a)-(b) describe the stored energy and ergotropy of the battery
with spin number of $N=2$ as functions of $\alpha$ and $\chi$, respectively.
Other parameters are $\eta=1$ and $J=h$. (c)-(d) are the same as (a)-(b), but
the spin number is $N=6$.
Fig. 11 further depicts the variation of effective space utilization rate and
$\mathcal{E}(\infty)/\Delta E(\infty)$ in the steady state with the number of
spin, under the three groups of optimal feedback conditions. These results can
reflect to some extent that our feedback scheme is independent of the number
of spins constituting the spin-chain quantum battery.
Figure 11: The value of $R(\infty)$ and $\mathcal{E}(\infty)/\Delta
E(\infty)$ of the battery with different spin number under the three groups of
optimal feedback parameters, where the measurement efficiency is $\eta=1$.
Figure 12: (a) The critical value $J_{c}$ of spin-spin interaction in energy
storage varies with the spin number. (b) The critical value $J_{c}$ of spin-
spin interaction in energy extraction (ergotropy) varies with the spin number.
Different curves correspond to different measuring efficiency $\eta$.
## Appendix D The charging process under imperfect measurement
Under imperfect measurement conditions ($\eta<1$), if the ground state of the
battery with $N=2$ is $\mid\downarrow\downarrow\rangle$ ($J<h/4$), the stored
energy of the battery in the steady state is
$\Delta
E_{1}(\infty)=\frac{2(h-2J)\vartheta(\eta+2\chi\eta\cos\alpha)}{2\chi^{2}+2\chi\eta\cos\alpha+\eta}+4(h-J)\vartheta^{2},$
(40)
where $\vartheta=\chi^{2}/(2\chi^{2}+2\chi\eta\cos\alpha+\eta)$. Since
$(h-2J)>0$ at this time, $\Delta E_{1}(\infty)$ can always be maximized by
selecting $\chi=1$, which means the feedback control is still valid under the
condition of imperfect measurement. While if the ground state of the system is
$(\mid\uparrow\downarrow\rangle-\mid\downarrow\uparrow\rangle)/\sqrt{2}$
($J>h/4$), the corresponding stored energy of the battery in the steady state
becomes
$\Delta E_{2}(\infty)=-h+4J+4J\vartheta^{2}+2(h-2J)\vartheta.$ (41)
The third and fourth terms in the r.h.s. of Eq. (41) are mutually restricted
for a large $J$, thus the feedback control may fail when $J$ exceeds a certain
value $J_{c}$ which can be determined as $J_{c}/h=(2-\eta)/2(1-\eta)$
according to $\Delta E_{2}(\infty)\mid_{(\chi)=1}=\Delta
E_{2}(\infty)\mid_{\chi=0}$.
In order to more intuitively reflect the influence of measurement efficiency
$(\eta)$ and spin number ($N$) on the critical coupling strength $J_{c}$, we
describe the variation of the critical value $J_{c}$ with the spin number
under different measurement efficiencies from the perspective of energy
storage and extraction in Figs. 12 (a) and 12 (b), respectively.
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|
# Lattice ground states for embedded-atom models in 2D and 3D
Laurent Bétermin Faculty of Mathematics, University of Vienna, Oskar-
Morgenstern-Platz 1, 1090 Vienna, Austria<EMAIL_ADDRESS>https://sites.google.com/site/homepagelaurentbetermin/ , Manuel Friedrich
Applied Mathematics, University of Münster, Einsteinstr. 62, D-48149 Münster,
Germany<EMAIL_ADDRESS>https://www.uni-
muenster.de/AMM/en/Friedrich/ and Ulisse Stefanelli Faculty of Mathematics,
University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria, Vienna
Research Platform on Accelerating Photoreaction Discovery, University of
Vienna, Währingerstraße 17, 1090 Wien, Austria, and Istituto di Matematica
Applicata e Tecnologie Informatiche E. Magenes \- CNR via Ferrata 1, 27100
Pavia, Italy<EMAIL_ADDRESS>http://www.mat.univie.ac.at/$\sim$stefanelli
###### Abstract.
The Embedded-Atom Model (EAM) provides a phenomenological description of
atomic arrangements in metallic systems. It consists of a configurational
energy depending on atomic positions and featuring the interplay of two-body
atomic interactions and nonlocal effects due to the corresponding electronic
clouds. The purpose of this paper is to mathematically investigate the
minimization of the EAM energy among lattices in two and three dimensions. We
present a suite of analytical and numerical results under different reference
choices for the underlying interaction potentials. In particular, Gaussian,
inverse-power, and Lennard-Jones-type interactions are addressed.
###### Key words and phrases:
Embedded-atom model, lattice energy minimization, Epstein zeta function.
###### 2010 Mathematics Subject Classification:
70G75, 74G65, 74N05
## 1\. Introduction
Understanding the structure of matter is a central scientific and
technological quest, cutting across disciplines and motivating an ever
increasing computational effort. First-principles calculations deliver
accurate predictions but are often impeded by the inherent quantum complexity,
as systems size up [30]. One is hence led to consider a range of
approximations. The minimization of empirical atomic pair-potentials
represents the simplest of such approximations being able to describe specific
properties of large-scaled atomic systems. Still, atomic pair-interactions
fall short of describing the basic nature of metallic bonding, which is multi-
body by nature, and often deliver inaccurate predictions of metallic systems.
The Embedded-Atom Model (EAM) is a semi-empirical, many-atom potential aiming
at describing the atomic structure of metallic systems by including a nonlocal
electronic correction. Introduced by Daw and Baskes [17], it has been used to
address efficiently different aspects inherent to atomic arrangements
including defects, dislocations, fracture, grain boundary structure and
energy, surface structure, and epitaxial growth. Proving capable of
reproducing experimental observations and being relatively simple to
implement, the Embedded-Atom Model is now routinely used in molecular dynamic
simulations [19, 28]. In particular, it has been applied in a variety of
metallic systems [22], including alkali metals Li, Na, K [20, 27, 38],
transition metals Fe, Ni, Cu, Pd, Ag, Pt, Au [14, 23, 27, 28], post-transition
metals Al, Pb [14, 25, 35], the metalloid Si [4], and some of their alloys
[14, 26].
In the case of a metallic system with a single atomic species, the EAM energy
is specified as
$\sum_{i}F(\overline{\rho}_{i})+\sum_{i\not=j}\phi(|x_{i}-x_{j}|)\quad\text{with}\quad\overline{\rho}_{i}=\sum_{j\not=i}\rho(|x_{i}-x_{j}|).$
Here, $\\{x_{i}\\}$ indicate atomic positions in ${\mathbb{R}}^{d}$ and the
long-range interaction potential
$\phi\colon\mathbb{R}_{+}:=(0,\infty)\to\mathbb{R}_{+}$ modulates atomic pair-
interactions. Atomic positions induce electronic-cloud distributions. The
function $\rho\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ models the long-range
electron-cloud contribution of an atom placed at $x_{j}$ on an atom placed at
$x_{i}$. The sum $\overline{\rho}_{i}$ describes the cumulative effect on the
atom placed at $x_{i}$ of the electronic clouds related to all other atoms.
Eventually, the function $F\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ describes
the energy needed to place (embed) an atom at position $x_{i}$ in the host
electron gas created by the other atoms at positions $\\{x_{j}\\}$.
Purely pair-interaction potentials can be re-obtained from the EAM model by
choosing $F=0$ and have been the subject of intense mathematical research
under different choices for $\phi$. The reader is referred to [13] for a
survey on the available mathematical results. The setting $F=0$ corresponds
indeed to the so-called Born-Oppenheimer approximation [30], which is well
adapted to the case of very low temperatures and is based on the subsequent
solution of the electronic and the atomic problem. As mentioned, this
approximation turns out to be not always appropriate for metallic systems at
finite temperatures [35, 8] and one is asked to tame the quantum nature of the
problem. This is however very challenging from the mathematical viewpoint and
rigorous optimality results for point configurations in the quantum setting
are scarce [12, 11]. The EAM model represents hence an intermediate model
between zero-temperature phenomenological pair-interaction energies and
quantum systems. Electronic effects are still determined by atomic positions,
but in a more realistic nonlocal fashion when $F$ is nonlinear, resulting in
truly multi-body interaction systems, see [17, 21, 35] and [19] for a review.
The aim of this paper is to investigate point configurations minimizing the
EAM energy. Being interested in periodic arrangements, we restrict our
analysis to the class of lattices, namely infinite configurations of the form
$L=\oplus_{i=1}^{d}\mathbb{Z}u_{i}$ where $\\{u_{i}\\}_{i=1}^{d}$ is a basis
of $\mathbb{R}^{d}$. This reduces the optimality problem to finite dimensions,
making it analytically and numerically amenable. In particular, the EAM
energy-per-atom of the lattice $L$ takes the specific form
$\mathcal{E}[L]=F\Big{(}\sum_{q\in
L\setminus\\{0\\}}\rho(|q|)\Big{)}+\sum_{q\in L\setminus\\{0\\}}\phi(|q|).$
In the classical pair-interaction case $F=0$, the lattice energy $\mathcal{E}$
has already received attention and a variety of results are available, see
[29, 32, 15, 5, 10, 9] and the references therein. Such results are of course
dependent on the choice of the potential $\phi$. Three reference choices for
$\phi$ are the Gaussian $\phi(r)=e^{-\pi\delta r^{2}}$ for $\delta>0$, the
inverse-power law $\phi(r)=r^{-s}$ for $s>d$, and the Lennard-Jones-type form
$\phi(r)=ar^{-\alpha}-br^{-\beta}$ for $d<\beta<\alpha$ and $a,\,b>0$. In the
Gaussian case, it has been shown by Montgomery [29] that, for all $\delta>0$,
the triangular lattice of unit density is the unique minimizer (up to
isometries) of $\mathcal{E}$ with $F=0$ among unit-density lattices. The same
can be checked for the the inverse-power-law case by a Mellin-transform
argument. More generally, the minimality of the triangular lattice of unit
density is conjectured by Cohn and Kumar in [15, Conjecture 9.4] to hold among
all unit-density periodic configurations. This fact is called universal
optimality and has been recently proved in dimension $8$ and $24$ for the
lattice $\mathsf{E}_{8}$ and the Leech lattice $\Lambda_{24}$, respectively
[16]. In the Lennard-Jones case, the minimality in 2d of the triangular
lattice at fixed density has been investigated in [11, 5], the minimality in
3d of the cubic lattice is proved in [7], and more general properties in
arbitrary dimensions have been investigated in [10]. A recap of the main
properties of the Lennard-Jones case is presented in Subsection 2.3. These
play a relevant role in our analysis.
In this paper, we focus on the general case $F\not=0$, when $F$ is nonlinear.
More precisely, we discuss the reference cases of embedding functions $F$ of
the form
$F(r)=r^{t}\log(\gamma r)\quad\text{or}\quad F(r)=r^{t}$
for $t,\,\gamma>0$. The first, logarithmic choice is the classical one chosen
to fit with the so-called Universal Binding Curve (see e.g., [31]) and
favoring a specific minimizing value $r_{0}>0$, see [2, 14]. The second,
power-law form favors on the contrary $r_{0}=0$ and allows for a particularly
effective computational approach. Let us mention that other choices for $F$
could be of interest. In particular, the form $F(r)=-c\sqrt{r}$, $c>0$, is
related to the Finnis-Sinclair model [21] and is discussed in Remark 4.3. Some
of our theory holds for general functions $F$, provided that they are
minimized at a sole value $r_{0}$. We call such functions of one-well type.
The electronic-cloud contribution function
$\rho\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ is assumed to be decreasing and
integrable. We specifically focus on the Gaussian and inverse-power law
$\rho(r)=e^{-\delta r^{2}}\quad\text{or}\quad\rho(r)=r^{-s}$
for $\delta>0$ and $s>d$, discussed, e.g., in [39] and [17, 21, 35],
respectively.
As for the pair-interaction potential
$\phi\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$, we assume a Lennard-Jones-type
form [3, 34] or an inverse-power law [17, 35], i.e.,
$\phi(r)=ar^{-\alpha}-br^{-\beta}\quad\text{or}\quad\phi(r)=r^{-\alpha}$
for $d<\beta<\alpha$ and $a,\,b>0$. Note that short-ranged potentials $\phi$
have been considered as well [17, 18].
Our main theoretical results amount at identifying minimizers in the specific
reference case of $F(r)=r\log r$ and $\rho(r)=r^{-s}$. More precisely, we find
the following:
* •
(Inverse-power law) If $\phi(r)=r^{-\alpha}$, the minimizers of $\mathcal{E}$
coincide with those of the Lennard-Jones potential $r\mapsto
r^{-\alpha}-r^{-s}$, up to rescaling (Theorem 4.1);
* •
(Lennard-Jones) If $\phi(r)=ar^{-\alpha}-br^{-\beta}$, under some
compatibility assumptions on the parameters, the minimizers of $\mathcal{E}$
coincide with those of the Lennard-Jones potential $r\mapsto
r^{-\alpha}-r^{-s}$ (Theorem 5.2).
Actually, both results hold for more general embedding functions $F$, see
(4.1) and Remarks 4.2–4.5. With this at hand, the problem can be reduced to
the pure Lennard-Jones case (i.e., $F=0$) which is already well understood. In
particular, in the two dimensional case we find that the triangular lattice,
up to rescaling and isometries, is the unique minimizer of $\mathcal{E}$ in
specific parameters regimes. These theoretical findings are illustrated by
numerical experiments in two and three dimensions. By alternatively choosing
the Gaussian $\rho(r)=e^{-\delta r^{2}}$, in two dimensions we additionally
observe the onset of a phase transition between the triangular and an
orthorhombic lattice, as $\delta$ decreases. In three dimensions, both in the
inverse-power-law case $\rho(r)=r^{-s}$ and in the Gaussian case
$\rho(r)=e^{-\delta r^{2}}$, the simple cubic lattice $\mathbb{Z}^{3}$ is
favored against the face-centered and the body-centered cubic lattice for $s$
or $\delta$ small, respectively.
In the power-law case $F(r)=r^{t}$, for $\rho$ of inverse-power-law type and
$\phi$ of Lennard-Jones type and specific, physically relevant choices of
parameters, one can conveniently reduce the complexity of the optimization
problem from the analytical standpoint. This reduction allows to explicitly
compute the EAM energy for any lattice of unit density, hence allowing to
investigate numerically minimality in two and three dimensions. Depending on
the parameters, the relative minimality of the triangular, square, and
orthorhombic lattices in two dimensions and the simple cubic, body-centered
cubic, and face-centered cubic lattices in three dimension is ascertained.
This is the plan of the paper: Notation on potentials and energies are
introduced in Subsections 2.1 and 2.2. The two subcases $F=0$ and $\phi=0$ are
discussed in Subsection 2.3 and in Section 3, respectively. In particular,
known results on Lennard-Jones-type interactions are recalled in Subsection
2.3. The inverse-power-law case $\phi(r)=r^{-\alpha}$ is investigated in
Section 4. The Lennard-Jones case $\phi(r)=ar^{-\alpha}-br^{-\beta}$ is
addressed theoretically and numerically in Section 5. In particular,
Subsection 5.1 contains the classical case $F(r)=r\log r$, and Subsection 5.2
discusses the power-law case $F(r)=r^{t}$.
## 2\. Notation and preliminaries
### 2.1. Lattices
For any dimension $d$, we write $\mathcal{L}_{d}$ for the set of all lattices
$L=\bigoplus_{i=1}^{d}\mathbb{Z}u_{i}$, where $\\{u_{i}\\}_{i=1}^{d}$ is a
basis of $\mathbb{R}^{d}$. We write $\mathcal{L}_{d}(1)\subset\mathcal{L}_{d}$
for the set of all lattices with unit density, which corresponds to
$|\det(u_{1},\ldots,u_{d})|=1$.
In dimension two, any lattice $L\in\mathcal{L}_{2}(1)$ can be written as
$L:=\mathbb{Z}\left(\frac{1}{\sqrt{y}},0\right)\oplus\mathbb{Z}\left(\frac{x}{\sqrt{y}},\sqrt{y}\right),$
for $(x,y)\in\mathcal{D}$, where
$\mathcal{D}=\big{\\{}(x,y)\in\mathbb{R}^{2}\,:\,0\leq x\leq
1/2,\,y>0;\,x^{2}+y^{2}\geq 1\big{\\}}$ (2.1)
is the so-called (half) fundamental domain for $\mathcal{L}_{2}(1)$ (see,
e.g., [29, Page 76]). In particular, the square lattice $\mathbb{Z}^{2}$ and
the triangular lattice with unit density, denoted by
$\mathsf{A}_{2}\in\mathcal{L}_{d}(1)$, are given by the respective choices
$(x,y)=(0,1)$ and $(x,y)=\left(1/2,{\sqrt{3}}/{2}\right)$, i.e.,
$\mathbb{Z}^{2}=\mathbb{Z}(1,0)\oplus\mathbb{Z}(0,1)\quad\text{ and
}\quad\mathsf{A}_{2}:=\sqrt{\frac{2}{\sqrt{3}}}\left[\mathbb{Z}(1,0)\oplus\mathbb{Z}\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\right].$
In dimension three, the fundamental domain of $\mathcal{L}_{3}(1)$ is much
more difficult to describe (see e.g., [36, Section 1.4.3]) and its
5-dimensional nature makes it impossible to plot compared to the 2-dimensional
$\mathcal{D}$ defined in (2.1). The Face-Centered Cubic (FCC) and Body-
Centered Cubic (BCC) lattices with unit density are respectively indicated by
$\mathsf{D}_{3}\in\mathcal{L}_{3}(1)$ and
$\mathsf{D}_{3}^{*}\in\mathcal{L}_{3}(1)$, and are defined as
$\displaystyle\mathsf{D}_{3}:=2^{-\frac{1}{3}}\left[\mathbb{Z}(1,0,1)\oplus\mathbb{Z}(0,1,1)\oplus\mathbb{Z}(1,1,0)\right];$
$\displaystyle\mathsf{D}_{3}^{*}:=2^{\frac{1}{3}}\left[\mathbb{Z}(1,0,0)\oplus\mathbb{Z}(0,1,0)\oplus\mathbb{Z}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right].$
###### Remark 2.1 (Periodic configurations).
All results in this paper are stated in terms of lattices, for the sake of
definiteness. Let us however point out that the same statements hold in the
more general setting of periodic configurations in dimensions
$d\in\\{8,24\\}$, on the basis of the recently proved optimality results from
[16]. In dimension $d=2$, universal optimality is only known among lattices,
see [29]. Still, the validity of the Cohn-Kumar conjecture (see [15,
Conjecture 9.4]) would allow us to consider more general periodic
configurations as well.
### 2.2. Potentials and energies
For any dimension $d$, let $\mathcal{S}_{d}$ be the set of all functions
$f\colon\mathbb{R}_{+}\to\mathbb{R}$ such that $|f(r)|={\rm O}(r^{-d-\eta})$
for some $\eta>0$ as $r\to\infty$. By
$\mathcal{S}_{d}^{+}\subset\mathcal{S}_{d}$ we denote the subset of
nonnegative functions. We say that a continuous function
$F\colon\mathbb{R}_{+}\to\mathbb{R}$ is a one-well potential if there exists
$r_{0}>0$ such that $F$ is decreasing on $(0,r_{0})$ and increasing on
$(r_{0},\infty)$.
For any $\phi\in\mathcal{S}_{d}$, we define the interaction energy
$E_{\phi}\colon\mathcal{L}_{d}\to\mathbb{R}$ by
$E_{\phi}[L]:=\sum_{q\in L\backslash\\{0\\}}\phi(|q|).$ (2.2)
If $\phi(r)=r^{-s}$, $s>d$, $E_{\phi}[L]$ actually corresponds to the Epstein
zeta function, which is defined by
$\displaystyle\zeta_{L}(s):=\sum_{q\in L\backslash\\{0\\}}\frac{1}{|q|^{s}}.$
(2.3)
For any function $F\colon\mathbb{R}_{+}\to\mathbb{R}$ and for any
$\rho\in\mathcal{S}_{d}^{+}$, we define the embedding energy
$E_{F,\rho}\colon\mathcal{L}_{d}\to\mathbb{R}$ by
$\displaystyle E_{F,\rho}[L]:=F(E_{\rho}[L])\quad\text{with}\quad
E_{\rho}[L]:=\sum_{q\in L\backslash\\{0\\}}\rho(|q|).$ (2.4)
Finally, for any $\phi\in\mathcal{S}_{d}$, any $\rho\in\mathcal{S}_{d}^{+}$,
and any $F\colon\mathbb{R}_{+}\to\mathbb{R}$, we define the _total energy_
$\mathcal{E}\colon\mathcal{L}_{d}\to\mathbb{R}$ by
$\displaystyle\mathcal{E}[L]:=E_{F,\rho}[L]+E_{\phi}[L]=F(E_{\rho}[L])+E_{\phi}[L].$
(2.5)
In the following, we investigate $\mathcal{E}$ under different choices of the
potentials $F$, $\rho$, and $\phi$. In some parts, we will require merely
abstract conditions on the potentials, such as a monotone decreasing $\rho$ or
a one-well potential $F$. In other parts, we will consider more specific
potentials. In particular, we will choose, for $\gamma,\delta,t,a,b>0$, $s>d$,
and $\alpha>\beta>d$,
$F(r)\in\\{r^{t},r^{t}\log(\gamma r)\\},\quad\rho(r)\in\\{r^{-s},e^{-\delta
r^{2}}\\},\quad\phi(r)\in\\{r^{-\alpha},ar^{-\alpha}-br^{-\beta}\\}.$
Note that the choice of $s$, $\delta$, $\alpha$, and $\beta$ implies that
$\phi\in\mathcal{S}_{d}$ and $\rho\in\mathcal{S}_{d}^{+}$, so that the sums in
(2.2) and (2.4) are well defined.
For any $L\in\mathcal{L}_{d}(1)$, any $\phi\in\mathcal{S}_{d}$, any
$\rho\in\mathcal{S}_{d}^{+}$, and any $F\colon\mathbb{R}_{+}\to\mathbb{R}$, we
define, if they uniquely exist, the following optimal scaling parameters for
the energies:
$\displaystyle\lambda^{\mathcal{E}}_{L}:=\mathop{\rm
argmin}\nolimits_{\lambda>0}\mathcal{E}[\lambda
L],\quad\lambda^{F,\rho}_{L}:=\mathop{\rm
argmin}\nolimits_{\lambda>0}E_{F,\rho}[\lambda
L],\quad\lambda^{\phi}_{L}:=\mathop{\rm
argmin}\nolimits_{\lambda>0}E_{\phi}[\lambda L].$ (2.6)
### 2.3. A recap on the Lennard-Jones-type energy
A classical problem is to study the $F=0$ case for a Lennard-Jones-type
potential
$\phi(r)=ar^{-\alpha}-br^{-\beta},\quad\alpha>\beta>d,\quad a,\,b>0.$ (2.7)
Let us recap some known facts in this case [5, 10], which will be used later
on. We start by reducing the minimization problem on _all_ lattices to a
minimization problem on lattices of _unit density only_. This is achieved by
computing the optimal scaling parameter of the energy $\lambda^{\phi}_{L}$ ,
see (2.6), for each $L\in\mathcal{L}_{d}(1)$, which in turn allows to find the
minimum of the energy among dilations of $L$. More precisely, in case (2.7),
for all $\lambda>0$ and all lattices $L\in\mathcal{L}_{d}(1)$, one has
$E_{\phi}[\lambda
L]=a\lambda^{-\alpha}\zeta_{L}(\alpha)-b\lambda^{-\beta}\zeta_{L}(\beta),$
where we use (2.3). (This energy was studied first in [5, Section 6.3].) Then,
we find the unique minimizer
$\displaystyle\lambda^{\phi}_{L}=\left(\frac{\alpha a\zeta_{L}(\alpha)}{\beta
b\zeta_{L}(\beta)}\right)^{\frac{1}{\alpha-\beta}},$ (2.8)
and therefore the energy is given by
$\min_{\lambda>0}E_{\phi}[\lambda
L]=E_{\phi}[\lambda^{\phi}_{L}L]=\frac{b^{\frac{\alpha}{\alpha-\beta}}\zeta_{L}(\beta)^{\frac{\alpha}{\alpha-\beta}}}{a^{\frac{\beta}{\alpha-\beta}}\zeta_{L}(\alpha)^{\frac{\beta}{\alpha-\beta}}}\left(\left(\frac{\beta}{\alpha}\right)^{\frac{\alpha}{\alpha-\beta}}-\left(\frac{\beta}{\alpha}\right)^{\frac{\beta}{\alpha-\beta}}\right)<0.$
The latter inequality follows from the fact that $\alpha>\beta$. Consequently,
for any lattices $L,\Lambda\in\mathcal{L}_{d}(1)$, we have that
$E_{\phi}[\lambda_{L}^{\phi}L]\leq
E_{\phi}[\lambda^{\phi}_{\Lambda}\Lambda]\iff\frac{\zeta_{L}(\alpha)^{\beta}}{\zeta_{L}(\beta)^{\alpha}}\leq\frac{\zeta_{\Lambda}(\alpha)^{\beta}}{\zeta_{\Lambda}(\beta)^{\alpha}}.$
This means that finding the lattice with minimal energy amounts to minimizing
the function
$\displaystyle L\mapsto
e^{*}(L):=\frac{\zeta_{L}(\alpha)^{\beta}}{\zeta_{L}(\beta)^{\alpha}}$ (2.9)
on $\mathcal{L}_{d}(1)$. This is particularly effective in dimension two where
for fixed $(\alpha,\beta)$ the minimizer can be found numerically by plotting
$L\mapsto\min_{\lambda}\mathcal{E}[\lambda L]$ in the fundamental domain
$\mathcal{D}$. Figure 1 shows the case $(\alpha,\beta)=(12,6)$, i.e., when
$\phi$ is the classical Lennard-Jones potential. The global minimum of
$E_{\phi}$ in $\mathcal{L}_{2}$ appears to be the triangular lattice
$\lambda^{\phi}_{\mathsf{A}_{2}}\mathsf{A}_{2}$.
For a certain range of parameters $(\alpha,\beta)$, this observation can be
rigorously ascertained. Indeed, for $d=2$, it is shown in [5, Theorem 1.2.B.]
that the global minimum of $E_{\phi}$ is uniquely achieved by a triangular
lattice $\lambda_{\mathsf{A}_{2}}^{\phi}\mathsf{A}_{2}$ if
$H(\alpha)<H(\beta),\quad\textnormal{where}\quad
H(t):=\frac{1}{2}\pi^{-t/2}\Gamma\left(\frac{t}{2}\right)t,$ (2.10)
and $\Gamma$ is the classical Gamma function
$\Gamma(r)=\int_{0}^{\infty}x^{r-1}e^{x}\,{\rm d}x$ for $r>0$. (In the sequel,
all statements on uniqueness are intended up to isometries, without further
notice.) In fact, under condition (2.10) one has that [5]
* •
$\mathsf{A}_{2}$ is the unique minimizer in $\mathcal{L}_{2}(1)$ of
$\displaystyle L\mapsto\lambda^{\phi}_{L}=\left(\frac{\alpha
a\zeta_{L}(\alpha)}{\beta b\zeta_{L}(\beta)}\right)^{\frac{1}{\alpha-\beta}}$,
* •
$\mathsf{A}_{2}$ is the unique minimizer in $\mathcal{L}_{2}(1)$ of $e^{*}$
defined in (2.9).
As pointed out in [5, Remark 6.18], it is necessary to choose
$2<\beta<\alpha<M\approx 9.2045818$ in order to obtain these optimality
results by using the method developed there. In particular, this means that
the following pairs of integer exponents can be chosen:
$(\alpha,\beta)\in\\{(4,3);(5,3);(6,3);(5,4);(6,4)\\}$. Note that the
classical Lennard-Jones potential $(\alpha,\beta)=(12,6)$ is not covered by
[5, Theorem 1.2.B.].
Figure 1. Contour plot of $L\mapsto
e^{*}(L)=\frac{\zeta_{L}(12)^{6}}{\zeta_{L}(6)^{12}}$ in the fundamental
domain $\mathcal{D}$. The triangular lattice $\mathsf{A}_{2}$ with coordinates
$(1/2,\sqrt{3}/2)$ appears to be the unique minimizer. Moreover,
$\mathbb{Z}^{2}$ with coordinates $(0,1)$ appears to be a saddle point.
We now ask ourselves what is the minimal scaling parameter $\lambda$ and the
corresponding lattice $L\in\mathcal{L}_{d}(1)$ for which $E_{\phi}[\lambda L]$
is minimized. Physically, this would correspond to identifying the first
minimum of $E_{\phi}$ starting from a high-density configuration by
progressively decreasing the density. We have the following.
###### Proposition 2.2 (Smallest volume meeting the global minimum).
Let $\phi$ be a Lennard-Jones-type potential as in (2.7). If
$L_{d}\in\mathcal{L}_{d}(1)$ is the minimizer of $L\mapsto\zeta_{L}(\beta)$ on
$\mathcal{L}_{d}(1)$ and $\lambda_{L_{d}}^{\phi}L_{d}$ is the unique global
minimizer of $E_{\phi}$ on $\mathcal{L}_{d}$, then $\lambda_{L_{d}}^{\phi}$ is
the unique minimizer of $L\mapsto\lambda^{\phi}_{L}$ on $\mathcal{L}_{d}(1)$.
###### Proof.
As discussed above, if $\lambda_{L_{d}}^{\phi}L_{d}$ is a global minimizer of
$E_{\phi}$ on $\mathcal{L}_{d}$, then $L_{d}$ minimizes the function $e^{*}$
defined in (2.9) on $\mathcal{L}_{d}(1)$. This yields
$\displaystyle\frac{\zeta_{L_{d}}(\alpha)^{\beta}}{\zeta_{L_{d}}(\beta)^{\alpha}}\leq\frac{\zeta_{L}(\alpha)^{\beta}}{\zeta_{L}(\beta)^{\alpha}}$
(2.11)
for all $L\in\mathcal{L}_{d}(1)$. We thus have
$\left(\frac{\zeta_{L}(\beta)\zeta_{L_{d}}(\alpha)}{\zeta_{L}(\alpha)\zeta_{L_{d}}(\beta)}\right)^{\beta}=\frac{\zeta_{L}(\beta)^{\alpha}\zeta_{L_{d}}(\alpha)^{\beta}}{\zeta_{L}(\alpha)^{\beta}\zeta_{L_{d}}(\beta)^{\alpha}}\left(\frac{\zeta_{L_{d}}(\beta)}{\zeta_{L}(\beta)}\right)^{\alpha-\beta}\leq\left(\frac{\zeta_{L_{d}}(\beta)}{\zeta_{L}(\beta)}\right)^{\alpha-\beta}.$
As we are assuming that $\zeta_{L_{d}}(\beta)\leq\zeta_{L}(\beta)$ for all
$L\in\mathcal{L}_{d}(1)$, we further get
$\frac{\zeta_{L}(\beta)\zeta_{L_{d}}(\alpha)}{\zeta_{L}(\alpha)\zeta_{L_{d}}(\beta)}\leq\left(\frac{\zeta_{L_{d}}(\beta)}{\zeta_{L}(\beta)}\right)^{\frac{\alpha-\beta}{\beta}}\leq
1,$ (2.12)
where we use that $\alpha>\beta$. In view of (2.8), this shows that
$\lambda^{\phi}_{L}\geq\lambda_{L_{d}}^{\phi}$ for all
$L\in\mathcal{L}_{d}(1)$. If $\lambda^{\phi}_{L}=\lambda_{L_{d}}^{\phi}$, then
we have a double equality in (2.12). This implies also equality in (2.11)
which is equivalent to $e^{*}(L)=e^{*}(L_{d})$. Therefore, it follows that
$L=L_{d}$ up to rotation, by uniqueness of the minimizer $L_{d}$ of $e^{*}$. ∎
Figure 2. Contour plot of
$L\mapsto(b/a)^{1/6}\lambda_{L}^{\phi}=\left(\frac{12\zeta_{L}(12)}{6\zeta_{L}(6)}\right)^{1/6}$,
see (2.8), in the fundamental domain $\mathcal{D}$. The triangular lattice
$\mathsf{A}_{2}$ with coordinates $(1/2,\sqrt{3}/2)$ is the unique minimizer
for any choice of $a,\,b>0$.
We refer to Figure 2 for an illustration in the two-dimensional case
$(\alpha,\beta)=(12,6)$. Note that in this case the global minimum is not
known. Still, the triangular lattice appears to be the first stable structure
reached by increasing the volume (decreasing the density). This is in
agreement with Figure 1 and Proposition 2.2. Recall that the triangular
lattice also minimizes $L\mapsto\zeta_{L}(\beta)$ on $\mathcal{L}_{d}(1)$, as
required in the statement of Proposition 2.2, see [29].
Notice that in dimension $d=3$ there is no rigorous result concerning the
minimizer of $E_{\phi}$ in $\mathcal{L}_{3}$. Only local minimality results
for cubic lattices $\mathbb{Z}^{3},\mathsf{D}_{3},\mathsf{D}_{3}^{*}$ have
been derived in [7]. Numerical investigations suggest that
$\lambda_{\mathsf{D}_{3}}^{\phi}\mathsf{D}_{3}$ is the unique minimizer of
$E_{\phi}$ in $\mathcal{L}_{3}$ for any values $\alpha>\beta>d$ of the
exponents, see, e.g., [40, Figure 5], [10, Figures 5 and 6] and [7, Conjecture
1.7]. Therefore, we can conjecture that $\mathsf{D}_{3}$ is the unique
minimizer of $L\mapsto\lambda_{L}^{\phi}$ in $\mathcal{L}_{3}(1)$ by
application of Proposition 2.2.
## 3\. Properties of the embedding energy $E_{F,\rho}$
In this section we focus on the properties of the embedding energy
$E_{F,\rho}$ given in (2.4). Although other choices for the potential $F$ may
been considered (see, e.g., [21, 19]), we concentrate ourselves on the one-
well case (see, e.g., [14] and references therein). In that case, it is clear
that the global minimum of $E_{F,\rho}$ in $\mathcal{L}_{d}$ can be achieved
for any $L$ by simply choosing $\lambda$ such that $E_{\rho}[\lambda
L]=r_{0}=\mathop{\rm argmin}\nolimits_{r>0}F(r)$. We now ask ourselves what is
the minimal scaling parameter $\lambda$ and the corresponding lattice
$L\in\mathcal{L}_{d}(1)$ for which $E_{F,\rho}[\lambda L]$ achieves $\min F$.
In other words, what is the minimizer of $L\mapsto\lambda^{F,\rho}_{L}$ in
$\mathcal{L}_{d}(1)$ (recall (2.6)). Physically, this would correspond to
reach the ground state of the embedding energy $\min F$ starting from a high-
density configuration by progressively decreasing the density.
###### Theorem 3.1 (Smallest volume meeting the global minimum).
Let $F\colon\mathbb{R}_{+}\to\mathbb{R}$ be a one-well potential and let
$\rho\in\mathcal{S}^{+}_{d}$ be strictly decreasing. Then,
$\lambda^{F,\rho}_{L}$ exists and $\min F$ is achieved by choosing
$\lambda^{F,\rho}_{L}L$ for all $L\in\mathcal{L}_{d}(1)$. Furthermore, if
$L_{d}$ is the unique minimizer in $\mathcal{L}_{d}(1)$ of $L\mapsto
E_{\rho}[\lambda_{L_{d}}^{F,\rho}L]$, then $L_{d}$ is the unique minimizer in
$\mathcal{L}_{d}(1)$ of $L\mapsto\lambda^{F,\rho}_{L}$.
###### Proof.
Let $r_{0}>0$ be the unique minimizer of $F$, namely, $F(r_{0})=\min F$. Given
any $L\in\mathcal{L}_{d}(1)$, the fact that $\rho\in\mathcal{S}_{d}^{+}$ is
strictly decreasing implies that $\lambda\mapsto E_{\rho}[\lambda L]$ is
strictly decreasing and goes to $0$ at infinity and to $\infty$ at $0$.
Therefore, there exists a unique $\lambda>0$ such that $E_{\rho}[\lambda
L]=r_{0}$. Such $\lambda$ obviously coincides with $\lambda_{L}^{F,\rho}$
given in (2.6). This shows the first part of the statement.
Suppose now that $L_{d}$ is the unique minimizer in $\mathcal{L}_{d}(1)$ of
$L\mapsto E_{\rho}[\lambda_{L_{d}}^{F,\rho}L]$. Assume by contradiction that
there exists $L\in\mathcal{L}_{d}(1)$, $L\neq L_{d}$, with
$\lambda^{F,\rho}_{L}\leq\lambda_{L_{d}}^{F,\rho}$. By using that
$\lambda\mapsto E_{\rho}[\lambda L]$ is decreasing, this would imply
$E_{\rho}[\lambda^{F,\rho}_{L}L]\geq
E_{\rho}[\lambda_{L_{d}}^{F,\rho}L]>E_{\rho}[\lambda_{L_{d}}^{F,\rho}L_{d}]=r_{0}=E_{\rho}[\lambda^{F,\rho}_{L}L],$
a contradiction. We thus deduce that
$\lambda_{L_{d}}^{F,\rho}\leq\lambda^{F,\rho}_{L}$ for all
$L\in\mathcal{L}_{d}(1)$, with equality if and only if $L=L_{d}$. ∎
We note that Theorem 3.1 can be applied to the choice $\rho(r)=r^{-s}$, $s>d$,
and the triangular lattice $\mathsf{A}_{2}$, the $\mathsf{E}_{8}$ lattice, or
the Leech lattice $\Lambda_{24}$ in dimensions $2$, $8$, and $24$,
respectively. In fact, these lattices are the unique minimizer of $L\mapsto
E_{\rho}[\lambda L]$ for all $\lambda>0$, see [29, 16].
Let us mention that, in this setting, asking $F$ to be one-well is not
restrictive. In fact, if $F$ is a strictly increasing (resp. decreasing)
function, no optimal scaling parameters $\lambda>0$ can be found since, for
any $L\in\mathcal{L}_{d}(1)$, $E_{F,\rho}[\lambda L]$ will be minimized for
$\lambda\to 0$ (resp. $\lambda\to\infty$).
## 4\. The EAM energy with inverse-power interaction $\phi(r)=r^{-\alpha}$
In this section, we study the energy $\mathcal{E}$ defined in (2.5) when
$\phi$ is given by the inverse-power interaction $\phi(r)=r^{-\alpha}$. The
main result of this section is the following.
###### Theorem 4.1 (EAM energy for inverse-power interaction).
For any $\alpha>s>d$, let $\rho(r)=r^{-s}$, let $\phi(r)=r^{-\alpha}$, and let
$F\in C^{1}(\mathbb{R}_{+})$. We assume that the functions
$\displaystyle g(r):=r^{1-{\alpha}/{s}}F^{\prime}(r)\quad\textnormal{and}\quad
h(r):=F(r)-\frac{s}{\alpha}rF^{\prime}(r)\quad\quad\text{for $r>0$}$ (4.1)
satisfy that $g$ is strictly increasing on $I:=\\{F^{\prime}<0\\}$, that
$g(I)=(-\infty,0)$, and that $h\circ g^{-1}$ is strictly decreasing on
$(-\infty,0)$. (Note that $g^{-1}$ exists on $(-\infty,0)$ and takes values in
$\mathbb{R}_{+}$.) Then, $\lambda^{\mathcal{E}}_{L}$ exists for all
$L\in\mathcal{L}_{d}(1)$ and the following statements are equivalent:
* •
$L_{d}$ is the unique minimizer in $\mathcal{L}_{d}(1)$ of $L\mapsto
e^{*}(L)=\frac{\zeta_{L}(\alpha)^{s}}{\zeta_{L}(s)^{\alpha}}$, see (2.9);
* •
$\lambda^{\mathcal{E}}_{L_{d}}L_{d}$ is the unique minimizer of $\mathcal{E}$
in $\mathcal{L}_{d}$;
* •
$\lambda_{L_{d}}^{\bar{\phi}}L_{d}$ is the unique minimizer in
$\mathcal{L}_{d}$ of $E_{\bar{\phi}}$ for $\bar{\phi}(r)=r^{-\alpha}-r^{-s}$,
see (2.7).
In particular, when $d=2$ and $H(\alpha)<H(s)$ where $H$ is defined by (2.10),
then the unique minimizer of $\mathcal{E}$ in $\mathcal{L}_{2}$ is the
triangular lattice $\lambda_{\mathsf{A}_{2}}^{\mathcal{E}}\mathsf{A}_{2}$.
Furthermore, if $L_{d}$ is the unique minimizer of $L\mapsto\zeta_{L}(s)$ in
$\mathcal{L}_{d}(1)$ as well as a minimizer of $e^{*}$ in
$\mathcal{L}_{d}(1)$, then $L_{d}$ is the unique minimizer of
$L\mapsto\lambda^{\mathcal{E}}_{L}$ in $\mathcal{L}_{d}(1)$, where
$\lambda^{\mathcal{E}}_{L}$ is defined in (2.6).
The gist of this result is the coincidence of the minimizers of $\mathcal{E}$
with those of $E_{\bar{\phi}}$ for $\bar{\phi}(r)=r^{-\alpha}-r^{-s}$ (up to
proper rescaling), under quite general choices of $F$. This results in a
simplification of the minimality problem for $\mathcal{E}$ as one reduces to
the study of minimality for the Lennard-Jones-type potential $\bar{\phi}$,
which is already well known, see Subsection 2.3. In particular, in two
dimensions and under condition $H(\alpha)<H(s)$, the unique minimizer is a
properly rescaled triangular lattice.
Before proving the theorem, let us present some applications to specific
choices of $F$.
###### Remark 4.2 (Application 1 - The classical case $F(r)=r\log r$).
We can apply this theorem to $F(r)=r^{t}\log(\gamma r)$ for $t\in(0,\alpha/s)$
and $\gamma>0$ which is a one-well potential with minimum attained at point
$r_{0}^{t}:=\frac{1}{\gamma}e^{-1/t}$. In particular, the case $F(r)=r\log r$
is admissible since $s<\alpha$. In fact, we have $I=(0,r_{0}^{t})$ and
$\displaystyle g(r)=r^{t-\alpha/s}\big{(}t\log(\gamma r)+1),\quad
g^{\prime}(r)=r^{t-\alpha/s-1}\left(\left(t-\frac{\alpha}{s}\right)(t\log(\gamma
r)+1)+t\right),$ $\displaystyle
h(r)=r^{t}\left(\left(1-\frac{ts}{\alpha}\right)\log(\gamma
r)-\frac{s}{\alpha}\right),\quad
h^{\prime}(r)=r^{t-1}\left(t\left(1-\frac{ts}{\alpha}\right)\log(\gamma
r)-\frac{2ts}{\alpha}+1\right).$
Since $g$ is strictly increasing on $(0,r_{1}^{t})$ for
$r_{1}^{t}:=\frac{1}{\gamma}e^{\frac{2ts-\alpha}{t(\alpha-ts)}}$ and
$r_{0}^{t}<r_{1}^{t}$ we have that $g$ is strictly increasing on $I$.
Moreover, $g(I)=(-\infty,0)$. On the other hand, $h$ is strictly decreasing on
$(0,r_{1}^{t})$. Therefore, also $h\circ g^{-1}$ is strictly decreasing on
$(-\infty,0)$. Hence, Theorem 4.1 applies.
###### Remark 4.3 (Application 2 - Finnis-Sinclair model).
Theorem 4.1 can also be applied to $F(r)=-c\sqrt{r}$ for $c>0$. This case is
known as the long-range Finnis-Sinclair model defined in [35], based on the
work of Finnis and Sinclair [21] on the description of cohesion in metals and
also used as a model to test the validity of machine-learning algorithms [24].
In this case, we obtain
$g(r)=-\frac{c}{2r^{\frac{\alpha}{s}-\frac{1}{2}}}\quad\textnormal{and}\quad
h(r)=c\sqrt{r}\left(\frac{s}{2\alpha}-1\right).$
Since $s<\alpha<2\alpha$, $g$ is strictly increasing on
$I=\\{F^{\prime}<0\\}=\mathbb{R}_{+}$, $g(I)=(-\infty,0)$, and $h$ is strictly
decreasing on $\mathbb{R}_{+}$. Therefore, Theorem 4.1 applies.
###### Remark 4.4 (Application 3 - inverse-power law).
Also the inverse-power law $F(r)=r^{-t}$ for $t>0$ satisfies the assumption of
the theorem. In fact, we have
$g(r)=-tr^{-t-{\alpha}/{s}}\quad\text{and}\quad
h(r)=\left(1+\frac{st}{\alpha}\right)r^{-t}.$
In particular, $g$ is strictly increasing on
$I=\\{F^{\prime}<0\\}=\mathbb{R}_{+}$ and $g(I)=(-\infty,0)$. Moreover, $h$ is
strictly decreasing on $\mathbb{R}_{+}$ and therefore also $h\circ g^{-1}$ is
strictly decreasing on $(-\infty,0)$.
###### Remark 4.5 (Application 4 - negative-logarithm).
We can apply Theorem 4.1 to the inverse-logarithmic case $F(r)=-\log r$.
Indeed, we compute
$g(r)=-r^{-\alpha/s}\quad\text{and}\quad h(r)=-\log r+\frac{s}{\alpha}.$
We hence have that $g$ is strictly increasing on
$I=\\{F^{\prime}<0\\}=\mathbb{R}_{+}$ and $g(I)=(-\infty,0)$. As $h$ is
strictly decreasing on $\mathbb{R}_{+}$, we have that $h\circ g^{-1}$ is
strictly decreasing on $(-\infty,0)$.
###### Proof of Theorem 4.1.
In view of (2.3) and (2.5), for any $\lambda>0$ and $L\in\mathcal{L}_{d}(1)$
we have that
$\mathcal{E}[\lambda
L]=F(\lambda^{-s}\zeta_{L}(s))+\lambda^{-\alpha}\zeta_{L}(\alpha).$
The critical points of $\lambda\mapsto\mathcal{E}[\lambda L]$ for fixed $L$
are the solutions of
$\partial_{\lambda}\mathcal{E}[\lambda
L]=-s\lambda^{-s-1}\zeta_{L}(s)\,F^{\prime}(\lambda^{-s}\zeta_{L}(s))-\alpha\lambda^{-\alpha-1}\zeta_{L}(\alpha)=0.$
(4.2)
This is equivalent to
$g(\lambda^{-s}\zeta_{L}(s))=-\frac{\alpha}{s}\frac{\zeta_{L}(\alpha)}{\zeta_{L}(s)^{\frac{\alpha}{s}}}=-\frac{\alpha}{s}e^{*}(L)^{\frac{1}{s}},\quad$
where $g$ is given in (4.1), and
$e^{*}(L)=\frac{\zeta_{L}(\alpha)^{s}}{\zeta_{L}(s)^{\alpha}}$ was defined in
(2.9). Since $g^{-1}$ is positive and strictly increasing on $(-\infty,0)$, we
have that the unique critical point is given by
$\displaystyle\lambda^{*}:=\left(\frac{\zeta_{L}(s)}{g^{-1}\left(-\frac{\alpha}{s}e^{*}(L)^{\frac{1}{s}}\right)}\right)^{\frac{1}{s}}.$
(4.3)
In view of (4.2), we also have that $\partial_{\lambda}\mathcal{E}[\lambda
L]\geq 0$ if and only if
$g(\lambda^{-s}\zeta_{L}(s))\leq-\frac{\alpha}{s}e^{*}(L)^{\frac{1}{s}}$,
which is equivalent to $\lambda\geq\lambda^{*}$. In particular,
$\lambda\mapsto\mathcal{E}[\lambda L]$ is decreasing on $(0,\lambda^{*})$ and
increasing on $(\lambda^{*},\infty)$. This shows that $\lambda^{*}$ is a
minimizer and thus $\lambda^{*}=\lambda^{\mathcal{E}}_{L}$, where
$\lambda^{\mathcal{E}}_{L}$ is defined in (2.6).
By using the fact that
$(\lambda^{\mathcal{E}}_{L})^{-\alpha}\zeta_{L}(\alpha)=-\frac{s}{\alpha}(\lambda^{\mathcal{E}}_{L})^{-s}\zeta_{L}(s)F^{\prime}((\lambda^{\mathcal{E}}_{L})^{-s}\zeta_{L}(s))$
from (4.2) and the identity $\lambda^{*}=\lambda^{\mathcal{E}}_{L}$, the
minimal energy among dilated copies $\lambda L$ of a given lattice $L$ can be
checked to be
$\displaystyle\mathcal{E}[\lambda^{\mathcal{E}}_{L}L]$
$\displaystyle=F\big{(}(\lambda^{\mathcal{E}}_{L})^{-s}\zeta_{L}(s)\big{)}+(\lambda^{\mathcal{E}}_{L})^{-\alpha}\zeta_{L}(\alpha)$
$\displaystyle=F\big{(}(\lambda^{\mathcal{E}}_{L})^{-s}\zeta_{L}(s)\big{)}-\frac{s}{\alpha}(\lambda^{\mathcal{E}}_{L})^{-s}\zeta_{L}(s)F^{\prime}\big{(}(\lambda^{\mathcal{E}}_{L})^{-s}\zeta_{L}(s)\big{)}$
$\displaystyle=h\big{(}(\lambda^{\mathcal{E}}_{L})^{-s}\zeta_{L}(s)\big{)}$
$\displaystyle=h\circ
g^{-1}\left(-\frac{\alpha}{s}e^{*}(L)^{\frac{1}{s}}\right),$
where $h$ is defined in (4.1). By assumption $h\circ g^{-1}$ is strictly
decreasing on $(-\infty,0)$. Hence, $L_{d}$ minimizes
$L\mapsto\mathcal{E}[\lambda^{\mathcal{E}}_{L}L]$ in $\mathcal{L}_{d}(1)$
(uniquely) if and only if $L_{d}$ minimizes $e^{*}$ (uniquely). This shows the
equivalence of the first two items in the statement. The equivalence to the
third item has already been addressed in the discussion before (2.9). The two-
dimensional case is a simple application of [5, Theorem 1.2.B.] which ensures
that $\mathsf{A}_{2}$ is the unique minimizer of $e^{*}$ in
$\mathcal{L}_{2}(1)$, as it has been already recalled in Subsection 2.3.
To complete the proof, it remains to show the final statement in $d$
dimensions. Assume that $L_{d}$ is the unique minimizer of
$L\mapsto\zeta_{L}(s)$ in $\mathcal{L}_{d}(1)$ as well as a minimizer of
$e^{*}$ in $\mathcal{L}_{d}(1)$. In this case, by using (4.3) and the identity
$\lambda^{*}=\lambda^{\mathcal{E}}_{L}$, it indeed follows that $L_{d}$ is the
unique minimizer of $L\mapsto\lambda^{\mathcal{E}}_{L}$ in
$\mathcal{L}_{d}(1)$, since $g^{-1}$ is positive and increasing on
$(-\infty,0)$. ∎
## 5\. The EAM energy with Lennard-Jones-type interaction
$\phi(r)=ar^{-\alpha}-br^{-\beta}$
We now move on to consider the full EAM energy $\mathcal{E}$ defined in (2.5)
for Lennard-Jones-type potentials $\phi$ as in (2.7). We split this section
into two parts. At first, we address the classical case $F(r)=r\log r$
analytically and numerically. Afterwards, we provide some further numerical
studies for the power law case $F(r)=r^{t}$.
### 5.1. The classical case $F(r)=r\log r$
We start with two theoretical results and then proceed with several numerical
investigations.
#### 5.1.1. Two theoretical results
The following corollary is a straightforward application of Theorem 3.1.
###### Corollary 5.1 (Existence of parameters for the optimality of
$\mathsf{A}_{2}$).
Let
$F(r)=r^{t}\log(\gamma
r),\quad\rho(r)=r^{-s},\quad\textnormal{and}\quad\phi(r)=ar^{-\alpha}-br^{-\beta},$
for $\gamma,t>0$, $s>2$, $\alpha>\beta>2$, and $a,b>0$. Then, given parameters
$(\alpha,\beta,\gamma,s,{t})$ such that $H(\alpha)<H(\beta)$, where $H$ is
defined in (2.10), one can find coefficients $a$ and $b$ such that the unique
global minimizer in $\mathcal{L}_{2}$ of $\mathcal{E}$ is the triangular
lattice $\lambda_{\mathsf{A}_{2}}\mathsf{A}_{2}$ where
$\lambda_{\mathsf{A}_{2}}=e^{\frac{t^{-1}+\log\gamma}{s}}\zeta_{\mathsf{A}_{2}}(s)^{\frac{1}{s}}.$
Moreover, $\mathsf{A}_{2}$ is the unique minimizer of
$L\mapsto\lambda^{\mathcal{E}}_{L}$ in $\mathcal{L}_{2}(1)$.
###### Proof.
We first remark that $F$ and $\rho$ satisfy the assumption of Theorem 3.1. By
recalling (2.3), (2.6) and using the fact that $\mathop{\rm argmin}\nolimits
F=\frac{1}{\gamma}e^{-1/t}$, we have
$\lambda_{\mathsf{A}_{2}}^{F,\rho}=e^{\frac{t^{-1}+\log\gamma}{s}}\zeta_{\mathsf{A}_{2}}(s)^{\frac{1}{s}},$
and
$E_{F,\rho}[\lambda_{\mathsf{A}_{2}}^{F,\rho}\mathsf{A}_{2}]=F((\lambda_{\mathsf{A}_{2}}^{F,\rho})^{-s}\zeta_{\mathsf{A}_{2}}(s))=\min
F$. On the other hand, we know from [5, Theorem 1.2] that $E_{\phi}$ is
uniquely minimized in $\mathcal{L}_{2}$ by
$\lambda_{\mathsf{A}_{2}}^{\phi}\mathsf{A}_{2}$ where
$\lambda_{\mathsf{A}_{2}}^{\phi}=\left(\frac{\alpha
a\zeta_{\mathsf{A}_{2}}(\alpha)}{\beta
b\zeta_{\mathsf{A}_{2}}(\beta)}\right)^{\frac{1}{\alpha-\beta}},$
see (2.8). Hence, if
$\lambda_{\mathsf{A}_{2}}^{F,\rho}=\lambda_{\mathsf{A}_{2}}^{\phi}$, then
$\lambda_{\mathsf{A}_{2}}^{F,\rho}\mathsf{A}_{2}=\lambda_{\mathsf{A}_{2}}^{\phi}\mathsf{A}_{2}$
is the unique minimizer of the sum of the two energies $E_{F,\rho}$ and
$E_{\phi}$. The identity
$\lambda_{\mathsf{A}_{2}}^{F,\rho}=\lambda_{\mathsf{A}_{2}}^{\phi}$ is
equivalent to equation
$\frac{a}{b}=\frac{\beta\zeta_{\mathsf{A}_{2}}(\beta)}{\alpha\zeta_{\mathsf{A}_{2}}(\alpha)}\zeta_{\mathsf{A}_{2}}(s)^{\frac{\alpha-\beta}{s}}e^{\frac{\alpha-\beta}{s}(t^{-1}+\log\gamma)}.$
For this choice of $a$ and $b$, we thus get that the unique global minimizer
in $\mathcal{L}_{2}$ of $\mathcal{E}$ is the triangular lattice
$\lambda^{\mathcal{E}}_{\mathsf{A}_{2}}\mathsf{A}_{2}$ with
$\lambda^{\mathcal{E}}_{\mathsf{A}_{2}}=\lambda_{\mathsf{A}_{2}}^{F,\rho}=\lambda_{\mathsf{A}_{2}}^{\phi}$.
The last statement follows by applying Proposition 2.2 to
$L_{d}=\mathsf{A}_{2}$. ∎
The drawback of the result is that it is not _generic_ in the sense that it
holds only for specific coefficients $a$ and $b$. We now give a result which
holds in any dimension for _all_ coefficients $a,\,b>0$, at the expense of the
fact that $\phi$ and $\rho$ need to have the same decay ${\rm O}(r^{-s})$. In
this regard, the result is in the spirit Theorem 4.1 but under the choice
$\phi(r)=ar^{-\alpha}-br^{-s}$.
###### Theorem 5.2 (EAM energy for Lennard-Jones-type interaction).
Let $F$ be as in Theorem 4.1 and additionally suppose that $F$ is convex and
in $C^{2}(\mathbb{R}_{+})$. Let
$\rho(r)=r^{-s},\quad\phi(r)=ar^{-\alpha}-br^{-s},\quad\text{ for }\
d<s<\alpha\quad\text{and}\quad a,\,b>0.$
Then, $\lambda^{\mathcal{E}}_{L}$ exists for all $L\in\mathcal{L}_{d}(1)$ and
the following statements are equivalent:
* •
$L_{d}$ is the unique minimizer of $L\mapsto
e^{*}(L)=\frac{\zeta_{L}(\alpha)^{s}}{\zeta_{L}(s)^{\alpha}}$, see (2.9);
* •
$\lambda_{L_{d}}^{\mathcal{E}}L_{d}$ is the unique minimizer of $\mathcal{E}$
in $\mathcal{L}_{d}$;
* •
$\lambda_{L_{d}}^{\phi}L_{d}$ is the unique minimizer in $\mathcal{L}_{d}$ of
$E_{{\phi}}$.
In particular, when $d=2$ and $H(\alpha)<H(s)$ where $H$ is defined by (2.10),
then the unique minimizer of $\mathcal{E}$ in $\mathcal{L}_{2}$ is the
triangular lattice $\lambda_{\mathsf{A}_{2}}^{\mathcal{E}}\mathsf{A}_{2}$.
Furthermore, if $L_{d}$ is the unique minimizer of $L\mapsto\zeta_{L}(s)$ in
$\mathcal{L}_{d}(1)$ as well as a minimizer of $e^{*}$ in
$\mathcal{L}_{d}(1)$, then $L_{d}$ is the unique minimizer of
$L\mapsto\lambda^{\mathcal{E}}_{L}$ in $\mathcal{L}_{d}(1)$.
###### Proof.
In view of (2.3), the energy $\mathcal{E}$ can be written as
$\mathcal{E}[L]=F(\zeta_{L}(s))+a\zeta_{L}(\alpha)-b\zeta_{L}(s)=a\big{(}\tilde{F}(\zeta_{L}(s))+\zeta_{L}(\alpha)\big{)},$
where $\tilde{F}(r)=a^{-1}(F(r)-br)$. In a similar fashion to (4.1), we define
$\tilde{g}(r):=r^{1-{\alpha}/{s}}\tilde{F}^{\prime}(r)=a^{-1}g(r)-\frac{b}{a}r^{1-{\alpha}/{s}},\quad\tilde{h}(r):=\tilde{F}(r)-\frac{s}{\alpha}r\tilde{F}^{\prime}(r)=a^{-1}h(r)-\frac{b}{a}\Big{(}1-\frac{s}{\alpha}\Big{)}r,$
where $g$ and $h$ are defined in (4.1). We first check that $\tilde{g}$ is
strictly increasing on $\tilde{I}:=\\{\tilde{F}^{\prime}<0\\}$. Indeed, since
$F$ (and hence $\tilde{F}$) is convex and $\alpha>s$, we get that
$\tilde{g}^{\prime}(r)=\left(1-\frac{\alpha}{s}\right)r^{-{\alpha}/{s}}\tilde{F}^{\prime}(r)+r^{1-{\alpha}/{s}}\tilde{F}^{\prime\prime}(r)\geq\left(1-\frac{\alpha}{s}\right)r^{-{\alpha}/{s}}\tilde{F}^{\prime}(r)>0$
for all $r\in\tilde{I}$. Since by assumption
$g(\\{F^{\prime}<0\\})=(-\infty,0)$ and
$\tilde{I}=\\{\tilde{F}^{\prime}<0\\}\supset\\{{F}^{\prime}<0\\}$, we find
$\tilde{g}(\tilde{I})=(-\infty,0)$. Eventually, $\tilde{h}\circ\tilde{g}^{-1}$
is strictly decreasing on $(-\infty,0)$, as well. We can hence apply Theorem
4.1 and obtain the assertion. ∎
###### Remark 5.3.
As a consequence of Remark 4.2, the previous result can be applied to
$F(r)=r\log r$. Already for this $F$, in the case of a more general Lennard-
Jones potential $\phi(r)=ar^{-\alpha}-br^{-\beta}$, the equation for the
critical points of $\lambda\mapsto\mathcal{E}[\lambda L]$ for a fixed lattice
$L$ is
$\log\lambda=\frac{a^{\prime}}{b^{\prime}}\lambda^{s-\alpha}-\frac{d^{\prime}}{b^{\prime}}\lambda^{s-\beta}+\frac{c^{\prime}}{b^{\prime}}$
for $a^{\prime}=\alpha a\zeta_{L}(\alpha)$, $b^{\prime}=s^{2}\zeta_{L}(s)$,
$c^{\prime}=s\zeta_{L}(s)(1+\log\zeta_{L}(s))$, and $d^{\prime}=\beta
b\zeta_{L}(\beta)$. This is generically not solvable in closed form when
$s\neq\beta$, and makes the computation of
$\mathcal{E}[\lambda^{\mathcal{E}}_{L}L]$ more difficult. This is why we
choose $s=\beta$ in the above result.
#### 5.1.2. Numerical investigation in 2d
We choose $s$ as parameter and fix $t=\gamma=a=b=1$, and $\alpha=12$,
$\beta=6$, i.e.,
$F(r)=r\log
r,\quad\rho(r)=r^{-s},\quad\phi(r)=\frac{1}{r^{12}}-\frac{1}{r^{6}}.$ (5.1)
We employ here a gradient descent method, which is rather computationally
intensive. Note that a more efficient numerical method will be amenable in
Subsection 5.2, as an effect of a different structure of the potentials.
Numerically, we observe the following (see Figure 3):
* •
For $s>s_{1}$, $s_{1}\approx 5.14$, the triangular lattice
$\lambda_{\mathsf{A}_{2}}^{\mathcal{E}}\mathsf{A}_{2}$ is apparently the
unique global minimizer of $\mathcal{E}$.
* •
For $s<s_{1}$, the energy does not seem to have a global minimizer.
Furthermore, for $s>s_{0}$, $s_{0}\approx 5.09$, we have checked (see Figure
4) that
$\min_{\lambda}\mathcal{E}[\lambda\mathbb{Z}^{2}]=\mathcal{E}[\lambda_{\mathbb{Z}^{2}}^{\mathcal{E}}\mathbb{Z}^{2}]>\mathcal{E}[\lambda_{\mathsf{A}_{2}}^{\mathcal{E}}\mathsf{A}_{2}]=\min_{\lambda}\mathcal{E}[\lambda\mathsf{A}_{2}],$
whereas the inequality is reversed if $s<s_{0}$.
Figure 3. Case (5.1) in two dimensions. Plot of
$L\mapsto\min_{\lambda}\mathcal{E}[\lambda L]$ in the fundamental domain
$\mathcal{D}$ for $s=6$ (up left), $s=5.3$ (up right), $s=5.15$ (down left)
and $s=5.07$ (down right). Figure 4. Case (5.1) in two dimensions. Plot of
$s\mapsto\min_{\lambda}\mathcal{E}[\lambda\mathbb{Z}^{2}]-\min_{\lambda}\mathcal{E}[\lambda\mathsf{A}_{2}]$
for $s\in[2.1,7]$.
We now replace $\rho$ by a Gaussian function. Namely, we consider the case
$F(r)=r\log r,\quad\rho(r)=e^{-\delta
r^{2}},\quad\phi(r)=\frac{1}{r^{12}}-\frac{1}{r^{6}}.$ (5.2)
Figure 5. Case (5.2) in two dimensions. Plot of
$\delta\mapsto\min_{\lambda}\mathcal{E}[\lambda\mathbb{Z}^{2}]-\min_{\lambda}\mathcal{E}[\lambda\mathsf{A}_{2}]$
for $\delta\in[0.1,5]$.
In this case, the triangular lattice
$\lambda_{\mathsf{A}_{2}}^{\mathcal{E}}\mathsf{A}_{2}$ still seems to be
minimizing $\mathcal{E}$ for large $\delta$, see Figure 6. More precisely:
* •
There exists $\delta_{0}\approx 1.04$ such that, for $\delta>\delta_{0}$, the
triangular lattice $\lambda_{\mathsf{A}_{2}}^{\mathcal{E}}\mathsf{A}_{2}$ is
the global minimizer of $\mathcal{E}$ in $\mathcal{L}_{2}$.
* •
For $\delta<\delta_{0}$, the global minimizer of $\mathcal{E}$ seems to move
(continuously) in $\mathcal{D}$ increasingly following the $y$-axis as
$\delta$ decreases to $0$. For instance,
* –
If $\delta=1$, then the minimizer is $(0,y_{1})$ where $y_{1}\approx 1.014$.
* –
If $\delta=0.95$, then the minimizer is $(0,y_{0.95})$ where $y_{0.95}\approx
1.665$.
* •
Furthermore, we have checked that, for $\delta>\delta_{0}$,
$\min_{\lambda}\mathcal{E}[\lambda\mathbb{Z}^{2}]=\mathcal{E}[\lambda_{\mathbb{Z}^{2}}^{\mathcal{E}}\mathbb{Z}^{2}]>\mathcal{E}[\lambda_{\mathsf{A}_{2}}^{\mathcal{E}}\mathsf{A}_{2}]=\min_{\lambda}\mathcal{E}[\lambda\mathsf{A}_{2}],$
whereas the inequality is reversed if $\delta<\delta_{0}$ (see Figure 5).
Figure 6. Case (5.2) in two dimensions. Plot of
$L\mapsto\min_{\lambda}\mathcal{E}[\lambda L]$ when $\delta=2$ (up left),
$\delta=1$ (up right) and $\delta=0.95$ (down left and right) in the
fundamental domain $\mathcal{D}$.
#### 5.1.3. Numerical investigation in 3d
Let us go back to case (5.1), now in three dimensions. We investigate the
difference of energies between the Simple Cubic (SC), Face-Centered Cubic
(FCC), and Body-Centered Cubic (BCC) lattices, namely,
$\mathbb{Z}^{3},\mathsf{D}_{3},\mathsf{D}_{3}^{*}$, as $s$ increases. Examples
of FCC and BCC metals are Al, Cu, Ag, Au, Ni, Pd, Pt, and Nb, Cr, V, Fe,
respectively [37]. Po is the only metal crystallizing in a SC structure [33].
Before giving our numerical results, let us remark that the lattices
$\mathbb{Z}^{3}$, $\mathsf{D}_{3}$, and $\mathsf{D}_{3}^{*}$ are critical
points of $\mathcal{E}$ in $\mathcal{L}_{3}(1)$. Moreover, recall the
following conjectures:
* •
Sarnak-Strombergsson’s conjecture (see [32, Equation (44)]): for all $s\geq
3/2$ (and in particular for $s>3$, so that $r\mapsto
r^{-s}\in\mathcal{S}_{3}^{+}$), $\mathsf{D}_{3}$ is the unique minimizer of
$L\mapsto\zeta_{L}(s)$ in $\mathcal{L}_{3}(1)$.
* •
The global minimizer of the Lennard-Jones energy $E_{\phi}$ is
$\lambda_{\mathsf{D}_{3}}^{\phi}\mathsf{D}_{3}$ (see e.g. [40, Figure 5] and
[7, Conjecture 1.7]).
We have numerically studied the following function
$s\mapsto\min_{\lambda>0}\mathcal{E}[\lambda L],\quad
L\in\\{\mathsf{D}_{3},\mathsf{D}_{3}^{*},\mathbb{Z}^{3}\\}$
for $s>3$, see Figure 7. We have found that there exist $s_{0}<s_{1}<s_{2}$
where $s_{0}\approx 5.4985$, $s_{1}\approx 5.576$, and $s_{2}\approx 5.584$
such that
* •
For $s\in(3,s_{0})$,
$\min_{\lambda>0}\mathcal{E}[\lambda\mathbb{Z}^{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}^{*}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}]$;
* •
For $s\in(s_{0},s_{1})$,
$\min_{\lambda>0}\mathcal{E}[\lambda\mathbb{Z}^{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}^{*}]$;
* •
For $s\in(s_{1},s_{2})$,
$\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathbb{Z}^{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}^{*}]$;
* •
For $s>s_{2}$,
$\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}^{*}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathbb{Z}^{3}]$.
It is remarkable that for small values of $s$ the simple cubic lattice
$\mathbb{Z}^{3}$ has lower energy with respect to the usually energetically
favored $\mathsf{D}_{3}$ and $\mathsf{D}_{3}^{*}$.
Figure 7. Case (5.1) in three dimensions. Plots of
$s\mapsto\min_{\lambda}\mathcal{E}[\lambda L]$ for $L=\mathsf{D}_{3}$ (red),
$L=\mathsf{D}_{3}^{*}$ (blue) and $L=\mathbb{Z}^{3}$ (black) on two different
intervals.
Consider now the Gaussian case (5.2) in three dimensions. The total energy
then reads
$\mathcal{E}[L]:=\theta_{L}(\delta)\log\theta_{L}(\delta)+\zeta_{L}(12)-\zeta_{L}(6),\quad\textnormal{where}\quad\theta_{L}(\delta):=\sum_{p\in
L\backslash\\{0\\}}e^{-\delta|p|^{2}}.$
In the following, we will call $\theta_{L}(\delta)$ the lattice theta function
with parameter $\delta>0$. Note however that under this name one usually
refers to such sum including the term for $p=0$ and with weight
$e^{-\delta\pi|p|^{2}}$.
We recall the following conjectures:
* •
Sarnak-Strombergsson’s conjecture (see [32, Equation (43)]): if $\delta<\pi$,
then $\mathsf{D}_{3}^{*}$ minimizes $L\mapsto\theta_{L}(\delta)$ in
$\mathcal{L}_{3}(1)$. If $\delta>\pi$, then $\mathsf{D}_{3}$ minimizes the
same lattice theta function in $\mathcal{L}_{3}(1)$ (with a coexistence phase
around $\pi$ actually).
* •
As mentioned before, the unique minimizer of the Lennard-Jones energy
$E_{\phi}$ in $\mathcal{L}_{3}$ is
$\lambda_{\mathsf{D}_{3}}^{\phi}\mathsf{D}_{3}$ (see e.g. [7] and [40, Figure
5]).
In Figure 8 we plot the functions
$\delta\mapsto\min_{\lambda>0}\mathcal{E}[\lambda L]$ for
$L\in\\{\mathsf{D}_{3},\mathsf{D}_{3}^{*},\mathbb{Z}^{3}\\}$. We numerically
observe that there exist $0<\delta_{1}<\delta_{2}<\delta_{3}$, where
$\delta_{1}\approx 1.13$, $\delta_{2}\approx 1.21$, and $\delta_{3}\approx
1.223$ such that
* •
for all $\delta\in(0,\delta_{1})$,
$\min_{\lambda>0}\mathcal{E}[\lambda\mathbb{Z}^{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}^{*}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}]$;
* •
for all $\delta\in(\delta_{1},\delta_{2})$,
$\min_{\lambda>0}\mathcal{E}[\lambda\mathbb{Z}^{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}^{*}]$;
* •
for all $\delta\in(\delta_{2},\delta_{3})$,
$\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathbb{Z}^{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}^{*}]$;
* •
for all $\delta>\delta_{3}$,
$\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathsf{D}_{3}^{*}]<\min_{\lambda>0}\mathcal{E}[\lambda\mathbb{Z}^{3}]$.
It is indeed important that the EAM energy favors $\mathsf{D}_{3}$ or
$\mathsf{D}_{3}^{*}$ for some specific choice of parameters. In fact, FCC and
BCC lattices are commonly emerging in metals. It is also remarkable that the
simple cubic lattice $\mathbb{Z}^{3}$ (up to rescaling) is favored with
respect to $\mathsf{D}_{3}$ or $\mathsf{D}_{3}^{*}$ for some other choice of
parameters. In [7], we were able to identify a range of densities such that
cubic lattices are locally optimal at fixed density, but it is the first time
– according to our knowledge – that such phenomenon is observed at the level
of the global minimizer.
Figure 8. Case (5.2) in three dimensions. Plot of
$\delta\mapsto\min_{\lambda}\mathcal{E}[\lambda L]$ for $L=\mathsf{D}_{3}$
(red), $L=\mathsf{D}_{3}^{*}$ (blue) and $L=\mathbb{Z}^{3}$ (black) on two
different intervals.
### 5.2. The power-law case $F(r)=r^{t}$
In this subsection, we study the case where $F(r)=r^{t}$, $t>0$. Although $F$
is not a one-well potential, this case turns out to be mathematically
interesting. Indeed, we are able to present a special case where we can
explicitly compute $\min_{\lambda}\mathcal{E}[\lambda L]$ for any
$L\in\mathcal{L}_{d}(1)$. As we have seen above, this dimension reduction is
extremely helpful when one looks for the ground state of $\mathcal{E}$ in
$\mathcal{L}_{d}$, especially for $d=2$, since we can plot
$L\mapsto\min_{\lambda}\mathcal{E}[\lambda L]$ in the fundamental domain
$\mathcal{D}$.
#### 5.2.1. A special power-law case
Let us now assume that
$F(r)=r^{t},\quad\rho(r)=r^{-s},\quad\phi(r)=ar^{-\alpha}-br^{-\beta},$
for $t>0$, $s>d$, $\alpha>\beta>d$, and $a,\,b>0$. Therefore, by (2.3) we
have, for any $\lambda>0$ and any $L\in\mathcal{L}_{d}(1)$, that
$\mathcal{E}[\lambda
L]=\lambda^{-st}\zeta_{L}(s)^{t}+a\lambda^{-\alpha}\zeta_{L}(\alpha)-b\lambda^{-\beta}\zeta_{L}(\beta).$
For a fixed lattice $L$, the critical points of
$\lambda\mapsto\mathcal{E}[\lambda L]$ are the solutions of the following
equation
$\displaystyle
b\beta\zeta_{L}(\beta)\lambda^{st+\alpha}-st\zeta_{L}(s)^{t}\lambda^{\alpha+\beta}-a\alpha\zeta_{L}(\alpha)\lambda^{st+\beta}=0.$
(5.3)
Solving this equation in closed form is impracticable out of a discrete set of
parameter values. Correspondingly, comparing energy values is even more
complicated than in the pure Lennard-Jones-type case, which is already
challenging when treated in whole generality.
Having pointed out this difficulty, we now focus on some additional
specifications of the parameters, allowing to proceed further with the
analysis. We have the following.
###### Theorem 5.4 (Special power-law case).
Let $\alpha,\beta,s$, and $t$ such that
$\displaystyle d<s,\quad
d<\beta<st<\alpha,\quad\textnormal{and}\quad\alpha=2st-\beta.$ (5.4)
Then, $\lambda^{\mathcal{E}}_{L}$ exists for all $L\in\mathcal{L}_{d}(1)$.
Moreover, $\lambda^{\mathcal{E}}_{L_{d}}L_{d}$ is a global minimizer in
$\mathcal{L}_{d}$ of $\mathcal{E}$, now reading
$\mathcal{E}[L]=\zeta_{L}(s)^{t}+a\zeta_{L}(\alpha)-b\zeta_{L}(\beta),$
if and only if $L_{d}$ is a minimizer in $\mathcal{L}_{d}(1)$ of
$\displaystyle e_{*}(L):$ $\displaystyle=-\frac{\displaystyle
C_{1}\frac{\zeta_{L}(s)^{2t}}{\zeta_{L}(\beta)}+C_{2}\zeta_{L}(s)^{t}\sqrt{c_{1}\frac{\zeta_{L}(s)^{2t}}{\zeta_{L}(\beta)^{2}}+c_{2}\frac{\zeta_{L}(\alpha)}{\zeta_{L}(\beta)}}+C_{3}\zeta_{L}(\alpha)}{\displaystyle\left(\sqrt{c_{1}}\frac{\zeta_{L}(s)^{t}}{\zeta_{L}(\beta)}+\sqrt{c_{1}\frac{\zeta_{L}(s)^{2t}}{\zeta_{L}(\beta)^{2}}+c_{2}\frac{\zeta_{L}(\alpha)}{\zeta_{L}(\beta)}}\right)^{\frac{\alpha}{\alpha-
st}}},$
where $C_{i},c_{j}$, $i\in\\{1,2,3\\}$, $j\in\\{1,2\\}$, are positive
constants defined by
$C_{1}:=\frac{st}{2b\beta}\left(\frac{st}{\beta}-1\right),\quad
C_{2}:=\frac{st}{\beta}-1,\quad
C_{3}:=a\left(\frac{\alpha}{\beta}-1\right),\quad
c_{1}:=\frac{s^{2}t^{2}}{4b^{2}\beta^{2}},\quad
c_{2}:=\frac{a\alpha}{b\beta}.$ (5.5)
###### Proof.
For any $L\in\mathcal{L}_{d}(1)$, any critical point of
$\lambda\mapsto\mathcal{E}[\lambda L]$ satisfies (see (5.3))
$\lambda^{st+\beta}\left(b\beta\zeta_{L}(\beta)\lambda^{\alpha-\beta}-st\zeta_{L}(s)^{t}\lambda^{\alpha-
st}-a\alpha\zeta_{L}(\alpha)\right)=0.$
Since $\lambda>0$, by writing $X=\lambda^{\alpha-st}$ and using (5.4) we want
to solve
$b\beta\zeta_{L}(\beta)X^{2}-st\zeta_{L}(s)^{t}X-a\alpha\zeta_{L}(\alpha)=0,\quad
X>0,$
for which the unique solution is
$X=\frac{st\zeta_{L}(s)^{t}+\sqrt{s^{2}t^{2}\zeta_{L}(s)^{2t}+4ab\alpha\beta\zeta_{L}(\alpha)\zeta_{L}(\beta)}}{2b\beta\zeta_{L}(\beta)}.$
Since $\alpha-st>0$ and $b\beta\zeta_{L}(\beta)>0$, we find that the critical
point is a minimizer and thus coincides with $\lambda^{\mathcal{E}}_{L}$
defined in (2.6). More precisely, we have
$\lambda^{\mathcal{E}}_{L}=\left(\frac{st\zeta_{L}(s)^{t}+\sqrt{s^{2}t^{2}\zeta_{L}(s)^{2t}+4ab\alpha\beta\zeta_{L}(\alpha)\zeta_{L}(\beta)}}{2b\beta\zeta_{L}(\beta)}\right)^{\frac{1}{\alpha-
st}}.$
We hence get, for any $L\in\mathcal{L}_{d}(1)$, that
$\displaystyle\min_{\lambda}\mathcal{E}[\lambda
L]=\mathcal{E}[\lambda^{\mathcal{E}}_{L}L]$
$\displaystyle=(\lambda^{\mathcal{E}}_{L})^{-st}\zeta_{L}(s)^{t}+a(\lambda^{\mathcal{E}}_{L})^{-\alpha}\zeta_{L}(\alpha)-b(\lambda^{\mathcal{E}}_{L})^{-\beta}\zeta_{L}(\beta)$
$\displaystyle=(\lambda^{\mathcal{E}}_{L})^{-\alpha}\left\\{\zeta_{L}(s)^{t}(\lambda^{\mathcal{E}}_{L})^{\alpha-
st}-b\zeta_{L}(\beta)(\lambda^{\mathcal{E}}_{L})^{\alpha-\beta}+a\zeta_{L}(\alpha)\right\\}$
$\displaystyle=(\lambda^{\mathcal{E}}_{L})^{-\alpha}\left\\{\zeta_{L}(s)^{t}(\lambda^{\mathcal{E}}_{L})^{\alpha-
st}-\frac{st\zeta_{L}(s)^{t}(\lambda^{\mathcal{E}}_{L})^{\alpha-
st}+a\alpha\zeta_{L}(\alpha)}{\beta}+a\zeta_{L}(\alpha)\right\\}$
$\displaystyle=(\lambda^{\mathcal{E}}_{L})^{-\alpha}\left\\{\zeta_{L}(s)^{t}\left(1-\frac{st}{\beta}\right)(\lambda^{\mathcal{E}}_{L})^{\alpha-
st}+a\zeta_{L}(\alpha)\left(1-\frac{\alpha}{\beta}\right)\right\\}$
$\displaystyle=(\lambda^{\mathcal{E}}_{L})^{-\alpha}\left\\{\zeta_{L}(s)^{t}\left(1-\frac{st}{\beta}\right)\left(\frac{st\zeta_{L}(s)^{t}+\sqrt{s^{2}t^{2}\zeta_{L}(s)^{2t}+4ab\alpha\beta\zeta_{L}(\alpha)\zeta_{L}(\beta)}}{2b\beta\zeta_{L}(\beta)}\right)+a\zeta_{L}(\alpha)\left(1-\frac{\alpha}{\beta}\right)\right\\}$
$\displaystyle=(\lambda^{\mathcal{E}}_{L})^{-\alpha}\left\\{\frac{st}{2b\beta}\left(1-\frac{st}{\beta}\right)\frac{\zeta_{L}(s)^{2t}}{\zeta_{L}(\beta)}+\left(1-\frac{st}{\beta}\right)\zeta_{L}(s)^{t}\sqrt{\frac{s^{2}t^{2}\zeta_{L}(s)^{2t}}{4b^{2}\beta^{2}\zeta_{L}(\beta)^{2}}+\frac{a\alpha\zeta_{L}(\alpha)}{b\beta\zeta_{L}(\beta)}}+a\left(1-\frac{\alpha}{\beta}\right)\zeta_{L}(\alpha)\right\\},$
where in the fourth line we have used the fact that
$\lambda^{\mathcal{E}}_{L}$ is a critical point of
$\lambda\mapsto\mathcal{E}[\lambda L]$, i.e.,
$b\beta\zeta_{L}(\beta)(\lambda^{\mathcal{E}}_{L})^{\alpha-\beta}-st\zeta_{L}(s)^{t}(\lambda^{\mathcal{E}}_{L})^{\alpha-
st}-a\alpha\zeta_{L}(\alpha)=0$. Note that by assumption we have
$1-\frac{st}{\beta}<0,\quad 1-\frac{\alpha}{\beta}<0.$
It follows that, defining the positive constants $C_{i},c_{j}$,
$i\in\\{1,2,3\\}$, $j\in\\{1,2\\}$, as in (5.5), that
$\displaystyle\min_{\lambda}\mathcal{E}[\lambda L]$
$\displaystyle=-(\lambda^{\mathcal{E}}_{L})^{-\alpha}\left\\{C_{1}\frac{\zeta_{L}(s)^{2t}}{\zeta_{L}(\beta)}+C_{2}\zeta_{L}(s)^{t}\sqrt{c_{1}\frac{\zeta_{L}(s)^{2t}}{\zeta_{L}(\beta)^{2}}+c_{2}\frac{\zeta_{L}(\alpha)}{\zeta_{L}(\beta)}}+C_{3}\zeta_{L}(\alpha)\right\\}$
$\displaystyle=-\frac{\displaystyle
C_{1}\frac{\zeta_{L}(s)^{2t}}{\zeta_{L}(\beta)}+C_{2}\zeta_{L}(s)^{t}\sqrt{c_{1}\frac{\zeta_{L}(s)^{2t}}{\zeta_{L}(\beta)^{2}}+c_{2}\frac{\zeta_{L}(\alpha)}{\zeta_{L}(\beta)}}+C_{3}\zeta_{L}(\alpha)}{\displaystyle\left(\sqrt{c_{1}}\frac{\zeta_{L}(s)^{t}}{\zeta_{L}(\beta)}+\sqrt{c_{1}\frac{\zeta_{L}(s)^{2t}}{\zeta_{L}(\beta)^{2}}+c_{2}\frac{\zeta_{L}(\alpha)}{\zeta_{L}(\beta)}}\right)^{\frac{\alpha}{\alpha-
st}}},$
which completes the proof. ∎
#### 5.2.2. Numerical investigations of the special power-law case in 2d and
3d
We let $t\in(0,9/d)$ vary and fix
$a=b=1,\quad\alpha=12,\quad\beta=6,\quad s=9/t,$
so that
$F(r)=r^{t},\quad\rho(r)=r^{-{9}/{t}},\quad\phi(r)=\frac{1}{r^{12}}-\frac{1}{r^{6}}.$
(5.6)
Note that (5.4) holds under these assumptions. In two dimensions, by testing
as $t\in(0,4.5)$ increases, we observe numerically the following:
* •
If $t\in(0,t_{1})$, $t_{1}\approx 1.605$, then $\mathsf{A}_{2}$ minimizes
$e_{*}$ (see Figures 9 and 10);
* •
If $t\in(t_{1},t_{2})$, where $t_{2}\approx 1.633$, then $\mathbb{Z}^{2}$ is a
local minimizer of $e_{*}$ but there seems to be no minimizer for $e_{*}$ (see
Figure 11);
* •
if $t\in(t_{2},4.5)$, there seems to be no minimizer for $e_{*}$, and
$\mathbb{Z}^{2}$ is a saddle point (see Figure 12).
Figure 9. Special power-law case (5.6) in two dimensions. Plot of $e_{*}$ on
the fundamental domain $\mathcal{D}$. For $t=1$ (left) and $t=1.5$ (right),
the minimizer of $e_{*}$ is the triangular lattice $\mathsf{A}_{2}$ given by
the point $(1/2,\sqrt{3}/2)$. Figure 10. Special power-law case (5.6) in two
dimensions. Plot of $e_{*}$ on the fundamental domain $\mathcal{D}$ for
$t=1.605$. The triangular lattice is the global minimizer of $e_{*}$ whereas
$\mathbb{Z}^{2}$ (given by the point $(0,1)$) is a local minimizer.
Similarly to the discussion of Subsection 5.1, for some choice of parameters,
a square lattice seems to be locally minimizing the EAM energy, at least
within the range of our numerical testing. In [6], we have identified a range
of densities for which a square lattice is optimal at fixed density. This
seems however to be the first occurrence of such minimality among all possible
lattices, without a density constraint. Indeed, when minimizing among all
lattices, the square lattice $\mathbb{Z}^{2}$ usually happens to be a saddle
point, see, e.g., Figure 1 for the Lennard-Jones case.
Figure 11. Special power-law case (5.6) in two dimensions. Plot of $e_{*}$ on
the fundamental domain $\mathcal{D}$ for $t=1.606$. The square lattice is a
local minimizer of $e_{*}$ which does not have any global minimizer. Still,
$\mathsf{A}_{2}$ is a local minimizer.
Figure 12. Special power-law case (5.6) in two dimensions. Plot of $e_{*}$ on
the fundamental domain $\mathcal{D}$. For $t=1.632$ (left) the square lattice
is a local minimizer of $e_{*}$ whereas $\mathsf{A}_{2}$ is a local maximizer.
For $t=2$ (right) it seems that $e_{*}$ does not have any local minimizer and
$\mathsf{A}_{2}$ stays a local maximizer. In both cases, there is no global
minimum.
We have numerically investigated the three-dimensional case as well, comparing
the energies of $L\in\\{\mathbb{Z}^{3},\mathsf{D}_{3},\mathsf{D}_{3}^{*}\\}$.
Figure 13 illustrates the numerical results. We observe that there exist
$t_{1},t_{2},t_{3}$, where $t_{1}\approx 1.5505$, $t_{2}\approx 1.5515$, and
$t_{3}\approx 1.5647$ such that:
* •
If $t\in(0,t_{1})$,
$e_{*}(\mathsf{D}_{3})<e_{*}(\mathsf{D}_{3}^{*})<e_{*}(\mathbb{Z}^{3})$;
* •
If $t\in(t_{1},t_{2})$,
$e_{*}(\mathsf{D}_{3})<e_{*}(\mathbb{Z}^{3})<e_{*}(\mathsf{D}_{3}^{*})$;
* •
If $t\in(t_{2},t_{3})$,
$e_{*}(\mathbb{Z}^{3})<e_{*}(\mathsf{D}_{3})<e_{*}(\mathsf{D}_{3}^{*})$;
* •
If $t\in(t_{3},3)$,
$e_{*}(\mathbb{Z}^{3})<e_{*}(\mathsf{D}_{3}^{*})<e_{*}(\mathsf{D}_{3})$.
When $t\to 0$, since $s=9/t\to\infty$ and $r^{t}\to 1$ for fixed $r>0$, it is
expected that the global minimizer of $\mathcal{E}$ in $\mathcal{L}_{3}$
converges to the one of $E_{\phi}$, which in turn is expected to be a FCC
lattice. This is supported by our numerics for $t<t_{1}$.
Figure 13. Special power-law case (5.6) in three dimensions. Plot of $t\mapsto
e_{*}(L)$ for $L=\mathsf{D}_{3}$ (red), $L=\mathsf{D}_{3}^{*}$ (blue) and
$L=\mathbb{Z}^{3}$ (black) for $t\in(0,3)$. The graph on the right is a close-
up of the two transitions at $t_{1}$ and $t_{2}$.
## Acknowledgments
MF and US are supported by the DFG-FWF international joint project FR
4083/3-1/I 4354. MF is also supported by the Deutsche Forschungsgemeinschaft
under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster:
Dynamics–Geometry–Structure. US and LB are supported by the FWF project F 65.
US is also supported by the FWF project P 32788.
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|
# Quasiperfect graph
Veronica Phan
###### Abstract.
A perfect graph is a graph which every induced subgraph has clique number
equal to chromatic number. In this paper, I will introduce a new family of
graphs, the quasiperfect graphs which generalizes the perfect graphs.
*Ho Chi Minh City; email<EMAIL_ADDRESS>
## 1\. introduction
All graphs in this paper are finite graphs. Let a graph $G=(V;E)$.
$\omega(G),\alpha(G),\chi(G)$ be the clique number, independent number,
chromatic number of $G$, respectively, $\overline{G}$ be the complement of
$G$, $G[S]$ be the subgraph induced in $G$ by $S$, $K_{n}$ be the complete
graph of order $n$, $K_{0}$ be the trivial graph.
A perfect graph is a graph which every induced subgraphs has clique number
equal to chromatic number [1, 2]. They are important objects in graph theory.
In this paper, I will introduce a new family of graphs, the quasiperfect
graphs which generalizes the perfect graph and show the quasiperfect graphs
also share some properties with the perfect graphs.
###### Definition 1.
The definition of quasiperfect graph is inductive. Let $K_{0}$ be
quasiperfect. Let a graph $G=(V,E)$, assume we define a graph is quasiperfect
or not for all graph have less vertices than $G$. $G$ is quasiperfect if and
only if:
$-$ There exist an independent set $PI$ such that it intersects all maximum
cliques, each vertex of $PI$ contained in a maximum clique and $G[V-PI]$ is
quasiperfect.
$-$ There exist a clique $PK$ such that it intersects all maximum independent
sets, each vertex of $PK$ contained in a maximum independent set and $G[V-PK]$
is quasiperfect..
We call the set $PI,PK$ as above prime independent set and prime clique
respectively.
## 2\. The properties of quasiperfect graphs
We want the quasiperfect graphs have the most basic property: clique number
equals chromatic number.
###### Theorem 1.
A quasiperfect graph $G=(V,E)$ has clique number equals chromatic number.
###### Proof.
We’ll prove by induction.
It’s trivial for $G=K_{0}$. Assume the statement is true for all induced
quasiperfect subgraph of $G$. We need to prove $G$ has clique number equals
chromatic number.
Let $PI$ be a prime independent set of $G$. Then we have $G[V-PI]$ is
quasiperfect, so by induction hypothesis, $\omega(G[V-I])=\chi(G[V-PI])$. We
have $PI$ intersects all maximum cliques, and for each cliques, there as most
$1$ vertex in $PI$ because $PI$ is independent, so
$\omega(G)=\omega(G[V-PI])+1$. We have a $\omega(G)$-coloring of $G$ by taking
a $\omega(G[V-PI])$-coloring of $G[V-PI]$, and color all vertices in $PI$ with
a new color. So $G$ clique number equals chromatic number. ∎
This theorem is the analog of _Weakly perfect graph theorem_ :
###### Theorem 2.
The complement of a quasiperfect graph $G=(V,E)$ is also quasiperfect.
###### Proof.
We’ll prove by induction.
It’s trivial for $G=K_{0}$. Assume the statement is true for all induced
quasiperfect subgraph of $G$. We need to prove $\overline{G}$ is quasiperfect.
The complement of a graph turn independent set into clique and vice versa. So
we easily see that prime independent set of $G$ is prime clique of
$\overline{G}$ and vice versa. So by Definition 1, you need to prove that
$\overline{G[V-PI]},\overline{G[V-PC]}$ is quasiperfect, which is true by
induction hypothesis and Definition 1. ∎
So we see that the quasiperfect graph have the clique number equal the
chromatic number, the complement of a quasiperfect graph is also quasiperfect
graph. They are also basic properties of perfect graphs as we want.
## 3\. Some families of graphs that are quasiperfect
We will show the quasiperfect graphs is the non-trivial generation of the
perfect graphs
### 3.1. The perfect graphs
We also hope that the perfect graphs is quasiperfect. Because all induced
subgraphs of the perfect graph is also perfect, we just need to proof there
exists a prime clique and prime independent set of perfect graph and the
result follow by induction. By taking complement, we just need to find a prime
clique. We inspired by the proof of _Weakly perfect graph theorem_ by Lovász
[3, 4].
Let $G$ be a perfect graph. For each vertex $v$ of $G$, we replace it by a
clique with $t_{v}$ vertices to create a new graph $G^{\prime}$, $t_{v}$ is
the number of maximum independent set contains $v$, if $t_{v}=0$, we just
delete $v$. Because all induced subgraphs of a perfect graph is also perfect
and replace a vertex by a clique create a new perfect graph so $G^{\prime}$ is
perfect, so it have clique number equal chromatic number. By the construction
of $G^{\prime}$, we can correspond each vertex of $G^{\prime}$ to a maximum
independent set of $G$, so we have a $I$-coloring of $G^{\prime}$ with $I$ is
the number of maximum independent set of $G$. We see that the set of all
vertices colored by the same color is a maximum independent set of
$G^{\prime}$, so this is an optimal coloring, so $\omega(G)=\chi(G)=I$. We
take a maximum clique $K^{\prime}$ of size $I$ in $G^{\prime}$, then
$K^{\prime}$ intersects all maximum independent set of $G^{\prime}$. Let $PK$
be the set of all vertices $v$ of $G$ such that there is a vertex
$v^{\prime}\in K^{\prime}$ created by $v$, then we easily have $PK$ is a prime
clique of $G$.
So the quasiperfect graphs is a generation of the perfect graphs. We can use
ideas from perfect graph to work with quasiperfect graph.
### 3.2. Graphs created from odd cycles
Now we create an imperfect graph which is quasiperfect. The easiest way is
creating from an odd cycles $C_{n}$ with $n\geq 5$. The chromatic number of
$C_{n}$ is $3$, so we need to add cliques of size $3$. Let
$v_{1}=v_{n+1},v_{2},...,v_{n}$ be vertices of $C_{n}$, with $v_{i},v_{i+1}$
are joined. We add new vertices $w_{k_{1}},w_{k_{2}},...,w_{k_{o}},1\leq
k_{p}\leq n$ and join $w_{k_{p}}$ with $v_{k_{p}},v_{k_{p}+1}$ to create a new
graph $G=(V,E)$.
We can choose prime clique $PK=\\{w_{k_{1}},v_{k_{1}},v_{k_{1}+1}\\}$ then
it’s easy to see that $G[V-PK]$ is a block graph (a graph which every
biconnected component is a clique), so is perfect and also quasiperfect.
Now we choose prime independent set $PI$. There are two cases:
$-$ If $o=n=2m+1$, we choose $PI=\\{w_{n},v_{2},v_{4},...,v_{2m}\\}$. It’s
easy to see that $PI$ is a prime independent set, and $G[V-PI]$ is a forest,
so is perfect and also quasiperfect.
$-$ If $o<n$, we taking the set $P=\\{v_{k_{1}},v_{k_{2}},...,v_{k_{o}}\\}$,
if there exists $l$ such that $v_{l-1},v_{l},v_{l+1}\in P$ and $v_{l},v_{l+1}$
are joined but $v_{l},v_{l-1}$ are not, delete $v_{l}$ from the set, in the
end we have a prime independent set $PI$ (because $o<n$). It’s easy to see
that $G[V-PI]$ is a forest, so is perfect and also quasiperfect.
So $G$ is quasiperfect and the quasiperfect graphs is a non-trivial generation
of the perfect graphs.
## 4\. Further remarks
The trick of replacing a vertex by a clique doesn’t work for quasiperfect
case. For example, we add a vertex to $C_{5}$ and join it to 2 vertices that
are joined, then replace all vertices of $C_{5}$ by large enough cliques of
the same size, it is easy to see that the result graph doesn’t have clique
number equal chromatic number. And we can proof that the trick only work for
perfect graph, so we can’t hope for any better generation.
It’s a natural question to ask which induced subgraphs of a quasiperfect graph
is quasiperfect? In the coloring in Theorem 1, we see that the prime
independent set is a set of all vertices have the same color, so we can hope
that, give a minimum coloring and a color $c$, remove all the vertices have
color $C$ from the quasiperfect creats a new quasiperfect graph.
There are some families of graphs base on algebraic objects have clique number
equal chromatic number such as the enhanced power graphs of groups, the
annihilating-ideal graphs of commutative rings [5, 6, 7]. We hope that those
families of graphs are also quaiperfect.
I’ve showed above that there are quasiperfect graphs that have odd cycle
$C_{n}$ with $n\geq 5$ as their induced subgraph. It’s a natural question to
ask for arbitrary graph $G=(V,E)$, is there a quasiperfect graphs
$G^{\prime}=(V^{\prime},E^{\prime})$ that has $G$ as its induced graph, and
for fixed value of $|V|$ how smallest $|V^{\prime}|$ can be?
It’s likely that the properties ”quasiperfect” is completely global, so by
research on the quasiperfect graphs (or even classification), we can learn
more about global property of the perfect graphs and other families of graphs
that are quasiperfect.
## 5\. acknowledgement
I would like to thank Professor Peter J. Cameron for endorsing me to submit
this paper.
## References
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|
# The Fact Selection Problem in
LLM-Based Program Repair
Nikhil Parasaram University College London
<EMAIL_ADDRESS>Huijie Yan† University College London
<EMAIL_ADDRESS>Boyu Yang† University College London
<EMAIL_ADDRESS>Zineb Flahy University College London
<EMAIL_ADDRESS>Abriele Qudsi University College London
<EMAIL_ADDRESS>Damian Ziaber University College London
<EMAIL_ADDRESS>Earl T. Barr University College London
<EMAIL_ADDRESS>Sergey Mechtaev University College London
<EMAIL_ADDRESS>
###### Abstract
Recent research has shown that incorporating bug-related facts, such as stack
traces and GitHub issues, into prompts enhances the bug-fixing capabilities of
large language models (LLMs). Considering the ever-increasing context window
of these models, a critical question arises: what and how many facts should be
included in prompts to maximise the chance of correctly fixing bugs? To answer
this question, we conducted a large-scale study, employing over 19K prompts
featuring various combinations of seven diverse facts to rectify 314 bugs from
open-source Python projects within the BugsInPy benchmark. Our findings
revealed that each fact, ranging from simple syntactic details like code
context to semantic information previously unexplored in the context of LLMs
such as angelic values, is beneficial. Specifically, each fact aids in fixing
some bugs that would remain unresolved or only be fixed with a low success
rate without it. Importantly, we discovered that the effectiveness of program
repair prompts is non-monotonic over the number of used facts; using too many
facts leads to subpar outcomes. These insights led us to define the fact
selection problem: determining the optimal set of facts for inclusion in a
prompt to maximise LLM’s performance on a given task instance. We found that
there is no one-size-fits-all set of facts for bug repair. Therefore, we
developed a basic statistical model, named Maniple, which selects facts
specific to a given bug to include in the prompt. This model significantly
surpasses the performance of the best generic fact set. To underscore the
significance of the fact selection problem, we benchmarked Maniple against the
state-of-the-art zero-shot, non-conversational LLM-based bug repair methods.
On our testing dataset of 157 bugs, Maniple repairs 88 bugs, 17% above the
best configuration.
###### Index Terms:
automated program repair, large language models, prompt engineering
${}^{\dagger}$${}^{\dagger}$footnotetext: These authors contributed equally to
this work.
## I Conclusion
In this paper, we explore the construction of effective prompts for LLM-based
APR, leveraging facts extracted from the buggy program and external sources.
Through a systematic investigation, we incorporate seven bug-relevant facts
into the prompts, notably including angelic values, a factor previously not
considered in this domain. Furthermore, we define the fact selection problem
and demonstrate that a universally optimal set of facts for addressing various
bugs does not exist. Building on this insight, we devise a bug-tailored fact
selection strategy enhancing the effectiveness of APR.
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|
# Predicting Individualized Effects of Internet-Based Treatment for Genito-
Pelvic Pain/Penetration Disorder: Development and Internal Validation of a
Multivariable Decision Tree Model
Anna-Carlotta Zarski
Friedrich-Alexander-Universität
Erlangen-Nürnberg &
Technical University of Munich
<EMAIL_ADDRESS>
&Mathias Harrer
Technical University of Munich &
Friedrich-Alexander-Universität
Erlangen-Nürnberg
<EMAIL_ADDRESS>
Paula Kuper
Otto-von-Guericke University Magdeburg &
Technical University of Munich
<EMAIL_ADDRESS>
&Antonia A. Sprenger
Otto-von-Guericke University Magdeburg &
Technical University of Munich
<EMAIL_ADDRESS>
Matthias Berking
Friedrich-Alexander-Universität
Erlangen-Nürnberg
<EMAIL_ADDRESS>
&David Daniel Ebert
Technical University of Munich
<EMAIL_ADDRESS>
###### Abstract
Genito-Pelvic Pain/Penetration-Disorder (GPPPD) is a common disorder but
rarely treated in routine care. Previous research documents that GPPPD
symptoms can be treated effectively using internet-based psychological
interventions. However, non-response remains common for all state-of-the-art
treatments and it is unclear which patient groups are expected to benefit most
from an internet-based intervention. Multivariable prediction models are
increasingly used to identify predictors of heterogeneous treatment effects,
and to allocate treatments with the greatest expected benefits. In this study,
we developed and internally validated a multivariable decision tree model that
predicts effects of an internet-based treatment on a multidimensional
composite score of GPPPD symptoms. Data of a randomized controlled trial
comparing the internet-based intervention to a waitlist control group
($N$=200) was used to develop a decision tree model using model-based
recursive partitioning. Model performance was assessed by examining the
apparent and bootstrap bias-corrected performance. The final pruned decision
tree consisted of one splitting variable, joint dyadic coping, based on which
two response clusters emerged. No effect was found for patients with low
dyadic coping ($n$=33; $d$=0.12; 95% CI: -0.57-0.80), while large effects
($d$=1.00; 95%CI: 0.68-1.32; $n$=167) are predicted for those with high dyadic
coping at baseline. The bootstrap-bias-corrected performance of the model was
$R^{2}$=27.74% (RMSE=13.22).
_K_ eywords Genito-Pelvic Pain/Penetration Disorder $\cdot$ Heterogeneity of
Treatment Effects $\cdot$ Clinical Prediction Model
## 1 Introduction
Genito-Pelvic Pain/Penetration-Disorder (GPPPD) is a sexual dysfunction
previously known as dyspareunia and vaginismus, that can be defined by genito-
pelvic pain, vaginal intercourse difficulties, pelvic floor muscle tension,
and fear of pain or vaginal penetration (American Psychiatric Association,
2013). Overlapping conditions include Vulvodynia and Provoked Vestibulodynia
(Bergeron et al., 2015). GPPPD adversely affects women’s sexuality, quality of
life, physical and mental health and relationships (Arnold et al., 2006;
Khandker et al., 2011; Pâquet et al., 2016; Thomtén, 2014). GPPPD symptoms are
experienced by 20.8% (1%-72%) of premenopausal women in the general population
(McCool et al., 2016) with a 12-month prevalence rate ranging from 4.9 to 10.9
depending on its associated impairment (Briken et al., 2020). Multiple
biopsychosocial factors influence the etiology of GPPPD with psychological
factors such as fear avoidance and pain catastrophizing playing an important
role in its maintenance (Thomtén et al., 2014; Thomtén & Linton, 2013).
Psychological interventions have been shown to be effective in reducing
genito-pelvic pain and associated distress in vaginismus and dyspareunia and
enabling sexual intercourse (Flanagan et al., 2015; Maseroli et al., 2018).
However, there is a lack of willingness to participate in psychological
interventions due to limited availability, fear of stigmatization, and
feelings of shame (Bergvall & Himelein, 2014; Bond et al., 2015; Donaldson &
Meana, 2011). Internet-based treatment approaches are considered particularly
suitable for sexual dysfunctions to make evidence-based treatment 1) more
easily accessible at any time from any location, 2) well scalable in the ratio
of invested resources to people reached, and 3) anonymous so that
stigmatization can be reduced (Ebert et al., 2018). Meta-analytic results
showed medium to large effects of internet-based treatment compared to control
conditions with regard to female sexual functioning (g=0.59, CI: 0.28–0.90,
I2=0%) and satisfaction (g=0.90, CI: 0.02–1.79, I2=82%) (Zarski et al., 2022).
For GPPPD in specific, we evaluated an internet-based treatment which resulted
in an improvement rate of 31% of women being able to successfully have sexual
intercourse compared to 13% in the control group and small to medium effects
on other GPPPD core symptom dimensions of genito-pelvic pain and coital and
noncoital fear of sexuality (d=0.40-0.74) (Zarski et al., 2021)
However, similar to other forms of treatment for various mental disorders
(Kessler et al., 2017), not all participants benefit to the same extent from
internet-based treatment for sexual dysfunctions. Although individuals with
mental disorders who do not receive treatment are at higher risk for
deterioration, about 25% of individuals show no changes, and 5-6% become worse
(Rozental et al., 2017, 2019). Differences in treatment outcomes can be
attributed to heterogeneity in personal and disorder characteristics (Barber,
2007; Delgadillo et al., 2016; DeRubeis et al., 2014). This heterogeneity may
result in different needs with regard to treatment. In order to enable
appropriate treatment modality allocation as well as adapt a chosen treatment
to individual needs, these varying effects should be identified (Kraemer et
al., 2002). To collect a risk profile at baseline could allow different
baseline conditions to be addressed in treatment, with the goal of promoting
treatment success.
Few existing studies have identified promising effect modifiers of
psychological treatment outcomes for GPPPD. Higher levels of pretreatment pain
intensity and catastrophizing have been shown to be associated with poorer
genito-pelvic pain outcomes after psychological treatment (Bergeron et al.,
2008; Brotto et al., 2015, 2020; Desrochers et al., 2010). Higher relationship
satisfaction, in turn, has been found to correlate with better treatment
outcome regarding sexual satisfaction and distress (Hummel et al., 2018;
Stephenson et al., 2013). In contrast, one study found high joint dyadic
coping and high levels in evaluation of dyadic coping associated with lower
odds of sexual intercourse (Zarski et al., 2017). Results on relationship
duration as predictor of pain outcomes were mixed (Brotto et al., 2020; N. O.
Rosen et al., 2021). Looking at sociodemographic variables, younger age
(Brotto et al., 2020; Zarski et al., 2017), and lower levels of education
(Zarski et al., 2017) have been associated with better treatment outcome for
GPPPD. With regard to psychological variables, lower anxiety in women has been
found to be associated with pain outcomes after treatment (Brotto et al.,
2020; N. O. Rosen et al., 2021). Additionally, women with higher levels of
childhood maltreatment receiving CBT-based couples therapy had poorer outcomes
in sexual satisfaction and sexual function at posttreatment compared to women
in a lidocaine condition (Charbonneau-Lefebvre et al., 2022).
Besides some high-quality moderator analysis, most studies, however, did not
include (1) specific a-priori statement about the intention of testing
moderators (2) evidence- or theory-based selection of moderators (3)
measurement of moderators prior to randomization (4) adequate quality of
measurements of moderators, and (5) specific test of interaction between
moderators and treatment (Pincus et al., 2011). Moreover, moderators were
almost exclusively evaluated on a variable-by-variable basis, and more complex
multivariate interactions were not taken into account. Modeling multivariate
interactions within a joint prognostic model could allow to better capture the
heterogeneity of treatment effects across patients (Dahabreh et al., 2016;
Kent et al., 2020). In this study, we therefore aim to develop and internally
validate a multivariate prognostic model predicting individualized treatment
effects of an Internet-based treatment for GPPPD.
## 2 Methods
Where applicable, this article adheres to the "transparent reporting of a
multivariable prediction model for individual prognosis or diagnosis (TRIPOD)"
statement (Collins et al., 2015).
### 2.1 Study Design
The data for this study came from a two-armed randomized controlled trial
($n_{\text{intervention}}$ = 100, $n_{\text{waitlist}}$ = 100) designed to
evaluate the efficacy of an internet-based treatment for GPPPD in comparison
to a waitlist control condition (study registration number: DRKS00010228,
ethical approval by University of Erlangen-Nürnberg: no. 324_15B). Further
information on the study and the intervention can be found in the published
study protocol, a case study, and the efficacy evaluation paper (Zarski,
Berking, & Ebert, 2018; Zarski, Berking, Hannig, et al., 2018; Zarski et al.,
2021).
### 2.2 Study Inclusion Criteria and Process
Participants were women 18 years and older who were unable to have sexual
intercourse for the last six months or longer and who were in a heterosexual
relationship. Additionally, pre-existing medical conditions causing GPPPD
symptoms had to have been ruled out before enrollment. In order to participate
in the study, individuals were required to have sufficient German language
skills, internet access, and provide informed consent. Those with 1) current
or lifetime psychosis or dissociative symptoms, 2) present substance
dependency or abuse, 3) present moderate or severe depression or bipolar
disorder, 4) current or lifetime posttraumatic stress disorder or
traumatization caused by sexual abuse, or 5) present treatment of GPPPD were
excluded from the study. After completion of the first assessment, a 1:1
randomization through an automated computer-based random integer generator
(randlist) took place and participants were allocated to either the
intervention group (IG) or the waitlist control group (WCG). The research
staff was blinded to the randomization until group allocation was successfully
completed.
### 2.3 Intervention
The 10-week online intervention consisted of 9 sessions in total, including an
extra booster session which took place four weeks after the program ended.
These sessions incorporated psychoeducation, communication exercises, non-
judgmental awareness, body exposure and genital self-exploration, attention-
focusing for pain-management, cognitive reconstructing, sensate focus, gradual
exposure of fingers and dilators, sexual intercourse preparation exercises,
and relapse prevention. Additionally, participants kept an online diary where
they monitored weekly transfer tasks as well as exercises in their daily life.
They also received encouraging text messages, personalized written feedback on
completed sessions, and practice reminders throughout the treatment program.
Specifically, if a module had failed to be completed within seven days, a
reminder from an eCoach was sent out.
### 2.4 Outcome measures
For this study, outcome data assessed in both the IG and WCG via self-report
after completion of treatment/12 weeks after randomization (T2) was used.
Potential moderators were assessed prior to randomization at baseline.
#### 2.4.1 Multidimensional composite primary outcome measure
As primary outcome, we built a composite measure to comprise the core
dimensions of GPPPD by aggregating scores of the following components: coital
and noncoital penetration ability, genito-pelvic pain and interference of
genital pain with sexual intercourse, fear of coitus and noncoital sexual
activity, and sexual satisfaction. The Primary Endpoint Questionnaire (PEQ)
was used to assess intercourse penetration behavior (1 item, scores: 0 [not
attempted or attempted, but unsuccessful] and 1 [attempted and sometimes
successful or attempted and always successful]) and noncoital self-insertion
behavior (3 items, $\alpha$=.71) (van Lankveld et al., 2006). Genito-pelvic
pain (3 items, $\alpha$=.90) and sexual satisfaction (3 items, $\alpha$=.75)
was measured using the respective subscales of the Female Sexual Functioning
Index (FSFI) (Berner et al., 2004; R. Rosen et al., 2000). Interference of
genital pain with sexual intercourse was assessed according to the Diagnostic
Guidelines for the Assessment of Genito-Pelvic Pain/Penetration Disorder (3
items, $\alpha$=.60) (Binik, 2010) and fear of coitus (3 items, $\alpha$=.78)
and noncoital sexual activity by the Fear of Sexuality Questionnaire (5 items,
$\alpha$=.82) (FSQ) (ter Kuile et al., 2007). Comparable scaling among the
included measures was achieved by 1) harmonizing measures so that higher
values indicated better GPPPD symptom severity, 2) building the same number of
quantiles over each scale, and 3) allocating scores in the same quantile to
the same value. For the dichotomous coital penetration scale, no coital
penetration was assigned to the lowest quantile and coital penetration to the
highest. A solution with eleven quantiles was chosen since the primary outcome
should sufficiently differentiate between participants. At the same time, the
chosen number of quantiles should neither overly exceed the smallest range of
the scales included nor increase the impact of the dichotomous coital
penetration scale beyond measure. When computing quantiles, skewness, mean and
standard deviation of the distribution of each scale were taken into account
using the R package sn (Azzalini, 2021). The new scores of each scale were
added to a sum score. Figure 1 shows the density plot of the aggregated
outcome. As sensitivity analyses, the aggregated outcome was recalculated
several times according to the leave-one-out principle, i.e., one scale score
at a time was excluded to build the outcome. Respective density plots and
histograms are shown in Figure 4 and 5 in the Appendix.
Figure 1: _Density plot of the aggregated outcome_
_Note._ Outcome at post-assessment based on imputed data.
#### 2.4.2 Participant-Level Moderators
In order to examine the potential moderating role of various baseline
variables relevant to GPPPD on the effect of the intervention, 39 potential
moderator variables were included: nine sociodemographic variables, i.e., age
(years), nationality (German/non-German), level of education
(low/middle/high), married (_yes/no_), children (_yes/no_), previous treatment
for GPPPD (_yes/no_), experience with psychotherapy (_yes/no_), online
training experience (_yes/no_); 14 variables related to sexual function, i.e.,
duration of GPPPD (years), lifelong GPPPD (_yes/no_), sexual intercourse
attempts in past 6 months, experience of sexual abuse (_yes/no_), control
(four items, $\alpha$=.72), catastrophic and pain (five items, $\alpha$=.73),
self-image (six items, $\alpha$=.81), genital incompatibility (two items,
$\alpha$=.66), and positive (five items, $\alpha$=.80) cognitions regarding
vaginal penetration assessed by the Vaginal Penetration Cognition
Questionnaire (VPCQ) (Klaassen & ter Kuile, 2009), sexual desire (two items,
$\alpha$=84), sexual arousal (four items, $\alpha$=93), lubrication (four
items, $\alpha$=94), orgasm (three items, $\alpha$=.93) assessed by the FSFI
(Berner et al., 2004; Rosen et al., 2000), noncoital insertion by the partner
(three items, $\alpha$=.60) assessed by the Primary Endpoint Questionnaire
(PEQ) (van Lankveld et al., 2006); 5 partnership-related variables, i.e.,
duration of partnership (years), delegated (two items, $\alpha$=92.), joint
(five items, $\alpha$=.76), and evaluation of (two items, $\alpha$=.87) dyadic
coping assessed by the Dyadic Coping Inventory (DCI) (Bodenmann, 2000;
Ledermann et al., 2010), satisfaction with partnership (9 items, $\alpha$=.79)
and partnership happiness (1 item) assessed by the Partnership Questionnaire
Short Form (PFB-k) (Kliem et al., 2012); 4 mental health variables, i.e.,
self-esteem assess by the Self-Esteem Scale (ten items, SES, $\alpha$=.90)
(von Collani & Herzberg, 2003), generalized anxiety disorder symptoms assessed
by the Generalized Anxiety Disorder Assessment (GAD-7, 7 items, $\alpha$=.81)
(Spitzer et al., 2006), trait anxiety assessed by the State Trait Anxiety
Inventory (STAI-T, 20 items, $\alpha$=.89) (Laux et al., 1981; Spielberger et
al., 1970), well-being assessed by the Well-Being Index (WHO-5, $\alpha$=.84;
Brähler et al., 2007); as well as the 7 GPPPD-related items used to build the
aggregated outcome.
### 2.5 Imputation
Missing data at baseline and post-test was imputed under the "missing at
random" assumption using random forest methodology (Stekhoven & Bühlmann,
2012). The R package missForest (Stekhoven, 2013) is a non-parametric missing
value imputation method that can handle mixed-type data on the basis of
Breiman’s random forests (Breiman, 2001). The imputation procedure consists of
the following principal steps: First, missing values are estimated using a
simple procedure (e.g., mean imputation) and the variables are sorted in
ascending order according to the number of missing values. Subsequently, a
random forest is fitted on the observed parts of the dataset to predict the
missing values variable by variable. This is repeated iteratively until a
stopping criterion is met (Stekhoven & Bühlmann, 2012). The imputation model
included all demographic and scale-level questionnaire scores collected at
baseline. Additionally, variables at post-test used to build the primary
outcome were included in the model. The number of trees in each forest was set
to 100.
### 2.6 Decision Tree Model
Model based-recursive partitioning (MOB) (Zeileis et al., 2008) was used to
examine potential treatment moderators using the R partykit package (Hothorn
et al., 2015). This method can be applied when it is assumed that a single
global model does not fit the data well. In such a case, MOB involved the
treatment effect to be tested for heterogeneity with respect to moderator
variables (via parameter stability tests). This results in a tree where each
node is associated with a local model consisting of subgroups of patients with
similar model trends (Zeileis et al., 2008).
### 2.7 Data Analysis
#### 2.7.1 Preselection of Moderator Candidates
Before starting MOB, potential moderator candidates are preselected in order
to reduce the number of variables in the final analysis model resulting in a
more parsimonious model. This was done using model-based random forest
analysis (Garge et al., 2013). It allows to calculate variable permutation
importance for the moderator variables through random forest methodology by
constructing multiple ($n$=300) model-based trees (Garge et al., 2013). For
each tree, a random subset of the splitting variables is sampled to construct
the tree, resulting in more stable and less sample-specific predictions. All
baseline variables except for the outcome and predictor variables are selected
as potential moderator variables. Potential moderator variables are ranked in
order of their contribution to the production of accurate predictions (Garge
et al., 2013).
#### 2.7.2 Model-Based Recursive Partitioning
For the subsequent model-based tree analysis, variables with a positive
variable importance are included as partitioning variables. The model was a
linear model in which the aggregated outcome at post-treatment was regressed
on the treatment indicator variable and the aggregated GPPPD symptom severity
outcome at baseline. The algorithm consists of several steps that are
performed iteratively over the model until no significant parameter
instability with respect to the moderators can be determined anymore (Zeileis
et al., 2008): 1) it fits the model to the observations in the current node,
then 2) it assesses parameter instability of the treatment effect with respect
to the covariates, 3) split points are computed that locally optimize the
objective function and then 4) splits the current node into child nodes. By
separating variable and cut-point selection, an unbiased variable selection is
enabled unlike in other tree-based methods (Fokkema et al., 2021). To avoid
overfitting, the significance level for parameter stability tests was set to
$\alpha$=0.05 and $P$ values were Bonferroni-corrected. The minimum number of
observations in each terminal node was set to 10, i.e., the default of ten
times the number of estimated parameters divided by the number of responses
per observation (Zeileis et al., 2008). Effect sizes were calculated in each
node as Cohen’s $d$ ($d$=0.2 small, $d$=0.5 medium, and $d$=0.8 large effects)
with 95% confidence intervals (CIs).
#### 2.7.3 Model Performance
Model performance was assessed using (adjusted) $R^{2}$ and root mean squared
errors (RMSE) to capture the (adjusted) proportion of variance explained by
the decision tree model and the mean difference between predicted and observed
values. In line with recommendations by Moons et al. (2019) and Steyerberg
(2019), bootstrap bias correction was applied to internally validate the
decision tree model. Via bootstrap bias correction, the proportion of excess
performance that is due to overfitting and does not reflect the true
population-based performance is quantified.
## 3 Results
### 3.1 Participant Characteristics
The women participating in this study were on average 28.75 (SD=8.89) years
old and over half had a university degree (53.50%, n=107). All women were in a
partnership and were not able to have sexual intercourse with vaginal
penetration due to GPPPD symptoms for at least the last 6 months prior to
study participants (see Table 1 and Table 2) (Zarski et al., 2021). At post-
treatment, 78% of participants in the IG and 92% in the WCG completed the
assessment.
Table 1: Participant characteristics and potential moderators at baseline.
Potential moderators | IG ($n=100$) | CG ($n=100$) | Total ($N=200$)
---|---|---|---
| | |
_Sociodemographic Variables_ | | |
Age (years), $M$ (SD), range | 29.46 (9.82) | 28.04 (7.84) | 28.75 (8.89)
German nationality, $n$ (%) | 89 (89.00) | 93 (93.00) | 182 (91.00)
Level of education, $n$ (%) | | |
\- Low | 0 | 4 (4.00) | 4 (2.00)
\- Middle | 42 (42.00) | 47 (47.00) | 89 (44.50)
\- High | 58 (58.00) | 49 (49.00) | 107 (53.50)
Married, $n$ (%) | 27 (27.00) | 24 (24.00) | 51 (25.50)
Have children, $n$ (%) | 13 (13.00) | 3 (3.00) | 16 (8.00)
Previous GPPPD treatment, $n$ (%) | 32 (32.00) | 32 (32.00) | 64 (32.00)
Psychotherapy experience, $n$ (%) | 36 (36.00) | 36 (36.00) | 72 (36.00)
Online training experience, $n$ (%) | 10 (10.00) | 4 (4.00) | 14 (7.00)
| | |
| | |
_Sexual Function Variables_ | | |
Duration of GPPPD (years), $M$ (SD)${}^{\text{a}}$ | 7.90 (7.15) | 8.13 (7.00) | 8.02 (7.06)
Lifelong GPPPD, $n$ (%) | 32 (32.00) | 44 (44.00) | 76 (38.00)
Intercourse attempts past 6 months | 6.55 (11.29) | 7.34 (20.87) | 6.95 (16.74)
Sexual abuse, $n$ (%)${}^{\text{b}}$ | 10 (10.99) | 9 (9.28) | 19 (10.11)
Control cognitions, $M$ (SD) | 2.62 (1.43) | 2.84 (1.46) | 2.73 (1.45)
Catastrophic and pain cognitions, $M$ (SD) | 4.16 (1.18) | 4.09 (1.30) | 4.12 (1.24)
Self-image cognitions, $M$ (SD) | 3.28 (1.37) | 3.47 (1.38) | 3.38 (1.37)
Genital incompatibility cognitions, $M$ (SD) | 3.05 (1.78) | 3.27 (1.77) | 3.16 (1.77)
Positive cognitions, $M$ (SD) | 2.26 (1.11) | 2.23 (1.22) | 2.25 (1.16)
Sexual desire, $M$ (SD) | 3.26 (1.06) | 3.35 (1.17) | 3.31 (1.11)
Arousal, $M$ (SD) | 4.03 (1.49) | 3.89 (1.55) | 3.96 (1.52)
Lubrication, $M$ (SD) | 3.99 (1.63) | 3.91 (1.73) | 3.95 (1.68)
Orgasm, $M$ (SD) | 3.96 (1.81) | 3.65 (1.82) | 3.81 (1.82)
Noncoital insertion by the partner, $M$ (SD) | 0.52 (0.64) | 0.49 (0.63) | 0.50 (0.64)
| | |
| | |
_Partnership-Related Variables_ | | |
Duration of partnership (years), $M$ (SD) | 6.72 (7.09) | 5.62 (4.59) | 6.17 (5.98)
Delegated dyadic coping, $M$ (SD) | 7.14 (2.02) | 7.00 (1.87) | 7.07 (1.94)
Joint dyadic coping, $M$ (SD) | 17.00 (4.04) | 17.26 (3.64) | 17.13 (3.84)
Evaluation of dyadic coping, $M$ (SD) | 7.37 (1.94) | 7.56 (1.63) | 7.47 (1.79)
Relationship quality, $M$ (SD) | 21.20 (4.21) | 20.86 (4.40) | 21.03 (4.30)
Happiness in the relationship, $M$ (SD) | 3.73 (0.97) | 3.64 (1.16) | 3.68 (1.07)
| | |
| | |
_Mental Health Variables_ | | |
Self-esteem, $M$ (SD) | 20.99 (5.71) | 20.55 (5.96) | 20.77 (5.82)
Generalized anxiety, $M$ (SD) | 6.94 (4.05) | 7.65 (4.05) | 7.30 (4.06)
Trait anxiety, $M$ (SD) | 50.03 (13.25) | 51.83 (14.47) | 50.93 (13.87)
Well-being, $M$ (SD) | 53.08 (17.18) | 47.48 (17.40) | 50.28 (17.47)
| | |
* •
_Note._ ${}^{\text{a}}$refers to a subsample of $n$ = 199,
${}^{\text{b}}$refers to a subsample of $n$ = 188, F2F: face-to-face
psychotherapy
Table 2: GPPPD-related variables at baseline and post-treatment.
GPPPD-Related Outcomes | IG ($n=100$) | CG ($n=100$) |
---|---|---|---
| | |
Intercourse penetration behavior T1, $n$, (%) | 0 | 0 |
Intercourse penetration behavior T2, $n$, (%) | 31 (31.00) | 13 (13.00) |
Noncoital self-insertion T1, $M$ (SD) | 1.11 (0.93) | 1.11 (0.92) |
Noncoital self-insertion T2, $M$ (SD) | 1.82 (0.85) | 1.17 (0.91) |
Genito-pelvic pain T1, $M$ (SD) | 1.61 (1.00) | 1.64 (1.06) |
Genito-pelvic pain T2, $M$ (SD) | 2.82 (1.36) | 1.84 (1.26) |
Genital pain interference T1, $M$ (SD) | 4.63 (0.45) | 4.63 (0.56) |
Genital pain interference T2, $M$ (SD) | 3.72 (0.97) | 4.37 (0.72) |
Coital fear T1, $M$ (SD) | 11.03 (2.86) | 11.12 (3.37) |
Coital fear T2, $M$ (SD) | 8.50 (3.20) | 9.93 (3.43) |
Noncoital fear T1, $M$ (SD) | 11.76 (4.39) | 11.42 (4.09) |
Noncoital fear T2, $M$ (SD) | 10.09 (3.55) | 11.70 (4.51) |
Sexual Satisfaction T1, $M$ (SD) | 3.35 (1.37) | 3.65 (1.33) |
Sexual Satisfaction T2, $M$ (SD) | 3.89 (1.12) | 3.57 (1.36) |
| | |
* •
_Note._ T1 = baseline, T2 = post-treatment, $n$ = sample size, $M$ = mean, SD
= standard deviation.
### 3.2 Decision Tree Model
The resulting regression-based tree depicts scatter plots for GPPPD symptom
severity as the aggregated outcome at baseline and at post-treatment as well
as box-and-whisker plots by treatment group with GPPPD symptom severity at
post-treatment as outcome measure. The data is partitioned at a joint dyadic
coping index of 13 which corresponds roughly to one standard deviation below
the sample mean ($M$ = 17.13; SD = 3.84).
### 3.3 Preselection of Moderator Candidates
Three potential moderators, namely noncoital insertion by the partner, self-
esteem, and interference of genital pain with sexual intercourse showed a
negative variable importance and were thus excluded from the pool of
partitioning variables. Variable importance was highest for joint dyadic
coping with a score above 1.0. Figure 6 in the Appendix illustrates the
variable permutation importance of potential moderator variables using model-
based random forest analysis.
### 3.4 Terminal Node Models
The final model-based tree is displayed in Figure 2 and the terminal node
specific regression coefficients in Table 3. For women with low joint dyadic
coping (Node 2; $N$ = 33), the effect of the intervention on GPPPD symptom
severity was negligible ($b$ = 4.08, $p$ = 0.139) while higher GPPPD symptom
severity at baseline was significantly associated with higher GPPPD symptom
severity scores at post-treatment ($b$ = 0.88, $p<$0.001). In contrast, women
with average-to-high joint dyadic coping (Node 3), a group comprising more
than three-quarters of the participants ($N$ = 167), profited significantly
from the intervention ($b$ = 13.24, $p<$0.001). Again, higher aggregated
outcome scores in GPPPD symptom severity at baseline were significantly
associated with higher aggregated outcome scores at post-treatment ($b$ =
0.43, $p<$0.001).
Figure 2: _Decision tree model; based on imputed data._
Accordingly, standardized mean treatment effects were highest for women with
average-to-high joint dyadic coping (Cohen’s $d$ = 1.00, 95%-CI: 0.68 - 1.32)
with a proportion of explained variance of $R^{2}$ = 30.36% (adjusted $R^{2}$
= 29.51). In women with low joint dyadic coping, the standardized mean
treatment effect size was Cohen’s $d$ = 0.12, 95%-CI: -0.57 to 0.80. A forest
plot of the subgroup-conditional effects is presented in Figure 3.
Taken together, the model suggests that women with average-to-high joint
dyadic coping will significantly benefit from the intervention in respect to
their GPPPD symptom severity while women with low joint dyadic coping will
probably not benefit. Assessing joint dyadic coping at an early stage might
thus be helpful for beneficial treatment assignment.
Figure 3: _Differential treatment effects in each terminal node resulting from
the model-based decision tree_
_Note._ d denotes Cohen’s $d$ in each terminal node, SE denotes respective
standard errors.
### 3.5 Model Performance
The proportion of variance explained by the decision tree model was $R^{2}$ =
38.64% (adjusted $R^{2}$ = 38.33%). After bootstrap bias correction, the
proportion of explained variance was reduced by about 10% ($R^{2}$ = 27.74%;
adjusted $R^{2}$ = 27.38%). The mean difference between predicted and observed
values was RMSE = 11.93. After bootstrap bias correction, this difference was
slightly increased (RMSE = 13.22). Both measures indicated substantial
predictive performance of the decision tree model even after bootstrap bias
correction.
### 3.6 Planned Further Analyses
To draw firm conclusions about this prediction model of individualized effects
of Internet-based treatment for GPPPD and to derive evidence-based practical
implications, external validation of the multivariable decision tree model is
needed. Therefore, the second aim of this study is to validate the tree-based
prediction model in a second, independent randomized-controlled trial on the
evaluation of internet-based treatment for GPPPD.
$\blacksquare$
## Funding
The project was supported by the "Gender & Diversity" funding of the
Friedrich-Alexander-Universität Erlangen-Nürnberg.
## Supplementary Information
Data is available upon reasonable request from the first author.
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## Appendix
Figure 4: _Leave-one-out density plots of the aggregated outcome if one scale
score at a time is excluded to build the outcome; based on imputed data._
Figure 5: _Leave-one-out histograms of the aggregated outcome if one scale
score at a time is excluded to build the outcome; based on imputed data._
Figure 6: _Variable permutation importance of potential moderator variables
using model-based random forest analysis; based on imputed data._ Table 3:
Regression coefficients of composite scores at post-treatment on composite
scores at baseline and group in terminal node models
Variable | Estimate | _SE_ | _t_ | _p_
---|---|---|---|---
Node 2 a | | | |
Constant | 3.435 | 3.982 | 0.863 | .395
Composite Score at Baseline | 0.877 | 0.122 | 7.189 | <.001
Group c | 4.084 | 2.686 | 1.520 | .139
Node 3 b | | | |
Constant | 18.694 | 3.301 | 5.663 | <.001
Composite Score at Baseline | 0.428 | 0.088 | 4.873 | <.001
Group c | 13.243 | 1.978 | 6.695 | <.001
* •
Note. Node 2 comprises participants with low, Node 3 participants with
average-to-high joint dyadic coping.
* •
${}^{\text{a}}$N = 33. ${}^{\text{b}}$N = 167. ${}^{\text{c}}$0 = waitlist
control group, 1 = intervention group.
|
# The Space Density of Intermediate Redshift, Extremely Compact, Massive
Starburst Galaxies
Kelly E. Whalen Department of Physics and Astronomy, Dartmouth College,
Hanover, NH 03755, USA Ryan C. Hickox Department of Physics and Astronomy,
Dartmouth College, Hanover, NH 03755, USA Alison L. Coil Center for
Astrophysics and Space Sciences, University of California, San Diego, La
Jolla, CA 92093, USA Aleksandar M. Diamond-Stanic Department of Physics and
Astronomy, Bates College, Lewiston, ME, 04240, USA James E. Geach Centre for
Astrophysics Research, University of Hertfordshire, Hatfield, Hertfordshire
AL10 9AB, UK John Moustakas Department of Physics and Astronomy, Siena
College, Loudonville, NY 12211, USA Gregory H. Rudnick Department of Physics
and Astronomy, University of Kansas, Lawrence, KS 66045, USA David S. N.
Rupke Department of Physics, Rhodes College, Memphis, TN, 38112, USA Paul H.
Sell Department of Astronomy, University of Florida, Gainesville, FL, 32611
USA Christy A. Tremonti Department of Astronomy, University of Wisconsin-
Madison, Madison, WI 53706, USA Julie D. Davis Department of Astronomy,
University of Wisconsin-Madison, Madison, WI 53706, USA Serena Perrotta
Center for Astrophysics and Space Sciences, University of California, San
Diego, La Jolla, CA 92093, USA Grayson C. Petter Department of Physics and
Astronomy, Dartmouth College, Hanover, NH 03755, USA
(Accepted for publication in ApJ)
###### Abstract
We present a measurement of the intrinsic space density of intermediate
redshift ($z\sim 0.5$), massive ($M_{*}\sim 10^{11}\ \text{M}_{\odot}$),
compact ($R_{e}\sim 100$ pc) starburst ($\Sigma_{SFR}\sim 1000\
\text{M}_{\odot}\ \text{yr}^{-1}\text{kpc}^{-1}$) galaxies with tidal features
indicative of them having undergone recent major mergers. A subset of them
host kiloparsec scale, $>1000\ \text{km}\ \text{s}^{-1}$ outflows and have
little indication of AGN activity, suggesting that extreme star formation can
be a primary driver of large-scale feedback. The aim for this paper is to
calculate their space density so we can place them in a better cosmological
context. We do this by empirically modeling the stellar populations of
massive, compact starburst galaxies. We determine the average timescale for
which galaxies that have recently undergone an extreme nuclear starburst would
be targeted and included in our spectroscopically selected sample. We find
that massive, compact starburst galaxies targeted by our criteria would be
selectable for $\sim 148^{+27}_{-24}$ Myr and have an intrinsic space density
$n_{\text{CS}}\sim(1.1^{+0.5}_{-0.3})\times 10^{-6}\ \ \text{Mpc}^{-3}$. This
space density is broadly consistent with our $z\sim 0.5$ compact starbursts
being the most extremely compact and star forming low redshift analogs of the
compact star forming galaxies in the early Universe as well as them being the
progenitors to a fraction of intermediate redshift post starburst and compact
quiescent galaxies.
galaxies: general — galaxies: evolution — galaxies: starburst — galaxies:
interactions
## 1 Introduction
Galaxy formation models within a $\Lambda$-Cold Dark Matter ($\Lambda$CDM)
framework that do not include feedback typically over-predict the present day
baryon fraction as well as the number of number density of galaxies on the
high and low mass ends of the local stellar mass function (SMF) (e.g., Croton,
2006; Kereš et al., 2009; Moster et al., 2010; Moustakas et al., 2013). This
implies that star formation over cosmic timescales is inefficient, which
requires that galaxy formation models inject energy into cooling clouds of
gas. This is typically done by invoking feedback from massive stars and active
galactic nuclei (AGNs) to heat and eject gas, thus reducing star formation
efficiency (e.g., Springel et al., 2005b; Di Matteo et al., 2005; Somerville &
Davé, 2015). Feedback as a driver of the cosmic star formation inefficiency is
supported by evidence of large-scale gas outflows and/or relativistic jets in
star forming and active galaxies (e.g. Veilleux et al., 2005; McNamara &
Nulsen, 2007; Fabian, 2012; Somerville & Davé, 2015).
In massive galaxies, feedback-driven outflows are often attributed to AGN
activity since dark matter halo mass, galaxy stellar mass, bulge mass, and
black hole mass all scale with one another (e.g., Ferrarese & Merritt, 2000;
Guo et al., 2010; Kormendy & Ho, 2013). However, cosmological galaxy formation
simulations show that the exclusion of stellar feedback in models leads to the
formation of galaxies that are $\sim 10$ times more massive than observed at a
given redshift, showing that stellar-driven feedback plays an integral role in
regulating star formation in massive galaxies (e.g., Springel et al., 2005b;
Hopkins et al., 2012). On small (giant molecular cloud) scales, feedback can
slow the local star formation rate by decreasing the gas surface density in a
region, but this alone is not sufficient to produce simulated galaxies whose
masses match those observed. Large-scale galactic wind-driven outflows where
$\dot{M}_{*,outflow}\sim\text{SFR}$ are necessary to be able to model galaxies
with masses that are consistent with observations (e.g., Veilleux et al.,
2005).
Constraining the importance of feedback-driven quenching is crucial to
understanding how massive galaxies form, especially at high redshift. Massive,
quiescent galaxies at $z>1.5$ are typically more compact than their local
counterparts by roughly a factor of 5 (e.g. Zirm et al., 2007; van Dokkum et
al., 2008; van der Wel et al., 2014). The likely progenitors of these massive,
compact quiescent galaxies are similarly compact star forming galaxies that
were formed in gas-rich mergers of disk galaxies and were then rapidly
quenched via some dissipative feedback (e.g., Barro et al., 2013; Stefanon et
al., 2013; van Dokkum et al., 2015). However, heavy dust obscuration coupled
with high redshift makes constraining the role of AGN vs. stellar-driven
feedback difficult with the typical UV signatures of outflows (e.g., van
Dokkum et al., 2015).
We have been studying a population of $z\sim 0.5$ massive, compact galaxies
which show signs of recent, extreme bursts of star formation and gas
depletion, similar to what we would expect as the progenitors to high-$z$
massive, quiescent galaxies (Tremonti et al., 2007; Diamond-Stanic et al.,
2012, 2021; Geach et al., 2013; Sell et al., 2014; Geach et al., 2014; Rupke
et al., 2019; Petter et al., 2020). Our sample of galaxies consists of sources
initially targeted as SDSS quasars, but subsequently classified as young post-
starburst galaxies due to their blue stellar continua, weak nebular emission
lines, and bright infrared photometry (Tremonti et al., 2007). Hubble Space
Telescope (HST) imaging showed that these galaxies have extremely compact
morphologies ($R_{e}\sim 100$ pc) with tidal features indicative of having
recently undergone a major merger event (see Figure 1) (Diamond-Stanic et al.,
2012; Sell et al., 2014). We also note that rings and diffraction spikes from
the HST PSF are visible in the images of our sources, showing that their
angular sizes are on the order of that of the PSF which further highlights
their compactness (Sell et al., 2014; Diamond-Stanic et al., 2021; Davis et
al., in prep). The sources in our sample can have SFR surface densities up to
$\sim 1000\ \text{M}_{\odot}\ \text{yr}^{-1}\text{kpc}^{-1}$ (Diamond-Stanic
et al., 2012; Sell et al., 2014), and lie below the $0.5<z<1$ size-mass
relations for star forming and quiescent galaxies (see Figure 2; Mowla et al.
2019; Diamond-Stanic et al. 2021). Spectroscopic observations show that these
galaxies host outflows with velocities $>1000\ \text{km}\ \text{s}^{-1}$ that
can extend to tens of kpc (Tremonti et al., 2007; Rupke et al., 2019; Davis et
al., in prep). There is also little evidence that these massive outflows are
primarily driven by AGN activity based on X-ray, IR, radio, and spectral line
diagnostics, meaning that extreme star formation can be responsible for gas
depletion in these galaxies (Diamond-Stanic et al., 2012; Sell et al., 2014;
Petter et al., 2020).
These galaxies are important because they allow us to directly observe the
effects of extreme star formation on gas kinematics in starburst and post-
merger galaxies. In merger-driven galaxy evolution scenarios, a major merger
event can trigger a strong burst of obscured star formation. Dissipative
feedback via AGN or starburst activity can then expel large amounts of gas and
dust from the galaxy, allowing it to passively evolve into a gas-poor massive
elliptical galaxy (e.g. Sanders et al., 1988; Lonsdale et al., 2006). The
objects we are studying can possibly be representative of galaxies that are
actively undergoing quenching, and might be an important phase for the
building up of a massive, quiescent elliptical population. However, this is
difficult to determine without knowing the space density of extreme compact
starburst galaxies like the ones we have been studying. We are broadly
defining our compact starbursts as massive, centrally concentrated galaxies
that have recently experienced a burst of star formation. The space density of
extreme massive, compact starbursts is strongly dependent on the timescales
upon which starburst events can be observed using our selection criteria.
The aim of this paper is to estimate the average amount of time sources in a
simulated galaxy population would be selected as extreme compact starburst
galaxies under our selection criteria, in addition to their space density. We
also place our galaxies into context with their high redshift compact star
forming analogs, compact quiescent galaxies, post starburst galaxies,
ultraluminous infrared galaxies (ULIRGs), the merger rate density, and
massive, quiescent galaxies within the same redshift interval (e.g. Sanders et
al., 1988; Lonsdale et al., 2006; Lotz et al., 2011; Barro et al., 2013; van
der Wel et al., 2014; Wild et al., 2016).
The outline of the paper is as follows: in Section 2 we discuss the selection
of the parent sample of galaxies. In Section 3 we discuss empirical model
construction and constraining model free parameters via an MCMC routine. In
Section 4 we discuss our implementation of the SDSS quasar selection function.
In Section 5 we calculate the average observability timescale and space
density for our population of compact starbursts. In Section 6 we place our
galaxies into cosmological context with other phases of merger-driven galaxy
evolution. We adopt a cosmology of $H_{0}=70.2\
\textrm{km}\textrm{s}^{-1}\textrm{Mpc}^{-1}$,
$\Omega_{M}=\Omega_{CDM}+\Omega_{b}=0.229+0.046=0.275$, and
$\Omega_{\Lambda}=0.725$ (Komatsu et al., 2011)
## 2 The observed sample
The selection criteria used for our sample will be detailed in Tremonti et al.
in prep, but we will give a brief summary in this section.
Our sample was originally selected with the objective to understand the role
galaxy-scale winds play in star formation quenching for massive, intermediate
redshift galaxies. The parent sample of galaxies we use in this work is drawn
from the Eighth Data Release of SDSS (York et al., 2000; Aihara et al., 2011).
We set out to select sources that were targeted as quasars (flagged either as
qso_hiz, qso_cap, qso_skirt, qso_mag_outlier), since the SDSS QSO sample
extends to fainter magnitudes than the main galaxy sample (Strauss et al.,
2002). Selecting sources that have been targeted as quasars allows our sample
to consist of objects that are massive and compact. The magnitude limits
ensure that our sources are massive, highly star forming, and not strongly
dust attenuated and the SDSS quasar selection algorithm requires that our
sources are either unresolved or that they are resolved but satisfy more
stringent color-magnitude cuts. This is described in more detail in Section
4.1.
We required that our sources were spectrally classified as galaxies with
apparent $16<i<20$. We selected sources within $0.4<z<0.9$ to ensure that the
MgII $\lambda\lambda 2796,2804$ line would be shifted into the optical so we
could use that as a probe of galactic winds. We also exclude sources that were
classified as distant red galaxies (legacy_target1 != DRG). Sources with
redshift warnings and bad quality plates were also thrown away. This initial
cut left us with a sample of 1198 galaxies.
We fit the SDSS spectra with a combination or simple stellar population
models, similar to Tremonti et al. (2004), and a type I quasar template. From
the spectral fitting, we calculated the fraction of light attributed to the
quasar model ($f_{qso}$). We also measured nebular emission and stellar
absorption line indices (following Kauffmann et al., 2003) for the sources in
our parent sample as well as the strength of the 4000 Å break
(D${}_{n}(4000)$) (Balogh et al., 1999). Our initial aim was target post
starburst galaxies (PSBs) by selecting galaxies with evidence of having gone
through a starburst event within the last 1 Gyr
($(\text{H}\delta_{A}+\text{H}\gamma_{A})/2$ OR $D_{n}(4000)<1.2.$), but with
little ongoing star formation within the last 10 Myr ([OII] 3727 Å equivalent
width (EW) $>-20$ Å). These cuts reduce our sample to 645 sources.
Lastly, our sample was limited to consisting of brighter galaxies with tighter
cuts on [OII] EW and including a cut on the measured quasar fraction to
further ensure that strong AGN were not included. The new cuts imposed were
[OII] 3727 ÅEW $>-15$ Å, and $f_{qso}<0.25$. We also require that apparent $g$
and $i$ magnitudes were brighter than $g<20$ or $i<19.1$. Although we select
for weak nebular emission to eliminate starbursts, many of our sources were
detected in WISE (Wright et al., 2010), and SED fitting through the mid-
infrared shows they can have SFRs$=20-500\ \text{M}_{\odot}\ \text{yr}^{-1}$
(Diamond-Stanic et al., 2012; Perrotta et al., 2021; Davis et al., in prep).
These cuts leave us with a sample of 121 galaxies. We take advantage of the
WISE detections for our sources and make an IR color cut of $W1-W2<0.8$ to
further limit AGN contamination (Stern et al., 2012; Hickox et al., 2017). The
WISE AGN cut leaves us with a population of 115 galaxies in what we are
considering to be our parent sample. We include this selection criteria in our
modeling of compact starburst galaxies to estimate the amount of time our
galaxies would be targeted and selected by this set of criteria. A full list
of targets is given in Table LABEL:tab:sample along with their redshifts,
stellar masses, and SDSS photometry.
In addition to the SDSS and WISE data for our parent sample, we also have
high-S/N ($\sim 15-30$ per pixel) spectra from the Blue Channel Spectograph on
the 6.5-m MMT (Angel et al., 1979), the Magellan Echellette (MagE; Marshall et
al. 2008) spectrograph on the Magellan Clay telescope, and the Low Resolution
Imaging Spectrometer (LRIS; Oke et al. 1995) on the Keck I telescope for 37 of
the sources in our parent sample. These observations and their data reduction
are detailed in Davis et al. (in prep), but broadly these observations were
done using 1” slits resulting in spectra with resolution $R\sim 600-4100$. We
refer to these 37 galaxies as the MgII sample.
## 3 Model construction
The aim of this work is to constrain the importance of massive, compact
starburst events in galaxy quenching at $z\sim 0.5$ by estimating the space
density of these objects. Here, we do this by constructing an empirical model
based on the galaxies we have in our sample and then evolving a large
simulated population of compact starbursts to estimate the timescales upon
which they would be targeted by our selection criteria. This process can be
broken down into two steps:
1. 1.
Construct a set of template distributions of stellar population parameters and
SFHs by fitting SDSS $ugriz$ model mags and W1, W2 photometry for the 115
galaxies in our sample with a Markov Chain Monte Carlo (MCMC; Metropolis et
al. 1953; Foreman-Mackey et al. 2013) fitter.
2. 2.
Use the posterior distribution of SFH parameters from step 1 to predict
luminous properties of a set of mock galaxies whose SFHs are consistent with
our observed sample. The luminous properties are computed using the Flexible
Stellar Population Synthesis models (FSPS; Conroy et al. 2009).
Since our small sample of galaxies consists of sources that are unresolved in
SDSS imaging, we have to make a number of assumptions about their underlying
stellar populations. First, we assume that the light from our compact
starburst galaxies can largely be broken down into two components: a young,
simple stellar population (SSP) that formed in a single, nuclear burst, and an
older component that has a star formation history representative of a massive,
star forming galaxy at $z\sim 0.5$. We note that there is likely clumpy star
formation occurring outside of the nuclear regions of our galaxies, but due to
their extremely compact HST morphologies it is fair to assume that the
contribution of these star forming regions to the total emitted light is
minimal compared to the large nuclear burst. We also assume that our galaxies
will only experience one burst of nuclear star formation and will then
passively evolve. Although HST observations (Sell et al., 2014) showed that
many of our sources have more than one core that could trigger a starburst
event, we note that these sources are still unresolved in SDSS so the burst
would not be localized to a particular core. This assumption is also
consistent with the single burst of star formation triggered by a merger event
seen in simulations (e.g. Springel et al., 2005a). Next, we naively assume
that since the nuclear burst component dominates the spectral energy
distribution (SED) of the total system, that the differences observed between
the galaxies in our sample can solely be attributed to differences in the
properties of the nuclear starburst. This assumption is consistent with the
galaxies in the MgII sample having very blue spectra and young ages as derived
from spectral modeling (e.g., Davis et al., in prep).
These assumptions allow us to construct a model that utilizes FSPS to simulate
the stellar populations for the nuclear starburst component as well as the
older, non-burst underlying stellar population. In our modeling framework, we
introduce four free parameters that are fit via an MCMC routine for each of
the galaxies in our sample: the age of the burst ($t_{age}$), the fraction of
total galaxy stellar mass formed in the nuclear burst ($f_{burst}$), the
optical depth for the dust around young stars formed in the nuclear burst
($\tau_{dust,1}$), and the total stellar mass of the system ($M_{*}$). We
separately calculate the $ugriz$, W1, W2, [OII] (3727 Å) fluxes for the
nuclear burst and non-burst components and their $f_{burst}$ weighted sum to
determine the SED and [OII] EW for the total simulated galaxy.
In this section, we describe the assumptions made in the FSPS modeling of both
the extended non-burst and nuclear starburst components as well as the MCMC
fitting we use to constrain values for the free parameters in our model.
For both, the non-burst and nuclear burst components, we make the following
assumptions. We assume a Chabrier (2003) initial mass function (imf_type = 1)
and $\log Z/Z_{\odot}=-0.3$ metallicity (logzsol = -0.3) using the $M_{*}-Z$
relation presented in Gillman et al. (2021) calibrated for solar
$12+\log(\text{O/H})=8.66$ and $Z_{\odot}=0.0121$. We set add_neb_emission =
true to allow for nebular emission from CLOUDY models (Byler et al., 2017). We
assume Charlot & Fall (2000) extinction (dust_type = 0) with dust_tesc = 7
($\log(t_{age}/\text{yr})$) (e.g Blitz & Shu, 1980; Charlot & Fall, 2000;
Conroy et al., 2009), where dust_tesc is the age in Charlot & Fall (2000)
extinction model at which stars are attenuated by $\tau_{dust,1}$ and
$\tau_{dust,2}$. We also set agb_dust = true since IR SEDs of star forming
galaxies are poorly fit without incorporating dust shells around AGB stars
(Villaume et al., 2015).
### 3.1 Modeling the extended, non-burst component
The photometric and morphological properties of the extended stellar
population are most important in the later stages of the compact starburst’s
evolution since the contribution of the nuclear burst wanes over time. Here,
we describe the assumptions we make in the FSPS modeling of the extended, non-
burst component. We initialize FSPS such that tage is the Hubble time (in Gyr)
at the redshift of a given galaxy, dust1 = 1, and dust2 = 0.5. We chose these
dust optical depths to ensure that the $ug$ photometry for the modeled
extended stellar component would be fainter than that of the reddest observed
sources in our sample, while being consistent with the recommended values
given in Charlot & Fall (2000). We explored the effects of changing tage and
the dust parameters for the extended components in the galaxies shown in
Figure 1 to ensure that our modeling is largely robust to extended component
assumptions and found that the results of our MCMC fitting do not change with
changing non-burst initial conditions.
A crucial piece to modeling the stellar population of the extended, non-burst
component is assuming a particular star formation history (SFH). HST images
show hints of a smooth, extended underlying stellar population (Diamond-Stanic
et al., 2021). The presence of tidal features in our HST observations suggests
that the galaxies in our sample have recently undergone merger events, and
their high star formation surface densities indicate that that these mergers
were likely gas rich (e.g., Diamond-Stanic et al., 2012; Sell et al., 2014).
Based on this, we assume that the extended, non-burst stellar populations have
a star formation history typical of actively star forming disk galaxies.
However, the SFHs of star forming disk galaxies are uncertain. There are many
possible SFHs that would be able to build up the tightly-correlated star
formation main sequence at late cosmic times (e.g. Oemler et al., 2017). For
simplicity, since young stars dominate the light output from a stellar
population we approximate the SFH as being flat over cosmic time to ensure
that the progenitor galaxies in the system were experiencing some degree of
star formation prior to merging. We do this by setting the FSPS SFH parameter
as a delayed-burst SFH (sfh = 4 in FSPS) but with the constant star formation
fraction set to 1.
We also note that we explored other SFHs that peaked at earlier cosmic times,
such as the dark matter halo mass dependent models constructed in Behroozi et
al. (2019), but our MCMC chains for these models were not able to reach
convergence. The inability for our chains to converge is consistent with the
fact that we do not believe that Behroozi et al. (2019)-like SFHs would be
physically representative of galaxies like those in our sample. For massive
($M_{*}\sim 10^{11}\ \text{M}_{\odot}$) galaxies like the ones in our sample,
this would suggest that our sources would have peaked in star formation at
$z\sim 2$ and then passively evolved until $z\sim 0.5$. This would imply that
the progenitors of our compact starbursts would be almost entirely be
quiescent, which is unlikely do to their high gas fractions. Therefore, we do
not include models like this in our analysis.
### 3.2 Modeling the nuclear burst
Recent observational evidence has shown that intermediate redshift, extreme
compact starburst galaxies are likely to exhibit flat age gradients, meaning
that their optical light is dominated by star formation that began and ended
in one uniform event (e.g., Setton et al., 2020). Since we expect all of the
stars formed in the nuclear burst to have formed at approximately the same
time, we model the starburst as a simple stellar population (SSP) in FSPS (sfh
= 0). This choice is consistent with very short burst durations we derive from
non-parametric SFH modeling of a subset of our sample with high S/N spectra
(Geach et al., 2018; Tremonti et al., in prep; Davis et al., in prep). This
work (detailed in Davis et al. (in prep)) is done by fitting the rest frame
UV-mid IR broadband photometry and high-resolution spectra simultaneously
using Prospector (Leja et al., 2019; Johnson et al., 2021). We also assume
that the dust in the vicinity of the nuclear starburst extincts some of the
light from the newly formed stars. We leave the age of the central burst
($\log t_{age}$) and the optical depth ($\tau_{burst}$) as free parameters
that will later be constrained with MCMC fits to the photometric data of the
sources in our observed sample. We set dust2 = $\tau_{burst}/2$ (e.g., Wild et
al., 2011). We similarly calculate SDSS $ugriz$ and WISE W1 & W2 magnitudes
for the nuclear bursts as we did for the extended, non-burst stellar
population.
### 3.3 Calculating PSF magnitudes
Once we have the model photometry for the extended, non-burst stellar
populations and their nuclear bursts, we can combine them to get the
photometry for the entire system. We start by converting the modeled apparent
AB magnitudes for the extended, non-burst stellar population and the burst
component to flux densities. The output magnitudes of FSPS are normalized to
$1\ \text{M}_{\odot}$ at every epoch, so we calculate the fluxes for our
galaxies and nuclei by multiplying their $1\ \text{M}_{\odot}$ flux densities
by their respective masses. We define the mass of the nuclear burst as
$M_{nuc}=f_{burst}\times M_{*}$ and $M_{host}=(1-f_{burst})\times M_{*}$. We
also leave $f_{burst}$ and $M_{*}$ as free parameters in our MCMC fitting in
addition to $\tau_{dust}$ and $\log t_{age}$ as described earlier.
For sources observed in SDSS, the QSO targeting pipeline takes a source’s
$ugriz$ PSF magnitudes as input rather than its de Vaucouleurs or exponential
disk model magnitudes (Richards et al., 2002). The output magnitudes from FSPS
are representative of model magnitudes, so we must first convert these to PSF
magnitudes before we run the SDSS QSO targeting algorithm on our modeled
sample. We do this by first assigning surface brightness profiles to both
components of the galaxy. For the extended, non-burst component, we assume a
$n=1$ Sérsic profile where the effective radius ($R_{\text{eff}}$) is taken
from the redshift-dependent star forming galaxy size-mass relation presented
in Mowla et al. (2019). Due to the nuclear starburst’s compact nature, we
assume a $n=4$ Sérsic profile where $R_{\text{eff}}$ is $\sim 300$ pc, as
motivated by observations (e.g., Geach et al., 2013; Sell et al., 2014).
Diamond-Stanic et al. (2021) showed that $R_{eff}<1$ kpc for the HST-observed
galaxies. We do not vary $R_{eff}$ for the nuclear components for our modeled
galaxies since $\sim 100$ pc scale starbursts would always be unresolved in
SDSS and are effectively observed as point sources.
We convert $R_{\text{eff}}$ for each component from kpc to arcsec using their
cosmological angular size distances and normalize the surface brightness
profiles ($I(r)$) for each component such that
$2\pi\int^{\infty}_{0}I_{comp}(r)rdr=f_{\nu,comp}.$
We then convolve these component surface brightness profiles with the SDSS PSF
in each photometric band. The full width half maxes (FWHMs) for the $ugriz$
bands are 1.53, 1.44, 1.32, 1.26, 1.29 arcsec, respectively. The convolved
burst and disk components are then added together to create a modeled total
galaxy surface brightness profile. We then fit this profile with a 2D-Gaussian
model of the SDSS PSF and integrate the Gaussian model fit to obtain PSF
fluxes in each respective band. The PSF fluxes are then converted to apparent
AB magnitudes so they could later (§4.1) be passed through the SDSS QSO
selection pipeline.
### 3.4 Constraining model free parameters with MCMC
We have constructed a 4-parameter model for the photometry and [OII] (3727 Å)
EW of intermediate-$z$ compact starbursts by utilizing FSPS. FSPS directly
outputs model mags and spectra of stellar populations. We calculate [OII]
(3727 Å) EW from the FSPS output spectrum using specutils (Earl et al., 2022).
As stated above, our compact starburst model is the sum of separately modeling
the host galaxy and nuclear burst contributions to the overall photometric and
spectral properties. In this model, we leave the age of the nuclear starburst
($\log\ t_{age}/\text{Myr}$), the burst fraction ($f_{burst}$), optical depth
of dust extincting young stellar light ($\tau_{dust}$), and the galaxy stellar
mass ($\log\ M_{*}/M_{\odot}$) as free parameters. Here we detail how we
constrain possible parameter values using MCMC fitting to the $ugriz$ and
W1/W2 photometry for our observed galaxies.
Figure 1: HST WFC3 cutouts of 6 representative galaxies in our sample that
overlap with those presented in Sell et al. (2014). We note that we omit
J0944+0930 and J1104+5946 from Sell et al. (2014) as they do not satisfy all
of our selection criteria. All of these galaxies show clear signs of tidal
disruptions, consistent with their extreme nuclear starbursts being triggered
by major merger events. Figure 2: Location of our galaxies (black star) within
the $0.5<z<1$ size-mass plane as presented in Mowla et al. (2019). Blue and
red points are van der Wel et al. (2014) star forming and quiescent galaxies,
respectively. The red, blue, and grey lines are the best fit size-mass
relations for the quiescent, star forming, and total CANDELS/3DHST galaxies in
Mowla et al. (2019). Our data point represents the average $R_{eff}$ and
$M_{*}$ for a subset of the MgII galaxies presented in Davis et al. (in prep).
Our sources are significanly more compact than other galaxies at similar $z$
and $M_{*}$.
#### 3.4.1 Parameter fitting
As discussed in Section 2, our collaboration has been studying a sample of 115
intermediate-$z$ compact starburst galaxies. Archival SDSS $ugriz$ and WISE W1
and W2 photometry are available for the full parent sample. For each of these,
we constrain the probability densities for $\log\ t_{age}$, $f_{burst}$,
$\tau_{dust}$, and $\log M_{*}$ using the ensemble adaptation of the
Metropolis-Hastings MCMC algorithm from the package, emcee (Metropolis et al.,
1953; Foreman-Mackey et al., 2013). Each step of our MCMC calculates the model
SDSS $ugriz$, WISE W1, and W2 photometry, and compares them to those for each
observed galaxy. For each galaxy, we run the MCMC such that the
autocorrelation time for each walker is $\sim 50$ times less than the run
time. For most of our galaxies this is $\sim 60,000$ steps. We use the emcee
ensemble stretch move with scale parameter $a=2$. We randomly initialize each
walker in the intervals
$0.5<\log\ t_{age}/\text{Myr}<2$ $0.05<f_{burst}<0.4$ $0.3<\tau_{dust}<1$
$10<\log M_{*}/M_{\odot}<11$
and allow them to explore the parameter space
$0.5<\log\ t_{age}/\text{Myr}<3$ $0.05<f_{burst}<0.65$ $0<\tau_{dust}<5$
$10<\log M_{*}/M_{\odot}<12$
such that it finds the parameter values that are most likely to minimize the
difference between the model and observed photometry.
Figure 3: Panel (a): Best fit SED for galaxy J0826+4305. The red points and
error bars are the observed photometry and $\pm 0.25$ magnitude uncertainty
region, respectively. The open black squares are the modeled photometry. The
blue, violet, and green curves are the modeled SED for the total galaxy
system, nuclear burst, and host galaxy, respectively. Panel (b): Triangle plot
of parameter posterior distributions for galaxy J0826+4305. We calculate the
mean and covariances of these posterior distributions to model them as
4D-Gaussian distributions. We then randomly draw sets of parameter values from
the Gaussian-modeled posterior to construct a mock population of compact
starbursts. Panel (c): Galaxy cutout as seen in Figure 1.
For each galaxy in our sample, we output the mean parameter values and their
covariance from MCMC-calculated posterior distributions. We use these mean
values and their covariances to model these posteriors as 4-dimensional
Gaussian distributions whose means and standard deviations are identical that
of the MCMC output. We do this to reduce noise later in our analysis since we
use these distributions to randomly draw sets of parameter values to model
mock galaxies based on the ones in our observed sample. The best fit SED and
parameter probability distributions for a constantly star forming host based
on the galaxy J0826+4305 can be seen in panels (a) and (b) Figure 3,
respectively.We also include these for J1713+2817, J2118+0017, J1506+6131,
J1558+3957, and J1613+2834 in Figures 9, 10, 11, 12, and 13, respectively. For
consistency with other studies of our objects, we note general agreement
between our best fit stellar masses and those presented in Sell et al. (2014)
for the galaxies that were included in both of our samples. This is shown in
Table 2.
For each of the 115 galaxies in our sample we randomly draw $\log\ t_{age}$,
$f_{burst}$, $\tau_{dust}$, and $\log M_{*}$ values from their respective
Gaussian-modeled posterior distributions taking into account the covariances
between each of the parameters, to model a population of galaxies with
properties similar to the observed source. We can then evolve these modeled
galaxies to estimate a distribution of selectable lifetimes for each of the
galaxies in our sample.
## 4 Modeling the targeting algorithm & selection function
The ultimate goal for our model is to be able to estimate the space density of
$z\sim 0.5$, massive, compact starburst galaxies. To do this, we need to
understand the timescales upon which these galaxies would be selected under a
set of targeting criteria. Here, we detail how we model the various components
of the selection function we use to identify sources in our sample.
### 4.1 The SDSS QSO targeting algorithm
All of the sources in our observed sample were initially targeted for SDSS
spectroscopy as QSOs based their bright magnitudes and blue colors. In order
to ensure that our modeled galaxies would satisfy these criteria, we need to
incorporate this selection into our modeled targeting function.
The SDSS QSO targeting algorithm identifies sources based on their location in
three-dimensional color space. This is the $(u-g)$-$(g-r)$-$(r-i)$ ($ugri$)
color cube for $z<3$ sources and $(g-r)$-$(r-i)$-$(i-z)$ ($griz$) cube for
galaxies at higher redshifts. The QSO catalog constructed from SDSS DR8
sources was selected using the Richards et al. (2002) targeting algorithm
111Python adaptation of Richards et al. (2002) QSO selection algorithm can be
found at www.github.com/ke27whal/sdss_qso_selection.. The SDSS quasar
selection function aims to identify sources that lie far from the region of
color space where stars are most likely to be found as well as for sources to
satisfy general color/magnitude cuts. All magnitudes referenced in the
targeting algorithm are PSF magnitudes. Since we are working with modeled data
that is free from observational uncertainty, we do not include the steps in
the algorithm that flag sources for having data with fatal errors.
Since quasars and local stars both exhibit bright apparent magnitudes and are
unresolved point sources, the algorithm needs to be able to differentiate
between them in color-color-color space. The algorithm makes use of the method
described in Newberg & Yanny (1997) that defines a “stellar locus” in color-
color-color space where stars are most likely to exist. The stellar locus is
constructed by analyzing the distribution of SDSS identified stars in color
space. To maintain generality, we will refer to the main coordinate system
describing the color-color-color cube as
$\langle\hat{x},\hat{y},\hat{z\rangle}$, where $\hat{x}$ is in the direction
of the bluest color axis and $\hat{z}$ in the direction of the reddest. The
locus construction algorithm begins by setting the endpoints of the stellar
distribution in color space and then iteratively calculating midpoints. This
process allows a local coordinate system
($\langle\hat{i}_{i},\hat{j}_{i},\hat{k}_{i}\rangle$) to be defined at each
locus point. At each locus point ($p_{i}$), $\ \hat{k}_{i}$ is defined as a
unit vector in the direction $\overrightarrow{p_{i+2}-p_{i}}$. As detailed in
Newberg & Yanny (1997), unit vectors $\hat{i}_{i}$, $\hat{j}_{i}$, and
$\hat{k}_{i}$ are given as
$\hat{k}_{i}\equiv k_{x}\hat{x}+k_{y}\hat{y}+k_{z}\hat{z},$
$\hat{j}_{i}\equiv(\hat{k}_{i}\times\hat{z})/|\hat{k}_{i}\times\hat{z}|=(k_{y}\hat{x}-k_{x}\hat{y})/\sqrt{k_{x}^{2}+k_{y}^{2}},$
$\hat{i}_{i}\equiv\hat{j}_{i}\times\hat{k}_{i}=[-k_{x}k_{z}\hat{x}-k_{y}k_{z}\hat{y}+(k_{x}^{2}+k_{y}^{2})\hat{z}]/\sqrt{k_{x}^{2}+k_{y}^{2}}.$
The cross section of the stellar locus is measured by fitting an ellipse
perpendicular to $\hat{k}_{i}$ at each point. The semi-major and semi-minor
axes of the ellipses are in the direction of unit vectors $\hat{l}_{i}$ and
$\hat{m}_{i}$, respectively, and are defined as
$\hat{l}_{i}\equiv\hat{i}_{i}\cos\theta_{i}+\hat{j}_{i}\sin\theta_{i},$
$\hat{m}_{i}\equiv-\hat{i}_{i}\sin\theta_{i}+\hat{j}_{i}\cos\theta_{i}$
where $\theta_{i}$ is the angle between the major axis of the ellipse and unit
vector $\hat{i}$. We adopted the locus point positions, $\theta_{i}$,
$\hat{k}_{i}$, $|\vec{l}_{i}|$, and $|\vec{m}_{i}|$ values from Richards et
al. (2002), and proceeded to construct right cylinders that define the
$4\sigma$ stellar locus probability region in color-color-color space. We also
incorporate the mid-$z$ inclusion region as the white dwarf/A star exclusion
regions detailed in Richards et al. (2002).
Sources targeted as quasars must also satisfy color and magnitude cuts in
addition to not belonging to the stellar locus. For low-$z$ sources in the
$ugri$ color cube, all objects must have apparent $i$-band magnitude
$15<i<19.1$ (Richards et al., 2002). Both extended and point source objects
are allowed to be selected as quasars, but they need to satisfy different sets
of criteria. Point source objects only need to fulfill the magnitude and
stellar locus cuts to be targeted. Extended sources are kept if they are
likely to contain an active nucleus. This is most likely when $(u-g)<0.9$, as
redder AGN would be at high-$z$ and would not be extended (Richards et al.,
2002; Adelman-McCarthy et al., 2006). This $(u-g)$ cut does not remove blue,
extended star forming galaxies, so a second cut of $l_{i}>0$ and $m_{i}>0$ is
applied where $l_{i}$ and $m_{i}$ are positions within the
$\langle\hat{k},\hat{l},\hat{m}\rangle$ coordinate space defined earlier. In
the high-$z$ $griz$ color cube, all outliers from the stellar locus with
$15<i<20.4$ are targeted as quasars. However, to avoid contamination from
low-$z$ quasars, sources are removed from the high-$z$ sample when all of the
following criteria are met;
$(g-r)<1.0,$ $(u-g)\geq 0.8,$ $i\geq 19.1\ \ \text{OR}\ \ (u-g)<2.5.$
We allow the sources in our sample to be targeted as either low-$z$ or
high-$z$ quasars since our observed sample contains a mixture of both target
types.
### 4.2 Spectroscopic/photometric selection
In addition to being blue, unresolved sources, the galaxies in our sample also
exhibit weak nebular emission characteristic of post starburst galaxies. As
mentioned earlier, we implement an emission line equivalent width (EW) cut on
[OII] (3727 Å) such that [OII] EW$>-15$ Å, consistent with that used for our
parent sample (Sell et al., 2014; Davis et al., in prep; Tremonti et al., in
prep). We also model the $g<20$ flux limit and $W1-W2<0.8$ WISE color cut that
we impose on our sample.
## 5 Estimating the space density
In this section, we discuss the various parameters that contribute to the
calculated compact starburst space density ($n_{CS}$) as well as the possible
sources of uncertainty. We estimate the space density in the redshift range
$0.4<z<0.9$ as
$n_{CS}\sim\frac{N_{targeted}}{f_{complete}}\cdot\frac{t_{cosmic}}{V_{0.4<z<0.9}}\cdot\frac{A_{sky}}{A_{SDSS}}\cdot\biggl{<}\frac{1}{t_{obs}}\biggr{>}.$
(1)
Here, $N_{targeted}$ is defined as the number of galaxies in our observed
sample of massive, compact starburst galaxies, $f_{complete}$ is the
completeness of the SDSS QSO catalog ($f_{complete}\sim 0.9$; Vanden Berk et
al. 2005), $V_{0.4<z<0.9}\ $ is the volume in Mpc-3 contained within the
redshift range $0.4<z<0.9$, $A_{SDSS}/A_{sky}$ is the fractional area of the
SDSS footprint relative to the area of the entire sky, $t_{cosmic}$ is the
amount of cosmic time in Myr contained in the redshift range $0.4<z<0.9$, and
$\langle 1/t_{obs}\rangle$ is the average of the inverse selectability
timescale in Myr. The only model-dependent factor in this calculation is the
amount of time our sources would be selected under a particular set of
targeting criteria, so we will spend the first part of this section focusing
on calculating this value.
It is also worth highlighting that the timescale we are calculating for our
sources is the amount of time these objects would be targeted under our set of
selection criteria. This is a separate quantity from the amount of physical
time galaxies might be undergoing an extremely compact starburst phase. The
physical timescale is also dependent on how we define these sources. A
unifying feature of the observed sources in our sample is that they are late-
stage major mergers that host extremely young stellar populations. It is
possible that some of them have quenched/are very recent PSBs and that others
are still forming stars. Broadly, we define our sources as galaxies that have
recently experienced an extreme nuclear burst of star formation. Calculating
the physical timescale for these sources would require much more detailed
modeling which is beyond the scope of this work. Our goal here is to estimate
the space density of objects that would be targeted by our selection criteria
at some point in their evolution.
### 5.1 Calculating observed lifetimes
Figure 4: Shown here are the modeled evolutionary tracks of the apparent
$i$-band and $g$-band SDSS magnitudes (panels (a) & (b)), [OII] equivalent
width (panel (c)), and WISE $W1-W2$ color (panel (d)) for a sub-sample of
modeled galaxies. The x-axis is age relative to the burst peak. The grey-
shaded rectangles represent the regions of parameter space that would not be
selected by the criteria placed on that given parameter. This is a schematic
representation— the full details of our source selection can be found in
Section 2.
For each of the 115 galaxies in our sample, we used SDSS $ugriz$ model mags
and WISE W1/W2 measured photometry to construct SEDs which were then fit by
our MCMC routine to obtain the posterior distributions for $\log
t_{age}/\text{Myr}$, $f_{burst}$, $\tau_{dust}$, and $\log\
M_{*,tot}/M_{\odot}$. These posterior distributions were then modeled as
4-dimensional Gaussian distributions and we output their covariance matrices.
For each of the 115 observed galaxies in our sample, we draw 200 sets of
parameters from the respective posterior distributions while taking into
account covariances between parameters. This gives us 115$\times$200 mock
galaxies which we then evolve. We evolve our modeled galaxies within the time
interval $-1<\log\ t_{age}/{\text{Myr}}<2.5$ in 1000 uniformly spaced steps.
We calculate [OII] EWs from the output FSPS spectrum using specutils (Earl et
al., 2022), as well as the photometry at each step to determine if the sources
would be targeted by our selection criteria at each time step. This allows us
to construct selected lifetime distributions for each of the 115 observed
galaxies in our sample. The evolutionary tracks for a subset of randomly
selected galaxies’ $i$ and $g$-band magnitudes, [OII] EWs, and $W1-W2$ colors,
as well as the selection limits on each respective parameter can be seen in
Figure 4. We note that Figure 4 does not include the SDSS QSO targeting
selection since that is a much more complicated set of criteria and would be
impossible to visually display. However, we do apply it in our target
selection.
Figure 5: Distribution of average selected lifetimes from the mock sample. We
find that extreme nuclear starbursts like the ones observed in our galaxies
would be selected for $\sim 148^{+27}_{-24}$ Myr, consistent with the burst
ages calculated in Davis et al. (in prep).
In the following section, we detail how we determine the space density of our
sources by randomly sampling with replacement the selected lifetime
distribution calculated by evolving mock galaxies. In short, we bootstrap by
generating 100,000 randomly sampled (with replacement) populations of 115 mock
galaxies. For each iteration, we randomly draw an array of 115 indices which
correlates to the various observed galaxies in our sample. We use the randomly
drawn indices to pull selected lifetimes from the corresponding selected
lifetime distributions. We then average these lifetimes to determine a
selectability timescale for that given mock population of galaxies. The
average selected lifetime distribution for the 100,000 samples of 115 mock
galaxies is shown in Figure 5. We find that on average, compact starburst
galaxies like the ones we observe would be selected under our set of targeting
criteria for $148^{+27}_{-24}$ Myr. This timescale is broadly consistent with
the average post-starburst peak age of $70\pm 106$ Myr calculated in Davis et
al. (in prep).
In our modeling, we find that our mock galaxies would be targeted soon after
the nuclear burst occurs, meaning that we can directly compare our
selectability timescale and the post-starburst peak SF ages in Davis et al.
(in prep). The light-weighted stellar ages of the MgII sample ranging from
$\sim$ 13-300 Myr) galaxies are consistent with the calculated selectability
timescale in this work. This is a good consistency check to ensure that our
modeling shows that galaxies in our observed sample would be selectable at
their best-fit stellar ages.
We next use the selectability timescales of our modeled compact starburst
galaxies to estimate their space density.
### 5.2 Calculating space density
As stated above, we estimate the space density in the redshift range
$0.4<z<0.9$ (Equation 1) by randomly sampling from our selected lifetime
distributions. To ensure that we sample a sufficiently large population of
mock galaxies, we iterate this part of the calculation 100,000 times.
For each of the 100,000 iterations, we randomly sample with replacement 115
galaxies from our mock sample. For each of the galaxies in that sample, we
randomly draw a $\log t_{obs}/\text{Myr}$ value from the observable lifetime
distribution that corresponds to that particular galaxy. In each iteration, we
use these $\log t_{obs}/\text{Myr}$ values to compute,
$\biggl{<}\frac{1}{t_{obs}}\biggr{>}=\frac{1}{N_{sim}}\sum_{i}^{N_{sim}}\
\bigg{(}\frac{1}{t_{obs,i}}\bigg{)},$ (2)
where $N_{sim}=115$. We then use this to calculate the space density for the
random population generated each iteration using the expression above.
Figure 6: Space density distribution calculated from our mock population of
galaxies. We estimate that the space density for our population of $0.4<z<0.9$
compact starburst galaxies is $(1.1^{+0.5}_{-0.3})\times 10^{-6}\
\text{Mpc}^{-3}$.
The resulting space density distribution (calculated using Equation 1) can be
seen in Figure 6. We estimate the space density of these massive, compact
starbursts to be $(1.1^{+0.5}_{-0.3})\times 10^{-6}\ \text{Mpc}^{-3}$ in the
redshift range $0.4<z<0.9$.
## 6 Cosmological context
One of the most interesting questions surrounding our sample of galaxies is
whether or not this type of compact starburst phase is characteristic in the
evolution of many, if not most, massive galaxies. A widely supported view of
galaxy formation and evolution is that mergers are responsible for building up
increasingly massive galaxies and for triggering starbursts and AGN activity
(e.g., Toomre, 1977; Sanders et al., 1988; Kauffmann et al., 1993; Mihos &
Hernquist, 1996; Hopkins et al., 2006, 2008; Lotz et al., 2011; Somerville &
Davé, 2015). Sanders et al. (1988) presented a basic framework in which the
collision of two gas-rich disk galaxies would funnel gas towards the center of
the system via tidal streams or shocks, thus creating a dusty, gas-rich
environment to foster rapid star formation (e.g. Lonsdale et al., 2006). This
dusty starburst stage would be selected as a ULIRG. As gas is fueling rapid
star formation, it is continuously being funneled into the nucleus and also
being accreted onto the black hole, thus also triggering AGN activity (e.g.
Hopkins et al., 2006, 2008). Within this framework, gas from the galaxy can be
expelled by a blowout phase driven by violent, dissipative feedback.
Figure 7: Comparison of the average timescales (in Gyr) upon which various
phases of massive galaxy evolution would be observable. The black star
represents the average selectability timescale for the modeled compact
starburst galaxies in our sample, and its error bar along the redshift axis
represents the size of the redshift range of our sources and the error bar
along the $t_{obs}$ axis is the statistical uncertainty calculated via
bootstrapping as described in Section 5.2 The grey, purple, and blue shaded
regions represent the range of observable timescales for galaxy mergers (Lotz
et al., 2011), ULIRGs (Farrah et al., 2003), and post starburst galaxies
(PSBs; Wild et al. 2016), respectively. We note that the timescales presented
for galaxy mergers and PSBs correspond to the amount of time a source would be
targeted under a set of selection criteria (similar to the value calculated
for our sources), while the timescale for ULIRGs reflects the amount of
physical time a source would experience star formation characteristic of the
ULIRG phase. We elaborate on how we obtain the timescale estimates for the
shaded regions in the text. It is clear that compact starburst galaxies like
the ones in our sample occur on relatively short lived timescales that are
comparable to that of ULIRG star formation. Figure 8: Comparison of the space
densities of various phases of massive galaxy evolution. The black star
represents the modeled space density for compact starburst galaxies like those
in our observed sample. Its error bar along the redshift axis represents the
size of the redshift range of our sources and the error bar along the space-
density axis is the statistical uncertainty calculated via bootstrapping as
described in Section 5.2. We note that there are additional systematic errors,
including uncertainty with model assumptions, which make this statistical
error a lower limit. The blue squares represent the space density evolution of
massive, compact star forming galaxies from the CANDELS survey (Barro et al.,
2013), the red points represent massive ($\log M_{*}/M_{\odot}\sim 11$),
compact quiescent galaxies (van der Wel et al., 2014), the green triangle
represents low-$z$ PSBs (Pattarakijwanich et al., 2016), and the purple
hexagon represents low-$z$ ULIRGs (Kim & Sanders, 1998). The grey, red,
purple, and green shaded regions depict the Lotz et al. (2011) observed merger
rate density, the Stott et al. (2013) observed merger rate density (calculated
using merger observability timescales), ULIRG space density (Magnelli et al.,
2011), and intermediate-$z$ PSB space density (Wild et al., 2016) ranges,
respectively. The Barro et al. (2013) points, Lotz et al. (2011) region, and
Stott et al. (2013) region have been adjusted to account that our sources have
masses $\log M_{*}/M_{\odot}>10.5$, while most of the other populations shown
include galaxies $\log M_{*}/M_{\odot}>10$. While only a relatively small
fraction of intermediate-$z$ major mergers will result in an extreme compact
starburst similar to those in our sample, it is likely that sources like ours
are the more extreme, lower-$z$ analogs to compact star forming galaxies more
common in the early Universe and are closely related to intermediate-$z$ PSBs.
The galaxies in our observed sample have many features that could tie them
into this evolutionary framework. We know that the galaxies for which we have
HST observations have disturbed morphological features such as tidal tails or
two nuclei, which is indicative of them having undergone a recent merger (e.g
Sell et al., 2014). In addition to having disturbed morphologies, our galaxies
host high velocity ionized and molecular gas outflows which can extend out to
kpc scales (e.g. Tremonti et al., 2007; Diamond-Stanic et al., 2012; Geach et
al., 2013, 2014; Sell et al., 2014; Geach et al., 2018) or even over 100 kpc
scales (Rupke et al., 2019).
In order to understand the evolutionary significance of extreme, compact star
formation events like those observed in our galaxies, we need to contextualize
their space density relative to that of various phases within massive galaxy,
merger-driven evolution. Our results are summarized in Figures 7 and 8, and we
discuss in greater detail within this section.
### 6.1 Evolution of massive compact galaxies
The sample of galaxies we have been studying is comparable to a high-$z$
population of similarly compact, massive forming galaxies. Massive, quiescent
galaxies in the Universe at $z>1.5$ are typically more compact than their
local counterparts by roughly a factor of 5 (e.g. Zirm et al., 2007; van
Dokkum et al., 2008; van der Wel et al., 2014). The progenitors of these
galaxies were likely compact star forming galaxies that were formed in gas-
rich mergers of disk galaxies and were then rapidly quenched via some
dissipative feedback, a formation scenario that is reminiscent of what we
expect for ULIRGs and quiescent galaxies in the lower-$z$ Universe (e.g.,
Barro et al., 2013; Stefanon et al., 2013; van Dokkum et al., 2015).
Barro et al. (2013) observed populations of compact quiescent and star forming
galaxies in the redshift range $\sim 1<z<3$ to understand the evolutionary
pathways that lead to the assembly of massive, compact quiescent galaxies we
see predominantly in the early Universe. We include their compact star forming
galaxy space density evolution as the blue squares in Figure 8 for comparison
with the intermediate-$z$ massive, compact starburst galaxies we are studying
(black star). We adjust the points from Barro et al. (2013) using redshift
appropriate stellar mass functions (Moustakas et al., 2013; Adams et al.,
2021) to account for the fact that their sample consists of sources with a
wider stellar mass distribution than our sample. The adjusted space density is
given as
$n_{\text{adjusted}}=n_{\text{literature}}\times\frac{\int^{\infty}_{\text{lim,
us}}\phi_{\text{SMF}}\ \text{d}\log M_{*}}{\int^{\infty}_{\text{lim,
lit}}\phi_{\text{SMF}}\ \text{d}\log M_{*}},$ (3)
where $n_{\text{literature}}$ is the literature space density calculated for a
larger mass range than our sample, and $\phi_{\text{SMF}}$ is the stellar mass
function. We use the Moustakas et al. (2013) and Adams et al. (2021) SMFs for
$z\leq 1.5$ and $z>1.5$, respectively. The Barro et al. (2013) compact star
forming galaxies have constant space densities at high redshift, but begin to
decline at $z<\sim 1.5$. This decline is consistent with the decline in galaxy
merger, star formation, and cold gas densities with decreasing redshift (e.g.,
Tacconi et al., 2010; Daddi et al., 2010; Tacconi et al., 2013; Madau &
Dickinson, 2014; Riechers et al., 2019).
We show in Figure 8 that the space density of our sources lies only slightly
below the space density evolution trend shown with the Barro et al. (2013)
compact star forming galaxies. We note that our galaxies are more extreme than
the Barro et al. (2013) sources as they are both more compact and more rapidly
star forming. This likely biases our compact starburst space density to be
slightly lower than that for the Barro et al. (2013) galaxies. It is possible
that our sources represent the low redshift analogs for an extreme subset of
compact starburst galaxies that are more prevalent in the early Universe.
Understanding how stellar feedback rapidly quenches star formation at
intermediate redshift is necessary to be able to build models for galaxy
formation and evolution in the early Universe when compact star formation
events were significantly more common. For compact star-forming galaxies in
the early Universe, it is difficult to observe the effects of feedback due to
their high redshift and the fact they are commonly obscured by dust, making it
nearly impossible to observe UV spectral signatures of outflows (e.g., van
Dokkum et al., 2015). The broad consistency between the space density of our
extreme, compact starburst galaxies and the Barro et al. (2013) sample allows
us to better understand how compact star formation might be a phase that
massive galaxies go through across a wide range of cosmic time.
Barro et al. (2013) also presented a schematic representation of how galaxies
evolve onto the local size-mass relation. Within this framework, compact star
forming galaxies will experience rapid quenching via AGN or star formation
feedback, resulting in a massive, compact quiescent galaxy population. Over
cosmic time, these sources will undergo minor and major mergers resulting in a
buildup of mass and size (e.g. Naab et al., 2009). If our sources are the low-
redshift analogs of early Universe compact star forming galaxies beginning
their quenching phase, we would expect that they would also end up as compact,
quiescent galaxies. We show the space density evolution from van der Wel et
al. (2014) for high-$z$, massive ($M_{*}\sim 10^{11}\ M_{\odot}$), compact
($R/(M_{*}/M^{11})^{0.75}<2.5$ kpc) galaxies as red points in Figure 8. The
space density of compact quiescent galaxies peaks just as that of compact star
forming galaxies begins to decline. It then wanes with decreasing redshift due
to size buildup via galaxy mergers. Within the lowest redshift bin, the van
der Wel et al. (2014) sources have a space density of $\sim 10$ larger than
that of our compact, starburst galaxies. It is also worth noting that the
compact quiescent galaxies would be considered to be “compact” for $\sim 2$
Gyr before minor mergers significantly contribute to size buildup (e.g., Naab
et al., 2009; Newman et al., 2012)— a timescale that is significantly longer
than the $\sim 100$ Myr timescale for which our sample would be targeted as
extremely compact starbursts (e.g. Barro et al., 2013). In addition to this,
the effective radii for the van der Wel et al. (2014) sources is significantly
larger than that of our nuclear starbursts. This could be due to the compact
quiescent radii being more linked to the stellar mass profiles, while ours
might be biased to small values because of mass-to-light ratio (M/L) effects.
However, Diamond-Stanic et al. (2021) showed that even accounting for M/L
effects that the stellar mass effective radius for our systems is on the order
of 0.1-0.5 kpc, which indicates that our population could be even smaller and
potentially more extreme than the compact quiescent galaxies in the van der
Wel et al. (2014) sample. All of this together suggests that a significant
fraction of massive, compact quiescent sources at intermediate redshift could
have recently gone through a starburst similar to what we observe for the
galaxies in our sample.
### 6.2 Comparison to post starburst galaxies
In order to get a full picture of the role intermediate-$z$, extremely compact
starbursts galaxies play in the buildup of a massive, quiescent population, we
also need to understand the evolutionary stages that follow their bursts. By
design of our selection criteria, the compact starburst galaxies in our sample
are similar to PSBs in that they have B and A-star dominated spectral features
and weak nebular emission. Understanding the population of PSBs in a similar
redshift interval as our sources would provide context for quenching
timescales as well as what the progenitors of PSBs might look like.
Wild et al. (2016) studied a population of massive, PSBs within $0.5<z<2$, and
determined that PSBs are a relatively short-lived, transitory phase in galaxy
evolution, likely lasting $\sim 0.1-1$ Gyr (see also Wild et al., 2009). This
timescale range was determined by modeling PSBs in both toy-model and
hydrodynamic simulations, and evolving them to determine the amount of time
they would be targeted as PSBs— a similar method to what we do here for our
compact starburst galaxies. The PSBs selectability timescale is given as the
blue region in Figure 7. Our compact starburst galaxies with selectability
timescales of $\sim 100$ Myr would be selected for $10-100$% of the time PSBs
would be selected by their respective selection criteria.
It would be expected that extremely compact starburst galaxies and PSBs would
have similar space densities within a given redshift range if they were two
evolutionary stages that were directly related to each other. In other words,
if compact starburst galaxies are the immediate progenitors to PSBs, they
should be found in similar abundances. This is what is seen in Figure 8. The
Wild et al. (2016) PSBs within the mass range $10.5<\log M_{*}/M_{\odot}<12.5$
show a decrease in space density with decreasing redshift. The lowest redshift
bin for the Wild et al. (2016) PSBs overlaps with the upper limits of the
redshift range probed for our compact starburst galaxies. The mass bin for
Wild et al. (2016) is consistent with that of our sources so we did not have
to correct for integrating the SMF within different mass intervals. Our
sources overlap within the margin of error with the estimated PSB space
density at the lowest redshift included in the Wild et al. (2016) sample.
The redshift evolution of the Wild et al. (2016) PSB space density is also
consistent with declining star formation and cold-gas densities over cosmic
time— properties that would also impact the frequency of extremely compact
bursts of star formation (e.g Madau & Dickinson, 2014; Riechers et al., 2019).
Since the cosmic SFR density sharply declines at low-$z$, we also want to
compare our compact starburst space density to that of low-$z$ PSBs to
determine if our calculated space density is consistent with the decline in
PSB space density on the interval $0<z<1$. We calculate the $z\sim 0.05$ PSB
space density by integrating the lowest-$z$ luminosity function presented in
Pattarakijwanich et al. (2016). This luminosity function is given per [5000 Å]
magnitude, a fiducial top hat filter used to calculate average $f_{\lambda}$
across $4950<\lambda/\text{\AA}<5100$ for the rest frame spectra of the PSBs
in their sample. In order to calculate a comparable space density from this,
we needed to construct a [5000 Å] mass-luminosity relation to determine our
bounds of integration. We did this by calculating [5000 Å] magnitudes from
SDSS spectra for the low-$z$ PSBs studied in French et al. (2018) using the
methodology described in Pattarakijwanich et al. (2016) and using MPA-JHU
stellar masses (Brinchmann et al., 2004; Tremonti et al., 2004). We then
integrated the Pattarakijwanich et al. (2016) luminosity function within
$10.5<\log M_{*}/\text{M}_{\odot}<11.5$, which corresponds to
$-23.3<[5000\text{\AA}]<-21.3$, to obtain a low-$z$ PSB space density of
$\sim(2.9^{+1.2}_{-1.3})\times 10^{-6}\ \text{Mpc}^{-3}$. This is given as the
green triangle in Figure 8. This is of the same order of magnitude of that for
our $z\sim 0.5$ compact starburst galaxies, which supports that a fraction of
the most extreme PSBs might have undergone an extremely compact starburst
phase like that observed in our galaxies.
### 6.3 Comparison to ULIRGs
Within the framework of merger-driven galaxy evolution, it is likely that
extremely compact starburst events are most relevant in the remnants of major,
gas-rich mergers. We also know that major, gas-rich mergers can trigger strong
bursts of dusty star formation which would be observed as a ULIRG with
$L_{FIR}>10^{12}\ L_{\odot}$. It is possible that sources like the massive,
extremely compact starburst galaxies in our sample could represent the
transition between the dust-obscured ULIRG and the beginning of a galaxy-scale
blowout. Here, we compare the selectability timescale and space density of our
compact starbursts to that of ULIRGs in order to contextualize their
importance in merger-driven galaxy evolution.
The timescales upon which a galaxy will experience ULIRG-like star formation
are poorly constrained. On the low end, SN-driven winds could cut the lifetime
of a single starburst in a ULIRG to 1-10 Myr (e.g., Thornley et al., 2000).
However, studies of ULIRGs with a wide variety of morphologies have allowed
the ULIRG lifetime to be estimated to be in the 0.1-1 Gyr range (e.g. Farrah
et al., 2001; Murphy et al., 2001; Farrah et al., 2003). It is possible that
this wide range of estimated ULIRG lifetimes is due to the fact that it is
likely that a ULIRG undergoes multiple large bursts of star formation,
allowing it to be selected as such on discontinuous time intervals (e.g.,
Bekki, 2001; Farrah et al., 2001). Farrah et al. (2003) analyzed a population
of 41 local ULIRGs and found that most of their sources would have lifetimes
$10\lesssim\text{Myr}\lesssim 40$. From all of the values quoted above, we
assume that the lifetime of a ULIRG is $\sim 1-100$ Myr, and show this range
as the purple shaded region in Figure 7. However, it is important to make the
distinction that these timescales are more strongly related to the physical
timescales of dusty star formation than to observable lifetimes caused by
respective selection criteria as discussed in other sections. The post-peak SF
ages for the MgII galaxies in our sample calculated in Davis et al. (in prep)
are better comparisons to the ULIRG lifetimes due to the fact that they are
tied more to the physical properties of the galaxies. As stated earlier, Davis
et al. (in prep) calculated the average post-peak SF age of $\sim 70$ Myr,
which is largely consistent our estimate that they would be able to be
targeted for $\sim 148^{+27}_{-24}$ Myr. These timescales are of a similar
order of magnitude to that of ULIRGs, which is largely unsurprising because
both types of systems are characterized by their energetic starbursts, albeit
ours are a bit more extreme.
We next compare our estimated compact starburst space density to that of
ULIRGs in a similar redshift interval. Koprowski et al. (2017) computed the
evolution of the far-IR luminosity function for galaxies out to $z\sim 5$. We
estimate the observed space density of ULIRGs by adopting the $0.5<z<1.5$ far-
IR luminosity function presented here. Integrating the luminosity function for
$L_{\text{IR}}>10^{12}\ L_{\odot}$ gives $n_{\text{ULIRG}}\sim 6\times
10^{-5}\ \text{Mpc}^{-3}$. This is shown as the purple shaded region in Figure
8, where the range of values is due to the uncertainty in the Schechter
function fit as described in Koprowski et al. (2017). We note that we do not
correct for differences in the mass distributions between the ULIRG sample and
our sources because ULIRG sample was luminosity selected. Similarly, Magnelli
et al. (2009) calculated the evolving far-IR luminosity function and space
density for ULIRGs for several redshift bins within the interval $0.4<z<1.3$.
For the $0.4<z<0.7$ and $0.7<z<1$ bins, $n_{\text{ULIRG}}\sim 3\times 10^{-5}\
\text{Mpc}^{-3}$ and $n_{\text{ULIRG}}\sim 2\times 10^{-5}\ \text{Mpc}^{-3}$,
respectively.
Comparing these values to our estimated compact starburst space density
($(1.1^{+0.5}_{-0.3})\times 10^{-6}\ \text{Mpc}^{-3}$) suggests that it is
possible that $\sim 3-8$% of intermediate-$z$ ULIRGs can experience a phase
similar to that observed in our sample of extremely compact starburst
galaxies. The physical timescales of ULIRGs and our compact starbursts are
driven by the same processes, and they are on the same order of magnitude,
while there is a factor $\sim 12-40$ difference in their space densities. It
is possible the sources in our sample represent a small fraction of the most
extreme population of ULIRGs that have the highest SFRs and/or are the most
compact.
We also compare the space density of our intermediate-$z$ massive, compact
starburst galaxies to that of low-$z$ ULIRGs, similar what we hae done in the
previous subsection for PSBs since we expect a sharp decline in the ULIRG
space density alongside that of the cosmic SFR density (e.g., Madau &
Dickinson, 2014). Kim & Sanders (1998) presented a luminosity function for
$0.05<z<0.2$ ULIRGs, and integrating the luminosity for $\log\
L_{\text{IR}}/\text{L}_{\odot}>12$ gives a space density of $\sim(4\pm
1)\times 10^{-7}\ \text{Mpc}^{-3}$. This is given as the purple hexagon in
Figure 8. Given that the space density of our intermediate-$z$, compact
starburst galaxies is calculated in a redshift range between that of the low
and intermediate-$z$ ULIRGs, this very steep decline in ULIRG space density
also suggests that a small fraction of ULIRGs could undergo a phase like that
observed in our galaxies as they evolve.
### 6.4 Comparison to $z\sim 0.5$ merger rate per co-moving unit volume
Since extremely compact starburst galaxies are likely formed by the merging of
gas-rich disk galaxies, it is important to characterize how many major mergers
could produce events like those observed in our sample of galaxies. This
requires having knowledge of the major merger rate over a given redshift
range. In the past few decades, much work has been done to constrain the
galaxy-galaxy merger rate throughout cosmic time. However, there are large
systematic uncertainties in this measurement that have prevented the reaching
of a consensus between theory and observations and even between different
observational techniques. Here, we summarize the most recent results in
calculating the $z\sim 0.5$ galaxy merger rate per co-moving unit volume and
use them to contextualize our compact starburst space density. To be more
concise, we will refer to the merger rate per co-moving unit volume as the
merger rate density for the rest of this paper.
A crucial piece of calculating the galaxy merger rate density is understanding
the timescales upon which a system would be identified as a major merger. This
is also the aspect of the calculation that contributes the most uncertainty to
the major merger rate density. The two main methods to identify merging
galaxies are to select systems with disturbed morphologies (e.g., Abraham et
al., 1994, 2003; Conselice, 2009; Lotz et al., 2008) or to search for systems
comprised of close pairs (e.g, Le Fèvre et al., 2000; Bluck et al., 2009).
Each of these methods probe different stages of the merger and are susceptible
to different biases. Close pair selection identifies sources before the merger
begins but morphological selection can detect systems before, during, and
after the merger occurs, allowing morphologically selected galaxy mergers to
be identifiable on different timescales than their close pair counterparts.
In Figure 7, we compare the selectability timescale calculated for our modeled
compact starburst galaxies (black star) to that of all galaxy mergers
presented in Lotz et al. (2008) (grey shaded region). The Lotz et al. (2011)
region reflects the range of timescales calculated for simulated systems with
mass ratios $1:10<\mu<1:1$ that were selected morphologically (for a detailed
review; Abraham et al. 1994, 2003; Lotz et al. 2011). We find that extreme
compact starburst events are selectable for a fraction of the amount of time
that a morphologically selected galaxy merger would be under its own
respective criteria.
Having constraints on galaxy merger timescales allows for the merger rate
density to be calculated. We show our calculated compact starburst space
density (black star) in conjunction with merger rate densities (grey and red
shaded regions) as well as the space densities of other phases of merger-
driven evolution in Figure 8. The grey shaded region represents the range of
the predicted observable merger rate densities calculated in Lotz et al.
(2011), and the red shaded region represents the observed range of merger rate
densities presented in Stott et al. (2013) which used Lotz et al. (2011)
predicted observable timescales. Both the Lotz et al. (2011) and Stott et al.
(2013) merger rate densities were calculated for samples containing galaxies
with $\log M_{*}/M_{\odot}>10$, while the compact starburst galaxies in our
sample are typically $\log M_{*}/M_{\odot}>10.5$. We therefore adjusted the
Lotz et al. (2011) and Stott et al. (2013) merger rate densities to ensure
that we are working within the same mass interval of the galaxy stellar mass
function (SMF) within the appropriate redshift range, as described above. We
also converted these merger rate densities to merger space densities by
assuming a typical merger timescale of 0.5 Gyr (Lotz et al., 2011).
We find that our estimated massive compact starburst space density is $\sim
200$ times smaller than the merger rate density within a similar redshift
interval, suggesting that only a small fraction of galaxy mergers would
trigger an extreme burst of compact star formation similar to our observed
sample. However, we reiterate that the Lotz et al. (2011) and Stott et al.
(2013) merger rates consider both major and minor mergers. It is likely that
these compact starburst events are triggered only by gas-rich major (mass
ratio 1:1 - 4:1) mergers which only make up a fraction of the total number of
mergers occurring across a given redshift range (e.g., Lin et al., 2010). This
suggests that although only a small fraction of all galaxy mergers might
result in extremely compact starbursts, that these could be a likely result of
a larger fraction of gas-rich major mergers.
### 6.5 Comparison to $z\sim 0.5$ massive, quiescent galaxies
Another way of understanding the role of compact starburst galaxies in the
buildup of quiescent galaxy populations is to compare their space density to
that of massive, quiescent galaxies within the same redshift range. Moustakas
et al. (2013) presented a detailed study of galaxies targeted in PRism Multi-
object Survey (PRIMUS) and provided contsraints on the evolution of the
stellar mass function from $0<z<1$. The galaxies in PRIMUS were sorted into
star forming and quiescent populations, and the evolution of their space
density was calculated across different stellar mass and redshift bins. For
quiescent PRIMUS galaxies in the mass range $10.5<\log M/M_{\odot}<11$, their
space density increases by $\sim 2\times 10^{-4}\ \text{Mpc}^{-3}$ from $z\sim
0.8$ to $z\sim 0.35$. The net decline in space density for star forming
galaxies in this redshift interval is $\sim 9\times 10^{-5}\ \text{Mpc}^{-3}$.
These changes in space density are comparable to the merger rate in this
redshift range and are a factor of $\sim 1000$ larger than our measured space
density of $n\sim(1.1^{+0.5}_{-0.3})\times 10^{-6}\ \text{Mpc}^{-3}$ for our
sample of massive, compact starburst galaxies. This is broadly consistent with
short-lived compact starbursts existing for $\sim 100$ Myr, evolving into
massive, quiescent galaxies which would exist on $\sim$Gyr timescales. It is
likely that this is a relatively rare phase of galaxy evolution within the
general population of massive, quiescent galaxies. However, it is possible
that the fraction of those that have also previously undergone extreme ULIRG
or PSB phases also could have experienced extremely, compact starbursts like
those in our sample.
## 7 Summary & Conclusions
In order to build up a population of quiescent galaxies, otherwise gas-rich
and star forming galaxies need to undergo some type of quenching process to
either disrupt or expel the gas in the system. Violent, dissipative feedback
in which either AGN activity or rapid star formation injects energy into the
ISM is an important process that impedes the formation of stars in a galaxy.
Observationally, feedback manifests as large-scale gas outflows being driven
from a galaxy.
Within the context of merger-driven galaxy evolution, we expect gas-rich
mergers of massive star forming galaxies to trigger dusty starburst events
that would then be followed by a blowout event in which nuclear gas and dust
is expelled from the system, therefore exposing the nuclear regions of the
galaxy. In this work, we have studied a population of 115 $z\sim 0.5$ massive
galaxies that are experiencing extreme, compact starburst events and outflows.
Resolved HST WFC3 observations of a subset of these show that they are merger
remnants, suggesting that these types of events could be an phase within a
simple merger-driven evolutionary pathway.
Our goal for this work was to determine how long galaxies like the ones we
observe would be selected under a certain set of selection criteria, to
estimate their space density, and to place them into cosmological context with
other evolutionary phases massive galaxies could experience. We do this by
empirically modeling the stellar populations of $z\sim 0.5$ massive, compact
starburst galaxies. Our model is dependent on four parameters: nuclear burst
age, burst mass fraction, optical depth of dust enshrouding newly formed
stars, and total galaxy stellar mass. These posterior distributions for these
parameter values are constrained for each of the 115 galaxies in our sample by
fitting the SDSS $ugriz$ and WISE W1/W2 photometry for the 151 galaxies in our
sample using an MCMC technique. We randomly draw sets of parameters from the
Gaussian models for the MCMC-calculated posterior distributions to assemble a
mock population of compact starburst galaxies. We evolve the modeled sources
to determine the timescales under which the galaxies we model would be
selected by our targeting criteria. We find that this timescale is
$148^{+27}_{-24}$ Myr and that the corresponding intrinsic space density is
$n_{\text{CS}}\sim(1.1^{+0.5}_{-0.3})\times 10^{-6}\ \text{Mpc}^{-3}$.
Our results, as summarized in Figure 8, suggest that our observed population
of extreme compact starburst galaxies could fit into an evolutionary scheme
described in Barro et al. (2013). At higher redshifts massive, compact star
forming galaxies are more common, and they are believed to be the progenitors
of massive, compact quiescent galaxies. Based on comparisons with the Barro et
al. (2013) sample of massive, compact galaxies it is likely that our sources
follow a similar life cycle in which a gas-rich major merger triggers a burst
of star formation. This starburst then drives massive, high velocity gas
outflows, thus rapidly quenching the galaxy. This galaxy would be observable
for $\sim 100$ Myr timescales as a PSB (e.g., Wild et al., 2016), and would
then evolve into a massive, compact, quiescent galaxy. Throughout cosmic time,
the massive, quiescent galaxy will undergo minor mergers, allowing it to grow
in both mass and size to become a typical quiescent galaxy consistent with the
mass-size relation of the massive quiescent galaxy population at z=0, which is
notably devoid of compact quiescent galaxies (e.g., Taylor et al., 2010).
Although it is more common for galaxies to experience this timeline earlier in
the Universe, our galaxies appear to be consistent with these trends within
their respective redshift interval. The space density of our massive, compact
starbursts suggests that they can contribute to the buildup of a fraction of
PSBs and massive, extreme compact quiescent galaxies within their epoch, which
in turn could contribute to the overall population of massive, quiescent
galaxies in the future.
We acknowledge support from the National Science Foundation (NSF) under a
collaborative grant (AST-1814233, 1813299, 1813365, 1814159 and 1813702) and
from the Heising-Simons Foundation grant 2019-1659.
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## Appendix A Auxillary MCMC Output
Table 1: Properties for the galaxies included in our sample. SDSS ID | $z$ | $\langle\log M_{*}/M_{\odot}\rangle$ | $\sigma_{\log M_{*}/M_{\odot}}$ | SDSS $u$ | SDSS $g$ | SDSS $r$ | SDSS $i$ | SDSS $z$ | WISE W1 | WISE W2
---|---|---|---|---|---|---|---|---|---|---
| | | | (AB) | (AB) | (AB) | (AB) | (AB) | (Vega) | (Vega)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
J1015+0004 | 0.417 | 11.0 | 0.07 | 22.03 | 20.71 | 19.25 | 18.95 | 18.77 | 15.83 | 15.38
J1109-0040 | 0.593 | 11.4 | 0.47 | 22.07 | 20.88 | 19.46 | 18.8 | 18.61 | 15.26 | 15.22
J1210+0030 | 0.441 | 11.1 | 0.08 | 21.88 | 20.87 | 19.37 | 19.02 | 18.79 | 15.87 | 15.3
J1341-0009 | 0.446 | 11.0 | 0.19 | 22.34 | 20.96 | 19.38 | 19.05 | 18.79 | 15.74 | 15.74
J1434-0052 | 0.461 | 11.3 | 0.51 | 23.45 | 21.04 | 19.29 | 18.66 | 18.31 | 14.86 | 14.64
J1440+0039 | 0.564 | 11.2 | 0.10 | 20.93 | 20.4 | 19.27 | 18.86 | 18.74 | 15.59 | 15.59
J1125-0145 | 0.519 | 10.9 | 0.27 | 19.6 | 19.33 | 18.69 | 18.48 | 18.39 | 14.84 | 14.65
J0745+3754 | 0.406 | 10.7 | 0.22 | 20.27 | 19.86 | 19.14 | 18.79 | 18.46 | 14.78 | 14.13
J0251-0657 | 0.406 | 11.1 | 0.27 | 22.91 | 21.14 | 19.39 | 18.88 | 18.57 | 15.38 | 15.19
J0905+5759 | 0.711 | 10.8 | 0.28 | 19.91 | 19.58 | 19.4 | 19.1 | 19.14 | 15.56 | 15.46
J1219+0336 | 0.451 | 11.0 | 0.21 | 20.15 | 19.52 | 18.79 | 18.53 | 18.33 | 14.99 | 14.56
J1232+0226 | 0.418 | 11.1 | 0.22 | 21.55 | 20.36 | 18.81 | 18.53 | 18.4 | 15.41 | 15.25
J1440+0107 | 0.456 | 10.9 | 0.23 | 20.63 | 20.26 | 19.38 | 18.97 | 18.76 | 15 | 14.53
J1441+0116 | 0.537 | 11.0 | 0.22 | 20.35 | 19.76 | 19.34 | 18.97 | 18.68 | 15.33 | 15.1
J0901+0314 | 0.459 | 10.6 | 0.23 | 19.55 | 19.29 | 18.82 | 18.7 | 18.57 | 15.22 | 15.01
J1107+0417 | 0.467 | 10.6 | 0.22 | 19.96 | 19.52 | 19.07 | 18.89 | 18.7 | 15.58 | 14.93
J1453+6022 | 0.406 | 10.9 | 0.15 | 20.49 | 20.04 | 19.02 | 18.78 | 18.55 | 15.61 | 15.33
J1506+6131 | 0.437 | 10.3 | 0.17 | 19.69 | 19.58 | 19.12 | 19.04 | 19.16 | 15.72 | 15.52
J1610+5104 | 0.469 | 11.1 | 0.07 | 22.1 | 20.93 | 19.35 | 18.92 | 18.76 | 15.68 | 15.51
J1635+4709 | 0.699 | 11.6 | 0.13 | 20.65 | 20.28 | 19.51 | 18.75 | 18.56 | 15.21 | 15.11
J2116-0634 | 0.728 | 11.3 | 0.18 | 20.74 | 20.02 | 19.72 | 19.2 | 19.05 | 15.51 | 15.55
J2311-0839 | 0.725 | 11.7 | 0.14 | 21.15 | 20.89 | 19.93 | 18.92 | 18.71 | 15.4 | 15.29
J2140+1209 | 0.751 | 11.1 | 0.25 | 20.63 | 20.19 | 19.85 | 19.31 | 19.1 | 15.57 | 14.98
J2256+1504 | 0.727 | 11.4 | 0.22 | 20.76 | 20.1 | 19.59 | 18.91 | 18.74 | 15.12 | 15.19
J2319+1435 | 0.422 | 10.5 | 0.36 | 22.62 | 21.07 | 19.42 | 19.01 | 18.78 | 15.77 | 15.44
J0826+4305 | 0.603 | 10.7 | 0.27 | 19.64 | 19.43 | 19.14 | 18.88 | 18.85 | 15.42 | 15.13
J0951+5514 | 0.402 | 11.3 | 0.11 | 20.65 | 20.01 | 18.91 | 18.51 | 18.15 | 14.85 | 14.37
J1235+6140 | 0.599 | 11.3 | 0.48 | 20.91 | 20.31 | 19.19 | 18.61 | 18.51 | 15.4 | 15.13
J1253+6256 | 0.536 | 10.4 | 0.17 | 19.69 | 19.64 | 19.3 | 19.25 | 19.22 | 16.16 | 15.68
J1506+5402 | 0.608 | 10.7 | 0.27 | 19.28 | 19.13 | 18.88 | 18.65 | 18.61 | 15.26 | 14.78
J1248+0601 | 0.632 | 11.2 | 0.18 | 20.89 | 20.33 | 19.49 | 18.98 | 18.85 | 15.77 | 15.64
J1117+5123 | 0.49 | 11.3 | 0.11 | 21.06 | 20.42 | 19.24 | 18.91 | 18.68 | 15.33 | 15.38
J1020+5331 | 0.457 | 11.0 | 0.31 | 22.53 | 20.68 | 19.21 | 18.88 | 18.69 | 15.77 | 15.62
J1401-0223 | 0.402 | 11.0 | 0.20 | 20.36 | 19.91 | 19.05 | 18.64 | 18.29 | 15.01 | 14.54
J0933+4135 | 0.441 | 10.7 | 0.25 | 19.07 | 18.97 | 18.46 | 18.39 | 18.24 | 15.14 | 14.59
J0939+4251 | 0.411 | 10.9 | 0.17 | 20.05 | 19.58 | 18.73 | 18.52 | 18.24 | 15.18 | 14.89
J1142+6037 | 0.568 | 11.5 | 0.29 | 20.86 | 20.13 | 18.81 | 18.29 | 18.17 | 15.05 | 14.79
J1713+2817 | 0.577 | 11.3 | 0.16 | 20.82 | 20.3 | 19.33 | 18.91 | 18.86 | 15.52 | 15.23
J1720+3017 | 0.684 | 11.6 | 0.10 | 21.25 | 20.67 | 19.75 | 18.89 | 18.78 | 15.45 | 15.47
J2118+0017 | 0.459 | 10.8 | 0.25 | 20.17 | 19.78 | 18.96 | 18.74 | 18.53 | 14.96 | 14.25
J0922+0452 | 0.476 | 11.1 | 0.08 | 21.22 | 20.34 | 18.99 | 18.79 | 18.57 | 15.7 | 15.49
J1052+0607 | 0.555 | 10.9 | 0.14 | 20.53 | 20.14 | 19.32 | 19 | 18.86 | 15.69 | 15.82
J1353+5300 | 0.408 | 11.3 | 0.18 | 20.43 | 19.84 | 18.81 | 18.38 | 18.12 | 14.65 | 14.2
J1436+5017 | 0.454 | 11.0 | 0.19 | 20.22 | 19.81 | 18.83 | 18.61 | 18.42 | 15.36 | 14.91
J1558+3957 | 0.402 | 10.6 | 0.23 | 19.37 | 19.07 | 18.54 | 18.44 | 18.24 | 15.17 | 14.55
J1604+3939 | 0.564 | 11.7 | 0.29 | 20.85 | 20.01 | 18.8 | 18.21 | 18.06 | 14.58 | 14.48
J0828+0336 | 0.572 | 11.0 | 0.11 | 20.9 | 20.3 | 19.3 | 18.98 | 18.94 | 16.09 | 15.77
J0808+2709 | 0.563 | 11.1 | 0.19 | 20.63 | 20.09 | 19.4 | 18.9 | 18.77 | 15.52 | 14.94
J1009+4336 | 0.519 | 10.9 | 0.26 | 19.6 | 19.37 | 18.79 | 18.56 | 18.38 | 14.98 | 14.62
J1133+0956 | 0.483 | 11.0 | 0.19 | 20.45 | 19.93 | 19.05 | 18.8 | 18.75 | 15.2 | 14.95
J0900+3212 | 0.496 | 11.3 | 0.22 | 20.18 | 19.84 | 18.96 | 18.5 | 18.14 | 14.67 | 14.49
J1330+4821 | 0.444 | 11.5 | 0.16 | 20.6 | 19.73 | 18.76 | 18.32 | 17.97 | 14.64 | 14.13
J1420+5313 | 0.742 | 11.8 | 0.20 | 20.72 | 20.39 | 19.84 | 19.01 | 18.67 | 14.68 | 14.61
J1556+4234 | 0.401 | 11.4 | 0.11 | 20.42 | 19.76 | 18.52 | 18.19 | 17.88 | 14.8 | 14.38
J1456+3849 | 0.421 | 10.8 | 0.23 | 19.84 | 19.48 | 18.88 | 18.49 | 18.16 | 14.44 | 13.8
J1459+3844 | 0.433 | 10.5 | 0.22 | 19.93 | 19.64 | 19.05 | 18.9 | 18.68 | 15 | 14.56
J1037+4048 | 0.439 | 11.1 | 0.16 | 22.17 | 20.94 | 19.4 | 18.98 | 18.74 | 15.74 | 15.56
J1248+4444 | 0.43 | 10.7 | 0.22 | 19.71 | 19.49 | 18.83 | 18.62 | 18.48 | 15.14 | 14.76
J1447+3650 | 0.414 | 11.0 | 0.06 | 22.71 | 20.89 | 19.17 | 18.84 | 18.72 | 15.7 | 15.58
J1520+3334 | 0.516 | 11.3 | 0.12 | 21.91 | 20.78 | 19.39 | 18.98 | 18.88 | 15.41 | 15.18
J1611+2650 | 0.483 | 11.4 | 0.13 | 20.97 | 20.18 | 18.97 | 18.67 | 18.45 | 14.92 | 14.62
J1039+4537 | 0.634 | 11.2 | 0.26 | 20.37 | 20 | 19.42 | 18.98 | 18.86 | 15.02 | 14.87
J1035+3854 | 0.422 | 11.0 | 0.29 | 22.44 | 20.8 | 19.26 | 18.96 | 18.75 | 15.85 | 15.59
J1052+4104 | 0.576 | 10.9 | 0.18 | 20.14 | 19.84 | 19.27 | 18.96 | 18.89 | 15.78 | 15.74
J1215+4233 | 0.479 | 11.2 | 0.33 | 22.22 | 20.86 | 19.21 | 18.82 | 18.58 | 15.48 | 15.22
J1244+4140 | 0.459 | 10.8 | 0.18 | 19.91 | 19.54 | 18.79 | 18.64 | 18.45 | 15.71 | 15.15
J0921+3251 | 0.73 | 11.1 | 0.41 | 26.53 | 15.87 | 14.65 | 16.92 | 16.62 | 15.01 | 14.96
J1012+1134 | 0.411 | 11.0 | 0.42 | 24.01 | 21.22 | 19.6 | 19.07 | 18.79 | 15.43 | 14.81
J1113+1119 | 0.628 | 11.6 | 0.63 | 20.57 | 17.48 | 17.04 | 16.68 | 16.5 | 15.49 | 15.43
J1232+0723 | 0.401 | 10.7 | 0.22 | 19.86 | 19.41 | 18.73 | 18.6 | 18.42 | 14.96 | 14.7
J1239+0731 | 0.542 | 11.0 | 0.18 | 20.51 | 20.12 | 19.29 | 18.95 | 18.85 | 15.62 | 15.49
J1415+4830 | 0.496 | 11.0 | 0.21 | 19.66 | 19.2 | 18.73 | 18.34 | 18.08 | 14.12 | 13.4
J1450+4621 | 0.782 | 11.6 | 0.15 | 20.6 | 20.09 | 19.66 | 18.89 | 18.85 | 15.24 | 15.23
J1658+2354 | 0.498 | 11.4 | 0.17 | 19.74 | 19.22 | 18.33 | 18.07 | 17.94 | 14.54 | 14.36
J0908+1039 | 0.502 | 11.0 | 0.23 | 19.77 | 19.45 | 18.74 | 18.47 | 18.27 | 14.89 | 14.6
J1119+1526 | 0.491 | 11.1 | 0.07 | 22.09 | 20.95 | 19.43 | 19.04 | 18.87 | 15.82 | 15.66
J0830+5552 | 0.526 | 11.0 | 0.25 | 20.19 | 19.85 | 19.16 | 18.81 | 18.56 | 14.82 | 14.49
J1435+0846 | 0.404 | 11.3 | 0.15 | 20.28 | 19.73 | 18.61 | 18.32 | 18.05 | 14.99 | 14.56
J0742+4844 | 0.431 | 11.0 | 0.14 | 20.79 | 20.19 | 19.03 | 18.83 | 18.59 | 15.6 | 15.19
J0752+1806 | 0.619 | 10.5 | 0.13 | 20.44 | 19.91 | 19.5 | 19.03 | 20.65 | 14.66 | 14.32
J0836+2526 | 0.531 | 10.8 | 0.23 | 20.75 | 20.29 | 19.28 | 18.94 | 18.8 | 16.12 | 15.82
J1016+3026 | 0.402 | 10.8 | 0.30 | 23 | 20.8 | 19.18 | 18.83 | 18.64 | 15.72 | 16.02
J1133+3958 | 0.487 | 11.1 | 0.21 | 19.7 | 19.29 | 18.52 | 18.29 | 18.15 | 14.74 | 14.46
J1229+3545 | 0.614 | 11.4 | 0.41 | 20.57 | 20 | 19 | 18.46 | 18.33 | 15.18 | 15.1
J1301+3615 | 0.573 | 11.3 | 0.19 | 20.51 | 20.01 | 19.13 | 18.68 | 18.59 | 15.05 | 14.91
J0901+2338 | 0.438 | 10.8 | 0.23 | 20.06 | 19.45 | 18.96 | 18.58 | 18.39 | 14.39 | 13.7
J0911+2619 | 0.471 | 11.0 | 0.24 | 20.18 | 19.71 | 18.95 | 18.58 | 18.36 | 14.5 | 13.92
J1403+2440 | 0.455 | 11.0 | 0.15 | 22.2 | 20.79 | 19.24 | 18.93 | 18.76 | 15.77 | 15.69
J1505+2312 | 0.417 | 11.0 | 0.07 | 22.6 | 20.89 | 19.33 | 18.98 | 18.71 | 15.78 | 15.39
J1548+1834 | 0.688 | 11.2 | 0.22 | 20.53 | 20.08 | 19.55 | 18.93 | 18.89 | 15.37 | 15.31
J1634+1729 | 0.491 | 10.9 | 0.23 | 20.71 | 20.1 | 19.35 | 19.04 | 18.78 | 15.22 | 14.54
J1635+1749 | 0.469 | 11.0 | 0.24 | 20.71 | 20.17 | 19.21 | 18.93 | 18.75 | 15.08 | 14.67
J1226+2753 | 0.427 | 10.3 | 0.14 | 19.14 | 19.14 | 18.83 | 18.81 | 18.78 | 15.87 | 15.46
J0936+2237 | 0.571 | 11.3 | 0.49 | 22.66 | 21.12 | 19.48 | 18.86 | 18.63 | 15.41 | 15.34
J1012+2258 | 0.504 | 11.5 | 0.20 | 20.74 | 20.13 | 18.81 | 18.43 | 18.21 | 14.89 | 14.69
J1000+2816 | 0.469 | 11.2 | 0.14 | 20.74 | 20.25 | 19.17 | 18.76 | 18.56 | 15.22 | 15.22
J0941+1827 | 0.569 | 11.5 | 0.14 | 21.02 | 20.26 | 19.1 | 18.61 | 18.36 | 15.2 | 14.97
J1005+1836 | 0.402 | 10.8 | 0.33 | 24.96 | 21.13 | 19.48 | 19.02 | 18.79 | 15.7 | 15.56
J0912+1523 | 0.747 | 11.7 | 0.29 | 20.91 | 20.37 | 19.59 | 18.64 | 18.4 | 15.23 | 15.06
J0900+1130 | 0.407 | 11.2 | 0.21 | 20.64 | 19.97 | 19.04 | 18.62 | 18.18 | 14.66 | 14.04
J1203+1807 | 0.595 | 11.4 | 0.38 | 22.37 | 21.25 | 19.73 | 18.96 | 18.82 | 15.41 | 15.38
J1205+1818 | 0.526 | 10.6 | 0.27 | 19.01 | 18.88 | 18.54 | 18.41 | 18.45 | 15.19 | 14.84
J1256+1826 | 0.424 | 11.0 | 0.39 | 22.52 | 21.02 | 19.35 | 18.88 | 18.6 | 15.42 | 15.27
J1248+1954 | 0.561 | 11.0 | 0.17 | 20.15 | 19.8 | 19.13 | 18.81 | 18.79 | 15.68 | 15.65
J1352+1653 | 0.533 | 11.3 | 0.32 | 22.07 | 21.07 | 19.47 | 18.88 | 18.63 | 15.57 | 15.36
J1400+1524 | 0.564 | 11.3 | 0.15 | 20.72 | 20.35 | 19.35 | 18.86 | 18.76 | 15.38 | 15.24
J1412+1635 | 0.454 | 11.1 | 0.39 | 22.5 | 20.91 | 19.3 | 18.88 | 18.57 | 15.5 | 15.21
J1412+1943 | 0.413 | 10.9 | 0.20 | 21.76 | 20.68 | 19.26 | 19 | 18.75 | 15.79 | 15.57
J1500+1739 | 0.577 | 10.7 | 0.27 | 19.68 | 19.38 | 19.04 | 18.82 | 18.76 | 15.15 | 14.82
J1516+1650 | 0.589 | 11.0 | 0.24 | 19.73 | 19.35 | 18.93 | 18.54 | 18.35 | 14.46 | 13.95
J1049+6433 | 0.454 | 11.4 | 0.38 | 21.79 | 20.36 | 18.78 | 18.35 | 18.13 | 15 | 14.71
J1528+0126 | 0.403 | 10.9 | 0.22 | 20.3 | 19.78 | 19.03 | 18.62 | 18.23 | 14.53 | 14.06
J0811+4716 | 0.516 | 11.0 | 0.11 | 21.16 | 20.69 | 19.55 | 19.2 | 18.92 | 15.93 | 15.87
J0827+2954 | 0.682 | 11.5 | 0.11 | 21.48 | 21.05 | 20.12 | 19.42 | 19.14 | 15.69 | 15.48
J1613+2834 | 0.449 | 11.0 | 0.24 | 20.26 | 19.76 | 18.94 | 18.69 | 18.42 | 14.84 | 14.25
Table 2: Comparison between our derived stellar masses and those presented in Sell et al. (2014). SDSS Name | $\langle\log M_{*}/M_{\odot}\rangle$ | $\log M_{*}/M_{\odot}$
---|---|---
| (This work) | (Sell et al., 2014)
(1) | (2) | (3)
J1506+6131 | $10.3^{+0.22}_{-0.15}$ | 10.2
J0826+4305 | $10.7\pm 0.29$ | 10.8
J2118+0017 | $10.8^{+0.22}_{-0.27}$ | 11.1
J1558+3957 | $10.6\pm 0.24$ | 10.6
J1613+2834 | $11.0^{+0.17}_{-0.24}$ | 11.2
Figure 9: Panel (a): Best fit SED for galaxy J01713+2817. The red points and
error bars are the observed photometry and $\pm 0.25$ magnitude uncertainty
region, respectively. The open black squares are the modeled photometry. The
blue, violet, and green curves are the modeled SED for the total galaxy
system, nuclear burst, and host galaxy, respectively. Panel (b): Triangle plot
of parameter posterior distributions for galaxy J01713+2817. We calculate the
mean and covariances of these posterior distributions to model them as
4D-Gaussian distributions. We then randomly draw sets of parameter values from
the Gaussian-modeled posterior to construct a mock population of compact
starbursts. Panel (c): Galaxy cutout as seen in Figure 1. Figure 10: Panel
(a): Best fit SED for galaxy J2118+0017. The red points and error bars are the
observed photometry and $\pm 0.25$ magnitude uncertainty region, respectively.
The open black squares are the modeled photometry. The blue, violet, and green
curves are the modeled SED for the total galaxy system, nuclear burst, and
host galaxy, respectively. Panel (b): Triangle plot of parameter posterior
distributions for galaxy J2118+0017. We calculate the mean and covariances of
these posterior distributions to model them as 4D-Gaussian distributions. We
then randomly draw sets of parameter values from the Gaussian-modeled
posterior to construct a mock population of compact starbursts. Panel (c):
Galaxy cutout as seen in Figure 1. Figure 11: Panel (a): Best fit SED for
galaxy J1506+6131. The red points and error bars are the observed photometry
and $\pm 0.25$ magnitude uncertainty region, respectively. The open black
squares are the modeled photometry. The blue, violet, and green curves are the
modeled SED for the total galaxy system, nuclear burst, and host galaxy,
respectively. Panel (b): Triangle plot of parameter posterior distributions
for galaxy J1506+6131. We calculate the mean and covariances of these
posterior distributions to model them as 4D-Gaussian distributions. We then
randomly draw sets of parameter values from the Gaussian-modeled posterior to
construct a mock population of compact starbursts. Panel (c): Galaxy cutout as
seen in Figure 1. Figure 12: Panel (a): Best fit SED for galaxy J1558+3957.
The red points and error bars are the observed photometry and $\pm 0.25$
magnitude uncertainty region, respectively. The open black squares are the
modeled photometry. The blue, violet, and green curves are the modeled SED for
the total galaxy system, nuclear burst, and host galaxy, respectively. Panel
(b): Triangle plot of parameter posterior distributions for galaxy J1558+3957.
We calculate the mean and covariances of these posterior distributions to
model them as 4D-Gaussian distributions. We then randomly draw sets of
parameter values from the Gaussian-modeled posterior to construct a mock
population of compact starbursts. Panel (c): Galaxy cutout as seen in Figure
1. Figure 13: Panel (a): Best fit SED for galaxy J1613+2834. The red points
and error bars are the observed photometry and $\pm$0.25 magnitude uncertainty
region, respectively. The open black squares are the modeled photometry. The
blue, violet, and green curves are the modeled SED for the total galaxy
system, nuclear burst, and host galaxy, respectively. Panel (b): Triangle plot
of parameter posterior distributions for galaxy J1613+2834. We calculate the
mean and covariances of these posterior distributions to model them as
4D-Gaussian distributions. We then randomly draw sets of parameter values from
the Gaussian-modeled posterior to construct a mock population of compact
starbursts. Panel (c): Galaxy cutout as seen in Figure 1.
|
# Benchmark Dataset for Precipitation Forecasting by Post-Processing the
Numerical Weather Prediction
Taehyeon Kim , Namgyu Ho∗, Donggyu Kim, Se-Young Yun
KAIST AI
Seoul, South Korea
{potter32, itsnamgyu, eaststar<EMAIL_ADDRESS>
Equally contributed.Mainly contributed to our open-source Python code
development.
###### Abstract
Precipitation forecasting is an important scientific challenge that has wide-
reaching impacts on society. Historically, this challenge has been tackled
using numerical weather prediction (NWP) models, grounded on physics-based
simulations. Recently, many works have proposed an alternative approach, using
end-to-end deep learning (DL) models to replace physics-based NWP models.
While these DL methods show improved performance and computational efficiency,
they exhibit limitations in long-term forecasting and lack the explainability.
In this work, we present a hybrid NWP-DL workflow to fill the gap between
standalone NWP and DL approaches. Under this workflow, the outputs of NWP
models are fed into a deep neural network, which post-processes the data to
yield a refined precipitation forecast. The deep model is trained with
supervision, using Automatic Weather Station (AWS) observations as ground-
truth labels. This can achieve the best of both worlds, and can even benefit
from future improvements in NWP technology. To facilitate study in this
direction, we present a novel dataset focused on the Korean Peninsula, termed
KoMet (Korea Meteorological Dataset), comprised of NWP outputs and AWS
observations. For the NWP model, the Global Data Assimilation and Prediction
Systems-Korea Integrated Model (GDAPS-KIM) is utilized. We provide analysis on
a comprehensive set of baseline methods aimed at addressing the challenges of
KoMet, including the sparsity of AWS observations and class imbalance. To
lower the barrier to entry and encourage further study, we also provide an
extensive open-source Python package for data processing and model
development. Our benchmark data and code are available at
https://github.com/osilab-kaist/KoMet-Benchmark-Dataset.
## 1 Introduction
Precipitation forecasting is an ardous problem of forecasting the specific
range of region rainfall based on current observations, from sources including
radar [47; 54], satellites [28], and rain gauges [54]. Forecasting plays a
vital role in society, aiding in a wide range of weather-dependant decision-
making, including large-scale crop management, marine services, and air
traffic control [72; 69; 37; 23]. A Numerical Weather Prediction (NWP) model
[64] is a general tool for weather forecasting, which involves calculating
complex physical processes pertaining to interactions across large time and
space scales, spanning the Earth’s atmosphere, ocean, land, and ice [36; 26;
4]. Despite continuous efforts to enhance NWP models [57], erroneous
predictions can occur due to the delicacy of the models with respect to the
noise in the initially observed state.
The weather and climate research community is becoming more aware of
contemporary deep learning (DL) technologies, and numerous attempts have been
made to apply them to the task of precipitation forecasting. Most DL-based
methods directly utilize radar observations to predict future precipitation.
Despite the lack of expert knowledge or explicit physics-based modeling, these
methods achieve notable performance while being computationally efficient [56;
1]. Ravuri et al. [47] facilitate the performance of observation-driven
methods with a deep generative model that learns probability distributions of
the data and allow for easy generation of samples from their learned
distributions. Espeholt et al. [16] extend the capability of precipitation
forecasting up to twelve hours ahead, surpassing the performance of existing
physics-based models widely used in the Continental United States. Despite its
significance for instantaneous forecasting, it is shown that pure observation-
driven DL models underperform those that are supplemented by NWP outputs when
lead time is larger than 6 hours, revealing the limitations of DL in long-term
forecasting. Furthermore, it still lacks explainability and explicit
consideration for physics constraints in the end-to-end DL framework [53].
This paper presents a novel benchmark dataset for a hybrid workflow that
combines the strengths of both NWP and DL approaches. Our new benchmark,
termed KoMet (Korea Meteorological dataset), consists of two parts:
predictions from an NWP model, specifically, the Global Data Assimilation and
Prediction Systems-Korea Integrated Model (GDAPS-KIM), and Automatic Weather
Station (AWS) observations, whose rainfall amounts are hourly monitored in
areas covering 13km around the center. The data is provided by the Korean
Meteorological Administration (KMA), for July and August of 2020 and 2021–two
months in which most precipitation in the Korean Peninsula occurs. Our dataset
takes into account of the characteristics of the Korean Peninsula whose
precipitation is difficult to predict due to congested spatio-temporal
variability owing to complex terrain and the Asian monsoon season [42; 9]. For
example, its complex topography leads to large variation in annual
precipitation in the southern part (1000 mm to 1800 mm) than the central part
(1100 mm to 1400 mm) [2]. Under the proposed workflow, GDAPS-KIM predictions
are used as inputs to a DL model which is trained to output refined
precipitation forecasts. AWS observations are used as ground-truth targets to
train the deep model. In nutshell, the overall goal is to post-process the
predictions from GDAPS-KIM using deep models, under the supervision of AWS
observations.
To date, many benchmark datasets have been introduced to facilitate existing
approaches in precipitation forecasting–those based on NWP and end-to-end DL
[70; 44]. We advocate an alternative approach, in which NWP output is post-
processed using DL models for a more refined prediction. This enables the
prediction to respect physics-related constraints and is partially explainable
by the injected NWP predictions. To catalyze future work, we also provide
extensive evaluation of several baseline models and learning methods over our
KoMet dataset. Our key contributions include:
* •
KoMet is the most comprehensive public dataset for DL post-processing of NWP
outputs. The spatial size of GDAPS-KIM is 50 x 65 covering the terrain and
surrounding sea whose each pixel covers 12km2 and 484 AWS (ground-truth
pixels) are installed in such resolution.
* •
Hybrid NWP-DL Baselines. We present and analyze a comprehensive set of
baseline methods for our proposed hybrid approach, and provide a suite of
Python modules designed to facilitate further research from the ML community.
#### Post-NWP Optimization.
We refer to the optimization problem implicit in the post-processing of NWP
model ouput for precipitation forecasting as post-NWP optimization, drawing a
connection to skillful precipitation optimization. Post-NWP optimization has
several key properties that differentiates it from a typical weather-forecast
optimization problem (Figure 1):
* •
Influential Variable Selection. In NWP, each pixel on grid has a list of
variables expressing the state regarding atmospheric feature (Table 1).
* •
Sensitivity to Hyperparameter Settings. NWP data can be regarded as tabular
data. In this sense, as Kadra et al. [25] observe, finding the optimal
cocktail with tuning the modern regularization techniques can boost the
performance comparable to the state-of-the-art models in end-to-end DL models.
* •
Robust Architectures. We expect a new dedicated architecture by considering
the spatial and temporal characteristics of our benchmark.
* •
Non-Learning Points of AWS. AWS is not installed at all locations due to
space and cost limitations. Therefore, its value for some regions are sparse,
which needs to be overcome for the supervision of NWP.
* •
Class-Imbalance. The distribution of precipitation amounts are highly
asymmetric. Despite rare occurrences of heavier rain, it can bring significant
real-world damages.
Figure 1: Overview of challenges for KoMet. The central part shows our main
workflow; as GDAPS-KIM simulations are released timely, a deep neural network
aims to post-process the model output for rain forecast. Challenges can be
categorized into five folds: (1) influential variable selection that selects
the key features for data pre-processing, (2) sensitivity to hyperparameter
settings, (3) robust architectures, (4) non-learning points on the ground-
truth labels, and (5) class-imbalance issues where the number of precipitation
areas are fewer than the number of other cases.
#### Organization.
The remainder of this paper is organized as follows. In Section 2, we discuss
the related literature on traditional NWP approaches, data assimilation, and
deep learning-based weather forecasting. In Section 3, we describe our
benchmark KoMet thoroughly. In Section 4, we formulate the problem settings
for the use of KoMet, and Section 5 exhibits the baseline experimental
results. Finally, Section 6 concludes the paper.
## 2 Related Works
#### Numerical Weather Prediction (NWP).
NWP models explain and predict the future changes of the atmospheric
conditions through solving dynamics and physics partial differential
equations. Since the advent of the NWP models, weather forecasting highly
relies on them according to their steady progress over time [33; 4]. NWP
applications have been involved into air quality [76], solar [41; 75; 67],
wind power [11; 8], wildfire [13], anomalies (i.e. severe weather events) [14;
35; 73; 50], and hydrological forecasting. The last of these, hydrological
forecasting, includes precipitation, humidity [15], drought [39; 62], and
evapotranspiration [34] forecasting. Despite its raving utility in real world,
NWP faces some endemic challenges [64; 4]. Firstly, the quality of NWP models
can be determined with how accurate the given observations of present
atmospheric states (initial condition) is [27; 58; 68; 38]. In addition, the
computational and power demands of NWP grow in lockstep with the resolution of
the forecast, creating a trade-off between forecast accuracy, which
necessitates higher levels of resolution, and forecasting time. Lastly, some
models utilize the machine learning approach for prediction while there
remains a lack of considering two types of uncertainty: aleatoric uncertainty
and epistemic uncertainty [12].
#### Data Assimilation (DA).
DA is the process of combining every possible source of information for the
accurate prediction of each status. Generally, DA serves as an initial
condition for computing the NWP in weather forecasting [27; 36]. Various
observation platforms have been evolving through weather station, radar, and
satellite [73; 63; 15; 5; 6] whose observation equipment is also becoming more
sophisticated. To reinforce model uncertainty, variational data assimilation
methods have been emerged as a line of works [66; 40]. Data assimilation
methods [66; 40] have been done on such observations. Three-dimensional
variational methods (3DVar) [10; 17; 31], four dimentional approaches (4DVar)
[43], ensemble Kalman filter [21; 32] are general examples of DA techniques
used in global NWP workflow.
#### Deep Learning-based Weather Forecasting.
Deep learning rises as the one way to mitigate the shortcomings of NWP. DL
enables its success by conjugating vast amount of available data as well as
geometrically increased computational power of tensor processing units (e.g.
GPUs or TPUs) [24]. Because deluge of meteorological data has been accumulated
since the establishment of weather stations, there are an increasing demand of
efforts to adopt modern DL techniques for weather forecasting [56; 53; 48;
51]. DL-based weather forecast approach can be categorized into two folds:
end-to-end DL approach and NWP-DL hybrid approach. End-to-end DL approaches
[30; 46; 49; 71; 74; 47] learn the representation from meteorological
observations using DNN without the access of physics knowledge. On the other
side, NWP-DL hybrid approaches [45; 71; 19; 65; 20; 22; 20] integrate
theoretical models (NWP) and data-driven models (DL), and thus encompass the
explainability as well as accurate expressivity.
Table 1: List of variables contained in the benchmark dataset. Pres data contains atmospheric features at different pressure levels, and Unis data contains some other features as well as those in Pres data but does not at certain pressure levels. † indicates the variable with integer type. Type | Long name | Short name | Description | Unit
---|---|---|---|---
Pres | U-component of wind | u | Wind in x/longitude-direction | ($ms^{-1}$)
V-component of wind | v | Wind in y/latitude direction | ($ms^{-1}$)
Temperature | T | Temperature | (K)
Relative humidity | rh liq | Humidity relative to saturation | (%)
Geopotential | hgt | Proportional to the height of a pressure level | ($m^{2}s^{-2}$)
Unis | Rain | rain | Rain | ($mm/h$)
Snow | snow | Snow | ($cm/h$)
Height of PBL | hpbl | Height of planetary boundary layer | (km)
Type of PBL† | pbltype | Type of planetary boundary layer | -
2m specfic humidity | q2m | Specific humidity at 2m height | (g/kg)
2m relative humidity | rh2m | Humidity relative to saturation at 2m height | (%)
2m temperature | t2m | Temperature at 2m height above surface | (K)
Surface temperature | tsfc | Surface temperature | (K)
10m u component of wind | u10m | Wind in x/longitude-direction at 10m height | ($ms^{-1}$)
10m v component of wind | v10m | Wind in y/latitude-direction at 10m height | ($ms^{-1}$)
Topography | topo | Topography | (m)
Pressure of surface | ps | Pressure of surface | (Pa)
## 3 KoMet Dataset
The KoMet dataset is comprised of predictions from GDAPS-KIM, a global
numerical weather prediction model operated by the Korea Meterological
Administration (KMA111https://www.kma.go.kr/eng/index.jsp), as well as AWS
observations which serve as ground-truth precipitation data.
### 3.1 Input: GDAPS-KIM
GDAPS-KIM is a global numerical weather prediction model designed to improve
the prediction accuracy of weather phenomena with a particular focus on the
Korean Peninsula [57]. GDAPS-KIM provides hourly predictions of various
atmospheric variables. This is achieved by assimilating observations from all
over the globe to compute the initial state, and subsequently solving dynamics
and physics equations to predict future states.
GDAPS-KIM prediction data is provided in the following format:
$\mathbf{X}\in\mathbb{R}^{T\times L\times C\times W\times H}$ where each
dimension corresponds to the origin time (T) at which the simulation took
place, lead time (L) between the origin time and the target prediction time,
the variable index (C), and the spatial dimensions of the prediction map (W,
H). Each GDAPS-KIM instance in KoMet dataset is a daily simulation executed at
00 UTC. We provide predictions made between July 1st and August 31st
(inclusive) of 2020 and 2021, i.e., $T=124$ days, when rainfall is most
intensive due to the seasonal characteristics of the region. We provide
predictions with lead times ranging from 0 to 89 hours, i.e., $L=90$. We
provide data on $C=122$ atmospheric variables for the Korean Peninsula and the
surrounding region lying within [32.94°N, 39.06°N] and [124.00°E, 132.00°E].
GDAPS-KIM has a resolution of 12 km $\times$ 12 km, and thus its spatial
dimension is 65$\times$50.
GDAPS-KIM variables can be categorized into two types: Pres variables and Unis
variables. While Pres variables include values for various altitudes
corresponding to specific air pressure levels (i.e., different isobaric
surfaces), Unis data does not. Each GDAPS-KIM instance has 5 Pres variable
types at 22 isobaric surfaces (22$\times$5=110 variables) and 12 Unis
variables (Table 1), thus it has 122 variables at each pixel. Despite its rich
variables, using all variables may not always be a silver bullet owing to the
high correlation among themselves and noisiness. Furthermore, as the
assimilated initial condition used for numerical prediction has significant
spatial- and temporal-variability, it is more prone to inaccuracies compared
to other regions, leading to cascading errors. In this paper, for the sake of
benchmark experiments, we use 12 variables222Unis 6 variables: rain, q2m,
rh2m, t2m, tsfc, ps; Pres 6 variables: T and rh_liq at 500, 700, 850 hpa. out
of 122 through the advice of KMA experts.
### 3.2 Ground-Truth: AWS Observations
Surface observations from AWS provide ground-truth data on hourly
precipitation. While AWS observation data includes various weather
measurements such as temperature, wind speed, and solar radiation, the
precipitation is only utilized for supervision. Unlike GDAPS-KIM predictions,
only several pixels have AWS observations as weather stations are sparsely
installed. Even for such stations, observations are occasionally omitted due
to miscellaneous reasons, leading to NaN values.
Raw AWS observation data is provided in the following format:
$\mathbf{Y}_{raw}\in\mathbb{R}^{h\times S}$ where each dimension corresponds
to the hour (h) at which the measurement of accumulated rainfall (from the
preceding hour) is taken and the weather station index (S). AWS observations
are recorded every hours at which GDAPS-KIM predictions are provided, with
some additional leeway. More precisely, hourly observations are ranging from
July 1st, 00:00 UTC to September 3rd, 23:00 UTC, totaling $h=3120$ hours, from
484 stations ($S=484$). To facilitate use of AWS data to train GDAPS-KIM post-
processing models, we use the location metadata of each weather station to
convert AWS data into the spatial grid format used in GDAPS-KIM. The format is
$\mathbf{Y}\in\mathbb{R}^{h\times W\times H}$ where each dimension corresponds
to the hour (h) at which the measurement of accumulated rainfall is taken, and
the spatial dimensions of the prediction map (W, H; identical to GDAPS-KIM).
We investigate the characteristics of AWS considering the sparsity of AWS
installation, temporal and spatial rainfall distributions. Figure 2 provides
the distributions of hourly and daily average of rainfall amounts on each
year. As Figure 2 shows, hourly rainfall events occur sporadically, and these
characteristics vary greatly from year to year. For the spatial distribution,
among the $65\times 50=3250$ pixels comprising the GDAPS-KIM grid, there are
only 484 pixels where ground-truth data is available, and even many of those
are concentrated to certain locations such as metropolitan areas (Figure 3).
These challenges can pose a significant challenge for training calibration
models, as the supervisory signal from ground-truth values is limited to
specific locations and time.
(a) Precipitation Ratio by Date (2020)
(b) Precipitation Ratio by Date (2021)
(c) Precipitation Ratio by Hour (2020)
(d) Precipitation Ratio by Hour (2021)
Figure 2: Temporal distribution of the rain ratio among all AWS observations.
‘Rain’ refers to hourly precipitation between 0.1mm and 10mm, while ‘heavy
rain’ refers to that above 10mm. In (c), (d), the left axis refers to the
ratio of ‘rain’ and the right axis is the ratio of ‘heavy rain’, in
percentages.
(a) No Rain Ratio (2020)
(b) Rain Ratio (2020)
(c) Heavy Rain Ratio (2020)
(d) No Rain Ratio (2021)
(e) Rain Ratio (2021)
(f) Heavy Rain Ratio (2021)
Figure 3: Spatial distribution of precipitation from AWS observations in South
Korea. ‘Rain’ refers to hourly precipitation between 0.1mm and 10mm, while
‘heavy rain’ refers to that above 10mm.
### 3.3 Dataset Interface
Table 2: Statistics in the KoMet benchmark. Rain rate (mm/h) | Proportion (%) | Rainfall Level
---|---|---
$[0.0,0.1)$ | 87.24 | No Rain
$[0.1,10.0)$ | 11.57 | Rain
$[10.0,\infty)$ | 1.19 | Heavy Rain
#### Inputs.
In our dataset, GDAPS-KIM predictions are provided in NumPy format. These
predictions are propagated to deep models, following normalization; each
variable is linearly scaled based on min-max values derived from the entire
dataset.
#### Outputs.
We formulate the precipitation forecast problem as a pointwise classification
task pertaining to three categorical classes: non-rain, rain, and heavy rain
(Table 2). Following this, the AWS observation data is pre-processed into 2D
array format according to the grid used in GDAPS-KIM, respectively. The
location of each station is determined within each grid based on the location
metadata of AWS stations and grid specifications for GDAPS-KIM. Of course, our
benchmark also supports predicting rainfall as a regression task or
classification tasks having more categories. However, the evaluation should be
based on the Table 2, which is generally used in South Korea [59].
### 3.4 Dataset Split
We split the data temporally into three non-overlapping datasets by repeatedly
using approximately 4 days for training followed by 2 days for validation and
2 days for testing. Separation by month or year could cause a shift in
distribution between datasets. With reference to Sønderby et al. [60], this
category of temporal split is utilized. Because the data scale is small, more
proportions are devoted to validation and testing compared to Sønderby et al.
[60].
Due to the overlap between the predicted instances at the same time from
different GDAPS-KIM simulations, it is not straightforward to perform this
split. For example, there are multiple predictions for April 21st, 2021
2:00PM, including a prediction ran at midnight the same day with lead time of
14 hours, and one ran at midnight of April 20th with 38 hour lead time, to
name a few. In this paper, we follow the strategy that divides among
prediction simulations, based on the point in time in which the predictions
were ran. Since predictions from GDAPS-KIM simulations are highly conditioned
on the initial state, our benchmark uses the aforementioned scheme to prevent
predictions from the same initial state to leak into different data splits.
## 4 Optimization for Post-Processing the NWP Outputs
#### Problem Formulation.
In this work, we consider the following optimization model:
$\displaystyle\min_{{\bm{w}}}\Big{\\{}\mathcal{L}({\bm{w}};\mathcal{D})\triangleq\mathbb{E}_{(\mathbb{X}_{t},\mathbb{Y}_{t})\sim\mathcal{D}}[\ell(\mathbb{X}_{t},\mathbb{Y}_{t};{\bm{w}})]\Big{\\}}\vspace{-10pt}$
(1)
where $\mathcal{L}$ is the objective function parameterized by ${\bm{w}}$ on
the dataset $\mathcal{D}$ having the input as the NWP outputs
$\mathbb{X}_{t}$, the corresponding rainy output $\mathbb{Y}_{t}$ at time $t$,
and $\ell$ is the classification loss between the output of the neural network
and the ground-truth. $\mathbb{X}_{t}$ is a sequence of NWP predictions
printed at time $t$,
${\bm{x}}_{(t,0)},{\bm{x}}_{(t,1)},\cdots,{\bm{x}}_{(t,L-1)}$ where
${\bm{x}}_{(t,i)}$ is the prediction at time $t+i$ based on a simulated NWP
output at time $t$, $L$ is the length of lead time which means the prediction
of additional hour per simulation, $\mathbb{Y}_{t}$ is a sequence of AWS
observations ${\bm{y}}_{t},{\bm{y}}_{t+1},\cdots,{\bm{y}}_{t+L-1}$. In this
approach, the prediction of the neural network at time $t$ is defined as the
following probabilistic forecast:
$\displaystyle
f(\tau;\mathbb{X}_{t_{0}},{\bm{w}},ws)=\mathbb{P}({\bm{y}}_{\tau}|{\bm{x}}_{(t,\Delta-
ws+1)},\cdots,{\bm{x}}_{(t,\Delta-1)},{\bm{x}}_{(t,\Delta)})\vspace{-10pt}$
(2)
where $f(\cdot;{\bm{w}})$ is the neural network parameterized by ${\bm{w}}$,
$t$ is the time at which the forecast is made, $\Delta=\tau-t_{0}$, $ws$ is
the window size which is the number of consequent observations used for
inference, and $\mathbb{P}(\cdot)$ is the probability matrix whose each pixel
in grid indicates the probability of class among {‘non-rain’, ‘rain’, ‘heavy
rain’}.
#### Algorithm Description.
Here, we describe one inference step (say the $\tau$-th; $\tau\geq t$) of the
proposed post-NWP optimization task. Firstly, the subset $\mathbb{X}_{t}$ is
reorganized with channel-wise concatenation: $[{\bm{x}}_{(t,\Delta-
ws+1)};\cdots;{\bm{x}}_{(t,\Delta-1)};{\bm{x}}_{(t,\Delta)}]\in\mathbb{R}^{(C*ws)\times
W\times H}$ where $W,H$ is the width and height of the grid map and $C$ is the
number of variables in each pixel–and thus each instance after reorganization
is 3D-shaped. We then create batches based on the modified $\mathbb{X}_{t}$
and perform forward and backward propagation. The model is trained over the
entire dataset until convergence, or pre-fixed total epochs.
#### Temporal Dependencies on Window Size & Lead Time.
The problem formulation of post-NWP optimization elicits two important
temporal parameters that influence the learning problem. First, the window
size of NWP predictions used for post-processing affects the richness of the
input information. While larger window size allows the post-processing model
to consider ample information, it may be more difficult to optimize the model.
Another important temporal component is lead time. Predictions from different
lead times exhibit distinct characteristics, leading to differences in
evaluation performance. Taking this into consideration, it may be beneficial
to train separate post-processing models for different lead times. For our
baseline analysis, we focus on a single model trained to handle all possible
lead times, as a representative baseline.
#### Evaluations.
For evaluation, the validity of forecasting algorithms are assessed using
diverse statistical metrics, which are commonly used for precipitation
forecasting. Following common practice in the multi-classification settings
[18], such metrics can be calculated based on the number of true positives
($TP_{k}$), false positives ($FP_{k}$), true negatives ($TN_{k}$), and false
negatives ($FN_{k}$) for some generic class $k$:
* •
Accuracy (ACC) returns an overall measure of how much the model is correctly
predicting on the entire set of data.
* •
Probability of Detection (POD) is a recall calculated as
$\frac{TP_{k}}{TP_{k}+FP_{K}}$.
* •
Critical Success Index (CSI) [14] is a categorical score that considers more
aspects of the confusion matrix similar with F1-score having the value as
$\frac{TP_{k}}{TP_{k}+FN_{k}+FP_{k}}$.
* •
False Alarm Ratio (FAR) [3] is the number of false alarms per the total number
of warnings or alarms, also known as the probability of false detection. It is
computed as $\frac{FP_{k}}{TP_{k}+FP_{k}}$
* •
Bias is defined as the ratio of the observed frequency of occurrence of a
phenomenon to the frequency of the occurrence predicted by the forecast model.
$\frac{TP_{k}+FP_{k}}{TP_{k}+FN_{k}}$. If it has a value greater than 1, it
means that the frequency of occurrence predicted by the forecasting model is
greater than the frequency of occurrence of the actual phenomenon, and
therefore more frequent prediction is made. The more accurate the forecast,
the closer this index is to 1.
## 5 Experiment
Table 3: Evaluation metrics of KoMet and baseline models for precipitation while 12 variables are utilized for the training. Best performances are marked in bold. | Rain | Heavy Rain
---|---|---
| Acc | POD | CSI | FAR | Bias | Acc | POD | CSI | FAR | Bias
GDAPS-KIM | 0.747 | 0.633 | 0.263 | 0.690 | 2.042 | 0.985 | 0.055 | 0.045 | 0.795 | 0.266
U-Net | 0.840 | 0.441 | 0.282 | 0.562 | 1.007 | 0.983 | 0.040 | 0.029 | 0.906 | 0.426
ConvLSTM | 0.869 | 0.387 | 0.296 | 0.444 | 0.696 | 0.986 | 0.007 | 0.006 | 0.889 | 0.059
MetNet | 0.854 | 0.468 | 0.314 | 0.512 | 0.959 | 0.986 | 0.013 | 0.012 | 0.838 | 0.079
Figure 4: CSI scores of GDAPS-KIM and baseline models for a class ‘rain’.
Scores are given for predictions corresponding to specific lead times, ranging
from 6 to 20 hours.
(a) Acc
(b) CSI
(c) Bias
Figure 5: Performance of the models according to the changes of sampling
strategies. Vanilla indicates baseline training strategy without resampling of
AWS targets. The metrics are averaged over 10 different random seeds and error
bars indicate the 95% confidence intervals.
(a) U-Net, window size
(b) ConvLSTM, window size
(c) MetNet, window size
(d) U-Net, weight decay
(e) ConvLSTM, weight decay
(f) MetNet, weight decay
Figure 6: (a), (b), (c): CSI performance according to changes in window size
in hours; (d), (e), (f): CSI performance according to changes in weight decay.
### 5.1 Benchmarking neural network architectures
Various models have been adopted or designed to effectively extrapolate future
weather status from past and current state. Here, we provide three baseline
architectures for precipitation forecasting: U-Net [52], ConvLSTM [47], and
MetNet [60; 16] (detailed explanations are provided in Appendix).
Table 3 shows the results for various lead times ranging from 6 to 87 hours,
according to the changes of the architectures. Compared to the statistics of
GDAPS-KIM for predicting class ‘rain’, the qualities of post-processed
predictions with the three networks are consistently improved while that of
MetNet is the best. In contrast, deep learning-based post-processing rather
hinders performance for ‘heavy rain’ prediction. In particular, MetNet faces
severe performance degradation. We leave this phenomenon as a challenge to be
addressed. Figure 4 illustrates the results of CSI scores for precipitation of
$\geq 0.1$ (mm/h) over lead time. Here, MetNet also outperforms other baseline
models.
### 5.2 Influential Variable Selection
As listed in Table 1, GDAPS-KIM contains many variables. Naively using all
available variables may impede generalization, therefore it is important to
select the most prominent variables. Variable selection experimentation can be
done in various ways, and we propose a baseline experiment in which the
variables are divided into Pres type and Unis type groups and considered as
manipulated variables and control variables, respectively. An example is
illustrated in the Appendix, and we observe that selecting certain significant
variables performs better than simply using more variables.
### 5.3 Sampling Strategies for Class-Imbalance Issue
As a solution to the class-imbalance issue, we provide two probabilistic data
sampling techniques for target transformation: (1) Under-Sampling for ‘No
Rain’ Points that uses only a fraction $p$ of the ‘no rain’ points used for
learning, to reduce the no precipitation points, and (2) Balancing for ‘Rain
Points’ that maintains the ratio between ‘no rain’ points and ‘rain’ points to
$1:p$ by under-sampling ‘no rain’ points, when there are excessive ‘no rain’
points.
Figure 5 depicts the performance of models trained with different sampling
strategies. Although the performance change differs depending on the
architecture, it can be confirmed that generalization can be further
facilitated if the data distribution is adjusted fairly. It is expected that
an approach that manipulates the loss function can also be effective [29; 7].
### 5.4 Sensitivity to Hyperparameter Settings
#### Window Size.
The first row of Figure 6 shows the CSI scores as the window size changes. In
‘rain’, the CSI score is relatively unchanged, while the ‘heavy rain’ case is
shown to be extremely sensitive.
#### Weight Decay.
The second row of Figure 6 shows the CSI scores according to the changes in
magnitude of weight decay. When comparing the performance of rain and heavy
rain, we see differing trends according to increase in weight decay.
The results above suggest the need for combinatorial optimization among a
variety of hyperparameters such as batch size, optimizer, beta of batch
normalization, and learning rate.
## 6 Conclusion & Future Work
In this paper, we present a new benchmark dataset, termed KoMet, for a hybrid
NWP-DL workflow that post-processes the rain prediction from South Korea’s NWP
models under the supervision of surface-level observations from AWS. Our work
engages in the solution of precipitation forecasting. We believe that our work
opens the door to develop robust and explainable forecasting services that
benefit the broad society. Our benchmark dataset will lower the barriers to
entry for those in the DL community to contribute to research in precipitation
forecasting.
## Acknowledgments and Disclosure of Funding
This work was supported by the Korea Meteorological Administration Research
and Development Program "Development of AI techniques for Weather Forecasting"
under Grant (KMA2021-00121) and Institute of Information & communications
Technology Planning & Evaluation (IITP) grant funded by the Korea
government(MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School
Program(KAIST)). We thank Jaehoon Oh and Sangmook Kim for discussing the
pipelines of precipitation forecasting and Yun Am Seo (Jeju National
University) and Min-Gee Hong (National Institute of Meteorological Sciences)
for discussing the data pre-processing.
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## Appendix A Overview of Appendix
In this supplementary material, we present additional details, results, and
experiments that are not included in the main paper due to the space limit.
## Appendix B Access to Dataset
* •
URL: https://github.com/osilab-kaist/KoMet-Benchmark-Dataset
* •
Dataset URL: https://www.dropbox.com/s/qachyygl2ouuy1v/KoMet.v1.0.tar.gz?dl=0
* •
Hosting and maintenance plan: The dataset is provided via Dropbox. Our GitHub
repository provides tools for data access as well as model training and
evaluation for our proposed workflow. We will continue to ensure public access
to our dataset, and possibly provide the dataset through multiple hosting
services, if the need arises.
* •
License: We include the MIT License about the use of public GitHub repository.
## Appendix C Ethics Statement
To address potential concerns, we describe the ethical aspect with respect to
security and the environment.
#### Security.
Weather information is an important factor in a country’s economic and
military security. Nevertheless, general users do not have access to up-to-
date NWP prediction data, so if access to live NWP data is well protected, the
security problem will be sufficiently prevented.
#### Environment.
While our approach brings more benefits in generalizing precipitation
forecasting, it still relies on NWP simulations which require vast amounts of
computation, and thus energy consumption. The same is true for training and
running deep learning models for post-processing. However, we believe that the
positive impact of accurate weather forecasts on energy consumption and
planning far outweighs these costs and will lead to more environmentally
conscious energy policies.
## Appendix D Limitations and Future Directions
In this section, we describe the limitations of our work and future directions
for further development.
#### Limitations.
Although we illustrate the advantages of our benchmark dataset, the bottleneck
lies in the accuracy of GDAPS-KIM as well as AWS. We only consider the summer
season of South Korea, specifically, July and August. We do not investigate
other NWP datasets. We also do not consider the precipitation nowcasting task,
with lead times of less than 3 hours, as this would limit the deep model’s
window size to less than 3, possibly hindering model performance.
#### Future Directions.
In future work, we aim to explore more robust training strategies under the
data scarcity scenario. Furthermore, thorough investigation is needed for the
selection of important variables in Table 1. In addition, we intend to develop
our methods in other seasons such as Spring, Fall, and Winter. Lastly, we plan
to explore methods to tackle precipitation nowcasting, using a limited window
size.
## Appendix E Additional Detailed Explanations
* •
Thresholds for Rain and Heavy Rain. KMA defines that rain occurs when the
rainfall is over 0.1mmh-1 and heavy rain occurs when the rainfall is over
10mmh-1. Literature based on the precipitation forecasting of the Korea
Peninsula follows this standard [59, 61].
* •
U-Net [52] is a model designed to solve the image segmentation problem in
biomedical images. During the propagation of the encoder part, important
features can be captured in a low-dimensional form. Applying this to the task
of NWP post-processing, U-Net is used to extract the meteorological features
from GDAPS-KIM and decode them into a feature map containing precipitation
predictions, similar to [47]. Note that [47] used radar observations as
inputs, rather than NWP predictions.
* •
Convolutional Long Short-Term Memory Neural Network (Conv-LSTM) [55, 56] is a
model that combines LSTM and convolutional operations, each designed to model
temporal and spatial relationships, respectively, within sequences of images.
This enables the model to identify relationships between sequential NWP
prediction maps to derive a refined predictions for precipitation. The model
structure consists of encoding, decoding and forecasting modules comprised of
stacks of ConvLSTM layers.
* •
MetNet. [60, 16] In MetNet, the spatial downsampler module first reduces the
size by passing the input through several convolutional layers. Next, the
ConvLSTM structure is used in the temporal encoder to create an output tensor
with spatial and temporal information for each pixel. Lastly, such propagated
feature map goes through the self-attention block of the Spatial Aggregator,
collects information in the global range, and finally passes through a
classifier to output the precipitation amount for each pixel as a probability
distribution.
## Appendix F Implementation Details
The exact implementation details and reproduction instructions for our
experiments are available on our GitHub repository. We explain important
details in the following paragraphs.
#### Data Preprocessing.
We refer to the settings in [16]. For the transformation of GDAPS-KIM, we only
apply Min-Max normalization to each variable. We employ various
transformations on the ground-truth AWS targets to mitigate the sparsity of
observations. In all of our experiments, we apply linear interpolation between
available observations to fill in the non-learning points, using functions
provided by the scipy package. This is illustrated in Figure 7. To tackle the
class imbalance issue, we study two sampling strategies in subsection 5.3.
Details of sampling strategies are explained in the following paragraphs.
(a) Original
(b) Interpolated
Figure 7: Visualization of AWS observations of precipitation before and after
interpolation, on April 23, 2020 06:00 UTC. The unit of precipitation is mm/h.
#### Under-Sampling for ‘No Rain’ Points.
As the ratio of ‘no rain’ is significantly higher than that of ‘rain’ and
‘heavy rain’, we consider a simple sampling strategy of randomly sampling a
subset of ‘no rain‘ points within each sample, at a rate of $p$. The remaining
‘no rain’ points are discarded during training, i.e., set to nan. We use a
sampling ratio of $p=0.20$.
#### Balancing for ‘Rain’ Points.
This technique considers the ratio of ‘rain’ and ‘heavy rain’ to ‘no rain’
within each feature map. Specifically, let $R$ be the number of points with
‘rain’ or ‘heavy rain’, and $N$ be the number of ‘no rain’ points. Given the
target rate $p$, we randomly sample ‘no rain’ points to match $p=N/R$ as
closely as possible. We also use a sampling ratio of $p=0.20$ for this method.
The specific strategy applied to each sample is as follows:
* •
Case 1. $R=0$: The sample is not used for training.
* •
Case 2. $(R×p)<N$: $(N-R×p)$ ‘no rain’ points are discarded from the sample.
* •
Case 3. $(R×p)>=N$: The sample is used as is.
#### Training Settings.
We use the Adam optimizer with the default learning rate of 0.001, and beta
parameters of 0.9 and 0.999. We train the models for 20 epochs and select the
best epoch based on the highest CSI performance on the validation set.
Reported performances are based on the test set, unless otherwise specified.
We use a window size of 3 and lead times ranging from 6 to 87 hours. We use
two Pres type variables (T, rh_liq) at three isobaric planes,{500hPa, 700hPa,
850hPa}; and six variables of Unis type (rain, q2m, rh2m, t2m, tsfc, ps).
These baseline settings are used in all experiments, unless otherwise stated.
#### Resources.
We run our experiments on NVIDIA GeForce RTX 3090 Ti GPUs. Each training run
takes approximately 30 minutes.
## Appendix G Additional Experiments
### G.1 Influential Variables
We conduct an ablation study to investigate the influence of each individual
variable. We start with a baseline set of variables, curated under the
guidance of weather forecasters in South Korea. This includes all baseline
variables used in our other experiments, as well as one additional Unis
variable: pbltype. We then measure the performance of our baseline U-Net model
when adding or subtracting individual variables used for training. We consider
the u, v and u10m, v10m variables jointly, as they both pertain to wind. For
Pres variables, we use values corresponding to three isobraic surfaces at
{500hPa, 700hPa, 850hPa}. We report our results in Table 4 and find that while
some metrics such as accuracy are relatively unaffected, others such as CSI
and bias vary depending on the choice of variables.
Table 4: Ablation study on the binary classification performance of our baseline U-Net model for ‘rain’ and ‘heavy rain’ depending on the selection of variables. Each row represents an addition (+) or a subtraction (-) of a single variable from our curated set. Type | Ablation | Rain | Heavy Rain
---|---|---|---
Acc | POD | CSI | FAR | Bias | Acc | POD | CSI | FAR | Bias
Curated | | 0.840 | 0.441 | 0.282 | 0.562 | 1.007 | 0.983 | 0.040 | 0.029 | 0.906 | 0.426
Pres | +uv | 0.859 | 0.385 | 0.280 | 0.493 | 0.759 | 0.983 | 0.042 | 0.031 | 0.896 | 0.407
-T | 0.857 | 0.388 | 0.279 | 0.503 | 0.782 | 0.984 | 0.037 | 0.030 | 0.874 | 0.296
-rh_liq | 0.854 | 0.444 | 0.302 | 0.515 | 0.915 | 0.984 | 0.018 | 0.015 | 0.930 | 0.264
+hgt | 0.851 | 0.392 | 0.272 | 0.530 | 0.834 | 0.983 | 0.046 | 0.035 | 0.880 | 0.388
Unis | +hpbl | 0.851 | 0.378 | 0.266 | 0.527 | 0.799 | 0.984 | 0.032 | 0.025 | 0.889 | 0.287
-pbltype | 0.840 | 0.483 | 0.300 | 0.558 | 1.093 | 0.983 | 0.041 | 0.031 | 0.891 | 0.378
+psl | 0.862 | 0.361 | 0.271 | 0.480 | 0.695 | 0.985 | 0.035 | 0.028 | 0.870 | 0.268
-q2m | 0.848 | 0.457 | 0.300 | 0.534 | 0.982 | 0.983 | 0.052 | 0.037 | 0.885 | 0.449
-rh2m | 0.838 | 0.489 | 0.301 | 0.561 | 1.112 | 0.983 | 0.065 | 0.046 | 0.860 | 0.462
-t2m | 0.846 | 0.460 | 0.299 | 0.540 | 1.000 | 0.983 | 0.044 | 0.032 | 0.892 | 0.409
-tsfc | 0.844 | 0.510 | 0.317 | 0.544 | 1.119 | 0.983 | 0.067 | 0.048 | 0.850 | 0.445
+uv10m | 0.843 | 0.447 | 0.288 | 0.553 | 0.999 | 0.984 | 0.049 | 0.039 | 0.837 | 0.299
+topo | 0.855 | 0.411 | 0.288 | 0.510 | 0.839 | 0.984 | 0.057 | 0.044 | 0.842 | 0.361
+ps | 0.852 | 0.445 | 0.300 | 0.521 | 0.927 | 0.985 | 0.026 | 0.022 | 0.884 | 0.227
### G.2 Lead Time on Heavy Rain Prediction
Figure 8 illustrates the results of CSI scores for precipitation of $\geq
10.0$ (mm/h) over lead time. Unlike the trends in rain prediction,
interestingly GDAPS-KIM outperforms other baseline DL models. It seems that
heavy rain prediction is a more ardous challenge to be addressed.
(a) Heavy Rain Figure 8: CSI scores of GDAPS-KIM and baseline models for
binary heavy rain classification. Scores are given for predictions
corresponding to specific lead times, ranging from 6 to 20 hours.
### G.3 Learning Rate
Figure 9 shows the CSI scores according to changes in learning rate. When
comparing the performance of ‘rain’ and ‘heavy rain’, we see different trends
based on increase in learning rate.
(a) U-Net
(b) ConvLSTM
(c) MetNet
Figure 9: CSI performance according to changes in learning rate.
### G.4 Learning Curve
We visualize the learning curves of the networks according to changes in
window size (Figure 10, Figure 11, Figure 12).
(a) U-Net, window size=1
(b) U-Net, window size=3
(c) U-Net, window size=6
Figure 10: U-Net: CSI learning curve according to window size.
(a) Conv-LSTM, window size=1
(b) Conv-LSTM, window size=3
(c) Conv-LSTM, window size=6
Figure 11: Conv-LSTM: CSI learning curve according to window size.
(a) MetNet, window size=1
(b) MetNet, window size=3
(c) MetNet, window size=6
Figure 12: MetNet: CSI learning curve according to window size.
### G.5 Generalization for Different Times of the Month
We evaluate the deep models trained on our benchmark for different times of
the year, ranging from July 2020 to July 2021. Our benchmark dataset covers
only summer for 2020 and 2021, and thus it harbors concerns about the
generalization of different seasons. While the data of other months is not
provided owing the limit of license, it is available for the analysis of CSI
distribution according to the lead time as well as month, as a substitute of
raw data.
Figure 13 and Figure 14 show the CSI distribution by lead time and month from
ConvLSTM and U-Net, respectively. For the lead time, both models has larger
CSI values of early lead time than those of latter and the overall curve has
inflection points at similar time compared to the curve of rain ratio. On the
other hand, the curve of heavy rain CSI is significantly similar with the
graph of heavy rain ratio distribution on lead time. For the month, it can be
seen that the annual average rain CSI is near 0.3, but the performance is
extremely low in winter in both models. This seems to be the case, however,
because the amount of rain in winter is close to zero. This phenomenon is more
pronounced in heavy rain.
(a) Rain CSI by lead time
(b) Heavy rain CSI by lead time
(c) Rain CSI by month
(d) Heavy rain CSI by month
Figure 13: CSI distribution of ConvLSTM outputs from July 2020 to July 2021.
Orange line indicates the rain CSI of the model and the blue dot line is the
distribution of observed rain ratio from AWS.
(a) Rain CSI by lead time
(b) Heavy rain CSI by lead time
(c) Rain CSI by month
(d) Heavy rain CSI by month
Figure 14: CSI distribution of U-Net outputs from July 2020 to July 2021.
Orange line indicates the rain CSI of the model and the blue dot line is the
distribution of observed rain ratio from AWS.
### G.6 Training on Other Multiple Categorical Classes
Our framework allows users to specify categorical classes in training as they
wish if 0.1 and 10 are included in the threshold list. By referring to the
other regions [47, 16, 56], we conduct an experiments with different
categorical classes while the evaluation is verified with the standard
threshold for rain and heavy rain used in our framework. Followings are the
threshold lists with the number of classes:
* •
V1: [0.1, 2.0, 10.0], 4 classes
* •
V2: [0.1, 5.0, 10.0], 4 classes
* •
V3: [0.1, 0.5, 5.0, 10.0], 5 classes
* •
V4: [0.1, 0.5, 5.0, 10.0, 30.0], 6 classes
Table 5 shows the performance according to the changes of the threshold list.
As the class of middle rain is newly created, it can be seen that the CSI can
be changed. Furthermore, when the number of classes in the middle rain is
further increased, it can be seen that the csi of the heavy rain increases
significantly with some degradation in rain CSI.
Table 5: Evaluation metrics of KoMet with the MetNet for precipitation while 12 variables are utilized for the training, according to the changes of the number of categorical classes during training. | Rain | Heavy Rain
---|---|---
| Acc | CSI | Bias | Acc | CSI | Bias
V1 | 0.862 | 0.293 | 0.769 | 0.986 | 0.011 | 0.079
V2 | 0.858 | 0.322 | 0.946 | 0.986 | 0.016 | 0.097
V3 | 0.871 | 0.270 | 0.581 | 0.987 | 0.048 | 0.210
V4 | 0.862 | 0.274 | 0.705 | 0.984 | 0.032 | 0.286
### G.7 Pointwise Architecture without the Use of Spatial Information
We conduct an experiment with methods that do not use spatial information for
NWP correction. While our benchmark architectures include both spatial encoder
and temporal encoder, it is important to check the needs of spatial neural
processing. Table 6 shows the results for various lead times from 6 to 87
hours, according to the presence of spatial kernel operator in neural
networks. Here, we design pointwise architecture having no spatial kernel.
Pointwise architecture with LSTM indicates the modified version of ConvLSTM
whose spatial encoder is replaced with pointwise architecture. For the NWP
correction, as Table 6 shows, using only pointwise operator lead to worse
optima than our benchmark models. In other words, it can be seen that the
methodology considering the spatial information is more effective.
Table 6: Evaluation metrics of KoMet and pointwise architectures for precipitation while 12 variables are utilized for the training. | Rain | Heavy Rain
---|---|---
| Acc | CSI | Bias | Acc | CSI | Bias
Pointwise Architecture | 0.871 | 0.195 | 0.346 | 0.987 | 0.001 | 0.009
Pointwise Architecture + LSTM | 0.866 | 0.253 | 0.575 | 0.987 | 0.006 | 0.060
ConvLSTM | 0.869 | 0.296 | 0.696 | 0.986 | 0.006 | 0.059
|
# Particle Mean Field Variational Bayes
Minh-Ngoc Tran Discipline of Business Analytics, The University of Sydney
Business School Paco Tseng ARC Centre for Data Analytics for Resources and
Environments (DARE) Robert Kohn School of Economics, UNSW Business School
###### Abstract
Mean Field Variational Bayes (MFVB) is one of the most computationally
efficient Bayesian inference methods. However, its use has been restricted to
models with conjugate priors or those that allow analytical calculations. This
paper proposes a novel particle-based MFVB approach that greatly expands the
applicability of the MFVB method. We establish the theoretical basis of the
new method by leveraging the connection between Wasserstein gradient flows and
Langevin diffusion dynamics, and demonstrate the effectiveness of this
approach using Bayesian logistic regression, stochastic volatility, and deep
neural networks.
Key words: Bayesian computation, optimal transport, Bayesian deep learning
## 1 Introduction
The main challenge of Bayesian statistics is to conduct inference of a
computationally intractable posterior distribution,
$\pi(x)\propto\exp{\left(-U(x)\right)}$, $x\in\mathbb{R}^{d}$, generally only
known up a normalising constant. To solve this problem, there are two main
classes of computational methods that provide different approaches to
approximate $\pi$. The first one is Markov chain Monte Carlo (MCMC) methods
(Metropolis et al.,, 1953; Hastings,, 1970; Robert and Casella,, 1999). For
many years, MCMC has been the standard approach for Bayesian analysis because
of its theoretical soundness. The method constructs a Markov chain to produce
simulation consistent samples from the target distribution $\pi$. A general
MCMC approach is the Metropolis-Hastings algorithm that generates a Markov
chain by first generating a proposed state from a proposal distribution, then
using an acceptance rule to decide whether to accept the proposal or stay at
the current state (Robert and Casella,, 1999).
Another, and often more efficient, class of MCMC methods is based on the
Langevin dynamics
$\text{\rm d}X_{t}=-\frac{1}{2}\nabla U\left(X_{t}\right)\text{\rm
d}t+\text{\rm d}B_{t}$ (1)
where $\left\\{B_{t}\right\\}_{t\geq 0}$ is the Brownian process on
$\mathbb{R}^{d}$. This stochastic differential equation (SDE) characterises
the dynamics of the process $\\{X_{t}\\}_{t\geq 0}$ whose distribution, under
some regularity conditions on the potential energy $U(x)$, converge to $\pi$
as the stationary distribution. In practice, however, it is necessary to work
with a discretisation of the SDE in (1), whose the distribution might not
converge to $\pi$ (Roberts and Tweedie,, 1996). The Metropolis-Hastings
acceptance rule is then needed to correct for the error produced by the
discretisation. This method is known as the Metropolis-adjusted Langevin
algorithm (MALA) (Besag,, 1994; Roberts and Tweedie,, 1996).
The Metropolis-Hastings acceptance rule is necessary to guarantee for
convergence of MCMC methods, but it also prevents the use of MCMC in big data
and big model settings. This is because calculating the Metropolis-Hastings
acceptance probability requires the full data (however, see, e.g. Quiroz et
al., (2019) for speeding up MCMC using data subsampling), and that this
probability can easily get close to zero when the dimension $d$ is high. Other
limitations of MCMC methods include the need for a sufficient burn-in period
for the generated Markov chain to be distributed as $\pi$, and the absence of
effective stopping criteria for checking convergence. These limitations can be
circumvented by the Variational Bayes method at the cost of some approximation
accuracy.
Variational Bayes (VB) (Waterhouse et al.,, 1996; Attias,, 1999; Blei et al.,,
2017) emerges as an alternative approach to inference in complex posterior
distributions with large datasets. More recently, it grown in popularity due
to its ability to scale up in terms of both the model complexity and data
size. Different to MCMC, the VB method proposes a family of distributions
$\mathcal{Q}$, called the variational family, and then identifies within
$\mathcal{Q}$ the closest variational distribution to $\pi$ with respect to
KL-divergence, i.e.,
$q^{*}\in\arg\min_{q\in\mathcal{Q}}\left\\{\text{KL}\left(q||\pi\right)\right\\}.$
(2)
A representative example is Mean-Field VB (MFVB) (sec. 3) which imposes a
factorisation structure on the variational distributions, i.e.
$\mathcal{Q}=\left\\{q=\prod q_{i}\right\\}$, and $q^{*}$, if it exists and is
unique, is called the mean-field approximation of $\pi$. An obvious limitation
of MFVB is that it fails to capture the posterior dependence between the
factorized blocks. Despite this limitation, MFVB has been widely used in
applications; see, e.g. Wand et al., (2011), Giordani et al., (2013), Wang and
Blei, (2013) and Zhang and Zhou, (2017). Implementation of MFVB relies on
conjugate priors and the ability to calculate the associated expectations
(sec. 3). As a result, it is challenging to apply standard MFVB for some
simple models such as Bayesian logistic regression.
Our work aims at extending the scope of MFVB to makes it widely applicable by
combining MFVB with the Langevin dynamics and circumventing the main issues of
either method while maintaining the strengths of both, that is providing a
scalable Bayesian inference algorithm with theoretical guarantees. The new
method, called Particle Mean field VB (PMFVB), leverages the Langevin dynamics
to bypass the limitations in standard MFVB, and employs the theory of
Wasserstein gradient flows to establish its theoretical guarantee. Wasserstein
gradient flows are a fundamental element of the Optimal Transport (OT) theory
(Ambrosio et al.,, 2005; Villani et al.,, 2009), that quantifies the
dissimilarity between probability measures and introduces a differential
structure into the space of probability measures. Inspired by fluid dynamics,
Jordan et al., (1998) introduce the concept of the gradient flow of a
functional defined on this space, which is a continuous curve of probability
measures along which the functional is optimised. It turns out that the
gradient flow of KL-divergence functional is identical to the Langevin
dynamics (sec. 2). This important connection between Wasserstein gradient
flows and Langevin dynamics, and Stochastic Differential Equation (SDE) theory
in general, provides the theoretical foundation for our PMFVB procedure.
Our contribution. We study the KL-divergence functional on the space of
factorised distributions $\mathcal{Q}$ equipped with the 2-Wasserstein
distance, and show that the MFVB optimisation problem (2) has an unique
solution $q^{*}$. We propose an algorithm for approximating the optimal mean
field distribution $q^{*}$ by particles that are moved by combining the
classical MFVB framework with the Langevin diffusion. We show that the
distribution of these particles converges to $q^{*}$. We also study the
posterior consistency of $q^{*}$ in terms of the data size. The numerical
performance of PMFVB is demonstrated using Bayesian logistic regression and
Stochastic Volatility - the statistical models that classical MFVB methods, to
the best of our knowledge, have not been successfully applied to. We then
develop a variant of PMFVB for inference in Bayesian deep learning, in which
modifications to standard PMFVB are introduced to make PMFVB suitable for deep
neural networks. For Bayesian inference in deep neural networks, Stochastic
Gradient Langevin Dynamics (SGLD) of Welling and Teh, (2011) is among the most
commonly used methods. We discuss connections between PMFVB and SGLD, and
numerically compare their performance using both simulated and real datasets.
Code implementing the examples is available at
https://github.com/VBayesLab/PMFVB.
Related work. Our work builds on recent advances in the Optimal Transport
theory and Langevin dynamics. The most closely related work to ours is Yao and
Yang, (2022) who, by focusing on a particular MFVB framework for statistical
models with local latent variables, combine Wasserstein gradient flows with
MFVB for dealing with the intractability of the optimal MFVB factorised
distribution. Our PMFVB framework is more general than earlier works and can
be applied to any statistical models including deep learning. Galy-Fajou et
al., (2021) develop a particle-based Gaussian approximation that uses a flow
of linear transformations for moving the particles; the resulting curve of
Gaussian distributions can be viewed as an approximation of the Wasserstein
gradient flow of the KL divergence functional. Lambert et al., (2022) study
convergence results of Gaussian Variational Inference using the theory of
Bures-Wasserstein gradient flow. They use particles to realise the flow of
Gaussian approximations, and also extend Gaussian approximation to mixtures of
Gaussians approximation.
The variant of PMFVB for neural networks is related to SGLD and its variants
(Welling and Teh,, 2011; Chen et al.,, 2014; Li et al.,, 2016; Kim et al.,,
2022).
Notation. $\nabla f(x)$ denotes the gradient vector of scalar function $f$
defined on $\mathbb{R}^{d}$. For a vector-valued function $v$ defined on
$\mathbb{R}^{d}$, its divergence is $\nabla\cdot v(x)=\sum_{j}\frac{\partial
v_{j}(x)}{\partial x_{j}}$. $\Delta f(x)$ is the Laplacian of $f$, $\Delta
f(x)=\nabla\cdot(\nabla f(x))=\sum_{j}\frac{\partial^{2}f(x)}{\partial
x_{j}^{2}}$. For a generic set ${\cal X}\subset\mathds{R}^{d}$, we denote by
$\mathcal{P}({\cal X})$ the set of probability measures on ${\cal X}$; and for
a measure $q$, with some abuse of notation, we will denote by $q(\text{\rm
d}x)$ and $q(x)$ the probability measure and density function, respectively.
## 2 Preliminaries
This section collects the preliminaries on optimal transport theory and
Langevin diffusion that are used in the paper.
### 2.1 Wasserstein space and gradient flow
Consider a generic set ${\cal X}\subset\mathbb{R}^{d}$, let ${\cal
P}_{2}(\mathcal{X})$ be the set of absolutely continuous probability measures
on ${\cal X}$ with finite second moments. For any
$p,q\in\mathcal{P}_{2}(\mathcal{X})$, let
$\displaystyle W_{2}(p,q)$ $\displaystyle=$
$\displaystyle\min_{\gamma\in\Gamma(p,q)}\Big{\\{}\Big{(}\int_{\mathcal{X}\times\mathcal{X}}\|x-y\|^{2}\gamma(dx\times
dy)\Big{)}^{1/2}\Big{\\}}$ (3) $\displaystyle=$
$\displaystyle\min_{T:T_{\\#}p=q}\Big{\\{}\Big{(}\int_{\mathcal{X}}\|x-T(x)\|^{2}p(dx)\Big{)}^{1/2}\Big{\\}}$
(4)
be the 2-Wasserstein metric on $\mathcal{P}_{2}(\mathcal{X})$. Here,
$\Gamma(p,q)$ denotes the set of joint probability measures on
$\mathcal{X}\times\mathcal{X}$ with the marginals $p$ and $q$, and
$T_{\\#}(p)$ is the push forward measure of $p$, i.e.
$T_{\\#}(p)(A)=p(x:T(x)\in A),\;\;\;A\subset\mathcal{X}.$
The existence of (3) and (4) and their equivalence is well studied; see, e.g.,
Ambrosio et al., (2005). It is well-known that, equipped with this metric,
$\mathcal{P}_{2}(\mathcal{X})$ becomes a metric space, often called the
Wasserstein space, denoted by $\mathbb{W}_{2}(\mathcal{X})$. The Wasserstein
space $\mathbb{W}_{2}(\mathcal{X})$ has many attractive properties (Ambrosio
et al.,, 2005; Villani et al.,, 2009) that make it possible to perform
calculus on this space. In particular, $\mathbb{W}_{2}(\mathcal{X})$ can
viewed as a Riemannian manifold (Otto,, 2001) whose rich geometry structure
can be exploited to efficiently solve optimisation problems such as (2).
Consider the functional $F(q)=\text{\rm KL}(q\|\pi)$ defined on
$\mathbb{W}_{2}(\mathcal{X})$, with some fixed measure
$\pi\in\mathbb{W}_{2}(\mathcal{X})$. Jordan et al., (1998) propose the
following iterative scheme, known as the JKO scheme, to optimize $F(q)$. Let
$q^{(0)}\in\mathbb{W}_{2}(\mathcal{X})$ be some initial measure and
$\epsilon>0$. At step $k\geq 0$, define
$q^{(k+1)}:=\arg\min_{q\in\mathbb{W}_{2}(\mathcal{X})}\Big{\\{}F(q)+\frac{1}{2\epsilon}W_{2}^{2}(q,q^{(k)})\Big{\\}}.$
(5)
Denote by $\frac{\delta F}{\delta q}(q):\mathbb{R}^{d}\mapsto\mathbb{R}$ a
first variation at $q$ of the functional $F$, i.e.
$\lim_{\epsilon\to 0}\frac{F(q+\epsilon\xi)-F(q)}{\epsilon}=\int\frac{\delta
F}{\delta q}(q)\text{\rm d}\xi$ (6)
for all $\xi\in\mathbb{W}_{2}(\mathcal{X})$ such that $F(q+\epsilon\xi)$ is
defined. The first variation, defined up to an additive constant,
characterises the change of $F$ at $q$. Let $v^{(k)}(x)=\nabla\frac{\delta
F}{\delta q}(q^{(k)})(x):\mathbb{R}^{d}\mapsto\mathbb{R}^{d}$; from (6), it
can be shown that
$v^{(k)}(x)=\nabla\log\pi(x)-\nabla\log q^{(k)}(x).$ (7)
Jordan et al., (1998) prove that (see also (Santambrogio,, 2015, Chapter 8)
and (Ambrosio et al.,, 2005, Chapter 10)), as $\epsilon\to 0$, the discrete-
time solution $\\{q^{(k)}\\}_{k=0,1,...}$ from (5) converges to the
continuous-time solution $\\{q_{t}\\}_{t\geq 0}$ of the continuity equation
$\frac{\partial q_{t}(x)}{\partial
t}+\nabla\cdot\big{(}q_{t}(x)v_{t}(x)\big{)}=0$ (8)
with $v_{t}(x)=\nabla\frac{\delta F}{\delta
q}(q_{t})(x)=\nabla\log\pi(x)-\nabla\log q_{t}(x)$. For
$\psi_{t}(x)=\log(q_{t}(x)/\pi(x))$, by noting that $\int({\text{\rm d}\log
q_{t}(x)}/{\text{\rm d}t})q_{t}(x)dx=0$, we have
$\displaystyle\frac{\text{\rm d}F(q_{t})}{\text{\rm
d}t}=\int\psi_{t}(x)\frac{\partial q_{t}(x)}{\partial t}dx$ $\displaystyle=$
$\displaystyle-\int\psi_{t}(x)\nabla\cdot(q_{t}(x)v_{t}(x))\text{\rm d}x$
$\displaystyle=$ $\displaystyle{\mathbb{E}}_{q_{t}}<\nabla\psi_{t},v_{t}>$
$\displaystyle=$
$\displaystyle-{\mathbb{E}}_{q_{t}}\big{(}\|v_{t}(x)\|^{2}\big{)}<0,$
which justifies that the curve $\\{q_{t}\\}_{t\geq 0}$, called the gradient
flow, minimizes the KL functional $F(q)$.
### 2.2 Langevin Monte Carlo diffusion
Let $\pi(\text{\rm d}x)$ be a target probability measure defined on ${\cal
X}\subset\mathbb{R}^{d}$ with density $\pi(x)$. The Langevin diffusion is the
stochastic process $\\{L_{t}\\}_{t\geq 0}$ governed by the SDE
$\displaystyle L_{0}$ $\displaystyle\sim$ $\displaystyle p_{0}$
$\displaystyle\text{\rm d}L_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\nabla\log\pi(L_{t})\text{\rm d}t+\text{\rm
d}B_{t},\;\;\;t>0$ (9)
where $B_{t}$ is the $d-$dimensional Brownian motion and $p_{0}$ is an initial
distribution on ${\cal X}\subset\mathbb{R}^{d}$. Under some regularity
conditions on $U(x)=-\log\pi(x)$, this SDE has an unique solution which is an
ergodic Markov process with the invariant distribution $\pi(\text{\rm d}x)$
(Pavliotis,, 2014, Chapter 4).
Let $q_{t}(x)$ be the probability density (w.r.t. the Lebesgue measure on
$\mathbb{R}^{d}$) of $L_{t}$; then $\\{q_{t}\\}_{t\geq 0}$ is the unique
solution of the Fokker–Planck equation (often called the forward Kolmogorov
equation in the probability literature)
$\displaystyle\frac{\partial q_{t}(x)}{\partial t}$ $\displaystyle=$
$\displaystyle-\nabla\cdot\Big{(}q_{t}(x)\nabla\log\pi(x)\Big{)}+\Delta
q_{t}(x),\;\;t>0$ (10) $\displaystyle q_{0}(x)$ $\displaystyle=$
$\displaystyle p_{0}(x);$ (11)
see Pavliotis, (2014), Chapter 4. One can easily check that
$\nabla\cdot\Big{(}q_{t}(x)\nabla\log\pi(x)\Big{)}-\Delta
q_{t}(x)=\nabla\cdot\big{(}q_{t}(x)v_{t}(x)\big{)},$
with $v_{t}(x)=\nabla\log\pi(x)-\nabla\log q_{t}(x)$; hence, the Fokker–Planck
equation (10) is identical to the continuity equation in (8). Therefore the
curve $\\{q_{t}\\}_{t\geq 0}$ induced by the Langevin dynamics can be viewed
as a gradient flow that minimises some sort of a discrepancy between $q_{t}$
and $\pi$; see, Dalalyan, (2017) and Cheng and Bartlett, (2018).
For a fixed $h>0$, consider the following Langevin Monte Carlo (LMC) diffusion
which is a time-continuous discretization approximation of (2.2)
$\text{\rm d}X_{t}^{h}=\frac{1}{2}\nabla\log\pi(X_{\tau(t)}^{h})\text{\rm
d}t+\text{\rm d}B_{t},\;\;\;t\geq 0,$ (12)
where $\tau(t):=kh$ if $t\in[kh,(k+1)h)$. Equation (12) implies that, at the
time points $kh$, $k=0,1,...$, we have
$X_{(k+1)h}^{h}=X_{kh}^{h}+\frac{h}{2}\nabla\log\pi(X_{kh}^{h})+\sqrt{h}\eta_{k},\;\;\eta_{k}\stackrel{{\scriptstyle
iid}}{{\sim}}N(0,I_{d}).$ (13)
Denote by $\mu_{t}^{h}$ the distribution of $X_{t}^{h}$, $t\geq 0$, from the
LMC diffusion (12). Cheng and Bartlett, (2018) prove the following lemma,
which says that the functional $F(\cdot)$ is reduced along the curve
$\\{\mu_{t}^{h}\\}_{t\geq 0}$.
###### Lemma 1.
[Cheng and Bartlett, 2018, Lemma 1] Suppose that $U(x)=-\log\pi(x)$ is
strongly convex and has a Lipschitz continuous gradient. That is, there exist
constants $c>0$ and $C>0$ such that
$cI_{d}\leq\nabla^{2}U(x)\leq CI_{d},\;\;\;\text{ for all }x\in{\cal X}.$
Let $F(\mu_{t}^{h})=\text{\rm KL}(\mu_{t}^{h}\|\pi)$. Then, when the step size
$h$ is sufficiently small
$\frac{\text{\rm d}F(\mu_{t}^{h})}{\text{\rm d}t}\leq 0,\;\;\;t>0.$ (14)
Although the Langevin dynamics in (2.2) converges to the invariant
distribution $\pi(\text{\rm d}x)$, its convergence rate is not optimal
(Pavliotis,, 2014). Many studies have aimed to improve the speed of
convergence. A simple yet effective method is to add a momentum term to the
drift coefficient $\nabla\log\pi(x)$; see, e.g., Hwang et al., (2005) and Kim
et al., (2022). We will use accelerated Langevin dynamics in our
implementation of the PMFVB method in Section 7.
## 3 Mean Field Variational Bayes
We are concerned with the problem of approximating a target probability
measure $\pi(\text{\rm d}x\times\text{\rm d}y)$ defined on $\Theta={\cal
X}\times{\cal Y}\subset\mathbb{R}^{d_{x}}\times\mathbb{R}^{d_{y}}$ with
density $\pi(x,y)$ (with respect to some reference measure such as the
Lebesgue measure). The methodology proposed in this paper can be easily
extended to cases of more than two blocks, $\Theta={\cal X}_{1}\times{\cal
X}_{2}\times\cdots\times{\cal X}_{k}$, $k>2$. MFVB approximates $\pi(\cdot)$
by a probability measure $q(\text{\rm d}x\times\text{\rm d}y)=q_{x}(\text{\rm
d}x)q_{y}(\text{\rm d}y)$, where $q_{x}(\text{\rm d}x)\in\mathcal{P}({\cal
X})$ and $q_{y}(\text{\rm d}y)\in\mathcal{P}({\cal Y})$. Consider the
following optimisation problem
$q^{*}\in\arg\min_{q\in\mathcal{P}({\cal X})\otimes\mathcal{P}({\cal
Y})}\Big{\\{}F(q)=\text{\rm KL}(q\|\pi)\Big{\\}}.$ (15)
We study in Section 5.1 conditions under which the problem in (15) is well-
defined and has a unique solution. Define
$q_{x}^{*}(x)\propto\exp\Big{(}{\mathbb{E}}_{q_{y}}\big{[}\log\pi(x,y)\big{]}\Big{)},\;\;\;\;q_{y}^{*}(y)\propto\exp\Big{(}{\mathbb{E}}_{q_{x}}\big{[}\log\pi(x,y)\big{]}\Big{)}.$
(16)
MFVB turns the optimisation problem (15) into the following coordinate-
descent-type problem:
$\text{given $q_{y}$ solve: }\min_{q_{x}\in\mathcal{P}({\cal
X})}\Big{\\{}\text{\rm KL}(q_{x}\|q_{x}^{*})\Big{\\}},$ (17)
and
$\text{given $q_{x}$ solve: }\min_{q_{y}\in\mathcal{P}({\cal
Y})}\Big{\\{}\text{\rm KL}(q_{y}\|q_{y}^{*})\Big{\\}}.$ (18)
Assuming that $q_{x}^{*}(x)$ and $q_{y}^{*}(y)$ have a standard form and that
the expectations in (16) can be computed, the solutions in (17)-(18) are
$q_{x}^{*}(x)$ and $q_{y}^{*}(y)$ respectively. These assumptions limit the
use of MFVB to simple cases. For example, even in the simple Bayesian logistic
regression model, MFVB cannot be used as the assumptions above are not
satisfied. The next section proposes a method for solving (15) without making
these assumptions, and Section 5.1 studies the theoretical properties of
$F(q)$ and its minimizer $q^{*}$.
## 4 Particle Mean Field Variational Bayes
We now present our particle MFVB procedure. The key idea is that whenever the
optimal solutions $q_{x}^{*}(x)$ and $q_{y}^{*}(y)$ of (17) and (18) are
unavailable in closed form, we use Langevin Monte Carlo diffusions to
iteratively approximate them. We assume below that both $q_{x}^{*}(x)$ and
$q_{y}^{*}(y)$ are intractable; in MFVB applications where the variational
distribution is factorized into $K$ blocks, $q=q_{1}\times
q_{2}\times\cdots\times q_{K}$, one only needs to use Langevin Monte Carlo
diffusions to approximate those optimal solutions $q_{k}^{*}$ that are
intractable. Particle MFVB works more efficiently when many but a few of the
optimal $q_{k}^{*}$ are tractable. Lemma 1 suggests that the KL functional
$F(q)$ decreases after each iteration, together with the result that $F(q)$
has the unique solution (c.f. Corollary 3), this justifies our particle MFVB
procedure. Theorem 4 provides a formal proof.
We use a set of particles $\\{X_{i}^{(t)},i=1,...,M\\}$ to approximate
$q_{x}^{*}(x)$ at iteration $t$, and $\\{Y_{i}^{(t)},i=1,...,M\\}$ to
approximate $q_{y}^{*}(y)$, $t\geq 1$. We note that, unlike Section 2 where
$t$ denotes continuous time, in this section $t$ denotes the $t$th iteration
in the PMFVB algorithm. Given $q_{y}^{(t)}(\text{\rm d}y)$ which is
approximated by $\widehat{q}_{y}^{(t)}(\text{\rm
d}y)=\frac{1}{M}\sum_{i}\delta_{Y_{i}^{(t)}}(\text{\rm d}y)$, a Langevin Monte
Carlo diffusion is used to approximate
$q_{x}^{*}(x)\propto\exp\Big{(}{\mathbb{E}}_{q_{y}^{(t)}}\big{[}\log\pi(x,y)\big{]}\Big{)}$:
$X_{i}^{(t+1)}=X_{i}^{(t)}+\frac{h_{x}}{2}{\mathbb{E}}_{q_{y}^{(t)}}\big{[}\nabla_{x}\log\pi(X_{i}^{(t)},y)\big{]}+\sqrt{h_{x}}\eta_{x,i},\;\;\;i=1,...,M$
(19)
with $\eta_{x,i}\sim N_{d_{x}}(0,I)$. The term
${\mathbb{E}}_{q_{y}^{(t)}}\big{[}\nabla_{x}\log\pi(X_{i}^{(t)},y)\big{]}$ can
be approximated using a subset of $\\{Y_{i}^{(t)},i=1,...,M\\}$ by
$\frac{1}{m}\sum_{k=1}^{m}\nabla_{x}\log\pi(X_{i}^{(t)},Y_{i_{k}}^{(t)})$,
where $\\{Y_{i_{k}}^{(t)},k=1,...,m\\}$ is a random subset of size $m$ from
$\\{Y_{i}^{(t)},i=1,...,M\\}$. That is,
$X_{i}^{(t+1)}=X_{i}^{(t)}+\frac{h_{x}}{2m}\sum_{k=1}^{m}\nabla_{x}\log\pi(X_{i}^{(t)},Y_{i_{k}}^{(t)})+\sqrt{h_{x}}\eta_{x,i},\;\;\;i=1,...,M.$
(20)
Similarly, given $q_{x}^{(t+1)}(\text{\rm d}x)$ approximated by the particles
$\\{X_{i}^{(t+1)},i=1,...,M\\}$, we use a Langevin Monte Carlo diffusion to
approximate
$q_{y}^{*}(y)\propto\exp\Big{(}{\mathbb{E}}_{q_{x}^{(t+1)}}\big{[}\log\pi(x,y)\big{]}\Big{)}$:
$Y_{i}^{(t+1)}=Y_{i}^{(t)}+\frac{h_{y}}{2m}\sum_{k=1}^{m}\nabla_{y}\log\pi(X_{i_{k}}^{(t+1)},Y_{i}^{(t)})+\sqrt{h_{y}}\eta_{y,i},\;\;\;i=1,...,M$
(21)
with $\eta_{y,i}\sim N_{d_{y}}(0,I)$. Algorithm 1 summarises the procedure.
Algorithm 1 Particle MFVB
1:procedure PMFVB($M,\epsilon,q^{0}_{x},q^{0}_{y}$)
2: Input: number of particles $M$, tolerance $\epsilon>0$, initial
distributions $q^{(0)}_{x}$ and $q^{(0)}_{y}$.
3: Initialise $X_{i}\sim q^{(0)}_{x}$ and $Y_{i}\sim q^{(0)}_{y},~{}i=1,\dots
M$. $t\leftarrow 0$.
4: while Not $S(\epsilon)$ do
5: Update $\\{X_{i}^{(t+1)},i=1,...,M\\}$ as in (20)
6: Update $\\{Y_{i}^{(t+1)},i=1,...,M\\}$ as in (21)
7: $t\leftarrow t+1$.
8: end while
9:end procedure
In Algorithm 1, $S(\epsilon)$ denotes a stopping rule as a function of some
tolerance $\epsilon>0$. The computational complexity in each iteration is
$O(mM)$. Once the subset $\\{Y_{i_{k}}^{(t)},k=1,...,m\\}$ has been selected,
the update in (20) can be paralellised across the particles $i$ as there is no
communication required between the particles. Similarly, the update in (21)
can also be parallelised.
We now discuss the stopping rule. When a validation data set is available, as
is typically the case in deep learning applications, one can use a stopping
rule based on the validation error and stop the iteration in Algorithm 1 if
the validation error, or a rolling window smoothed version of it, no longer
decreases. This stopping rule is recommended in Section 7 where PMFVB is used
for training deep neural networks. Alternatively, the update in Algorithm 1
can be stopped using the lower bound. Let $\pi(x,y)=\widetilde{\pi}(x,y)/C$
with $C$ the normalising constant. Then,
$\text{\rm KL}(q\|\pi)=-\mathcal{L}(q)+\log C,$
where $\mathcal{L}(q)$ is the lower bound term
$\mathcal{L}(q)=\int\log\widetilde{\pi}(x,y)q(\text{\rm d}x,\text{\rm
d}y)+H(q),\;\;\;H(q)=-\int\log q(x,y)q(\text{\rm d}x,\text{\rm d}y).$
The entropy term $H(q)$ encourages the spread of the particles to avoid their
collapse to a degenerate distribution. However, as the LMC diffusion already
spreads the particles by adding a Gaussian noise to them (c.f. (13)), hence
circumventing convergence to a degenerate measure. At the $t$th iteration,
given the $M$ particles $\\{X_{i}^{(t)},Y_{i}^{(t)}\\}_{i=1}^{M}$
approximating $q$, we suggest approximating the entropy $H(q)$ by
$-(1/M)\sum_{i=1}^{M}\log(1/M)=\log(M)$. The lower bound term at the $t$th
iteration is approximated by
$\widehat{\mathcal{L}}=\frac{1}{M}\sum_{i}\log\widetilde{\pi}(X_{i}^{(t)},Y_{i}^{(t)})+\log(M).$
(22)
## 5 Theoretical analysis of particle MFVB
In order to avoid technical complications, we will assume in this section that
${\cal X}$ and ${\cal Y}$ are compact sets in $\mathbb{R}^{d_{x}}$ and
$\mathbb{R}^{d_{y}}$, respectively. All proofs of the theorems and corollaries
are in the Appendix.
### 5.1 Properties of functional $F(q)$ on the Wasserstein space
We first study the properties of the KL functional $F(q)$ on the Wasserstein
space
$\mathcal{Q}=\mathbb{W}_{2}(\mathcal{X})\otimes\mathbb{W}_{2}(\mathcal{Y})$.
To the best of our knowledge, there is no previous work studying the
theoretical properties of the MFVB problem (15). By limiting $P({\cal
X})\otimes\mathcal{P}({\cal Y})$ to the Wasserstein space
$\mathbb{W}_{2}(\mathcal{X})\otimes\mathbb{W}_{2}(\mathcal{Y})$, the theorem
below shows that the optimal MFVB distribution exists and is unique.
###### Theorem 2.
Assume that ${\cal X}$ and ${\cal Y}$ are compact sets and that $\pi(x,y)$ is
continuous in both $x$ and $y$. Then,
* (i)
$F(q)$ is lower semi-continuous (w.r.t. the weak convergence on ${\cal Q}$).
* (ii)
$F(q)$ is convex.
###### Corollary 3.
Under the assumptions in Theorem 2, $F(q)$ has an unique minimizer on ${\cal
Q}$.
### 5.2 Convergence of the particle MFVB algorithm
As the number of particles $M\to\infty$, Algorithm 1 defines a sequence of
measures $\\{q^{(t)}(\text{\rm d}x\times\text{\rm d}y)=q_{x}^{(t)}(\text{\rm
d}x)q_{y}^{(t)}(\text{\rm d}y),\ t=1,2,...\\}$ on ${\cal Q}$. The theorem
below shows that the KL functional $F(q^{(t)})$ is non-increasing over $t$,
and hence $q^{(t)}$ converges to the solution $q^{*}$.
###### Theorem 4.
Assume that, for each fixed $y$, $U(x)=-\log\pi(x,y)$ is strongly convex and
has a Lipschitz continuous gradient w.r.t. $x$. That is, there exist constants
$c_{1}>0$ and $C_{1}>0$ such that
$c_{1}I_{d_{x}}\leq\nabla^{2}U(x)\leq C_{1}I_{d_{y}},\;\;\;\text{ for all
$x$}.$
Similarly, for each fixed $x$, $V(y)=-\log\pi(x,y)$ is strongly convex and has
a Lipschitz continuous gradient w.r.t. $y$. That is, there exist constants
$c_{2}>0$ and $C_{2}>0$ such that
$c_{2}I_{d_{y}}\leq\nabla^{2}V(y)\leq C_{2}I_{d_{y}},\;\;\;\text{ for all
$y$}.$
Then, if the step sizes $h_{x}$ and $h_{y}$ are sufficiently small and
$M\to\infty$, $F(q^{(t)})$ is non-increasing over $t$, and $q^{(t)}$ converges
to the unique minima $q^{*}$ of $F(q)$.
### 5.3 Posterior consistency of $q^{*}$
This section studies the properties of the particle MFVB solution
$q^{*}=q_{n}^{*}$ in the context of Bayesian inference where the target
$\pi(x,y)=\pi_{n}(\theta)$ is a posterior distribution
$\pi_{n}(\theta)=p(\theta|X^{(n)})$
of a model parameter $\theta\in\mathbb{R}^{d}$, and $X^{(n)}$ denotes the data
of size $n$. Is the PMFVB approximation $q_{n}^{*}$ posterior consistent? i.e.
does $q_{n}^{*}(\text{\rm d}\theta)$ concentrate on a small neighborhood of
the true parameter as the sample size $n\to\infty$? We look at this problem
from the frequentist point of view where we assume that the true data
generating parameter exists. To study this question, we first define some
notation.
Let $p_{\theta}^{(n)}(\text{\rm d}X^{(n)})$ be the distribution of data
$X^{(n)}$ under the model, parameterized by $\theta\in\Theta$. Assume that
$X^{(n)}$ is generated under some true parameter $\theta_{0}\in\Theta$, and
denote by $p_{0}^{(n)}(\text{\rm d}X^{(n)})$ the true underlying distribution
of $X^{(n)}$. Let $\pi_{0}(\text{\rm d}\theta)\in\mathcal{P}(\Theta)$ be the
prior distribution of $\theta$. The posterior distribution is
$\pi_{n}(\text{\rm d}\theta)\propto\pi_{0}(\text{\rm
d}\theta)p_{\theta}^{(n)}(X^{(n)}).$
We shall study the asymptotic behavior of the PMFVB variational posterior
$q_{n}^{*}=\arg\min_{q\in\mathcal{Q}}\text{\rm KL}\big{(}q\|\pi_{n}\big{)}$
with the space of probability measures $\mathcal{Q}$ having the factorised
form in Section 3, $\mathcal{Q}={\cal P}({\cal X})\otimes{\cal P}({\cal Y})$,
$\Theta={\cal X}\times{\cal Y}$. We note that it is unnecessary to equip
$\mathcal{Q}$ with the Wassterstein distance in this section.
The asymptotic behavior and convergence rate of the posterior
$\pi_{n}(\text{\rm d}\theta)$ are well studied in the Bayesian statistics
literature; see, e.g., Ghosal et al., (2000). The asymptotic behavior of the
conventional VB approximation was studied recently in Zhang and Gao, (2019)
and Alquier and Ridgway, (2019), who study the conditions on the variational
family (also the prior $\pi_{0}$ and the likelihood $p_{\theta}^{(n)}$) to
characterize the convergence properties of the variational posterior. The
predicament with conventional VB is that the variational family should not be
too large, so that the VB optimization is solvable; but it should not be too
small either, so that the variational posterior can still achieve some sort of
posterior consistency. It turns out that the variational family $\mathcal{Q}$
in our particle MFVB is general enough; using the results in Zhang and Gao,
(2019), we will show that the PMFVB approximation $q_{n}^{*}$ enjoys the
posterior consistency as the posterior $\pi_{n}(\text{\rm d}\theta)$ does.
We need the following assumptions on the prior $\pi_{0}$ and likelihood
$p_{\theta}^{(n)}$.
###### Assumption 5.
Let $\varepsilon_{n}$ be a sequence of positive numbers such that
$\varepsilon_{n}\to 0$ and $n\varepsilon_{n}^{2}\geq 1$, and $C_{1},C_{2}$ and
$C$ are constants such that $C>C_{2}+2$.
* (A1)
Testing condition: For any $\varepsilon>\varepsilon_{n}$, there exist a set
$\Theta_{n}(\varepsilon)\subset\Theta$ and a test function
$\phi_{n}=\phi_{n}(X^{(n)})\in[0,1]$ such that
${\mathbb{E}}_{p^{(n)}_{0}}(\phi_{n})\leq\exp(-Cn\varepsilon^{2})\;\;\text{and}\;\;\sup_{\begin{subarray}{c}\theta\in\Theta_{n}(\varepsilon),\\\
\|\theta-\theta_{0}\|_{2}^{2}>C_{1}\varepsilon^{2}\end{subarray}}{\mathbb{E}}_{p^{(n)}_{\theta}}(1-\phi_{n})\leq\exp(-Cn\varepsilon^{2})$
* (A2)
Prior mass conditions:
$\pi_{0}\big{(}\Theta_{n}(\varepsilon)^{c}\big{)}\leq\exp(-Cn\varepsilon^{2})$
and, for some $\rho>1$,
$\pi_{0}\Big{(}\theta:D_{\rho}(p^{(n)}_{0}\|p^{(n)}_{\theta})\leq
C_{2}n\varepsilon_{n}^{2}\Big{)}>0,\;\;\;\text{for any $n$}.$
* (A3)
Smoothness:
$|\log\pi_{0}(\theta)-\log\pi_{0}(\theta^{\prime})|\leq
C_{3}\|\theta-\theta^{\prime}\|_{2},\;\;\;\forall\theta,\theta^{\prime}\in\Theta$
for some $C_{3}>0$, and
$|\log p_{\theta}^{(n)}(X^{(n)})-\log p_{\theta^{\prime}}^{(n)}(X^{(n)})|\leq
C_{4}(X^{(n)})\|\theta-\theta^{\prime}\|_{2},\;\;\;\forall\theta,\theta^{\prime}\in\Theta$
with $C_{5}:={\mathbb{E}}_{p^{(n)}_{0}}\big{[}C_{4}(X^{(n)})\big{]}<\infty$.
Here, $D_{\rho}(p\|q)$ denotes the $\rho$-Rényi divergence between two
probability measures $p$ and $q$,
$D_{\rho}(p\|q)=\begin{cases}\frac{1}{\rho-1}\log\int\left(\frac{\text{\rm
d}p}{\text{\rm d}q}\right)^{\rho}\text{\rm d}q,&\rho\not=1,\\\ \text{\rm
KL}(p\|q),&\rho=1.\end{cases}$
Assumptions (A1) and (A2) are standard in the Bayesian statistics literature
(Ghosal et al.,, 2000; Zhang and Gao,, 2019), and are used to characterize the
convergence rate of the posterior distribution.
Assumption (A1) states that, restricted to a subset $\Theta_{n}(\varepsilon)$
of $\Theta$, there exists a test function that is able to distinguish the true
probability measure from the complement of its neighborhood. Assumption (A2)
requires that the prior distribution concentrates on $\Theta_{n}(\varepsilon)$
and puts a positive mass on “good” values of $\theta$ in the sense that
$p^{(n)}_{\theta}$ is close enough to the true measure $p^{(n)}_{0}$ in terms
of the $\rho$-Rényi divergence. Under these assumptions, one can prove that
the convergence rate of the posterior $\pi_{n}(\text{\rm d}\theta)$ to the
true data generating measure is $\varepsilon_{n}^{2}$ (Ghosal et al.,, 2000;
Zhang and Gao,, 2019). Typically, $\varepsilon_{n}=1/\sqrt{n}$. Assumption
(A3) requires some smoothness of the prior and likelihood with respect to
$\theta$. The theorem below shows that $q_{n}^{*}$ is posterior consistent.
###### Theorem 6.
Suppose that conditions (A1), (A2) and (A3) are satisfied. Then, for any
$\epsilon>0$,
$q_{n}^{*}\Big{(}\|\theta-\theta_{0}\|_{2}^{2}>\epsilon\Big{)}=o(1)\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0,\;\;\;\;p^{(n)}_{0}-a.s.$ (23)
## 6 Numerical examples
### 6.1 Bayesian logistic regression
Although Bayesian logistic regression is a benchmark model in statistics, is
not straightforward to use the classical MFVB method here because of the lack
of a conjugate prior. This section demonstrates that it is straightforward to
use the PMFVB method for Bayesian logistic regression. Consider the model
$\beta\sim
N(0,\sigma_{0}^{2}I_{d}),\;\;\;y_{i}\sim\text{Binomial}\big{(}1,\sigma(x_{i}^{\top}\beta)\big{)},\;\;\;\sigma(x_{i}^{\top}\beta)=\frac{1}{1+\exp(-x_{i}^{\top}\beta)},$
(24)
where $\beta=(\beta_{0},\beta_{1},\beta_{2},\beta_{3})^{\top}$ and
$x_{i}=(1,x_{i,1},x_{i,2},x_{i,3})^{\top}$. We generated a dataset of size
$n=200$ from (24) with $d=4$ and $\sigma_{0}^{2}=4$. The likelihood is
$L(\beta)=\prod_{i=1}^{n}\big{(}\sigma(x_{i}^{\top}\beta)\big{)}^{y_{i}}\big{(}1-\big{(}\sigma(x_{i}^{\top}\beta)\big{)}\big{)}^{1-y_{i}}$
with posterior
$\pi(\beta)\propto p(\beta)L(\beta).$
We consider the PMFVB procedure with the model parameters $\beta$ factorized
into two blocks $\theta_{1}=(\beta_{0},\beta_{1})^{\top}$ and
$\theta_{2}=(\beta_{2},\beta_{3})^{\top}$. Write
$x_{i}^{(1)}=(1,x_{i,1})^{\top}$ and $x_{i}^{(2)}=(x_{i,2},x_{i,3})^{\top}$.
The gradient of the log posterior with respect to $\theta_{1}$ and
$\theta_{2}$ is given by
$\nabla_{\theta_{1}}\log\pi(\theta_{1},\theta_{2})=\sum_{i=1}^{n}\left(y_{i}-\sigma\left({x_{i}^{(1)}}^{\top}\theta_{1}+{x_{i}^{(2)}}^{\top}\theta_{2}\right)\right)x_{i}^{(1)}-\frac{1}{\sigma_{0}^{2}}\theta_{1}$
and
$\nabla_{\theta_{2}}\log\pi(\theta_{1},\theta_{2})=\sum_{i=1}^{n}\left(y_{i}-\sigma\left({x_{i}^{(1)}}^{\top}\theta_{1}+{x_{i}^{(2)}}^{\top}\theta_{2}\right)\right)x_{i}^{(2)}-\frac{1}{\sigma_{0}^{2}}\theta_{2},$
respectively.
The PMFVB algorithm maintains a set of $M$ particles
$\\{\theta_{1,i}^{(t)},\theta_{2,i}^{(t)}\\}_{i=1,...,M}$ over iterations
$t\geq 0$. At iteration $t+1$, according to (20), the first block of the
particles is updated as
$\theta_{1,i}^{(t+1)}=\theta_{1,i}^{(t)}+\frac{h}{2m}\sum_{k=1}^{m}\nabla_{\theta_{1}}\log\pi(\theta_{1,i}^{(t)},\theta_{2,i_{k}}^{(t)})+\sqrt{h}N(0,I_{2}),$
(25)
where $\\{\theta_{2,i_{k}}^{(t)}\\}_{k=1,...,m}$ is a random subset of size
$m$ from $\\{\theta_{2,i}^{(t)}\\}_{i=1,...,M}$, and $h>0$ is a step size. The
second block of particles is updated as
$\theta_{2,i}^{(t+1)}=\theta_{2,i}^{(t)}+\frac{h}{2m}\sum_{k=1}^{m}\nabla_{\theta_{2}}\log\pi(\theta_{1,i_{k}}^{(t+1)},\theta_{2,i}^{(t)})+\sqrt{h}N(0,I_{2}),$
(26)
where $\\{\theta_{1,i_{k}}^{(t+1)}\\}_{k=1,...,m}$ is a random subset from
$\\{\theta_{1,i}^{(t+1)}\\}_{i=1,...,M}$. The PMFVB procedure for
approximating the posterior $\pi(\beta)$ iterates between (25) and (26) until
stopping.
Below, we implemented the PMFVB algorithm using $M=3000$ particles, and the
optimisation time took $18$ seconds. In comparison, the MCMC (Halmitonian
Monte Carlo) was conducted using PyMC with standard setup, 10,000 samples and
1,000 burn-in period, and the sampling procedure took 54 seconds. Both
algorithms were run on the same Dell Optiplex 7490 AIO (i7-11700) computer.
The trace plot in Figure 1 shows the lower bound (22) over the iterations.
Figure 1: The trace plot of the PMFVB lower bound in Bayesian logistic
regression.
Figure 2 plots the marginal posteriors using kernel density estimation based
on the PMFVB particles and Hamiltonian MC, and shows similar results.
(a) $\beta_{0}$
(b) $\beta_{1}$
(c) $\beta_{2}$
(d) $\beta_{3}$
Figure 2: The estimated posterior density curves and rug plots of Bayesian
logistic regression: the green is of the PMFVB and black of HMC (generated
using PyMC).
### 6.2 Stochastic volatility
This section applies the PMFVB method to Bayesian inference in the Stochastic
Volatility (SV) model (Taylor,, 1982). Let $\\{y_{t},\ t=1,2,...\\}$ be an
asset return time series. The SV model is
$\displaystyle y_{t}$ $\displaystyle\sim$ $\displaystyle
N\big{(}0,e^{x_{t}}\big{)},\;\;t=1,2,...,T$ (27) $\displaystyle x_{t}$
$\displaystyle\sim$ $\displaystyle N\big{(}\mu(1-\phi)+\phi
x_{t-1},\sigma^{2}\big{)},\;\;t=2,...,T,\;\;x_{1}\sim
N\big{(}\mu,\sigma^{2}/(1-\phi^{2})\big{)},$ (28)
with $\mu\in\mathbb{R},\phi\in(-1,1)$ and $\sigma^{2}>0$ being the model
parameters. Write $\theta=(\mu,\phi,\sigma^{2})$. Following Kim et al.,
(1998), we use the prior $\mu\sim N(0,\sigma_{0}^{2})$ with
$\sigma_{0}^{2}=10$, $\tau=(1+\phi)/2\sim\text{Beta}(a_{0},b_{0})$ with
$a_{0}=20$ and $b_{0}=1.5$, and $\sigma^{2}\sim\text{inverse-
Gamma}(\alpha_{0},\beta_{0})$ with $\alpha_{0}=2.5$ and $\beta_{0}=0.025$. It
is challenging to perform Bayesian inference for the SV model because the
likelihood $p(y|\theta)$ is a high-dimensional integral over the latent
variables $x=x_{1:T}$. A number of Bayesian methods are available for
estimating the SV model including SMC2 (Chopin et al.,, 2013; Gunawan et al.,,
2022) and fixed-form Variational Bayes (Tran et al.,, 2020). However, to the
best of our knowledge, MFVB has never been successfully used for this model.
To apply the PMFVB for SV, we use the following factorized variational
distribution
$q(\theta,x)=q(\mu)q(\sigma^{2})q(\phi,x).$ (29)
This factorization leads to an analytical update for $q(\mu)$ and
$q(\sigma^{2})$, and we only need one LMC procedure to update $q(\phi,x)$, see
Appendix A for the derivation. One could also use
$q(\theta,x)=q(\theta)q(x),$ (30)
but then two LMC procedures are needed to update $q(\theta)$ and $q(x)$ as
$q(\theta)$ cannot be updated analytically.
We generate a return time series of $T=500$ observations from the SV model
(27)-(28) with $\mu=1$, $\phi=0.8$ and $\sigma=0.5$. Figure 3 plots the
posterior densities for $\theta$ estimated by the PMFVB and SMC methods. We
use 500 particles in both methods. The CPU times taken by PMFVB and SMC were
8.2 and 29.3 minutes, respectively. The running time for SMC depends on many
factors such as the number of particles used in the particle filter for
estimating the likelihood and the number of Markov moves. We select these
numbers following their typical use in the literature; see, e.g., Gunawan et
al., (2022). Figure 3 shows that the PMFVB estimates for $\mu$ and $\phi$ are
almost identical to that of SMC except the estimate of $\sigma^{2}$, where
PMFVB underestimates the posterior variance. This is the well-known problem
for the MFVB method due to the factorization (29) it imposes on the
variational distribution. Section 8 suggests a possible solution.
Figure 3: The posterior density curves for the SV model parameters estimated
by PMFVB (solid line) and SMC (dashed line). The right panel shows the
smoothed lower bound in PMFVB over a window of size 50.
## 7 Particle MFVB for Bayesian neural networks
This section presents a variant of the PMFVB approach for Bayesian inference
in big models like deep neural networks. One of the most commonly used method
for Bayesian inference in deep learning is perhaps the Stochastic Gradient
Langevin Dynamics (SGLD) method of Welling and Teh, (2011) and its variants.
Let $\theta\in\mathbb{R}^{d}$ be the model parameters and $\pi(\text{\rm
d}\theta)$ their posterior distribution. The SGLD algorithm is based on the
discretised Langevin diffusion (2.2)
$\theta^{(t+1)}=\theta^{(t)}+\frac{h}{2}\widehat{\nabla\log\pi\big{(}\theta^{(t)}\big{)}}+\sqrt{h}\eta_{t},\;\;\;\eta_{t}\stackrel{{\scriptstyle
i.i.d}}{{\sim}}N(0,I_{d}),$ (31)
where $\widehat{\nabla\log\pi(\theta)}$ is an unbiased estimator of the
gradient $\nabla\log\pi(\theta)$, often computed from a data mini-batch. There
is a large literature aiming at improving SGLD by exploiting the curvature
structure of the log-target density function. For example, Li et al., (2016)
propose a preconditioned Stochastic Gradient Langevin Dynamics (pSGLD) that
rescales the gradient of the log target density by a diagonal matrix learnt
using the second moments of the previous gradients. Kim et al., (2022)
introduce several SGLD schemes that add an adaptive drift to the noisy log-
gradient estimator. See also Girolami and Calderhead, (2011) and Chen et al.,
(2014).
### 7.1 The algorithm
We introduce three refinements to make the PMFVB approach computationally
efficient in big-data and complex-model situations.
First, we choose the updating block randomly in each iteration $t$ and for
each particle $i$. Let $\iota=\\{1\leq j_{1}<j_{2}<...j_{m}\leq d\\}$ be an
index subset of size $m$ from $\\{1,2,...,d\\}$. We denote by
$\theta(\iota)=(\theta_{j_{1}},...,\theta_{j_{m}})$ the sub-vector obtained
from $\theta$ corresponding to the index set $\iota$; and by
$\theta(\setminus\iota)$ the vector from $\theta$ after removing the
components in $\theta(\iota)$. For iteration $t$ and for each particle $i$,
the index set $\iota_{i}$ is randomly selected (we suppress the dependence on
$t$ for notational simplicity); the corresponding block of $m$ components
$\theta_{i}^{(t)}(\iota_{i})$ of $\theta_{i}^{(t)}$ is updated via LMC
$\theta_{i}^{(t+1)}(\iota_{i})=\theta_{i}^{(t)}(\iota_{i})+\frac{h}{2}{\mathbb{E}}_{q_{\theta(\setminus\iota)}^{(t)}}\Big{[}\nabla_{\theta(\iota_{i})}\log\pi\big{(}\theta_{i}^{(t)}(\iota_{i}),\theta(\setminus\iota_{i})\big{)}\Big{]}+\sqrt{h}\eta_{i},$
(32)
with $\eta_{i}\sim N_{m}(0,I)$. Here, $q_{\theta(\setminus\iota_{i})}^{(t)}$
denotes the marginal distribution of the particles $\theta_{i}^{(t)}$ w.r.t.
the $\theta(\setminus\iota_{i})$ components.
Second, we approximate the expectation term
${\mathbb{E}}_{q_{\theta(\setminus\iota_{i})}^{(t)}}[\cdot]$ in (32) by
$\nabla_{\theta(\iota_{i})}\log\pi\big{(}\theta_{i}^{(t)}(\iota_{i}),\bar{\theta}^{(t)}(\setminus\iota_{i})\big{)}$,
where $\bar{\theta}^{(t)}$ is the sample mean of the particles
$\\{\theta_{i}^{(t)},i=1,...,M\\}$. This leads to a significant reduction in
computational time, compared to an alternative that averages over a subset of
particles as in (20).
Third, it is desirable to incorporate an adaptive SGLD scheme into the LMC
update in PMFVB. Our work uses the ADAM-based adaptive-drift SGLD scheme of
Kim et al., (2022). Algorithm 2 summarises the method.
Algorithm 2 Particle MFVB for neural networks
1:procedure PMFVB-NN($h$, $\beta_{1},\beta_{2},a,\lambda,\epsilon$)
2: Input: step size $h$, smoothing weights $\beta_{1},\beta_{2}$, tolerance
$\epsilon>0$, scale factor $a$ and $\lambda>0$.
3: Initialise particles $\theta_{i}^{(0)}\sim p_{0},~{}i=1,...,M$, $m_{0}=0$,
$v_{0}=0$, $t=0$.
4: while Not $S(\epsilon)$ do
5: $g_{t}\leftarrow m_{t}\oslash\sqrt{v_{t}+\lambda}$ and
$\bar{\theta}^{(t)}\leftarrow\frac{1}{M}\sum_{i=1}^{M}\theta^{(t)}_{i}$
6: for $i$ in $[M]$ do
7: $\iota_{i}\leftarrow\\{1\leq j_{1}<j_{2}<...j_{m}\leq d\\}$,
$j_{i}\in\\{1,...,d\\}$ are randomly selected.
8: Update:
$\tilde{g}_{t}:=\Big{(}\nabla_{\theta(\iota_{i})}\log\pi\big{(}\theta_{i}^{(t)}(\iota_{i}),\bar{\theta}^{(t)}(\setminus\iota_{i})\big{)}+ag_{t}(\iota_{i})\Big{)}$
$\displaystyle\theta_{i}^{(t+1)}(\iota_{i})$
$\displaystyle\leftarrow\theta_{i}^{(t)}(\iota_{i})+\frac{h}{2}\tilde{g}_{t}+\sqrt{h}\eta_{i},~{}\eta_{i}\sim
N_{m}(0,I)$ (33) $\displaystyle\theta_{i}^{(t+1)}$
$\displaystyle\leftarrow\big{(}\theta_{i}^{(t+1)}(\iota_{i}),\theta_{i}^{(t)}(\setminus\iota_{i})\big{)}$
9: $g^{(t)}_{i}\leftarrow\nabla_{\theta}\log\pi(\theta_{i}^{(t+1)})$, new
gradient
10: Update the adaptive drift
$\displaystyle\bar{g}_{t}\leftarrow\frac{1}{M}\sum_{i=1}^{M}g^{(t)}_{i},$
$\displaystyle~{}~{}\bar{v}_{t}\leftarrow\sqrt{\frac{1}{M}\sum_{i}(g^{(t)}_{i}-\bar{g}_{t})\odot(g^{(t)}_{i}-\bar{g}_{t})}$
$\displaystyle m_{t+1}\leftarrow\beta_{1}m_{t}+(1-\beta_{1})\bar{g}_{t},$
$\displaystyle~{}~{}v_{t+1}\leftarrow\beta_{2}v_{t}+(1-\beta_{2})\bar{v}_{t}.$
11: end for
12: $t=t+1$.
13: end while
14:end procedure
In Algorithm 2, $\oslash$ and $\odot$ denote the component-wise division and
multiplication operators, respectively.
The implementation below follows Kim et al., (2022) and sets $\beta_{1}=0.9$,
$\beta_{2}=0.99$, $a=100$ and $\lambda=10^{-8}$. The block size $m$ is 10% of
the length $d$ of $\theta$. The step size $h$ is typically $0.001$.
### 7.2 Applications
#### 7.2.1 Bayesian neural networks for regression
Consider the neural network regression model
$y_{i}=\eta(x_{i},w)+\epsilon_{i},\;\;\epsilon_{i}\sim
N(0,\sigma^{2}),\;\;i=1,...,n,$ (34)
where $\eta(x_{i},w)$ is the output from a neural network with the input
$x_{i}$ and weights $w=\\{w_{j},j=1,...,d_{w}\\}$. Neural networks are prone
to overfitting and the Bayesian framework provides a principled way for
dealing with this problem by placing a regularization prior on the weights. We
consider the following adaptive ridge-type regularization prior on the weights
$\displaystyle w_{j}|\tau_{j}$ $\displaystyle\sim$ $\displaystyle
N(0,\tau_{j}),\;\;j=1,...,d_{w},$ (35) $\displaystyle\tau_{j}$
$\displaystyle\sim$ $\displaystyle\text{Inverse Gamma}(\alpha_{0},\beta_{0}).$
We set $\alpha_{0}=1$ and $\beta_{0}=0.01$ in all the examples below. For the
variance $\sigma^{2}$, we use the improper prior $p(\sigma^{2})\sim
1/\sigma^{2}$ (Park and Casella,, 2008). The model parameters include
$\theta=\big{(}w,\tau=\\{\tau_{j},j=1,...,d_{w}\\},\sigma^{2}\big{)}$. Given
the different roles of $w$, $\tau$ and $\sigma^{2}$, it is natural to use the
following factorized variational distribution in PMFVB
$q(\theta)=q(w)q(\tau)q(\sigma^{2}).$ (36)
It can be seen from (16) that the optimal MFVB distribution for $\tau$ is
$q(\tau)=\prod_{j}q(\tau_{j})$, where $q(\tau_{j})$ has an Inverse-Gamma
density with scale $\alpha_{0}+1/2$ and rate $\beta_{0}+\langle
w_{j}^{2}\rangle/2$. Here, as common in the MFVB literature,
$\langle\cdot\rangle$ denotes the expectation with respect to the variational
distribution $q$. The optimal MFVB distribution for $\sigma^{2}$ is Inverse-
Gamma with scale $n/2$ and shape
$\frac{1}{2}\langle\sum_{i=1}^{n}(y_{i}-\eta(x_{i},w))^{2}\rangle$. The terms
$\langle w_{j}^{2}\rangle$ and
$\langle\sum_{i=1}^{n}(y_{i}-\eta(x_{i},w))^{2}\rangle$ can be approximated
using the $w$-particles. The optimal MFVB distribution for the weights $w$ has
the log-density
$\log q(w)=-\frac{1}{2}\sum_{j=1}^{d_{w}}\langle\frac{1}{\tau_{j}}\rangle
w_{j}^{2}-\frac{1}{2}\langle\frac{1}{\sigma^{2}}\rangle\sum_{i=1}^{n}(y_{i}-\eta(x_{i},w))^{2},$
(37)
which indicates that $q(\text{\rm d}w)$ cannot be updated analytically. Note
that the expectation terms $\langle\frac{1}{\tau_{j}}\rangle$ and
$\langle\frac{1}{\sigma^{2}}\rangle$ can be computed analytically. Based on
(37), a Langevin MC procedure in Algorithm 2 is used to approximate the
distribution $q(\text{\rm d}w)$.
#### A simulation study
We simulate data from the following non-linear model (Tran et al.,, 2020)
$y=5+10x_{1}+\frac{10}{x_{2}^{2}+1}+5x_{3}x_{4}+2x_{4}+5x_{4}^{2}+5x_{5}+2x_{6}+\frac{10}{x_{7}^{2}+1}+5x_{8}x_{9}+5x_{9}^{2}+5x_{10}+\epsilon,$
where $\epsilon\sim{\cal N}(0,1)$, $(x_{1},...,x_{20})^{\top}$ are generated
from a multivariate normal distribution with mean zero and covariance matrix
$(0.5^{|i-j|})_{i,j}$; the last ten variables are not in the regression. The
training data has 100,000 observations, the validation and test datasets each
have 10,000 observations.
We compare the predictive performance of PMFVB with Gaussian VB of Tran et
al., (2020), SGLD of Welling and Teh, (2011), preconditioned SGLD of Li et
al., (2016) and the ADAM-based adaptive drift SGLD of Kim et al., (2022). We
use 300 particles in PMFVB. For the SGLD methods, one must first transform
$\tau$ and $\sigma^{2}$ into an unconstrained space, then apply the Langevin
MC to jointly sample both the weights $w$, the transformed $\sigma^{2}$ and
the latent (transformed) $\tau$. The dimension of this LMC is double that of
the LMC used in the PMFVB method where one needs to sample $w$ only. We set
the tuning parameters in the SGLD methods following the suggestions in the
papers proposing the methods.
The performance metrics include the best validation-data partial predictive
score (PPS) and the test-data PPS,
$\text{PPS}=-\frac{1}{|D|}\sum_{(x,y)\in D}p(y|x,\widehat{\theta})$
where $D$ is the validation and test data, respectively, $\widehat{\theta}$ is
the posterior mean estimate of the model parameters. We also report the test-
data MSE and the CPU running time (in minutes). For each method, the Bayesian
neural network model is trained until the validation PPS no longer decreases
after 100 iterations. Table 1 summarizes the results, which are based on 10
different runs for each algorithm. The PMFVB method achieves the best
predictive performance; it is also stable across different runs as reflected
by the standard deviations. In terms of the running time, the precondioned
SGLD is the most efficient. We note, however, that the PMFVB is parallelisable
and its running time can be greatly reduced if multiple-core computers are
used.
Method | Validation PPS | Test PPS | Test MSE | CPU
---|---|---|---|---
Gaussian VB | 1.2305 (0.0678) | 1.2513 (0.0714) | 4.3795 (0.6384) | 18.78
SGLD | 1.3751 (0.1060) | 1.3881 (0.1055) | 4.9997 (1.1608) | 2.74
Precondioned SGLD | 1.0527 (0.1274) | 1.0676 (0.1335) | 3.1517 (0.8158) | 1.36
Adam SGLD | 1.1849 (0.0485) | 1.2021 (0.0487) | 3.7987 (0.4874) | 4.26
PMFVB | 0.8239 (0.0642) | 0.8598 (0.0640) | 2.0631 (0.2603) | 16.29
Table 1: Simulation data: Predictive performance in term of the partial
predictive score (PPS), the mean squared error (MSE) and CPU time (in
minutes), averaged over 10 different runs. The numbers in brackets are the
standard deviation across the replicates. The best scores for each performance
metric are highlighted in bold. The structure of the neural net is
(20,20,20,1), i.e., one input layer of 20 units, two hidden layers each of 20
units and one output unit.
#### The HILDA data
The Household, Income and Labour Dynamics in Australia (HILDA) Survey111The
HILDA Survey was conducted by the Australian Government Department of Social
Services (DSS). The findings and views reported in this paper, however, are
those of the authors and should not be attributed to the Australian
Government, DSS, or any of DSS’ contractors or partners. data consists of many
household-based variables about economic and personal well-being. We apply the
Bayesian neural network model to predict the Income, using 43 covariates many
of which are dummy variables used to encode the categorical variables. The
dataset is randomly divided into a training set of 14,010 observations for
fitting the model, a validation set of of 1751 observations for stopping, and
a test set of 1751 observations for final performance evaluation. We use a
neural network with three hidden layers, each having 50 units, and use the
same algorithm settings as in the previous simulation example. Table 2
summarizes the result which shows that the PMFVB algorithm achieves the best
predictive performance together with the highest stability (across different
runs).
Method | Validation PPS | Test PPS | Test MSE | CPU
---|---|---|---|---
Gaussian VB | $-0.0957$ (0.0071) | $-0.1474$ (0.0143) | 0.2739 (0.0078) | 10.62
SGLD | $-0.1272$ (0.0035) | $-0.1850$ (0.0051) | 0.2541 (0.0026) | 2.40
Precondioned SGLD | $-0.1065$ (0.0141) | $-0.1621$ (0.0190) | 0.2612 (0.0037) | 5.42
Adam SGLD | $-0.1321$ (0.0035) | $-0.1897$ (0.0022) | 0.2516 (0.0011) | 1.57
PMFVB | ${\bf-0.1331}$ (0.0019) | ${\bf-0.1932}$ (0.0014) | 0.2498 (0.0007) | 9.07
Table 2: HILDA data: Predictive performance in term of the partial predictive
score (PPS), the mean squared error (MSE) and CPU time (in minutes), averaged
over 10 different runs. The numbers in brackets are the standard deviation
across the replicas. The best scores for each performance metric are
highlighted in bold. The structure of the neural net is (43,50,50,50,1), i.e.
three hidden layers each of 50 units.
#### 7.2.2 Bayesian neural networks for classification
Consider the neural network binary classification model
$\displaystyle y_{i}$ $\displaystyle\sim$
$\displaystyle\text{Binomial}\big{(}1,p(x_{i},w)\big{)},$ $\displaystyle
p(x_{i},w)$ $\displaystyle=$
$\displaystyle\frac{1}{1+\exp\big{(}-\eta(x_{i},w)\big{)}},\;\;i=1,...,n$
where $\eta(x_{i},w)$ is the output from a neural network with the input
$x_{i}$ and weights $w=\\{w_{j},j=1,...,d_{w}\\}$. We use the same
regularisation prior as in the regression (35). The model parameters are
$\theta=\big{(}w,\tau=\\{\tau_{j},j=1,...,d_{w}\\}\big{)}$. We consider the
following factorization in the PMFVB
$q(\theta)=q(w)q(\tau).$
The optimal MFVB distribution for $\tau$ is $q(\tau)=\prod_{j}q(\tau_{j})$,
where $q(\tau_{j})$ has an Inverse-Gamma density with scale $\alpha_{0}+1/2$
and rate $\beta_{0}+\langle w_{j}^{2}\rangle/2$. The term $\langle
w_{j}^{2}\rangle$ can be approximated from the $w$-particles. The optimal MFVB
distribution for the weights $w$ has the log-density
$\log q(w)=-\frac{1}{2}\sum_{j=1}^{d_{w}}\langle\frac{1}{\tau_{j}}\rangle
w_{j}^{2}+\sum_{i=1}^{n}\Big{(}y_{i}\eta(x_{i},w)-\log(1+e^{\eta(x_{i},w)})\Big{)}.$
(38)
The expectation term $\langle\frac{1}{\tau_{j}}\rangle$ can be computed
analytically. Based on (38), a Langevin MC procedure in Algorithm 2 is used to
approximate the distribution $q(\text{\rm d}w)$.
#### The census data
The census dataset obtained from the U.S. Census Bureau is available on the
UCI Machine Learning Repository. The task is to classify whether a person’s
income is over $50K per year, based on 14 covariates including age, work
class, race. Using dummy variables to represent the categorical variables,
there are a total of 103 input variables in the Bayesian neural network model.
The full dataset of 45,221 observations is divided into a training set (53%),
validation set (14%) and test set (33%). We use 200 particles in the PMFVB
method. Table 3 summarizes the results which show that the PMFVB obtains the
best predictive performance.
Method | Validation PPS | Test PPS | Test MCR | CPU
---|---|---|---|---
SGLD | 0.3153 (0.0046) | 0.4304 (0.0348) | 0.1990 (0.0033) | 3.83
Precondioned SGLD | 0.3133 (0.0012) | 0.4407 (0.0251) | 0.2034 (0.0051) | 3.47
Adam SGLD | 0.3109 (0.0011) | 0.4422 (0.0211) | 0.2097 (0.0079) | 3.20
PMFVB | 0.3052 (0.0008) | 0.4217 (0.0069) | 0.1958 (0.0030) | 18.30
Table 3: Census data: Predictive performance in term of the partial predictive
score (PPS), miss-classification rate (MCR), and CPU time (in minutes),
averaged over 10 different runs. The numbers in brackets are the standard
deviation across the runs. The best scores for each performance metric are
highlighted in bold. The structure of the neural net is (103,100,100,1).
## 8 Discussion
We propose a particle-based MFVB procedure for Bayesian inference, which
extends the scope of classical MFVB, is widely applicable and enjoys
attractive theoretical properties. The new method can also be used for
training Bayesian deep learning models.
The main limitation of MFVB methods including PMFVB is the use of factorized
variational distributions, which might fail to capture the dependence
structure between the blocks of variables. This limitation can be mitigated
using the reparametrisation method of Tan, (2021). Write the target
distribution as $\pi(x,y)=\pi_{x}(x)\pi_{y|x}(y|x)$. Let $b(x)$ and $H(x)$ be
the gradient and minus Hessian of $\log\pi_{y|x}(y|x)$ at some point
$y=y_{0}$. Assume that $H(x)$ is positive definite, consider the
transformation $\widetilde{y}=H(x)^{1/2}\big{(}y-\mu(x)\big{)}$ with
$\mu(x)=H(x)^{-1}b(x)+y_{0}$. The joint density of $x$ and $\widetilde{y}$ is
$\widetilde{\pi}(x,\widetilde{y})=|H(x)|^{-1/2}\pi_{x}(x)\pi_{y|x}\big{(}H(x)^{-1/2}\widetilde{y}+\mu(x)|x\big{)}.$
The motivation is that, if $\pi_{y|x}(y|x)\sim
N\big{(}\mu(x),H(x)^{-1}\big{)}$, then $\widetilde{y}\sim N(0,I)$ and is
independent of $x$. In general, we can expect that the dependence between $x$
and $\widetilde{y}$ is reduced and is much less than the dependence between
$x$ and $y$. Tan, (2021) considers this reparametrisation approach in Gaussian
Variational Bayes and documents significant improvement in posterior
approximation accuracy. Coupling the reparametrisation approach with PMFVB
could lead to an efficient technique for Bayesian inference. This research is
in progress.
## Appendix A: Derivation for the SV example
The joint posterior of $\theta$ and latent $x$ is
$\displaystyle\pi(\theta,x)$ $\displaystyle\propto$ $\displaystyle
p(\theta)p(x|\theta)p(y|x,\theta)$ $\displaystyle=$ $\displaystyle
p(\mu)p(\phi)p(\sigma^{2})N\big{(}x_{1}|\mu,\frac{\sigma^{2}}{1-\phi^{2}}\big{)}\prod_{t=2}^{T}N\big{(}x_{t}|\mu(1-\phi)+\phi
x_{t-1},\sigma^{2}\big{)}\prod_{t=1}^{T}N\big{(}y_{t}|0,e^{x_{t}}\big{)},$
where $p(\mu)=N(\mu|0,\sigma^{2})$,
$p(\phi)\propto\big{(}(1+\phi)/2\big{)}^{a_{0}-1}\big{(}(1-\phi)/2\big{)}^{b_{0}-1}$,
and
$p(\sigma^{2})\propto(\sigma^{2})^{-(1+\alpha_{0})}\exp(-\beta_{0}/\sigma^{2})$.
Using (16), the optimal MFVB distribution $q(\mu)$ is
$N(\mu_{q},\sigma_{q}^{2})$ with $\mu_{q}=B/A$ and $\sigma_{q}^{2}=1/A$, where
$A=\frac{1}{\sigma_{0}^{2}}+\frac{\alpha_{\sigma^{2}}}{\beta_{\sigma^{2}}}\langle
1-\phi^{2}\rangle+(T-1)\frac{\alpha_{\sigma^{2}}}{\beta_{\sigma^{2}}}\langle(1-\phi)^{2}\rangle$
and
$B=\frac{\alpha_{\sigma^{2}}}{\beta_{\sigma^{2}}}\langle(1-\phi^{2})x_{1}\rangle+(T-1)\frac{\alpha_{\sigma^{2}}}{\beta_{\sigma^{2}}}\langle(1-\phi)\sum_{t=2}^{T}(x_{t}-\phi
x_{t-1})\rangle,$
with $\alpha_{\sigma^{2}}$ and $\beta_{\sigma^{2}}$ given below. Recall that
$\langle\cdot\rangle$ denotes the expectation with respect to the variational
distribution $q$. The optimal MFVB distribution $q(\sigma^{2})$ is Inverse-
Gamma$(\alpha_{\sigma^{2}},\beta_{\sigma^{2}})$, where
$\alpha_{\sigma^{2}}=\alpha_{0}+T/2$ and
$\beta_{\sigma^{2}}=\beta_{0}+\frac{1}{2}\Big{\langle}(1-\phi^{2})\big{[}(x_{1}-\mu_{q})^{2}+\sigma_{q}^{2}\big{]}+\sum_{t=2}^{T}\big{[}(x_{t}-\mu_{q}(1-\phi)-\phi
x_{t-1})^{2}+(1-\phi)^{2}\sigma_{q}^{2}\big{]}\Big{\rangle}.$
All the expectations $\langle\cdot\rangle$ in the expressions above are with
respect to $q(\phi,x)$, which can be estimated from the $(\phi,x)$-particles.
The logarithm of the optimal MFVB density for $(\phi,x)$ is
$\displaystyle\log q(\phi,x)$ $\displaystyle=$ $\displaystyle\Big{\langle}\log
p(\phi)+\log p(x_{1}|\theta)+\sum_{t=2}^{T}\log
p(x_{t}|x_{t-1},\theta)+\sum_{t=1}^{T}\log p(y_{t}|x_{t})\Big{\rangle}+C$
$\displaystyle=$
$\displaystyle\Big{\langle}(a_{0}-1)\log(1+\phi)+(b_{0}-1)\log(1-\phi)+\frac{1}{2}\log(1-\phi^{2})-\frac{1-\phi^{2}}{2\sigma^{2}}(x_{1}-\mu)^{2}$
$\displaystyle-\sum_{t=2}^{T}\frac{1}{2\sigma^{2}}\big{(}x_{t}-\mu(1-\phi)-\phi
x_{t-1}\big{)}^{2}-\sum_{t=1}^{T}\big{(}\frac{x_{t}}{2}+\frac{1}{2}y_{t}^{2}e^{-x_{t}}\big{)}\Big{\rangle}+C,$
where $C$ is a constant independent of $\phi$ and $x$. For the LMC step, we
need the gradient $\nabla_{\phi}\log q(\phi,x)$ and $\nabla_{x}\log
q(\phi,x)$.
$\displaystyle\nabla_{\phi}\log q(\phi,x)$ $\displaystyle=$
$\displaystyle\frac{a_{0}-1}{1+\phi}-\frac{b_{0}-1}{1-\phi}-\frac{\phi}{1-\phi^{2}}+\phi\frac{\alpha_{\sigma^{2}}}{\beta_{\sigma^{2}}}\big{[}(x_{1}-\mu_{q})^{2}+\sigma_{q}^{2}\big{]}$
$\displaystyle+\frac{\alpha_{\sigma^{2}}}{\beta_{\sigma^{2}}}\sum_{t=2}^{T}\big{[}(x_{t-1}-\mu_{q})(x_{t}-\mu_{q})-\phi(x_{t-1}-\mu_{q})^{2}+(1-\phi)\sigma_{q}^{2}\big{]},$
$\nabla_{x_{1}}\log
q(\phi,x)=-\frac{\alpha_{\sigma^{2}}}{\beta_{\sigma^{2}}}(1-\phi^{2})(x_{1}-\mu_{q})-\frac{1}{2}+\frac{y_{1}^{2}}{2}e^{-x_{1}}+\phi\frac{\alpha_{\sigma^{2}}}{\beta_{\sigma^{2}}}\big{(}x_{2}-(1-\phi)\mu_{q}-\phi
x_{1}\big{)},$ $\nabla_{x_{t}}\log
q(\phi,x)=-\frac{1}{2}+\frac{y_{t}^{2}}{2}e^{-x_{t}}-\frac{\alpha_{\sigma^{2}}}{\beta_{\sigma^{2}}}\big{(}x_{t}-(1-\phi)\mu_{q}-\phi
x_{t-1}\big{)}+\phi\frac{\alpha_{\sigma^{2}}}{\beta_{\sigma^{2}}}\big{(}x_{t+1}-(1-\phi)\mu_{q}-\phi
x_{t}\big{)},$
for $t=2,...,T-1$, and finally,
$\nabla_{x_{T}}\log
q(\phi,x)=-\frac{1}{2}+\frac{y_{T}^{2}}{2}e^{-x_{T}}-\frac{\alpha_{\sigma^{2}}}{\beta_{\sigma^{2}}}\big{(}x_{T}-(1-\phi)\mu_{q}-\phi
x_{T-1}\big{)}.$
## Appendix B: Technical proofs
###### Proof of Theorem 2.
Consider two measures in ${\cal Q}={\mathbb{W}}_{2}({\cal
X})\otimes{\mathbb{W}}_{2}({\cal Y})$: $q^{(1)}(\text{\rm d}x\times\text{\rm
d}y)=q_{x}^{(1)}(\text{\rm d}x)q_{y}^{(1)}(\text{\rm d}y)$ and
$q^{(2)}(\text{\rm d}x\times\text{\rm d}y)=q_{x}^{(2)}(\text{\rm
d}x)q_{y}^{(2)}(\text{\rm d}y)$. Then,
$W_{2}^{2}(q^{(1)},q^{(2)})=W_{2}^{2}(q_{x}^{(1)},q_{x}^{(2)})+W_{2}^{2}(q_{y}^{(1)},q_{y}^{(2)}).$
(39)
With a generic measure $q(\text{\rm d}x\times\text{\rm d}y)=q_{x}(\text{\rm
d}x)q_{y}(\text{\rm d}y)\in{\cal Q}$, write $F(q)$ as
$F(q)=F_{x}(q_{x})+F_{y}(q_{y})+F_{xy}(q)$ (40)
where
$F_{x}(q_{x})=\int_{{\cal X}}q_{x}(x)\log q_{x}(x)\text{\rm
d}x,\;\;\;F_{y}(q_{y})=\int_{{\cal Y}}q_{y}(y)\log q_{y}(y)\text{\rm d}y$
and
$F_{xy}(q)=\int_{{\cal X}\times{\cal
Y}}\big{(}-\log\pi(x,y)\big{)}q(x,y)\text{\rm d}x\times\text{\rm d}y.$
To show that $F(q)$ is lower semi-continuous (l.s.c), consider a sequence of
measures $\\{q^{n}(\text{\rm d}x\times\text{\rm d}y)=q_{x}^{n}(\text{\rm
d}x)q_{y}^{n}(\text{\rm d}y)\\}_{n\geq 1}\subset{\mathbb{W}}_{2}({\cal Q})$
weakly converging to $q^{*}(\text{\rm d}x\times\text{\rm
d}y)=q_{x}^{*}(\text{\rm d}x)q_{y}^{*}(\text{\rm d}y)$, i.e.
$W_{2}(q^{n},q^{*})\to 0,~{}\text{ as }n\to\infty.$
Then (39) implies that
$W_{2}(q_{x}^{n},q_{x}^{*})\to 0,\;\;\;\;\text{ and
}\;\;\;\;\;W_{2}(q_{y}^{n},q_{y}^{*})\to 0,$
hence
$q_{x}^{n}\stackrel{{\scriptstyle
w}}{{\longrightarrow}}q_{x}^{*},\;\;\;\;\text{ and
}\;\;\;\;\;q_{y}^{n}\stackrel{{\scriptstyle w}}{{\longrightarrow}}q_{y}^{*}.$
By Proposition 7.7 of Santambrogio, (2015), $F_{x}(q_{x})$ and $F_{y}(q_{y})$
are l.s.c; hence we have
$\liminf_{n}F_{x}(q_{x}^{n})\geq F_{x}(q_{x}^{*}),\;\;\;\;\text{ and
}\;\;\;\;\;\liminf_{n}F_{y}(q_{y}^{n})\geq F_{y}(q_{y}^{*}).$ (41)
As $\log\pi(x,y)$ is continuous, and hence bounded, on the compact set ${\cal
X}\times{\cal Y}$, by the definition of weak convergence, we have that
$\lim_{n}F_{xy}(q^{n})=F_{xy}(q^{*}).$ (42)
From (41)-(42),
$\liminf_{n}F(q^{n})\geq F(q^{*}),$ (43)
proving that $F(q)$ is l.s.c.
We now show that $F(q)$ is convex. Consider two measures
$q^{(1)},q^{(2)}\in{\cal Q}$ and any $t\in(0,1)$. Because $f(z)=z\log(z)$ is
convex, $F_{x}(q_{x})=\int f(q_{x})\text{\rm d}x$ and $F_{y}(q_{y})=\int
f(q_{y})\text{\rm d}y$ are convex. Also, $F_{xy}(q)$ is linear and hence
convex. Therefore,
$\displaystyle F(tq^{(1)}+(1-t)q^{(2)})$ $\displaystyle=$ $\displaystyle
F_{x}(tq_{x}^{(1)}+(1-t)q_{x}^{(2)})+F_{y}(tq_{y}^{(1)}+(1-t)q_{y}^{(2)})+F_{xy}(tq^{(1)}+(1-t)q^{(2)})$
$\displaystyle\leq$ $\displaystyle
t\big{(}F_{x}(q_{x}^{(1)})+F_{y}(q_{y}^{(1)})+F_{xy}(q^{(1)})\big{)}+(1-t)\big{(}F_{x}(q_{x}^{(2)})+F_{y}(q_{y}^{(2)})+F_{xy}(q^{(2)})\big{)}$
$\displaystyle=$ $\displaystyle tF(q^{(1)})+(1-t)F(q^{(2)}).$
∎
###### Proof of Corollary 3.
As ${\cal X}$ and ${\cal Y}$ are compact, by the Prokhorov theorem, ${\cal
P}({\cal X})$ and ${\cal P}({\cal Y})$ are compact (w.r.t. to the weak
convergence, and also w.r.t the Wasserstein metric). As ${\cal P}_{2}({\cal
X})\subset{\cal P}({\cal X})$, for any sequence of measures $\\{\mu_{n}\\}$ in
${\cal P}_{2}({\cal X})$, there must exist a subsequence $\\{\mu_{n_{k}}\\}$
weakly converging to some measure $\mu\in{\cal P}({\cal X})$. As ${\cal X}$ is
compact, $\int_{{\cal X}}|x|^{2}\mu(\text{\rm d}x)<\infty$, hence $\mu\in{\cal
P}_{2}({\cal X})$. This implies that ${\cal P}_{2}({\cal X})$ is compact.
Similarly, ${\cal P}_{2}({\cal Y})$ is compact, and therefore the product
space ${\cal Q}={\mathbb{W}}_{2}({\cal X})\otimes{\mathbb{W}}_{2}({\cal Y})$
is compact. Recall that ${\mathbb{W}}_{2}({\cal X})$ (res.
${\mathbb{W}}_{2}({\cal Y})$) is ${\cal P}_{2}({\cal X})$ (res. ${\cal
P}_{2}({\cal Y})$) equipped with the Wasserstein distance. From Theorem 2,
$F(q)$ is l.s.c on the compact space ${\cal Q}$, by the Weierstrass theorem,
there exists $q^{*}\in{\cal Q}$ such that $F(q^{*})=\min\\{F(q):q\in{\cal
Q}\\}$. The uniqueness of $q^{*}$ is implied by the fact that $F(q)$ is
convex. ∎
###### Proof of Theorem 4.
Given $q_{y}^{(t)}$, define
$q_{x}^{*}(x)=\exp\Big{(}{\mathbb{E}}_{q_{y}^{(t)}}\big{[}\log\pi(x,y)\big{]}+C(q_{y}^{(t)})\Big{)},$
where $C(q_{y}^{(t)}$ is the normalising constant. We have that
$\displaystyle F(q^{(t)})=\text{\rm KL}(q_{x}^{(t)}q_{y}^{(t)}\|\pi)$
$\displaystyle=$ $\displaystyle\int
q_{x}^{(t)}(x)q_{y}^{(t)}(y)\log\frac{q_{x}^{(t)}(x)q_{y}^{(t)}(y)}{\pi(x,y)}\text{\rm
d}x\times\text{\rm d}y$ $\displaystyle=$ $\displaystyle\int
q_{x}^{(t)}(x)\Big{(}\log
q_{x}^{(t)}(x)-{\mathbb{E}}_{q_{y}^{(t)}}\big{(}\log\pi(x,y)\big{)}\Big{)}\text{\rm
d}x+E(q_{y}^{(t)})$ $\displaystyle=$ $\displaystyle\text{\rm
KL}(q_{x}^{(t)}\|q_{x}^{*})+E(q_{y}^{(t)})+C(q_{y}^{(t)})$
with $E(q_{y}^{(t)})=\int q_{y}^{(t)}(y)\log q_{y}^{(t)}(y)\text{\rm d}y$. If
the step size $h_{x}$ is sufficiently small, Lemma 1 guarantees that
$\text{\rm KL}(q_{x}^{(t+1)}\|q_{x}^{*})\leq\text{\rm
KL}(q_{x}^{(t)}\|q_{x}^{*});$
hence
$F(q_{x}^{(t+1)}q_{y}^{(t)})\leq F(q_{x}^{(t)}q_{y}^{(t)})=F(q^{(t)}).$ (44)
Given $q_{x}^{(t+1)}$, define
$q_{y}^{*}(y)=\exp\Big{(}{\mathbb{E}}_{q_{x}^{(t+1)}}\big{[}\log\pi(x,y)\big{]}+C(q_{x}^{(t+1)})\Big{)}.$
We have that
$F(q_{x}^{(t+1)}q_{y}^{(t)})=\text{\rm
KL}(q_{y}^{(t)}\|q_{y}^{*})+E(q_{x}^{(t+1)})+C(q_{x}^{(t+1)}).$
Lemma 1 guarantees that
$\text{\rm KL}(q_{y}^{(t+1)}\|q_{y}^{*})\leq\text{\rm
KL}(q_{y}^{(t)}\|q_{y}^{*})$
and hence
$F(q^{(t+1)})=F(q_{x}^{(t+1)}q_{y}^{(t+1)})\leq F(q_{x}^{(t+1)}q_{y}^{(t)}).$
(45)
From (44)-(45), $F(q^{(t)})$ is reduced over $t$. By Corollary 3, $q^{(t)}$
must converge to the unique minimizer $q^{*}$ of $F(q)$. ∎
###### Proof of Theorem 6.
Under the conditions (A1) and (A2), by Theorem 1 of Zhang and Gao, (2019),
${\mathbb{E}}_{p_{0}^{(n)}}\Big{[}{\mathbb{E}}_{q_{n}^{*}}\Big{(}\|\theta-\theta_{0}\|_{2}^{2}\Big{)}\Big{]}=O(\varepsilon_{n}^{2}+\gamma_{n}^{2})$
(46)
with
$\gamma_{n}^{2}=\frac{1}{n}\min_{q\in\mathcal{Q}}{\mathbb{E}}_{p_{0}^{(n)}}\Big{[}\text{\rm
KL}\big{(}q\|\pi_{n}\big{)}\Big{]}.$ (47)
Denote by
$p^{(n)}(X^{(n)})=\int p_{\theta}^{(n)}(X^{(n)})\pi_{0}(\text{\rm d}\theta)$
the marginal likelihood. For any $q\in{\cal Q}$, we have
$\displaystyle\gamma_{n}^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{n}{\mathbb{E}}_{p_{0}^{(n)}}\Big{[}\int\log\frac{q(\theta)p^{(n)}(X^{(n)})}{\pi_{0}(\theta)p_{\theta}^{(n)}(X^{(n)})}q(\text{\rm
d}\theta)\Big{]}$ (48) $\displaystyle=$ $\displaystyle\frac{1}{n}\text{\rm
KL}(q\|\pi_{0})+\frac{1}{n}\int\Big{(}p_{0}^{(n)}(X^{(n)})\int\log\frac{p^{(n)}(X^{(n)})}{p_{\theta}^{(n)}(X^{(n)})}q(\text{\rm
d}\theta)\Big{)}\text{\rm d}X^{(n)}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\text{\rm
KL}(q\|\pi_{0})+\frac{1}{n}{\mathbb{E}}_{q}\Big{(}\int
p_{0}^{(n)}(X^{(n)})\log\frac{p_{0}^{(n)}(X^{(n)})}{p_{\theta}^{(n)}(X^{(n)})}\text{\rm
d}X^{(n)}-\int
p_{0}^{(n)}(X^{(n)})\log\frac{p_{0}^{(n)}(X^{(n)})}{p^{(n)}(X^{(n)})}\text{\rm
d}X^{(n)}\Big{)}$ $\displaystyle=$ $\displaystyle\frac{1}{n}\text{\rm
KL}(q\|\pi_{0})+\frac{1}{n}{\mathbb{E}}_{q}\Big{(}\text{\rm
KL}\big{(}p_{0}^{(n)}\|p_{\theta}^{(n)}\big{)}-\text{\rm
KL}\big{(}p_{0}^{(n)}\|p^{(n)}\big{)}\Big{)}$ $\displaystyle\leq$
$\displaystyle\frac{1}{n}\text{\rm
KL}(q\|\pi_{0})+\frac{1}{n}{\mathbb{E}}_{q}\Big{(}\text{\rm
KL}\big{(}p_{0}^{(n)}\|p_{\theta}^{(n)}\big{)}\Big{)}.$
Select $q(\theta)=N(\theta_{0},1/nI_{d})$, and estimate the first term in
(48).
$\frac{1}{n}\text{\rm KL}(q\|\pi_{0})\leq\frac{1}{n}|{\mathbb{E}}_{q}\log
q(\theta)|+\frac{1}{n}|{\mathbb{E}}_{q}\log\pi_{0}(\theta)|.$
We have that
$\frac{1}{n}|{\mathbb{E}}_{q}\log
q(\theta)|\leq\frac{1}{n}\big{(}\frac{d}{2}\log(2\pi)+\frac{d}{2}\log
n+\frac{d}{2}\big{)}=o(1).$
By Assumption (A3),
$\displaystyle\frac{1}{n}|{\mathbb{E}}_{q}\log\pi_{0}(\theta)|$
$\displaystyle\leq$
$\displaystyle\frac{1}{n}\Big{(}|\log\pi_{0}(\theta_{0})|+{\mathbb{E}}_{q}|\log\pi_{0}(\theta)-\log\pi_{0}(\theta_{0})|\Big{)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{n}\Big{(}|\log\pi_{0}(\theta_{0})|+C_{3}{\mathbb{E}}_{q}\|\theta-\theta_{0}\|_{2}\Big{)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{n}\Big{(}|\log\pi_{0}(\theta_{0})|+C_{3}\sqrt{\frac{d}{n}}\Big{)}=o(1).$
Therefore,
$\frac{1}{n}\text{\rm KL}(q\|\pi_{0})=o(1).$ (49)
We now estimate the second term in (48). By Assumption (A3),
$\displaystyle\frac{1}{n}{\mathbb{E}}_{q}\Big{(}\text{\rm
KL}\big{(}p_{0}^{(n)}\|p_{\theta}^{(n)}\big{)}\Big{)}$ $\displaystyle\leq$
$\displaystyle\frac{1}{n}{\mathbb{E}}_{q}\Big{(}{\mathbb{E}}_{p_{0}^{(n)}}\big{|}\log
p_{\theta}^{(n)}(X^{(n)})-\log p_{\theta_{0}}^{(n)}(X^{(n)})\big{|}\Big{)}$
(50) $\displaystyle\leq$
$\displaystyle\frac{1}{n}C_{5}{\mathbb{E}}_{q}\big{(}\|\theta-\theta_{0}\|_{2}\big{)}$
$\displaystyle\leq$ $\displaystyle\frac{1}{n}C_{5}\sqrt{\frac{d}{n}}=o(1).$
Equations (48), (49) and (50) imply that $\gamma_{n}^{2}=o(1)$. Therefore,
from (46) and noting that $\varepsilon_{n}^{2}=o(1)$, we have that
${\mathbb{E}}_{p_{0}^{(n)}}\Big{[}{\mathbb{E}}_{q_{n}^{*}}\Big{(}\|\theta-\theta_{0}\|_{2}^{2}\Big{)}\Big{]}=o(1).$
By Markov’s inequality
${\mathbb{E}}_{p_{0}^{(n)}}\Big{[}q_{n}^{*}\big{(}\|\theta-\theta_{0}\|_{2}^{2}>\epsilon\big{)}\Big{]}\leq\frac{{\mathbb{E}}_{p_{0}^{(n)}}\Big{[}{\mathbb{E}}_{q_{n}^{*}}\Big{(}\|\theta-\theta_{0}\|_{2}^{2}\Big{)}\Big{]}}{\epsilon}\to
0,$
implying
$q_{n}^{*}\Big{(}\|\theta-\theta_{0}\|_{2}^{2}>\epsilon\Big{)}\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}0,\;\;\;\;p^{(n)}_{0}-a.s.$
∎
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|
$(s,a,s')$ is in $\IOAtrans(A)$, it means that $A$ can move from state
$s$ to state $s'$ by executing action $a$.
There is also an equivalence relation $\IOAtask(A)$ on the output and internal actions, which is used for enforcing fairness conditions—the basic idea is that in a fair execution some action in each equivalence class must be executed eventually (a more accurate definition will be given below).
The I/O automaton model carries with it a lot of specialized jargon.
We'll try to avoid it as much as possible. One thing that will be
difficult to avoid in reading [186] is the notion of a
signature, which is just the tuple $\IOAsig(A) = (\IOAin(A), \IOAout(A),
\IOAint(A))$ describing the actions of an automaton $A$.
§.§ Enabled actions
An action $a$ is enabled in some state $s$ if $\IOAtrans(A)$
contains at least one transition $(s,a,s')$. Input actions are
always enabled—this is a requirement of the model. Output
and internal actions—the “locally controlled” actions—are not
subject to this restriction. A state $s$ is quiescent if
only input actions are enabled in $s$.
§.§ Executions, fairness, and traces
An execution of A is a sequence $s_{0} a_{0} s_{1} a_{1}
\dots{}$ where each triple $(s_{i}, a_{i} s_{i+1})$ is in
$\IOAtrans(A)$. Executions may be finite or infinite; if finite, they must end in a state.
A trace of $A$ is a subsequence of some execution consisting
precisely of the external (i.e., input and output) actions, with
states and internal actions omitted. If we don't want to get into the
guts of a particular I/O automaton—and we usually don't, unless we
can't help it because we have to think explicitly about states for
some reason—we can describe its externally-visible behavior by just giving its set of traces.
§.§ Composition of automata
Composing a set of I/O automata yields a new super-automaton whose
state set is the Cartesian product of the state sets of its components
and whose action set is the union of the action sets of its
components. A transition with a given action $a$ updates the states
of all components that have $a$ as an action and has no effect on the states of other components. The classification of actions into the three classes is used to enforce some simple compatibility rules on the component automata; in particular:
* An internal action of a component is never an action of another component—internal actions are completely invisible.
* No output action of a component can be an output action of another component.
* No action is shared by infinitely many
components.[Note that infinite (but countable) compositions
are permitted.] In practice this means that no action can be an input action of infinitely many components, since the preceding rules mean that any action is an output or internal action of at most one component.
All output actions of the components are also output actions of the composition. An input action of a component is an input of the composition only if some other component doesn't supply it as an output; in this case it becomes an output action of the composition. Internal actions remain internal (and largely useless, except for bookkeeping purposes).
The equivalence relation is the union of the relations for the components: this turns out to give a genuine
equivalence relation on output and internal actions precisely because the first two compatibility rules hold.
Given an execution or trace $X$ of a composite automaton that includes
$A$, we can construct the corresponding execution or trace $X|A$ of
$A$ which just includes the states of $A$ and the actions visible to
$A$ (events that don't change the state of $A$ drop out). The
definition of composition is chosen so that $X|A$ is in fact an
execution/trace of $A$ whenever $X$ is.
§.§ Hiding actions
Composing $A$ and $B$ continues to expose the outputs of $A$ even if
they line up with inputs of $B$. While this may sometimes be desirable, often we want to shove such internal communication under the rug. The model lets us do this by redefining the signature of an automaton to make some or all of the output actions into internal actions.
§.§ Fairness
I/O automata come with a built-in definition of execution!fairfair
executionfair executions, where an execution of $A$ is
fair if, for each equivalence class $C$ of actions in
* the execution is finite and no action in $C$ is enabled in the final state, or
* the execution is infinite and there are infinitely many
occurrences of actions in $C$, or
* the execution is infinite and there are infinitely many states
in which no action in $C$ is enabled.
If we think of $C$ as corresponding to some thread or process, this
says that $C$ gets infinitely many chances to do something in an
infinite execution, but may not actually do them if it gives ups and
stops waiting (the third case). The finite case essentially says that
a finite execution isn't fair unless nobody is waiting at the end.
The motivation for this particular definition is that it guarantees
(a) that any finite execution can be extended to a fair execution and
(b) that the restriction $X|A$ of a fair execution or trace $X$ is also fair.
Fairness is useful e.g. for guaranteeing message delivery in a message-passing system: make each message-delivery action its own task class and each message will eventually be delivered; similarly make each message-sending action its own task class and a process will eventually send every message it intends to send. Tweaking the task classes can allow for possibilities of starvation, e.g. if all message-delivery actions are equivalent then a spammer can shut down the system in a “fair” execution where only his (infinitely many) messages are delivered.
§.§ Specifying an automaton
The typical approach is to write down preconditions and effects for
each action (for input actions, the preconditions are empty). An
example would be the spambot in Algorithm <ref>.
IOAinputActioninput actionend action
IOAoutputActionoutput actionend action
IOApreconditionpreconditionend precondition
IOAeffectseffectsend effects
$\IOAspambotState ← m$
$\IOAspamAction = m$
none (keep spamming)
Spambot as an I/O automaton
(Plus an initial state, e.g. $\IOAspambotState = ⊥$, where $⊥$ is not a possible message, and a task partition, of which we will speak more below when we talk about liveness properties.)
§ HIGH-LEVEL VIEW: TRACES
When studying the behavior of a system, traces are what we really care about, and we want to avoid talking about states as much as possible. So what we'll aim to do is to get rid of the states early by computing the set of traces (or fair traces) of each automaton in our system, then compose traces to get traces for the system as a whole. Our typical goal will be to show that the resulting set of traces has some desirable properties, usually of the form (1) nothing bad happens (a safety property); (2) something good eventually happens (a liveness property); or (3) the horribly complex composite automaton representing this concrete system acts just like that nice clean automaton representing a specification (a simulation).
Very formally, a trace property specifies both the signature of the automaton and a set of traces, such that all traces (or perhaps fair traces) of the automata appear in the set. We'll usually forget about the first part.
Tricky detail: It's OK if not all traces in $P$ are generated by $A$
(we want $\IOAtrace(A) \subseteq P$, but not necessarily $\IOAtrace(A)
= P$). But $\IOAtrace(A)$ will be pretty big (it includes, for
example, all finite sequences of input actions) so hopefully the fact
that $A$ has to do something with inputs will tell us something useful.
§.§ Example
A property we might demand of the spambot above (or some other
abstraction of a message channel) is that it only delivers messages
that have previously been given to it. As a trace property this says
that in any trace $t$, if $t_{k} = \IOAspamAction(m)$, then $t_{j} =
\IOAsetMessageAction(m)$ for some $j < k$. (As a set, this is just
the set of all sequences of external spambot-actions that have this
property.) Call this property $P$.
To prove that the spambot automaton given above satisfies $P$, we
might argue that for any execution $s_{0}a_{0}s_{1}a_{1}\dots{}$, that
$s_{i} = m$ in the last $\IOAsetMessageAction$ action preceding
$s_{i}$, or $⊥$ if there is no such action. This is easily proved by
induction on $i$. It then follows that since $\IOAspamAction(m)$ can
only transmit the current state, that if $\IOAspamAction(m)$ follows
$s_{i} = m$ that it follows some earlier $\IOAsetMessageAction(m)$ as claimed.
However, there are traces that satisfy $P$ that don't correspond to
executions of the spambot; for example, consider the trace
$\IOAsetMessageAction(0) \IOAsetMessageAction(1) \IOAspamAction(0)$.
This satisfies $P$ (0 was previously given to the automaton
$\IOAspamAction(0)$), but the automaton won't generate it because the
0 was overwritten by the later $\IOAsetMessageAction(1)$ action. Whether this is indicates a problem with our automaton not being nondeterministic enough or our trace property being too weak is a question about what we really want the automaton to do.
§.§ Types of trace properties
§.§.§ Safety properties
$P$ is a safety property if
* $P$ is nonempty.
* $P$ is prefix-closed, i.e. if $xy$ is in $P$ then $x$
is in $P$.
* P is limit-closed, i.e. if $x_{1}, x_{1}x_{2},
x_{1}x_{2}x_{3}, \dots{}$ are all in $P$, then so is the infinite sequence obtained by taking their limit.
Because of the last restrictions, it's enough to prove that $P$ holds
for all finite traces of $A$ to show that it holds for all traces (and
thus for all fair traces), since any trace is a limit of finite
traces. Conversely, if there is some trace or fair trace for which
$P$ fails, the second restriction says that $P$ fails on any finite
prefix of $P$, so again looking at only finite prefixes is enough. The spambot property mentioned above is a safety property.
Safety properties are typically proved using invariants, properties that are shown by induction to hold in all reachable states.
§.§.§ Liveness properties
$P$ is a liveness property of $A$ if any finite sequence of
actions in $\IOAacts(A)$ has an extension in $P$. Note that liveness
properties will in general include many sequences of actions that
aren't traces of $A$, since they are extensions of finite sequences
that $A$ can't do (e.g. starting the execution with an action not
enabled in the initial state). If you want to restrict yourself only
to proper executions of $A$, use a safety property. (It's worth
noting that the same property $P$ can't do both: any $P$ that is both a liveness and a safety property includes all sequences of actions because of the closure rules.)
Liveness properties are those that are always eventually satisfiable;
asserting one says that the property is eventually satisfied. The
typical way to prove a liveness property is with a progress
a function $f$ on states that (a) drops by at least 1 every time
something that happens infinitely often happens (like an action from
an always-enabled task class) and (b) guarantees $P$ once it reaches 0.
An example would be the following property we might demand of our
spambot: any trace with at least one $\IOAsetMessageAction(\dots{})$
action contains infinitely many $\IOAspamAction(\dots{})$ actions.
Whether the spambot automaton will satisfy this property (in
fair traces) depends on its task partition. If all
$\IOAspamAction(\dots{})$ actions are in the same equivalence class,
then any execution with at least one will have
some (…) action enabled at all times thereafter,
so a fair trace containing a can't be finite
(since spam is enabled in the last state) and if infinite contains
infinitely many spam messages (since spam messages of some sort are
enabled in all but an initial finite prefix). On the other hand, if
$\IOAspamAction(m_1)$ and $\IOAspamAction(m_2)$ are not equivalent in
$\IOAtask(A)$, then the spambot doesn't satisfy the liveness property:
in an execution that alternates $\IOAsetMessageAction(m_1)
\IOAsetMessageAction(m_2) \IOAsetMessageAction(m_1)
\IOAsetMessageAction(m_2) \dots$ there are infinitely many states in
which $\IOAspamAction(m_1)$ is not enabled, so fairness doesn't
require doing it even once, and similarly for $\IOAspamAction(m_2)$.
§.§.§ Other properties
Any other property $P$ can be expressed as the intersection of a
safety property (the
closure of $P)$ and a liveness property (the union of $P$ and the set
of all finite
sequences that aren't prefixes of traces in $P$). The intuition is
that the safety property prunes out the excess junk we threw into the
liveness property to make it a liveness property, since any sequence
that isn't a prefix of a trace in $P$ won't go into the safety
property. This leaves only the traces in $P$.
Example: Let $P = \{ 0^{n}1^{\infty} \}$ be the set of traces where we
eventually give up on our pointless 0-action and start doing only
1-actions forever. Then $P$ is the intersection of the safety
property $S = \{ 0^{n}1^{m} \} \cup P$ (the extra junk is from
prefix-closure) and the liveness property $L = \{ 0^{n}11^{m}0x | x in
\{0,1\}^{*} \} \cup P$. Property $S$ says that once we do a 1 we
never do a 0, but allows finite executions of the form $0^{n}$ where
we never do a 1. Property $L$ says that we eventually do a 1-action, but that we can't stop unless we later do at least one 0-action.
§.§ Compositional arguments
The product of trace properties $P_{1}, P_{2} \dots{}$ is
the trace property $P$ where $T$ is in $P$ if and only if $T|\IOAsig(P_{i})$ is
in $P_{i}$ for each $i$. If the $\{A_{i}\}$ satisfy corresponding
propertties $\{P_{i}\}$ individually, then their composition satisfies
the product property. (For safety properties, often we prove
something weaker about the $A_{i}$, which is that each $A_{i}$
individually is not the first to violate $P$—i.e., it can't leave
$P$ by executing an internal or output action. In an execution where
inputs by themselves can't violate $P$, $P$ then holds.)
Product properties let us prove trace properties by smashing together
properties of the component automata, possibly with some restrictions
on the signatures to get rid of unwanted actions. The product
operation itself is in a sense a combination of a Cartesian product
(pick traces $t_{i}$ and smash them together) filtered by a
consistency rule (the smashed trace must be consistent); it acts much
like intersection (and indeed can be made identical to intersection if
we treat a trace property with a given signature as a way of
describing the set of all $T$ such that $T|\IOAsig(P_{i})$ is in
§.§.§ Example
Consider two spambots $A_1$ and $A_2$ where we identify the
$\IOAspamAction(m)$ operation of $A_1$ with the
$\IOAsetMessageAction(m)$ operation of $A_2$; we'll call this combined
action $\IOAspamAction_1(m)$ to distinguish it from the output actions
of $A_2$. We'd like to argue that the composite automaton $A_1+A_2$
satisfies the safety property (call it $P_{m}$) that any occurrence of
$\IOAspamAction(m)$ is preceded by an occurrence of
$\IOAsetMessageAction(m)$, where the signature of $P_{m}$ includes
$\IOAsetMessageAction(m)$ and $\IOAspamAction(m)$ for some specific
$m$ but no other operations. (This is an example of where trace property signatures can be useful without being limited to actions of any specific component automaton.)
To do so, we'll prove a stronger property $P'_{m}$, which is
$P_{m}$ modified to include the $\IOAspamAction_1(m)$ action in its
signature. Observe that $P'_m$ is the product of the safety
properties for $A_1$ and $A_2$ restricted to $\IOAsig(P'_m)$, since
the later says that any trace that includes $\IOAspamAction(m)$ has a
previous $\IOAspamAction_1(m)$ and the former says that any trace that
includes $\IOAspamAction_1(m)$ has a previous
$\IOAsetMessageAction(m)$. Since these properties hold for the
individual $A_1$ and $A_2$, their product, and thus the restriction
$P'_m$, holds for $A_1+A_2$, and so $P_{m}$ (as a further restriction) holds for $A_1+A_2$ as well.
Now let's prove the liveness property for $A_1+A_2$, that at least one
occurrence of $\IOAsetMessageAction$ yields infinitely many
actions. Here we let
$L_{1} = \{ \text{at least one \IOAsetMessageAction action} ⇒
\text{infinitely many $\IOAspamAction_1$ actions} \}$ and
$L_{2} = \{
\text{at least one $\IOAspamAction_1$ action}
\text{infinitely many \IOAspamAction actions} \}$.
The product of these properties is all sequences with (a) no
actions or (b) infinitely many actions, which is what we want. This product holds if the individual
properties $L_{1}$ and $L_{2}$ hold for $A_1+A_2$, which will be the
case if we set $\IOAtask(A_1)$ and $\IOAtask(A_2)$ correctly.
§.§ Simulation arguments
Show that $\IOAtraces(A)$ is a subset of $\IOAtraces(B)$ (possibly
after hiding some actions of $A$) by showing a
simulation relation
$f:\IOAstates(A) \rightarrow \IOAstates(B)$ between states of $A$ and
states of $B$. Requirements on $f$ are
* If $s$ is in $\IOAstart(A)$, then $f(s)$ includes some element
of $\IOAstart(B)$.
* If $(s,a,s')$ is in $\IOAtrans(A)$ and $s$ is reachable, then
for any reachable $u$ in $f(s)$, there is a sequence of actions $x$
that takes $u$ to some $v$ in $f(s')$ with $\IOAtrace(x) =
\IOAtrace(a)$.
Using these we construct an execution of $B$ matching (in trace) an
execution of $A$ by starting in $f(s_{0})$ and applying the second
part of the definition to each action in the $A$ execution (including the hidden ones!)
§.§.§ Example
A single spambot $A$ can simulate the conjoined spambots $A_1+A_2$.
Proof: Let $f(s) = (s,s)$. Then $f(⊥) = (⊥, ⊥)$ is a start
state of $A_1+A_2$. Now consider a transition $(s,a,s')$ of $A$; the
action a is either (a) $\IOAsetMessageAction(m)$, giving $s' = m$;
here we let $x = \IOAsetMessageAction(m) \IOAspamAction_1(m)$ with
$\IOAtrace(x) = \IOAtrace(a)$ since $\IOAspamAction_1(m)$ is internal
and $f(s') = (m,m)$ the result of applying $x$; or (b) $a =
\IOAspamAction(m)$, which does not change $s$ or $f(s)$; the matching
$x$ is $\IOAspamAction(m)$, which also does not change $f(s)$ and has the same trace.
A different proof could take advantage of $f$ being a relation by
defining $f(s) = \{ (s,s') | s' \in \IOAstates(A_2) \}$. Now we don't
care about the state of $A_2$, and treat a $\IOAsetMessageAction(m)$
action of $A$ as the sequence $\IOAsetMessageAction(m)$ in $A_1+A_2$
(which updates the first component of the state correctly) and treat a
$\IOAspamAction(m)$ action as $\IOAspamAction_1(m) \IOAspamAction(m)$
(which updates the second component—which we don't care about—and
has the correct trace.) In some cases an approach of this sort is
necessary because we don't know which simulated state we are heading
for until we get an action from $A$.
Note that the converse doesn't work: $A_1+A_2$ don't simulate $A$,
since there are traces of $A_1+A_2$ (e.g. $\IOAsetMessageAction(0)
\IOAspamAction_1(0) \IOAsetMessageAction(1) \IOAspamAction(0)$) that
don't restrict to traces of $A$. See <cit.> for a more
complicated example of how one FIFO queue can simulate two FIFO queues
and vice versa (a situation called bisimulation).
Since we are looking at traces rather than fair traces, this kind of
simulation doesn't help much with liveness properties, but sometimes
the connection between states plus a liveness proof for $B$ can be used
to get a liveness proof for $A$ (essentially we have to argue that $A$
can't do infinitely many action without triggering a $B$-action in an
appropriate task class). Again see <cit.>.
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|
# Handling missing data in large healthcare dataset:
a case study of unknown trauma outcomes
E. M. Mirkes<EMAIL_ADDRESS>Department of Mathematics, University of
Leicester, Leicester, LE1 7RH, UK T.J. Coats<EMAIL_ADDRESS>Emergency
Medicine Academic Group, Department of Cardiovascular Sciences, University of
Leicester, Leicester, LE1 7RH, UK J. Levesley<EMAIL_ADDRESS>Department of
Mathematics, University of Leicester, Leicester, LE1 7RH, UK A. N. Gorban
<EMAIL_ADDRESS>Department of Mathematics, University of Leicester, Leicester,
LE1 7RH, UK
###### Abstract
Handling of missed data is one of the main tasks in data preprocessing
especially in large public service datasets. We have analysed data from the
Trauma Audit and Research Network (TARN) database, the largest trauma database
in Europe. For the analysis we used 165,559 trauma cases. Among them, there
are 19,289 cases (13.19%) with unknown outcome. We have demonstrated that
these outcomes are not missed ‘completely at random’ and, hence, it is
impossible just to exclude these cases from analysis despite the large amount
of available data. We have developed a system of non-stationary Markov models
for the handling of missed outcomes and validated these models on the data of
15,437 patients which arrived into TARN hospitals later than 24 hours but
within 30 days from injury. We used these Markov models for the analysis of
mortality. In particular, we corrected the observed fraction of death. Two
naïve approaches give 7.20% (available case study) or 6.36% (if we assume that
all unknown outcomes are ‘alive’). The corrected value is 6.78%. Following the
seminal paper of Trunkey (1983) the multimodality of mortality curves has
become a much discussed idea. For the whole analysed TARN dataset the
coefficient of mortality monotonically decreases in time but the stratified
analysis of the mortality gives a different result: for lower severities the
coefficient of mortality is a non-monotonic function of the time after injury
and may have maxima at the second and third weeks. The approach developed here
can be applied to various healthcare datasets which experience the problem of
lost patients and missed outcomes.
###### keywords:
Missed data , Big data , Data cleaning , Mortality , Markov models , Risk
evaluation
## 1 Introduction
Enthusiasm for the use of big data in the improvement of health service is
huge but there is a concern that without proper attention to some specific
challenges the mountain of big data efforts will bring forth a mouse [1]. Now,
there is no technical problem with “big” in healthcare. Electronic health
records include hundreds of millions of outpatient visits and tens of millions
of hospitalizations, and these numbers grow exponentially. The main problem is
in quality of data.
“Big data” very often means “dirty data” and the fraction of data inaccuracies
increases with data volume growth. Human inspection at the big data scale is
impossible and there is a desperate need for intelligent tools for accuracy
and believability control.
The second big challenge of big data in healthcare is missed information.
There may be many reasons for data incompleteness. One of them is in health
service “fragmentation”. This problem can be solved partially by the national
and international unification of the electronic health records (see, for
example, Health Level Seven International (HL7) standards [13] or discussion
of the template for uniform reporting of trauma data [37]). However, some
fragmentation is unavoidable due to the diverse structure of the health
service. In particular, the modern tendency for personalization of medicine
can lead to highly individualized sets of attributes for different patients or
patient groups. There are several universal technologies for the handling of
missing data [40, 41, 36, 46, 23, 22, 14]. Nevertheless, the problem of
handling missed values in large healthcare datasets is certainly not
completely solved. It continues to attract the efforts of many researchers
(see, for example, [8]) because the popular universal tools can lead to bias
or loss of statistical power [21, 51]. For each system, it is desirable to
combine various existing approaches for the handling of missing data (or to
invent new ones) to minimize the damage to the results of data analysis. For
the best possible solution, we have to take into account the peculiarities of
each database and to specify the further use of the cleaned data (it is
desirable to understand in advance how we will use the preprocessed data).
In our work we analyze missed values in the TARN database [54]. We use the
preprocessed data for:
* 1.
the evaluation of the risk of death,
* 2.
the identification of the patterns of mortality,
* 3.
approaching several old problems like the Trunkey hypothesis about the
trimodal distribution of trauma mortality [55].
The ‘two stage lottery’ non-stationary Markov model developed in the sequel
can be used for the analysis of missing outcomes in a much wider context than
the TARN database and could be applied to the handling of data gaps in
healthcare datasets which experience the problem of transferred and lost
patients and missing outcomes.
In this paper we analyze the unknown outcomes. The next task will be the
analysis of missed data in the most common “input” attributes.
## 2 Data set
There are more than 200 hospitals which send information to TARN (TARN
hospitals). This network is gradually increasing. Participation in TARN is
recommended by the Royal College of Surgeons of England and the Department of
Health. More than 93% of hospitals across England and Wales submit their data
to TARN. TARN also receives data from Dublin, Waterford (Eire), Copenhagen,
and Bern.
We use TARN data collected from 01.01.2008 (start of treatment) to 05.05.2014
(date of discharge). The database contains 192,623 records and more than 200
attributes. Sometimes several records correspond to the same trauma case
because the patients may be transferred between TARN hospitals. We join these
records. The resulting database includes data of 182,252 different trauma
cases with various injuries.
Figure 1: The groups of the patients for analysis of mortality. FOD in the
group ‘Available W30D’ can be calculated from the data directly. Mortality in
the group ‘OUT30’ will be evaluated on the basis of the non-stationary Markov
model. The group of 16,693 patients which arrived (were transferred from other
institutions) to TARN hospitals later than 24 hours after injury was excluded
from the mortality analysis. Its subgroup ‘IN30’ of 15,437 patients is used
for validation of the Markov model for ‘OUT30’ group. The subgroups with
age$<65$ and age$\geq 65$ should be separated because for age$\geq 65$ the
following traumas are excluded from the database: Acetabulum fractures (AIS
8562xx), Pelvic/Acetabulum fractures (AIS 8563xx), Pelvic ring fractures (AIS
8561xx), Pubic rami and Femoral neck fractures (AIS 85316x).
16,693 records correspond to patients, who arrived (transferred from other
institutions) to TARN hospitals later than 24 hours after injury. This sample
is biased, for example the Fraction Of Dead outcomes (FOD) for this sample is
3.34% and FOD for all data is 6.05%. This difference is very significant for
such a big sample. (If all the outcomes in a group of the trauma cases are
known then we use the simple definition of FOD in the group: the ratio of the
number of registered deaths in this group to the total number of patients
there. Such a definition is not always applicable. The detailed and more
sophisticated analysis of this notion follows in the next section.) We remove
these 16,693 trauma cases from analysis but use them later for validation of
the “mortality after transfer” model. Among them, there are 15,437 patients
who arrived at a TARN hospital within 30 days after injury. We call this group
‘IN30’ for short (Fig. 1).
As a result we have 165,559 records for analysis (‘Main group’). This main
group consists of two subgroups: 146,270 patients from this group approached
TARN during the first day of injury and remained in TARN hospitals or
discharged to a final destination during the first 30 days after injury. We
call this group the ‘Available within 30 days after injury’ cases (or
‘Available W30D’ for short). The other 19,289 patients have been transferred
within 30 days after injury to a hospital or institution (or unknown
destination) who did not return data to the TARN system. We call them
‘Transferred OUT OF TARN within 30 days after injury’ or just ‘OUT30’ (Fig.
1).
The patients with the non-final discharge destinations ‘Other Acute hospital’
and ‘Other institution’ were transferred from a TARN hospital to a hospital
(institution) outside TARN and did not return to the TARN hospitals within 30
days after injury.
The database includes several indicators for evaluation of the severity of the
trauma case, in particular, Abbreviated Injury Scale (AIS), Injury Severity
Score (ISS) and New Injury Severity Score (NISS). For a detailed description
and comparison of the scores we refer readers to reviews [30, 32]. The
comparative study of predictive ability of different scores has a long history
[19, 43, 7, 42]. The scores are used for mortality predictions and are tested
on different datasets [52, 29, 53, 4].
## 3 Definitions and distributions of outcomes
The widely used definition of the endpoint outcome in trauma research is
survival or death within 30 days after injury [9, 4, 48].
A substantial number of TARN in-hospital deaths following trauma occur after
30 days: there are 957 such cases (or 8% of TARN in-hospital death) among
11,900 cases with ‘Mortuary’ discharge destination. This proportion is
practically the same in the main group (165,559 cases): 894 deaths after 30
days in hospital (or 7.9%) among 11,347 cases with ‘Mortuary’ discharge
destination.
Death later than 30 days after injury may be considered as caused by co-
morbidity rather than the direct consequence of the injury [4]. These later
deaths are not very interesting from the perspective of an acute trauma care
system (as we cannot influence them), but they might be very interesting from
the perspective of a geriatric rehabilitation centre or of an injury
prevention program for elderly patients.
On the other hand, when “end of acute care” is used as an outcome definition
then a significant portion of deaths remains unnoticed. For example, in the
3332 trauma cases treated in the Ulleval University Hospital (Oslo, Norway,
2000-2004) 18% of deaths occurred after discharge from the hospital [48].
The question of whether it is possible to neglect trauma caused mortality
within 30 days after trauma for the patients with the discharge destination
‘Home’, ‘Rehabilitation’ and other ‘recovery’ outcomes is not trivial [48].
Moreover, here are two questions:
* 1.
How do we collect all the necessary data after discharge within 30 days after
trauma – a technical question?
* 2.
How do we classify the death cases after discharge within 30 days after
trauma; are they consequences of the trauma or should they be considered as
comorbidity with some additional reasons?
The best possible answer to the first question requires the special
combination of technical and business process to integrate data from different
sources. The recent linkage from TARN to the Office for National Statistics
(ONS) gives the possibility to access the information about the dates of death
in many cases. It is expected that the further data integration process will
recover many gaps in the outcome data.
The last question is far beyond the scope of data management and analysis and
may be approached from different perspectives. Whether or not the late deaths
are important in a model depends on the question being asked. From the data
management perspective, we have to give the formal definition of the outcome
in the terms of the available database fields. It is impossible to use the
standard definition as survival or death within 30 days after injury because
these data are absent. We define the outcome ‘Alive W30D’ for the TARN
database being as close to the standard definition as it is possible.
In the TARN database discharge destinations ‘Home (own)’, ‘Home (relative or
other carer)’, ‘Nursing Home’, and ‘Rehabilitation’ are considered as final.
If we assume that these trauma cases have the outcome ‘Alive W30D’ then we
loose some cases of death. From the acute care perspective these cases can be
considered as irrelevant. Let us accept this definition. There still remain
many cases with unknown outcome. For analysis of these cases we introduce the
outcome category ‘Transferred’. In this category we include the cases which
left the TARN registry to a hospital or other institution outside TARN, or to
an unknown destination within 30 days. The relations between the discharge
destinations and these three outcomes are presented in Table 1.
Table 1: Distribution of outcomes in the main group (W30D means within 30
days after injury).
Subgroup | Alive W30D | Dead W30D | Unknown | Total
---|---|---|---|---
Available W30D | 135,733 | 10,537 | 0 | 146,270
OUT30 | 0∗ | 0∗ | 19,279 | 19,289
Total | 135,733 | 10,537 | 19,289 | 165,559
∗No known survival or deaths.
As we can see from Table 1, 19,289 trauma cases (or 11.35% of all cases) have
unknown outcome. The first standard question is: can we delete these data and
apply available case analysis? For this purpose we have to consider these
outcome data as “Missing Completely at Random” (MCAR) [39, 40, 36, 46]. This
is definitely not the case. The group with unknown outcomes is exactly the
‘OUT30’ group. The probability of belonging to this group depends, for
example, on the severity of injury (which can be measured, by the maximal
severity, by NISS, by GCS or by another severity score). The $\chi^{2}$ test
of independence shows that transfer depends on the severity with $p$-value
$p<10^{-300}$ (this is the probability that such a strong dependence might
appear by chance).
One can consider all these cases as alive because these patients have been
alive at the point of discharge from TARN hospitals. If we consider all
transferred as alive then the FOD is 6.35%. If we delete all the transferred
patients (study only the Available W30D group) then the FOD is 7.2%. If we
test this hypothesis on 15,437 patients of the group ‘IN30’ transferred to
TARN hospitals from outside the network within 30 days after injury then we
find that the nonzero mortality for them (3.10%).
The data table with known outcomes is necessary for further machine learning
and the main goal is outcome prediction and risk evaluation.
We choose to remove the OUT30D group from data table but simultaneously to
adjust the weights of the retained cases to compensate for the removal. The
information about the OUT30D cases will be used in the construction of the
weights. It is necessary to evaluate the mortality of the patients transferred
from TARN before removing their records and reweighting of the rest. In the
next section we develop, identify and validate Markov models for the analysis
of the mortality of transferred patients.
Another method for handling missed outcomes is multiple imputation of the
outcomes (about multiple imputations see, for example, [22]). Both methods use
similar stochastic models of mortality and transfer. The large number of cases
allows us to use the reweighting approach. A significant majority of the
evaluated weights are between 0.9 and 1.1 (see Section 6).
## 4 Non-stationary Markov model for the analysis of missing outcomes
### 4.1 Structure of model
Figure 2: a) The basic Markov model of mortality (‘recovery/death lottery’)
with two absorbing states (states from which patients do not leave), ‘D’
(death) and ‘R’ (recovery). b) The ‘lottery of transfer’ (from the TARN
network) with one absorbing state ‘L’ (‘left’). The transition probabilities
$\alpha=\alpha(t,s)$, $\nu=\nu(t,s)$ and $\mu(t,s)$ depend on the time after
injury $t$ and on the state of the patient on the first day after trauma
presented by the values of attributes $s$. Figure 3: The Markov model of
mortality and transfer from TARN hospitals to hospitals outside TARN for the
limit case of ‘advanced transfer’, when the lottery of transfer (Fig. 2 b)
occurs every day before the lottery of survival (Fig. 2 a). It has six states:
‘H’ (an alive patient in a TARN hospital), ‘L’ (an alive patient in a hospital
outside TARN), ‘D’ (death in a TARN hospital), ‘${\rm D_{L}}$’ (death in a
hospital outside TARN), ‘R’ (recovery of a patient in a TARN hospital) and
‘${\rm R_{L}}$’ (recovery of a patient in a hospital outside TARN). Four of
them are absorbing: ‘D’, ‘${\rm D_{L}}$’, ‘R’, and ‘${\rm R_{L}}$’. The
transitions from H to ${\rm D_{L}}$ and ${\rm R_{L}}$ are superpositions of
the same day transitions: ${\rm H\to L\to D_{L}}$ and $\rm H\to L\to R_{L}$
Figure 4: The Markov model of mortality and transfer from TARN hospitals to
hospitals outside TARN for the limit case of ‘retarded transfer’, when the
lottery of transfer (Fig. 2 b) occurs every day after the lottery of survival
(Fig. 2 a). It has the same states as the model with advanced transfer (Fig.
3) but different transition probabilities.
We propose a system of Markov models for evaluation of mortality in trauma
datasets. In these models each day each patient can participate in two
‘lotteries’ (Fig. 2). The first lottery (recovery/death), Fig. 2 a, has three
outcomes: ‘R’ (recovery), ‘D’ (death), and ‘H’ (remains in a TARN hospital).
The second lottery (of transfer), Fig. 2 b, has two outcomes: ‘H’ (remains in
a TARN hospital) and ‘L’ (transfer from the TARN hospital to a hospital or
‘other institution’ outside TARN). The probabilities of outcomes depend on the
time from the injury $t$ and on the state of the patient after injury $s$. It
is important to stress that $s$ in our models characterizes the state of the
patient on the first day after trauma and may include severity, type of injury
(blunt/penetrating), localization of traumas, age, gender, airway status,
systolic and diastolic blood pressure, etc, but cannot change in time.
The description of state $s$ may vary in the level of detail depending on the
available information. We have fitted and tested two models based on the
severity of trauma: the maximal severity model and the (binned) NISS model. In
Section 5 we demostrate that it is necessary to refine the model and to
include the age group in $s$ for low severities. For different purposes the
mortality model can include more detail.
The lotteries (Fig. 2) do not commute. We consider two limit cases: ‘advanced
transfer’ (Fig. 3) and ‘retarded transfer’ (Fig. 4). In models with advanced
transfer the lottery of transfer Fig. 2 b) each day precedes the lottery of
recovery/death (Fig. 2 a). In models with retarded transfer, conversely, the
lottery of recovery/death precedes the lottery of transfer.
These two models are important because many other much more general Markov
models are between them in the following exact sense. It is a very strong
assumption that every day there are two steps only: the recovery/death lottery
and the transfer lottery. It may be more realistic to assume that every day
there are many ‘fractional steps’ of recovery/death and of transfer from TARN
and the result of the day is the aggregate result of all of these fractional
steps. Assume that the events of recover, death and transfer are sampled for
every day after injury $t$ from a number $M$ consecutive random choices with
probabilities $\alpha_{i},\nu_{i}$ for recovery/death and $\mu_{i}$ for
transfer out of TARN ($i=1,\ldots,M$), and this chain of choices is Markovian
(the choices for a patient do not depend on the previous choices directly but
only on the current state, H, R or L). It is non-stationary because the
transition probabilities depend on time. They are different for different days
after injury.
This sequence of choices is displayed as a sequence of fractional steps:
$\begin{split}\mbox{recovery/death}_{1}&\to\mbox{transfer}_{1}\to\ldots\\\
&\to\mbox{recovery/death}_{M}\to\mbox{transfer}_{M}.\end{split}$
The probability of in-TARN death in the above model of sequential choice, on a
given day after trauma is
$\nu_{1}+\nu_{2}(1-\alpha_{1}-\nu_{1})(1-\mu_{1})+\ldots+\nu_{M}\prod_{i=1}^{M-1}(1-\alpha_{i}-\nu_{i})(1-\mu_{i}).$
Similarly, the probability for recovery is
$\alpha_{1}+\alpha_{2}(1-\alpha_{1}-\nu_{1})(1-\mu_{1})+\ldots+\alpha_{M}\prod_{i=1}^{M-1}(1-\alpha_{i}-\nu_{i})(1-\mu_{i}).$
Finally, the probability of transfer to a hospital outside of TARN is
$\begin{split}\mu_{1}(1-\alpha_{1}-\nu_{1})&+\mu_{2}(1-\mu_{1})(1-\alpha_{1}-\nu_{1})(1-\alpha_{2}-\nu_{2})\\\
&+\ldots+\mu_{M}\prod_{i=1}^{M-1}(1-\mu_{i})\prod_{j=1}^{M}(1-\alpha_{j}-\nu_{j}){.}\end{split}$
The probabilities $\alpha_{i},\nu_{i}$ for the fractional steps should be
consistent with the daily probabilities $\alpha,\nu$: if there is no transfer
then the resulting probabilities of recovery or death should be the same:
$\begin{split}&\alpha_{1}+\alpha_{2}(1-\alpha_{1}-\nu_{1})+\ldots+\alpha_{M}\prod_{i=1}^{M-1}(1-\alpha_{i}-\nu_{i})=\alpha,\\\
&\nu_{1}+\nu_{2}(1-\alpha_{1}-\nu_{1})+\ldots+\nu_{M}\prod_{i=1}^{M-1}(1-\alpha_{i}-\nu_{i})=\nu.\\\
&\mbox{{Also,}
}\prod_{i=1}^{M}(1-\alpha_{i}-\nu_{i})=1-\alpha-\nu.\end{split}$ (1)
Similarly, for $\mu_{i}$ we get the conditions
$\begin{split}\mu_{1}+\mu_{2}(1-\mu_{1})+\ldots+\mu_{M}\prod_{i=1}^{M-1}(1-\mu_{i})&=\mu\\\
\mbox{ {and} }\prod_{i=1}^{M}(1-\mu_{i})&=1-\mu{.}\end{split}$ (2)
###### Proposition 1.
The probability of in-TARN death in the described model of sequential choice
for every day after trauma is between the probabilities for the Markovian
model with advanced transfer (Fig. 3) and the Markovian model with retarded
transfer (Fig. 4):
$\begin{split}\nu(1-\mu)\leq\nu_{1}&+\nu_{2}(1-\alpha_{1}-\nu_{1})(1-\mu_{1})+\ldots\\\
&+\nu_{M}\prod_{i=1}^{M-1}(1-\alpha_{i}-\nu_{i})(1-\mu_{i})\leq\nu.\end{split}$
(3)
###### Proof.
According to conditions (1), (2),
$\begin{split}&\nu(1-\mu)\\\
&=\left[\nu_{1}+\nu_{2}(1-\alpha_{1}-\nu_{1})+\ldots+\nu_{M}\prod_{i=1}^{M-1}(1-\alpha_{i}-\nu_{i})\right]\\\
&\quad\times\prod_{i=1}^{M}(1-\mu_{i}).\end{split}$ (4)
Notice that for every $j$ ($1\leq j\leq M$),
$\prod_{i=1}^{M}(1-\mu_{i})\leq\prod_{i=1}^{j}(1-\mu_{i})$
because $0\leq 1-\mu_{i}\leq 1$ for all probabilities $\mu_{i}$. Therefore,
$\nu_{j}\prod_{i=1}^{j}(1-\alpha_{i}-\nu_{i})\prod_{k=1}^{M}(1-\mu_{k})\leq\nu_{j}\prod_{i=1}^{j}(1-\alpha_{i}-\nu_{i})(1-\mu_{i})$
and the following inequality holds
$\begin{split}&\left[\nu_{1}+\nu_{2}(1-\alpha_{1}-\nu_{1})+\ldots+\nu_{M}\prod_{i=1}^{M-1}(1-\alpha_{i}-\nu_{i})\right]\\\
&\times\prod_{i=1}^{M}(1-\mu_{i})\\\
&\leq\nu_{1}+\nu_{2}(1-\alpha_{1}-\nu_{1})+\ldots+\nu_{M}\prod_{i=1}^{M-1}(1-\alpha_{i}-\nu_{i}).\end{split}$
(5)
The left inequality in (3) is proven. The right inequality in (3) follows from
condition (1) because for every product
$\nu_{j}\prod_{i=1}^{j}(1-\alpha_{i}-\nu_{i})(1-\mu_{i})\leq\nu_{j}\prod_{i=1}^{j}(1-\alpha_{i}-\nu_{i}){.}$
∎
The proofs of the following propositions are very similar
###### Proposition 2.
The probability of in-TARN recovery in the described model of sequential
choice for every day after trauma is between the probabilities for the
Markovian model with advanced transfer (Fig. 3) and the Markovian model with
retarded transfer (Fig. 4):
$\begin{split}\alpha(1-\mu)\leq\alpha_{1}&+\alpha_{2}(1-\alpha_{1}-\nu_{1})(1-\mu_{1})+\ldots\\\
&+\alpha_{M}\prod_{i=1}^{M-1}(1-\alpha_{i}-\nu_{i})(1-\mu_{i})\leq\alpha{.}\end{split}$
(6)
$\square$
###### Proposition 3.
The probability of transfer outside TARN in the described model of sequential
choice for every day after trauma is between the probabilities for the
Markovian model with advanced transfer (Fig. 3) and the Markovian model with
retarded transfer (Fig. 4):
$\mu(1-\alpha-\nu)\leq\sum_{j=1}^{M}\mu_{j}(1-\alpha_{j}-\nu_{j})\prod_{i=1}^{j-1}(1-\alpha_{i}-\nu_{i})(1-\mu_{i})\leq\mu{.}$
(7)
$\square$
### 4.2 Transition probabilities and their evaluation
In the above models (Figs. 3 and 4), death and recovery of the transferred
patients have the same probabilities as for the patients of TARN hospitals.
These probabilities are defined by the state of the patient $s$ and by the
time after injury. Of course, in reality there is often a hope that the
transfer will improve the situation and the probability of death will decrease
for the same state of the patient. Nevertheless, in this paper we will neglect
the changes of probabilities after transfer (just because we have no
sufficient reason for such a change). Of course, these models could be
extended to include the changes of mortality for transferred patients, if
necessary.
Another question is the definition of $s$. Which attributes should be included
in the ‘state’ for the models (Figs. 3, 4)? To motivate this choice, we should
take into account two considerations:
1. 1.
The models will be used to analyse data with unknown outcomes. Trauma cases
with missed outcomes make up 10-12% of the dataset. Therefore, an error of 10%
in mortality for data with unknown outcomes will cause an error of $\sim$1% in
mortality for the whole dataset and it is possible to use relatively coarse
models (see below).
2. 2.
The description of the state $s$ should include attributes whose values are
known for a significant majority of cases. This is especially important
because for cases with unknown outcomes many of the attributes are often also
unknown (a more detailed analysis of data with missed attributes is presented
in the next section).
Formally, there are many possibilities for defining $s$. It could include the
initial state after trauma (characteristics of injury and coma status, for
example), age, gender, the current state ($t$ days after trauma), fragments of
history, etc. For our purposes, we select, identify and compare three coarse
models:
1. 1.
The coarsest model $s=\emptyset$.
2. 2.
The maximal severity model, $s=$the maximal severity score (an integer from 1
to 6).
3. 3.
The binned NISS model with seven bins: NISS=1-3, 4-8, 9, 10-16, 17-24, 25-35,
36+; $s$ is the bin number (7 values). The bins for $s=2,\ldots,7$ have
approximately equal depth whereas the bin with $s=1$ (NISS=1-3) is much
smaller.
We observe that the cases with maximal severity 1 (or NISS=1-3, which is the
same) are very special. First of all, the age distributions in this group for
the ‘Available W30D’ and the ‘OUT30’ subgroups are very different (Fig. 5). If
we do not take into account this difference then we overestimate mortality in
this group. The necessary refinement of the model with isolation of elderly
patients with low severity of trauma is presented in Section 5.
Figure 5: Age distributions for two groups of low severity cases (NISS bin
1-3). The age distribution for the low severity patients in TARN (‘Available
W30D’ AND NISS=1-3) for age binned in five bins (0-5.5; 5.5-15.5; 15.5-54.5;
54.5-74.5; $>$74.5) has clear maximum for elderly patients (age $>$74.5),
whereas the absolute majority of the the low severity patients which left TARN
without registered outcome (‘OUT30’ AND NISS=1-3) belong to the group with age
15.5-54.5.
Our approach may be combined with any stochastic model for early outcome
prediction (see, for example, [29, 47, 5]).
For the finite set of $s$ values, evaluation of all the coefficients
$\alpha(t,s)$, $\nu(t,s)$, and $\mu(t,s)$ is a particular case of a standard
statistical problem of proportion estimate for each given value of $s$; we use
the Wilson score interval (CI) [56]:
$\frac{1}{1+\frac{z^{2}}{n}}\left[\hat{p}+\frac{z^{2}}{2n}\pm
z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}+\frac{z^{2}}{4n^{2}}}\right],$ (8)
where $\hat{p}$ is the coefficient estimate, $z$ is the error percentile
($z=1.96$ for the 95% confidence interval), and $n$ is the number of degrees
of freedom (for a dataset without weights this is just the sample size).
For the coarsest model the fraction of patients transferred outside TARN is
11.65%. This is just the fraction of patients transferred (within 30 days
after injury) in Table 1. The 95% CI (8) for this fraction is 11.5–11.8%. For
the maximal severity (Table 2) and the binned NISS (Table 3) models the
fraction of patients transferred outside TARN depends on $s$ (bins) and the CI
in each bin is larger than for the total fraction in the coarsest models.
Nevertheless, the CIs for different bins in these models do not intersect (the
only exclusion is the CI for the smallest bin, maximal severity 6, in the
maximal severity model, Table 2). In particular, this means that the
probability of transfer outside TARN hospitals depends strongly on the trauma
severity.
Table 2: Sizes of bins and fractions of transfer out of TARN (within 30 days after injury) for the maximal severity models. Max severity | OUT30 | Total | Fraction of OUT30 | 95% CI
---|---|---|---|---
1 | 1,905 | 3,005 | 63.39% | 61.66–65.10%
2 | 3,094 | 35,109 | 8.81% | 8.52–9.11%
3 | 6,203 | 77,518 | 8.00% | 7.81–8.20%
4 | 4,535 | 29,603 | 15.32% | 14.91–15.73%
5 | 3,542 | 20,175 | 17.56% | 17.04–18.09%
6 | 10 | 149 | 6.71% | 3.88–11.72%
Table 3: Sizes of bins and fractions of patients transferred to a hospital or institution (or unknown destination) (within 30 days after injury) for the binned NISS models. NISS bin | OUT30 | Total | Fraction of OUT30 | 95% CI
---|---|---|---|---
1-3 | 1,905 | 3,005 | 63.39% | 61.66–65.10%
4-8 | 2,078 | 24,982 | 8.32% | 7.98–8.67%
9 | 2,159 | 36,722 | 5.88% | 5.64–6.12%
10-16 | 2,710 | 29,237 | 9.27% | 8.94–9.61%
17-24 | 2,882 | 25,074 | 11.49% | 11.11–11.89%
25-35 | 3,603 | 23,557 | 15.29% | 14.84–15.76%
36+ | 3,952 | 22,982 | 17.20% | 16.71–17.69%
For each value of $s$ and time after injury $t$ the following quantities are
found for the analysed dataset:
* 1.
$H(t,s)$ – the number of patients in state $s$ registered as alive in a TARN
hospital at any time during day $t$ after injury (in this number we include
the patients which have stayed at a TARN hospital during day $t$ after injury,
the patients who have died on this day in a TARN hospital, have been
discharged, or have been transferred outside TARN on this day);
* 2.
$\Delta D(t,s)$ – the number of patients in state $s$ who died in TARN
hospitals on day $t$ after injury;
* 3.
$\Delta R(t,s)$ – the number of patients in state $s$ who recovered
(discharged to one of the final recovery destinations) in TARN hospitals on
day $t$ after injury;
* 4.
$\Delta L(t,s)$ – the number of patients in state $s$ who transferred out of
TARN hospitals to other hospitals, institutions or unknown destinations on day
$t$ after injury.
Just for control, the following identity should hold: $H(t+1,s)=H(t,s)-\Delta
D(t,s)-\Delta R(t,s)-\Delta L(t,s)$ because state $s$ in our models does not
change in time.
For the model with advanced transfer from TARN hospitals the coefficients are
defined following the scheme presented in Fig. 3:
$\begin{split}&\mu(t,s)=\frac{\Delta L(t,s)}{H(t,s)};\;\nu(t,s)=\frac{\Delta
D(t,s)}{(1-\mu(t,s))H(t,s)};\;\\\ &\alpha(t,s)=\frac{\Delta
R(t,s)}{(1-\mu(t,s))H(t,s)}.\end{split}$ (9)
For the model with retarded transfer from TARN hospitals the coefficients are
defined following the scheme presented in Fig. 4:
$\begin{split}&\nu(t,s)=\frac{\Delta
D(t,s)}{H(t,s)};\;\alpha(t,s)=\frac{\Delta R(t,s)}{H(t,s)};\;\\\
&\mu(t,s)=\frac{\Delta L(t,s)}{(1-\alpha(t,s)-\nu(t,s))H(t,s)}.\end{split}$
(10)
### 4.3 Evaluation of FOD
Each model provides us with the corrected FOD. We use the basic assumption
that the probability of dying at time $t$ after injury depends on $s$ but is
the same inside and outside TARN. For each $t$ and $s$ we define the specific
cumulative FOD (scFOD$(t,s)$) as the fraction of patients with state $s$ who
died during the time interval $[1,t]$:
$\begin{split}\mbox{scFOD}(t,s)=\nu(1,s)&+\nu(2,s)(1-\alpha(1,s)-\nu(1,s))+\ldots\\\
&+\nu(t,s)\prod_{i=1}^{t-1}(1-\alpha(i,s)-\nu(i,s)).\end{split}$ (11)
The cumulative FOD at time $t$ (cFOD$(t)$) for the whole model (for all $s$
together) is
$\mbox{cFOD}(t)=\frac{\sum_{s}\mbox{scFOD}(t,s)H(1,s)}{H_{0}}{,}$ (12)
where $H_{0}=\sum_{s}H(1,s)$ is the total number of patients in our dataset
(in our case study, $H_{0}=165,559$).
The functions cFOD$(t)$ and scFOD$(t,s)$ for all $s$, grow monotonically with
$t$.
If we define the final outcome as survival or death within 30 days after
injury then the target value is FOD= cFOD(30).
Let us compare two following naïve approaches to the handling of missing
outcomes with the Markov models we have created.
* 1.
Available case analysis. Just delete all of the 19,289 cases with the outcome
‘Transferred OUT OF TARN within 30 days after injury’ from the dataset. In the
remaining cases all outcomes are known and the FOD is the ratio
$\frac{\mbox{Dead (W30D)}}{\mbox{Total}}$ in the reduced dataset.
* 2.
Consider all transferred patients as alive. In this case, the total number of
patients does not change and the FOD is the ratio $\frac{\mbox{Dead
(W30D)}}{\mbox{Total}}$, where the number ‘Dead (W30D)’ is the same but the
number ‘Total’ is calculated for the whole original dataset (Table 1).
###### Remark 1.
If we apply available case analysis then none of the numbers $\Delta D(t,s)$
and $\Delta R(t,s)$ change but the numbers $H(t,s)$ of the patients in TARN
will decrease for all $t$ and $s$ (or do not change if there is nothing to
delete). The corresponding mortality coefficients $\nu(t,s)$ they will be
larger then the coefficients (9), (10) for all the Markov models considered
before. This means that the MCAR (Missing Completely At Random) approach to
missed outcomes always overestimates mortality, while the second naïve
approach (‘Consider all transferred patients as alive’) always underestimates
mortality.
We have created six Markov models for mortality of transferred patients. They
differ by the state variable $s$ (the coarsest model without $s$, the maximal
severity model with six states and the binned NISS model with seven states)
and by the order of the ‘recovery/death’ and ‘transfer’ lotteries (Fig. 2). In
Table 4 we compare the mortality evaluated by these models, and by the two
naïve models. We can see that the difference between all of our Markov models
is not significant; we cannot reject the hypothesis that they coincide with
any one of them ($p$-value is between 0.20 and 0.56). Both of the naïve models
differ significantly from all of the six Markov models. The difference between
the naïve models is also significant. The interval of mortality predicted by
the Markov models is $(6.77\%,6.91\%)$. The average of the six Markovian
predictions is 6.84%. None of the Markov model predictions differ
significantly from this average. Both of the naïve predictions are
significantly different.
Table 4: FOD for different models; the $p$-value is the probability that the difference between FOD and 6.85% (FOD for the coarsest model with advanced transfer) appears by chance. Model | Alive | Dead | FOD | $p$-value
---|---|---|---|---
Available case study | 135,733 | 10,537 | 7.20% | $1.3\times 10^{-8}$
All transferred are alive | 155,022 | 10,537 | 6.36% | $5.0\times 10^{-15}$
Coarsest advanced | 154,217 | 11,342 | 6.85% | 1.00
Coarsest retarded | 154,350 | 11,209 | 6.77% | 0.20
Max severity, advanced | 154,120 | 11,439 | 6.91% | 0.34
Max severity, retarded | 154,266 | 11,293 | 6.82% | 0.41
NISS binned, advanced | 154,145 | 11,414 | 6.89% | 0.48
NISS binned, retarded | 154,292 | 11,267 | 6.81% | 0.57
### 4.4 Validation of the models on the excluded trauma cases: patients
transferred to TARN (‘IN30’)
For each type of model the coefficients $\mu$, $\alpha$ and $\nu$ are
evaluated using the dataset of 165,559 patients entering TARN in the first day
of injury (Fig 1, Main Group). Let us test the models with evaluated
coefficients we have described here on data we have not used before. These
data consist of the 16,693 cases who came to TARN hospitals more than one day
after injury, which we deleted from the original set before modelling. This is
a special and biased sample, ‘IN30’ (see Fig. 1). We now apply the models
developed and identified in the previous subsections to analyse this sample.
We expect that there should be some similarity between the groups of patients
transferred from TARN (‘OUT30’) and the patients transferred to TARN (‘IN30’)
(Fig. 1). We do not expect quantitative coincidence of the results for the
groups ‘OUT30’ and ‘IN30’ because there is no precise symmetry between the
patients moved to TARN and the patients moved from TARN. The hospitals in TARN
are those with a special interest in trauma - in particular the large major
trauma centers, so the transfers in (mainly for acute specialist care) will
not be the same as those transferred out (mainly for complex rehabilitation,
or special geriatric care, etc.).
Therefore, the estimated behavior of the mortality of the group transferred
from TARN can be qualitatively validated using the observed mortality in the
group who moved to TARN.
We consider survival during the first 30 days. Hence we have to use the
records which correspond to this period only. There are 15,437 such records
among the 16,693 in ‘IN30’.
In these estimates of the FOD we explicitly use the empirical fluxes into and
from TARN hospitals. For each $t,s$ we have the following quantities:
* 1.
$L_{\rm in}(t,s)$ – the number of patients in state $s$ which came to TARN on
day $t$ after injury;
* 2.
$L_{\rm out}(t,s)$ – the number of patients in state $s$ from ‘IN30’ which
were transferred from TARN on day $t$ after injury.
* 3.
$h_{\rm IN30}(t,s)$ – the number of patients in IN30 in state $s$ on day $t$
after injury.
* 4.
$D_{\rm IN30}(t,s)$ – the number of deaths in TARN of the patients from IN30
in state $s$ by day $t$ after injury (cumulative).
* 5.
$R_{\rm IN30}(t,s)$ – the number of patients in ‘IN30’ in state $s$ who
recovered by day $t$ after injury (cumulative).
We use the values $L_{\rm in}(t,s)$ and $L_{\rm out}(t,s)$ from the database,
evaluate $h_{\rm IN30}(t,s)$, $D_{\rm IN30}(t,s)$, and $R_{\rm IN30}(t,s)$ for
every model and then compare the resulting outcomes (evaluated numbers of
death in TARN of the patients from ‘IN30’ within 30 days of injury,
$\sum_{s}D_{\rm IN30}(30,s)$) to empirical data from TARN records.
For each model with advanced transfer the variables $h_{\rm IN30}(t,s)$,
$D_{\rm IN30}(t,s)$, and $R_{\rm IN30}(t,s)$ are evaluated by recurrence
formulas:
$\begin{split}h_{\rm IN30}(t+1,s)&=[h_{\rm IN30}(t,s)+L_{\rm in}(t+1,s)\\\
-&L_{\rm out}(t+1,s)][1-\alpha(t+1,s)-\nu(t+1,s)];\\\ R_{\rm
IN30}(t+1,s)&=R_{\rm IN30}(t,s)+\alpha(t+1,s)\\\ \times[&h_{\rm
IN30}(t,s)+L_{\rm in}(t+1,s)-L_{\rm out}(t+1,s)];\\\ D_{\rm
IN30}(t+1,s)&=D_{\rm IN30}(t,s)+\nu(t+1,s)\\\ \times[&h_{\rm IN30}(t,s)+L_{\rm
in}(t+1,s)-L_{\rm out}(t+1,s)]{,}\end{split}$ (13)
with initial condition
$h_{\rm IN30}(0,s)=R_{\rm IN30}(0,s)=D_{\rm IN30}(0,s)=0.$
For each model with retarded transfer the variables $h_{\rm IN30}(t,s)$,
$D_{\rm IN30}(t,s)$, and $R_{\rm IN30}(t,s)$ are evaluated by recurrence
formulas:
$\begin{split}h_{\rm IN30}(t+1,s)=&h_{\rm
IN30}(t,s)[1-\alpha(t+1,s)-\nu(t+1,s)]\\\ &+L_{\rm in}(t+1,s)-L_{\rm
out}(t+1,s);\\\ R_{\rm IN30}(t+1,s)=&R_{\rm IN30}(t,s)+\alpha(t+1,s)h_{\rm
IN30}(t,s);\\\ D_{\rm IN30}(t+1,s)=&D_{\rm IN30}(t,s)+\nu(t+1,s)h_{\rm
IN30}(t,s){,}\end{split}$ (14)
with initial condition
$h_{\rm IN30}(0,s)=R_{\rm IN30}(0,s)=D_{\rm IN30}(0,s)=0.$
For each model, the coefficients $\alpha(t,s)$ and $\nu(t,s)$ are evaluated
using the previously analysed dataset (without IN30) by formulas (9) and (10).
The results are presented in Table 5.
Table 5: Comparison of the models with the empirical data about patients from ‘IN30’. Model | Alive | Dead | Total | FOD | CI 95
---|---|---|---|---|---
Empirical data | 13,038.00 | 417.00 | 13,455.00 | 3.10% | 2.82–3.41%
Coarsest advanced | 12,834.55 | 620.45 | 13,455.00 | 4.61% | 4.27–4.98%
Coarsest retarded | 12,933.67 | 521.33 | 13,455.00 | 3.87% | 3.56–4.21%
Max severity, advanced | 12,824.90 | 630.10 | 13,455.00 | 4.68% | 4.34–5.05%
Max severity, retarded | 12,920.71 | 534.29 | 13,455.00 | 3.97% | 3.65–4.31%
NISS binned, advanced | 12,885.93 | 569.07 | 13,455.00 | 4.23% | 3.90–4.58%
NISS binned, retarded | 12,971.22 | 483.78 | 13,455.00 | 3.60% | 3.29–3.92%
We can see that all the models overestimate mortality in ‘IN30’. The available
case analysis demonstrates the worst performance (the relative error exceeds
100% of empirical mortality). Models with retarded transfer perform better in
this test than the models with advanced transfer. The NISS binned model with
retarded transfer is the best (the relative error in prediction of FOD is 16%
of the empirical data and, at least, the 95% confidence intervals for the
result of this model and for the empirical data intersect). There exist
further possibilities for improving the models presented but already the
relative error of $16\%$ for ‘IN30’ in the estimation for the total database
will give the input in the relative error in the FOD $\lesssim$1% (or absolute
error $\lesssim$0.07%). That is much better than the errors of the available
case evaluations or of the approach ‘all are alive’ to the evaluation of
mortality of transferred patients.
## 5 Model refinement
We use a coarse model based on the severity of trauma for the evaluation of
FOD in the group ‘OUT30’. The reason for selection of such a coarse model is
that a fraction of cases in this ‘OUT30’ cohort is relatively small with
respect to the‘Available W30D’ cases. As we can see from Table 3, this
fraction is relatively small in all cells except small severities (NISS bin
1-3). For refinement of the Markov model for this cell, we compare the age
structure of the ‘Available W30D’ and the ‘OUT30’ fractions of this severity
bin (Fig. 5). We see that the fraction of elderly patients with low severities
in TARN hospitals is high, whereas for patients transferred from TARN this
fraction is much lower. Mortality in the group of patients 74.5+ is much
higher than in the adult group, therefore the model overestimates mortality in
the low severity states. To refine the model let us use two cells for low
severity: ‘NISS 1-3 y’ (NISS bin 1-3 and age $<54.5$) and ‘NISS 1-3 o’ (NISS
bin 1-3 and age $>54.5$). This refined model gives a significantly different
FOD for NISS 1-3. In the cell ‘NISS 1-3 y’ the corrected FOD is 0.54% and in
the cell ‘NISS 1-3 o’ it is 4.08% (almost eight times greater). The corrected
overall FOD for NISS 1-3 is 1.42% versus 2.68% in the NISS retarded model
without the above refinement.
The effect of the refinement on the FOD for trauma cases is less because the
fraction of traumas with NISS severity 1-3 is relatively small (2.0%). For the
refined model with retarded transfer the FOD for transferred patients
decreases from 3.79% (retarded transfer NISS model) to 3.59% and the total
fraction of death is changed from 6.81% to 6.78% (compare to Table 4).
## 6 Weighting adjustment of death cases for further analysis
Single imputation of missed values does not reflect the uncertainty in data
properly. From the probabilistic point of view, a datapoint with missed values
should be considered as a conditional probability distribution of the form
$\mathbf{P}(\mbox{missed values }|\mbox{ known values}).$
Two approaches utilize this idea the multiple imputation and weighting
adjustment.
In the multiple imputation approach several replicas of the database are
created, which differ in the imputed values [40, 41, 23, 51]. The distribution
of this values should reflect the conditional means and conditional variances
of the imputing attributes. It is not completely clear, how many imputations
should be generated. Rubin claims that “typically as few as five multiple
imputations (or even three in some cases) is adequate under each model for
nonresponse” [41]. Nevertheless, more recently, Graham et al produced
practical recommendations for selection of number of imputations $m$ and
demonstrated that a reasonable choice is $m\geq 20$ and for some cases $m=100$
is not enough [23]. The multiple imputation algorithms are implemented in the
standard statistical software [38]. Sterne et al [51] discussed use and misuse
of imputation in epidemiological and clinical research and tried to produce a
standard for reporting of handling of missed data in medical research.
The weighting adjustment approach substitutes a datapoint with missed values
by a set of additional weights on the complete datapoints [33, 26, 34]. The
simplest version of this approach is the cell weighting adjustment. This
follows the assumption that complete datapoints within a cell represent the
incomplete datapoints within that cell. An incomplete datapoint within the
cell is substituted by the equidistribution on the complete datapoints there.
Of course, cell weighting can inflate the variances for large cells. In this
section, we use cell weighting adjustments for the handling of missed
outcomes. Cells are defined by state $s$ and the outcome.
We will use the database for evaluation of the death risk for trauma patients.
The ‘Main Group’ selected for further analysis includes the ‘OUT30’ subgroup
with 19,289 data cases transferred from TARN hospitals within 30 days after
injury (Fig 1). The targeted outcome (alive or dead within 30 days after
injury) is unknown for these patients. Data without outcome cannot be used for
risk evaluation and should be deleted. Let us call the result of deletion the
truncated database. It is demonstrated in the previous sections that the
simple removal of the cases with unknown outcome shifts the risk estimates;
the proportion of Dead and Alive outcomes in the truncated database differs
from reality and the risk is overestimated (the pessimistic evaluation). This
bias may be compensated by reweighting of the cases with known outcomes. There
are 146,270 such ‘Available W30D’ cases. In this subsection we estimate
weights $w(t,s)$ that should be assigned to the cases of death on day $t$
after injury with state $s$ to hold the probability of death for the truncated
database. For the estimation of the proper FOD that should be kept we use the
Markov model of mortality based on binned NISS with delayed transfer out of
TARN (after selection dead and recovered patient, see Fig. 4). This model
demonstrates the best verification results (Table 5) and is the most plausible
from the common sense point of view.
According to the model, the probability of the patient in state $s$ dying on
day $t$ after injury is evaluated as
$p_{d}(t,s)=\frac{\Delta D(t,s)+\Delta D_{L}(t,s)}{H_{0}(s)}{,}$
where $H_{0}(s)=H(1,s)$ is the initial number of patients in state $s$ on the
first day after injury. For the truncated data with weights this probability
is evaluated as the ratio of the sums with weights:
$p_{d}^{w}(t,s)=\frac{w(t,s)\Delta D(t,s)}{H_{0}^{w}(s)},$ (15)
where
${H_{0}^{w}(s)}=H(31,s)+R(30,s)+\sum_{t=1}^{30}w(t,s)\Delta D(t,s)$ (16)
and the superscript $w$ corresponds to the truncated dataset with weights. The
numbers $H(t,s)$, $R(t,s)$ and $\Delta D(t,s)$ are the same for the original
and truncated datasets.
The probability of dying within 30 days from injury is evaluated as the
proportion of deaths (we use the model to find $D_{L}(30,s)$)
$p_{d}(s)=\frac{D(30,s)+D_{L}(30,s)}{H_{0}(s)}{.}$
For the truncated database $p_{d}(s)$ is evaluated as the proportion of
weighted deaths:
$p^{w}_{d}(s)=\frac{\sum_{t=1}^{30}w(t,s)\Delta D(t,s)}{H_{0}^{w}(t,s)}.$
This should be the same number. Therefore, the weighted sum of deaths for the
truncated database is:
$\sum_{t=1}^{30}w(t,s)\Delta
D(t,s)=\frac{p_{d}(s)}{1-p_{d}(s)}(H(31,s)+R(30,s)){.}$
The last expression in the brackets is just the number of ‘Alive within 30
days’ outcomes. Immediately we get
$H_{0}^{w}(s)={\frac{1}{1-p_{d}(s)}(H(31,s)+R(30,s))}.$
The formula for the calculation of the weights of death cases in the truncated
database is
$w(t,s)=\frac{p_{d}(t,s)H_{0}^{w}(s)}{\Delta D(t,s)}.$ (17)
The weighting procedure changes the number of effective degrees of freedom can
affect the statistical power of the dataset but for the TARN dataset this
change is rather minor. For example, for the standard problem of the
evaluation of the confidence interval in the proportion estimate the number of
degrees of freedom $n_{w}$ in the weighted database with weights $w_{i}$ is
$n_{w}=\frac{\left(\sum_{i}w_{i}\right)^{2}}{\sum_{i}{w_{i}^{2}}}.$ (18)
For our dataset $n_{w}=143,574.85$ and the number of Available W30D records is
146,270 (Fig. 1). The difference of degrees of freedom for the non-weighted
and weighted datasets is less than 2%.
## 7 FOD and patterns of mortality
The models we have developed alow us to evaluate the FOD for various groups of
patients. The rich TARN data give us the chance of studying various special
groups and detailed stratifications of the trauma cases: by the severities of
various injuries in combined traumas, by the age of patients, and by time
(day) after trauma. Each example below is supplemented by a medical
commentary.
### 7.1 Example: FOD as function of age
The age distribution of trauma cases and the dependence of FOD on age are
shown in Fig. 6. Here we find surprisingly high accuracy of the piecewise
linear approximation of FOD for adult and elderly patients with a jump in the
slope at $\mbox{age}\approx 62$.
Figure 6: Age distribution of trauma cases in ‘Available W30D’ group and the
FOD (corrected) as a function of age. The piecewise linear segmentation of
FOD(age) has an obvious break point at $\mbox{age}\approx 62$.
The number of cases per year in the dataset drops down at age 65 because for
age$\geq 65$ some traumas are excluded from the database (see Fig. 1).
#### Medical commentary
The increase in mortality with age is well established. Previous versions of
the standard trauma outcome prediction system had two different models with an
age cutoff at 55 years. More recent models have age as a weighted continuous
variable with an interaction term between gender and age. There has been a
dramatic change in the trauma population over the last 10 years, with a rapid
increase in the number of older patients with major injury. Understanding the
effects of age on trauma care and adapting to a changing population will be a
key challenge for trauma systems in the developed world over the next 10
years.
### 7.2 Example: FOD of combined traumas of various severity
Evaluation of the severity of combined traumas is a classical problem. The
very popular solution is NISS – sum of squares of three maximal severities,
$s_{1}^{2}+s_{2}^{2}+s_{3}^{2}$ ($s_{1}\geq s_{2}\geq s_{3}$) (see, for
example [29, 53, 52]). The best severity score should give the best evaluation
of mortality. This is a basic and rather old idea for defining and comparing
trauma indices [44]. Of course, it is possible to use three (or more)
severities together as a multi-dimensional trauma severity index (‘severity
profile’ [43]) but the combination in one index may be beneficial from
different points of view.
The simplest method of combination is:
* 1.
Calculate FOD for every combination of severities for combined traumas for a
large database;
* 2.
Either use this FOD instead of the severity score
* 3.
Or find and use a convenient analytic approximation for this FOD (smoothed
FOD).
Of course, such evaluation of probabilities for several input attributes were
used by many authors and compared to other approaches [4, 5]. In this paper,
we use TARN database and evaluate FOD of combined traumas as a function of
three input attributes, three biggest severity scores $s_{1}\geq s_{2}\geq
s_{3}$ (like in NISS).
We use the dataset of 146,270 ‘Available W30D’ patients approached TARN during
the first day of injury and remained in TARN or were discharged to a final
destination within the first 30 days after injury (Fig. 1).
Using our models, we calculate estimates with weights which take into account
modeled mortality/survival of the patients transferred from TARN and other
patients with unknown outcomes. Results for the maximal severity $s_{1}=5$ are
presented in Table 6. The available case analysis gives qualitatively the same
results, hence, the effects we observe are not generated by the reweighting
procedure.
Table 6: FOD for the maximal severity $s_{1}=5$ and various $s_{2}$ and $s_{3}$ for data after reweighting. | $s_{3}$
---|---
$s_{2}$ | 0 | 1 | 2 | 3 | 4 | 5
0 | 0.3590 | | | | |
1 | 0.2324 | 0.2906 | | | |
2 | 0.1566 | 0.1496 | 0.0791 | | |
3 | 0.2466 | 0.2064 | 0.1315 | 0.1439 | |
4 | 0.2579 | 0.2881 | 0.1643 | 0.2105 | 0.3113 |
5 | 0.4073 | 0.5668 | 0.4067 | 0.3666 | 0.4140 | 0.5908
The results presented in Table 6 seem to be counterintuitive: FOD for combined
injuries with severities $s_{1}=5$ and $1\leq s_{2}\leq 4$ are less than FOD
for $s_{2}=s_{3}=0$ and the same maximal severity $s_{1}=5$. Similar non-
monotonic behavior is observed for other values of the maximal severities.
Elementary estimates demonstrate that the probability $p$ of obtaining these
(or larger) deviations to below from the FOD for single injuries ($s_{1}=5$,
$s_{2}=s_{3}=0$) for all cases with $1\leq s_{2}\leq 4$ simultaneously is less
than $10^{-10}$. The number of cases used for these estimates are given in
Table 7. If the second severity coincides with the maximal one,
$s_{2}=s_{1}=5$ then the FOD is larger than for single traumas.
Table 7: Number of cases for the maximal severity $s_{1}=5$ and various $s_{2}$ and $s_{3}$. | $s_{3}$
---|---
$s_{2}$ | 0 | 1 | 2 | 3 | 4 | 5
0 | 1,376 | | | | |
1 | 276 | 101 | | | |
2 | 302 | 163 | 332 | | |
3 | 577 | 243 | 645 | 1,580 | |
4 | 349 | 140 | 203 | 2,653 | 2,301 |
5 | 387 | 102 | 95 | 807 | 2,159 | 1,842
It may be convenient to have formulas for estimation of FOD. This smoothed FOD
(${\rm sFOD}_{s_{1}}$) is found for $s_{1}=2,\ldots,5$ as a linear combination
of $s_{2,3}$ and $s^{2}_{2,3}$ (19). For $s_{1}=1$ the simple formulas do not
have much sense and we have to use a refined model with the inclusion of age
(Sec. 5) The number of cases is not sufficient for good approximation for this
extended model. For $s_{1}=6$ the number of cases is not sufficient and we use
three bins for trauma severities marked by the values of the coarse-grained
variable $\hat{s}$: $0\leq s_{2}\leq 2$ ($\hat{s}_{2}=0$, 48 cases), $3\leq
s_{2}\leq 4$ ($\hat{s}_{2}=1$, 53 cases), and $5\leq s_{2}\leq 6$
($\hat{s}_{2}=2$, 38 cases). ${\rm sFOD}_{6}$ is presented as a quadratic
function of $\hat{s}_{2}$.
$\begin{split}{\rm sFOD}_{2}=&0.01910+0.02124s_{2}+0.00037s_{3}\\\
&-0.01054s_{2}^{2}-0.00084s_{3}^{2};\\\ {\rm
sFOD}_{3}=&0.02202+0.00256s_{2}-0.00238s_{3}\\\
&+0.00099s_{2}^{2}+0.00101s_{3}^{2};\\\ {\rm
sFOD}_{4}=&0.06571-0.02075s_{2}-0.03116s_{3}\\\
&+0.00706s_{2}^{2}+0.01086s_{3}^{2};\\\ {\rm
sFOD}_{5}=&0.35899-0.13335s_{2}-0.10879s_{3}\\\
&+0.02963s_{2}^{2}+0.02748s_{3}^{2};\\\ {\rm
sFOD}_{6}=&0.80297-0.08750\hat{s}_{2}+0.06102\hat{s}_{2}^{2}.\end{split}$ (19)
All the coefficients are estimated using weighted least squares method. The
weight of the severities combination $(s_{1},s_{2},s_{3})$ is defined as the
sum of weights of the corresponding trauma cases.
#### Medical commentary
The complete outcome dataset derived from this work allows all patients to be
included in the analysis of the effect of combined injuries. The counter-
intuitive results from this analysis (some combinations of injuries seem to
have better outcomes than a single injury of the same severity) provides a
fertile area for further work. It may be that the explanation is technical,
within the way that the continuum of human tissue destruction from trauma is
reduced to a simple 5 point scale. Each point on the scale is actually a band
that covers a range of tissue damage. There might also be a true physiological
explanation for the lower lethality of combined injuries, as each injury
absorbs some of the force of impact. The same concept is used in Formula 1,
where the cars are designed to break into pieces, with each piece absorbing
some of the impact. In humans there is a well known concept that the face can
act as a ‘crumple zone’ and mitigate effect of force on the brain. The effect
of injury combinations shown in Table 6 is a novel finding that requires
further analysis.
### 7.3 Example. Time after trauma, non-monotone and multimodal mortality
coefficients
In the early 1980s a hypothetical statement was published that the deaths from
trauma have a trimodal distribution with the following peaks: immediate, early
and late death [3, 35]. This concept was clearly articulated in a popular
review paper in Scientific American [55]. The motivation for this hypothesis
is simple: Trunkey [55] explains that the distribution of death is the sum of
three peaks: “The first peak (‘Immediate deaths’) corresponds to people who
die very soon after an injury; the deaths in this category are typically
caused by lacerations of the brain, the brain stem, the upper spinal cord, the
heart or one of the major blood vessels. The second peak (‘Early deaths’)
corresponds to people who die within the first few hours after an injury; most
of these deaths are attributable to major internal hemorrhages or to multiple
lesser injuries resulting in severe blood loss. The third peak (‘Late
deaths’) corresponds to people who die days or weeks after an injury; these
deaths are usually due to infection or multiple organ failure.”
Strictly speaking, the sum of three peaks does not have to be a trimodal
distribution. Many groups have published refutations of trimodality: they did
not find the trimodal distribution of death. In 1995, Sauaia et al reported
that the “greater proportion of late deaths due to brain injury and lack of
the classic trimodal distribution” [45]. Wyatt et al could not find this
trimodal distribution in data from the Lothian and Borders regions of Scotland
between 1 February 1992 and 31 January 1994 [57]. They hypothesised that this
may be (partly) due to improvements in care.
Recently, more data has become available and many such reports have been
published [12, 27, 6]. The suggestion that the improvement in care has led to
the destruction of the second thd third peaks has been advanced a number of
times [27]. In 2012, Clark et al performed an analysis of the distribution of
survival times after injury using interval censored survival models [10]. They
considered the trimodal hypothesis of Trunkey as an artifact and provide
arguments that the observed (in some works) second peak is a result of
differences in the definition of death.
Figure 7: Daily coefficient of mortality – evaluated probability of a patient
to die on day $t$ under condition that he/she survived during days $1\div
t-1$: a) for NISS=1-8, b) for all dataset. The coefficient is filtered by
moving 5-day average starting from the 3rd day. The mortality coefficients are
evaluated with the Markov models with retarded transfer. Data for age$<65$ and
age$\geq 65$ are represented separately.
K. Søreide et al analysed the time distribution from injury to death
stratified by cause of death. They demonstrated that the trimodal structure
may be, probably, extracted from data but its manifestation is model–dependent
(see Fig. 6 in [50]). There were several discussion papers published:
“Trimodal temporal distribution of fatal trauma – Fact or fiction?” [2, 28].
The trimodal hypothesis was tested on TARN data [31]. It was demonstrated that
“the majority of in hospital trauma deaths occur soon after admission without
further peaks in mortality”. We reproduce the same results, indeed. But TARN
database, the largest European trauma database, allows us to make a stratified
analysis of mortality and the preliminary results demonstrate the richness of
the possible patterns of death.
Let us test the famous Trunkey hypothesis. In Fig. 7 the daily mortality
coefficients are presented for low severities (a) (NISS severities 1-8, 27,987
cases in database, 508 death in TARN, 3,983 patients transferred from TARN
within 30 days after injury), and for the whole database (b). For the
prediction of death in the ‘OUT30’ group we used the model with retarded
transfer.
The non-monotonicity and peaks in the mortality for low severities of injury
are illustrated in Fig. 7. Further analysis of these patterns should involve
other attributes such as the age of the patient and the type and localization
of the injury.
#### Medical commentary
It has been widely accepted that the Trunkey trimodal distribution was a
theoretical concept designed to illustrate the different modes of dying
following injury. Previous analysis of trauma data has looked at all patients
and has not shown any mortality peaks, however this new analysis shows that
there are peaks (patterns) if subgroups are studied. The underlying clinical
or patient factors are not immediately obvious, but future analysis giving a
better understanding of patterns of death could act as a stimulus to look for
the clinical correlates of these patterns - with the potential to find
modifiable factors. The pattern of death in various subgroups as shown in
Figure 7 is a novel finding that requires further analysis.
## 8 Discussion
Handling of data with missed outcomes is one of the first data cleaning tasks.
For many healthcare datasets, the problem of lost patients and missed outcomes
(in 30 days, in six months or any other period of interest) is important.
There are two main approaches for solving this problem:
1. 1.
To find the lost patients in other national and international databases;
2. 2.
To recover the distribution of the missed outcomes and all their correlations
using statistical methods, data mining and stochastic modelling.
Without any doubt the first approach is preferable if it is available: it is
better to have complete information when it is possible. Nevertheless, there
may be various organizational, economical and informational restrictions. It
may be too costly to find the necessary information, or this information may
be unavailable or even does not exist in databases. If there are only small
number of lost cases (dozens or even hundreds) then they may be sought
individually. However if there are thousands of losses then we need either a
data integration system with links to appropriate databases like the whole NHS
and ONS data stores (with the assumption that the majority of the missed data
may be taken from these stores) or a system of models for the handling of
missed data, or both because we might not expect all missed data to be found
in other databases.
In the TARN dataset, which we analyse in this paper the outcome is unavailable
for 19,289 patients. The available case study paradigm cannot be applied to
deal with missed outcomes because they are not missed ‘completely at random’.
Non-stationary Markov models of missed outcomes allow us to correct the
fraction of death. Two naïve approaches give 7.20% (available case study) or
6.36% (if we assume that all unknown outcomes are ‘alive’). The corrected
value is 6.78% (refined model with retarded transfer). The difference between
the corrected and naïve models is significant, whereas the difference between
different Markov corrections is not significant despite the large dataset.
Non-stationary Markov models for unknown outcomes can utilize any scheme of
predictive models with using any set of available attributes. We demonstrate
the construction of such models using maximal severity model, binned NISS
model and binned NISS supplemented by the age structure at low severities. We
use weighting adjustment to compensate for the effect of unknown outcomes. The
large TARN dataset allows us to use this method without significant damage to
the statistical power.
Analysis of mortality for a combination of injuries gives an unexpected
result. If $s_{1}\geq s_{2}\geq s_{3}$ are the three maximal severities of
injury in a trauma case then the expected mortality (FOD) is not a monotone
function of $s_{3}$, $s_{3}$, under given $s_{1}$. For example, for
$s_{1}=4,5$ expected FOD first decreases when $s_{2,3}$ grow from 0 to 1-2 and
then increases when $s_{2}$ approaches $s_{1}$.
Following the seminal Trunkey paper [55], multimodality of the mortality
curves is a widely discussed problem . For the complete TARN dataset the
coefficient of mortality monotonically decreases in time but stratified
analysis of the mortality gives a different result: for lower severities FOD
is a non-monotonic function of the time after injury and may have maxima at
the second and third weeks after injury. Perhaps, this effect may be
(partially) related to geriatric traumas.
We found that the age distribution of trauma cases is strongly multimodal
(Fig. 6). This is important for healthcare planning.
The next step should be the handling of missed values of input attributes in
the TARN database Firstly, we should follow the “Guidelines for reporting any
analysis potentially affected by missing data” [51], report the number of
missing values for each variable of interest, and try to “clarify whether
there are important differences between individuals with complete and
incomplete data”. Already preliminary analysis of the patterns in the
distribution of the missed input data in the TARN dataset demonstrates that
the gaps in data are highly correlated and need further careful analysis.
Secondly, we have to test and compare various methods of handling missing
input attributs in the TARN database.
It is not necessary to analyse all attributes in the database for mortality
prediction and risk evaluation. It is demonstrated that there may exist an
optimal set of input attributes for mortality prediction in emergency medicine
and additional variables may even reduce the value of predictors [20].
Therefore, before the analysis of imputation efficiency, it is necessary to
select the set of most relevant variables of interest.
The models developed in this case study can be generalized in several
directions. Firstly, for trauma datasets, different attributes could be
included in the ‘state’ $s$ for the non-stationary Markov models (Figs. 3, 4).
We did not explore all such possibilities but have studied just simple models
of the maximal severity and binned NISS. An example of model refinement with
inclusion of age in the state variable $s$ is presented in Section 5.
Secondly, the ‘two stage lottery’ non-stationary Markov model could be used as
a general solution applicable to any health dataset where ‘TRANSFER IN’ or
‘TRANSFER OUT’ is a feature. Transfer between hospitals is common in
healthcare, therefore, we expect that models of this type will be useful for
all large healthcare data repositories.
## 9 Summary
1. 1.
The Trauma Audit and Research Network (TARN) have collected the largest
European trauma database. We have analysed 192,623 cases from the TARN
database. We excluded from the analysis 16,693 patients (8.67%), who arrived
into TARN hospitals later than 24 hours after injury. The other 146,270
patients (75.94%) approached TARN during the first day of injury and remained
in TARN or discharged to a final destination within 30 days of injury. 19,289
patients (13.19%) from this group transferred from TARN to another hospital or
institution (or unknown destination) within 30 days of injury. For this
subgroup the outcome is unknown.
2. 2.
Analysis of the missed outcomes demonstrated that they cannot be considered as
misses ‘completely at random’. Therefore, the analysis of available cases is
not applicable for the TARN database. Special efforts are needed to handle
data with missed outcomes.
3. 3.
We have developed a system of non-stationary Markov models for the handling of
missed outcomes and validated these models on the data arising from patients
who moved to TARN (and excluded from the model fitting). We have analysed
mortality in the TARN database using the Markov models which we have developed
and also validated.
4. 4.
The results of analysis were used for weighting adjustment in the available
cases database (reweighting of the death cases). The database with adjusted
weights can be used for further data mining tasks and will keep the proper
fraction of deaths.
5. 5.
The age distribution of trauma cases is essentially multimodal, which is
important for healthcare planning.
6. 6.
Our analysis of the mortality coefficient in the TARN database demonstrates
that (i) for complex traumas the fraction of death is not a monotone function
of all severities of injuries and (ii) for lower severities the fraction of
death is not a monotonically decreasing function of time after injury and may
have intermediate peaks in the second and third weeks after injury.
7. 7.
The approach developed here can be applied to various healthcare datasets
which have the problem of lost patients, inter–hospitals transfer and missing
outcomes.
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Evgeny Mirkes (Ph.D., Sc.D.) is a Research Fellow at the University of
Leicester. He worked for Russian Academy of Sciences, Siberian Branch, and
Siberian Federal University (Krasnoyarsk, Russia). His main research interests
are biomathematics, data mining and software engineering, neural networks and
artificial intelligence. He led and supervised many medium-sized projects in
data analysis and development of decision-support systems for computational
diagnosis and treatment planning.
Timothy J. Coats (FRCS (Eng), MD, FCEM) is a Professor of Emergency Medicine
at the University of Leicester. Chair FAEM Research Committee 2000-2009, Chair
Trauma Audit and Research Network (TARN), Chair NIHR Injuries and Emergencies
National Specialist Group. Research Interests: Diagnostics and monitoring in
Emergency Care, Coagulation following injury, Predictive modeling of outcome
following injury.
Jeremy Levesley (Ph.D, FIMA) is a Professor in the Department of Mathematics
at the University of Leicester. His research area is kernel based
approximation methods in high dimensions, in Euclidean space and on manifolds.
He is interested in developing research at the interface of mathematics and
medicine, and sees interpretation of medical data sets as a key future
challenge for mathematics.
Alexander N. Gorban (Ph.D., Sc.D., Professor) holds a Personal Chair in
applied mathematics at the University of Leicester since 2004. He worked for
Russian Academy of Sciences, Siberian Branch (Krasnoyarsk, Russia), and ETH
Zürich (Switzerland), was a visiting Professor and Research Scholar at Clay
Mathematics Institute (Cambridge, MA), IHES (Bures-sur-Yvette, Île de France),
Courant Institute of Mathematical Sciences (New York), and Isaac Newton
Institute for Mathematical Sciences (Cambridge, UK). His main research
interests are dynamics of systems of physical, chemical and biological
kinetics; biomathematics; data mining and model reduction problems.
|
# Collisional effects on the electrostatic shock dynamics in thin-foil targets
driven by an ultraintense short pulse laser
A Sundström1, L Gremillet2,3, E Siminos4 and I Pusztai1 1 Department of
Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden 2 CEA,
DAM, DIF, F-91297 Arpajon, France 3 Université Paris-Saclay, CEA, LMCE,
F-91680 Bruyères-le-Châtel, France 4 Department of Physics, Gothenburg
University, SE-412 96 Göteborg, Sweden<EMAIL_ADDRESS>
###### Abstract
We numerically investigate the impact of Coulomb collisions on the ion
dynamics in high-$Z$, solid density caesium hydride and copper targets,
irradiated by high-intensity ($I\approx 2{-}5\times
10^{20}\,\mathrm{Wcm^{-2}}$), ultrashort (${\sim}10\,\mathrm{fs}$), circularly
polarized laser pulses, using particle-in-cell simulations. Collisions
significantly enhance electron heating, thereby strongly increasing the speed
of a shock wave launched in the laser-plasma interaction. In the caesium
hydride target, collisions between the two ion species heat the protons to
${\sim}100{-}1000\,\mathrm{eV}$ temperatures. However, in contrast to previous
work (A.E. Turrell et al., 2015 _Nat. Commun._ 6 8905), this process happens
in the upstream only, due to nearly total proton reflection. This difference
is ascribed to distinct models used to treat collisions in dense/cold plasmas.
In the case of a copper target, ion reflection can start as a self-amplifying
process, bootstrapping itself. Afterwards, collisions between the reflected
and upstream ions heat these two populations significantly. When increasing
the pulse duration to $60\,\mathrm{fs}$, the shock front more clearly
decouples from the laser piston, and so can be studied without direct
interference from the laser. The shock wave formed at early times exhibits
properties typical of both hydrodynamic and electrostatic shocks, including
ion reflection. At late times, the shock is seen to evolve into a hydrodynamic
blast wave.
Originally published in Plasma Phys. Control. Fusion 62 085015, © The Authors,
2020
Original Content from this work may be used under the terms of the Creative
Commons Attribution 4.0 licence. Any further distribution of this work must
maintain attribution to the author(s) and the title of the work, journal
citation and DOI.
doi:10.1088/1361-6587/ab9a62
## 1 Introduction
The use of lasers to accelerate ions is a field of intense research [1], with
many demonstrated or envisioned applications, such as imaging of
electromagnetic fields in plasmas [2, 3], creation of warm dense matter [4, 5,
6], production of intense neutron sources [7], material testing [8, 9],
laboratory astrophysics [10], and ion-beam therapy [11, 12]. Among the few
laser-based ion acceleration mechanisms considered so far, including the
extensively studied, and particularly robust, target normal sheath
acceleration (TNSA), collisionless shock acceleration (CSA) is of particular
interest due to its potential to produce a relatively narrowly peaked ion
energy spectrum [13, 14, 15, 16, 17, 18]. Collisionless shocks also play a
role in particle energization in astrophysical plasmas [19, 20].
As the shock front passes by, the plasma is rapidly compressed and directional
kinetic energy is converted into thermal energy. This can take place either
through collisional processes, such as in hydrodynamic shocks – relevant in,
e.g., inertial fusion plasmas [21, 22] and relativistic laser-plasma
experiments [23] – or collisionless mechanisms, involving longitudinal
electrostatic fields generated by space charge effects from shock compression
[14]. Collisionless shocks can also hinge upon self-generated magnetic fields,
such as those resulting from the Weibel instability [24, 25], yet such shocks,
of turbulent character, develop at Mach numbers much larger than those of the
laminar electrostatic shocks that we shall address here [26]. In relativistic
laser-plasma interactions, electrostatic shocks can arise either from the
forward push exerted by the laser’s ponderomotive force (or “laser piston”)
[14] in the radiation pressure acceleration (RPA) regime, or from electron
pressure gradients in nonuniform plasmas [17]. While “collisionless shocks”,
as the name suggests, are sustained through collective collisionless plasma
processes, Coulomb collisions may play a role in their dynamics. Indeed, a
finite collisionality, while affecting the shock, does not necessarily disrupt
it [27].
Although the effect of collisions is often deemed negligible in high-intensity
laser-plasma interactions, due to the high particle energies at play, it can
become important when using solid or near-solid density targets, especially if
they contain elements of high atomic numbers. In this paper, we consider two
scenarios where collisions play an important role: one has basic science
interest while the other is relevant for high energy density applications. We
also present cases with parameters in between, to clarify how changes in laser
and target parameters affect the ion dynamics, and in particular the
properties of the resulting electrostatic shocks. In all cases, we will
consider a circularly polarized femtosecond ($10{-}60\,\mathrm{fs}$) laser
pulse.
The first case we consider is motivated by the work by Turrell, Sherlock &
Rose [28] (hereafter referred to as TSR), where it was reported that inter-
species collisions in a caesium hydride (CsH) target induce ultrafast
collisional ion heating, and essentially affect the shock dynamics. We find
significantly different results compared to what is reported by TSR, even
though we study essentially the same physical setup. Importantly, we do not
observe the occurrence of ultrafast proton heating downstream of the shock, as
most of the protons are reflected, and as such, there is no appreciable inter-
species friction in the downstream. As we will discuss, this discrepancy is
likely due to a different behaviour, at the high densities considered, of the
different collision algorithms employed by TSR and us.
The other case we address was first considered in a recent study of ours [29]
investigating ionization and collisional electron heating effects in solid
copper targets, relevant for warm-dense-matter generation. Here, we focus on
the ion dynamics and examine the impact on the generated shock of the
increased electron density in copper compared to CsH. We also assess the
sensitivity of the ion dynamics to the laser parameters and target thickness.
When using circular laser polarization, collisions dominate the electron
heating, which, in turn, results in the formation of a stronger electrostatic
shock compared to a purely collisionless simulation. In the scenarios with
copper, the evolution of the shock is studied, from a hybrid
hydrodynamical–electrostatic shock, through a gradual dissipation of its
energy, to the transition to a hydrodynamical blast wave. In particular, the
onset of shock ion-reflection is found to be self-amplifying. Collisional
friction between the upstream and reflected ions heats the upstream ion
population, which enhances the fraction of reflected ions.
## 2 Simulation study
In this paper, we investigate two different target materials, caesium hydride
(CsH) and pure copper (Cu), both at their respective solid densities. We
perform one-dimensional (1D) particle-in-cell (PIC) simulations with the
Smilei PIC code [30] (version 4.1), which has a collision module that has been
benchmarked [31] in the high-density/low-temperature regimes relevant for this
paper. In all cases, we use a circularly polarized (CP),
$\lambda=800\,\mathrm{nm}$ wavelength laser with a Gaussian temporal profile.
The simulation box consists of 51200 cells over a length of
$20\,\mathrm{\textrm{\textmu}{m}}$ (resolution $\Delta{x}=0.39\,\mathrm{nm}$),
and a $4^{\rm th}$ order interpolation shape function is employed. The use of
a high-order shape function ensures good energy conservation despite the Debye
length in our collisionless simulation being somewhat lower than the mesh
size. The electrons are initialized at a temperature of
$T_{\mathrm{e},0}=1{-}10\,\mathrm{eV}$ and the ions at a temperature of
$0.1{-}1.0\,\mathrm{eV}$.
Both target materials contain a highly charged, $Z^{*}$, ion species, such
that the effect of collisions is significant. This high collisionality turns
out to be of crucial importance for the electron heating. Since CP is used,
the target electrons are energized through inverse Bremsstrahlung rather than
from the strongly inhibited $j{\times}B$ [32] or vacuum heating [33, 34]
mechanisms. In our recent work [29], we showed that collisional electron
heating produces well-thermalized electron populations with temperatures in
the ${\sim}1{-}10\,\mathrm{keV}$-range.
The use of the CsH target was inspired by the work by TSR [28]. As a target
material, CsH could be of interest for laser acceleration of protons since it
contains hydrogen volumetrically, like a plastic target. An advantage of this
material over plastic, though, is the much higher ionization degree ($Z^{*}$)
that can be reached, hence enhancing collisional effects. Although practically
challenging, due to the high chemical reactivity of CsH and difficulties in
the target fabrication, it would, in principle, be possible to use this
material in an experiment.
The CsH target is composed of an equal number mixture of protons and caesium
ions. The charge state of the Cs ions is set to a fixed value of $Z^{*}=27$,
corresponding to full ionization of the three outermost shells. The resulting
quasi-neutral electron density is ${n_{\mathrm{e},0}=250\,n_{\mathrm{c}}}$,
where $n_{\mathrm{c}}=\epsilon_{0}m_{\mathrm{e}}\omega^{2}/e^{2}\approx
1.7\times 10^{21}\,\mathrm{cm^{-3}}$ is the critical density ($\epsilon_{0}$
is the vacuum permittivity, $m_{\mathrm{e}}$ is the electron mass, $\omega$ is
the laser frequency and $e$ is the elementary charge), corresponding to a
collisionless skin depth of $l_{\mathrm{s}}=8.0\,\mathrm{nm}$ which is well
resolved. The target thickness is $300\,\mathrm{nm}$, as in the simulations of
TSR.
Copper, on the other hand, lacks the embedded protons but is, from a practical
standpoint, much more readily available as a target material. Copper is also
relatively highly charged, and hence presents a collisionality comparable to
CsH. The lack of embedded protons111The copper is also modelled without any
proton contamination layer on the surfaces. While such a contamination layer
would affect the TNSA process, and somewhat the laser absorption, it is not
expected to have a significant impact on the shock dynamics, that is the focus
of this paper. makes copper less suitable for volumetric proton acceleration,
but its high collisionality could be beneficial for other applications, such
as warm-dense-matter generation [29]. In the simulations, the copper ions are
initialized with three fully ionized atomic shells ($Z^{*}=27$), and at solid
density (corresponding to $n_{\mathrm{e},0}=1307\,n_{\mathrm{c}}$ and
$l_{\mathrm{s}}=3.5\,\mathrm{nm}$). This choice is informed by simulation
results for a copper target including field and collisional ionization
processes, analyzed in Ref. [29], showing that the average $Z^{*}$ rapidly
reaches this value, then it stagnates, due to a significant jump in ionization
energy beyond the three atomic shells. We found that retaining the ionization
dynamics has no significant impact on the ion dynamics.
With the copper targets, two different target thicknesses and two different
laser parameters were considered. The thinner target is $300\,\mathrm{nm}$
thick, as in the CsH simulations, which has the advantage of quicker heating
and homogenization compared to a thicker target. The thicker
($2.5\,\mathrm{\textrm{\textmu}{m}}$) target, on the other hand, can be more
suitable for warm-dense-matter applications: A high energy density will be
maintained over a longer time since hydrodynamic expansion takes longer to
reach the interior of a thicker target. We note that at the high densities and
ionization degrees considered here, the useful lifetime of the target can also
be affected by radiative losses, dominantly through Bremsstrahlung at the
temperatures of interest. We find, however, that for our parameters, the
radiative cooling time is typically of several picoseconds, so that
Bremsstrahlung losses should not greatly impact the plasma dynamics during the
integration time (${\leq}1\,\mathrm{ps}$) of our simulations. For the same
reason, internal radiative energy transport was also not modelled in the
simulation.
We considered two different sets of laser parameters: an amplitude of
$a_{0}=15$ ($I\approx 5\times 10^{20}\,\mathrm{Wcm^{-2}}$) and full-width-at-
half-maximum (FWHM) duration of $10\,\mathrm{fs}$, as well as $a_{0}=10$
($I\approx 2\times 10^{20}\,\mathrm{Wcm^{-2}}$) and a FWHM duration of
$60\,\mathrm{fs}$. The former is used with both the CsH and Cu thin targets,
and the latter is used for both the thin and thick Cu targets. The use of
thicker targets goes along with increased integration times, allowing a larger
number of fast particles to reach the domain boundaries. To keep them inside
the domain, the thicker target is initialized with its front at
$x=7.5\,\mathrm{\textrm{\textmu}{m}}$ compared to the other targets located at
$1\,\mathrm{\textrm{\textmu}{m}}$.
For an accurate modelling of Coulomb collisions, employing the relativistic
PIC algorithm of [31] (to be further discussed in Sec. 3.2), a relatively high
number of particles per cell is needed. In the thinner target, 500 macro-
particles per species per cell was used, while in the thicker target, the
particle number was reduced somewhat to 400 macro-particles per species per
cell. Resolution tests, with halved particle number or halved spatial
resolution (with same total number of particles), for the Cu thin target
simulation show that the simulations are numerically converged.
## 3 Ion dynamics in the CsH target
Motivated by the previous work by TSR, we performed a similar set of
simulations in CsH. However, despite virtually identical setups, our results
differ significantly from the ones by TSR.
### 3.1 Comparison of collisional and collisionless results
Figure 1: Electron distributions at (left) and after the laser peak intensity
(middle and right), with (top) and without (bottom) collisions, using CP.
Green curve: electron density. Please note the different momentum scales for
the collisional and collisionless simulations.
The primary effect of the strong target collisionality is to significantly
enhance electron heating through inverse Bremsstrahlung [29]. As an
illustration of the collisional electron heating, Fig. 1 shows the electron
phase space of the collisional (top row) and collisionless (bottom row) CsH
simulations at three successive times: during peak laser intensity at
$t=21\,\mathrm{fs}$, right after the laser pulse has ended at
$t=45\,\mathrm{fs}$, and even later at $t=70\,\mathrm{fs}$. In figures
hereafter, the phase space distribution functions $f$ are normalized to the
maximum value of each respective _initial_ Maxwellian distribution, $f_{\rm
max}$.
The electrons in the target front layer are energized in the transverse
($y$-$z$) plane by the laser electric field. Then collisions scatter their
momentum into the longitudinal direction, as seen through the large spread in
$p_{x}$ near the plasma front in the $t=21\,\mathrm{fs}$ frame of the
collisional distribution. Collisions then entail a fast thermalization of the
electrons to a Maxwellian distribution, yielding a bulk temperature of
$T_{\mathrm{e}}\approx 10\,\mathrm{keV}$ that corresponds to an ion-acoustic
speed of
$c_{\mathrm{s}}\approx(Z^{*}_{{\mathrm{Cs}}}T_{\mathrm{e}}/m_{{\mathrm{Cs}}})^{1/2}\approx
1.5\times 10^{-3}c$, where $c$ is the speed of light in vacuum.
The electron density is also indicated in Fig. 1 (green solid curve, right
axis). Compared to the collisionless case, the collisional simulation shows
smoother spatial structures likely due to a combination of higher temperature,
collisional dissipation and dispersion of non-linear waves. The Debye length
is $\lambda_{\mathrm{D}}\approx 1\,\mathrm{nm}$ and
$\lambda_{\mathrm{D}}\approx 0.1\,\mathrm{nm}$ in the collisional and
collisionless cases, respectively. The collisional electron density profile
also shows signs of an electrostatic shock wave: a density jump moving away
from the target front is visible in the $t=45\,\mathrm{fs}$ and
$70\,\mathrm{fs}$ panels. In the collisionless case, the density profile
exhibits two peaks in both time frames. The rightmost density jump is due to
the leading edge of the radiation-pressure-accelerated Cs ions (Fig. 2), while
the leftmost density peak corresponds to an electrostatic shock, which, due to
the low electron temperature, is too slow for its propagation to be noticeable
over the displayed time and length scales.
Figure 2: Proton (top frame) and caesium ion (bottom frame) distributions in
the $300\,\mathrm{nm}$ CsH target at peak laser intensity
($t=21\,\mathrm{fs}$) and after the pulse has passed ($t=45\,\mathrm{fs}$ and
$70\,\mathrm{fs}$), with (upper panels) and without (lower panels) collisions,
using CP. The longitudinal electric field is also plotted (turquoise solid
line, right axes). Note the different electric field scales between the
collisional and collisionless panels.
In Fig. 2, the evolution of the ion distributions in the collisional and
collisionless CsH targets are shown. The top frame shows the proton, the lower
one the Cs ion phase spaces, with the upper (lower) rows in both frames
corresponding to the collisional (collisionless) simulations, at times
$t=21\,\mathrm{fs}$, $t=45\,\mathrm{fs}$ and $t=70\,\mathrm{fs}$. At
$t=21\,\mathrm{fs}$, the difference between the collisional and collisionless
simulations is quite small; in both cases, the protons and Cs ions are pushed
by the laser piston. However, due to the lower charge-to-mass ratio of the Cs
ions compared to the protons, the Cs ions react more slowly to the radiation
pressure (RP) induced electrostatic field (at $x\approx
1\,\mathrm{\textrm{\textmu}{m}}$) than the protons, as seen by the almost four
times higher velocity reached by the protons ($p_{x}/mc\approx 0.02$) at
$t=21\,\mathrm{fs}$. Owing to the short pulse duration ($10\,\rm fs$), the Cs
ions do not have enough time to react to RP before the pulse ends.
Also shown in Fig. 2 is the longitudinal electric field, $E_{x}$ (turquoise
curve), normalized to $m_{\mathrm{e}}c\omega/e\approx 4.013\times
10^{12}\,\mathrm{V/m}$. The charge separation during the RPA phase creates a
strong longitudinal electric field, visible as a positive spike in $E_{x}$
close to $x=1\,\mathrm{\textrm{\textmu}{m}}$ in the $t=21\,\mathrm{fs}$
panels. Note that the peaks of the RP field are cut off in the display. The
collisionless RP field reaches a normalized amplitude of
$eE_{x}/(m_{\mathrm{e}}c\omega)=8.6$, while the field in the collisional
simulation reaches only $5.6$. However, the RP field in the collisional
simulation has a wider spatial extent. When the electric field is integrated,
the potential drop across the RP field is $e\phi\approx 220\,\mathrm{keV}$ and
$280\,\mathrm{keV}$ in the collisionless and collisional cases, respectively.
Thus, collisions do not affect the RPA process significantly, as apparent from
the comparison of the collisional and collisionless panels at
$t=21\,\mathrm{fs}$ in Fig. 2.
With collisions, the electrostatic structure caused by RPA transforms into an
electrostatic shock, as evidenced by the single strong oscillation of $E_{x}$
and modulations in the downstream ion distribution in the $t=45\,\mathrm{fs}$
and $70\,\mathrm{fs}$ frames of Fig. 2. A close inspection of the
collisionless simulation reveals the same behaviour (although barely visible
in Fig. 2), indicating that an electrostatic shock has also formed there.
However, due to the high electron temperature from collisional heating, the
shock is much stronger and faster in the collisional case. In absolute units,
the average shock velocity between $t=45\,\mathrm{fs}$ and $70\,\mathrm{fs}$
was $v_{\mathrm{sh}}/c\approx 4.3\times 10^{-3}$ and $v_{\mathrm{sh}}/c\approx
0.9\times 10^{-3}$ in the collisional and collisionless simulations,
respectively. Yet, the higher electron temperature in the collisional target
($T_{\mathrm{e}}\approx 10\,\mathrm{keV}$ vs. $T_{\mathrm{e}}\approx
0.2\,\mathrm{keV}$) leads to a lower Mach number ($\mathcal{M}\approx 2.9$ vs.
$\mathcal{M}\approx 4$). The low shock speed in the collisionless simulation
implies that the shock-reflected ions have a significantly lower energy
compared to those originating from the initial burst of the RPA. In both the
collisional and collisionless cases, given its limited energy reservoir
provided by the ultrashort ($10\,\mathrm{fs}$) laser pulse, the shock wave
steadily loses its energy, as seen by the declining field amplitude and the
sloped reflected ion structure in the proton and Cs phase spaces (i.e. the
shocks are losing speed).
Another consequence of the efficient inverse Bremsstrahlung electron heating
is that the collisional simulation displays TNSA at the target rear boundary,
whereas it is virtually nonexistent in the collisionless simulation, as
evident at $t=70\,\mathrm{fs}$ in Fig. 2. Due to the use of CP, the electrons
are weakly energized in the collisionless case, hence quenching TNSA. In the
collisional simulation, the TNSA protons attain energies slightly lower than
the RPA protons at the final simulation time.
In the collisional case, we also see that the reflected and upstream proton
and Cs ion populations are being significantly heated, in contrast to their
collisionless counterparts. By fitting Maxwellians to the proton distribution
in the range $x=1.15{-}1.18\,\mathrm{\textrm{\textmu}{m}}$ (close to, but
still beyond direct influence from the shock front) at time
$t=70\,\mathrm{fs}$, the upstream proton population is found to have already
been heated to $T_{\mathrm{p}}^{\rm(u)}=120\,\mathrm{eV}$, while the reflected
protons are at a temperature of $T_{\mathrm{p}}^{\rm(r)}=750\,\mathrm{eV}$. We
recall that the initial ion temperature was $0.1\,\mathrm{eV}$. Simulations in
which various types of collisions (e.g., proton–Cs or ion–electron) have been
selectively switched off (not shown here), reveal that the heating of the
reflected ions proceeds from their friction with the background Cs ions, while
the upstream ions are mainly collisionally heated by the fast electrons.
Figure 3: Proton (top panel) and Cs ion (bottom panel) distributions in the
shock frame of reference at $t=70\,\mathrm{fs}$, together with the shock
electrostatic potential, $e\phi/T_{\mathrm{e}}$ (blue solid line, right axes),
using $T_{\mathrm{e}}=10\,\mathrm{keV}$. Also shown are contours of constant
energy, $\mathcal{E}=mv^{2}/2+eZ(\phi-\phi_{\max})$ (black, dashed or dotted
lines). The black dashed line corresponds to $\mathcal{E}=0$ at the potential
peak, $\phi_{\max}$.
To get a more detailed picture of the vicinity of the electrostatic shock
front, close-ups of the proton (top) and Cs (bottom) distributions, at
$t=70\,\mathrm{fs}$, are displayed in Fig. 3. The distributions have been
shifted to the shock rest frame (at velocity $v_{\mathrm{sh}}/c\approx
3.1\times 10^{-3}$), relative to the position of the potential maximum,
$x_{\mathrm{sh}}$; the velocities are normalized to the ion-acoustic sound
speed, $c_{\mathrm{s}}$. The electrostatic potential,
$\phi(x)=-\int_{x_{0}}^{x}E_{x}(x^{\prime})\,\mathrm{d}{x^{\prime}}$, where
$x_{0}$ is such that $\phi$ averages to zero in the range
$8\leq(x-x_{\mathrm{sh}})/\lambda_{\mathrm{D}}\leq 10$222In an idealized
electrostatic shock, $\phi\to 0$ as $x\to\infty$. However, in practice, the
electrostatic potential presents spatial variation even well upstream of the
shock front, from sources other than the shock, which motivates this averaging
procedure. The choice of the $x$-range to average over is somewhat arbitrary,
but it is chosen reasonably close to the shock front, while sufficiently
outside the shock width., is also plotted (blue line), along with
corresponding constant energy contours (black dashed or dotted lines). The
black dashed line represents the constant energy contour which has zero
(shock-frame) kinetic energy at the peak of $\phi$. This line is an
approximate boundary between the reflected and passing ions; in a steady
state, this would be a separatrix. The top frame shows that almost all protons
are located within the reflected region of phase space. Meanwhile, only around
$5{-}10\%$ of the Cs ions are reflected, and accordingly, the upstream Cs
distribution mostly lies below the passing–reflected boundary. The difference
in ion reflection between the two ion species is due to their different
charge-to-mass ratios [35].
The electrostatic potential is seen to oscillate downstream of the shock (left
side in Fig. 3), which creates regions of ion trapping. In a perfectly steady-
state and collisionless electrostatic shock, these regions would be empty, as
there would be no means for the ions to cross the separatrix. However, due to
the slowly decreasing amplitude and speed of the shock, the trapping regions
experience a steady influx of Cs ions. These adiabatic effects are likely more
important here than collisional scattering [27]. While the Cs ions mainly
enter the trapping regions from the leftmost potential hump in Fig. 3, almost
no protons pass the shock front and hence only few protons ever enter the
trapped region. The protons trapped in those regions are mostly remnants of
the protons left behind the main RPA (seen to the left of the shock front in
the $t=45\,\mathrm{fs}$ frame of Fig. 2).
### 3.2 Ultrafast ion heating revisited
The theoretical study of TSR [28] predicts that an ultrafast collisional ion
heating may take place in plasmas composed of light and heavy ion species.
This result is born out by 1D collisional PIC simulations performed with the
Epoch [36] code, considering a CsH target almost identical to that in the
current paper. The authors ascribe the observed ultrafast heating to
collisional friction between the protons and Cs ions as they experience a
differential acceleration in the electrostatic field of the shock.
The CsH setup presented in this paper is almost identical to that of TSR –
apart from a mere $1\%$ difference in the electron density, the laser
polarization and the increased resolution in our case. We have also run a
Smilei simulation with exactly the same physical parameters (including linear
polarization and numerical resolution) as TSR. As regards the ion dynamics,
this simulation yields results virtually identical to the CsH simulation
presented in Fig 2 (therefore, they are not presented here separately).
However, none of our simulations reproduce the main findings of TSR, namely,
the collisional downstream proton heating ascribed to inter-species ion
friction and the absence of ion reflection. By contrast, our simulations
indicate that the collisional interaction between the ion species does not
inhibit the proton reflection; in fact, as shown in e.g. Fig. 3, nearly all
protons are reflected, and these are subsequently heated through collisional
friction through the ambient (upstream) Cs ions. The proton heating is
strongest in the _reflected_ ion population.
The PIC results of TSR are interpreted by a two-fluid model retaining the
momentum and energy moments of the Fokker-Planck equation, assuming Maxwellian
distributions. It provides steady-state expressions for the longitudinal
derivatives of the temperatures and velocities of the two fluid species, which
are then integrated over the spatial width of the shock front. The energy
input to the system comes from an electric field term representing the
electrostatic shock field. Importantly, the possibility of ion reflection is
ruled out by construction: protons are forced to pass through the barrier and
gain all the available potential energy, which is consistent with their
simulation results, but not with ours. In the collisionless case the protons
should clearly be reflected due to their higher charge to mass ratio than that
of Cs. The only way to avoid proton reflection is if a very strong friction
between the two species pulls the ions across the potential barrier. This,
however, requires a much stronger collisional coupling than what we observe.
Thus, we believe that the difference between TSR’s results and ours is (at
least partly333Modifications to the implementation of the Sentoku & Kemp [37]
collision model in Epoch over time make a direct comparison to TSR difficult.)
a consequence of the different collision algorithms used. The version of the
Epoch [36] code used by TSR was equipped with a collision module based on the
algorithm proposed by Sentoku & Kemp [37] (SK), while Smilei employs the
scheme developed by Pérez et al. [31] (NYP), which generalizes the Nanbu &
Yonemura scheme [38, 39] to the relativistic regime. Both collision models are
designed to reproduce the Fokker–Planck limit, where small-angle collision
events dominate, which is relevant for high-temperature and/or low-density
plasmas. However, at the high plasma densities considered here, which are
susceptible to quantum degeneracy and coupled plasma effects, corrections must
be made to avoid unphysically high collision frequencies444It should be
emphasized though, that these PIC simulations are intrinsically classical, and
as such, a self-consistent treatment of quantum effects is clearly outside
their scope. Thus the extensions of any binary collision model to dense/cold
plasma regions are ad-hoc models designed to reproduce plasma-averaged
collisional properties expected from advanced warm-dense-matter or condensed-
matter models [40].. This is also a major point where the SK and NYP
algorithms differ.
In the high-density/low-temperature regime, the SK model forces the effective
temperature of the interacting species to stay above the Fermi temperature, in
order to emulate the Fermi-degenerate regime. This leads to the maximum
collision frequency
$\hat{\nu}_{\alpha\beta}^{\rm(SK)}=m_{\mathrm{e}}Z^{*}_{\beta}e^{4}\log\Lambda/(12\pi^{3}\epsilon_{0}^{2}\hbar^{3})$
between two particles of species $\alpha$ and $\beta$ [37, eq. (10)]. By
contrast, drawing from the prescription of Ref. [41] for coupled plasmas, the
NYP model applies a lower bound on the collisional mean free path, which can
never get smaller than the mean inter-particle distance $r_{\beta}\sim(4\pi
n_{\beta}/3)^{-1/3}$. This yields the saturated collision frequency
$\hat{\nu}_{\alpha\beta}^{\rm(NYP)}=(4\pi
n_{\beta}/3)^{1/3}(T_{\alpha}/2m_{\alpha})^{1/2}$ [31, sec. I-C].
At the considered ion density ($n_{{\mathrm{Cs}}}=1.5\times
10^{28}\,\mathrm{m^{-3}}$) and a representative ion temperature of
$T_{\mathrm{i}}=100\,\mathrm{eV}$, we find
$\nu_{\mathrm{p}\,{\mathrm{Cs}}}^{\rm(Spitzer)}\approx 1.5\times
10^{15}\,\mathrm{s^{-1}}$, and $\lambda_{\mathrm{p}\,{\mathrm{Cs}}}\approx
0.06\,\mathrm{nm}$ that is significantly smaller than the inter-atomic
distance ${\sim}0.2\,\mathrm{nm}$. Thus the dense-plasma limit of NYP should
hold under such conditions but not the degenerate SK limit. One thus obtains
that
$\nu_{\mathrm{p}\,{\mathrm{Cs}}}^{\rm(SK)}=\nu_{\mathrm{p}\,{\mathrm{Cs}}}^{\rm(Spitzer)}\approx
1.5\times 10^{15}\,\mathrm{s^{-1}}$ is more than $5$ times larger than the
dense-plasma value $\nu_{\mathrm{p}\,{\mathrm{Cs}}}^{\rm(NYP)}\approx
2.7\times 10^{14}\,\mathrm{s^{-1}}$. This discrepancy is only strengthened
when considering Cs–Cs collisions. Again at $n_{{\mathrm{Cs}}}=1.5\times
10^{28}\,\mathrm{m^{-3}}$ and $T_{\mathrm{i}}=100\,\mathrm{eV}$, one finds
$\nu_{{\mathrm{Cs}}\,{\mathrm{Cs}}}^{\rm(Spitzer)}\approx 6.6\times
10^{16}\,\mathrm{s^{-1}}$, which is over three orders of magnitude larger than
the dense-plasma NYP value,
$\nu_{{\mathrm{Cs}}\,{\mathrm{Cs}}}^{\rm(NYP)}\approx 2.4\times
10^{13}\,\mathrm{s^{-1}}$. Regarding the electron–Cs ion collisions, one has
$\nu_{\mathrm{e}\,{\mathrm{Cs}}}^{\rm(SK)}=\nu_{\mathrm{e}\,{\mathrm{Cs}}}^{\rm(Spitzer)}\approx
6.3\times 10^{16}\,\mathrm{s^{-1}}$ at $T_{\mathrm{e}}=100\,\mathrm{eV}$, but
this corresponds to a mean free path
$\lambda_{\mathrm{e}\,{\mathrm{Cs}}}\approx 0.06\,\mathrm{nm}\ll
r_{{\mathrm{Cs}}}$, so again the dense-plasma limit applies, which gives
$\nu_{\mathrm{e}\,{\mathrm{Cs}}}^{\rm(NYP)}\approx 1.2\times
10^{16}\,\mathrm{s^{-1}}$. The difference is even larger at the lower
temperatures associated with the early-time interaction.
Moreover, the SK and NYP schemes handle colliding particles with non-equal
statistical weights differently, which impacts the accuracy of energy
conservation. However, that is likely not the cause of the diverging
simulation results, since the number of computational particles is large in
both cases, so as to limit statistical noise.
A recent simulation study [42] of dense ($n_{\mathrm{e}}=60n_{\mathrm{c}}$)
plasmas driven at relativistic laser intensities, comparing the results of the
SK and NYP555Since version 4.17 (June 2019) Epoch has the full NYP algorithm
implemented as well. modules in Epoch and the NYP module of Smilei, confirms
that the SK model indeed results in stronger effective collisionality. In
addition, a good agreement between Epoch and Smilei was found when both
employed the NYP algorithm.
Which of these two treatments of collisions in dense/cold plasmas is more
physically correct is still a debated issue. Therefore, along with further
numerical investigation, experimental verification should be sought for in
order to determine the parameter regions of validity, and accuracy, of the two
collision algorithms. Our results suggest that such differentiation between
the algorithms is possible using laser-plasma experiments in multi-species,
dense plasmas, such as the CsH case presented here. A good benchmarking test
would be to compare ion energy spectra in cases where collisions are
sufficiently strong to suppress ion reflection according to SK but not
according to NYP. Such experiments might need to control the target density
profile on the rear side, e.g. through laser ablation, in order to suppress
TNSA, and make the shock accelerated ion population clearer. A potentially
suitable experiment has recently been performed [18], but it would require
further investigation to see whether the accuracy of the two models can be
assessed from the obtained data (which clearly showed ion reflection).
## 4 Ion dynamics in copper targets
Figure 4: Copper ion phase-space distribution in the $300\,\mathrm{nm}$ thick
Cu target from the collisional simulation, at times $t=21\,\mathrm{fs}$,
$t=45\,\mathrm{fs}$ and $70\,\mathrm{fs}$.
We will now turn to the pure copper simulations, first considering similar
target and laser parameters to the CsH case, and subsequently changing these
parameters one by one. The two main differences compared to CsH are the lack
of multi-species effects and the ${\sim}5$ times higher electron density
(assuming $Z^{*}=27$). However, just as in the CsH target, the primary effect
of collisions in the Cu plasma is the inverse Bremsstrahlung-type electron
heating. The bulk electrons are heated to $T_{\mathrm{e}}\approx
3.7\,\mathrm{keV}$, corresponding to a sound speed of $c_{\mathrm{s}}\approx
1.3\times 10^{-3}c$.
Figure 4 shows the collisional Cu ion phase-space distribution, at peak laser
intensity ($t=21\,\mathrm{fs}$), close after the laser irradiation
($t=45\,\mathrm{fs}$), and even later in time ($t=70\,\mathrm{fs}$). Similarly
to Cs, the Cu ions have a rather low charge-to-mass ratio ($Z^{*}/A=0.42$) and
do not have time to fully respond to the laser piston during the short-pulse
irradiation. Again, the initial perturbation from the laser piston transforms
into an electrostatic shock, yet it is losing energy faster than in the CsH
target.
Since the copper plasma does not contain any protons, all of the reflected
charge, needed to sustain the shock, consists of Cu ions. Owing to their high
charge ($Z^{*}=27$), the collisional interaction between the reflected and the
upstream ions is stronger than in the collisional CsH case, resulting in a
noticeable heating of these two populations, as seen in the collisional Cu ion
distributions at $45\,\mathrm{fs}$ and $70\,\mathrm{fs}$ in Fig. 4. Some
heating is observed in the collisional proton and Cs ion distributions of Fig.
2 as well, but significantly weaker than in the copper plasma.
Figure 5: Copper ion phase-space distribution in the $300\,\mathrm{nm}$ thick
Cu target from the collisional simulation, with an $a_{0}=10$ and
$60\,\mathrm{fs}$ duration laser pulse. The distribution is shown at times
$t=90\,\mathrm{fs}$, $t=150\,\mathrm{fs}$ and $200\,\mathrm{fs}$. Figure 6:
Temporal evolution of the normalized potential drop across the shock front,
$\hat{\phi}=e\phi/T_{\mathrm{e}}$, and shock Mach number, $\mathcal{M}$, for
the thin copper target, long pulse collisional simulation. The blue line
represents a moving average (over three data points) of $\hat{\phi}$. The
vertical lines indicate the time of peak (solid) and half (dotted) laser
intensity. For both the normalization of the potential and for the ion
acoustic sound speed, an electron temperature of
$T_{\mathrm{e}}=4\,\mathrm{keV}$ was used as a representative value.
We have also studied a scenario wherein the copper plasma is illuminated by a
laser pulse of longer duration ($60\,\mathrm{fs}$ FWHM) and lower intensity
($a_{0}=10$). The Cu ion phase-space distribution from the collisional
simulation is displayed in Fig. 5, shown at times $t=90\,\mathrm{fs}$ (at peak
laser intensity), $t=150\,\mathrm{fs}$ (close to the end of the pulse) and
$t=200\,\mathrm{fs}$. Like in the two previous setups, an electrostatic shock
is generated. It forms out of a perturbation that detaches from the laser
piston already as early as $t\approx 60\,\mathrm{fs}$, before the pulse has
reached half its maximum intensity. It displays electrostatic shock-like
properties, such as a sharp rise in lab-frame ion velocity in conjunction with
a steep electrostatic potential barrier, but it lacks any ion reflection, as
seen in the $t=90\,\mathrm{fs}$ frame of the collisional simulation.
Figure 6 shows the evolution of the normalized666Using a fixed value of
$T_{\mathrm{e}}=4\,\mathrm{keV}$, derived from Maxwellian fits to the electron
energy spectrum (whole plasma). The measured electron temperature stays fairly
close to this value during the entire duration of the pre-shock and the
electrostatic shock. potential jump $\hat{\phi}=e\phi/T_{\mathrm{e}}$ and
Mach number $\mathcal{M}$ of the shock from the time of detachment from the
laser piston to its demise. The transition to a fully developed, ion-
reflecting, electrostatic shock occurs when
$\hat{\phi}\gtrsim\mathcal{M}^{2}/2$, which is at around $t\approx
90\,\mathrm{fs}$. The longer pulse duration and more gradual increase in
intensity, detaches the onset of shock reflection from RPA.
The peaks in $\hat{\phi}$ and $\mathcal{M}$ are followed by a more gradual
decrease in the Mach number and in the shock potential peak, starting at
around $t\approx 110\,\mathrm{fs}$. The vertical lines in Fig. 6 represent the
time of peak (solid) and half (dotted) laser intensity. The peaks thus occur
before the laser intensity has halved. The delayed peaks in shock speed and
potential relative to the peak laser intensity likely originate from the fact
that the interaction has reached a stage where the laser is no longer able to
supply more power than the energy dissipation rate of the shock.
The reflection of ions appears as a process bootstrapping itself. After the
first few ions have been reflected, collisional heating between the upstream
and reflected ions cause a broadening of the longitudinal momentum
distribution of the upstream ions, leading to more ions entering the reflected
region of phase-space. This upstream heating is seen in the collisional
$t=150\,\mathrm{fs}$ frame of Fig. 5. Towards the end of the simulation, the
upstream and reflected ion populations start to merge into each other, after
which the determination of the shock speed relative to the upstream population
becomes unreliable. The shock ends somewhat abruptly when it collides with the
rarefaction wave emanating from the back of the target, which occurs at
roughly $t\approx 250\,\mathrm{fs}$.
Figure 7: Distribution of copper ions in a $2.5\,\mathrm{\textrm{\textmu}{m}}$
thick target, with collisions, at times $t=110\,\mathrm{fs}$,
$t=150\,\mathrm{fs}$ and $500\,\mathrm{fs}$. Note that the initial position of
the target front is at $x=7.5\,\mathrm{\textrm{\textmu}{m}}$, different from
the other simulations presented. Figure 8: Spatial profiles of the copper ion
temperature and density at $t=500\,\mathrm{fs}$ and $t=1000\,\mathrm{fs}$.
Both profiles display a sharp jump; the temperature jump by about a factor of
$2.6$, while the density jumps by about a factor $2.0$.
As our final setup, we switch to a $2.5\,\mathrm{\textrm{\textmu}{m}}$ copper
target, driven by an $a_{0}=10$ and $60\,\mathrm{fs}$ FWHM duration pulse.
Those parameters may be of interest to warm-dense-matter studies [29]. The
simulation results are shown in Fig. 7. Despite the significant increase in
target areal density, the measured electron temperature still reaches
$T_{\mathrm{e}}\approx 3.5\,\mathrm{keV}$, thus the initial evolution of the
shock is very similar to that in the thin-target simulation. Indeed, the Mach
number and shock potential evolve as those displayed in Fig. 6, both
qualitatively and quantitatively (when accounting for a time shift of
${\simeq}15\,\mathrm{fs}$ corresponding to the different target position). The
initial shock wave displays characteristic features of an electrostatic shock,
such as ion reflection and a velocity modulation in the downstream. However,
it also shows signs of a collisional shock, such as isotropization of the
downstream ion distribution (i.e. the longitudinal and transverse temperatures
are comparable to each other). The shock can therefore be claimed to be in a
hybrid regime between a collisionless electrostatic shock and a hydrodynamic
shock.
Since the target is now significantly thicker, the shock wave has time to
further dissipate its energy, and the ion reflection terminates at $t\approx
300\,\mathrm{fs}$, well before the shock front encounters the rarefaction wave
from the back of the target. As the shock steadily loses speed – and the
electrostatic potential drop across the shock front decreases – a point is
reached when the electrostatic potential barrier is too weak to cause ion
reflection (in fact, the electric field reaches the level of statistical
noise). However, even though the ion reflection is absent, the steady
propagation of a clear shock front structure in phase space is clearly visible
inside the target, in the $t=500\,\mathrm{fs}$ panel of Fig. 7. There are
corresponding discontinuities in the ion temperature, $T_{{\mathrm{Cu}}}$, and
density, $n_{{\mathrm{Cu}}}$, profiles: Figure 8 shows that
$T_{{\mathrm{Cu}}}$ and $n_{{\mathrm{Cu}}}$ jump by a factor of $2.6$ and
$2.0$, respectively. Since the laser no longer exerts radiation pressure on
the target front side, the latter rapidly expands towards the vacuum as a
rarefaction wave propagates into the shocked plasma. At $t=500\,\mathrm{fs}$,
this rarefaction wave has caught up with the shock front to create a weakly
supersonic ($\mathcal{M}\approx 1.3$), planar blast wave [43, Sec. 4.3], which
slowly decays away (compare the ion temperature and density jumps at
$t=500\,\mathrm{fs}$ and $t=1000\,\mathrm{fs}$).
To study how the qualitative features of the shock dynamics depend on laser
parameters, further simulations have been performed; $a_{0}$ ranging from $2$
to $14$ and the pulse FWHM duration ranging from $15\,\mathrm{fs}$ to
$120\,\mathrm{fs}$. In addition, two simulations have been run with $a_{0}=7$
and $14$, with the respective pulse durations varied such that the pulse
energy would stay the same as in the case presented above ($a_{0}=10$ and
$60\,\mathrm{fs}$ FWHM duration). We could identify two qualitatively
different regimes for the ion dynamics. At lower intensities, the pulse is not
strong enough to initiate ion reflection at any point; instead, a shock-like
structure similar to the one displayed at $t=110\,\mathrm{fs}$ in Fig. 7 is
launched, and is sustained for several hundred femtoseconds, with its speed
and amplitude decaying rather slowly. This behaviour is observed here for
$a_{0}\leq 7$, and also in the simulation with $a_{0}=7$ and
$120\,\mathrm{fs}$ FWHM duration. The latter indicates that that both the
laser intensity and energy are important for the onset of ion reflection.
At higher intensities, the behaviour is qualitatively similar to the one shown
in Fig. 7 – ion reflection is initiated, followed by a gradual loss of energy,
until ion reflection no longer occurs, and the shock turns into a
collisionally sustained blast wave. However, the time scale for this
transition to happen depends on the laser parameters: both higher intensity
and shorter pulses result in an earlier onset of the ion reflection, as well
as a faster transition into a blast wave. The reason for the faster demise of
ion reflection may be linked to the rapid collisional heating of the upstream
ions by the reflected ions. A hotter upstream favours ion reflection, thus
hastening the shock dissipation. Remarkably, the ions in the downstream of the
blast wave are heated to several tens of $\mathrm{keV}$ temperatures, in the
first ${\sim}100\,\mathrm{fs}$ after the ion reflection has ended. For
instance, the temperature recorded in Fig. 8 is ${\sim}20\,\mathrm{keV}$ at
$t=500\,\mathrm{fs}$. In the case of $a_{0}=14$ and FWHM duration of
$30\,\mathrm{fs}$, the downstream ion temperature reaches
${\sim}60\,\mathrm{keV}$ at $t=300\,\mathrm{fs}$, then dropping down to
${\sim}30\,\mathrm{keV}$ at $t=500\,\mathrm{fs}$. Unlike the heating scenario
put forward by TSR [28], the heating process of the downstream heavy ions
revealed by our simulations does not involve inter-species friction induced in
the shock electrostatic potential.
Another trend observed in the scans is that shorter duration pulses generate
faster shock evolution, i.e. a faster onset of ion refection, as well as a
faster decay into a blast wave. This is also likely linked to the interaction
of the laser piston and the plasma. Shorter laser pulses are quicker to reach
their maximum intensity. The laser piston may therefore reach sufficient
strength to reflect ions, before any pre-shock perturbation (e.g. as that in
the $t=110\,\mathrm{fs}$ panel in Fig. 7) would have time to form and overtake
the piston. The early onset of ion reflection then leads to a rapid transition
to a blast wave, as discussed in the previous paragraph.
In relation to the transition from hybrid shock to a blast wave, we note that
the end of the ion reflection is accompanied by an increase in the width of
the shock front, from $\Delta x_{\mathrm{sh}}\sim 1.6\,\mathrm{nm}$ (i.e., a
few times the Debye length $\lambda_{\mathrm{D}}\approx 0.3\,\mathrm{nm}$) to
$6{-}9\,\mathrm{nm}$. This width is about an order of magnitude larger than
the collisional ion mean free path, here estimated as the inter-atomic
distance, $\lambda_{\rm mfp}\approx 0.25\,\mathrm{nm}$. Our finding is
consistent with previous estimates of the width of weakly supersonic
($\mathcal{M}\approx 2$) hydrodynamic shocks ($\Delta{x}_{\mathrm{sh}}\approx
20\lambda_{\rm mfp}$) [22, 44].
Finally, the robustness of the ion dynamics observed in the thick copper
target has been tested against possible multidimensional effects on the laser-
driven electron energization and subsequent ion dynamics, through a two-
dimensional simulation of the thick copper target, detailed further in [29].
This simulation reveals that the situation studied here is sufficiently
collisional that the shock does not suffer from transverse modulations.
## 5 Conclusions
Using particle-in-cell simulations, we have numerically investigated the
impact of Coulomb collisions on the ion dynamics in high-$Z^{*}$, solid
density caesium hydride and copper targets, irradiated by high-intensity
($I\approx 2{-}5\times 10^{20}\,\mathrm{Wcm^{-2}}$), ultrashort
($10{-}60\,\mathrm{fs}$), circularly polarized laser pulses.
In all cases collisional absorption through inverse Bremsstrahlung heats the
electrons up to $3{-}10\,\mathrm{keV}$ temperatures throughout the target,
while the use of CP reduces the creation of high-energy electrons.
Subsequently, collisions quickly relax the electrons to a Maxwellian
distribution. The impact of the laser pulse launches an electrostatic shock
wave. In all cases studied here, the collisionally enhanced electron heating
results in faster shock waves, with higher potential drops across the shock
front, than in the corresponding collisionless simulations.
In the CsH target, the different charge-to-mass ratios of the hydrogen and
caesium ions result in strong proton reflection. In contrast to the results of
TSR [28], we do not observe a large number of protons passing through the
shock front and get heated via collisional friction with the Cs ions. Instead,
inter-species friction results in the _reflected_ ions being heated up to
${\sim}\mathrm{keV}$ temperatures. The difference in proton reflection between
our results and those of TSR appears to be a consequence of distinct collision
models in the dense/cold plasmas where the Spitzer theory no longer applies.
This suggests that laser plasma experiments, using targets containing a highly
charged species and protons volumetrically, may be utilized to differentiate
between numerical collision models.
In pure Cu targets, the collisional coupling between the reflected and
upstream ions is stronger, causing an appreciable heating of these two. Also,
the higher density of both ions and electrons causes a faster decay of the
shock in the CsH target. When turning to a somewhat lower-intensity, but
longer-duration laser pulse, the initial stages of the shock launching process
become more decoupled from the laser pulse and the RPA. Here, the shock forms
already prior to the on-target laser peak. However, the shock front continues
to accelerate until about ${\sim}20\,\mathrm{fs}$ after the on-target laser
peak. Because of the quick launch of the electrostatic shock, the maximum
energy of the accelerated ions has less sharp temporal variation, since there
is no transition from the RPA ions to the CSA ions. Yet, the shock initially
lacks ion reflection, the onset of which appears to be bootstrapping itself
via heating of the upstream ions by the reflected ones.
Lastly, we increased the target thickness in order to follow the electrostatic
shock evolution over a longer duration, and to become more relevant to high-
energy-density-physics applications. While the shock wave is at no point
purely electrostatic, as it exhibits some features of hydrodynamic shocks, we
observe the shock speed and potential drop to decay until the shock loses its
capability to reflect ions. At this stage, the electrostatic potential drop
across the shock front has also disappeared, and a rarefaction wave launched
from the target front side has overtaken the shock front, turning it into a
weakly supersonic ($\mathcal{M}\approx 1.3$) collisional blast wave. This
formation is capable of locally heating up the downstream ions to tens of
$\mathrm{keV}$ temperatures for a duration of about ${\sim}100\,\mathrm{fs}$.
The authors are grateful for fruitful discussions with L. Hesslow and T.
Fülöp, as well as to M. Grech and F. Pérez for support with Smilei, and A. E.
Turrel for providing inputs for simulations presented in [28]. This project
has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme under grant
agreement No 647121, the Swedish Research Council (grant no. 2016-05012), and
the Knut och Alice Wallenberg Foundation. The simulations were performed on
resources provided by the Swedish National Infrastructure for Computing (SNIC)
at Chalmers Centre for Computational Science and Engineering (C3SE) and High
Performance Computing Center North (HPC2N).
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|
# Measurement Error Mitigation via Truncated Neumann Series
Kun Wang<EMAIL_ADDRESS>Institute for Quantum Computing, Baidu Research,
Beijing 100193, China Yu-Ao Chen<EMAIL_ADDRESS>Institute for Quantum
Computing, Baidu Research, Beijing 100193, China Xin Wang
<EMAIL_ADDRESS>Institute for Quantum Computing, Baidu Research, Beijing
100193, China
###### Abstract
Measurements on near-term quantum processors are inevitably subject to
hardware imperfections that lead to readout errors. Mitigation of such
unavoidable errors is crucial to better explore and extend the power of near-
term quantum hardware. In this work, we propose a method to mitigate
measurement errors in computing quantum expectation values using the truncated
Neumann series. The essential idea is to cancel the errors by combining
various noisy expectation values generated by sequential measurements
determined by terms in the truncated series. We numerically test this method
and find that the computation accuracy is substantially improved. Our method
possesses several advantages: it does not assume any noise structure, it does
not require the calibration procedure to learn the noise matrix a prior, and
most importantly, the incurred error mitigation overhead is independent of
system size, as long as the noise resistance of the measurement device is
moderate. All these advantages empower our method as a practical measurement
error mitigation method for near-term quantum devices.
## I Introduction
Quantum computers hold great promise for a variety of scientific and
industrial applications McArdle _et al._ (2020); Cerezo _et al._ (2020);
Bharti _et al._ (2021); Endo _et al._ (2021). However, in the current stage
noisy intermediate-scale quantum (NISQ) computers Preskill (2018) introduce
significant errors that must be dealt with before performing any practically
valuable tasks. Errors in a quantum computer are typically classified into
quantum gate errors and measurement errors. For quantum gate errors, various
quantum error mitigation techniques have been proposed to mitigate the damages
caused by errors on near-term quantum devices Temme _et al._ (2017); Endo
_et al._ (2018); Li and Benjamin (2017); McClean _et al._ (2017, 2020);
McArdle _et al._ (2019); Bonet-Monroig _et al._ (2018); He _et al._ (2020);
Giurgica-Tiron _et al._ (2020); Kandala _et al._ (2019); Endo _et al._
(2021); Sun _et al._ (2021); Czarnik _et al._ (2020). For measurement
errors, experimental works have demonstrated that measurement errors in
quantum devices can be well understood in terms of classical noise models Chow
_et al._ (2012); Kandala _et al._ (2019); Chen _et al._ (2019), which is
recently rigorously justified Geller (2020). Specifically, a $n$-qubit noisy
measurement device can be characterized by a noise matrix $A$ of size
$2^{n}\times 2^{n}$. The element in the $\bm{x}$-th row and $\bm{y}$-th
column, $A_{\bm{x}\bm{y}}$, is the probability of obtaining a outcome $\bm{x}$
provided that the true outcome is $\bm{y}$. If one has access to this
stochastic matrix, it is straightforward to classically reverse the noise
effects simply by multiplying the probability vector obtained from
experimental statistics by this matrix’s inversion. However, there are several
limitations of this matrix inversion approach: (i) The complete
characterization of $A$ requires $2^{n}$ calibration experiment setups and
thus is not scalable. (ii) The matrix $A$ may be singular for large $n$,
preventing direct inversion. (iii) The inverse $A^{-1}$ is hard to compute and
might not be a stochastic matrix, indicating that it can produce negative
probabilities.
Several approaches have been proposed to deal with these issues Maciejewski
_et al._ (2020); Tannu and Qureshi (2019); Nachman _et al._ (2019); Hicks
_et al._ (2021); Bravyi _et al._ (2020); Geller and Sun (2020); Murali _et
al._ (2020); Kwon and Bae (2020); Funcke _et al._ (2020); Zheng _et al._
(2020); Maciejewski _et al._ (2021); Barron and Wood (2020). For example,
Ref. Chen _et al._ (2019); Maciejewski _et al._ (2020) elucidated that the
quality of the measurement calibration and the number of measurement samples
affected the performance of measurement error mitigation methods dramatically.
Motivated by the unfolding algorithms in high energy physic, Ref. Nachman _et
al._ (2019); Hicks _et al._ (2021) used the iterative Bayesian unfolding
approach to avoid pathologies from the matrix inversion. Ref. Bravyi _et al._
(2020) introduced a new classical noise model based on the continuous time
Markov processes and proposed an error mitigation approach that cancels errors
using the quasiprobability decomposition technique Pashayan _et al._ (2015);
Temme _et al._ (2017); Howard and Campbell (2017); Endo _et al._ (2018);
Takagi (2020); Jiang _et al._ (2020); Regula _et al._ (2021). However, most
of these works make an explicit assumption on the physical noise model and
require the calibration procedure to learn the stochastic matrix $A$, and thus
is not scalable in general. Recently, Ref. Berg _et al._ (2020) proposed a
noise model-free measurement error mitigation method that forces the bias in
the expectation value to appear as a multiplicative factor that can be
removed.
In this work, we propose a measurement error mitigation method motivated by
the Neumann series, applicable for any quantum algorithms where the
measurement statistics are used for computing the expectation values of
observables. The idea behind this method is to cancel the measurement errors
by utilizing the noisy expectation values generated by sequential
measurements, each determined by a term in the truncated Neumann series. The
method is deliberately simple, does not make any assumption about the actual
physical noise model, and does not require calibrating the stochastic matrix a
priori.
The paper is organized as follows. Section II describes the quantum task of
computing expectation values and explains how the noisy measurement incurred
bias to the results. Section III presents the error mitigation technique via
truncated Neumann series. Section IV reports the experimental demonstration of
our error mitigation method. The Appendices summarize technical details used
in the main text.
## II Computing the expectation value
Let $\rho$ be an $n$-qubit quantum state generated by a quantum circuit. Most
of the quantum computing tasks end with computing the expectation value
$\operatorname{Tr}[O\rho]$ of a given observable $O$ within a prefixed
precision $\varepsilon$, by post-processing the measurement outcomes of the
quantum state. This task is the essential component of multifarious quantum
algorithms, notable practical examples of which are variational quantum
eigensolvers Peruzzo _et al._ (2014); McClean _et al._ (2016), quantum
approximate optimization algorithm Farhi _et al._ (2014), and quantum machine
learning Biamonte _et al._ (2017); Havlíček _et al._ (2019).
For simplicity, we assume that the observable $O$ is diagonal in the
computational basis and its elements take values in the range $[-1,1]$, i.e.,
$\displaystyle
O=\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})|{\bm{x}}\rangle\\!\langle{\bm{x}}|,\quad|O(\bm{x})|\leq
1,$ (1)
where $O(\bm{x})$ is the $\bm{x}$-th diagonal element of $O$ and $|\alpha|$ is
the absolute value of $\alpha$. Note that we adopt the convention that the
diagonal elements are indexed from $0$. Consider $M$ independent experiments
where in each round we prepare the state $\rho$ using the same quantum circuit
and measure each qubit in the computational basis (see, e.g., Fig. 1). Let
$\bm{s}^{m}\in\\{0,1\\}^{n}$ be the measurement outcome observed in the $m$-th
round. We further define the empirical mean value
$\displaystyle\eta^{(0)}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{M}\sum_{m=1}^{M}O(\bm{s}^{m}).$ (2)
Let $\operatorname{vec}(\rho)$ be the $2^{n}$-dimensional column vector formed
by the diagonal elements of $\rho$. Then Bravyi _et al._ (2020)
$\displaystyle E^{(0)}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{E}[\eta^{(0)}]=\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\operatorname{vec}(\rho)=\operatorname{Tr}[O\rho],$
(3)
where $\mathbb{E}[X]$ is the expectation of the random variable $X$. Eq. (3)
implies that $\eta^{(0)}$ is an unbiased estimator of
$\operatorname{Tr}[O\rho]$. What’s more, the standard deviation
$\sigma(\eta^{(0)})\leq 1/\sqrt{M}$. By Hoeffding’s inequality Hoeffding
(1963), $M=2\log(2/\delta)/\varepsilon^{2}$ would guarantee that
$\displaystyle\operatorname{Pr}\\{|\eta^{(0)}-\operatorname{Tr}[O\rho]|\leq\varepsilon\\}\geq
1-\delta,$ (4)
where $\operatorname{Pr}\\{\cdot\\}$ is the event’s probability, $\delta$ is
the specified confidence, and all logarithms are in base $2$ throughout this
paper.
Figure 1: Computing the expectation value $\operatorname{Tr}[O\rho]$ with the
ideal measurement device (left) and the noisy measurement device (right) not .
However, measurement devices on current quantum hardware inevitably suffer
from hardware imperfections that lead to readout errors, which are manifested
as a bias toward the expectation values we aim to compute (cf. the right side
of Fig. 1). As previously mentioned, in the most general scenario, these
errors are modeled by a $2^{n}\times 2^{n}$ noise matrix $A$. If there were no
measurement error at all, $A$ is the identity matrix $I$. The off-diagonal
elements of $A$ completely characterize the readout errors. By definition, $A$
is column-stochastic in the sense that the elements of each column are non-
negative and sum to $1$.
Suppose now that we adopt the same procedure for computing $\eta$ in (2),
where we perform $M$ independent experiments and collect the measurement
outcomes. Denote by $\bm{s}^{m,1}\in\\{0,1\\}^{n}$ the outcome observed in the
$m$-th round, where the superscript $1$ indicates that the noisy measurement
is applied. As (2), we define
$\displaystyle\eta^{(1)}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{M}\sum_{m=1}^{M}O(\bm{s}^{m,1}).$ (5)
We prove in Appendix A that
$\displaystyle E^{(1)}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{E}[\eta^{(1)}]=\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|A\operatorname{vec}(\rho),$
(6)
indicating that $\eta^{(1)}$ is no longer an estimator of
$\operatorname{Tr}[O\rho]$. Comparing Eqs. (3) and (6), we find that in the
ideal case, the sampled probability distribution approximates
$\operatorname{vec}(\rho)$ due to the weak law of large numbers, while in the
noisy case, the sampled probability distribution approximates
$A\operatorname{vec}(\rho)$, leading to a bias in the estimator.
## III Error mitigation via truncated Neumann series
A direct approach to eliminate the measurement errors from
$A\operatorname{vec}(\rho)$ is to apply the inverse matrix $A^{-1}$. However,
this approach is resource-consuming and only feasible when $n$ is small. To
deal with this difficulty, we simulate the effect of $A^{-1}$ using a
truncated Neumann series. That is, $A^{-1}$ is approximated by a linear
combination of the terms $A^{k}$ for different $k$, with carefully chosen
coefficients. This idea has previously been applied for linear data detection
in massive multiuser multiple-input multiple-output wireless systems Wu _et
al._ (2013).
Define the _noise resistance_ of the noise matrix $A$ as
$\displaystyle\xi\mathrel{\mathop{\mathchar
58\relax}}=2\left(1-\min_{\bm{x}\in\\{0,1\\}^{n}}\left\langle{\bm{x}}\right|A\left|{\bm{x}}\right\rangle\right).$
(7)
By definition, $1-\xi/2$ is the minimal diagonal element of $A$. Intuitively,
$\xi/2$ characterizes the noisy measurement device’s behavior in the worst-
case scenario since it is the maximal probability for which the true outcome
should be $\bm{x}$ yet the actual outcome is not $\bm{x}$. In the following,
we assume $\xi<1$, which is equivalent to the condition that the minimal
diagonal element of $A$ is larger than $0.5$. This assumption is reasonable
since otherwise the measurement device is too noisy to be applied from the
practical perspective. Under this assumption, the stochastic matrix $A$ is
nonsingular and the Neumann series implies that (Stewart, 1998, Theorem 4.20)
$\displaystyle A^{-1}$ $\displaystyle=\sum_{k=0}^{\infty}(I-A)^{k}$ (8a)
$\displaystyle=\sum_{k=0}^{K}(I-A)^{k}+\mathcal{O}((I-A)^{K+1})$ (8b)
$\displaystyle=\sum_{k=0}^{K}c_{K}(k)A^{k}+\mathcal{O}((I-A)^{K+1}),$ (8c)
where for arbitrary non-negative integers $0\leq k\leq K$, the coefficient
function is defined as
$\displaystyle c_{K}(k)\mathrel{\mathop{\mathchar
58\relax}}=(-1)^{k}\binom{K+1}{k+1},$ (9)
and $\binom{n}{k}$ is the binomial coefficient. Intuitively, Eq. (8) indicates
that one may approximate the inverse matrix $A^{-1}$ using the first $K$
Neumann series terms, if the behavior of the remaining terms
$\mathcal{O}((I-A)^{K+1})$ can be bounded. We show that this is indeed the
case in the measurement error mitigation task. More specifically, using the
first $K+1$ terms in the expansion (8) of $A^{-1}$, we obtain the following.
###### Theorem 1.
For arbitrary positive integer $K$, it holds that
$\displaystyle\left|\operatorname{Tr}[O\rho]-\sum_{k=1}^{K+1}c_{K}(k-1)E^{(k)}\right|\leq\xi^{K+1},$
(10)
where
$\displaystyle E^{(k)}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|A^{k}\operatorname{vec}(\rho).$
(11)
The proof is given in Appendix B. As evident from Theorem 1, the noise
resistance $\xi$ of the noise matrix $A$ determines the number of terms
required in the truncated Neumann series to approximate $A^{-1}$ to the
desired precision. What is more, since $\xi<1$, the approximation error decays
exponentially in terms of $K$. By the virtue of (6), $E^{(k)}$ can be viewed
as the noisy expectation value generated by a noisy measurement device whose
corresponding noise matrix is $A^{k}$. Let
$\overline{E}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{k=1}^{K+1}c_{K}(k-1)E^{(k)}$. Theorem 1 inspires a systematic
way to estimate the expectation value $\operatorname{Tr}[O\rho]$ in two steps.
Figure 2: Experimental setup for estimating $E^{(4)}$, in which the noisy
measurement device (box in blue) is executed $4$ times sequentially.
Firstly, we choose $K$ for which the RHS. of (10) evaluates to $\varepsilon$,
yielding the optimal truncated number
$\displaystyle K=\left\lceil\frac{\log\varepsilon}{\log\xi}-1\right\rceil.$
(12)
Such a choice guarantees that $\overline{E}$ is $\varepsilon$-close to the
expectation value $\operatorname{Tr}[O\rho]$. Secondly, we compute the
quantity $\overline{E}$ by estimating each term $E^{(k)}$ and computing the
linear combination according to the coefficients $c_{K}$. Since $\overline{E}$
itself is only an $\varepsilon$-estimate of $\operatorname{Tr}[O\rho]$, it
suffices to approximate $\overline{E}$ within an error $\varepsilon$.
Motivated by the relation between $\eta^{(1)}$ and $E^{(1)}$ (see the
discussions and calculations in obtaining (5)), we declare that each $E^{(k)}$
can be estimated via the following procedure:
1. 1.
Generate a quantum state $\rho$.
2. 2.
Using $\rho$ as input, execute the noisy measurement device $k$ times
_sequentially_ and collect the outcome produced by the final measurement
device, i.e., the $k$-th measurement device.
3. 3.
Repeat the above two steps $M$ rounds and collect the measurement outcomes.
4. 4.
Define an average analogous to (5) and output it as an estimate of $E^{(k)}$.
We elaborate thoroughly on the concept of sequential measurement in Appendix C
and show that the classical noise model describing the sequential measurement
repeating $k$ times is effectively characterized by the stochastic matrix
$A^{k}$. For a sequential measurement repeating $k$ times, one can think of
the rightmost $k-1$ measurements as implementing the calibration subroutine
since they accept the computational basis states as inputs. To some extent,
this is a _dynamic_ calibration procedure where we do not statically enumerate
all computational bases as input states but dynamically prepare the input
states based on the output information of the target state from the first
measurement device. For illustrative purpose, we demonstrate in Fig. 2 the
experimental setup for estimating the noisy expectation value $E^{(4)}$, where
the measurement device is repeated four times in each round. We summarize the
whole procedure in the following Algorithm 1.
Algorithm 1 Error mitigation via truncated Neumann series
0: Quantum circuit generating the $n$-qubit state $\rho$, the $n$-qubit
quantum observable $O$, the $n$-qubit noisy measurement device, noise
resistance $\xi$, probability tolerance $\delta$, precision parameter
$\varepsilon$.
0: $\eta$, as an estimate of $\operatorname{Tr}[O\rho]$.
1: Compute $K=\left\lceil\log\varepsilon/\log\xi-1\right\rceil$;
2: Compute $\Delta=\binom{2K+2}{K+1}-1$;
3: Compute $M=\lceil 2(K+1)\Delta\log(2/\delta)/\varepsilon^{2}\rceil$;
4: for $k=1,\cdots,K+1$ do
5: for $m=1,\cdots,M$ do
6: Run the quantum circuit to generate $\rho$;
7: Execute the measurement device $k$ times sequentially;
8: Obtain the measurement outcome $\bm{s}^{m,k}$;
9: end for
10: Compute $\eta^{(k)}=\frac{1}{M}\sum_{m=1}^{M}O(\bm{s}^{m,k})$;
11: end for
12: Compute $\eta=\sum_{k=1}^{K+1}c_{K}(k-1)\eta^{(k)}$, where $c_{K}$ is
defined in (9);
13: Output $\eta$.
We claim that the output $\eta$ of Algorithm 1 approximates the expectation
value $\operatorname{Tr}[O\rho]$ pretty well, as captured by the following
proposition.
###### Proposition 2.
The output $\eta$ of Algorithm 1 satisfies
$\displaystyle\operatorname{Pr}\left\\{|\operatorname{Tr}[O\rho]-\eta|\leq
2\varepsilon\right\\}\geq 1-\delta.$ (13)
Proof of the proposition is given in Appendix D. Intuitively, Eq. (13) says
that the output $\eta$ of Algorithm 1 estimates the ideal expectation value
$\operatorname{Tr}[O\rho]$ with error $2\varepsilon$ at a probability greater
than $1-\delta$. Analyzing Algorithm 1, we can see that we ought to expand the
Neumann series to the $K$-th order, where $K$ is computed via (12), and
estimate the $K+1$ noisy expectation values $E^{(1)},\cdots,E^{(K+1)}$
individually. For each expectation, we need $M$ copy of quantum states. As so,
the total number of quantum states consumed is given by
$\displaystyle M(K+1)$
$\displaystyle=2(K+1)^{2}\Delta\log(2/\delta)/\varepsilon^{2}$
$\displaystyle\approx 4^{K}\log(2/\delta)/\varepsilon^{2}.$ (14)
In other words, our error mitigation method increases the number of quantum
states that is required to achieve the given precision $\varepsilon$ by a
factor of $4^{K}$ compared with the case of the ideal measurement. In Fig. 3,
we plot the optimal truncated number $K$ (12) as a function of the noise
resistance $\xi$, the error tolerance parameter is fixed as
$\varepsilon=0.01$. One can check from the figure that $K\leq 10$ whenever the
noise resistance satisfies $\xi\leq 0.657$ (Equivalently, the minimal diagonal
element of $A$ is larger than $0.67$). That is to say, the incurred error
mitigation overhead $4^{K}$ is independent of the system size, so long as the
noise resistance $\xi$ is moderate, in the sense that it is below a certain
threshold (say $0.657$). On the other hand, the number of noisy quantum
measurements applied in Algorithm 1 is given by
$\displaystyle\left(\sum_{k=1}^{K+1}k\right)M\approx
2(K+1)^{3}\Delta\log(2/\delta)/\varepsilon^{2}.$ (15)
Compared to (14), our method has used more number of measurements than the
number of quantum states by a multiplier $K+1$. We remark that both costs are
roughly characterized by the prominent factor $4^{K}$.
Figure 3: The optimal truncated number $K$ (12) as a function of the noise
resistance $\xi$, where the precision parameter is $\varepsilon=0.01$.
### III.1 Discussion on the noise resistance
In practical applications, $\xi$ can be obtained from the specifications of
NISQ devices. For example, in IBM quantum devices, the specifications are
often reported $2-10\%$ Kwon and Bae (2020). If such information is not
available, we may perform calibration to obtain $A$ first and then compute
$\xi$, which is still resource-efficient compared to computing the inverse
matrix $A^{-1}$.
When defining $\xi$ in (7), we do not consider any structure of $A$. If
certain noise model is assumed, the calculation of $\xi$ can be simplified. In
the following, we consider the tensor product noise model and show that the
noise resistance can be compute analytically. Assume $A$ is a tensor product
of $n$ $2\times 2$ stochastic matrices, i.e.,
$\displaystyle A_{\operatorname{tp}}=\begin{bmatrix}1-\alpha_{1}&\beta_{1}\\\
\alpha_{1}&1-\beta_{1}\end{bmatrix}\otimes\cdots\otimes\begin{bmatrix}1-\alpha_{n}&\beta_{n}\\\
\alpha_{n}&1-\beta_{n}\end{bmatrix},$ (16)
where $\alpha_{i}$ and $\beta_{i}$ are error rates describing the $i$-th
qubit’s readout errors $0\to 1$ and $1\to 0$, respectively. One can show that
$\displaystyle\xi(A_{\operatorname{tp}})=2\left(1-\prod_{i=1}^{n}\min\\{1-\alpha_{i},1-\beta_{i}\\}\right).$
(17)
Specially, if $\alpha_{i},\beta_{i}\ll 1$, then $\xi\approx
2(1-1/e^{\gamma})$, where $\gamma\mathrel{\mathop{\mathchar
58\relax}}=\sum_{i=1}^{n}\max\\{\alpha_{i},\beta_{i}\\}$ is called the noise
strength in Bravyi _et al._ (2020).
## IV Experimental results
We apply the proposed error mitigation method to the following illustrative
example and demonstrate its performance. Consider the input state
$\rho=|{\Phi}\rangle\\!\langle{\Phi}|$, where
$\displaystyle\left|{\Phi}\right\rangle\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\sqrt{2^{n}}}\sum_{i=0}^{2^{n}-1}\left|{i}\right\rangle,$
(18)
which is the maximal superposition state. The observable $O$ is a tensor
product of Pauli $Z$ operators, i.e., $O=Z^{\otimes n}$. The ideal expectation
value is $\operatorname{Tr}[O\rho]=0$. We choose $n=8$ and randomly generate a
noise matrix $A^{\ast}$ whose noise resistance satisfies $\xi(A^{\ast})\approx
0.657$ (as so the noise matrix is moderate). We repeat the procedure for
producing the noisy expectation value $\eta^{(1)}$ and Algorithm 1 for
producing the mitigated expectation value $\eta$ a total number of $1000$
times. Note that all these experiments assume the same noise matrix $A$, and
the parameters are chosen as $\varepsilon=\delta=0.01$. The obtained
expectation values are scatted in Fig. 4. It is easy to see from the figure
that the noisy measurement device, characterized by the noise matrix
$A^{\ast}$, incurs a bias $\approx-0.007$ to the estimated expectation values.
On the other hand, the error mitigated expectation values distributed evenly
around the ideal value $0$ within a distance of $0.01$ with high probability.
As evident from Fig. 4, several mitigated expectation values fall outside the
expected region. These statistical outcomes match our conclusion in
Proposition 2, validating the correctness and performance of the proposed
error mitigation method.
Fig. 5 shows the (noisy and mitigated) expectation values estimated via the
above procedure as a function of the number of qubits. In our numerical setup,
for an experiment whose number of qubits $n$ is less than $8$, its
corresponding noise matrix is obtained by partially tracing out the rightmost
$8-n$ qubit systems from $A^{\ast}$. The entire experiment for each $n$ was
repeated $1000$ times in order to estimate the error bars. The reason that the
noisy estimates behave well for single and two qubits is that the underlying
noise matrices are close to the identity in the total variation distance
Maciejewski _et al._ (2020). It can be seen that the cross-talk noise
presented in the noisy measurement device severely distorts the estimated
expectation value while our error mitigation method is insensitive to this
kind of error.
Figure 4: $1000$ noisy estimates $\eta^{(1)}$ (blue triangles) and error
mitigated estimates $\eta$ (red dots) for the ideal expectation value
$\operatorname{Tr}[O\rho]=0$. Here, the number of qubits is $8$. Figure 5:
Average expectation values for $1\leq n\leq 8$ qubits obtained with (red dots)
and without (blue triangles) the error mitigation method. Each error bar is
estimated by repeating the experiment $1000$ times.
## V Conclusions
We have introduced a scalable method to mitigate measurement errors in
computing expectation values of quantum observables, an essential building
block of numerous quantum algorithms. The idea behind this method is to
approximate the inverse of the noise matrix determined by the noisy
measurement device using a small number of the Neumann series terms. Our
method via the truncated Neumann series outperforms the exact matrix inversion
method by significantly reducing the resource costs in time and samples of
quantum states while only slightly degrading the error mitigation performance.
In particular, our method works for any classical noise model and does not
require the calibration procedure to learn the noise matrix a prior. Most
importantly, the incurred error mitigation overhead is independent of the
system size, as long as the noise resistance of the noisy measurement device
is moderate. This property is beneficial and will be more and more important
as the quantum circuit sizes increase. We have numerically tested this method
and found that the computation accuracy is substantially improved. We believe
that the proposed method will be useful for experimental measurement error
mitigation in NISQ quantum devices.
## Acknowledgements
We thank Runyao Duan for helpful suggestions. We would like to thank Zhixin
Song for collecting the experiment data.
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## Appendix A Proof of Eq. (6)
###### Proof.
By the definition of $\eta^{(1)}$, we have
$\displaystyle\eta^{(1)}=\frac{1}{M}\sum_{m=1}^{M}O(\bm{s}^{m,1})=\frac{1}{M}\sum_{m=1}^{M}\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\bm{s}^{m,1}\rangle=\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\left(\frac{1}{M}\sum_{m=1}^{M}|\bm{s}^{m,1}\rangle\right).$
(19)
The expectation value can be evaluated as
$\displaystyle E^{(1)}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{E}[\eta^{(1)}]$
$\displaystyle=\mathbb{E}\left[\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\left(\frac{1}{M}\sum_{m=1}^{M}|\bm{s}^{m,1}\rangle\right)\right]$
(20)
$\displaystyle=\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\mathbb{E}\left[\frac{1}{M}\sum_{m=1}^{M}|\bm{s}^{m,1}\rangle\right]$
(21)
$\displaystyle=\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|A\operatorname{vec}(\rho).$
(22)
∎
## Appendix B Proof of Theorem 1
###### Proof.
First of all, notice that
$\displaystyle\left|\operatorname{Tr}[O\rho]-\sum_{k=1}^{K+1}c_{K}(k-1)E^{(k)}\right|$
$\displaystyle=\left|\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\operatorname{vec}(\rho)-\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\left(\sum_{k=1}^{K+1}c_{K}(k-1)A^{k}\operatorname{vec}(\rho)\right)\right|$
(23)
$\displaystyle=\left|\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\left(I-\sum_{k=1}^{K+1}c_{K}(k-1)A^{k}\right)\operatorname{vec}(\rho)\right|$
(24)
$\displaystyle=\left|\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\left(I-A\left(\sum_{k=1}^{K+1}c_{K}(k-1)A^{k-1}\right)\right)\operatorname{vec}(\rho)\right|$
(25)
$\displaystyle=\left|\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\left(I-A\left(\sum_{k=0}^{K}c_{K}(k)A^{k}\right)\right)\operatorname{vec}(\rho)\right|$
(26)
$\displaystyle=\left|\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\left(I-A\left(\sum_{k=0}^{K}(I-A)^{k}\right)\right)\operatorname{vec}(\rho)\right|$
(27)
$\displaystyle=\left|\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|\left(I-\left(I-(I-A)^{K+1}\right)\right)\operatorname{vec}(\rho)\right|$
(28)
$\displaystyle=\left|\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|(I-A)^{K+1}\operatorname{vec}(\rho)\right|,$
(29)
where (27) follows from (8) and (28) follows from the closed-form formula of a
geometric series. Now we show that the quantity in (29) can be bounded from
above. Define the induced matrix $1$-norm of a $m\times n$ matrix $B$ as
$\displaystyle\left\lVert B\right\rVert_{1}\mathrel{\mathop{\mathchar
58\relax}}=\max_{1\leq j\leq n}\sum_{i=1}^{n}|B_{ij}|\equiv\max_{1\leq j\leq
n}\sum_{i=1}^{n}|\left\langle{i}\right|B\left|{j}\right\rangle|,$ (30)
which is simply the maximum absolute column sum of the matrix. Let
$\rho(\bm{y})$ is the $\bm{y}$-th diagonal element of the quantum state
$\rho$. Consider the following chain of inequalities:
$\displaystyle\left|\sum_{\bm{x}\in\\{0,1\\}^{n}}O(\bm{x})\langle\bm{x}|(I-A)^{K+1}\operatorname{vec}(\rho)\right|$
$\displaystyle=\left|\sum_{\bm{x}\in\\{0,1\\}^{n}}\sum_{\bm{y}\in\\{0,1\\}^{n}}O(\bm{x})\rho(\bm{y})\langle\bm{x}|(I-A)^{K+1}|\bm{y}\rangle\right|$
(31a)
$\displaystyle\leq\sum_{\bm{x}\in\\{0,1\\}^{n}}\sum_{\bm{y}\in\\{0,1\\}^{n}}|O(\bm{x})|\cdot\rho(\bm{y})\cdot\left|\langle\bm{x}|(I-A)^{K+1}|\bm{y}\rangle\right|$
(31b)
$\displaystyle\leq\sum_{\bm{x}\in\\{0,1\\}^{n}}\sum_{\bm{y}\in\\{0,1\\}^{n}}\rho(\bm{y})\left|\langle\bm{x}|(I-A)^{K+1}|\bm{y}\rangle\right|$
(31c)
$\displaystyle=\sum_{\bm{y}\in\\{0,1\\}^{n}}\rho(\bm{y})\sum_{\bm{x}\in\\{0,1\\}^{n}}\left|\langle\bm{x}|(I-A)^{K+1}|\bm{y}\rangle\right|$
(31d)
$\displaystyle\leq\sum_{\bm{y}\in\\{0,1\\}^{n}}\rho(\bm{y})\lVert(I-A)^{K+1}\lVert_{1}$
(31e) $\displaystyle=\lVert(I-A)^{K+1}\lVert_{1}$ (31f)
$\displaystyle\leq\lVert I-A\lVert_{1}^{K+1}$ (31g) $\displaystyle=\xi^{K+1},$
(31h)
where (31c) follows from the assumption that $|O(\bm{x})|\leq 1$ (cf. Eq.
(1)), (31e) follows from the definition of induced matrix $1$-norm, (31f)
follows from the fact that $\rho$ is a quantum state and thus
$\sum_{\bm{y}}\rho(\bm{y})=1$, (31g) follows from the submultiplicativity
property of the induced matrix norm, and (31h) follows from Lemma 3 stated
below. We are done. ∎
###### Lemma 3.
Let $A$ be a column stochastic matrix of size $d\times d$. It holds that
$\displaystyle\left\lVert I-A\right\rVert_{1}=\xi(A),$ (32)
where $\xi(A)$ is defined in (7).
###### Proof.
Since $A$ is column stochastic, $I-A$ has non-negative diagonal elements and
negative off-negative elements. Thus
$\displaystyle\left\lVert I-A\right\rVert_{1}$ $\displaystyle=\max_{1\leq
j\leq d}\left(1-A_{jj}+\sum_{i\neq j}A_{ij}\right)$ (33)
$\displaystyle=\max_{1\leq j\leq d}\left(1-A_{jj}+1-A_{jj}\right)$ (34)
$\displaystyle=2\max_{1\leq j\leq d}\left(1-A_{jj}\right)$ (35)
$\displaystyle=2-2\min_{1\leq j\leq d}A_{jj}$ (36)
$\displaystyle=\mathrel{\mathop{\mathchar 58\relax}}\xi(A),$ (37)
where the second line follows from the fact that $A$ is column stochastic. ∎
## Appendix C Sequential measurements
In the Appendix, we prove that the classical noise model describing the
sequential measurement repeating $k$ times is effectively characterized by the
stochastic matrix $A^{k}$. We begin with the simple case $k=2$. Since the
noise model is classical and linear in the input, it suffices to consider the
computational basis states as inputs. As shown in Fig. 6, we apply the noisy
quantum measurement device two times sequentially on the input state
$|{\bm{x}}\rangle\\!\langle{\bm{x}}|$ in computational basis where
$\bm{x}\in\\{0,1\\}^{n}$. Assume the measurement outcome of the first
measurement is $\bm{y}$ and the measurement outcome of the second measurement
is $\bm{z}$, where $\bm{y},\bm{z}\in\\{0,1\\}^{n}$. Assume that the noise
matrix associated with this sequential measurement is $A^{\prime}$. That is,
the probability of obtaining the outcome $\bm{z}$ provided the true outcome is
$\bm{x}$ is given by $A^{\prime}_{\bm{z}\bm{x}}$. Practically, we input
$|{\bm{x}}\rangle\\!\langle{\bm{x}}|$ to the first noisy measurement device
and obtain the outcome $\bm{y}$. The probability of this event is
$A_{\bm{y}\bm{x}}$, by the definition of the noise matrix. Similarly, we input
$|{\bm{y}}\rangle\\!\langle{\bm{y}}|$ to the second noisy measurement device
and obtain the outcome $\bm{z}$. The probability of this event is
$A_{\bm{z}\bm{y}}$. Inspecting the chain $\bm{x}\to\bm{y}\to\bm{z}$, we have
$\displaystyle
A^{\prime}_{\bm{z}\bm{x}}=\sum_{\bm{y}\in\\{0,1\\}^{n}}A_{\bm{y}\bm{x}}A_{\bm{z}\bm{y}}=A^{2}_{\bm{z}\bm{x}}.$
(38)
The above analysis justifies that the classical noise model describing the
sequential measurement repeating $2$ times is effectively characterized by the
stochastic matrix $A^{2}$. The general case can be analyzed similarly.
Figure 6: Apply the noisy quantum measurement device two times sequentially on
the input state $|{\bm{x}}\rangle\\!\langle{\bm{x}}|$ where
$\bm{x}\in\\{0,1\\}^{n}$. The measurement outcome of the first measurement is
$\bm{y}$ and the measurement outcome of the second measurement is $\bm{z}$.
Mathematically, quantum measurements can be modeled as quantum-classical
quantum channels (Wilde, 2016, Chapter 4.6.6) where they take a quantum system
to a classical one. Experimentally, the implementation of quantum measurement
is platform-dependent and has different characterizations. For example, the
fabrication and control of quantum coherent superconducting circuits have
enabled experiments that implement quantum measurement Naghiloo (2019). Based
on the outcome data, experimental measurements are typically categorized into
two types: those only output classical outcomes and those output both
classical outcomes and quantum states. That is, besides the usually classical
outcome sequences, the measurement device will also output a quantum state on
the computational basis corresponding to the classical outcome. For the former
type, we can implement the sequential measurement via the _qubit reset_ Egger
_et al._ (2018); Magnard _et al._ (2018); Yirka and Subasi (2020) approach,
by which we mean the ability to re-initialize the qubits into a known state,
usually a state in the computational basis, during the course of the
computation. Technically, when the $i$-th noisy measurement outputs an outcome
sequence $\bm{s}^{i}\in\\{0,1\\}^{n}$, we use the qubit reset technique to
prepare the computational basis state
$|\bm{s}^{i}\rangle\\!\langle\bm{s}^{i}|$ and feed it to the $(i+1)$-th noisy
measurement (cf. Fig. 6). In this case, the noisy measurement device can be
reused. For the latter type, the sequential measurement can be implemented
efficiently: when the $i$-th noisy measurement outputs a classical sequence
and a quantum state on the computational basis, we feed the quantum state to
the $(i+1)$-th noisy measurement.
## Appendix D Proof of Proposition 2
###### Proof.
By definition,
$\displaystyle\eta=\sum_{k=1}^{K+1}c_{K}(k-1)\eta^{(k)}=\frac{1}{M}\sum_{k=1}^{K+1}\sum_{m=1}^{M}c_{K}(k-1)O(\bm{s}^{m,k})=\frac{1}{M(K+1)}\sum_{k=1}^{K+1}\sum_{m=1}^{M}(K+1)c_{K}(k-1)O(\bm{s}^{m,k}).$
(39)
Introducing the new random variables $X_{m,k}\mathrel{\mathop{\mathchar
58\relax}}=(K+1)c_{K}(k-1)O(\bm{s}^{m,k})$, we have
$\displaystyle\eta=\frac{1}{M(K+1)}\sum_{k=1}^{K+1}\sum_{m=1}^{M}X_{m,k}.$
(40)
Intuitively, Eq. (40) says that $\eta$ can be viewed as the empirical mean
value of the set of random variables
$\displaystyle\left\\{X_{m,k}\mathrel{\mathop{\mathchar
58\relax}}m=1,\cdots,M;k=1,\cdots,K+1\right\\}.$ (41)
First, we show that the absolute value of each $X_{m,k}$ is upper bounded as
$\displaystyle|X_{m,k}|=|(K+1)c_{K}(k-1)O(\bm{s}^{m,k})|\leq(K+1)|c_{K}(k-1)||O(\bm{s}^{m,k})|\leq(K+1)|c_{K}(k-1)|,$
(42)
where the second inequality follows from the assumption of $O$ (cf. Eq. (1)).
Then, we show that $\eta$ is an unbiased estimator of the quantity
$\sum_{k=1}^{K+1}c_{K}(k-1)E^{(k)}$:
$\displaystyle\mathbb{E}[\eta]$
$\displaystyle=\mathbb{E}\left[\frac{1}{M(K+1)}\sum_{k=1}^{K+1}\sum_{m=1}^{M}X_{m,k}\right]$
(43a)
$\displaystyle=\mathbb{E}\left[\frac{1}{M}\sum_{k=1}^{K+1}\sum_{m=1}^{M}c_{K}(k-1)O(\bm{s}^{m,k})\right]$
(43b)
$\displaystyle=\sum_{k=1}^{K+1}c_{K}(k-1)\left(\sum_{\bm{x}}O(\bm{x})\langle\bm{x}|\mathbb{E}_{M}\left[\frac{1}{M}\sum_{m=1}^{M}|\bm{s}^{m,k}\rangle\right]\right)$
(43c)
$\displaystyle=\sum_{k=1}^{K+1}c_{K}(k-1)\left(\sum_{\bm{x}}O(\bm{x})\langle\bm{x}|A^{k}\operatorname{vec}(\rho)\right)$
(43d) $\displaystyle=\sum_{k=1}^{K+1}c_{K}(k-1)E^{(k)},$ (43e)
where the last equality follows from (11). Eqs. (42) and (43) together
guarantee that the prerequisites of the Hoeffding’s inequality hold. By the
Hoeffding’s equality, we have
$\displaystyle\Pr\left\\{\left|\eta-\sum_{k=1}^{K+1}c_{K}(k-1)E^{(k)}\right|\geq\varepsilon\right\\}$
$\displaystyle\leq
2\exp\left(-\frac{2M^{2}(K+1)^{2}\varepsilon^{2}}{4\sum_{k=1}^{K+1}\sum_{m=1}^{M}((K+1)c_{K}(k))^{2}}\right)$
(44)
$\displaystyle=2\exp\left(-\frac{2M^{2}(K+1)^{2}\varepsilon^{2}}{4M(K+1)^{3}\left(\sum_{k=0}^{K}[c_{K}(k)]^{2}\right)}\right)$
(45) $\displaystyle=2\exp\left(-\frac{M\varepsilon^{2}}{2(K+1)\Delta}\right),$
(46)
where $\Delta\mathrel{\mathop{\mathchar
58\relax}}=\sum_{k=1}^{K+1}[c_{K}(k)]^{2}=\binom{2K+2}{K+1}-1$. Solving
$\displaystyle
2\exp\left(-\frac{M\varepsilon^{2}}{2(K+1)\Delta}\right)\leq\delta$ (47)
gives
$\displaystyle M\geq 2(K+1)\Delta\log(2/\delta)/\varepsilon^{2}.$ (48)
To summarize, choosing $K=\left\lceil\log\varepsilon/\log\xi-1\right\rceil$
and $M=\lceil 2(K+1)\Delta\log(2/\delta)/\varepsilon^{2}\rceil$, we are able
obtain the following two statements
$\displaystyle\Pr\left\\{\left|\eta-\sum_{k=1}^{K+1}c_{K}(k-1)E^{(k)}\right|\geq\varepsilon\right\\}\leq\delta,$
(49)
$\displaystyle\left|\operatorname{Tr}[O\rho]-\sum_{k=1}^{K+1}c_{K}(k-1)E^{(k)}\right|\leq\varepsilon,$
(50)
where the first one is shown above and the second one is proved in Theorem 1.
Using the union bound and the triangle inequality, we conclude that $\eta$ can
estimate the ideal expectation value $\operatorname{Tr}[O\rho]$ with error
$2\varepsilon$ at a probability greater than $1-\delta$. ∎
|
# Stabilizer Tensor Networks: universal quantum simulator on a basis of
stabilizer states
Sergi Masot-Llima Barcelona Supercomputing Center, Barcelona 08034, Spain
Universitat de Barcelona, Barcelona 08007, Spain Artur Garcia-Saez Barcelona
Supercomputing Center, Barcelona 08034, Spain Qilimanjaro Quantum Tech,
Barcelona 08007, Spain
###### Abstract
Efficient simulation of quantum computers relies on understanding and
exploiting the properties of quantum states. This is the case for methods such
as tensor networks, based on entanglement, and the tableau formalism, which
represents stabilizer states. In this work, we integrate these two approaches
to present a generalization of the tableau formalism used for Clifford circuit
simulation. We explicitly prove how to update our formalism with Clifford
gates, non-Clifford gates, and measurements, enabling universal circuit
simulation. We also discuss how the framework allows for efficient simulation
of more states, raising some interesting questions on the representation power
of tensor networks and the quantum properties of resources such as
entanglement and magic, and support our claims with simulations.
Simulation of quantum computing is crucial for two main reasons: driving
science in fields like condensed matter physics [1, 2] or quantum chemistry
[3, 4, 5], as long as we do not have large, error-corrected devices, and
testing quantum advantage claims [6, 7, 8] made by cutting-edge devices [9,
10]. To simulate efficiently beyond a few dozen qubits, we must find
alternative characterisations due to the exponential growth of brute force
approaches. Thus, a large effort is put towards identifying which states are
easy and why, given the absence of a universal description of simulablity.
Resource theories [11] are a useful tool for this task: they characterize the
operations that are easy to do (free operations) in a certain framework. We
are particularly interested in entanglement [12] and stabilizer rank [13, 14],
for their relation to tensor networks (TN) and the stabilizer formalism,
respectively.
The interest in relating different resources, particularly these two, is not
new. Previous research has found some of these states present maximal
entanglement, at least in the bipartite sense [15], although most types of
entangled states are not achievable with these circuits. Recently, magic in
Matrix Product States (MPS), a type of TN, has also been characterized and
looked into [16, 17, 18], and it is noteworthy that separable states with a
lot of magic are complex in the stabilizer formalism, even though they are
trivial to simulate with resource theories of entanglement. This means that
these resources are, in some sense, orthogonal, as depicted in Fig. 1a. In
this article, we unify simulation strategies for entanglement and magic by
using a special basis [19] in conjunction with tensor networks, as shown in
Fig. 1b, and we focus on how the proposed method can simulate arbitrary
circuits.
Figure 1: Showcase of the stabilizer formalism. In a), we classify states
under two different resources. For the resource of entanglement, in blue,
(non-stabilizerness, orange), axis y=0 (x=0) represents the free states, while
its adjacent region represents states with low amounts of entanglement (non-
stabilizerness), which are classically simulatable with tensor networks
(stabilizer tableaus). They are simultaneously simulatable with stabilizer
tensor networks, in green, and can be characterized with a different resource
$R$. In b) we show how stabilizer TN joins the other methods: the tableau
formalism (b1) encodes a stabilizer state and a set of destabilizer generators
which are used to form a basis $\mathcal{B}(\mathcal{S},\mathcal{D})$ for the
Hilbert space. The first $n$ rows of the tableau encode the decomposition of
$n$ destabilizer generators, and rows $n+1$, … , $2n$ encode $n$ stabilizer
generators. An extra column $r$ indicates the phase of each generator, since
$S$ and $-S$ stabilize different states. The amplitudes of decomposing a state
$\ket{\psi}$ into $\mathcal{B}(\mathcal{S},\mathcal{D})$ are stored in a
tensor network (green, b2).
Entanglement as a resource for simulation is usually characterized by
bipartite entanglement between sectors of the system. This applies to methods
such as circuit cutting [20], entanglement forging [21] or tensor networks
[22]. These simulations rely on limited entanglement, mostly between close
neighbors [23], or on a hierarchical structure of entanglement [24, 25]. Free
operations consist of single-qubit (local) gates and classical communication
[26, 27]. In the extreme case of no entanglement, the simulation cost of a
system grows linearly with its size; in more complex cases, systems can adhere
to an area law [28] that allows TN methods like DMRG [29] to perform efficient
simulations with great success. Tensor networks encode high-dimensional
tensors into the product of smaller, low-dimensional ones, and are in general
advantageous whenever long-range correlations are restricted. The tensors can
be connected through bonds in various geometries, and networks with more
complexity and expressive power will entail higher performance costs. We focus
on a 1D MPS structure to encode the amplitudes of a quantum state:
$\mathcal{T}^{i_{1}i_{2}\dots i_{n}}=\sum_{k_{1}k_{2}\dots
k_{n-1}}(T_{1})^{i_{1}}_{k_{1}}(T_{2})^{i_{2}k_{1}}_{k_{2}}(T_{3})^{i_{3}k_{2}}_{k_{3}}\dots(T_{N})^{i_{n}k_{n-1}}$
(1)
In this structure, the dimension $\chi$ of a given bond $k$ corresponds to the
entropy between the two subsystems it connects, as measured by the Schmidt
rank [30]. We call this $\chi$ the bond dimension. A separable state can be
encoded into a TN with $\chi=1$, whereas a state with mostly local
entanglement (AKLT state [31]) needs $\chi=2$, and a maximally entangled state
requires up to $\chi=2^{n/2}$. The bond dimension can also be artificially
limited at the cost of precision.
The stabilizer tableau formalism [32], on the other hand, can simulate
efficiently any circuit composed only by Clifford gates with a classical
computer. The states that can be prepared under these constraints are known as
stabilizer states. In this context, the set $\mathcal{C}$ of Clifford gates
are the free operations, and non-Clifford gates increase the stabilizer rank
[33], which constitutes the resource. However, resource theories of stabilizer
states are typically studied with other measures such as magic [34],
stabilizer Rényi entropy [35] or Wigner positivity [36] due to their
interesting properties. This simulation formalism is based on a stabilizer set
$\mathcal{S}$ of Pauli operators ($\mathcal{P}_{n}$). It uniquely defines a
state that fulfills $S\ket{\psi_{\mathcal{S}}}=\ket{\psi_{\mathcal{S}}}$ for
any $S\in\mathcal{S}$. A Pauli operator P can be described with two boolean
vectors and a phase, $P=\alpha\cdot(x_{1}x_{2}\dots
x_{n})\cdot(z_{1}z_{2}\dots z_{n})$, meaning we only need $2n+1$ boolean
values to represent one. Also, a set of $n$ generators of $\mathcal{S}$ are
enough to fully define the group. This means a tableau of $n\times(2n+1)$
boolean entries stores all the information about $\mathcal{S}$ (see Fig. 1,
b1). Since $\mathcal{P}_{n}$ is closed under the action of any gate
$C\in\mathcal{C}$, a Clifford circuit can be simulated by finding the new
tableau after applying each gate $C$ in it. An efficient ($O(n^{2})$ time)
approach to update the tableaus is known [37], and it works by also storing
the generators $d_{i}$ of the destabilizer group $\mathcal{D}$ – these
operators fulfill $\\{d_{i},s_{i}\\}=0,[d_{i},d_{j}]=0$ and $[d_{i},s_{j}]=0$
for any $i\neq j$.
In the following equations, we use the notation $d_{\hat{i}}$ for a generic
destabilizer in $\mathcal{D}$, defined by $\hat{i}$ and the generators $d_{i}$
as $d_{\hat{i}}=d_{1}^{i_{1}}\dots d_{n}^{i_{n}}$; the same follows for
stabilizers $s_{\hat{i}}$. Paired with the stabilizer state
$\ket{\psi_{\mathcal{S}}}$, they define the set
$\\{d_{\hat{i}}\ket{\psi_{\mathcal{S}}}\\}_{\hat{i}}$, which forms a basis
$\mathcal{B}(\mathcal{S},\mathcal{D})$[19] of the Hilbert space
$\mathcal{H}^{n}$:
$\ket{\psi}=\sum_{i=0}^{2^{n}}\nu_{i}d_{\hat{i}}\ket{\psi_{\mathcal{S}}}.$ (2)
This is shown and proven in Lemma 1. We encode these amplitudes on an MPS
using $\ket{\nu}=\sum_{i}\nu_{i}\ket{i}$, and show how they change when
applying any unitary gate or measurement with the following update rules:
1. 1.
Clifford gate $G$: Update the stabilizer basis
$\mathcal{B}(\mathcal{S},\mathcal{D})$ by conjugating with $G$, following the
rules in the tableau formalism (see [37] or Appendix D) for the update
$\ket{\psi_{\mathcal{S}}}\rightarrow
G\ket{\psi_{\mathcal{S}}}=\ket{\psi_{\tilde{\mathcal{S}}}}$. This gives a new
basis $\mathcal{B}(\tilde{\mathcal{S}},\tilde{\mathcal{D}})$:
$G\ket{\psi}=\sum_{i}\nu_{i}Gd_{i}\ket{\psi_{\mathcal{S}}}=\sum_{i}\nu_{i}\tilde{d}_{i}\ket{\psi_{\tilde{\mathcal{S}}}},$
(3)
and leaves the coefficient state $\ket{\nu}$ unchanged
$G\ket{\nu}=\ket{\nu}.$ (4)
2. 2.
Non-Clifford gate $\mathcal{U}$: Find the decomposition
$\mathcal{U}=\sum_{i}\phi_{i}\delta_{\hat{d}_{i}}\sigma_{\hat{s}_{i}}$, then
modify $\ket{\psi}$ as:
$\mathcal{U}\ket{\psi}=\sum_{i,j}((-1)^{j\cdot{\hat{s}_{i}}}\phi_{i}\nu_{j})\>d_{j+\hat{d}_{i}}\ket{\psi_{\mathcal{S}}}.$
(5)
When the decomposition only has two terms, this is equivalent to a rotation on
$\ket{\nu}$:
$\ket{\nu^{\prime}}=\cos(\theta)I-i\sin(\theta)X_{I_{x}}Y_{I_{y}}Z_{I_{z}}\ket{\nu}.$
(6)
The basis $\mathcal{B}(\mathcal{S},\mathcal{D})$ stays unchanged.
3. 3.
Measurement of observable $O$: Find the decomposition
$O=\alpha\delta_{\hat{d}}\sigma_{\hat{s}}$ and the value of
$\braket{\mathcal{O}}$ with:
$\braket{\mathcal{O}}=\alpha\braket{\nu}{X_{\hat{d}}Z_{\hat{s}}}{\nu}.$ (7)
Now choose an outcome $m\in\\{+,-\\}$ with probability
$p_{+}=\frac{I+\braket{O}}{2}$, $p_{-}=1-p_{+}$, and let $k$ be the position
of the first 1 in $\hat{d}$. Then update the stabilizer basis to
$\mathcal{B^{\prime}}(\mathcal{S^{\prime}},\mathcal{D^{\prime}})$ following
the rules for a measurement in the original formalism, and find
$\ket{\psi^{\prime}}=(I+m\mathcal{O})/2\ket{\psi}$ as:
$\ket{\psi^{\prime}}=\sum_{i}\delta_{i_{k},0}\left(\frac{1}{\sqrt{2}}\nu_{\hat{i}}+m\frac{\alpha(-1)^{\hat{i}\cdot\hat{s}}}{\sqrt{2}}\nu_{\hat{i}+\hat{d}}\right)d_{\hat{i}}\ket{\psi_{\mathcal{S^{\prime}}}},$
(8)
which equals to a projection and rotation on $\ket{\nu}$:
$\ket{\nu^{\prime}}=\ket{0}\bra{0}_{k}\left(\frac{1}{\sqrt{2}}I+m\frac{\alpha(-i)^{|I_{y}|}}{\sqrt{2}}X_{I_{x}}Y_{I_{y}}Z_{I_{z}}\right)\ket{\nu}.$
(9)
When simulating a circuit, we use a decomposition into CNOT and single qubit
rotations, as is usually done in real devices. This ensures that all non-
Clifford gates conform to the particular case of Eq. 6. The measured
observables $O$ are decomposed into (de)stabilizers, which we distinguish from
the generators of the basis $d_{\hat{i}}$ by writing $\sigma_{\hat{s}}$
($\delta_{\hat{d}}$) instead. In Annex A we explain the rules in more detail
and prove two more Lemmas that justify the equations shown here. To compute
the change of basis for Clifford gates, we employ already known [37] efficient
methods to update the tableau. Since the amplitudes do not change, they
preserve $\chi$. Non-Clifford gates and measurements, on the other hand, can
introduce correlations between amplitudes that increase $\chi$ and make
calculations more expensive. Consequently, $\chi$ constitutes our resource and
the free operations include all Clifford gates and also non-Clifford gates
$\mathcal{U}$ such that Eq. 6 is a local rotation on $\ket{\nu}$. We prove
this in corollary 2.1, in the annex.
The key ingredient to stabilizer tensor networks is allowing the basis to
change beyond local rotations. The tableau algorithm replaces the
computational basis with a basis of stabilizer states, which can have some
entanglement, and forgoes the correspondence between qubits and tensors. In a
way, entanglement is transferred from the tensor network $\ket{\nu}$
representation into the basis, at the cost of single qubit gates potentially
becoming entangling on the amplitudes of $\ket{\nu}$. In general, this only
happens if the circuit already contained entangling gates, and thus that part
of the circuit was entangling to begin with. Therefore, we argue that the
formalism does not generate fictitious entanglement. Instead, we say we store
potential entanglement in the basis.
We mentioned several resources linked to non-stabilizerness. Among those,
stabilizer rank has a direct link to our formalism. For an arbitrary state
$\ket{\psi}$, its stabilizer rank is the smallest $\xi$ that allows a
decomposition into stabilizer states $\ket{\psi_{S}}$:
$\ket{\psi}=\sum_{i=1}^{\xi}\alpha_{i}\ket{\psi_{S}^{i}}.$ (10)
Stabilizer states have $\xi=1$. The structure on the basis states we use does
not mean that a low stabilizer rank translates into a simple $\ket{\nu}$, as
the necessary stabilizer states might not be simultaneously in
$\mathcal{B}(\mathcal{S},\mathcal{D})$. We can define a pseudo-stabilizer rank
$\tilde{\xi}$ as the amount of non-zero coefficients in $\ket{\nu}$. This is
obviously an upper bound to $\xi$, but a thorough characterization of how
these quantities relate is left for future work.
Let us demonstrate that our formalism can efficiently simulate two different
scenarios: low entanglement and low stabilizer rank. In the original tableau
formalism [37], it was already shown how we can simulate any circuit and
encode any state in the $n$ qubit Hilbert space with a superposition of
tableaus. However, this is akin to a brute-force simulation with the
statevector approach, and grows exponentially with the number $t$ of non-
Clifford gates. Instead, our formalism can take advantage of all the tools
that have been developed for tensor networks simulations. Consider the state
$\ket{T}^{n}$, which can be prepared with:
$\ket{T}^{n}=\prod_{i=1}^{n}T_{i}\prod_{i=1}^{n}H_{i}\ket{0}^{\otimes n}.$
(11)
The first layer of Hadamards, which are Clifford gates and only updates the
tableau, sets the stabilizer basis to $s_{i}=X_{i},d_{i}=Z_{i}$. In this
basis, each $T$-gate on qubit $i$
$T_{i}=\cos(\frac{\pi}{8})I-i\sin(\frac{\pi}{8})Z_{i}=\cos(\frac{\pi}{8})I-i\sin(\frac{\pi}{8})d_{i},$
(12)
fulfills the criteria for a free operation, so the resulting state is
represented by a trivial MPS with $\chi=1$. Notice that, in this case, the
pseudo-stabilizer rank is maximal $\tilde{\xi}=2^{n}$. With the conventional
generalization of tableaus, we would need $2^{n}$ copies, as each $T$-gate
duplicates the number of necessary tableaus. This is not the best that can be
achieved: the stabilizer rank of $\ket{T}^{n}$ for small $n$ has been shown to
be low [38], meaning an optimal decomposition requires fewer tableaus (and
also $\xi<<\tilde{\xi}$). However, a general method to find these
decompositions is not known. Most importantly, the growth of $\xi$ with $n$ is
expected to be exponential [39] unless quantum computing is completely
simulatable (even though a supralinear lower bound has not been found [33]),
whereas stabilizer tensor networks can represent these states efficiently for
any $n$. On the other hand, some stabilizer states have been shown to have
maximum bipartite entanglement [15], as mentioned earlier. These states can be
prepared with a Clifford circuit, so in a simulation with stabilizer tensor
networks they will be an element of the basis
$\mathcal{B}(\mathcal{S},\mathcal{D})$, and therefore trivial to represent
with $\xi=\tilde{\xi}=1$, despite being expensive with a regular MPS.
In addition to these examples, there likely exists a different resource $R$
that captures the power of the approach, defining whether a state can be
efficiently represented or not with a single metric as illustrated in Fig. 1a.
This resource $R$ must be related to the two discussed resources, in the sense
that both low entanglement or low stabilizer rank imply low $R$, because we
have seen that we can simulate either case. However, since stabilizer states
can be very entangled, and separable states can have high non-stabilizerness,
it follows that high entanglement does not imply high $R$, nor does low
stabilizer rank imply high $R$, indicating that it isn’t trivially connected
to these resources.
Beyond the cases of complete stabilizerness or no entanglement, the efficiency
should persist when these resources are present in a low amount for the
formalism to be useful. Our discussion so far highlights two advantages in
that regard. First, notice that we can always process Clifford gates directly
on $\ket{\nu}$ instead of changing the basis so that the TN behaves
traditionally. This means any state simulatable with tensor networks is also
feasible in our approach. Additionally, we have seen that Clifford gates don’t
change $\ket{\nu}$, independently of its $\chi$, so we can always move freely
in the space of states with fixed stabilizer rank.
Figure 2: Average and maximum increase of entanglement after applying a single
T-gate on a random Clifford tableau, measured with $log(\chi^{\prime})$ where
$\chi^{\prime}$ is the maximum bond dimension of the MPS in the stabilizer TN.
The average is done over $\sim n^{2}$ uniformly sampled Clifford circuits. In
the inset, the distribution of $log(\chi^{\prime})$ for $n=40$.
Nonetheless, a gate that entangles $\ket{\psi}$ by a certain amount could in
principle become a more entangling gate on $\ket{\nu}$; alternatively, a gate
that does not increase stabilizer rank by much could also add a lot of
entanglement to $\ket{\nu}$. To show this is not the case, it suffices to look
at single-qubit rotations $\mathcal{R}$ for both cases, due to the
decomposition we employ. When $\theta$ is not a multiple of $\pi/4$, this
rotation is also a good example of an operation that only increases stabilizer
rank slightly (Eq. 12). We can bind the growth of $\chi$ in our MPS after
applying $\mathcal{R}$ by using Eq. 6 (which is equivalent to a CNOT cascade
[40]). The worst-case scenario for an MPS is $\chi^{\prime}=2^{4}\chi$, as
proven in Annex B, although our simulations (Fig. 2) show that, on average, it
is only $\sim 2^{2.46}$, and that it does not grow as $n\rightarrow\infty$.
For other TN structures with more connectivity, this bound can decrease.
Regardless, since it is bounded, we can ensure efficient simulation with a low
amount of non-Clifford single-qubit rotations. Notice that the worst-case
scenario only happens if the circuit applies CNOT gates before $\mathcal{R}$,
which were free in the stabilizer TN and stored potential entanglement in the
basis.
Conclusions and Outlook: We have presented a new approach to circuit
simulation, unifying two different frameworks each with its characterizing
resource. We have also shown in which instances it offers an advantage, and
identified its free operations for the characterization of a resource
theoretic description. In addition, developing the formalism has identified
several interesting research directions. First, one could decide to not always
use a Clifford gate to update the stabilizer basis of the TN, and apply it
directly to $\ket{\nu}$ instead. The criteria used for this decision would
directly affect the growth of the resource in the simulation. Also, the
storage and retrieval of the potential entanglement in
$\mathcal{B}(\mathcal{S},\mathcal{D})$ is possible with entangling Clifford
gates, but changing the basis also changes $\ket{\nu}$. This means that,
during a simulation, we cannot trivially decrease the cost of the next non-
stabilizer gate without potentially increasing $\chi$ of the current TN. A
thorough study of how to optimally allocate this resource to decrease the cost
of the simulation is left for future work.
Bounding $\chi$ of the MPS and checking the accuracy of results on states with
different amounts of entanglement, magic, or other resources is a strong
candidate to characterize the resource $R$, which relates non-trivially to
both entanglement and stabilizer rank. In general, being able to relate $\chi$
to a magnitude other than bipartite entanglement also opens up the field of
tensor networks to the use of other resources. The evident locality of $\chi$
is an obstacle for which resources can be used in this way, making a link
between magic and $\chi$ of our vector $\ket{\nu}$ a specially interesting
objective. We can also use stabilizer TNs as a practical tool to bound the
simulation hardness of a circuit based on the amount of T-gates it contains, a
research that is usually restricted to theoretical approaches [41]. The method
described in this article is available in a Python implementation
111https://github.com/bsc-quantic/stabilizer-TN that can simulate any circuit.
Thus, one can look into improving the efficiency of the implementation. An
obvious candidate is integration with the handling of tableaus as done by STIM
[43], the most performant Python approach to stabilizer simulation.
Acknowledgements: We want to thank Ema Puljak, Berta Casas and Axel Pérez-
Obiol for their comments on the manuscript, together with all members of BSC’s
Quantic group for their suggestions and support.
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## Annex
## Appendix A Lemmas and proofs
Here we show and prove the lemmas that generate the update rules in the main
text. We also include extra comments that make the notation and intuition
behind the formalism clearer.
###### Lemma 1.
For a given stabilizer basis $\mathcal{B}(S,D)$, any state $\ket{\psi}$ in an
n-dimensional Hilbert space $\mathcal{H}^{n}$ can be described as
$\ket{\psi}=\sum_{i}\nu_{i}d_{\hat{i}}\ket{\psi_{S}}$, where
$\hat{i}=(i_{1}\dots i_{n})$, $\nu_{i}$ are complex coefficients fulfilling
$\sum_{i}|\nu_{i}|^{2}=1$, and $d_{\hat{i}}=d_{1}^{i_{1}}\cdot
d_{2}^{i_{2}}\cdot\dots\cdot d_{n}^{i_{n}}$ with respect to the destabilizer
generators $d_{i}\in D$.
While this property underlies the use of stabilizers in error-correction, and
thus can be deduced with their formalism, it can also be seen very concisely
entirely within the formalism used in this paper.
Proof: We show that all $d_{\hat{i}}\ket{\psi_{S}}$ are i) normalized and ii)
mutually orthogonal, so they form an orthonormal basis, then that iii) the
space they generate is the same dimension as the full n-dimensional Hilbert
space.
* •
i) The basis states are normal:
$\bra{\psi_{S}}d_{\hat{i}}d_{\hat{i}}\ket{\psi_{S}}=\braket{\psi_{S}}{\psi_{S}}=1$,
using that $\delta_{\hat{i}}\in\mathcal{P}^{n}$ implies
$(\delta_{\hat{i}})^{2}=Id$.
* •
ii) The basis states are orthogonal: If we take two different states
$d_{\hat{i}}\ket{\psi_{S}}$, $d_{\hat{j}}\ket{\psi_{S}}$, then there is a
stabilizer generator $d_{k}$ from D that such that $i_{k}=1,j_{k}=0$ or
$i_{k}=0,j_{k}=1$. Taking the first case without loss of generality, the
stabilizer generator $s_{k}$ anticommutes with $d_{\hat{i}}$ and commutes with
$d_{\hat{j}}$, so that
$\bra{\psi_{S}}d_{\hat{i}}d_{\hat{j}}\ket{\psi_{S}}=\bra{\psi_{S}}d_{\hat{i}}d_{\hat{j}}s_{k}\ket{\psi_{S}}=-\bra{\psi_{S}}d_{\hat{i}}s_{k}d_{\hat{j}}\ket{\psi_{S}}=-\bra{\psi_{S}}s_{k}d_{\hat{i}}d_{\hat{j}}\ket{\psi_{S}}=-\bra{\psi_{S}}d_{\hat{i}}d_{\hat{j}}\ket{\psi_{S}}$
(13)
Therefore $\bra{\psi_{S}}d_{\hat{i}}d_{\hat{j}}\ket{\psi_{S}}=0$.
* •
iii) The basis generates a space of dimension $2^{n}$: since there are $n$
destabilizers $d_{i}$, $\hat{i}=(i_{1}\dots i_{n})$ can take $2^{n}$ different
values, so the basis $\\{d_{\hat{i}}\ket{\psi_{S}}\\}_{\hat{i}}$ has that many
elements.
$\hfill\square$
Other than the basic structure, we need to understand how arbitrary gates
modify $\ket{\psi}$ and $\ket{\nu}$. First, we check how different gates are
decomposed into the gates of the basis $\mathcal{B}(\mathcal{S},\mathcal{D})$.
Then, we find which operations they correspond to within this formalism. We
can see from the definition of the stabilizer basis (Eq. 2) that a
destabilizer $\delta_{\hat{j}}=d_{1}^{j_{1}}\dots d_{n}^{j_{n}}$ takes us from
one element of the basis to another.
$\delta_{\hat{j}}\ket{\psi}=\delta_{\hat{j}}\sum_{i=0}^{2^{n}-1}\nu_{i}d_{\hat{i}}\ket{\psi_{\mathcal{S}}}=\sum_{i=0}^{2^{n}-1}\nu_{i}d_{\hat{i}+\hat{j}}\ket{\psi_{\mathcal{S}}}=\sum_{i=0}^{2^{n}-1}\nu_{i+j}d_{\hat{i}}\ket{\psi_{\mathcal{S}}}.$
(14)
On the other hand, the multiplication of a stabilizer
$\sigma_{\hat{j}}=s_{1}^{j_{1}}\dots s_{n}^{j_{n}}$ introduces a sign
depending on the element of the basis due to anticommutation. Notice that,
because $d_{i}$ only anticommutes with $s_{i}$, checking for commutativity is
as simple as doing the inner product of their boolean vectors
$\hat{d}\cdot\hat{s}$.
$\sigma_{\hat{j}}\ket{\psi}=\sum_{i=0}^{2^{n}-1}\nu_{i}\sigma_{\hat{j}}d_{\hat{i}}\ket{\psi_{\mathcal{S}}}=\sum_{i=0}^{2^{n}-1}\nu_{i}(-1)^{\hat{i}\cdot\hat{j}}d_{\hat{i}}\sigma_{\hat{j}}\ket{\psi_{\mathcal{S}}}=\sum_{i=0}^{2^{n}-1}(-1)^{\hat{i}\cdot\hat{j}}\nu_{i}d_{\hat{i}}\ket{\psi_{\mathcal{S}}}.$
(15)
These are equivalent to $X$ and $Z$ operations, respectively, on the
computational basis. Therefore, on $\ket{\nu}$ we have:
$\delta_{\hat{d}}\ket{\psi}=X_{\hat{d}}\ket{\nu}\quad,\quad\sigma_{\hat{s}}\ket{\psi}=Z_{\hat{s}}\ket{\nu}.$
(16)
Since $\mathcal{S}\cup\mathcal{D}$ are a basis for $\mathcal{P}^{n}$, we can
decompose any operator as:
$\mathcal{U}=\sum_{i}\phi_{i}\delta_{\hat{d}_{i}}\sigma_{\hat{s}_{i}}.$ (17)
The previous observations tell us how to apply each factor individually, but
any decomposition with more than one term is more complicated. Observe that
the difference between the update on $\ket{\psi}$ and on $\ket{\nu}$ is only
the changing basis, therefore the transformation to $\ket{\nu}$ must also be a
unitary operation. This means that our tensor network representation can use
the same tools as with circuit simulation, even if the equivalency is not
trivial. The following lemma shows us one useful instance.
###### Lemma 2.
For a given stabilizer basis $\mathcal{B}(S,D)$, any unitary that can be
decomposed in the form
$\mathcal{U}=\phi_{1}\delta_{\hat{d}_{1}}\sigma_{\hat{s}_{1}}+\phi_{2}\delta_{\hat{d}_{2}}\sigma_{\hat{s}_{2}},$
(18)
is equivalent, in the stabilizer tensor network formalism, to a change of
basis with Clifford gates $\delta_{\hat{d}_{1}}\sigma_{\hat{s}_{1}}$ followed
by a single multi-qubit rotation over the X,Y and Z axes on $\ket{\nu}$:
$\mathcal{R}_{X_{I_{x}}Y_{I_{y}}Z_{I_{z}}}(2\theta)=\cos(\theta)I-i\sin(\theta)X_{I_{x}}Y_{I_{y}}Z_{I_{z}},$
(19)
with $\theta=\arccos{(Re(\phi_{1}))}$. Using $\circ_{h}$ 222Hadamard
multiplication is an element wise multiplication of two tensors $a$ and $b$ of
the same shape, such that the entries of the result $c$ follow: $c_{i_{1}\dots
i_{n}}=(a\circ_{\textit{h}}b)_{i_{1}\dots i_{n}}=a_{i_{1}\dots i_{n}}\cdot
b_{i_{1}\dots i_{n}}$. , the chosen axes $I_{x}$,$I_{y}$ and $I_{z}$ related
to
$\delta_{\hat{d}_{1}},\sigma_{\hat{s}_{1}},\delta_{\hat{d}_{2}},\sigma_{\hat{s}_{2}}$
as follows:
$I_{y}=(\hat{d}_{1}+\hat{d}_{2})\otimes_{h}(\hat{s}_{1}+\hat{s}_{2})\quad,\quad
I_{x}=(\hat{d}_{1}+\hat{d}_{2})+I_{y}\quad,\quad
I_{z}=(\hat{s}_{1}+\hat{s}_{2})+I_{y}$ (20)
Proof: To make notation a bit easier to read, we refer to operators
$\sigma_{\hat{s}_{i}}$,$\delta_{\hat{d}_{j}}$ as $\delta_{i},\sigma_{j}$ and
use $\delta_{i}\cdot\sigma_{j}$ to mean $\hat{s}_{i}\cdot\hat{d}_{j}$, except
where there might be ambiguity. This helps us keep track of the equations
without remembering which operator carries which index. Remember that
$\delta\cdot\sigma=1$ when the operators anticommute and $0$ when they
commute, and also that any (de)stabilizer is hermitian. It can be checked that
unitarity implies the following conditions:
$\begin{split}\mathcal{U}^{\dagger}\mathcal{U}=I\iff&(\phi_{1}^{*}\sigma_{1}^{\dagger}\delta_{1}^{\dagger}+\phi_{2}^{*}\sigma_{2}^{\dagger}\delta_{2}^{\dagger})(\phi_{1}\delta_{1}\sigma_{1}+\phi_{2}\delta_{2}\sigma_{2})=\phi_{1}^{*}\phi_{1}I+\phi_{2}^{*}\phi_{2}I+\phi_{2}^{*}\phi_{1}\sigma_{2}\delta_{2}\delta_{1}\sigma_{1}+\phi_{1}^{*}\phi_{2}\sigma_{1}\delta_{1}\delta_{2}\sigma_{2}=\\\
&=(\phi_{1}^{*}\phi_{1}+\phi_{2}^{*}\phi_{2})I+(\phi_{2}^{*}\phi_{1}(-1)^{(\sigma_{2}\cdot\delta_{2}+\delta_{1}\cdot\sigma_{2}+\delta_{2}\cdot\sigma_{1}+\sigma_{1}\cdot\delta_{1})}+\phi_{1}^{*}\phi_{2})\sigma_{1}\delta_{1}\delta_{2}\sigma_{2}=I\\\
\iff&\begin{cases}\phi_{1}\phi_{1}^{*}+\phi_{2}\phi_{2}^{*}=1\\\
\phi_{2}^{*}\phi_{1}(-1)^{(\delta_{1}+\delta_{2})\cdot(\sigma_{1}+\sigma_{2})}=-\phi_{1}^{*}\phi_{2}\end{cases}.\end{split}$
(21)
The first condition tells us that we can rewrite the coefficients with
trigonometric functions. We can also factorize the complex phase like
$\phi_{1}=e^{i\varphi_{1}}\cos(\theta)$ and
$\phi_{2}=e^{i\varphi_{1}}e^{i\omega}\sin(\theta)$, and ignore the term
$e^{i\varphi_{1}}$ as a global phase. Substituting this in the second
condition:
$\begin{split}e^{-i\omega}\sin(\theta)\cos(\theta)(-1)^{(\delta_{1}+\delta_{2})\cdot(\sigma_{1}+\sigma_{2})}&=-\cos(\theta)e^{i\omega}\sin(\theta)\rightarrow\\\
\rightarrow
e^{2i\omega}=(-1)^{(\delta_{1}+\delta_{2})\cdot(\sigma_{1}+\sigma_{2})+1}\rightarrow
e^{i\omega}&=\pm(-i)^{(\delta_{1}+\delta_{2})\cdot(\sigma_{1}+\sigma_{2})+1}.\end{split}$
(22)
We can also rewrite the unitary as:
$\mathcal{U}=(\cos(\theta)I+e^{i\omega}\sin(\theta)\delta_{2}\sigma_{2}\>\delta_{1}\sigma_{1})\>\delta_{1}\sigma_{1}=(\cos(\theta)I+e^{i\omega}\sin(\theta)(-1)^{\delta_{1}\cdot\sigma_{2}}\delta_{2}\delta_{1}\>\sigma_{2}\sigma_{1})\>\delta_{1}\sigma_{1}.$
(23)
As we have seen in Eqs. 14,15, we treat a (de)stabilizer operator as a set of
Z (X) gates. Therefore, the Pauli operator $\delta_{1}\sigma_{1}$ to the right
is strictly a Clifford update (in fact, with only X,Y and Z gates) that we can
apply first. It’s left to check that the remaining factor behaves like a
rotation.
$\tilde{\mathcal{U}}=\cos(\theta)I+e^{i\omega}\sin(\theta)(-1)^{\delta_{1}\cdot\sigma_{2}}\delta_{2}\delta_{1}\>\sigma_{2}\sigma_{1}$
(24)
Similarly, applying $\sigma_{\hat{s}_{i}}$ ($\delta_{\hat{d}_{j}}$) is
equivalent to the transformation $Z_{\hat{s}_{i}}$ ($X_{\hat{d}_{j}}$), and
doing two such transformations consecutively is as simple as adding the
boolean vectors: $Z_{\hat{s}_{j}}Z_{\hat{s}_{i}}=Z_{\hat{s}_{j}+\hat{s}_{i}}$.
With $I_{x}$,$I_{y}$ and $I_{z}$ defined as above, one can check that
$\begin{split}I_{x}+I_{y}=\hat{\delta}_{1}+\hat{\delta}_{2}&\rightarrow
X_{I_{x}+I_{y}}=\delta_{2}\delta_{1}\\\
I_{y}+I_{z}=\hat{\sigma}_{1}+\hat{\sigma}_{2}&\rightarrow
Z_{I_{y}+I_{z}}=\sigma_{2}\sigma_{1}\\\
I_{y}=(\hat{\delta}_{1}+\hat{\delta}_{2})\circ_{\textit{h}}(\hat{\sigma}_{1}+\hat{\sigma}_{2})&\rightarrow|I_{y}|=\sum_{a\in
I_{y}}a=(\hat{\delta}_{1}+\hat{\delta}_{2})\cdot(\hat{\sigma}_{1}+\hat{\sigma}_{2})\end{split},$
(25)
which is almost the form in Eq.19. Notice that $I_{x}$ has the unique 1s of
$\hat{d}_{1}+\hat{d}_{2}$, with $0$s elsewhere, $I_{z}$ those of
$\hat{s}_{1}+\hat{s}_{2}$, whereas $I_{y}$ has the common $1$s between
$\hat{d}_{1}+\hat{d}_{2}$ and $\hat{s}_{1}+\hat{s}_{2}$, with the Hadamard
product (element-wise multiplication) enabling the closed form description of
$I_{y}$. Since $X\cdot Z=-iY$, we can rewrite 19 as
$\mathcal{R}_{X_{I_{x}}Y_{I_{y}}Z_{I_{z}}}(2\theta)=\cos(\theta)I+\sin(\theta)(-i)^{|I_{y}|+1}X_{I_{x}+I_{y}}Z_{I_{y}+I_{z}}.$
(26)
Putting eq.25 and eq.22 together means that the sinus term has the correct
phase to be a unitary, so the proposed transformation is indeed a rotation and
its coefficients relate to the original unitary as stated. Since the sign can
always be changed with the angle of rotation, we are done. $\hfill\square$
The values $v_{i}$ of the vectors $I_{x}$ ($I_{y}$,$I_{z}$) indicate that we
rotate qubit $i$ over $X$ ($Y$,$Z$) if $v_{i}=1$, and we do nothing if
$v_{i}=0$. Notice that this gate can be implemented with a cascade of CNOT
gates on the affected qubits and a single qubit rotation, plus the appropriate
basis changes from $X$ to $Y,Z$ [40]. In particular, we implement the rotation
on the innermost affected qubit and add CNOT cascades to each side. And
example can be seen in Fig. 4. Lemma 2 is very useful because the basic
$R_{X}$, $R_{Y}$ and $R_{Z}$ gates have this form, so we know how to update
with any non-clifford single qubit gate. With the $\ket{\nu}$ notation, lemma
2 can be summarized with:
$\mathcal{U}\ket{\psi}=(\phi_{1}\delta_{1}\sigma_{1}+\phi_{2}\delta_{2}\sigma_{2})\ket{\psi}=\mathcal{R}_{X_{I_{x}}Y_{I_{y}}Z_{I_{z}}}(2\theta)\ket{\nu}$
(27)
In the algorithm, we have to check the value of $\delta_{1}\cdot\sigma_{2}$ to
get the sign of the transformation angle right.
We also have a corollary that tells us what the free operations are, which is
needed to identify the resource:
###### Corollary 2.1.
In the context of stabilizer simulation as a resource theory, the free
operations depend on the basis $\mathcal{B}(S,D)$ and correspond to
$\mathcal{U}=\text{cos}(\theta/2)\alpha\delta\sigma+i\text{sin}(\theta/2)d_{i}s_{i}\alpha\delta\sigma$,
where $d_{i}\in D$, $s_{i}\in S$ are generators and $P=\alpha\delta\sigma$ is
the decomposition of a Pauli matrix $P$ into $\mathcal{B}(S,D)$.
Proof: Since it fits the conditions of lemma 2, we apply first the Pauli
operator $P$, which consists only of Clifford gates. Then, since
$d_{i}s_{i}\ket{\psi_{S}}=d_{i}\ket{\psi_{S}}$ is an element of the basis,
using 20, we see that $\cos(\theta/2)+i\sin(\theta/2)d_{i}s_{i}$ is equivalent
to a single qubit rotation $R_{X}(-\theta)$ on $\ket{\nu}$, which does not
increase its bond dimension. $\hfill\square$
For an observable $\mathcal{O}=\alpha\delta_{\hat{n}}\sigma_{\hat{m}}$ we
have:
$\begin{split}\braket{\psi}{\mathcal{O}}{\psi}=\bra{\psi_{\mathcal{S}}}\sum_{j}\nu_{\hat{j}}^{*}d_{\hat{j}}^{*}\>\alpha\delta_{\hat{n}}\sigma_{\hat{m}}\sum_{i}\nu_{\hat{i}}d_{\hat{i}}\ket{\psi_{\mathcal{S}}}&=\sum_{i,j}\alpha\nu_{\hat{j}}^{*}\nu_{\hat{i}}(-1)^{{\hat{m}}\cdot{\hat{i}}}\braket{\psi_{\mathcal{S}}}{d_{\hat{j}}d_{\hat{n}}d_{\hat{i}}}{\psi_{\mathcal{S}}}=\\\
=\sum_{i,j}\alpha\nu_{\hat{j}}^{*}\nu_{\hat{i}}(-1)^{{\hat{m}}\cdot{\hat{i}}}\delta^{\prime}_{\hat{j},\hat{i}+\hat{n}}&=\alpha\sum_{i}(-1)^{{\hat{m}}\cdot{\hat{i}}}\nu^{*}_{\hat{i}+\hat{n}}\nu_{\hat{i}},\end{split}$
(28)
where $\delta^{\prime}$ is a Kronecker delta. This is much simpler on
$\ket{\nu}$:
$\braket{\psi}{\mathcal{O}}{\psi}=\alpha\braket{\nu}{X_{\hat{n}}Z_{\hat{m}}}{\nu}.$
(29)
Notice that the $\alpha$ phase comes from forcing a specific decomposition on
$\mathcal{B}(\mathcal{S},\mathcal{D})$, which might mean we have to write
$XZ=-iY$ instead of $Y$ directly; it does not mean we allow non-physical
observables, that is, with non-real expected values. A measurement of this
observable $\mathcal{O}$ projects the state $\ket{\psi}$:
$\ket{\psi}\rightarrow\frac{I\pm\mathcal{O}}{2}\ket{\psi}.$ (30)
The sign is $+$ ($-$) when projecting to the positive $\ket{\psi_{+}}$
(negative $\ket{\psi_{-}}$) eigenstate. We can calculate
$\braket{\mathcal{O}}$ with Eq. 29 and randomly decide the output with
probability $p=\frac{1+\braket{\mathcal{O}}}{2}$ for $\ket{\psi_{+}}$ and
$1-p=\frac{1-\braket{\mathcal{O}}}{2}$ for $\ket{\psi_{-}}$. Then the
following lemma shows how to update the coefficients.
###### Lemma 3.
For a given stabilizer basis $\mathcal{B}(S,D)$ and an observable
$\mathcal{O}$ that decomposes as
$\mathcal{O}=\alpha\delta_{\hat{n}}\sigma_{\hat{m}},$ (31)
the projection $\braket{I\pm\mathcal{O}}{2}$ onto the positive (negative)
eigenstate is equivalent to the following non-unitary operation on
$\ket{\nu}$:
$P_{k}\cdot\tilde{\mathcal{R}}_{X_{I_{x}}Y_{I_{y}}Z_{I_{z}}}=P_{k}\cdot\left(\frac{1}{\sqrt{2}}I\pm\frac{\alpha(-i)^{|I_{y}|}}{\sqrt{2}}X_{I_{x}}Y_{I_{y}}Z_{I_{z}}\right),$
(32)
where $k$ is the position of the first $1$ in $\hat{n}$, $P_{k}$ is the
projector $\ket{0}\bra{0}$ on qubit $k$, and the choice of rotation axes is
given by $\delta_{\hat{n}},\sigma_{\hat{m}}$ as
$I_{y}=\hat{n}\circ_{h}\hat{m}\quad,\quad I_{x}=\hat{n}+I_{y}\quad,\quad
I_{z}=\hat{m}+I_{y}$ (33)
The resulting state $\ket{\psi^{\prime}}$ is a valid quantum state when
renormalized as
$\sqrt{\frac{2}{1\pm\braket{\psi}{\mathcal{O}}{\psi}}}\ket{\psi^{\prime}}$.
Proof: We can expand the projection of $\frac{I\pm\mathcal{O}}{2}$ into:
$\frac{I\pm\mathcal{O}}{2}\ket{\psi}=\frac{1}{2}\sum_{\hat{i}}(I\pm\alpha\delta_{\hat{n}}\sigma_{\hat{m}})\nu_{\hat{i}}d_{\hat{i}}\ket{\psi_{\mathcal{S}}}=\frac{1}{2}\sum_{\hat{i}}(\nu_{\hat{i}}\pm\alpha(-1)^{\hat{i}\cdot\hat{m}}\nu_{\hat{i}+\hat{n}})d_{\hat{i}}\ket{\psi_{\mathcal{S}}}.$
(34)
Then we must consider the update to the stabilizer basis. When $\hat{n}=0$, we
are projecting onto a stabilizer of $\ket{\psi_{\mathcal{S}}}$ and there is no
update to $\mathcal{B}(\mathcal{S},\mathcal{D})$. In this case we are directly
left with:
$\frac{I\pm\mathcal{O}}{2}\ket{\psi}=\frac{1}{2}\sum_{\hat{i}}\nu_{\hat{i}}(1\pm\alpha(-1)^{\hat{i}\cdot\hat{m}})d_{\hat{i}}\ket{\psi_{\mathcal{S}}}\quad\text{if
}\hat{n}=0.$ (35)
In the case $\delta_{\hat{n}}\neq 0$, it was shown in [19] how to update the
basis in terms similar to our Eq. 34. Adapting those results to our notation
for $\ket{\nu}$, we get the new basis
$\mathcal{B^{\prime}}(\mathcal{S}^{\prime},\mathcal{D}^{\prime})$ and :
$\frac{I\pm\mathcal{O}}{2}\ket{\psi}=\sum_{\hat{i}}\left[\frac{1}{2}(\pm\alpha(-1)^{\hat{i}\cdot\hat{m}})^{\hat{i}_{k}}\nu_{\hat{i}}\right]\>d_{\hat{i}+\hat{i}_{k}\cdot\hat{n}}^{\prime}\ket{\psi_{\mathcal{S}^{\prime}}}\quad\text{if
}\hat{n}\neq 0,$ (36)
where $k$ is the position of the first $1$ in $\hat{n}$ and $\hat{i}_{k}$ the
$k^{th}$ element of $\hat{i}$. Notice that $\hat{i}_{k}=0$ implies
$(\hat{i}+\hat{n})_{k}=1$, that is,
$d_{\hat{i}+\hat{i}_{k}\cdot\hat{n}}=d_{\hat{i}}$ and
$d_{(\hat{i}+\hat{n})+(\hat{i}+\hat{n})_{k}\cdot\hat{n}}=d_{(\hat{i}+\hat{n})+\hat{n}}$,
so the coefficient
$\frac{1}{2}(\alpha(-1)^{\hat{i}\cdot\hat{m}})^{\hat{i}_{k}}\nu_{\hat{i}}$
stays in $d_{\hat{i}}\ket{\psi_{\mathcal{S}}}$ and
$\frac{1}{2}(\alpha(-1)^{(\hat{i}+\hat{n})\cdot\hat{m}})^{(\hat{i}+\hat{n})_{k}}\nu_{\hat{i}+\hat{n}}$
moves to
$d_{(\hat{i}+\hat{n})+\hat{n}}\ket{\psi_{\mathcal{S}}}=d_{\hat{i}}\ket{\psi_{\mathcal{S}}}$,
leaving $\nu_{\hat{i}+\hat{n}}$ empty; for $\hat{i}_{k}=1$ both coefficients
concentrate on $\nu_{\hat{i}+\hat{n}}$ and leave $\nu_{\hat{i}}$ empty
instead: the measurement halves the non-zero coefficients whenever
$\hat{n}\neq 0$. This means we can rewrite the above as:
$\frac{1\pm\mathcal{O}}{2}\ket{\psi}=\sum_{\hat{i}}\delta_{\hat{i}_{k},0}\left(\frac{1}{\sqrt{2}}\nu_{\hat{i}}\pm\frac{\alpha(-1)^{\hat{i}\cdot\hat{m}}}{\sqrt{2}}\nu_{\hat{i}+\hat{n}}\right)d_{\hat{i}}\ket{\psi_{\mathcal{S}}}\quad\text{if
}n\neq 0.$ (37)
where the Kronecker delta filters the non-zero coefficients. Defining
$\hat{i}_{k}\equiv 0$ when $\hat{n}=0$, we see the only difference between Eq.
35 and Eq. 37 is a factor of $\sqrt{2}$. Since we have to normalize at the end
anyway, we can rejoin both cases and proceed with Eq. 37. In terms of
$\ket{\nu}$, we can prepare the superposition on all states, which looks
almost like a rotation:
$\tilde{\mathcal{R}}\ket{\nu}=\tilde{\mathcal{R}}\sum_{i}\nu_{\hat{i}}=\sum_{i}\frac{1}{\sqrt{2}}\nu_{\hat{i}}\pm\frac{\alpha(-1)^{\hat{i}\cdot\hat{m}}}{\sqrt{2}}\>\nu_{\hat{i}+\hat{n}},$
(38)
and remove the duplicate coefficients afterwards with a projection on the
$\ket{0}$ state of qubit $k$:
$P_{k}=I_{0}\otimes\dots I_{k-1}\otimes\begin{pmatrix}1&0\\\
0&0\end{pmatrix}\otimes I_{k+1}\dots\otimes I_{n}\equiv\ket{0}\bra{0}_{k},$
(39)
so that
$\frac{I+\mathcal{O}}{2}\ket{\nu}=P_{k}\>\tilde{\mathcal{R}}\ket{\nu}$. This
transformation is similar to the rotation in lemma 2, but removing the $i$
phase. We can reuse the reasoning there to find:
$\tilde{\mathcal{R}}_{X_{I_{x}}Y_{I_{y}}Z_{I_{z}}}=\frac{1}{\sqrt{2}}I\pm\frac{\alpha(-i)^{|I_{y}|}}{\sqrt{2}}X_{I_{x}}Y_{I_{y}}Z_{I_{z}},$
(40)
where $I_{x},I_{y},I_{z}$ are related to $\delta_{\hat{n}},\sigma_{\hat{m}}$
exactly as in Eq. 33. Now, $I_{x}$ has the unique 1s of $\hat{n}$, with $0$s
elsewhere, $I_{z}$ those of $\hat{m}$, whereas $I_{y}$ has the common $1$s
between $\hat{n}$ and $\hat{n}$. Although it is not a unitary operation, it’s
equivalent to a projection, so the output is not normalized but is otherwise a
valid state. To find the normalization term, we reuse Eq. 28:
$\braket{\psi}{\mathcal{O}}{\psi}=\alpha\sum_{\hat{i}}(-1)^{\hat{i}\hat{m}}\nu^{*}_{\hat{i}+\hat{n}}\nu_{\hat{i}}=\alpha^{*}\sum_{\hat{i}}(-1)^{\hat{i}\hat{m}}\nu^{*}_{\hat{i}}\nu_{\hat{i}+\hat{n}}.$
(41)
The second equality is a consequence of $\braket{\psi}{\mathcal{O}}{\psi}$
being real. Now:
$\begin{split}&\qquad\qquad\qquad\qquad\qquad\mathcal{N}^{2}=\left(\bra{\psi_{\mathcal{S}}}\frac{1\pm\mathcal{O}^{\dagger}}{2}\right)\left(\frac{1\pm\mathcal{O}}{2}\ket{\psi_{\mathcal{S}}}\right)=\\\
&=\sum_{\hat{i},\hat{j}}\bra{\psi_{\mathcal{S}}}d_{\hat{j}}\left(\frac{1}{\sqrt{2}}\nu_{\hat{j}}^{*}\pm\frac{\alpha^{*}(-1)^{\hat{j}\cdot\hat{m}}}{\sqrt{2}}\nu_{\hat{j}+\hat{n}}^{*}\right)\delta_{\hat{j}_{k},0}\delta_{\hat{i}_{k},0}\left(\frac{1}{\sqrt{2}}\nu_{\hat{i}}\pm\frac{\alpha^{*}(-1)^{\hat{i}\cdot\hat{m}}}{\sqrt{2}}\nu_{\hat{i}+\hat{n}}\right)d_{\hat{i}}\ket{\psi_{\mathcal{S}}}=\\\
&=\sum_{\hat{i},\hat{j}}\delta_{\hat{j}_{k},0}\delta_{\hat{i}_{k},0}\left(\frac{1}{\sqrt{2}}\nu_{\hat{j}}^{*}\pm\frac{\alpha^{*}(-1)^{\hat{j}\cdot\hat{m}}}{\sqrt{2}}\nu_{\hat{j}+\hat{n}}^{*}\right)\left(\frac{1}{\sqrt{2}}\nu_{\hat{i}}\pm\frac{\alpha^{*}(-1)^{\hat{i}\cdot\hat{m}}}{\sqrt{2}}\nu_{\hat{i}+\hat{n}}\right)\braket{\psi_{\mathcal{S}}}{d_{\hat{j}}d_{\hat{i}}}{\psi_{\mathcal{S}}}=\\\
&=\sum_{\hat{i},\hat{j}}\delta_{\hat{j}_{k},0}\delta_{\hat{i}_{k},0}\left(\frac{1}{2}\nu_{\hat{j}}^{*}\nu_{\hat{i}}\pm\frac{\alpha^{*}(-1)^{\hat{j}\cdot\hat{m}}}{2}\nu_{\hat{j}+\hat{n}}^{*}\nu_{\hat{i}}\pm\frac{\alpha(-1)^{\hat{i}\cdot\hat{m}}}{2}\nu_{\hat{j}}^{*}\nu_{\hat{i}+\hat{n}}+\frac{|\alpha|^{2}}{2}v_{\hat{j}+\hat{n}}^{*}v_{\hat{i}+\hat{n}}\right)\delta_{\hat{i},\hat{j}}=\\\
&\quad=\sum_{\hat{i}}\delta_{\hat{i}_{k},0}\left(\frac{1}{2}|\nu_{\hat{i}}|^{2}+\frac{|\alpha|^{2}}{2}v_{\hat{i}+\hat{n}}^{*}v_{\hat{i}+\hat{n}}\pm\frac{\alpha^{*}(-1)^{\hat{i}\cdot\hat{m}}}{2}\nu_{\hat{i}+\hat{n}}^{*}\nu_{\hat{i}}\pm\frac{\alpha(-1)^{\hat{i}\cdot\hat{m}}}{2}\nu_{\hat{i}}^{*}\nu_{\hat{i}+\hat{n}}\right)=\\\
&\qquad\qquad\qquad\quad=\frac{1}{2}\sum_{\hat{i}}\delta_{\hat{i}_{k},0}|\nu_{\hat{i}}|^{2}+\frac{1}{2}\sum_{\hat{i}}\delta_{\hat{i}_{k},0}|\nu_{\hat{i}+\hat{n}}|^{2}\pm\sum_{\hat{i}}\delta_{\hat{i}_{k},0}\braket{\psi}{\mathcal{O}}{\psi}=\\\
&\qquad\qquad\qquad=\frac{1}{2}\sum_{\hat{i}}(\delta_{\hat{i}_{k},0}+\delta_{(\hat{i}+\hat{n})_{k},0})|\nu_{i}|^{2}\pm\frac{1}{2}\braket{\psi}{\mathcal{O}}{\psi}=\frac{1\pm\braket{\psi}{\mathcal{O}}{\psi}}{2}.\end{split}$
(42)
We used that $\hat{i}_{k}=0\leftrightarrow(\hat{i}+\hat{n})_{k}=1$ again, so
the sum of $(\delta_{\hat{i}_{k},0}+\delta_{(\hat{i}+\hat{n})_{k},0})$ is
always $1$. Because $(\hat{i}+\hat{n})+\hat{n}=\hat{i}$, each contribution to
the sum appears twice, so the Kronecker delta selects one of each pair and is
thus equivalent to the factor $\frac{1}{2}$ in front of the expected value.
This equation proves we have a valid quantum state in all cases with a
renormalization term of:
$\mathcal{N}=\sqrt{\frac{1\pm\braket{\psi}{\mathcal{O}}{\psi}}{2}}.$ (43)
$\hfill\square$
We can implement the rotation we found with a CNOT cascade similarly to Eq.
19, but instead of a central $R_{X}$ rotation we need the following one-qubit
(non-unitary) operation:
$\tilde{\mathcal{R}}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&\pm\alpha(-i)^{|I_{y}|}\\\
\pm\alpha(-i)^{|I_{y}|}&1\end{pmatrix}.$ (44)
## Appendix B Entangling power of an arbitrary gate
There is not a unique way to define the entangling power of a gate. There are
many notions of multipartite entanglement that one can use, and the
entanglement of the final state depends on the initial state. This means we
have to look at an average over all possible states or a worst/best case,
which can also complicate the calculation of the chosen metric. One of the
first proposals [45] used the average linear entropy over all possible product
states $\ket{\psi_{A}}\otimes\ket{\phi_{B}}$ on a preset bipartition $A,B$ of
the whole space:
$e(U)\coloneqq\overline{E(U\ket{\psi_{A}}\otimes\ket{\phi_{B}})}^{\psi_{A},\phi_{B}}.$
(45)
This approach has appeared specially useful when studying entangling power on
mixed states [46]. Others are based on unitary evolution and focus on the norm
of an infinitesimal transformation, such as [47, 48]. Metrics other than
entanglement have also been used, such as quantum discord [49]. Since we deal
with an MPS, we focus on the bipartite entanglement case, and instead of the
average, we find the maximum bond dimension $\chi^{\prime}$ needed in our MPS
after applying a gate on an MPS that had maximum bond dimension $\chi$. We use
the Schmidt decomposition of unitaries [50], which has been used previously to
characterize arbitrary gates [51] and tells us we can decompose any unitary as
$\mathcal{U}=\sum_{i=1}^{k}s_{i}A_{i}\otimes B_{i},$ (46)
where $s_{i}\geq 0$, $\sum_{i}|s_{i}|^{2}=1$ and $A_{i},B_{i}$ are an
orthogonal operator basis [52]. When applying the rotation $\mathcal{R}$ in
Eq. 6, we can focus on any arbitrary bond of our MPS by decomposing it as in
Fig. 3. This way, we only need to check how the gates that cross the chosen
bond affect $\chi$.
Figure 3: Possible decomposition of the rotation 6 showcasing how the gates
affect $\chi$ for different bonds. All gates that can be grouped into a
unitary on partition A (B) are irrelevant. The bond in blue starts with
$\chi_{b}$ and each CNOT can increase it by at most a factor of $2$, reaching
$\chi_{b}^{\prime}\leq 4\chi_{b}$ after the transformation. For the orange
bond, starting with $\chi_{o}$, the implementation of a CNOT across far away
qubits on an MPS requires that we apply a SWAP gate at each line crossing,
which increases $\chi$ by at most $4$, so that at the end we have
$\chi_{o}^{\prime}\leq 16\chi_{o}$. This argument is independent of the
initial bonds.
With the Schmidt decomposition [30] of the initial state $\ket{\psi}$ as:
$\ket{\psi}=\sum_{i=1}^{\chi}\lambda_{i}\ket{\psi_{A}^{i}}\otimes\ket{\psi_{B}^{i}},$
(47)
limited to rank $\chi$, applying a two qubit gate with Schmidt number $k$
means we get that $\chi^{\prime}$ is at most:
$\mathcal{U}_{k}\ket{\psi}=\sum_{j=1}^{k}\sum_{i=1}^{\chi}(s_{j}\lambda_{i})A_{j}\ket{\psi_{A}^{i}}\otimes
B_{j}\ket{\psi_{B}^{i}}.=\sum_{i=1}^{k\chi}\tilde{\lambda}_{i}\ket{\phi_{A}^{i}}\otimes\ket{\phi_{B}^{i}}$
(48)
The final form is still a valid Schmidt decomposition thanks to the
orthonormality of $A_{i},B_{i}$ and $\sum_{i}s_{k}^{2}=1$. A CNOT gate has
Schmidt number $k=2$ [52], so the set of two CNOTs that are applied to a
particular bond can increase at most $\chi^{\prime}\leq 4\chi$. Counter to
intuition, the worst-case scenario when using an MPS is not an update that
affects all qubits, but one that affects qubits that are far apart: a CNOT
gate over tensors that are not neighbours is implemented with SWAPs on our
MPS. A SWAP gate has Schmidt rank 4, so Eq. 48 gives the bound
$\chi^{\prime}\leq 16\chi$ instead, as stated in the main text and fitting the
simulations in 2. Since this maximum is a consequence of SWAP gates, TN
geometries other than MPS that adapt to the connectivity of the simulated
circuit can reduce the bound to $4\chi$; this entails a bigger complexity in
the TN contraction, as is the case in general for higher dimensional networks.
## Appendix C Stabilizer TN update example
It is useful to illustrate an example of a coefficient update with a non-
Clifford gate in terms of $\ket{\nu}$. We take an arbitrary basis
$\mathcal{B}(\mathcal{S},\mathcal{D})$ for $5$ qubits and a unitary that
decomposes as
$\mathcal{U}=\frac{\sqrt{3}}{2}\delta_{\hat{d}_{1}}\sigma_{\hat{s}_{1}}+\frac{1}{2}\delta_{\hat{d}_{2}}\sigma_{\hat{s}_{2}}=\left(\cos(\pi/6)+\sin(\pi/6)\delta_{\hat{d}_{2}}\sigma_{\hat{s}_{2}}\delta_{\hat{d}_{1}}\sigma_{\hat{s}_{1}}\right)\delta_{\hat{d}_{1}}\sigma_{\hat{s}_{1}}=\left(\cos(\pi/6)-\sin(\pi/6)(\delta_{\hat{d}_{2}}\delta_{\hat{d}_{1}})(\sigma_{\hat{s}_{2}}\sigma_{\hat{s}_{1}})\right)\delta_{\hat{d}_{1}}\sigma_{\hat{s}_{1}}.$
(49)
Let us assume that the vector representation of $\delta,\sigma$ is:
$\begin{split}\left.\begin{array}[]{c}\hat{d}_{1}=(1,1,0,0,0)\\\
\hat{d}_{2}=(1,0,0,1,0)\end{array}\right\\}&\rightarrow\hat{d}_{2}\hat{d}_{1}=(0,1,0,1,0)\\\
\left.\begin{array}[]{c}\hat{s}_{1}=(0,0,0,1,0)\\\
\hat{s}_{2}=(0,0,1,0,0)\end{array}\right\\}&\rightarrow\hat{s}_{2}\hat{s}_{1}=(0,0,1,1,0)\end{split},$
(50)
which also implies $\hat{d}_{1}\cdot\hat{s}_{2}=1$. Then using Eqs. 14,15 and
the $\ket{\nu}$ notation we can rewrite
$\mathcal{U}\ket{\nu}=\left(\cos(\pi/6)-\sin(\pi/6)(X_{1}X_{3})(Z_{2}Z_{3})\right)X_{0}X_{1}Z_{3}\ket{\nu}=\left(\cos(\pi/6)+i\sin(\pi/6)X_{1}Y_{3}Z_{2}\right)X_{0}X_{1}Z_{3}\ket{\nu}$
(51)
We see that this fits Eq. 6. Then the coefficients can be updated with the
corresponding multiqubit rotation, which we show in Fig. 4. We also show the
decomposition that we have used in our Python implementation, which uses two
cascades centred on the middle qubit instead of a single CNOT cascade.
Figure 4: Example of coefficient update for the unitary described in Eq. 49.
Horizontal lines are ”qubit” sites in the traditional MPS tensor network
representation of a quantum state. Gates are applied from left to right. The
resulting TN of this example is more entangled than the initial $\ket{\nu}$.
Most circuit simulations compile an input gate set into a specific set of
gates, since practical realizations of quantum computers are similarly bound
by a limited set of native gates. Our simulation approach can handle any
circuit with a $\\{CNOT,R_{X},R_{Y},R_{Z}\\}$ decomposition, so it’s
compatible with most circuits despite the limitations of lemma 2. Other
characterizations are possible and still compatible with the stabilizer TN
framework, but the implementation of unitaries of arbitrary decomposition is
left for future work.
## Appendix D Tableau update rules
Our formalism relies on the update rules for the original tableau:
$\left(\begin{array}[]{ccc|ccc|c}x_{1,1}&\cdots&x_{1,n}&z_{1,1}&\cdots&x_{1,n}&r_{1}\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
x_{n,1}&\cdots&x_{n,n}&z_{n,1}&\cdots&z_{n,n}&r_{n}\\\ \hline\cr
x_{n+1,1}&\cdots&x_{n+1,n}&z_{n+1,1}&\cdots&x_{n+1,n}&r_{n+1}\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
x_{2n,1}&\cdots&x_{2n,n}&z_{2n,1}&\cdots&z_{2n,n}&r_{2n}\\\
\end{array}\right).$ (52)
This section follows [37] exactly. It is included to give a self-contained
explanation of our formalism’s update rules, since the original update rules
are also used. Considering that a measurement over $X$ or $Y$ basis can be set
to a Clifford operator followed by a $Z$ basis measurement, the basic
operations we need are, using always base 2:
1. 1.
CNOT operator with control qubit $a$ and target qubit $b$: For every row $i$,
update entries $\textit{x}_{ib}$, $\textit{z}_{ia}$, $\textit{r}_{i}$ as
$r_{i}:=r_{i}\oplus x_{ia}z_{ib}(x_{ib}\oplus z_{ia}\oplus 1)\quad;\quad
x_{ib}:=x_{ib}\oplus x_{ia}\quad;\quad z_{ia}:=z_{ia}\oplus z_{ib}.$ (53)
2. 2.
Hadamard operator on qubit $a$: For every row $i$, update entries
$\textit{x}_{ia}$, $\textit{z}_{ia}$, $\textit{r}_{i}$ as
$r_{i}:=r_{i}\oplus x_{ia}z_{ia}\quad;\quad x_{ia}:=z_{ia}\quad;\quad
z_{ia}:=x_{ia}.$ (54)
3. 3.
Phase operator on qubit $a$: For every row $i$, update entries
$\textit{z}_{ia}$, $\textit{r}_{i}$ as
$r_{i}:=r_{i}\oplus x_{ia}z_{ia}\>;\>z_{ia}:=z_{ia}\oplus x_{ia}.$ (55)
4. 4.
Measurement over $Z$ basis on qubit $a$: The update is different whether the
measurement commutes with the current stabilizers. If it does, we do not need
to update the tableau. Then, for each row $i\in\\{1\dots 2n\\}$, we do the
operation rowsum(i,t) on an auxiliary row $t$ that starts with all zeroes, and
the phase of $t$ tells us if we measure 0 or 1. Otherwise, it must
anticommute, so the outcome $r$ is random with equal probability, but we must
change the tableau. To do so, we choose the row $i$ of one of the
anticommuting stabilizers $H$ (those with $x_{ia}=1$), do the operation
rowsum(i,h) for all $h\in H\setminus\\{i\\}$, and finally we add the
observable $Z_{a}$ to the list of stabilizers at $i$ and store the former
stabilizer $i$ as a destabilizer at row $i-n$.
While not technically an operation that appears in the circuit, we also need
to define what this ”rowsum” operation does:
1. 5.
rowsum(a,b): Sets generator $a$ to $a+b$, that is, $x_{aj}=x_{aj}\oplus
x_{bj}$ and $z_{aj}=z_{aj}\oplus z_{bj}$ for all $j\in\\{1\dots n\\}$, while
properly changing its phase too. To do so, we need a function
$g(x_{1},z_{1},x_{2},z_{2})$ that returns the exponent of the phase we get
from multiplying $x_{1}z_{1}\cdot x_{2}z_{2}$. That is:
$\begin{split}x_{1}=z_{1}=0&\rightarrow g=0\\\ x_{1}=z_{1}=1&\rightarrow
g=z_{2}-x_{2}\\\ x_{1}=1,z_{1}=0&\rightarrow g=z_{2}(2x_{2}-1)\\\
x_{1}=0,z_{1}=1&\rightarrow g=x_{2}(1-2z_{2})\end{split}.$ (56)
Then the phase $r_{a}$ is:
$\begin{split}r_{a}=0\text{ if
}2r_{a}+2r_{b}+\sum_{j=1}^{n}g(x_{bj},z_{bj},x_{aj},z_{aj}\equiv 0\>(\text{mod
}4)\\\ r_{a}=1\text{ if
}2r_{a}+2r_{b}+\sum_{j=1}^{n}g(x_{bj},z_{bj},x_{aj},z_{aj}\equiv 2\>(\text{mod
}4)\\\ \end{split}.$ (57)
|
# Infrared and Optical Detectability of Dyson Spheres at White Dwarf Stars
B. Zuckerman Department of Physics and Astronomy, University of California,
Los An geles, CA 90095-1562, USA
###### Abstract
It has been hypothesized that advanced technological civilizations will
construct giant space colonies and supporting infrastructures to orbit about
their home stars. With data from recent satellites that operate at infrared
and optical wavelengths (Spitzer, WISE, TESS, Kepler), in company with a few
modest assumptions, it is now possible to begin to constrain observationally
the frequency of such space-based civilizations in our Milky Way Galaxy.
Key words: white dwarfs – infrared: stars – astrobiology
## 1 Introduction
The number of technological civilizations in our Milky Way Galaxy is a topic
of endless debate and speculation. It is generally agreed that, if such
civilizations are abundant, then they must be long lived. Thus, such
civilizations will ultimately have to deal with the evolution of their stars,
first into red giants and then into white dwarfs. Two possible paths would be
mass migration to another star (Zuckerman 1985; Hansen & Zuckerman 2021) or
remaining at the evolving star and accommodating to the changes in luminosity
and stellar wind flux (Gertz 2020). These are not mutually exclusive and both
paths could be taken by an advanced civilization.
One plausible picture of an advanced civilization that accommodates stellar
evolution – rather than entirely fleeing from it – would include numerous
giant space colonies and other giant constructs in orbit in sphere- or ring-
like configurations about a white dwarf or an old main sequence star. This
type of technological civilization is often referred to as a Dyson sphere or
ring, after Freeman Dyson who first described it (Dyson 1960). One motivation
for giant constructs would be to supply energy to the space colonies,
especially those in orbit around white dwarfs, because white dwarfs are of low
and declining luminosity.
Whatever the detailed arrangement, one anticipates that energy collection
devices would be situated close to stars so as to minimize their size. Their
temperatures would probably lie in the range 300 to 1000 K and they would
radiate primarily between a few and 10 $\mu$m wavelengths (Wright et al 2014).
This emission would be in addition to (an excess above) emission from the star
itself. It seems unlikely that carbon/water based life could ever exist at
temperatures much above 300 K. But should such life evolve (redesign itself)
into something very different, then perhaps the space colonies themselves
would reside at temperatures substantially above 300 K.
The focus of the present paper is the detectability of artificial constructs
in orbit around white dwarf stars, although main sequence stars are also
considered (Section 6). The past two decades have witnessed the birth of the
study of planetary systems around white dwarfs via detection of elements
heavier than helium in their photospheres, detection of infrared emission from
orbiting dust particles and optical spectral lines from orbiting gas, and
occasional discovery of orbiting planets and brown dwarfs via either their IR
emission or a transit in front of a white dwarf. The photospheric heavy
elements are seen in at least 25% of all white dwarfs and is a marker for the
presence of orbiting dust particles and gas. The dust can reveal itself
directly via excess IR emission above that emitted by the white dwarf
photosphere, but such excess is seen in only a few percent of all white dwarfs
(Rocchetto et al. 2015, Farihi 2016, Wilson et al. 2019). Even less common is
excess IR emission emitted by a cool brown dwarf or cool white dwarf. Thus, in
order to identify a Dyson sphere/ring at a white dwarf, one must eliminate
dust particles and brown and white dwarfs as the carrier of excess IR
emission.
In addition to absorbing radiation emitted by the white dwarf, the space
constructs will sometimes pass (transit) between the star and our telescopes,
thus causing a dip in the received optical light. Arnold (2005) discussed the
transit light-curve signatures of very large artificial objects. Agol (2011)
considered transits of Earths in the habitable zones of white dwarfs. The
first examples of cool obects transiting white dwarfs have recently been
obtained with the Zwicky Transient Facility and NASA’s TESS satellite
(Vanderbosch et al 2020 & 2021; Vanderburg et al 2020; van Roestel et al
2021).
Zuckerman (1985) showed that a few percent of the technological civilizations
in our Milky Way galaxy may already have experienced the evolution of their
star from main sequence to white dwarf (see Section 4 below). Thus, if
technological civilizations are common, as some believe, then finding one in
orbit around a white dwarf could be quite plausible.
In the following we refer to Dyson spheres/rings as DSRs. The primary purpose
of the present paper is to consider and compare the observational limits that
can currently be placed on the number (frequency) of DSRs at white dwarf stars
via their infrared emission and/or transit frequency. Many guesstimates for
the number of technological civilizations in the Milky Way can be found in the
literature (see Section 2). But now, thanks to advances in discovery and
analysis of extrasolar planetary systems, it is possible to frame the question
in a more quantitative way: On what fraction of planets in the habitable zone
of sun-like stars do long-lived technological civilizations arise and
eventually build a DSR?
In the following section we define what is meant by the number of
technological civilizations in our Galaxy. In Section 3 we consider the
carriers of excess IR emission at white dwarf stars. Section 4 is a discussion
of how infrared bright a DSR would have to be to have already been detected at
a white dwarf star and what constraints such measurements place on the
frequency of DSRs. Section 5 is a consideration of transit detectability of
DSRs and how it compares with detection via infrared emission. Section 6
describes the current observational situation at main sequence stars. Section
7 compares the advantages and disadvantages of white dwarfs compared to main
sequence stars when one is searching for a DSR.
## 2 What is meant by the number of technological civilizations in the
Galaxy?
In order to give context to limits on the frequency of DSRs in the Galaxy, one
should have some idea of how many long-lived technological civilizations “N”
might be anticipated. The range suggested in the literature is huge, extending
from zero (Hart 1975) to as many as 10 billion (F. Drake as quoted in Tarter
2001). Other estimates include: 50,000 to 1,000,000 (Shklovskii & Sagan 1966),
one billion (Oliver & Billingham 1971, Goldsmith & Owen 1992), and 10 million
(Sagan 1980). As a consequence of the discovery during the past few decades
that there are more planets than stars in the Milky Way, many persons would
likely expect a (very) large value for N.
In each of the above examples, N refers to the number of independently arising
technologies and does not include the possibility that a technological
civilization might expand to occupy one or more additional star systems. The
relevance for the present paper of any such migrations depends strongly on the
motivations for migration.
If the only motivation for interstellar travel is to escape from one’s home
planetary system so as to avoid the evolution of the home star to red giant
and white dwarf (Zuckerman 1985, Hansen & Zuckerman 2021), then one would
never expect a technological civilization to build a DSR around any white
dwarf other than the white dwarf that their home star evolves into. Because it
is hard to imagine any reason why a technological civilization would want to
migrate to a white dwarf and then construct a DSR, it is probably safe to
assume that the frequency of DSRs at white dwarf stars will depend only on the
number of independently arising technological civilizations. In any event, in
the unlikely event that migration to white dwarfs does occur frequently, then
this would only serve to increase the likelihood of finding a DSR at a white
dwarf.
By contrast, interstellar travel and colonization might increase the number of
DSRs that exist at main sequence stars. As one example, if curiosity about
biology on other worlds is an important motivation for interstellar travel
(e.g., Zuckerman 2019), then some DSRs at main sequence stars might be
constructed by a technological civilization that first arose at a different
star.
In the discussion to follow we assume that N is equal to the number of
independently arising technological civilizations in the Milky Way. We want to
estimate what constraints existing and potential future infrared and optical
observations of white dwarf and main sequence stars place on the frequency of
DSRs and thus on N.
## 3 White dwarfs with excess infrared emission
A few percent of white dwarf stars display excess infrared emission –
characteristic temperatures $\sim$1000 K – above their photospheres (e.g.,
Rocchetto et al 2015). From the first discovery, such emission has generally
been attributed to either orbiting dust particles or an orbiting brown dwarf
(Zuckerman & Becklin 1987). It is now understood that the carrier of the IR
emission is almost always orbiting dust particles, with but few examples of
(spatially unresolved) brown dwarf or cool white dwarf companions (e.g., see
Introduction in van Roestal et al 2021). A summary of studies of IR emission
from white dwarfs can be found in the Introduction and in Table 1 in Xu et al
(2020). When dust is responsible for an IR excess, then, with no reliably
known counter examples, “pollution” of the white dwarf photosphere by elements
heavier than helium is also observed. The generally accepted model is that the
material contained in the dust particles is accreted onto the white dwarf
(Veras 2016 & 2021; Farihi 2016; Jura & Young 2014, Zuckerman & Young 2018).
The resulting level of photospheric pollution for stars with excess IR
emission is typically large with respect to the average pollution level seen
in all polluted white dwarfs. The parent body for the dust is usually a rocky
asteroid or asteroids (Jura 2003 & 2008) that have been torn apart by the
strong gravitational field of the white dwarf.
Thus, if heavy elements are detected in the photosphere of a white dwarf with
excess IR emission, then it is likely that the excess is due to dust particles
and not to a brown dwarf or a DSR. Because excess IR emission carried by dust
particles is less likely for cool (cooling times $>$1000 Myr) white dwarfs
(e.g., Table 3 in Rocchetto et al 2015), statistically, white dwarfs with
excess IR emission and relatively low temperatures $<$8000 K could more likely
host a DSR. But some white dwarfs with temperatures as low as $\sim$6000 K
(Debes et al 2019; Gentile Fusillo et al 2019) and 4200 K (Hollands et al
2021) appear to host excess IR emission likely due to dust. A typical
blackbody temperature of the dust at white dwarfs is $\sim$1000 K, but a few
are known with temperatures as low as 300-400 K (Table 3 in Rocchetto et al
2015). As mentioned above, these are plausible temperatures for a DSR.
WD2328+107 probably has an IR excess (Wilson et al 2019) and has a progenitor
main sequence mass similar to that of the Sun (Rocchetto et al 2015), but no
evidence for heavy elements in its atmosphere. Still, the putative excess IR
emission might be carried by a brown dwarf or a cool white dwarf companion
(Wilson et al 2019). A JWST observation should be able to clarify the
situation.
Based on the above considerations, with the possible exception of WD2328+017,
it seems unlikely that any currently known example of excess IR emission would
be a good DSR candidate. Elimination of dust as the carrier of IR emission
requires, at a minimum, the absence of photospheric pollution and silicate
emission features in the 10 $\mu$m spectrum; such features are usually present
(Jura et al 2009) when dust is present.
## 4 The Infrared Detectability of Dyson Spheres at white dwarf stars
What limits can currently be placed on the frequency of DSRs at white dwarfs?
While it might be possible to discover a narrowband signal generated by an
extraterrestrial intelligence (ETI) from a white dwarf system simply by
accident, there have not been any published radio or optical searches of white
dwarfs meant specifically to detect ETI. Thus, the limits on N described below
are the first to be obtained for white dwarfs.
We frame the following discussion in two ways. In the first, we estimate upper
limits to N that are implied by Spitzer Space Telescope surveys of white
dwarfs in two limits: (1) all N technological civilizations in the Galaxy
arise on stars with masses similar to that of the Sun (see Equation 2), and
(2) technological civilizations arise with equal probability on stars with
masses less than or about equal to that of the Sun. A second way of
approaching the matter does not involve N directly, but rather asks the
question: if a certain fraction, $\alpha$, of G-type stars in the Galaxy
contain an appropriate rocky planet in the habitable zone such that life might
originate, then what fraction of these potentially habitable planets
eventually evolve a technological civilization that builds a DSR? Estimates
for $\alpha$ have been derived from Kepler data (see below).
Zuckerman (1985) calculated the number of technological civilizations that
have originated on planets that orbit around main sequence stars that have
evolved to white dwarfs in a time that is less than the age of our Galaxy
which he, and we, assume to be 1010 years. We redo these calculations here,
but using more recent information.
Sollima (2019) considers a variety of star formation histories. For the
purposes of the present paper, assumption of a constant birthrate during the
past 1010 years is adequate. For stars with masses about equal to and greater
than a solar mass, following Sollima, we assume that dQ/dt, the rate of
formation of stars in the Galaxy with masses between M and M+dM, is given by :
$\frac{dQ}{dt}\ =\ 1.55\frac{dM}{M^{2.55}}$ (1)
In the above and below expressions, Q is simply a number, but we cannot label
it with “N” because historically and, as above, N has been reserved to denote
the number of technological civilizations in the Galaxy The exponent 2.55 is
the mean of the range of power law slopes, 2.4-2.7, suggested by Sollima. And,
following Iben (1984), we have normalized to a Galactic birthrate of one star
per year with a mass greater than M⊙. Sollima gives a history of the evolution
of the power law exponent going back to Salpeter (1955) who adopted 2.35 over
the range of stellar masses of interest here.
The white dwarf initial-final mass relation for progenitor stars with main
sequence masses between 0.85 and 7.5 M⊙ is given in Cummings et al (2018).
Mamajek
(2021)111http://www.pas.rochester.edu/$\sim$emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt
has compiled an extensive list of the properties of main sequence stars, in
particular, spectral type vs. stellar luminosity and mass. We assume that
stars with main sequence lifetimes that are shorter than 4.5 x $10^{9}$ years
(M $>$1.25 M⊙) do not last sufficiently long to permit a technological
civilization to evolve on an orbiting planet. And we assume a minimum main
sequence mass of 0.95 M⊙ to allow some time for early production of metals
with which to make rocky planets. Therefore, the interesting range of stellar
masses lies between 0.95 and 1.25 M⊙. According to Mamajek, these masses
correspond to spectral types G7 and F6, respectively, and with luminosities
0.74 and 2.69 L⊙ Thus, stars with nearly the mass of the Sun have main-
sequence lifetimes that vary like M-3.8.
The number of stars in the Milky Way born in the mass range 0.95 to 1.25 M⊙
during the past $10^{10}$ years is:
$Q_{\rm i}\ =\ 10^{10}\int_{0.95}^{1.25}\frac{1.55\,dM}{M^{2.55}}\ =\
3.75\times 10^{9}\,{\rm stars.}$ (2)
The number of these stars still on the main sequence is:
$Q_{\rm a}\ =\
\int_{0.95}^{1.25}1.55\,\frac{dM}{M^{2.55}}\int_{0}^{10^{10}/M^{3.8}}dt\ =\
2.95\times 10^{9}\,{\rm stars.}$ (3)
So the number of stars that have “died” and become white dwarfs is:
$Q_{\rm i}-Q_{\rm a}\ =\ 8\times 10^{8}$ (4)
Now we want to estimate how many technological civilizations (N*) might have
existed on planets orbiting these 8 x $10^{8}$ white dwarfs. Let’s first
assume that all N technological civilizations in the Galaxy originated around
stars in the main sequence mass range 0.95 to 1.25 M⊙. Then
N*max = (8 x $10^{8}$)N/(3.75 x $10^{9}$) = 0.2 N
If, for example, N = $10^{9}$, so N*max = 2 x $10^{8}$ is distributed among 8
x $10^{8}$ white dwarfs. Then 25% of all white dwarfs could potentially
display a DSR.
How many white dwarfs have been searched for excess IR emission and to what
level of sensitivity? And how many of these had masses on the main sequence in
the range 0.95 to 1.25 M⊙? The relevant satellites are Spitzer and the Wide-
field Infrared Survey Explorer (WISE). Various published papers describe
Spitzer surveys for excess IR emission of white dwarfs. Table A1 in Rocchetto
et al (2015) lists 134 white dwarfs of which 30 had progenitors with main
sequence masses $\leq$1.25 M⊙. Wilson et al. (2019) carried out a more
extensive survey (of 217 white dwarfs) that included the Rocchetto stars as a
subset. Assuming the same fraction of low mass progentors among the 217 stars
as among the 134, yields a total of about 50 white dwarfs in the relevant mass
range. Figure 10 in Farihi (2016) shows 108 white dwarfs with photospheres
that are polluted with heavy elements that were searched with Spitzer for IR
excess emission during its first 7 cycles. Most of these stars are cooler than
the stars in the Rocchetto/Wilson sample, so there is little overlap. Wilson
et al note that only about one in 30 white dwarfs with polluted photospheres
exhibits an IR excess in 3-4 $\mu$m IRAC photometry. Putting all of these
considerations together, suggests that the Farihi sample of 108 stars contains
about 25 that lack detectable excess IR emission and had main sequence masses
that fall in the relevant range (less than 1.25 M⊙). Yet another major white
dwarf survey was by Mullally et al (2007), with $\sim$100 searched. Finally
there were some smaller Spitzer surveys, for example by Farihi et al (2008) of
17 white dwarfs.
In total, it appears that Spitzer surveyed at least 100 white dwarfs that had
masses on the main sequence in the range 0.95 to 1.25 M⊙ and with no evidence
for excess IR emission. Table 3 in Rocchetto et al (2015) indicates that the
typical search sensitivity – excess IR luminosity divided by bolometric
luminosity – was about 0.1%, for excess emission at temperatures between about
300 and 1700 K. Thus, the considerations that follow apply only for DSRs of
fractional IR luminosity greater than 0.1%.
As noted just above, if all technological civilizations originated on stars
with main sequence spectral types between G7 and F6, and if N is equal to one
billion, then about 25% of the 100 Spitzer-observed white dwarfs with these
spectral types and without orbiting dust or a cool companion could potentially
display a DSR. Yet none have been reported in the literature where an IR
excess exists that could not definitely or probably be attributed to dust or
to a companion.
If N = 100 million, then we might have expected to detect 2.5 DSRs among the
100 relevant Spitzer white dwarfs. Given statistics of small numbers, these
few DSRs might not be included among the 100 observed white dwarfs. So, for
the situation where all technological civilizations in the Milky Way orginate
on main sequence stars with spectral types betwen G7 and F6, and where a DSR
around the white dwarf that these stars evolve into has a fractional excess IR
luminosity of at least 0.1%, then the implication is that N is less than 100
million.
Based on the IMF for stellar masses less than a solar mass deduced by Sollima
(2019), G7 through F6 stars comprise about 10% of all stars with main sequence
lifetimes that are long enough to allow evolution of life to technology. If we
assume that the N technological civilizations arise with equal probability
around all such stars with no preference for stellar mass, then the
implication from white dwarfs observed with Spitzer is that N is less than
$10^{9}$.
In summary, Spitzer observations of white dwarfs indicate that the upper limit
to N is about 300 million, plus or minus a few hundred million; the better
($<$$10^{8}$) and poorer ($<$$10^{9}$) bounds to N apply, respectively, in the
cases where technological civilizations preferentially form on planets at
G-type stars or, alternatively if they form with equal probabilites on planets
that orbit stars with masses less than or equal to that of the Sun. Again, the
technological civilization must have contructed a DSR with fractional IR
luminosity of at least 0.1% for these limits to apply (see Section 7 for
consideration of whether this is a plausible DSR luminosity).
From Equation 4 we see that about a billion F6 through G7 stars that were on
the main sequence are now white dwarfs. Studies of Kepler and other data bases
by Zink & Hansen (2019) and by Bryson et al. (2021) suggest that about 30% of
G-type stars are orbited by a potentially habitable planet, or about 300
million such planets that orbit the white dwarfs of interest here. If as many
as one in 30 of these planets spawns life that eventually evolves to a state
where it constructs a DSR with luminosity at least 0.1% that of its host white
dwarf, then in a sample of 100 white dwarfs we might have expected to see a
DSR. Thus, fewer than 3% of the habitable planets that orbit sun-like stars
host life that evolves to technology, survives to the white dwarf stage of
stellar evolution, and builds a DSR with fractional IR luminosity of at least
0.1%.
The WISE mid-infrared survey of the sky (Wright et al 2010) was sufficiently
sensitive to yield significant limits for N for main sequence stars (see
Section 6). But for the much fainter white dwarfs, clear identification of
excess IR emission is plagued by confusion (e.g., Dennihy et al 2020; Xu et al
2020) with other sources of IR emission. Hopefully, future studies with the
WISE database will enable these data to contribute to our understanding of the
frequency of DSRs at white dwarfs.
## 5 Optical Detectability of Transits of Dyson Spheres at white dwarf stars
As noted in the Introduction, a DSR, or part of it, might be detected as it
transits between its star and a space-based telescope such as TESS or Kepler.
In the present section we consider the relative sensitivity of the optical
transit and infrared excess detection techniques. As noted in Section 4, the
current infrared sensitivity limit – excess IR luminosity due to a DSR divided
by bolometric luminosity – is about 0.1%. This limit depends on total solid
angle blocked by the entire entourage of artificial constructs as seen from
the surface of the white dwarf, but with no constraints on the size of
individual structures. In contrast, the minimal requirement for detection by
an optical transit depends on the size of the largest construct, while the
total solid angle blocked by the DSR can be far less than 0.1%. We ask whether
a DSR might be detected via transits with Kepler K2 (Howell et al 2014) or
with TESS (Ricker et al. 2015).
van Sluijs & Van Eylen (2018) investigated the sensitivity of K2 to substellar
objects that transit white dwarfs. K2 targets were observed with either long
(30 min) or short (1 min) exposure times; about 1000 non-composite, confirmed,
white dwarfs were observed with the long cadence mode and about 300 with the
short. For a 7000 K white dwarf (cooling time to reach this temperature
$\sim$1.5 Gyr (Dufour et al. 2017), a black body with an orbital semi-major
axis equal to a solar radius would have a temperature of 500 K. The transit
time of a small object with this semi-major axis is about 1 min. So, for a
transiting object of given size the detection efficiency is larger in the
short cadence mode (see Figure 2 in van Sluijs & Van Eylen (2018). The
diameters of the smallest objects for which the detection efficiencies are as
large as $\sim$10% are comparable to that of Ceres ($\sim$1000 km). If a
structure has an orbital semi-major axis of a solar radius, then the
probability that its orbit would give rise to a transit as seen from Earth
would be $\sim$0.01. If there are ”n” large structures that lie in
sufficiently different planes, then the transit probability would increase by
a factor of n.
If a large construct were closer to the white dwarf than a solar radius,
either hotter or with high albedo (light shields), then the transit
probability could be a few percent. If the white dwarf were cooler than 7000 K
then the orbital semi-major axes of constructs of a given temperature could be
even smaller. But for a cooling time of $\sim$6 Gyr the white dwarf would
still be $\sim$5000 K, and thus a transit probability only a factor of two
larger for a structure of a given temperature.
To have been detected with K2, a DSR would have to include a large structure
(diameter $>$1000 km) with an appropriate orbital inclination with respect to
our line of sight. No such transiting objects were found with K2.
No summary of TESS studies of white dwarfs is yet available in the literature.
TESS observes a given region of the sky for $\sim$27 days with a number of
cadences. Andrew Vanderburg (personal communication, 2022) is leading a
collaboration that should eventually observe, with 2 minute cadence, about
10,000 white dwarfs; to date about 5000 have been observed. Because of TESS’
large (21”) pixels, background objects in addition to the white dwarf of
interest are often included in the white dwarf pixel. After correction for the
presence of such objects in the white dwarf pixel, the scatter in the measured
flux is typically about 10% for a white dwarf with Gaia G magnitude = 16 (A.
Vanderburg, personal communication 2022). Based on statistics given in
Gentile-Fusillo et al (2021) and in Dufour et al (2017) one anticipates about
2500 white dwarfs will be this bright or brighter. With a representative 3
hour orbital period, a given structure will transit a few 100 times in 27
days, after which one can construct a final phase folded light curve with flux
scatter of about 1% for each of the 2500 white dwarfs. Thus, a signal-to-noise
ratio of, say, 10 in a 27 day average would require a transit depth of
$\sim$10%. As with K2, such a deep transit requires DSR constucts with
diameter of order 1000 km. The full TESS sample of 10,000 white dwarfs should
contain $\sim$800 with G mag = 15 or brighter, so for these brightest stars a
transit depth of a few % for detection. Ultimately, when the final TESS white
dwarf sample is observed and analyzed, the detection limits quoted in this
paragraph may turn out to be too conservative – because a significant fraction
of the final sample should eventually be observed with 20 sec cadence, and/or
for longer than 27 days (specifically multiples of 27 by a factor of 2 or
more) (A. Vanderburg, personal communication 2022).
As in Section 4, it is possible to derive constraints on the value of N with
use of the Kepler K2 and TESS databases (when the latter becomes available).
However, such constraints will apply only to DSRs that include at least one
structure with diameter $\sim$1000 km and will be sensitive to the number of
such constructs and the inclination of their orbital planes relative to our
line of sight. Should transits with small fractional dips be detected, it
might be possible to distinguish natural from artificial objects by study of
the transit shape (e.g., Arnold 2005; Sandford & Kipping 2019). However, this
would likely require an additional factor of $\sim$10 in signal-to-noise ratio
(G. Marcy, personal communication 2022).
## 6 The frequency of technological civilizations at main sequence stars
Evidence of ETI at main sequence stars can be in the form of excess IR
emission from a DSR or as radiation beamed toward Earth at radio, IR, or
optical wavelengths for the purpose of communication. Each technique has its
own advantages and disadvantages. Concerning beamed transmissions, at radio
wavelengths one representative major project is “SETI Observations of
Exoplanets with the Allen Telescope Array” (Harp et al 2016). A representative
search at optical wavelengths would be “A Search for Laser Emission with
Megawatt Thresholds from 5600 FGKM Stars” (Tellis & Marcy 2017). The website
technosearch.seti.org gives a comprehensive list of (perhaps all) radio and
optical search programs. Based on the discussion in Sections 4 and 7, perhaps
it would be worthwhile to devote more time to white dwarfs as targets in such
searches. Searches for beamed transmissions usually assume a signal that is so
narrow in frequency space and/or has a timing profile such that it cannot be
produced by a “natural” source. One huge advantage for a search for a beamed
transmission, compared to an IR or transit search for a DSR, is that once one
has ruled out terrestrial interference, there are no false positives to
contend with.
Mid-infrared satellites have observed numerous main sequence stars in both
pointed (Spitzer) and scanning mode (WISE). In all cases two important
questions are: (1) what is the minimum fractional IR luminosity (LIR/L*) due
to a DSR that can be detected, and (2) if an IR excess is detected, can it be
shown with certainty that it must be coming from a DSR rather than from some
”natural” source such as dust or a brown dwarf. Main sequence stars are much
brighter than white dwars, so many more can be searched for a DSR in a
brightness limited sample (see, e.g., Cotten & Song 2016). Nonetheless, as
outlined in Section 7, in some ways white dwarfs are better targets.
Wright et al (2014) outline a plan to search with Spitzer and WISE at mid-
infrared wavelengths for the waste energy of advanced technological
civilizations. They review the idea of Kardashev Type I, II, and III
technological civilizations; these correspond, respectively, to technologies
that command planetary, circumstellar and galactic energy sources. Homo
sapiens are a Type I civilization and DSRs correspond to Type II. However, the
focus of the Wright et al paper is on Type III civilizations that, they say,
are “very rare in the local universe”.
There are two published studies relevant to the frequency of Type II
civilizations (= DSRs): Kennedy & Wyatt (2013) and Moor et al. (2021). Kennedy
& Wyatt surveyed $\sim$24,000 A8 to K5 type main sequence stars for excess IR
emission in the 12 $\mu$m (W3) channel of WISE. Their goal was to discover
stars that are orbited by warm (T $>$ 200 K) dusty disks with ages of at least
8 Myr. There is no mention in their paper of DSRs or Kardashev civilizations.
Perhaps this is why the Kennedy & Wyatt paper is not cited by Wright et al
(2014).
Only about two dozen warm IR excess stars emerged from this extensive study;
many of these were previously known (see Table 1 in Kennedy & Wyatt (2013).
Important for the present study, none of the Table 1 stars satisfies both the
age ($>$4.5 Gyr) and spectral type (F6-G7) of interest to us here, but with
the possible exception of HD 154593. However, from a Gaia proper motion
anomaly, Kervella et al (2019) deduce that HD 154593 has a cool unseen
companion. C. Melis (personal communication, 2021) has carried out an
extensive search of the literature over a much broader range of stellar phase
space than covered by Kennedy & Wyatt for stars that would satisfy our
constraints for age and spectral type; Melis found no such stars with a
reliable IR excess.
Thus, none of the Kennedy & Wyatt sample of 24,000 stars shows evidence for a
DSR; only a fraction of the 24,000 satisfy our age and spectral type
constraints. Their Figure 1 of spectral types suggests that about 12,000 stars
are in the range F6-G7. Their Figure 2 gives the age distribution; $\sim$20%
are older than 4.6 Gyr. Thus we can say that none of $\sim$2400 stars with the
appropriate age and spectral type show evidence of a DSR. Similar to the white
dwarfs discussed in Section 4, the absence of a DSR at these 2400 main
sequence stars is for fractional IR luminosities – excess IR luminosity
divided by bolometric luminosity – of about 0.1% or greater. In both classes
of target stars one is dealing with a brightness limited sample. But even
though the white dwarfs are fainter than the main sequence stars, for the
former sample the IR data are from Spitzer, in contrast to the less sensitive
WISE data used for the main sequence sample.
From Equation 3 we see that about 3 billion F6 through G7 stars are now on the
main sequence. As noted in Section 4, studies of Kepler and other data bases
by Zink & Hansen (2019) and by Bryson et al. (2021) suggest that about 30% of
G-type stars have a potentially habitable planet, or about 109 such planets.
If a million (106) of these evolve to a technological civilization that
constructs a DSR, then one should have been found in the Kennedy & Wyatt
sample of 2400 stars. Thus, we can say that at most about one in a 1000 G-type
stars with habitable planets harbors a planet with life that eventually
evolves to a state where it constructs a DSR with luminosity at least 0.1%
that of its host star.
It would be worthwhile to analyze the Moor et al. (2021) database for DSRs.
But this is not possible to do without more detailed information on stellar
spectral types and age than is presented in their paper. We can, however, say
that none of the warm IR excess stars identified by Moor et al appears to be a
good candidate for a DSR.
The above discussion and that in Section 4 assume that all N civilizations
arise independently and do not colonize other stars. Hansen & Zuckerman (2021)
argue that many such migrations will be motivated by stellar evolution and
will be to M-type stars. If so, then M stars would represent a plausible host
for DSRs. However, the false positive rate for excess IR emission at M stars
that are cross correlated with the WISE database is large (Silverberg et al
2018) and, at any rate, no one has ever confidentally identified an M star of
age $\sim$5 Gyr or greater that possesses an IR excess.
## 7 Discussion
In Sections 4 and 6 we appraised the occurrence frequency of DSRs at white
dwarfs and at main sequence stars based on various observations and
assumptions. Here we compare the relative advantages and shortcomings involved
when white dwarfs or main sequence stars are utilized as determinants of DSR
frequency. The IR surveys referred to in Sections 4 and 6, have similar
sensitivities to DSRs ($\sim$0.1%) when the luminosity of a DSR is measured as
a fraction of the bolometric luminosity of a central star; we will call this
fraction $\tau$. At this point in time, no evidence exists of a DSR of this
fractional brightness or larger at any of the surveyed white dwarfs and main
sequence stars.
No doubt, motivation is a major factor that determines whether an advanced
technological civilization would want to construct a DSR. As noted in the
Introduction, coping with stellar evolution is one unavoidable motivation.
Interior planets will be destroyed during the giant (AGB) phase of stellar
evolution, while surviving outer planets will become exceedingly cold as the
white dwarf cools. Thus, arguably, eventually all organisms will have migrated
from planetary surfaces to artificial space colonies. If so, then any
civilization that orbits a white dwarf must produce a DSR. A question then is,
what is a likely value for $\tau$?
One can envision reasons why $\tau$ at a white dwarf would be quite large. Far
more organisms can live in an ensemble of space colonies than can live on a
planet such as Earth. Thus, the value of $\tau$ depends on the numbers of
organisms, how much space each one would desire, and how much energy would be
required per capita. Needless to say, from our youthful, naive, vantage point
we simply don’t know the answers to such questions. About all that one might
say is that most Homo sapiens would surely prefer to live in a huge space
colony, the bigger the better. If this motivation is retained even in an
advanced long-lived civilization, then $\tau$ could be quite large; easily
0.1%. And, as noted in Section 5, really large constructs might produce
detectable transit signals.
Thus, $\tau$ = 0.1% would seem to be a significant search sensitivity, and
upper limits to the frequency of DSRs with this luminosity at white dwarfs,
meaningful. Unfortunately, given their low apparent brightness, as noted in
Section 4, only of order 100 white dwarfs with main sequence progenitors in
the appropriate mass range were examined with Spitzer. Given this small sample
size, we estimated that at least 3% of all habitable planets at G-type main
sequence stars must spawn life that eventually evolves to technology in order
for at least one to have been detected with Spitzer at a white dwarf. This
seems a bit “optimistic”.
Because a few 1000 G-type main sequence stars with appropriate age and
spectral type were examined with WISE, as noted in Section 6, only one in a
1000 or so G-type stars needs to spawn life that eventually develops
technology for a DSR with $\tau$$>$0.1% to have been detected in the Kennedy &
Wyatt (2013) survey. However, main sequence stars suffer at least two
disadvantages as target stars when compared to white dwarfs; one disadvantage
would be less motivation to build a substantial DSR because one’s home planet
remains a good abode for life. In our own solar system, if a sunshield is
constructed and employed at the inner Earth-Sun Lagrange point – to counter
the increasing solar luminosity – then Earth could remain quite habitable for
a few Gyr more into the future. Perhaps a more important consideration would
be the greater luminosity, say about a factor of 1000, of the Sun compared to
a typical white dwarf. For a DSR with the same $\tau$ and temperature around
the Sun as one around a white dwarf, a DSR at the former would have to have
1000 times the area of one at the latter. While there is sufficient material
in the asteroid belt to build such an extensive DSR, would the motivation to
do so exist? For transits of main sequnce stars by structures with
temperatures in the range 300 to 1000 K, the orbital period would be much
longer than around white dwarfs, thus relatively few transits per year. For a
given structure, the probability of proper alignment of orbital plane and line
of sight to Earth would be small and its required cross section would be
larger than that of Ceres.
## 8 CONCLUSIONS
We have considered the detection of a Dyson Sphere or Ring (DSR) at a white
dwarf star via its infrared emission or via a transit between our telescopes
and the star. Of order 100 white dwarfs of appropriate mass were observed in
the infrared with the Spitzer Space Telescope; no plausible DSR candidate has
been identified. We also considered DSRs at main sequence stars; a few 1000 of
appropriate age were observed with the Wide-field Infrared Survey Explorer
(WISE) and no plausible DSR candidate was identified. These results, along
with a few reasonable assumptions, can be used to place limits on the number
of technological civilizations in the Galaxy or, alternatively, on the
fraction of habitable planets that orbit solar type stars and on which a
technological civilization eventually emerges and subsequently constructs a
DSR.
We discussed transit surveys of white dwarfs with the Kepler 2 and TESS
missions. These could, in principle, detect a DSR that includes at least one
structure with diameter $\sim$1000 km. Detection of a DSR via its infrared
excess emission depends on the total solid angle blocked by the entire
entourage of artificial constructs as seen from the surface of the white
dwarf, but does not require the existence of any really large individual
structure. In contrast, the minimal requirement for detection by an optical
transit depends on the size of the largest construct, while the total solid
angle blocked by a DSR can be far less than is required for detection of an
infrared excess.
Both the white dwarf and main sequence studies were sensitive to DSRs whose
infrared luminosity “$\tau$” is about 0.1% that of the bolometric luminosity
of their central star. Regarding the number of number “N” of technological
civilizations in the Milky Way Galaxy, based on the white dwarf studies, if
all N originate at solar-type main sequence stars and construct DSRs with
$\tau$ at least as large as 0.1%, then N $<$100 million. If technological
civilizations emerge with equal probability around all stars of solar mass or
less, then N$<$$10^{9}$.
An alternative way to interpret the observational results is to ask: what
fraction of potentially habitable planets that orbit solar-type stars spawn
living organisms that eventually evolve to a technology that then constructs a
DSR with $\tau$ at least as large as 0.1%? From the white dwarf studies we
estimate that this fraction is at most 3%. From the main sequence studies the
upper limit to this fraction could be as small as 0.1%, but with a few
potentially significant caveats. Additional examination of the existing
databases would be worthwhile to ensure that no DSR was missed.
Substantial progress to improve the current limits, or to detect a DSR, will
require a new mid-infrared space telescope. The diameters of WISE, Spitzer,
and JWST are 40, 85, and 650 cm, respectively. JWST with its long slew time
and numerous Galactic and extragalactic objects waiting in line to be observed
is not the right telescope with which to search for DSRs.
The median Gaia G-magnitude of the Roccetto et al (2015) white dwarf sample
was 15.4. According to Gentillo-Fusillo et al (2019), there are about 5000
white dwarfs within 200 pc of Earth and brighter than magnitude 17. Given that
about 25% of white dwarfs have masses in the (low mass) range of relevance for
the possible existence of a DSR (as discussed in Section 4), a scanning
telescope operated in a similar way to WISE, but with the diameter of Spitzer,
could improve by an order of magnitude the limits on DSR frequency derived in
this paper. Such a telescope would yield much new science in general and would
not be too costly. The 30-m class ground-based telescopes currently envisioned
or in construction could be used to vet any potential DSR candidates.
Future telescopes, such as LSST, may be suitable for detection of transits of
DSRs at white dwarfs. But individual constructs would have to have diameters
$\sim$1000 km, i.e. comparable to that of Ceres (Cortés & Kipping 2019). ESA’s
planned PLATO spacecraft, if successful, should be at least a few times more
senstitive than TESS for detection of very shallow transits at white dwarfs
and thus able to detect constructs with diameters a few times smaller than
1000 km.
The author is grateful to Andrew Vanderburg and Beth Klein for a variety of
assistance, to referee Geoffrey Marcy for many useful suggestions, and to Carl
Melis, Jay Farihi, Jill Tarter, and Brad Hansen for helpful advice. This
research was supported in part by grants to UCLA from NASA and the NSF. We
have made use of NASA’s Astrophysics Data System.
DATA AVAILABILITY No new data were generated or analysed in support of this
research.
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|
#
Anomalies and phases of
strongly-coupled chiral gauge theories:
recent developments
Stefano Bolognesi(1,2), Kenichi Konishi(1,2), Andrea Luzio(3,2)
(1)Department of Physics ”E. Fermi”, University of Pisa,
Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy
(2)INFN, Sezione di Pisa, Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy
(3)Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56127 Pisa, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
After many years of investigations, our understanding of the dynamics of
strongly-coupled chiral gauge theories is still quite unsatisfactory today.
Conventional wisdom about strongly-coupled gauge theories, successfully
applied to QCD, is not always as useful in chiral gauge theories. Recently
some new ideas and techniques have been developed, which involve concepts of
generalized symmetries, of gauging a discrete center symmetry, and of
generalizing the ’t Hooft anomaly matching constraints to include certain
mixed symmetries. This new development has been applied to chiral gauge
theories, leading to many interesting, sometimes quite unexpected, results.
For instance, in the context of generalized Bars-Yankielowicz and generalized
Georgi-Glashow models, these new types of anomalies give a rather clear
indication in favor of the dynamical Higgs phase, against confining, flavor
symmetric vacua.
Another closely related topics is strong anomaly and the effective low-energy
action representing it. It turns out that they have significant implications
on the phase of chiral gauge theories, giving indications consistent with the
findings based on the generalized anomalies.
Some striking analogies and contrasts between the massless QCD and chiral
gauge theories seem to emerge from these discussions. The aim of this work is
to review these developments.
###### Contents
1. 1 Introduction
2. 2 Computation of the mixed anomalies
1. 2.1 Gauging a 1-form ${\mathbbm{Z}}_{k}\subset{\mathbbm{Z}}_{N}$ center symmetry
2. 2.2 Color-flavor locked ${\mathbbm{Z}}_{N}$ center symmetry: Master formula
3. 2.3 Comments on the paper [78]
4. 2.4 Comments on the papers [79, 80]
5. 2.5 Higgs phase and anomaly-matching
3. 3 Physics of models with ${\mathbbm{Z}}_{k}\subset{\mathbbm{Z}}_{N}$ center symmetry
1. 3.1 Self-adjoint antisymmetric tensor matter
1. 3.1.1 $SU(6)$ gauge group
2. 3.1.2 $SU(N)$ models
2. 3.2 Adjoint QCD
3. 3.3 QCD with ”tensor quarks”
4. 3.4 Chiral theories with $\tfrac{N-4}{k}$ $\psi^{\\{ij\\}}$’s and $\tfrac{N+4}{k}$ ${\bar{\chi}_{[ij]}}$’s
4. 4 Generalized anomalies and phases of the generalized BY and GG models
1. 4.1 Bars-Yankielowicz models
2. 4.2 Georgi-Glashow models
5. 5 Strong anomaly and phases of chiral gauge theories
1. 5.1 $U(1)_{A}$ problem and the $\theta$ dependence in QCD
1. 5.1.1 ${\cal N}=1$ supersymmetric theories
2. 5.2 $\psi\eta$ model and strong-anomaly
3. 5.3 Strong anomaly: the generalized BY models
4. 5.4 Strong anomaly and the $\chi\eta$ model
5. 5.5 Generalized GG models and strong anomaly
6. 5.6 Strong anomaly in the chiral gauge theories considered in Sec. 3
1. 5.6.1 $SU(6)$ model with a single fermion in a self-adjoint antisymmetric representation
2. 5.6.2 Adjoint QCD with $N_{\rm c}=N_{\rm f}=2$
6. 6 Summary and discussion
7. A Chirally symmetric confining $\psi\eta$ model
8. B Higgs phase of the $\psi\eta$ model
9. C Symmetric confining phase for the $\chi\eta$ model
10. D Higgs phase of the $\chi\eta$ model
11. E Confining symmetric phase of the BY models
12. F Higgs phase in the BY models
13. G Confining symmetric phase of the GG models
14. H Higgs phase in the GG models
## 1 Introduction
One of the mysteries of the world we live in is the fact that it has a
nontrivial chiral property. The macroscopic structures such as biological
bodies have often approximately left-right symmetric forms, but not exactly.
At the molecular level, $O(10^{-6}\,{\rm cm})$, the structure of DNA possesses
a definite chiral spiral form. At the microscopic scales of the fundamental
interactions, $O(10^{-14}\,{\rm cm})$, the left-handed and right-handed quarks
and leptons have distinct couplings to the $SU(3)\times SU(2)_{L}\times
U(1)_{Y}$ gauge bosons. The parity violation in the ”weak interactions”
processes, though it was first considered somewhat weird, later found a
natural explanation [2]: the fundamental entity of our world is a set of Weyl
fermions in the $\big{(}\tfrac{1}{2},0\big{)}$ representation of the Lorentz
group. If such fermions are in a generic complex representation of the gauge
group, the resulting theory will break parity. In other words, the origin of
parity violation is to be traced to the type of building blocks our world is
made of, rather than to a peculiar property of the Fermi interactions.
Most of grand unification schemes such as those based on $SU(5),SO(10)$ or
$E_{7}$ groups are also all based on chiral gauge theories.
The Glashow-Weiberg-Salam (GWS) $SU(2)_{L}\times U(1)_{Y}$ theory (as well as
its GUT generalizations) is a weakly coupled theory and, as such, is well
understood within the framework of perturbation theory. But this also means
that the theory should be regarded, at best, as a very good low-energy
effective theory. In particular, it is unlikely that the gauge symmetry
breaking sector described by a potential term for the Higgs scalar, though
phenomenologically quite successful, is a self-consistent, fundamental
description. Nevertheless, attempts to replace it by new, QCD-like strongly-
coupled gauge theories (such as Technicolor, Extended technicolor, Walking
technicolor, etc.) have not been entirely successful so far.
On the other hand, our understanding of strongly-coupled chiral gauge theories
is today uncomfortably limited [3]-[22]. This is in a striking contrast to the
case of vector-like gauge theories, for which we have an extensive literature.
They include some general theorems [23], [24], lattice simulations [25]-[28],
the effective Lagrangians [29]-[37], the conventional ’t Hooft anomaly
analysis [3], the powerful exact results in ${\cal N}=2$ supersymmetric
theories [38]-[44], space compactification with semi-classical approximation
[45]-[47], and so on.
These theoretical tools are unfortunately unavailable for the analysis of
strongly-coupled chiral gauge theories, with a few exceptions such as the
large $N$ approximation, the ’t Hooft anomaly matching constraints, and some
general considerations based on the renormalization group. Taken together,
they yield some useful but not very detailed knowledge about the dynamics and
phases of the chiral gauge theories. Partial list of the papers on these
efforts are found in [3]-[22]. This kind of situation is certainly limiting
rather severely our capability of finding the place for chiral gauge theories
in the context of a realistic theory of the fundamental interactions beyond
the standard model, e.g., in the context of composite models for the quarks
and leptons, the composite Higgs boson models, the composite dynamical models
for dark matter, and so on. The need for new ideas for making true progress in
this research field is badly felt.
In our view, a little hope for a breakthrough comes from the very recent ideas
involving the concept of generalized symmetries [48]-[51]. Pure Yang-Mills
theories and QCD-like theories have been analyzed by using new, stronger,
version of ’t Hooft anomaly matching constraints, involving $0$-form and
$1$-form symmetries together [52]-[70].
The generalized symmetries are symmetries which do not act on local field
operators, as the conventional symmetries, but only on extended objects, such
as closed lines and surfaces. As the corresponding gauge functions are now
1-form, 2-form fields (in the standard gauging of global symmetries the gauge
transformation parameters which appear in the exponent are just - 0-form -
functions of the spacetime), these new types of symmetries are sometimes
called 1-form-, 2-form-, etc symmetries.
A familiar example of the 1-form symmetry is the ${\mathbbm{Z}}_{N}$ center
symmetry in Euclidean $SU(N)$ Yang-Mills theory at finite temperature, which
acts on the Polyakov loop. The unbroken (or broken) center symmetry by the
vacuum expectation value (VEV) of the Polyakov loop, is a valid criterion for
confinement (or de-confinement) phase. Similarly, the area law / perimeter law
of the VEV of the Wilson loops can be considered as the criterion for
confinement / Higgs phase. Note however that the center symmetry
${\mathbbm{Z}}_{N}$ as used this way is a global 1-form symmetry. The main
input of the new development [52]-[70] is the idea of gauging 1-form
symmetries.
In fact, these generalized symmetries are symmetries of the systems being
considered, even if they act in a way different from the conventional symmetry
action. We are free to decide to ”gauge” these new types of symmetries.
Anomalies one encounters in doing so are obstructions of gauging a symmetry,
which is by definition a ’t Hooft anomaly. And as in the usual requirement of
the ultraviolet (UV) - infrared (IR) ”anomaly matching”, a similar conditions
arise in gauging the generalized (higher-form) symmetries together with some
conventional (”0-form”) symmetries, which have come to be known in recent
literature as a ”mixed ’t Hooft anomaly”. As will be seen below, these
constraints carry significant information on the dynamics of wide classes of
chiral gauge theories as well, which is our main interest.
Very recently, the present authors have realized [96] that the strong anomaly,
which plays a prominent role in the solution of the so-called $U(1)_{A}$
problem in QCD, can also be significant in the study of the phases of chiral
gauge theories, such as those discussed in this review. A key observation is
that the well-known low-energy strong-anomaly effective action for QCD, which
reproduces the effects of the strong anomaly in the low-energy effective
action, has a nontrivial implication on the symmetry breaking pattern itself.
For some reason these ideas have not been applied much to the study of
strongly-interacting chiral gauge theories until now.
It is found that, quite remarkably, the considerations based on the strong
anomaly yield similar indications on the phase of the chiral gauge theories,
as those found by applying the generalized ’t Hooft anomalies. Even though
they are arguments fully independent of each other, the agreement of the
results should not probably be considered entirely accidental. Indeed they
both originate from the proper treatment of the strong anomaly on various
$U(1)$ symmetries present in the theory.
The rest of the work is organized as follows. In Sec. 2 the procedure of the
computing anomalies associated with the gauging of certain 1-form discrete
symmetry is discussed, as this constitutes one of the main theoretical tools
of our analysis. The detail of the discussion is further divided in two parts.
The first part, Sec. 2.1, concerns models with 1-form center symmetry
${\mathbbm{Z}}_{k}\subset{\mathbbm{Z}}_{N}\subset SU(N)$ which does not act on
the matter fermions. These models have ordinary (0-form) discrete symmetries
also, call ${\mathbbm{Z}}_{\ell}$, which are nonanomalous remnants of
anomalous $U(1)$ symmetries.
The second class of models contain some matter fermions in the fundamental or
antifundamental representation of $SU(N)$. Normally one would conclude that
1-form center symmetry ${\mathbbm{Z}}_{N}$ is simply absent in such models.
However, as explained in Sec. 2.2, it turns out that it is still possible to
define a color-flavor locked 1-form center symmetry ${\mathbbm{Z}}_{N}$.
In Sec. 3 and Sec. 4, applications to various chiral gauge theories of these
new ’t Hooft anomaly constraints are explored. In Sec. 3, physics of various
chiral gauge theories of the first kind are discussed, by using the general
results of Sec. 2.1. Sec. 4 is dedicated to the applications of the formulas
found in Sec. 2.2 to two large classes of chiral gauge theories, the so-called
generalized Bars-Yankielowicz (BY) and Georgi-Glashow (GG) models.
After discussing the new, generalized anomalies and their implications in
various kinds of chiral gauge theories, we explore in Sec. 5 the implications
of the strong anomaly on the phases of the same classes of chiral gauge
theories, studied in Sec. 2 $\sim$ Sec. 4.
We conclude in Sec. 6 by summarizing the results found, and by discussing
interesting analogies and contrasts between the dynamics of massless QCD and
chiral gauge theories. A clearer picture of infrared dynamics of many
strongly-coupled chiral gauge theories seems to emerge.
Appendices A \- H are a collection of tables summarizing the massless fermions
and their quantum numbers in various possible phases of BY and GG models.
## 2 Computation of the mixed anomalies
Gauging of a discrete (1-form) center symmetry and calculating anomalies
induced by it in some otherwise nonanomalous global discrete symmetry \- a
generalized ’t Hooft anomaly - is the central theme of the work reviewed here.
Let us go through the basic elements of the analysis and enlist main formulas
needed to get to the physics results discussed in the subsequent sections. For
more general introduction and theoretical considerations on the generalized
symmetries the reader can consult the original literature [50]-[71].
We need to distinguish two different classes of models: the first concerns the
systems where the fermions do not transform under a ${\mathbbm{Z}}_{k}$ ($k$
is a divisor of $N$) subgroup of the $SU(N)$ center. These systems possess a
${\mathbbm{Z}}_{k}$ 1-form symmetry (the ”center symmetry”), which acts
naturally on fundamental Wilson loops. Their analysis is relatively
straightforward. In the second class of system the fundamental fermions
transform non-trivially under the center of the gauge group,
${\mathbbm{Z}}^{\rm c}_{N}$, and only the diagonal combination
${\mathbbm{Z}}_{N}\subset{\mathbbm{Z}}^{\rm c}_{N}\times G_{\rm f}$ (being
$G_{\rm f}$ the flavor symmetry group) leaves them invariant. The study of
these models requires a careful determination of the global structure of the
symmetry group involved.
### 2.1 Gauging a 1-form ${\mathbbm{Z}}_{k}\subset{\mathbbm{Z}}_{N}$ center
symmetry
First consider $SU(N)$ theories with an exact center
${\mathbbm{Z}}_{k}\subset{\mathbbm{Z}}_{N}$ symmetry, $k$ being a divisor of
$N$, under which the matter fermions do not transform. Examples are: pure
$SU(N)$ YM theory or the adjoint QCD, where $k=N$, or various models with
fermions neutral with respect to some ${\mathbbm{Z}}_{k}$, see Sec. 3.
The procedure was formulated in [51] building upon some earlier results
[48]-[50], and used in [52] for $SU(N)$ Yang-Mills theory at $\theta=\pi$. The
methods have been further developed and found other areas of applications
[53]-[68].
1-form center symmetry can be simply understood in the formalism of principal
bundles. Here the gauge and the fermions fields are defined locally on open
patches $U_{i}$ of our spacetime. These local definitions are glued together
by $SU(N)$ valued transitions functions, $g_{ij}:U_{i}\cap U_{j}\rightarrow
SU(N)$. In particular,
$\psi_{i}(x)=R(g_{ij}(x))\psi_{j}(x)\quad x\in U_{i}\cap U_{j}\;,$ (2.1)
where $\psi_{i}$ and $\psi_{j}$ are the local expressions of the field $\psi$
(which transform in the representation R) in the patches $U_{i}$ and $U_{j}$.
To require that the theory is an $SU(N)$ theory (i.e. the fundamental Wilson
loops are meaningful) enforces the cocycle condition,
$g_{ij}g_{jk}g_{ki}=\mathbbm{1}\;,$ (2.2)
in the triple intersection, $U_{ijk}=U_{i}\cap U_{j}\cap U_{k}$.
In this language a global 1-form symmetry transformation multiplies the
transition functions $g_{ij}$ by $\mathbbm{Z}_{k}$ elements, $z_{ij}$ (one for
each simple intersection, $U_{ij}=U_{i}\cap U_{j}$), which satisfy their own
consistency condition
$z_{ij}z_{jk}z_{ki}=1\;.$ (2.3)
This transformation is a symmetry of the system if it does not spoil the
equation (2.1), i.e. if $\mathbbm{Z}_{k}$ does not act on fermions. However,
it can act non-trivially on fundamental Wilson loops.111It acts on non-
contractible Wilson loop, therefore global 1-form gauge transformations are
indexed by elements of $H^{1}(\mathcal{M},\mathbbm{Z}_{k})$. The $z_{ij}$
implement a Čech version of this cohomology group.
If one relaxes the cocycle consistency condition, allowing
$z_{ij}z_{jk}z_{ki}=z_{ijk}\in\mathbbm{Z}_{N}\;,$ (2.4)
one obtains a gauge 1-form symmetry transformation. In this case the condition
(2.2) does not make sense, and must be replaced by
$g_{ij}g_{jk}g_{ki}=B_{ijk}\in\mathbbm{Z}_{k}\;,$ (2.5)
where the new data, $B_{ijk}$, are (a discretized version of) a 2-form
connection.222Similarly, the $B_{ijk}$ are representatives of the second Čeck
cohomology group, $H^{2}(M,\mathbbm{Z}_{k})$. The closeness of $B$ can be seen
on quadruple overlaps. This construction defines an
$\frac{SU(N)}{\mathbbm{Z}_{k}}$ gauge bundle. If one consider the $B_{ijk}$
data dynamical, summing on them in the functional integral, one obtains a
$\frac{SU(N)}{\mathbbm{Z}_{k}}$ gauge theory.
In [49] and [51] a useful construction is presented that reproduces this
gauging in terms of continuous fields. We adopt this description, which is
reviewed below briefly.
The rough idea is to replace the discrete $\mathbbm{Z}_{k}$ 1-form symmetry
with a $U(1)$ 1-form symmetry, at the price of introducing other new degrees
of freedom. Gauge fixing these new degrees of freedom, one can gauge-fix most
of the continuous 1-form symmetry. What remains is the discrete 1-form
symmetry.
As a first step, one must introduce a pair of $U(1)$ 2-form and 1-form333In
most part of this review a compact differential-form notation is used. For
instance, $a\equiv T^{\rm c}A_{\mu}^{\rm c}(x)\,dx^{\mu}$; $F=da+a^{2}$ ;
$F^{2}\equiv F\wedge
F=\frac{1}{2}F^{\mu\nu}F^{\rho\sigma}dx_{\mu}dx_{\nu}dx_{\rho}dx_{\sigma}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}d^{4}x=F^{\mu\nu}{\tilde{F}}_{\mu\nu}d^{4}x$
, and so on. gauge fields $\big{(}B_{\rm c}^{(2)},B_{\rm c}^{(1)}\big{)}$
such that [51]
${k}B^{(2)}_{\mathrm{c}}=dB^{(1)}_{\mathrm{c}}\;,$ (2.6)
satisfying
$\displaystyle B^{(2)}_{\mathrm{c}}\to
B^{(2)}_{\mathrm{c}}+d\lambda_{\mathrm{c}}\;,\qquad B^{(1)}_{\mathrm{c}}\to
B^{(1)}_{\mathrm{c}}+{k}\lambda_{\mathrm{c}}\;.$ (2.7)
$\lambda_{\mathrm{c}}$ is a 1-form gauge function. The $SU(N)$ gauge field $a$
is embedded into a $U(N)$ field,
$\widetilde{a}=a+\frac{1}{k}B^{(1)}_{\mathrm{c}},$ (2.8)
and one requires invariance under $U(N)$ gauge transformation. The gauge field
tensor $F(a)$ is replaced by
$F(a)\to{\tilde{F}}({\tilde{a}})-B^{(2)}_{\mathrm{c}}\;.$ (2.9)
This fixes the manner these $\mathbb{Z}_{k}^{\rm c}$ gauge fields are coupled
to the standard gauge fields $a$. The matter fields must also be coupled to
the $U(N)$ gauge fields, so that the 1-form gauge invariance, (2.7) is
respected. For a Weyl fermion $\psi$ in the representation $R$ this can be
done having the kinetic term as
${\bar{\psi}}\,\gamma^{\mu}\left(\partial+R({\tilde{a}})-\frac{{\cal
N}(R)}{k}B_{\rm c}^{(1)}\right)_{\mu}P_{L}\,\psi\;,$ (2.10)
where $R({\tilde{a}})$ is the appropriate matrix form for the representation;
${\cal N}(R)$ is the $N$-ality of $R$. $P_{L}$ is the projection operator on
the left-handed fermions.
We introduce an external $U(1)_{\psi}$ gauge field $A_{\psi}$ to study the
anomaly, e.g., of a $U(1)_{\psi}$ symmetry $\psi\to e^{i\alpha}\psi$, or of a
discrete subgroup of it, and couple it to the fermion as
${\bar{\psi}}\,\gamma^{\mu}\left(\partial+R({\tilde{a}})-\frac{{\cal
N}(R)}{k}B_{\rm c}^{(1)}+A_{\psi}\right)_{\mu}P_{L}\,\psi\;.$ (2.11)
The standard anomaly calculation for $\psi\to
e^{i\alpha}\psi\simeq\psi+i\alpha\psi$, gives
$\delta S_{\delta
A_{\psi}^{(0)}}=\frac{2\,T(R)}{8\pi^{2}}\int{\mathrm{tr}}F^{2}\,\delta\alpha=2\,T(R)\,{\mathbbm{Z}}\,\delta\alpha\;.$
(2.12)
${\mathbbm{Z}}$ is the integer instanton number, and it leads to the well-
known result that a discrete subgroup
${\mathbbm{Z}}_{2T(R)}\subset U(1)_{\psi}$ (2.13)
remains. $T(R)$ is twice the Dynkin index,
$\mathrm{tr}\,T^{a}T^{b}=\delta^{ab}D(R)\;,\qquad
D\Big{(}\raisebox{-3.0pt}{\yng(1)}\Big{)}=\frac{1}{2}\;,\qquad T(R)\equiv
2\,D(R)\;.$ (2.14)
With $\big{(}B_{\rm c}^{(2)},B_{\rm c}^{(1)}\big{)}$ fields in Eq. (2.11),
$U(1)_{\psi}$ symmetry can further be broken due to the replacement,
${\mathrm{tr}}F^{2}\to{\mathrm{tr}}\big{(}{\tilde{F}}-B^{(2)}_{\rm
c}\big{)}^{2}\;.$ (2.15)
Indeed,
$\frac{1}{8\pi^{2}}\int_{\Sigma_{4}}{\mathrm{tr}}\big{(}{\tilde{F}}-B^{(2)}_{\rm
c}\big{)}^{2}=\frac{1}{8\pi^{2}}\int_{\Sigma_{4}}\big{\\{}{\mathrm{tr}}{\tilde{F}}^{2}-N(B^{(2)}_{\rm
c})^{2}\big{\\}}\;:$ (2.16)
we recall that $B^{(2)}_{\rm c}$ is Abelian, $\propto{\mathbbm{1}}_{N}$, and
that ${\mathrm{tr}}{\tilde{F}}=N\,B^{(2)}_{\rm c}$. The first term is an
integer. The second is
$-\frac{N}{8\pi^{2}}\int_{\Sigma_{4}}\big{(}B^{(2)}_{\rm
c}\big{)}^{2}=-\frac{N}{8\pi^{2}k^{2}}\int_{\Sigma_{4}}dB^{(1)}_{\rm c}\wedge
dB^{(1)}_{\rm c}=\frac{N}{k^{2}}\,{\mathbbm{Z}}\;,$ (2.17)
which is generally fractional. This explains the origin of various
0-form-1-form (mixed) anomalies and the consequent stronger anomaly conditions
in many models discussed in Sec. 3.
### 2.2 Color-flavor locked ${\mathbbm{Z}}_{N}$ center symmetry: Master
formula
Subtler situations present themselves, when a gauge theory of our interest
contains matter Weyl fermions in the fundamental, or antifundamental,
representation of the gauge group $SU(N)$. Ordinarily, this means that the
center symmetry is lost, leaving no possibilities of gauging the 1-form
${\mathbbm{Z}}_{N}$ center symmetry. Actually, in order to consider the ’t
Hooft anomalies one must externally gauge also the flavor symmetry group,
$G_{\rm f}$. Having done so, in the systems of our interest $SU(N)_{\rm
c}\times G_{\rm f}$ is found not to act faithfully. In particular, there is a
$\mathbbm{Z}_{N}$ subgroup that leaves all the fields invariant. In other
words, there is a $\mathbbm{Z}_{N}$ ”color-flavor-locked” 1-form
symmetry.444This hinges upon a quite remarkable property of the generalized
symmetries, that they are all Abelian. This reflects the fact it is not
possible to define time ordering between two extended operators, hence
impossible to define equal-time commutators between them. In the case of
particles, how the (equal-time) commutators can arise in the operator
formalism, as a limit of time-ordered products taken in different orders, is
best explained in Feynman’s book on Path-Integral formulation of quantum
mechanics [72]. Similarly to the previous case, gauging of this 1-form
symmetry allows us to gauge the faithful symmetry group of the system,
$\frac{SU(N)_{\rm c}\times G_{\rm f}}{\mathbbm{Z}_{N}}$.555One should keep in
mind that there are ”more” configurations in $\frac{SU(N)}{\mathbbm{Z}_{k}}$
($\frac{SU(N)_{\rm c}\times G_{\rm f}}{\mathbbm{Z}_{N}}$) than in the $SU(N)$
($SU(N)\times G_{\rm f}$) gauge bundles, i.e. any of the latter always belong
to the former, but not the other way around.
To introduce this kind of systems, and to discuss the method of analysis
developed, we consider the concrete example of an $SU(N)$ gauge theory with
matter left-handed fermions in the reducible, complex representation,
$\yng(2)\oplus(N+4)\,{\bar{{\yng(1)}}}\,$ (2.18)
that is,
$\displaystyle\psi^{\\{ij\\}}\,,\quad\eta_{i}^{B}\,,\qquad{\footnotesize
i,j=1,2,\ldots,N\;,\quad B=1,2,\ldots,N+4}\;,$ (2.19)
(which is the simplest of the so-called Bars-Yankielowicz models). This model
will be referred to as the ”$\psi\eta$” model below.
The symmetry group of the model is
$G_{\rm f}=SU(N+4)\times U(1)_{\psi\eta}\;,$ (2.20)
where $U(1)_{\psi\eta}$ indicates the anomaly-free combination of
$U(1)_{\psi}$ and $U(1)_{\eta}$, associated with the two types of matter Weyl
fermions of the theory. In this model, the conventional ’t Hooft anomaly
matching discussion allows, apparently, a confining phase, with no condensates
and with full unbroken global symmetry, and with some simple set of massless
composite fermions - ”baryons” - saturating the anomaly matching equations,
see Appendix A. Notably, the anomaly constraints are also consistent with a
dynamical Higgs phase, in which the color and (part of) the flavor symmetry
are dynamically broken by certain bi-fermion condensates, see Appendix B.
The situation is analogous in a large class of chiral gauge theories, to be
discussed in Sec. 4 below. Clearly, the conventional ’t Hooft anomaly matching
requirement is not powerful enough to discriminate among possible (confining
or dynamical Higgs) vacua.
To go beyond the conventional (perturbative) ’t Hooft anomaly analyses, it is
necessary to consider the global properties of the symmetry groups, not only
the algebra. For even $N$ the true symmetry group of the model is found to be
[70]:
$SU(N)_{\rm color}\times G_{\rm f}\;,\qquad G_{\rm f}=\frac{SU(N+4)\times
U(1)_{\psi\eta}\times(\mathbb{Z}_{2})_{F}}{\mathbb{Z}_{N}\times\mathbb{Z}_{N+4}}\;,$
(2.21)
and not (2.20), where $(\mathbb{Z}_{2})_{F}$ is the fermion parity,
$\psi,\eta\to-\psi,-\eta$.
Indeed, as promised, there is a subgroup of $SU(N)_{\rm c}\times SU(N+4)\times
U(1)_{\psi\eta}\times(\mathbbm{Z}_{2})_{F}$,
${\mathbbm{Z}}_{N}=SU(N)\cap\\{U(1)_{\psi\eta}\times(\mathbb{Z}_{2})_{F}\\}\;,$
(2.22)
which leaves the matter fields invariant.666There is another independent
subgroup, $\mathbbm{Z}_{N+4}$, which does not act on matter filed, leading to
another $\mathbbm{Z}_{N+4}$ 1-form center symmetry. In [70] the effects of
gauging this flavor center symmetry and the resulting mixed anomalies in the
$\psi\eta$ model have also been taken into account. None of the main results
however were found to depend on it. Here for simplicity we consider only the
gauging of the color-flavor locked center symmetry ${\mathbbm{Z}}_{N}$,
together with $U(1)_{\psi\eta}$ and $({\mathbbm{Z}}_{2})_{F}$. The gauge
transformation with
$\mathrm{e}^{\frac{2\pi\mathrm{i}}{N}}\in\mathbb{Z}_{N}\subset SU(N)$,
$\psi\to\mathrm{e}^{\frac{4\pi\mathrm{i}}{N}}\psi\;,\;\qquad\eta\to\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}}\eta\;,$
(2.23)
can be undone by the following $(\mathbb{Z}_{2})_{F}\times U(1)_{\psi\eta}$
transformation:
$\psi\to(-1)\,\mathrm{e}^{\mathrm{i}\frac{N+4}{2}\frac{2\pi}{N}}\psi=\mathrm{e}^{-\mathrm{i}\frac{N}{2}\frac{2\pi}{N}}\,\mathrm{e}^{\mathrm{i}\frac{N+4}{2}\frac{2\pi}{N}}\psi\;,\qquad\eta\to(-1)\,\mathrm{e}^{-\mathrm{i}\frac{N+2}{2}\frac{2\pi}{N}}\eta=\mathrm{e}^{\mathrm{i}\frac{N}{2}\frac{2\pi}{N}}\,\mathrm{e}^{-\mathrm{i}\frac{N+2}{2}\frac{2\pi}{N}}\eta\;.$
(2.24)
A relevant fact is that the odd elements of $\mathbb{Z}_{N}$ belong to the
disconnected component of $U(1)_{\psi\eta}\times(\mathbb{Z}_{2})_{F}$ whereas
the even elements belong to the connected component of the identity.
The presence of a subgroup which acts trivially means that there is a 1-form
global symmetry. Again, in the discrete language introduced before, it acts on
transition functions. In particular, if $g^{\rm c}_{ij}$, $u_{ij}$ and
$q_{ij}$ are the transition functions for $SU(N)$, $U(1)_{\psi\eta}$ and
$(\mathbbm{Z}_{2})_{F}$, one may introduce some $\mathbbm{Z}_{N}$ transitions
functions (a $\mathbbm{Z}_{N}$ gauge field), $z_{ij}$, and transform
$g_{ij}\rightarrow z_{ij}g_{ij}\;,\quad
u_{ij}\rightarrow(z_{ij})^{-1}u_{ij}\;,\quad\text{and}\quad
q_{ij}\rightarrow(z_{ij})^{-\frac{N}{2}}\;q_{ij}\;.$ (2.25)
If one drops the cocycle condition for $z_{ij}$, one gauges the 1-form
symmetry. In this case one must introduce also the 2-form connection 777Again,
an element of $H^{2}(M,\mathbbm{Z}_{N})$, $B_{ijk}\in\mathbbm{Z}_{N}$.,
described by the new data $B_{ijk}\in\mathbbm{Z}_{N}$, which are read from the
transition functions
$g_{ij}g_{jk}g_{ki}=B_{ijk}\;,\quad u_{ij}u_{jk}u_{ki}=(B_{ijk})^{-1}\;,\quad
q_{ij}q_{ji}q_{ki}=(B_{ijk})^{-\frac{N}{2}}\;.$ (2.26)
This definition assures that all fields (matter, gauge) are well defined in
the triple intersections.
Again, let us turn to the continuous language. As a first step, we have to
gauge $U(1)_{\psi\eta}\times(\mathbbm{Z}_{2})_{F}$, introducing
1. 1.
$A$: $U(1)_{\psi\eta}$ 1-form gauge field,
2. 2.
$A_{2}^{(1)}$: $({\mathbbm{Z}}_{2})_{F}$ 1-form gauge field,
in addition to the dynamical color gauge $SU(N)$ field, $a$.
The gauging of 1-form discrete $\mathbb{Z}_{N}$ symmetry is done by
introducing
$NB_{\mathrm{c}}^{(2)}=dB_{\mathrm{c}}^{(1)}\;.$ (2.27)
These 2-form gauge fields must be coupled to the $SU(N)$ gauge fields $a$ and
$U(1)_{\psi\eta}\times(\mathbb{Z}_{2})_{F}$ gauge fields ($A$ and
$A_{2}^{(1)}$) appropriately. We first embed $a$ in a $U(N)$ gauge field
$\widetilde{a}$ as
$\widetilde{a}=a+\frac{1}{N}B^{(1)}_{\mathrm{c}}$ (2.28)
and requiring the invariance
$\displaystyle B_{\mathrm{c}}^{(2)}$ $\displaystyle\to
B_{\mathrm{c}}^{(2)}+\mathrm{d}\lambda_{\mathrm{c}}\;,\qquad
B_{\mathrm{c}}^{(1)}\to B_{\mathrm{c}}^{(1)}+N\lambda_{\mathrm{c}}\;,$
$\displaystyle\widetilde{a}$
$\displaystyle\to\widetilde{a}+\lambda_{\mathrm{c}}\;.$ (2.29)
In these equations $\lambda_{\mathrm{c}}$ is a properly normalized $U(1)$
gauge field, which satisfies its own Dirac quantization condition.
To reproduce (2.25) correctly in this continuous language, $U(1)_{\psi\eta}$
and $(\mathbb{Z}_{2})_{F}$ gauge fields must also transform,
$A\to A-\lambda_{\mathrm{c}}\;,\qquad A_{2}^{(1)}\to
A_{2}^{(1)}+\frac{N}{2}\lambda_{\mathrm{c}}\;.\;$ (2.30)
The last equation needs a comment, as $A^{(1)}_{2}$ is a $\mathbbm{Z}_{2}$
gauge field, while $\lambda_{\mathrm{c}}$ is a $U(1)$ gauge field. To be
precise, it is more correct to proceed as it has been done with the $SU(N)$
gauge field. In particular one should write a $U(1)$ gauge connection
$\tilde{A}_{2}=A_{2}+\frac{1}{2}B^{(1)}_{c},$ (2.31)
and impose
$\tilde{A}_{2}\rightarrow\tilde{A}_{2}+\frac{N}{2}\lambda_{\mathrm{c}}\;.$
(2.32)
As before $a$ is not a globally defined $SU(N)$ gauge field while $\tilde{a}$
is a correctly normalized $U(N)$ gauge field, and now $\tilde{A}_{2}$ is a
correctly normalized $U(1)$ field.
One has now an $\frac{SU(N)}{{\mathbbm{Z}}_{N}}$ connection rather than
$SU(N)$. It implies that
$\frac{1}{2\pi}\int_{\Sigma_{2}}B_{\mathrm{c}}^{(2)}=\frac{n_{1}}{N}\;,\qquad
n_{1}\in{\mathbbm{Z}}_{N}\;,$ (2.33)
in a closed two-dimensional subspace, ${\Sigma_{2}}$. On a topologically
nontrivial four dimensional spacetime of Euclidean signature which contains
such sub-spaces one has
$\frac{1}{8\pi^{2}}\int_{\Sigma_{4}}(B_{\mathrm{c}}^{(2)})^{2}=\frac{n}{N^{2}}\;,$
(2.34)
where $n\in{\mathbbm{Z}}_{N}$.
The fermion kinetic term with the background gauge fields is determined by the
minimal coupling procedure as
$\displaystyle\overline{\psi}\gamma^{\mu}\left(\partial+\mathcal{R}_{\mathrm{S}}(\widetilde{a})+\frac{N+4}{2}A+\tilde{A}_{2}\right)_{\mu}P_{\mathrm{L}}\psi\;$
$\displaystyle+\,\overline{\eta}\gamma^{\mu}\left(\partial+\mathcal{R}_{\mathrm{F}^{*}}(\widetilde{a})-\frac{N+2}{2}A-\tilde{A}_{2}\right)_{\mu}P_{\mathrm{L}}\eta\;.$
(2.35)
(with an obvious notation). Note that each of the kinetic terms is invariant
under (2.29) and (2.30).
The 1-form gauge invariance of our system can be made completely manifest, by
rewriting the above as
$\displaystyle\overline{\psi}\gamma^{\mu}\left(\partial+[\mathcal{R}_{\mathrm{S}}(\widetilde{a})-\frac{2}{N}B_{\mathrm{c}}^{(1)}]+\frac{N+4}{2}[A+\frac{1}{N}B_{\mathrm{c}}^{(1)}]+[\tilde{A}_{2}-\frac{1}{2}B_{\mathrm{c}}^{(1)}]\right)_{\mu}P_{\mathrm{L}}\psi\;$
$\displaystyle+\,\overline{\eta}\gamma^{\mu}\left(\partial+[\mathcal{R}_{\mathrm{F}^{*}}(\widetilde{a})+\frac{1}{N}B_{\mathrm{c}}^{(1)}]-\frac{N+2}{2}[A+\frac{1}{N}B_{\mathrm{c}}^{(1)}]-[\tilde{A}_{2}-\frac{1}{2}B_{\mathrm{c}}^{(1)}]\right)_{\mu}P_{\mathrm{L}}\eta\;.$
(2.36)
Written this way, the expression inside each square bracket is invariant under
(2.29) and (2.30). This leads to the gauge field strength for the $\psi$ and
$\eta$, in the form used in the analysis à la Stora-Zumino descent procedure
[73, 74], in [70, 71]. The final answer of course does not depend on the
rewriting of the kinetic terms as (2.36); the original form (2.35) is
perfectly adequate for the calculation of the anomaly in a more
straightforward approach explained below.
Under the fermion parity, such as $\psi\to-\psi$, $\eta\to-\eta$ in the
$\psi\eta$ model, the contribution to the
$(\mathbbm{Z}_{2})_{F}-[{\mathbbm{Z}}_{N}]^{2}$ anomaly from a fermion in the
representation ${R}$ is given by the phase in the partition function,
$c_{2}\,\frac{1}{8\pi^{2}}\int_{\Sigma_{4}}{\mathrm{tr}}_{R}\left[\big{(}F(\tilde{a})\big{)}^{2}\right]\,(\pm\pi)$
(2.37)
where $c_{2}$ is the ${\mathbbm{Z}}_{2}$ charge of the fermion. In the case of
the $\psi\eta$ model, for instance, $c_{2}(\psi)=1$, $c_{2}(\eta)=-1$, see
(2.35).
Now
$\displaystyle{\mathrm{tr}}_{R}\left[\big{(}F(\tilde{a})\big{)}^{2}\right]$
$\displaystyle=$
$\displaystyle{\mathrm{tr}}_{R}\left[\big{(}F(\tilde{a})-B^{(2)}_{\rm
c}+B^{(2)}_{\rm c}\big{)}^{2}\right]$ (2.38) $\displaystyle=$
$\displaystyle{\mathrm{tr}}\left[\left(\mathcal{R}_{R}\big{(}F(\tilde{a})-B^{(2)}_{\rm
c}\big{)}+{\cal N}(R)B^{(2)}_{\rm c}{\mathbbm{1}}_{d(R)}\right)^{2}\right]$
$\displaystyle=$
$\displaystyle{\mathrm{tr}}\left[\mathcal{R}_{R}\big{(}F(\tilde{a})-B^{(2)}_{\rm
c})^{2}+{\cal N}(R)^{2}\big{(}B^{(2)}_{\rm
c}\big{)}^{2}{\mathbbm{1}}_{d(R)}\right]\;.$
$\mathcal{R}_{R}$ is the matrix form for the representation $R$ and ${\cal
N}(R)$ its $N$-ality, and we used the fact that
${\mathrm{tr}}_{R}\big{(}F(\tilde{a})-B^{(2)}_{\rm c}\big{)}=0\;,$ (2.39)
valid for an $SU(N)$ element in a general representation.
${\mathbbm{1}}_{d(R)}$ is the $d(R)\times d(R)$ unit matrix ($d(R)$ is the
dimension of the representation $R$). One finds
$\displaystyle{\mathrm{tr}}_{R}\left[\big{(}F(\tilde{a})\big{)}^{2}\right]$
$\displaystyle=$ $\displaystyle
D(R)\,{\mathrm{tr}}_{F}\left[\big{(}F(\tilde{a})-B^{(2)}_{\rm
c}\big{)}^{2}\right]+d(R){\cal N}(R)^{2}\big{(}B^{(2)}_{\rm c}\big{)}^{2}=$
(2.40) $\displaystyle=$ $\displaystyle
D(R)\,{\mathrm{tr}}_{F}\left[F({\tilde{a}})\right]^{2}+\left[-D(R)\cdot
N+d(R){\cal N}(R)^{2}\right]\big{(}B^{(2)}_{\rm c}\big{)}^{2}\;,$
where $D(R)$ is twice the Dynkin index $T_{R}$, (2.14). Note that
$\frac{1}{8\pi^{2}}\int_{\Sigma_{4}}{\mathrm{tr}}_{F}\left[F({\tilde{a}})^{2}\right]\in{\mathbbm{Z}}\;:$
(2.41)
the first term in Eq. (2.40) corresponds to the conventional instanton
contribution to the $({\mathbbm{Z}}_{2})_{F}$ anomaly. In all models of
interest here, however, the sum of the instanton contribution from the
fermions is of th form,
$({\rm an}\,{\rm even}\,{\rm
integer})\times\frac{1}{8\pi^{2}}\int_{\Sigma_{4}}{\mathrm{tr}}_{F}\left[F({\tilde{a}})^{2}\right]\,\times({\pm\pi})=2\pi{\mathbbm{Z}}\;,$
(2.42)
which is trivial.
The fact that $({\mathbbm{Z}}_{2})_{F}$ anomaly is absent in the standard
instanton analysis because of a (nonvanishing) even coefficient, and of the
quantized instanton flux, but not because of an algebraic cancellation from
different fermions, is of utmost importance. Indeed, the gauging of the 1-form
${\mathbbm{Z}}_{N}$ by the introduction of the 2-form gauge fields
$B^{(2)}_{\rm c}$ basically amounts to the fractionalization of the instanton
flux à la ’t Hooft, (2.34), and as a consequence, a nonvanishing mixed anomaly
involving $({\mathbbm{Z}}_{2})_{F}$ can appear.
Thus the non-vanishing mixed $({\mathbbm{Z}}_{2})_{F}-[{\mathbbm{Z}}_{N}]^{2}$
anomaly comes only from the second term of Eq. (2.40), containing the 2-form
gauge field,
$\boxed{\Delta S^{({\rm Mixed}\,{\rm anomaly})}=(\pm\pi)\cdot\sum_{\rm
fermions}c_{2}\,\Big{(}d(R){\cal N}(R)^{2}-N\cdot
D(R)\Big{)}\,\frac{1}{8\pi^{2}}\int_{\Sigma_{4}}\big{(}B^{(2)}_{\rm
c}\big{)}^{2}\;.}$ (2.43)
This is the master formula.
The formula (2.43) gives the result for the mixed anomaly in all models
considered in [70, 71] at once. For instance, for the $\psi\eta$ model, one
gets
$\displaystyle\Delta S^{({\rm Mixed}\,{\rm anomaly})}$ (2.44) $\displaystyle=$
$\displaystyle\frac{\pm\pi}{N^{2}}\left[\left(\frac{N(N+1)}{2}\cdot
4-N(N+2)\right)-(N+4)\left(N\cdot 1-N\cdot 1\right)\right]=\pm\pi\;,$
which means that there is a $({\mathbbm{Z}}_{2})_{F}$ anomaly in the presence
of the ${\mathbbm{Z}}_{N}$ gauging. More precisely, there is an obstruction
for gauging simultaneously ${\mathbbm{Z}}_{N}$, $U(1)_{\psi\eta}$ and
$(\mathbb{Z}_{2})_{F}$, by keeping the equivalence
${\mathbbm{Z}}_{N}\subset
SU(N)\sim{\mathbbm{Z}}_{N}\subset\\{U(1)_{\psi\eta}\times(\mathbb{Z}_{2})_{F}\\}\;.$
(2.45)
One can repeat the same calculation in the possible confining phase without
symmetry breaking, Appendix A. The result is very simple: there is no such
anomaly. This can be traced back to the fact that the massless baryons are
$SU(N)_{\rm c}$ singlet, and any appearance of $B^{(1)}_{\rm c}$ in their
covariant derivative simply cancel. Clearly this mismatch of anomaly forbids
confinement without symmetry breaking.
The same cannot be said in the dynamical Higgs scenario, see Appendix B, as
the color group is broken.
Even though, for concreteness, we discussed above the particularly simple
model, the $\psi\eta$ model, the master formula found above is actually
applicable to any theory, after the correct symmetry is found and after the
fermion kinetic terms, invariant under the 1-form ${\mathbbm{Z}}_{N}$ gauge
symmetry are written down. The results for the
$({\mathbbm{Z}}_{2})_{F}-[{\mathbbm{Z}}_{N}]^{2}$ anomaly in the $\chi\eta$
model as well as all other generalized Bars-Yankielowicz and Georgi-Glashow
models [70], [71] discussed below in Sec. 4 indeed follow straightforwardly
from the master formula (2.43) this way. See Sec. 4 below.
### 2.3 Comments on the paper [78]
In a recent paper [78] the $\psi\eta$ model and $\chi\eta$ model (in our
notation) are studied, and the authors claim that “there is no
$({\mathbbm{Z}}_{2})_{F}$ anomaly”, of the type discussed in the previous
section. Such a statement is however intrinsically ambiguous. It is unclear
whether the authors’ claim is that there is no $({\mathbbm{Z}}_{2})_{F}$
symmetry, or that there is one but is nonanomalous.
Indeed, their argument in their Sec. 2.1 seems to indicate the former; but as
we have made explicit here in (2.21) [71], an independent
$({\mathbbm{Z}}_{2})_{F}$ symmetry exists, but only in the
$SU(N)/{\mathbbm{Z}}_{N}$ gauge theory, not in the original $SU(N)$ theory.
Their comments in Sec. 2.5 also seem to be in line with the first. But the
fact that the $({\mathbbm{Z}}_{2})_{F}$ symmetry we are interested in here
coincides with the angle $2\pi$ space rotation, is well known, and it has been
taken into account in our papers [70, 71]. Any $({\mathbbm{Z}}_{2})_{F}$
anomaly could be cancelled by a space rotation, so in that sense, there would
never be a $({\mathbbm{Z}}_{2})_{F}$ anomaly. But this is not the point. As
there is no a priori guarantee that the Lorentz invariance cannot be
dynamically broken, a $({\mathbbm{Z}}_{2})_{F}$ anomaly arising from the gauge
dynamics cannot be allowed, if the Lorentz invariance is to be maintained.
Their discussion in Sec. 3 about the Higgs phase in these models does not
contain anything new, as compared to what we have discussed about the Higgs
phase, see [71], and Sec. 2.5 and in Appendices B, D, F, H, in this work, for
the $\chi\eta$, $\psi\eta$ models and for all other BY and GG models.
The main point of the paper [78] seems to be in Sec. 2.2, which apparently
leads to the second conclusion, that there is a $({\mathbbm{Z}}_{2})_{F}$
symmetry but is non anomalous. They argue that, by choosing the normalization
of the $\mathbbm{Z}_{2}$ gauge field as
$\int_{\Sigma}dA_{2}^{(1)}=2\pi\,{\mathbbm{Z}}\;,$ (2.46)
(a formula in the line below Eq. (2.13) of [78]), the relation
$2A_{2}^{(1)}-B^{(1)}_{\mathrm{c}}-B^{(1)}_{\mathrm{f}}=dA_{2}^{(0)}\;,$
(2.47)
leads to
$2\,\mathrm{d}A_{2}^{(1)}-{N}B^{(2)}_{\mathrm{c}}-(N+4)B^{(2)}_{\mathrm{f}}=0\;:$
(2.48)
by taking the derivatives of the both sides, hence to the constraints on the
fluxes of $\mathbbm{Z}_{2}$ and $\mathbbm{Z}_{N+4}$ gauge fields,
$\int_{\Sigma_{2}}N\,B^{(2)}_{\mathrm{c}}+\int_{\Sigma_{2}}(N+4)\,B^{(2)}_{\mathrm{f}}=4\pi
k\;,\qquad k\in{\mathbbm{Z}}\;.$ (2.49)
If one chooses not to introduce the 1-form gauging $B^{(2)}_{\mathrm{f}}$ (as
we did in [71]) one would simply get
$\int_{\Sigma_{2}}N\,B^{(2)}_{\mathrm{c}}=4\pi k\;,\qquad
k\in{\mathbbm{Z}}\;,$ (2.50)
and our anomaly (2.43) would indeed disappear. The rest of Sec. 2 in [78] all
follows from the normalization, (2.46).
However, (2.46) means that their background $\mathbbm{Z}_{2}$ gauge field
corresponds to
$\psi\to\psi\;,\qquad\eta\to\eta\;,$ (2.51)
i.e., no transformation (the trivial element of $\mathbbm{Z}_{2}$). The fact
that one finds no anomaly in such a background is certainly correct, but it is
not what one is interested in.
The correct normalization for a $\mathbbm{Z}_{2}$ gauge field is the one we
have adopted,
$\oint A_{2}^{(1)}=\frac{2\pi m}{2}\;,\qquad m\in{\mathbbm{Z}}\;,$ (2.52)
that (for the nontrivial element) corresponds to the holonomy,
$\psi\to-\psi\;,\qquad\eta\to-\eta\;,$ (2.53)
This leads to (we ignore the ${\mathbbm{Z}}_{N+4}$ gauge field)
$\oint dx^{\mu}\big{(}2A_{2}^{(1)}-B^{(1)}_{\mathrm{c}}\big{)}_{\mu}=\oint
dA_{2}^{(0)}=2\pi n\;,\qquad n\in{\mathbbm{Z}}\;,$ (2.54)
and
$\int_{\Sigma_{2}}N\,B^{(2)}_{\mathrm{c}}=2\pi k\;,\qquad
k\in{\mathbbm{Z}}\;,$ (2.55)
and this leads to the anomaly, (2.43).
In other words, our assumption is that it is possible to choose the smallest
cycle of $B^{(2)}$ compatible with the Dirac flux quantization for $B^{(1)}$,
i.e.
$\int_{\Sigma}B^{(2)}=\frac{1}{N}\int_{\Sigma}dB^{(1)}=\frac{2\pi}{N}\;,$
(2.56)
without any topological obstruction. In the discrete language, the analogous
assumption is to be able to choose $B_{ijk}$ to be any element in
$H^{2}(\mathcal{M},\mathbbm{Z}_{N})$.
Actually, if one insists on working with the theory on a smooth manifold,
without any topological defect for the $\mathbbm{Z}_{2}$ gauge field $A_{2}$,
the assumption made above cannot be maintained, as pointed out by ourselves
[70]. And this seems to be the point on which the authors of [78] are trying
to make a clean mathematical statement.
However, $A_{2}$ is not a proper $(\mathbbm{Z}_{2})_{F}$ gauge field, as $a$
is not an $SU(N)$ gauge field. In particular its cocycle condition in triple
overlap might fail, leading to curvature-like insertions at discrete points.
Moreover, the 1-form gauging of $\mathbbm{Z}_{N}$ invalidates the naive Dirac
quantization condition for $A_{2}$, as it can be checked directly: if one
separates a 2D cycle $\Sigma$ through the curve $\gamma$ in
$\Sigma_{1}\cup\Sigma_{2}$, one obtains
$\displaystyle\int_{\Sigma}dA_{2}$ $\displaystyle=$
$\displaystyle\int_{\Sigma_{1}}dA_{2}+\int_{\Sigma_{2}}dA_{2}=\int_{\gamma}(A_{2})_{\Sigma_{1}}-\int_{\gamma}(A_{2})_{\Sigma_{2}}$
(2.57) $\displaystyle=$
$\displaystyle\frac{N}{2}\int_{\gamma}\lambda_{12}=k\pi\;,\qquad
k\in{\mathbbm{Z}}$
(see (2.30)) as $\lambda_{12}$ (1-form gauge transition functions) 888The
simple fact that the $\mathbbm{Z}_{2}$ gauge field transforms non-trivially
and changes from a patch to another, means that gauging of 1-form symmetry has
been appropriately implemented. Indeed, thanks to (2.57), a Wilson loop of the
form $e^{\oint A_{2}}$ is not 1-form gauge invariant, i.e., it is not a proper
line operator. This is where we differ from part of the analysis of [78]. is
a $\mathbbm{Z}_{N}$ 1-form gauge field, satisfying
$\oint_{\gamma}\lambda=\frac{2\pi}{N}\;.$ (2.58)
This leads to the more general flux quantization (2.56), and allows to insert
an odd number of flux insertion in the surface.
To recapitulate, in half of [78] the authors argue that there is no
$(\mathbbm{Z}_{2})_{F}$ symmetry; in the other half, they discuss the
background $(\mathbbm{Z}_{2})_{F}$-${\mathbbm{Z}}_{N}$ 1-form gauge fields,
corresponding however to the trivial element of the 1-form
$(\mathbbm{Z}_{2})_{F}$ transformation, finding no anomaly.
### 2.4 Comments on the papers [79, 80]
Two interesting papers appeared recently, which discuss the $\psi\eta$ and
$\chi\eta$ models. The authors of [79, 80] start from the ${\cal N}=1$
supersymmetric version of the models, and introduce a particular (“anomaly-
mediated”) supersymmetry breaking perturbation. In the second paper (on the
$\psi\eta$ model) this is done by making use of the known (Seiberg-) duality
for this systems, at the origin of the moduli space of this model. In the
$\psi\eta$ model, with $SU(N)$ gauge group and with a global symmetry group
$G_{\rm f}=SU(N+4)\times U(1)_{\psi\eta}\;,$ (2.59)
the authors claim [80] that for $N\geq 21$ the global symmetry is broken to
$SO(N)$, with no massless composite fermions, whereas for $N<21$ the system
flows into a conformal fixed point in the IR.
For the $\chi\eta$ model, with odd $N$, they argue [79] that the global
symmetry $SU(N-4)$ is spontaneously broken to $Sp(N-5)$. For $N$ even the
unbroken symmetry is claimed to be $Sp(N-4)$.
We shall not go into the details and merits of their analyses, but will make
only a few general comments on their use of the supersymmetric models as the
starting point of the analysis. First of all, in supersymmetric version of
these models, there are often nontrivial quantum moduli space of vacua (vacuum
degeneracies, or flat directions), whereas in the nonsupersymmetric chiral
models we are studying here the vacuum is always unique and strongly coupled.
It is a nontrivial question which point in the moduli space of the
supersymmetric theory (apart from which perturbation to use) is the correct
one to choose, to start the analysis 999This subtle problem is discussed in
[81] in a slightly different but basically similar context, of perturbing a
${\cal N}=2$ supersymmetric model to ${\cal N}=0$ (nonsupersymmetric) model,
in the attempt of finding out the correct infrared dynamics of the non
supersymmetric $SU(2)$ theories with different number of (adjoint) flavors. .
Secondly, all bifermion condensates such as $\langle\psi\eta\rangle$ (in the
$\psi\eta$ model) and $\langle\chi\eta\rangle$ and $\langle\chi\chi\rangle$
(in the $\chi\eta$ model), which are analogue of the quark condensate in QCD,
and play the central roles in the (candidate) Higgs vacua of these models, are
forbidden in supersymmetric version of the models, as can be easily proven by
use of supersymmetric Ward-Takahashi identities [82]. In other words, these
condensates are absent, unless supersymmetry is dynamically broken, which does
not occur in general, supersymmetric chiral gauge theories [82]. Also, in
supersymmetric models, the global symmetry breaking occurs due to the
condensation of scalar fields, which do not exist in nonsupersymmetric
theories. Because of all this, the infrared dynamics of supersymmetric and
nonsupersymmetric theories are usually very different, even though the gauge
group and the global symmetry group are the same. A strong bifermion
condensates such as $\langle\psi\eta\rangle\sim\Lambda^{3}$ or
$\langle\chi\eta\rangle\sim\Lambda^{3}$ are intrinsically nonperturbative
effects. They cannot be found via small perturbations in a theory in which
they vanish by symmetries.
The crucial question whether or not a phase transition occurs when the
supersymmetry breaking mass parameters introduced reach some critical values,
seems to be unanswered in [79, 80].
### 2.5 Higgs phase and anomaly-matching
As said above, it is possible to satisfy the standard ’t Hooft anomaly
matching also in the Higgs phase, see Appendix B. Even though these results
are known from the earlier work [6]-[19] and in [71], the remarkable way it
works, as compared to the matching equations in the ”confining vacua”, is
perhaps not generally known. The Higgs phase of these chiral theories are, in
general, described by massless NG bosons together with some massless fermions.
These fermions saturate the conventional ’t Hooft anomaly triangles with
respect to the unbroken flavor symmetries. The way they do is, however, quite
remarkable, and in our view, truly significant. As can be seen from Table 3,
Table 5, and in similar Tables 7, 8, 10 and 11 for the generalized BY and GG
models (in Appendices B, D, F, H), the set of fermions remaining massless in
UV and those in the IR are identical in their quantum numbers, charges, and
multiplicities. Therefore, the matching of anomalies (in the unbroken global
symmetries) is completely automatic, and natural. No arithmetic equations need
be solved. We may further argue that this way the system ”solves” ’t Hooft’s
anomaly-matching conditions in the true sense. Note that this solution (Higgs
phase vacua, with given sets of condensates) is stable, in the sense that any
extra (1-form) gauging or possible new mixed anomalies would not introduce any
new constraints: the matching continues to be automatic.
## 3 Physics of models with ${\mathbbm{Z}}_{k}\subset{\mathbbm{Z}}_{N}$
center symmetry
In this section we review the study of symmetry breaking, implied by the
various mixed anomalies of the type,
${\mathbbm{Z}}_{\ell}^{(0)}-\big{[}{\mathbbm{Z}}_{k}^{(1)}\big{]}^{2}$, where
${\mathbbm{Z}}_{\ell}^{(0)}$ is some 0-form (ordinary) discrete symmetry, and
${\mathbbm{Z}}_{k}^{(1)}$ is a 1-form symmetry, based on a subgroup,
${\mathbbm{Z}}_{k}\subset{\mathbbm{Z}}_{N}$ of the color $SU(N)$ center. The
method of analysis has already been explained in Sec. 2.1.
${\mathbbm{Z}}_{\ell}$ and ${\mathbbm{Z}}_{k}$ depend on the particular model
considered, but as we will see, many interesting models can be analyzed this
way [69].
### 3.1 Self-adjoint antisymmetric tensor matter
We start with a class of $SU(N)$ gauge theories ($N$ even) where the matter
fermions are in the $\frac{N}{2}$ rank fully antisymmetric representation.
There is a 1-form ${\mathbbm{Z}}_{\frac{N}{2}}^{\rm c}$ center symmetry
present, and we wish to know if some mixed ’t Hooft anomaly with the 0-form
(ordinary) symmetries might arise.
#### 3.1.1 $SU(6)$ gauge group
Our first example is an $N=6$ theory with $N_{\rm f}$ flavors of Weyl fermions
in the representation
${\underline{20}}\,=\,\yng(1,1,1)\ .$ (3.1)
This ($SU(6)$) is the simplest nontrivial case of interest, as we will see.
Moreover, if $N_{f}\leq 10$ the theory is asymptotic free, and discussions
basically similar to those below can be worked out [69].
There is a $U(1)_{\psi}$ global symmetry, in all these cases, broken by the
instantons to a global discrete ${\mathbbm{Z}}_{6N_{\rm f}}^{\psi}$ symmetry,
which is further broken by the 1-form gauging to ${\mathbbm{Z}}_{2N_{\rm
f}}^{\psi}$. This last step is due to a mixed ’t Hooft anomaly, that is, there
is an obstruction to gauging such a ${\mathbbm{Z}}_{\frac{N}{2}}^{\rm c}$
discrete center symmetry, together with the global ${\mathbbm{Z}}_{6N_{\rm
f}}^{\psi}$ symmetry.
Below, we will take $N_{\rm f}=1$. The model was studied first in [59] and
then in [69].
This model has a nonanomalous ${\mathbbm{Z}}_{6}^{\psi}$ symmetry,
${\mathbbm{Z}}_{6}^{\psi}\,:\quad\psi\to e^{\frac{2\pi i}{6}j}\psi\ ,\qquad
j=1,2,\dots,6\ .$ (3.2)
which is a subset of $U(1)_{\psi}$. The system has also an exact center
symmetry acting Wilson loops as
${\mathbbm{Z}}_{3}^{\rm c}\,:\quad e^{i\oint A}\to e^{\frac{2\pi
i}{6}k}e^{i\oint A}\ ,\qquad k=2,4,6\ ,$ (3.3)
which does not act on $\psi$.
By introducing the ${\mathbbm{Z}}_{3}^{\rm c}$ gauge fields,
${3}B^{(2)}_{\mathrm{c}}=dB^{(1)}_{\mathrm{c}}\;,$ (3.4)
use of (2.15)-(2.17) gives, for the anomalous ${\mathbbm{Z}}_{6}^{\psi}$
transformation,
$\left(\frac{6}{8\pi^{2}}\int{\mathrm{tr}}{\tilde{F}}^{2}-\frac{6N}{8\pi^{2}}\int(B_{\rm
c}^{(2)})^{2}\right)\delta A_{\psi}^{(0)}\;,\qquad\delta
A_{\psi}^{(0)}=\frac{2\pi{\mathbbm{Z}}_{6}^{\psi}}{6}\;.$ (3.5)
The first term in (3.5) is trivial, as
$\frac{1}{8\pi^{2}}\int{\mathrm{tr}}{\tilde{F}}^{2}\in{\mathbbm{Z}}\,,\qquad
A_{\psi}=dA_{\psi}^{(0)}\;,$ (3.6)
(which is the standard gauge anomaly, reducing
$U(1)_{\psi}\longrightarrow{\mathbbm{Z}}_{6}^{\psi}$).
Due to the second term in (3.5), $\delta A_{\psi}^{(0)}$ gets now multiplied
by ($N=6$ here)
$-\frac{6N}{8\pi^{2}}\int\big{(}B_{\rm
c}^{(2)}\big{)}^{2}=-6N\Big{(}\frac{1}{3}\Big{)}^{2}{\mathbbm{Z}}=-6\,\frac{2}{3}{\mathbbm{Z}}\
.$ (3.7)
We see the reduction of the global chiral ${\mathbbm{Z}}_{6}^{\psi}$ symmetry
$\delta A_{\psi}^{(0)}=\frac{2\pi\ell}{6}\ ,\qquad\ell=1,2,\ldots,6$ (3.8)
to its subgroup ($\ell=3,6$),
${\mathbbm{Z}}_{6}^{\psi}\longrightarrow{\mathbbm{Z}}_{2}^{\psi}\ .$ (3.9)
Thus it is not possible that the vacuum of this system is confining, has a
mass gap, and with no condensates breaking the ${\mathbbm{Z}}_{6}^{\psi}$
symmetry.
What are the implications of (3.9) on the phase of the theory? First of all,
it implies a threefold vacuum degeneracy, under the assumption that the system
confines (with mass gap) and that no massless fermions are present in the
infrared, on which $\frac{{\mathbbm{Z}}_{6}^{\psi}}{{\mathbbm{Z}}_{2}^{\psi}}$
can act. A natural assumption is that there are some condensates which
”explain” such a reduction of the symmetry in the infrared. However, physics
depends on which condensates form. The simplest assumption is that a bi-
fermion condensate
$\langle\psi\psi\rangle\sim\Lambda^{3}\neq 0$ (3.10)
forms. As $\psi\in{\underline{20}}$ a scalar bi-fermion composite may be in
one of the irreducible representations of $SU(6)$, appearing on the right hand
side of the composition-decomposition
$\yng(1,1,1)\otimes\yng(1,1,1)=\yng(1,1,1,1,1,1)\oplus\yng(2,1,1,1,1)\oplus+\ldots\;.$
(3.11)
The most attractive channel is the first, ${\underline{1}}$, it vanishes by
the Fermi statistics. This leaves us with the second best possibility that
$\psi\psi$ in the adjoint representation acquires a VEV (dynamical Higgs
mechanism) [4, 18, 19].
Note that even though such a condensate should necessarily be understood as a
gauge-dependent form of some gauge invariant VEV, it unambiguously determines
the breaking of global discrete chiral symmetry as (3.9), where the broken
symmetry $\frac{{\mathbbm{Z}}^{\psi}_{6}}{{\mathbbm{Z}}^{\psi}_{2}}$ acts on
the degenerate vacua, permuting them. The reason for this is that as the
global symmetry group ${\mathbbm{Z}}_{6}^{\psi}$ commutes with the color
$SU(6)$ a gauge transformation cannot undo the nontrivial transformation of
the condensate under ${\mathbbm{Z}}_{6}^{\psi}$.
Of course, four-fermion, gauge-invariant condensates such as
$\langle\psi\psi\psi\psi\rangle\neq 0\;,\qquad{\rm
or}\qquad\langle{\bar{\psi}}{\bar{\psi}}\psi\psi\rangle\neq 0\;,$ (3.12)
could also form, first of which also breaks ${\mathbbm{Z}}^{\psi}_{6}$ as in
(3.9).101010We however do not share the view expressed in [59] that the gauge
non-invariance of the bi-fermion composite $\psi\psi$ means
$\langle\psi\psi\rangle=0\;;$ $\langle\psi\psi\psi\psi\rangle\neq 0$.
The bi-fermion $\psi\psi$ condensate being in the adjoint representation, it
is possible that physics in the infrared is described by full dynamical
Abelianization [18, 19]. The low-energy theory could be an Abelian $U(1)^{5}$
theory. In such a case, although the infrared theory may look trivial, there
is a remnant of the ${\mathbbm{Z}}_{6}$ symmetry of the UV theory. Domain
walls which connect the three vacua would exist, and nontrivial infrared $3$D
physics can appear there.
$SU(6)$ models with $N_{\rm f}\geq 2$ have been studied also. Physics
implications from the mixed anomaly turn out to depend quite nontrivially on
the value of $N_{\rm f}$[69].
#### 3.1.2 $SU(N)$ models
We next consider $SU(N)$ ($N$ general, even) theory, with left-handed fermions
$\psi$ in the self-adjoint, totally antisymmetric representation. It exhibits
some interesting features of the generalized anomalies.
The first coefficient of the beta function is
$b_{0}=\frac{11N-2N_{\rm f}T_{R}}{3}\ .$ (3.13)
The twice Dynkin index is given by
$2\,T_{R}={{N-2}\choose{\frac{N-2}{2}}}\ .$ (3.14)
See Table 3.15 for $2T_{R}$ and $d(R)$ for some even values of $N$ .
$\begin{tabular}[]{|c |c cccc| }\hline\cr$N$&$4$&$6$&$8$&$10$&$12$\\\
\hline\cr$2\,T_{R}$&$2$&$6$&$20$&$70$&$252$\\\
$d(R)$&$6$&$20$&$70$&$252$&$924$\\\ \hline\cr\end{tabular}\ .$ (3.15)
We limit ourselves to asymptotically free theories ($N\leq 10$).
The system has an exact 1-form symmetry:
${\mathbbm{Z}}_{\frac{N}{2}}^{\rm c}\,:\quad e^{i\oint A}\to e^{\frac{2\pi
i}{N}k}\,e^{i\oint A}\ ,\qquad k=2,4,\ldots N\ ,$ (3.16)
as well as a global discrete symmetry:
${\mathbbm{Z}}_{2T_{R}}\,:\quad\psi\to e^{\frac{2\pi i}{2T_{R}}j}\,\psi\
,\qquad j=1,2,\ldots 2T_{R}\,.$ (3.17)
By introducing a 1-form gauge fields $\big{(}B_{\rm c}^{(2)},B_{\rm
c}^{(1)}\big{)}$
$\frac{N}{2}B_{\rm c}^{(2)}=dB_{\rm c}^{(1)}\ $ (3.18)
(see Sec. 3), one arrives at the conclusion that the phase of the partition
function is transformed by
$-\frac{2\pi k}{2T_{R}}2T_{R}\frac{4}{N}{\mathbbm{Z}}=-{2\pi
k}\frac{4}{N}{\mathbbm{Z}}\ ,\qquad k=1,2,\dots,2T_{R}\ ,$ (3.19)
under ${\mathbbm{Z}}_{2T_{R}}$. In other words, ${\mathbbm{Z}}_{2T_{R}}$
become in general anomalous. The consequence of this mixed anomaly however
depends on $N$ nontrivially:
(i)
For $N=4$, the mixed anomaly vanishes:
$\frac{4}{N}=1\ .$ (3.20)
(ii)
For $N=4\ell$, $\ell\geq 2$, instead,
$\frac{4}{N}=\frac{1}{\ell}\ ,$ (3.21)
and the discrete symmetry breaking takes the form,
${\mathbbm{Z}}_{2T_{R}}^{\psi}\longrightarrow{\mathbbm{Z}}_{\frac{2T_{R}}{\ell}}^{\psi}\;.$
(3.22)
For $N=4\ell$, we note that $2T_{R}$ is an integer multiple of $\ell$.
(iii)
Finally, for $N=4\ell+2$,
$2T_{R}\cdot\frac{4}{N}=2T_{R}\cdot\frac{2}{2\ell+1}\ ,$ (3.23)
thus the breaking of the discrete symmetry is
${\mathbbm{Z}}_{2T_{R}}^{\psi}\longrightarrow{\mathbbm{Z}}_{\frac{2T_{R}}{2\ell+1}}^{\psi}\;.$
(3.24)
Just for a check, for $N=4\ell+2$, $2\ell+1$ is a divisor of $2T_{R}$ (see
Appendix of [69]).
The systematics of different cases, (i) $\sim$ (iii), above, can be nicely
understood in terms of the properties of the fractional instantons (torons) of
this model. See [69].
### 3.2 Adjoint QCD
$SU(N)$ theories with $N_{\rm f}$ Weyl fermions $\lambda$ in the adjoint
representation, are sometimes called ”the adjoint QCD”. These systems have
been extensively studied by using different methods: by semi-classical
analysis [83], by direct lattice simulations [84], and more recently, by
studying the mixed anomalies [52, 53, 56]. See also [85], and more recent work
[60, 61].
Here the color ${\mathbbm{Z}}_{N}^{\rm c}$ center 1-form symmetry is exact,
which can be fully gauged. The system has also a nonanomalous ordinary
($0$-form) discrete chiral symmetry,
${\mathbbm{Z}}_{2N_{\rm f}N}^{\lambda}:\quad\lambda\to e^{\frac{2\pi
i}{2N_{\rm f}N}k}\lambda\ ,\qquad k=1,2,\ldots,2N_{\rm f}N\;,$ (3.25)
as is well known. A set of gauge fields are introced:
* •
$A_{\lambda}$: ${\mathbbm{Z}}_{2N_{\rm f}N}^{\lambda}$ 1-form gauge field, for
(3.25);
* •
$B^{(2)}_{\rm c}$: $\mathbb{Z}_{N}^{\rm c}$ 2-form gauge field.
${\mathbbm{Z}}_{2N_{\rm f}N}^{\lambda}$ is found to induces a phase change in
the partition function,
$\left(\frac{2NN_{\rm
f}}{8\pi^{2}}\int{\mathrm{tr}}{\tilde{F}}^{2}-\frac{2N^{2}N_{\rm
f}}{8\pi^{2}}\int(B_{\rm c}^{(2)})^{2}\right)\delta A_{\lambda}^{(0)}\ ,$
(3.26) $\delta A_{\lambda}^{(0)}\in\frac{2\pi i}{2NN_{\rm f}}{\mathbbm{Z}}\;.$
(3.27)
The first term conserves ${\mathbbm{Z}}_{2NN_{\rm f}}^{\lambda}$; the second
term
$\Delta S(\delta A_{\lambda}^{(0)})\in\frac{2\pi i}{N}{\mathbbm{Z}}\ ,$ (3.28)
breaks the chiral discrete symmetry further as
${\mathbbm{Z}}_{2NN_{\rm f}}^{\lambda}\longrightarrow{\mathbbm{Z}}_{2N_{\rm
f}}^{\lambda}$ (3.29)
as found in [52, 53, 56].
The case of $SU(2)$, $N_{\rm f}=2$, is of particular interest. In this case,
the discrete chiral symmetry ${\mathbbm{Z}}_{8}^{\lambda}$ is broken by the
1-form ${\mathbbm{Z}}_{2}^{\rm c}$ gauging as
${\mathbbm{Z}}_{8}^{\lambda}\longrightarrow{\mathbbm{Z}}_{4}^{\lambda}\ .$
(3.30)
The invariance of the standard $SU(2)$ theory
$\lambda\to e^{\pm\frac{2\pi i}{8}}\lambda\ ,$ (3.31)
becomes anomalous.
What is the implication of these results on the physics in the infrared? A
familiar lore about the infrared dynamics of this system is (for instance, see
[85]) that a condensate
$\langle\lambda^{\\{I}\lambda^{J\\}}\rangle\neq 0\ ,\qquad SU(2)_{\rm
f}\longrightarrow SO(2)_{\rm f}\;$ (3.32)
($I,J=1,2$ being the flavor $SU(2)_{\rm f}$ indices) forms. That would lead to
four-fold degenerate vacua, and in each of them, two NG bosons.
In an interesting work [56] Anber and Poppitz have proposed that the system
instead may develop a four-fermion condensate,
$\langle\lambda\lambda\lambda\lambda\rangle\neq 0\ ,\quad{\rm
with}\quad\langle\lambda\lambda\rangle=0\;.$ (3.33)
Such a condensate breaks ${\mathbbm{Z}}_{8}^{\lambda}$ spontaneously to
${\mathbbm{Z}}_{4}^{\lambda}$, leaving only doubly degenerate $SU(2)_{\rm f}$
symmetric vacua (and no NG bosons). Massless baryons of spin $\tfrac{1}{2}$
$B\sim\lambda\lambda\lambda\;$ (3.34)
(which is necessarily a doublet of the unbroken $SU(2)_{\rm f}$) should appear
in the infrared spectrum to saturate all the conventional ’t Hooft and Witten
anomaly matching conditions. The action of the broken
${\mathbbm{Z}}_{8}^{\lambda}/{\mathbbm{Z}}_{4}^{\lambda}$ is a permutation
between the two degenerate vacua,
$\langle\lambda\lambda\lambda\lambda\rangle\to-\langle\lambda\lambda\lambda\lambda\rangle\
.$ (3.35)
As usual in an anomaly-matching discussion one can tell some dynamical
scenario is consistent, but not that such a vacuum is necessarily realized. It
remains to establish which between the familiar $SO_{f}(2)$ symmetric vacuum
and the proposed $SU(2)_{\rm f}$ symmetric one, is actually realized.
The adjoint QCD with general $N$ reduces to ${\cal N}=1$ supersymmetric Yang-
Mills theory, for the spacial case of $N_{\rm f}=1$. A great number of results
on nonperturbative aspects are known [37, 86, 82, 87] there. Note that in this
case (3.29) leads to an $N$ fold vacuum degeneracy, in agreement with the
well-known Witten index of pure ${\cal N}=1$ $SU(N)$ Yang-Mills theory.
One may also start from the ${\cal N}=2$ supersymmetric $SU(2)$ Yang-Mills
theory, where many exact results for the infrared physics are known [38, 39,
44]. It can be deformed to ${\cal N}=1$ theory by an adjoint-scalar mass
perturbation, which yields a confining, chiral symmetry breaking vacua. Exact
calculation of gauge fermion condensates $\langle\lambda\lambda\rangle$ from
this viewpoint can be found in [88, 89]. The pure ${\cal N}=2$ theory could
also be perturbed directly to ${\cal N}=0$ [81]. In principle, such an
approach can give hints about $N_{\rm f}=2$ adjoint QCD, even though it is not
a simple task to identify correctly the vacuum which can be reached by such a
deformation.
### 3.3 QCD with ”tensor quarks”
We now move to theories with matter fermions in $N_{\rm f}$ pairs of
$\psi,{\tilde{\psi}}=\yng(2)\oplus{\bar{\yng(2)}}$ (3.36)
or
$\psi,{\tilde{\psi}}=\yng(1,1)\oplus{\bar{\yng(1,1)}}\;.$ (3.37)
For reference, the standard QCD quarks are in $\yng(1)\oplus{\bar{\yng(1)}}$.
The first beta-function coefficient is
$b_{0}=\frac{11N-2N_{\rm f}(N\pm 2)}{3}\ .$ (3.38)
As the $k=\frac{N}{2}$ element of the center ${\mathbbm{Z}}_{N}$ does not act,
there is a
${\mathbbm{Z}}_{2}^{\rm c}\subset{\mathbbm{Z}}_{N}^{\rm c}$ (3.39)
center symmetry.111111This model has been considered by Cohen [90], in
particular in relation with such a center symmetry, and concerning the
possible existence of an order parameter for confinement. Also there is a
discrete axial symmetry subgroup
${\mathbbm{Z}}_{2N_{\rm f}(N\pm 2)}^{\psi}\,:\quad\psi\to e^{\tfrac{2\pi
i}{2N_{\rm f}(N\pm 2)}}\,\psi\ ,\quad{\tilde{\psi}}\to e^{\tfrac{2\pi
i}{2N_{\rm f}(N\pm 2)}}\,{\tilde{\psi}}\,,$ (3.40)
respected by instantons. The $\pm$ refer respectively to two types of models,
Eq. (3.36) and Eq. (3.37).
Let us study for simplicity the $N_{\rm f}=1$ theory: the analysis is similar
to the cases discussed in the preceding sections. The anomaly is given by
$\left(-\frac{2(N\pm 2)}{8\pi^{2}}{\mathrm{tr}}{\tilde{F}}^{2}+\frac{2N(N\pm
2)}{8\pi^{2}}(B_{\rm c}^{(2)})^{2}\right)\delta A_{\psi}^{(0)}$ (3.41)
where
$\delta A_{\psi}^{(0)}\in\frac{2\pi}{2(N\pm 2)}{\mathbbm{Z}}_{2(N\pm 2)}\;.$
(3.42)
Now
$\frac{2(N\pm 2)}{8\pi^{2}}\int{\mathrm{tr}}{\tilde{F}}^{2}\in 2(N\pm
2){\mathbbm{Z}}\ ,$ (3.43)
means that the first term of (3.41) is trivial. By using
$\frac{1}{8\pi^{2}}\int(B_{\rm c}^{(2)})^{2}=\frac{1}{4}{\mathbbm{Z}}\ ,$
(3.44)
the second term gives an anomaly
$A=2\pi\frac{N}{4}\,{\mathbbm{Z}}\;.$ (3.45)
We find therefore no anomaly for $N=4\ell$; for $N=4\ell+2$, the (1-form)
gauging breaks the discrete symmetry as
${\mathbbm{Z}}_{2(N\pm 2)}^{\psi}\longrightarrow{\mathbbm{Z}}_{N\pm 2}^{\psi}\
.$ (3.46)
The assumption of the ”quark condensate”
$\langle\psi{\tilde{\psi}}\rangle\neq 0\;,$ (3.47)
is consistent with (3.46). The bi-fermion condensate (3.47) however breaks the
discrete symmetry as
${\mathbbm{Z}}_{2(N\pm 2)}^{\psi}\longrightarrow{\mathbbm{Z}}_{2}^{\psi}\ ,$
(3.48)
i.e., stronger than suggested by (3.46).
### 3.4 Chiral theories with $\tfrac{N-4}{k}$ $\psi^{\\{ij\\}}$’s and
$\tfrac{N+4}{k}$ ${\bar{\chi}_{[ij]}}$’s
Our next theoretical laboratory is chiral $SU(N)$ gauge theories with matter
fermions in a complex representation, $\tfrac{N-4}{k}$ $\psi^{\\{ij\\}}$’s and
$\tfrac{N+4}{k}$ ${\bar{\chi}_{[ij]}}$, or
$\frac{N-4}{k}\,\,\yng(2)\oplus\frac{N+4}{k}\,\,{\bar{\yng(1,1)}}\ .$ (3.49)
$k$ is a common divisor of $(N-4,N+4)$ and $N\geq 5$. Asymptotic freedom
requirement
$11N-\frac{2}{k}(N^{2}-8)>0\ ,$ (3.50)
is compatible with various possible choices for $(N,k)$. We studied two simple
models in [69]:
(i)
$(N,k)=(6,2)$: $SU(6)$ with
$\yng(2)\oplus 5\,\,{\bar{\yng(1,1)}}\ ;$ (3.51)
(ii)
$(N,k)=(8,4)$: $SU(8)$ with
$\yng(2)\oplus 3\,\,{\bar{\yng(1,1)}}\ .$ (3.52)
Below we review the results of the analysis of the first model, $SU(6)$ theory
with matter fields in $21$ $\oplus$ $5$ $\otimes$ ${15}$∗. The implications of
the mixed anomalies turn out to be quite subtle even for this simple model, as
will be seen.
Classically the symmetry group is
$SU(5)\times U(1)_{\psi}\times U(1)_{\chi}\ .$ (3.53)
The anomalies:
$\displaystyle U(1)_{\psi}-[SU(6)]^{2}$ $\displaystyle=$
$\displaystyle\frac{T_{\tiny\yng(2)}}{T_{\tiny\yng(1)}}=N+2=8\ ,$
$\displaystyle U(1)_{\chi}-[SU(6)]^{2}$ $\displaystyle=$
$\displaystyle\frac{5T_{\tiny\bar{\yng(1,1)}}}{T_{\tiny\yng(1)}}=5(N-2)=20\ ,$
(3.54)
fixes the charges of the nonanomalous $U(1)_{\psi\chi}\subset
U(1)_{\psi}\times U(1)_{\chi}$ symmetry:
$(Q_{\psi},Q_{\chi})=(5,-2)\ .$ (3.55)
There are also unbroken discrete groups:
$U(1)_{\psi}\longrightarrow{\mathbbm{Z}}_{8}^{\psi}\ ,\qquad
U(1)_{\chi}\longrightarrow{\mathbbm{Z}}_{20}^{\chi}\ .$ (3.56)
By studying the overlap of
${\mathbbm{Z}}_{8}^{\psi}\times{\mathbbm{Z}}_{20}^{\chi}$ and
$U(1)_{\psi\chi}$ one arrives at
$\frac{U(1)_{\psi\chi}\times{\mathbbm{Z}}_{8}^{\psi}\times{\mathbbm{Z}}_{20}^{\chi}}{{\mathbbm{Z}}_{40}}\sim
U(1)_{\psi\chi}\times{\mathbbm{Z}}_{4}\;$ (3.57)
(see (3.63) and (3.64) below). Taking furthermore the color center and
$SU_{f}(5)$ center into account, the true anomaly-free symmetry group is:
$\frac{SU(5)\times U(1)_{\psi}\times U(1)_{\chi}}{{\mathbbm{Z}}_{6}^{\rm
c}\times{\mathbbm{Z}}_{5}^{f}}\longrightarrow\frac{SU(5)\times
U(1)_{\psi\chi}\times{\mathbbm{Z}}_{4}}{{\mathbbm{Z}}_{6}^{\rm
c}\times{\mathbbm{Z}}_{5}^{f}}\ .$ (3.58)
From the consideration of the standard ’t Hooft anomaly analysis and by the
impossibility of finding an appropriate set of massless baryons [69], one
concludes that if the system confines the global symmetry (3.57) must be
broken spontaneously, at least partially. Therefore the question is whether
the 1-form gauging of a center symmetry can tell us anything useful.
First of all, one finds that both ${\mathbbm{Z}}_{8}^{\psi}$ and
${\mathbbm{Z}}_{20}^{\chi}$ are broken by the 1-form ${\mathbbm{Z}}_{2}^{\rm
c}$ gauging:
${\mathbbm{Z}}_{8}^{\psi}\longrightarrow{\mathbbm{Z}}_{4}^{\psi}\
,\qquad{\mathbbm{Z}}_{20}^{\chi}\longrightarrow{\mathbbm{Z}}_{10}^{\chi}\ .$
(3.59) $\delta A_{\psi}^{(0)}=\frac{2\pi k}{8}\ ,\quad k=2,4,\ldots,8\
,\qquad\delta A_{\chi}^{(0)}=-\frac{2\pi\ell}{20}\ ,\quad\ell=2,4,\ldots,20\
.$ (3.60)
In order for the system to ”match” the reduction of the symmetry (3.59) in the
infrared, some condensates are expected to form. A more careful analysis is,
however, needed to find out which bifemion condensates actually occur in the
infrared, in order to be consistent with the systematics of the mixed-
anomalies.
The division by ${\mathbbm{Z}}_{40}$,
$\psi\to e^{5i\alpha}\psi\ ,\qquad\chi\to e^{-2i\alpha}\chi\ ,$ (3.61)
$\alpha=\frac{2\pi k}{40}\ ,\qquad k=1,2,\ldots,40\ ,$ (3.62)
in the global symmetry group, (3.57), is relevant. The quotient
${\mathbbm{Z}}_{4}\sim\frac{{\mathbbm{Z}}_{20}\times{\mathbbm{Z}}_{8}}{{\mathbbm{Z}}_{40}}$
(3.63)
also forms a subgroup acting as
$\psi\rightarrow e^{2\pi i\frac{2k}{8}}\psi=e^{2\pi
i\frac{k}{4}}\psi\;,\qquad\chi\rightarrow e^{-2\pi
i\frac{5k}{20}}\chi=e^{-2\pi i\frac{k}{4}}\chi\;,$ (3.64)
or
$\delta A_{\psi}^{(0)}=\frac{2\pi k}{4}\ ,\qquad\delta
A_{\chi}^{(0)}=-\frac{2\pi k}{4}\ ,\qquad k=1,2,3,4\,.$ (3.65)
The subtlety is that ${\mathbbm{Z}}_{40}$ remains nonanomalous, even after
1-form gauging of ${\mathbbm{Z}}_{2}^{\rm c}$:
$-\left(8\cdot\frac{2\pi k}{8}-20\cdot\frac{2\pi
k}{20}\right)\,\frac{1}{8\pi^{2}}\left[\int{\mathrm{tr}}{\tilde{F}}^{2}-6\,(B_{\rm
c}^{(2)})^{2}\right]=0\;.$ (3.66)
At the same time, ${\mathbbm{Z}}_{4}$ itself is affected by the gauging of the
center ${\mathbbm{Z}}_{2}^{\rm c}$ symmetry. We find [69] the generalized
anomaly:
$-3\cdot 2\pi
k\,\frac{1}{8\pi^{2}}\left[\int{\mathrm{tr}}{\tilde{F}}^{2}-6\,(B_{\rm
c}^{(2)})^{2}\right]=2\pi k\cdot\left({\mathbbm{Z}}+3\cdot
6\cdot\frac{\mathbbm{Z}}{4}\right)\;.$ (3.67)
Thus ${\mathbbm{Z}}_{4}$ is reduced to ${\mathbbm{Z}}_{2}$ ($k=2,4$).
These are the fates of the discrete symmetries
${\mathbbm{Z}}_{20}\times{\mathbbm{Z}}_{8}\sim{\mathbbm{Z}}_{40}\times{\mathbbm{Z}}_{4}$
(3.68)
under the gauged 1-form center symmetry ${\mathbbm{Z}}_{2}^{\rm c}$. What do
they imply on possible condensates such as
$\psi\chi\;,\qquad\psi\psi,\qquad\chi\chi\;$ (3.69)
? The MAC criterion might suggest formation of condensates in one or more of
the channels
$\displaystyle
A:\qquad\psi\left(\raisebox{-2.0pt}{\yng(2)}\right)\,\psi\left(\raisebox{-2.0pt}{\yng(2)}\right)\quad{\rm
forming}\quad\,\raisebox{-6.0pt}{\yng(2,2)}\;;$ $\displaystyle
B:\qquad\chi\left(\bar{\raisebox{-9.0pt}{\yng(1,1)}}\right)\,\chi\left(\bar{\raisebox{-9.0pt}{\yng(1,1)}}\right)\qquad\
\ {\rm forming}\quad\bar{\raisebox{-12.0pt}{\yng(1,1,1,1)}}\;;$ $\displaystyle
C:\qquad\psi\left(\raisebox{-2.0pt}{\yng(2)}\right)\,\chi\left(\bar{\raisebox{-9.0pt}{\yng(1,1)}}\right)\quad\
\ {\rm forming~{}adjoint~{}representation}\,\,.$ (3.70)
The one-gluon exchange strengths corresponding to these scalar composites are
proportional to $\frac{16}{6},\frac{28}{6},\frac{32}{6}$, respectively. Of
these, the last is the most attractive, and thus one might be tempted to
assume that the only condensate in the system is
$\langle(\psi\chi)_{\rm adj}\rangle\neq 0\;.$ (3.71)
However, the mixed anomaly analysis as sketched here shows that at least two
different types of condensates must form in the infrared. For instance the
${\mathbbm{Z}}_{4}\to{\mathbbm{Z}}_{2}$ breaking would not be accounted for if
(3.71) were the only condensate. See [69] for more details. One is led to
conclude that two or all of the condensates (3.70), are generated by the
system.
## 4 Generalized anomalies and phases of the generalized BY and GG models
In this section we review the generalized anomaly in a large class of chiral
gauge theories, BY (Bars-Yankielowicz) and GG (Georgi-Glashow) models. The
procedure for computing the anomaly against gauging color-flavor locked 1-form
${\mathbbm{Z}}_{N}$ symmetry has been exhibited in Sec. 2.2, by using the
simplest of this class of models, $\psi\eta$ model. The master formula found
there, however, can be applied to any of the BY and GG models.
### 4.1 Bars-Yankielowicz models
The BY models are $SU(N)$ gauge theories with Weyl fermions
$\displaystyle\psi^{ij}\,,\quad\eta_{i}^{A}\,\,,\quad\xi^{i,a}$ (4.1)
in
$\yng(2)\oplus(N+4+p)\,{\bar{{\yng(1)}}}\;\oplus p\,{{{\yng(1)}}}\;.$ (4.2)
The indices run as
$\displaystyle\footnotesize i,j=1,\ldots,N\;,\quad A=1,\ldots,N+4+p\;,\quad
a=1,\ldots,p\;.$ (4.3)
The $\psi\eta$ model corresponds to $p=0$. The number of the extra fundamental
pairs $p$ is limited by $\frac{9}{2}N-3$ before asymptotic freedom (AF) is
lost. The classical symmetry group is
$SU(N)_{\mathrm{c}}\times U(1)_{\psi}\times U(N+4+p)_{\eta}\times
U(p)_{\xi}\;.$ (4.4)
Strong anomaly breaks the symmetry group (4.4) to
$\displaystyle p=0:\quad SU(N)_{\mathrm{c}}\times SU(N+4)_{\eta}\times
U(1)_{\psi\eta}\;,$ $\displaystyle p=1:\quad SU(N)_{\mathrm{c}}\times
SU(N+5)_{\eta}\times U(1)_{\psi\eta}\times U(1)_{\psi\xi}\;,$ $\displaystyle
p>1:\quad SU(N)_{\mathrm{c}}\times SU(N+4+p)_{\eta}\times SU(p)_{\xi}\times
U(1)_{\psi\eta}\times U(1)_{\psi\xi}\;,$ (4.5)
where the anomaly-free combinations are:
$U(1)_{\psi\eta}:\qquad\psi\to\mathrm{e}^{\mathrm{i}(N+4+p)\alpha}\psi\;,\quad\eta\to\mathrm{e}^{-\mathrm{i}(N+2)\alpha}\eta\;,$
(4.6)
with $\alpha\in\mathbbm{R}$, and
$U(1)_{\psi\xi}:\qquad\psi\to\mathrm{e}^{\mathrm{i}p\beta}\psi\;,\quad\xi\to\mathrm{e}^{-\mathrm{i}(N+2)\beta}\xi\;,$
(4.7)
with $\beta\in\mathbbm{R}$. The choice of these two unbroken $U(1)$’s is
arbitrary. For example another $U(1)_{\eta\xi}$
$U(1)_{\eta\xi}:\qquad\eta\to\mathrm{e}^{\mathrm{i}p\gamma}\eta\;,\quad\xi\to\mathrm{e}^{-\mathrm{i}(N+4+p)\gamma}\xi\;,$
(4.8)
with $\gamma\in\mathbbm{R}$ may be chosen.
Table 1 summarizes the charges.
| $SU(N)_{\rm c}$ | $SU(N+4+p)$ | $SU(p)$ | ${U}(1)_{\psi\eta}$ | ${U}(1)_{\psi\xi}$
---|---|---|---|---|---
$\psi$ | ${\yng(2)}$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $N+4+p$ | $p$
$\eta$ | $(N+4+p)\cdot{\bar{\yng(1)}}$ | $N\cdot{\yng(1)}$ | $N(N+4+p)\,\cdot(\cdot)$ | $-(N+2)$ | $0$
$\xi$ | $p\cdot{{\yng(1)}}$ | $Np\,\cdot(\cdot)$ | $N\cdot{{\yng(1)}}$ | $0$ | $-(N+2)$
Table 1: The multiplicity, charges and the representations. $(\cdot)$ is a
singlet representation.
The standard ’t Hooft anomaly matching study, based on the (perturbative)
symmetry group, (4.5), led to the observation that the anomaly triangles
associated with (4.5) can be all matched in the infrared, assuming
confinement, no condensate formation, and assuming a simple sets of massless
composite fermions - baryons - saturating all the anomaly triangles. This is
highly non trivial, as seen in the summary given in Appendix E.
At the same time, the anomaly-matching equations are consistent with dynamics
Higgs phase also, where certain bi-fermion condensates form, which break the
color and part of the flavor symmetries dynamically, leaving still some non-
trivial unbroken chiral symmetry in the infrared. See the review in Appendix
F.
It is thus important to find out whether or not new, mixed type of anomalies
are present in the theories, and whether more stringent anomaly constraints
arise, capable of discriminating these two dynamical possibilities.
As already emphasized, the study of the mixed anomalies require clarifying the
global structure of the symmetry group, beyond their local properties, (4.5).
The result of a detailed analysis done in [71] is that the true symmetry group
of the BY model is
$SU(N)_{\rm c}\times\frac{SU(N+4+p)\times SU(p)\times{\cal
H}}{\mathbb{Z}_{N}\times\mathbb{Z}_{N+4+p}\times\mathbb{Z}_{p}}\;,$ (4.9)
where
${\cal H}=U(1)_{1}\times U(1)_{2}\times({\mathbbm{Z}}_{2})_{F}\;,$ (4.10)
when $p$ and $N$ are both even. That is, it has two disconnected components.
$U(1)_{1}$ and $U(1)_{2}$ being any two out of $U(1)_{\psi\eta}$,
$U(1)_{\psi\xi}$, and $U(1)_{\eta\xi}$.
When $p$ and/or $N$ is odd, instead,
${\cal H}=U(1)_{1}\times U(1)_{2}\;:$ (4.11)
it has only one connected component. In these cases symmetry group is
connected, and perturbative (algebra) aspects of the ’t Hooft anomaly
triangles exhaust the UV-IR anomaly matching conditions. See [70]. Thus the
most interesting BY models are those with $p$ and $N$ both even, to which we
turn now.
The fact that
${\mathbbm{Z}}_{N}\subset U(1)_{\psi\eta}\times
U(1)_{\psi\xi}\times({\mathbbm{Z}}_{2})_{F}\;$ (4.12)
for $N$, $p$ both even, can be shown explicitly, by choosing
${\alpha}=\frac{2\pi}{N}\;,\quad{\beta}=-\frac{2\pi}{N}\;.$ (4.13)
We couple the system to the appropriate background gauge fields,
* •
$A_{\psi\eta}$: $U(1)_{\psi\eta}$ 1-form gauge field,
* •
$A_{\psi\xi}$: $U(1)_{\psi\xi}$ 1-form gauge field,
* •
$A_{2}$: $(\mathbb{Z}_{2})_{F}$ 1-form gauge field,
* •
${\tilde{a}}$: $U(N)_{\rm c}$ 1-form gauge field,
* •
$B^{(2)}_{\mathrm{c}}$: $\mathbb{Z}_{N}$ 2-form gauge field.
Under the 1-form gauge transformation the fields transform as
$\displaystyle B^{(2)}_{\mathrm{c}}$ $\displaystyle\to
B^{(2)}_{\mathrm{c}}+{\mathrm{d}}\lambda_{\mathrm{c}}\;,\qquad\
\,B^{(1)}_{\mathrm{c}}\to B^{(1)}_{\mathrm{c}}+{N}\lambda_{\mathrm{c}}\;,$
$\displaystyle{\tilde{a}}$ $\displaystyle\to{\tilde{a}}+\lambda_{\rm
c}\;,\qquad\qquad\
{\tilde{F}}({\tilde{a}})\to{\tilde{F}}({\tilde{a}})+d\lambda_{\rm c}\;,$
$\displaystyle A_{\psi\eta}$ $\displaystyle\to A_{\psi\eta}-\lambda_{\rm
c}\;,$ $\displaystyle A_{\psi\xi}$ $\displaystyle\to A_{\psi\xi}+\lambda_{\rm
c}\;,$ $\displaystyle A_{2}$ $\displaystyle\to A_{2}+\frac{N}{2}\lambda_{\rm
c}\;.$ (4.14)
The fermion kinetic terms are:
$\displaystyle\overline{\psi}\gamma^{\mu}\left(\partial+\mathcal{R}_{\mathrm{S}}(\widetilde{a})+\frac{N+4+p}{2}A_{\psi\eta}+\frac{p}{2}A_{\psi\xi}+A_{2}\right)_{\mu}P_{\mathrm{L}}\psi+$
$\displaystyle\overline{\eta}\gamma^{\mu}\left(\partial+\mathcal{R}_{\mathrm{F}^{*}}(\widetilde{a})-\frac{N+2}{2}A_{\psi\eta}-A_{2}\right)_{\mu}P_{\mathrm{L}}\eta+$
$\displaystyle\overline{\xi}\gamma^{\mu}\left(\partial+\mathcal{R}_{\mathrm{F}}(\widetilde{a})-\frac{N+2}{2}A_{\psi\xi}+A_{2}\right)_{\mu}P_{\mathrm{L}}\xi\;.$
(4.15)
Knowing the $({\mathbbm{Z}}_{2})_{F}$ charges $+1$, $-1$, $+1$ for the
fermions $\psi$ $\eta$ and $\xi$, respectively, and their representations
under $SU(N)$, the master formula (2.43) gives straightforwardly the result
for the mixed anomaly: the result is
$N^{2}\,{1\over
8\pi^{2}}\int_{\Sigma^{4}}(B^{(2)}_{\mathrm{c}})^{2}\,\frac{1}{2}\delta
A_{2}^{(0)}=N^{2}\times\frac{\mathbbm{Z}}{N^{2}}\,({\pm\pi})=\pm\pi\times{\mathbbm{Z}}\;:$
(4.16)
a $({\mathbbm{Z}}_{2})_{F}-[{\mathbbm{Z}}_{N}]^{2}$ mixed anomaly.
One finds no $({\mathbbm{Z}}_{2})_{F}$ anomaly in the IR, if the system is in
the symmetric vacuum of Appendix. E. Such an inconsistency would be avoided,
if one assumed instead that the system is in the dynamical Higgs phase
(Appendix. F), as the color-flavor locked 1-form symmetry would be
spontaneously broken.
### 4.2 Georgi-Glashow models
The GG models have matter fermions in
$\displaystyle\chi^{ij}\,,\quad\eta_{i}^{A}\,\,,\quad\xi^{i,a}\;,$ (4.17)
i.e., in
$\yng(1,1)\oplus(N-4+p)\,{\bar{{\yng(1)}}}\;\oplus p\,{{{\yng(1)}}}\;,$ (4.18)
where
$\displaystyle\footnotesize i,j=1,\ldots,N\;,\quad A=1,\ldots,N-4+p\;,\quad
a=1,\ldots,p\;.$ (4.19)
The simplest of the GG models (with $p=0$) - call it the $\chi\eta$ model -
can be analyzed following the same steps taken in the case of the $\psi\eta$
model, in Sec. 2.2. The (true) symmetry group of the $\chi\eta$ model is
$SU(N)\times G_{\rm f}\;,\qquad G_{\rm f}=\frac{SU(N-4)\times
U(1)_{\chi\eta}\times(\mathbb{Z}_{2})_{F}}{\mathbb{Z}_{N}\times\mathbb{Z}_{N-4}}\;.$
(4.20)
for even $N$. $U(1)_{\chi\eta}$ and $(\mathbb{Z}_{2})_{F}$ act as
$U(1)_{\chi\eta}\;:\qquad\psi\to e^{i\tfrac{N-4}{2}\beta}\psi\;,\qquad\eta\to
e^{-i\tfrac{N-2}{2}\beta}\eta\;;$ (4.21)
$(\mathbb{Z}_{2})_{F}\;:\qquad\chi,\eta\to-\chi,-\eta\;.$ (4.22)
The division by ${\mathbb{Z}}_{N}$ in (4.20) is due to the equivalence
relation
$(\mathrm{e}^{\mathrm{i}\beta},(-1)^{n})\sim(\mathrm{e}^{\mathrm{i}(\beta-\frac{2\pi}{N})},(-1)^{n}\mathrm{e}^{\mathrm{i}\frac{2\pi}{N}\frac{N}{2}})\,,$
(4.23)
meaning that $U(1)_{\chi\eta}$ gauge field $A$ and $(\mathbb{Z}_{2})_{F}$
gauge field $A_{2}^{(1)}$ have charge $-1$ and $\frac{N}{2}$, respectively.
By introducing the gauge fields
* •
$A_{2}$: $({\mathbbm{Z}}_{2})_{F}$ 1-form gauge field,
* •
$A$: $U(1)=U(1)_{\chi\eta}$ 1-form gauge field,
* •
$B^{(2)}_{\mathrm{c}}$: $\mathbb{Z}_{N}$ 2-form gauge field,
the analysis follows step by step that done in the $\psi\eta$ model. The
result of the calculation, by use of the master formula (Sec. 2.2), is that
there is a mixed $(\mathbbm{Z}_{2})_{F}-[{\mathbbm{Z}}_{N}]^{2}$ anomaly in
the UV,
$\Delta S^{(4)}_{\rm UV}=\pm i\pi{\mathbbm{Z}}\;:$ (4.24)
the partition function changes sign under (4.22).
Of course, the ”massless baryons” lead to no anomalies in the infrared. The
conclusion is that in the confining phase (see Appendix G) the mixed
$(\mathbbm{Z}_{2})_{F}-[{\mathbbm{Z}}_{N}]^{2}$ anomaly does not UV-IR match.
In other words such a symmetric confining phase cannot be the correct vacuum
of the system. There is no difficulty for the dynamical Higgs phase. The fact
that in this particular (and only) case of the simplest of the GG model
($p=0$), the $\chi\eta$ model, the confining, no-condensate phase (Appendix C)
and the dynamical Higgs phase (Appendix D) happen to have the same global
symmetry of the massless sector does not necessarily imply that these phases
are the same phase (see a more detailed discussion on this issue in [96]).
More general GG models with $p\neq 0$ have also been analysed in detail, in
[70]. The analysis is similar to that done for the $\chi\eta$ model and for
the general $BY$ models reviewed above. The conclusion is that the confining,
symmetric vacuum (Appendix G) is not consistent with the implications of the
generalized anomalies. The Higgs phase (Appendix H) seems to be perfectly
consistent with these new constraints.
## 5 Strong anomaly and phases of chiral gauge theories
Very recently the present authors have realized [96] that strong anomaly,
which plays a prominent role in the solution of the so-called $U(1)_{A}$
problem in QCD, can also be significant in the study of the phases of chiral
gauge theories, such as those discussed so far in this review. More
concretely, a key observation is that the well-known low-energy strong-anomaly
effective action for QCD, which reproduces the effects of the strong anomaly
in the low-energy effective action, has a nontrivial implication on the
symmetry breaking pattern itself. Note that there is a subtlety in this
argument, as the well-known QCD effective sigma model action already assumes
the standard chiral symmetry breaking into vectorlike residual symmetries (see
(5.2) below).
The criterion we adopt to study chiral gauge theories of unknown low-energy
symmetries and phases, is that it should be possible to write a low-energy
strong-anomaly local effective action by using the low-energy degrees of
freedom (NG bosons and/or massless composite fermions) present in the assumed
phase.
The simple form of such a low-energy action we will find in the dynamical
Higgs phase, in contrast to the impossibility of writing analogous terms with
massless baryons only (in the confining phase), provides another, independent,
indication that the first type of phase (dynamical Higgs phase, characterized
by certain bi-fermion condensates) represents a more plausible IR phase of
this class of models.
Before considering the chiral gauge theories we discussed in the previous
sections from this new angle, we first review quickly what is known about the
$U(1)_{A}$ problem in the standard QCD [91, 92].
### 5.1 $U(1)_{A}$ problem and the $\theta$ dependence in QCD
In QCD the $U(1)_{A}$ symmetry is broken by the strong anomaly, and also
spontaneously broken by the quark condensate
$\langle{\bar{\psi}_{L}}\psi_{R}\rangle\sim-\Lambda^{3}\neq 0\;,$ (5.1)
which breaks the nonanomalous chiral symmetry to its vectorlike subgroup,
$SU(N_{\rm f})_{L}\times SU(N_{\rm f})_{R}\to SU(N_{\rm f})_{V}\;.$ (5.2)
At this point one expects, besides the NG bosons of the $SU(N_{\rm f})_{A}$
symmetry (the pions), another NG boson relative to $U(1)_{A}$, ($\eta$ or
$\eta^{\prime}$)121212Here $\eta,\eta^{\prime}$ are the $SU(2)$ or $SU(3)$
singlet pseudoscalar mesons of the real world, as in the Particle Data
Booklet. The reader will not confuse them with the Weyl fermion in the chiral
$\psi\eta$ or $\chi\eta$ models being studied here., which would get mass due
to the strong anomaly. The chiral Lagrangian allows us to understand how this
works qualitatively, and quantitatively in the large $N$ limit.
To reproduce the effects of the strong anomaly, the authors of [30]-[34] add
to the standard chiral Lagrangian
$L_{0}=\frac{F_{\pi}^{2}}{2}\mathop{Tr}\nolimits\,\partial_{\mu}U\partial^{\mu}U^{\dagger}+\mathop{Tr}\nolimits
M\,U+{\rm h.c.}+\ldots\;,\qquad U\equiv{\bar{\psi}}_{R}\psi_{L}$ (5.3)
($F_{\pi}$ is the usual pion-decay constant), a new term
${\hat{L}}=\frac{i}{2}q(x)\,\log\det
U/U^{\dagger}+\frac{N}{a_{0}F_{\pi}^{2}}q^{2}(x)-\theta\,q(x)\;,$ (5.4)
where $q(x)$ is the topological density
$q(x)=\frac{g^{2}}{32\pi^{2}}F_{\mu\nu}^{a}{\tilde{F}}^{a,\mu\nu}\;,$ (5.5)
and $a_{0}$ is some constant of the order of unity. The variation of $\hat{L}$
under the action of $U(1)_{A}$
$\psi_{L}\to e^{i\alpha}\psi_{L}\;,\quad\psi_{R}\to
e^{-i\alpha}\psi_{R}\;,\quad U\to e^{2i\alpha}\,U$ (5.6)
reproduces the variation of the phase of the partition function,
$\Delta S=2N_{\rm f}\alpha\int
d^{4}x\frac{g^{2}}{32\pi^{2}}F_{\mu\nu}^{a}{\tilde{F}}^{a,\mu\nu}\;,$ (5.7)
due to the strong anomaly.
The expression (5.4) can be manipulated as shown in [30]-[34]: integrating
away the topological charge $q(x)$ one obtains an equivalent expression
${\hat{L}}=-\frac{F_{\pi}^{2}\,a_{0}}{4N}(\theta-\frac{i}{2}\log\det
U/U^{\dagger})^{2}\;,$ (5.8)
which is well defined as
$\langle U\rangle\propto{\mathbf{1}}\neq 0\;.$ (5.9)
Expanding (5.8) around this VEV,
$U\propto
e^{i\tfrac{\pi^{a}t^{a}}{F_{\pi}}+i\tfrac{\eta\,t^{0}}{F_{\pi}^{(0)}}}={\mathbf{1}}+i\frac{\pi^{a}t^{a}}{F_{\pi}}+i\frac{\eta\,t^{0}}{F_{\pi}^{(0)}}+\ldots\;,$
(5.10)
one finds the mass term for the would-be NG boson, $\eta$.
The idea here is to reverse the logics: one can actually argue that the
presence of such an effective action needed for reproducing the strong anomaly
implies a nonvanishing condensate, $\langle
U\rangle=\langle\bar{\psi}_{R}\psi_{L}\rangle\neq 0$, and hence, indirectly,
also the spontaneous breaking of nonanomalous chiral symmetry, (5.2),
affecting the low-energy physics.
Even if in QCD there is a powerful direct argument [23] for such a vectorlike
symmetry-breaking pattern, it is interesting to note that the requirement of
faithfully representing the $U(1)_{A}$ anomaly in the infrared seems to imply
the same conclusion.
It is this kind of consideration that has led recently the present authors to
apply an analogous argument to chiral gauge theories [96], by requiring the
effective low-energy theory to be able to express the strong anomaly
appropriately. The following (Sec. 5.2-Sec. 5.5) is the review of some of the
results found.
#### 5.1.1 ${\cal N}=1$ supersymmetric theories
Before proceeding to the discussion of chiral gauge theories, let us make a
brief comment on ${\cal N}=1$ supersymmetric models. In the context of ${\cal
N}=1$ supersymmetric gauge theories, the strong-anomaly effective action is
derived by using the so-called Veneziano-Yankielowicz (VY) and Affleck-Dine-
Seiberg (ADS) superpotentials [35, 36, 37]. They correctly reproduce in the
infrared effective theory the effects of instantons and supersymmetric Ward-
Takahashi identities, and embody the anomaly of [93, 94]. This last one, known
as Konishi anomaly, has direct implications on the vacuum properties of the
theory under consideration. It is a straightforward consequence of the strong
anomaly, via supersymmetry 131313The Konishi anomaly can also be viewed as
representing an anomalous supersymmetry transformation law for some composite
fields [93, 94].. The VY and ADS superpotentials are indeed crucial in
determining the infrared dynamics and phases of the ${\cal N}=1$
supersymmetric gauge theories. For a review, see for instance [95].
### 5.2 $\psi\eta$ model and strong-anomaly
Let us apply a similar idea, i.e. of writing an effective action which
reproduces the strong anomaly of the UV theory in the low energy theory, to
one of the simplest chiral gauge theories, the $\psi\eta$ model (see Sec.
2.2). Let us remind ourselves briefly of the symmetries of the model. At the
infinitesimal level the quantum symmetry group of $\psi\eta$ is
$SU(N+4)_{\eta}\times U(1)_{\psi\eta}\;,$ (5.11)
while any combinations of $U(1)_{\psi}\times U(1)_{\eta}$ different form
$U(1)_{\psi\eta}$ is broken by a strong anomaly. The low-energy effective
action must capture this strong anomaly correctly.
In Sec. 2.2 it was shown that the $\psi\eta$ model cannot confine maintaining
the full global symmetry unbroken. The system instead can break the gauge
symmetry dynamically (as well as part of the global symmetry), and a color-
flavor locking condensate forms:
$\langle\psi^{ij}\eta^{A}_{j}\rangle=\begin{cases}c_{\psi\eta}\,\delta^{iA}\;,\qquad
A=1,\dots N\;,\\\ 0\;,\qquad A=N+1,\dots N+4\;.\end{cases}$ (5.12)
Unlike ${\bar{\psi}_{R}}\psi_{L}$ in QCD,
${\tilde{\phi}}=\sum_{k,j}^{N}\psi^{kj}\eta_{j}^{k}\;,$ is not gauge
invariant. It is convenient to re-express this condensate in a gauge invariant
form, i.e.
$\det U\;,\qquad U_{k\ell}\equiv\psi^{kj}\eta_{j}^{\ell}\;.$ (5.13)
Such a gauge invariant condensate is fully equivalent to (5.12). It causes the
breaking
$SU(N+4)\times U(1)\rightarrow SU(N)_{{}_{\rm cf}}\times SU(4)\times
U(1)^{\prime}\;,$ (5.14)
(Appendix B). $U(1)^{\prime}$ is the unbroken combination of $U(1)_{\psi\eta}$
and $U(1)_{D}$, where $U(1)_{D}$ is $U(1)\subset SU(N+4)$ generated by
$T_{D}={\rm diag}(4\cdot\mathbbm{1}_{N\times N},-N\cdot\mathbbm{1}_{4\times
4})$.
At this point it is useful to look into the NG boson sector of the theory,
which leads to an apparent puzzle. From the symmetry breaking one expects to
find $8N$ nonabelian NG bosons relative to $\frac{SU(N+4)}{SU(N)\times
SU(4)\times U(1)_{D}}$, interpolated in a gauge invariant fashion by
$\phi^{A}=(\psi^{ij}\eta_{j}^{a})^{*}(T^{A})^{a}_{b}(\psi^{ik}\eta^{b}_{k})\;.$
(5.15)
Here $T_{A}$ are the $8N$ broken generators that connect the $N$ dimensional
subspace (where $SU(N)$ acts) and the $4$ dimensional one (where $SU(4)$
acts).
The problem emerges when one considers $U(1)$ NG boson(s). There is certainly
a physical massless NG boson, living in
$\frac{U(1)_{D}\times U(1)_{\psi\eta}}{U(1)^{\prime}}\;.$ (5.16)
The gauge-invariant field that interpolates it can be taken as the (imaginary
part of the) condensate $\det\;U$ itself. However, with the condensate (5.12)
alone, there is no space for another possible NG boson, associated with the
symmetry breaking of an anomalous $U(1)$ symmetry (any generic combination of
$U(1)_{\psi}$ and $U(1)_{\eta}$ other than $U(1)_{\psi\eta}$ is in fact
spontaneously broken by ${\langle\psi\eta\rangle}$). This would-be NG boson
would get a mass from the strong anomaly, but in any case it needs to be
described by an interpolating field (which?). Another related fact is that
there is actually a particular anomalous symmetry (a special combination of
$U(1)_{\psi}$ and $U(1)_{\eta}$)
$U(1)_{A}:\ \begin{cases}\psi\rightarrow e^{i\alpha}\psi\;,\\\ \eta\rightarrow
e^{-i\alpha}\eta\;,\end{cases}$ (5.17)
which is not spontaneously broken by the $\psi\eta$ condensate. How would such
a symmetry manifest itself in the infrared? These are the first hints that the
description in terms of the condensate $\psi\eta$ (or $\det\;U$) is not a
complete one.
Another reason to look for other condensates is that it is not possible to
write an effective Lagrangian which realizes all the (nonanomalous) global
symmetries, with the composite field $\det U$ alone. For further details see
[96].
With these considerations in mind, let us construct the correct form of the
strong-anomaly effective action systematically. We start from the very
beginning,
${\cal L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+{\cal L}^{\rm fermions}\;,$ (5.18)
${\cal L}^{\rm
fermions}=-i\overline{\psi}{\bar{\sigma}}^{\mu}\left(\partial+\mathcal{R}_{\mathrm{S}}(a)\right)_{\mu}\psi\;-i\overline{\eta}{\bar{\sigma}}^{\mu}\left(\partial+\mathcal{R}_{\mathrm{F}^{*}}(a)\right))_{\mu}\eta\;,$
(5.19)
where $a$ is the $SU(N)$ gauge field, and the matrix representations
appropriate for $\psi$ and $\eta$ fields are indicated with $\cal
R_{\mathrm{S}}$ and $\cal R_{\mathrm{F}^{*}}$. We change the variables by
${\cal L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+{\cal L}^{\rm
fermions}+\mathop{Tr}\nolimits[(\psi\eta)^{*}U]+{\rm{\rm
h.c.}}+{B}\,(\psi\eta\eta)^{*}+{\rm{\rm h.c.}}\;,$ (5.20)
where $U$ is the composite scalars of $N\times(N+4)$ color-flavor mixed matrix
form,
$\mathop{Tr}\nolimits[(\psi\eta)^{*}U]\equiv(\psi^{ij}\eta_{j}^{m})^{*}U^{im}\;,$
(5.21)
and ${B}$ are the baryons $B\sim\psi\eta\eta$,
$B^{mn}=\psi^{ij}\eta_{i}^{m}\eta_{j}^{n}\;,\qquad m,n=1,2,\ldots,N+4\;,$
(5.22)
antisymmetric in $m\leftrightarrow n$. In writing down the lagrangian (5.20)
we have anticipated the fact that these baryon-like composite fields, present
in the Higgs phase together with the composite scalars $~{}\psi\eta$ (see
Appendix B), are also needed to write down the strong-anomaly effective
action. This allows us to dodge the problem about the NG bosons (and to break
$U(1)_{A}$), as we are going to explain.
Integrating $\psi$ and $\eta$ out, one gets
${\cal L}^{\rm
eff}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\mathop{Tr}\nolimits({\cal
D}U)^{\dagger}{\cal
D}U-i\overline{B}\,{\bar{\sigma}}^{\mu}\partial_{\mu}{B}-V\;.$ (5.23)
The potential $V$ is assumed to be such that its minimum is of the form:
$\langle U^{im}\rangle=\,c_{\psi\eta}\,\Lambda^{3}\delta^{im}\;,\qquad\quad\
i,m=1,2,\dots N\;,$ (5.24)
and contains the strong anomaly term,
$V=V^{(0)}+{\hat{L}}_{\rm an}\;.$ (5.25)
${\hat{L}}_{\rm an}$ of the form,
${\hat{L}}_{\rm an}={{\rm
const}}\,\left[\log\left(\epsilon\,{B}{B}\,{\det}U\right)-\log\left(\epsilon\,{B}{B}\det
U\right)^{\dagger}\right]^{2}\;$ (5.26)
which is analogue of (5.8) in QCD. The argument of the logarithm
$\epsilon\,{B}{B}\,\det
U\equiv\epsilon^{m_{1},m_{2},\ldots,m_{N+4}}\epsilon^{i_{1},i_{2},\ldots,i_{N}}{B}_{m_{N+1},m_{N+2}}{B}_{m_{N+3},m_{N+4}}U_{i_{1}m_{1}}U_{i_{2}m_{2}}\ldots
U_{i_{N}m_{N}}\;,$ (5.27)
is invariant under the full (nonanomalous) symmetry,
$SU(N)_{\rm c}\times SU(N+4)\times U(1)_{\psi\eta}\;$ (5.28)
as it should be. Moreover it contains $N+2$ $\psi$’s and $N+4$ $\eta$’s, the
correct numbers of the fermion zeromodes in the instanton background: it
corresponds to a ’t Hooft’s instanton n-point function, e.g.,
$\langle\psi\eta\eta(x_{1})\psi\eta\eta(x_{2})\psi\eta(x_{3})\ldots\psi\eta(x_{N+2})\rangle\;.$
(5.29)
This effective Lagrangian is well defined only if the argument of the
logarithm takes a VEV. In particular it is natural to assume
$\langle\epsilon^{(4)}{B}{B}\rangle\neq 0\;,\qquad\langle\det U\rangle\neq
0\;,$ (5.30)
where
$\epsilon^{(4)}{B}{B}=\epsilon_{\ell_{1}\ell_{2}\ell_{3}\ell_{4}}{B}^{\ell_{1}\ell_{2}}{B}^{\ell_{1}\ell_{2}}\;,\qquad\ell_{i}=N+1,\ldots,N+4\;.$
(5.31)
As
$\langle\det U\rangle\propto{\mathbf{1}}_{N\times N}\;$ (5.32)
takes up all flavors up to $N$ (the flavor $SU(N+4)$ symmetry can be used to
orient the symmetry breaking this way), ${B}{B}$ must be made of the four
remaining flavors, as in (5.31). These baryons were not among those considered
in the earlier studies [12, 71], but assumed to be massless here, and
indicated as $B^{[A_{2}B_{2}]}$ in Table 3. This is possible because these
fermions do not have any perturbative anomaly with respect to the unbroken
symmetry group, $SU(N)\times SU(4)\times U(1)$: ’t Hooft anomaly matching
considerations cannot tell if they are massive or not, either the two options
are possible.
Now we see how the apparent puzzle about the NG bosons hinted at above is
solved. We can define the interpolating fields of the two NG boson by
expanding the condensates,
$\displaystyle\det U=\langle\det
U\rangle+\ldots\propto{\mathbf{1}}+\frac{i}{F_{\pi}^{(0)}}\,\phi_{0}+\ldots\;;$
$\displaystyle\epsilon^{(4)}{B}{B}=\langle\epsilon^{(4)}{B}{B}\rangle+\ldots\propto{\mathbf{1}}+\frac{i}{F_{\pi}^{(1)}}\,\phi_{1}+\ldots\;,$
(5.33)
(here $F_{\pi}^{(0)}$ and $F_{\pi}^{(1)}$ are some constants with dimension of
mass). Clearly in general the physical NG boson and the anomalous would-be NG
boson will be interpolated by two linear combinations of $\phi_{0}$ and
$\phi_{1}$. The effective Lagraingian allows us to fix these linear
combinations.
Indeed, as the effective Lagrngian (5.26) is invariant under the nonanomalous
symmetry group, and in particular $U(1)_{\psi\eta}$ and $U(1)_{D}$ do not act
on $\epsilon\;BB\;\det U$, the NG boson which appears in the strong-anomaly
effective action as the fluctuation of $\epsilon BB\;\det U$,
${\tilde{\phi}}\equiv
N_{\pi}\left[\frac{1}{F_{\pi}^{(0)}}\,\phi_{0}+\frac{1}{F_{\pi}^{(1)}}\,\phi_{1}\right]\;,\qquad
N_{\pi}=\frac{F_{\pi}^{(0)}F_{\pi}^{(1)}}{\sqrt{\big{(}F_{\pi}^{(0)}\big{)}^{2}+\big{(}F_{\pi}^{(1)}\big{)}^{2}}}\;,$
(5.34)
cannot be the massless physical one: it is the would-be NG boson relative to
the anomalous symmetry. Indeed the effective action provides a mass term for
this NG boson.
The orthogonal combination
${\phi}\equiv
N_{\pi}\left[\frac{1}{F_{\pi}^{(1)}}\phi_{0}-\frac{1}{F_{\pi}^{(0)}}\phi_{1}\right]\;,$
(5.35)
i.e. the interpolating field of the physical NG boson living in the coset
(5.16), remains massless.
Before we have included in the low-energy description some massless baryon
which are neither required, nor excluded by the ’t Hooft anomaly matching. Now
one can see their ultimate fate using the strong-anomaly effective Lagrangian.
In particular (5.26) contains a 4-fermion coupling between these baryons,
which, plugging the VEVs (5.30), provides a mass term for them.
The last remark is that the strong-anomaly effective action does not depend on
the absolute value of the condensates, $BB$ and $\det U$, separately. This
simply means that these symmetry considerations alone cannot determine the
mechanism of condensation. In particular, even if $\langle\det U\rangle\neq 0$
is somehow expected, a VEV for $BB$ is more surprising, and probably due to
residual dipole interactions between the baryons. However a more in-depth
study on how these two flat directions are lifted by quantum effects is needed
to precisely understand how these two condensate form.
### 5.3 Strong anomaly: the generalized BY models
As the solution given above on the $\psi\eta$ model is notably subtle, one
might wonder whether a similar mechanism is at work in the generalized Bars-
Yankielowicz models, an $SU(N)$ gauge theory with Weyl fermions
$\displaystyle\psi^{ij}\,,\quad\eta_{i}^{A}\,\,,\quad\xi^{i,a}$ (5.36)
in the direct-sum representation
$\yng(2)\oplus(N+4+p)\,{\bar{{\yng(1)}}}\;\oplus p\,{{{\yng(1)}}}\;.$ (5.37)
Also in this case (Ref. [71] and Appendix C) the conventional ’t Hooft anomaly
matching equations allow a chirally symmetric confining vacuum, with massless
baryons
${({B}_{1})}^{[AB]}=\psi^{ij}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad{({B}_{2})}^{a}_{A}=\bar{\psi}_{ij}\bar{\eta}^{i}_{A}\xi^{j,a}\;\qquad{({B}_{3})}_{\\{ab\\}}=\psi^{ij}{\bar{\xi}}_{i,a}{\bar{\xi}}_{j,b}\;,$
(5.38)
(the first is anti-symmetric in $A\leftrightarrow B$ and the third is
symmetric in $a\leftrightarrow b$), saturating all conventional ’t Hooft
anomaly triangles. The study of the generalized anomaly in Sec. 4.1 has
however shown that such a vacuum is not consistent.
A dynamical Higgs phase with condensates
$\displaystyle\langle
U^{iB}\rangle=\langle\psi^{ij}\eta_{i}^{B}\rangle=\,c_{\psi\eta}\,\Lambda^{3}\delta^{jB}\neq
0\;,\qquad j,B=1,\dots,N\;,$ $\displaystyle\langle
V^{aA}\rangle=\langle\xi^{i,a}\eta_{i}^{A}\rangle=\,c_{\eta\xi}\,\Lambda^{3}\delta^{N+4+a,A}\neq
0\;,$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
a=1,\dots,p\;,\quad A=N+5,\dots,N+4+p\;,$ (5.39)
and with symmetry breaking
$\displaystyle SU(N)_{\mathrm{c}}\times SU(N+4+p)_{\eta}\times
SU(p)_{\xi}\times U(1)_{\psi\eta}\times U(1)_{\psi\xi}$
$\displaystyle\xrightarrow{\langle\xi\eta\rangle,\langle\psi\eta\rangle}SU(N)_{{\rm
cf}_{\eta}}\times SU(4)_{\eta}\times SU(p)_{\eta\xi}\times
U(1)_{\psi\eta}^{\prime}\times U(1)_{\psi\xi}^{\prime}\;,$ (5.40)
is fully consistent with the gauging of the center symmetry (Ref. [71] and
Sec. 4.1).
A strong-anomaly effective action for the BY theories can be constructed in a
way similar to the $\psi\eta$ model. Instead of (5.27), one has now
$\displaystyle\epsilon\left({{B}_{1}}{{B}_{1}}\det U\det
V\right)\equiv\epsilon^{m_{1},m_{2},\ldots,m_{N+4+p}}\epsilon^{i_{1},i_{2},\ldots,i_{N}}\epsilon^{k_{1},k_{2},\ldots,k_{p}}\times$
$\displaystyle\times$
$\displaystyle{B}_{1}^{[m_{N+1},m_{N+2}]}{B}_{1}^{[m_{N+3},m_{N+4}]}U^{i_{1}m_{1}}U^{i_{2}m_{2}}\ldots
U^{i_{N}m_{N}}V^{m_{N+5}k_{1}}\ldots V^{m_{N+4+p}k_{p}}\;.$
The rest of the analysis can be completed by closely following that of the
$\psi\eta$ model discussed in Sec. 5.2. We skip the details of the analysis.
Let us note only that the strong anomaly effective action with such a
logarithm, is perfectly consistent with, and perhaps implies, the condensates,
(5.39): i.e., that the system is in dynamical Higgs phase, (5.40). It is,
instead, not possible to rewrite a strong-anomaly effective action with
logarithmic argument (LABEL:insteadofBis), in terms of massless composite
fermions (5.38) alone.
### 5.4 Strong anomaly and the $\chi\eta$ model
It is an interesting exercise to apply the same reasoning about the strong
anomaly to the $\chi\eta$ model. We will find that there are good analogies
with the $\psi\eta$ case studied above, but also quite significant
differences.
In this model any combination of $U(1)_{\chi}$ and $U(1)_{\eta}$, except
$U(1)_{\chi\eta}$ (see Table 5), is anomalous, therefore some term similar to
(5.4) (in QCD) should appear.
In the dynamical Higgs scenario for the $\chi\eta$ model, there are two bi-
fermon condensates,
$\langle\chi^{ij}\eta_{j}^{m}\rangle=c_{\chi\eta}\,\delta^{im}\,\Lambda^{3}\;,\qquad
i,m=1,2,\ldots,N-4\;,$ (5.42)
and
$\langle\chi\chi\rangle\neq 0\;.$ (5.43)
This implements a two-step breaking,
$\displaystyle SU(N)\times SU(N-4)\times U(1)_{\chi\eta}$
$\displaystyle\xrightarrow{\langle\chi\eta\rangle}$ $\displaystyle
SU(N-4)_{\rm cf}\times SU(4)_{\rm c}\times U(1)^{\prime}$ (5.44)
$\displaystyle\xrightarrow{\langle\chi\chi\rangle}$ $\displaystyle
SU(N-4)_{\rm cf}\times U(1)^{\prime}\;.$
As before, in order to construct a fully consistent effective action, one
should keep the full nvariance of the original theory, either spontaneously
broken or not. To do so, it is convenient to re-express the condensates (5.42)
in a gauge invariant way. The answer is to write a single gauge-invariant
condensate
$\displaystyle U$ $\displaystyle=$ $\displaystyle\epsilon_{i_{1}i_{2}\ldots
i_{N}}\epsilon_{m_{1}m_{2}\ldots
m_{N-4}}(\chi\eta)^{i_{1}m_{1}}(\chi\eta)^{i_{2}m_{2}}\ldots(\chi\eta)^{i_{N-4}m_{N-4}}\chi^{i_{N-3}i_{N-2}}\chi^{i_{N-1}i_{N}}$
(5.45) $\displaystyle\sim$
$\displaystyle\epsilon\,(\chi\eta)^{N-4}(\chi\chi)\;,$
which encodes both of the two (gauge depending) ones.
This choice suggests that the correct strong-anomaly effective action for the
$\chi\eta$ model is
$\frac{i}{2}q(x)\log\epsilon(\chi\eta)^{N-4}(\chi\chi)+{\rm h.c.}\;,$ (5.46)
where, again, $q(x)$ is the topological density defined in Eq. (5.5). Clearly
it is by construction invariant under the whole (nonanomalous) symmetry group
$SU(N)_{\rm c}\times SU(N-4)\times U(1)_{\chi\eta}\;.$ (5.47)
Let us comment briefly some features that suggest that this is indeed the
correct result.
* •
The argument of the logarithmic function here matches the correct number of
the fermion zeromodes in the instanton background ($N_{\chi}=N-2$ and
$N_{\eta}=N-4$);
* •
In contrast, there is no way of writing the strong anomaly effective action
(5.46) in terms of the ”baryons”, $B\sim\chi\eta\eta$, of the assumed
confining, chirally symmetric phase (Appendix C). No combination of the
baryons can saturate the correct number of the fermion zeromodes, cfr (5.45).
This anomaly effective action (5.46) agrees with the one proposed by Veneziano
[7] for the case of $SU(5)$, and generalizes it to all $SU(N)$ $\chi\eta$
models. A key observation we share with [7] and generalizes to models with any
$N$, is that this strong anomaly effective action, which should be there in
the low-energy theory to reproduce correctly the (anomalous and nonanomalous)
symmetries of the UV theory, implies nonvanishing condensates,
$\langle\chi\eta\rangle\neq 0\;,\qquad\langle\chi\chi\rangle\neq 0\;,$ (5.48)
i.e., that the system is in dynamical Higgs phase, Appendix D.
Up to now the story has been very similar to the one about the $\psi\eta$
model. However there are some differences. Differently from the $\psi\eta$
model, where the baryon condensate mush enter in the strong-anomaly effective
action, here the structure of the effective action simplifies, and no baryon
is needed. Moreover, contrary to the $\psi\eta$ model, the $\chi\eta$ system
has no physical $U(1)$ NG boson: it is eaten by a color $SU(N)$ gauge boson.
However the counting of the broken and unbroken $U(1)$ symmetries is basically
similar in the two models. Of the two nonanomalous symmetries ($U(1)_{\rm c}$
and $U(1)_{\chi\eta}$), a combination remains a manifest physical symmetry,
and the other becomes the longitudinal part of a color gauge boson. Still
another, anomalous, $U(1)$ symmetry exists, which is any combination of
$U(1)_{\chi}$ and $U(1)_{\eta}$ other than $U(1)_{\chi\eta}$. This symmetry is
also spontaneously broken, hence it must be associated with a NG boson, even
though it will get mass by the strong anomaly.
As in the $\psi\eta$ model, one can describe this situation explicitly, by
expanding the composite $\chi\eta$ and $\chi\chi$ fields around their VEV’s,
$\displaystyle(\det U)^{\prime}=\langle(\det
U)^{\prime}\rangle+\ldots\propto{\mathbf{1}}+i\frac{1}{F_{\pi}^{(0)}}\,\phi_{0}^{\prime}+\ldots\;;$
$\displaystyle\chi\chi=\langle\chi\chi\rangle+\ldots\propto{\mathbf{1}}+i\frac{1}{F_{\pi}^{(1)}}\,\phi_{1}^{\prime}+\ldots\;,$
(5.49)
where $(\det U)^{\prime}$ is defined in the $N-4$ dimensional color-flavor
mixed space, and
$\chi\chi\equiv\epsilon_{i_{1},i_{2},i_{3},i_{4}}\chi^{i_{1}i_{2}}\chi^{i_{3}i_{4}}\;,\qquad
N-3\leq i_{j}\leq N\;.$ (5.50)
Now one can see that the strong-anomaly effective action (5.46) gives mass to
${\tilde{\phi}}^{\prime}\equiv
N_{\pi}\left[\frac{1}{F_{\pi}^{(0)}}\phi_{0}^{\prime}+\frac{1}{F_{\pi}^{(1)}}\phi_{1}^{\prime}\right]\;,\qquad
N_{\pi}=\frac{F_{\pi}^{(0)}F_{\pi}^{(1)}}{\sqrt{(F_{\pi}^{(0)})^{2}+(F_{\pi}^{(1)})^{2}}}\;,$
(5.51)
whereas an orthogonal combination
${\phi}^{\prime}\equiv
N_{\pi}\left[\frac{1}{F_{\pi}^{(1)}}\phi_{0}^{\prime}-\frac{1}{F_{\pi}^{(0)}}\phi_{1}^{\prime}\right]\;$
(5.52)
remains massless. The latter corresponds to the potential NG boson which is
absorbed by the color $T_{\rm c}$ gauge boson.
### 5.5 Generalized GG models and strong anomaly
Let us now turn to the generalized GG models [71], i.e. $SU(N)$ gauge theories
with Weyl fermions
$\displaystyle\chi^{[ij]}\,,\quad\eta_{i}^{A}\,\,,\quad\xi^{i,a}$ (5.53)
in the direct-sum representation,
$\yng(1,1)\oplus(N-4+p)\,{\bar{{\yng(1)}}}\;\oplus p\,{{{\yng(1)}}}\;.$ (5.54)
It turns out that the simple structure of the strong-anomaly effective action
(5.46), which does not need bi-baryon condensate, works out also in this case:
$\frac{i}{2}q(x)\log\epsilon\chi\chi\det\,(\chi\eta)\det(\xi\eta)+{\rm
h.c.}\;,$ (5.55)
where a shorthand notation
$\displaystyle\epsilon\chi\chi\det\,(\chi\eta)\det(\xi\eta)=$ $\displaystyle=$
$\displaystyle\epsilon_{i_{1}i_{2}\ldots i_{N}}\,\epsilon_{k_{1}k_{2}\ldots
k_{p}}\,\epsilon_{m_{1}m_{2}\ldots m_{N-4+p}}$ $\displaystyle\times$
$\displaystyle(\chi\eta)^{i_{1}m_{1}}(\chi\eta)^{i_{2}m_{2}}\ldots(\chi\eta)^{i_{N-4}m_{N-4}}(\chi^{i_{N-3}i_{N-2}}\chi^{i_{N-1}i_{N}})(\xi\eta)^{m_{N-3}k_{1}}\ldots(\xi\eta)^{m_{N-4+p}k_{p}}\;$
has been used. The strong anomaly action (5.55) requires the condensates
$\displaystyle\langle\chi^{ij}\eta_{i}^{A}\rangle={\rm
const}.\,\Lambda^{3}\delta^{jA}\neq 0\;,\qquad j=1,\dots,N-4\;,\quad
A=1,\dots,N-4\;,$ $\displaystyle\langle\xi^{i,a}\eta_{i}^{B}\rangle={\rm
const}.\,\Lambda^{3}\delta^{N-4+a,B}\neq 0\;,\qquad a=1,\dots,p\;,\quad
B=N-3,\dots,N-4+p\;,$
and
$\langle\chi^{j_{1},j_{2}}\chi^{j_{3},j_{4}}\rangle={\rm
const}.\,\epsilon^{j_{1},j_{2},\ldots,j_{4}}\Lambda^{3}\neq 0\;,\qquad
j_{1},\ldots,j_{4}=N-3,N-2,\ldots,N\;.$ (5.58)
Hearteningly, these are exactly the set of condensates expected to occur in
the dynamical Higgs phase of the GG models (Ref. [71] and Appendix H).
### 5.6 Strong anomaly in the chiral gauge theories considered in Sec. 3
Up to now we have analysed the implications of the strong anomaly in models
discussed in Sec. 2.2 and in Sec. 4. In these models, being able to
discriminate between different types of phases (confinement with unbroken
global symmetry versus dynamical Higgs phase) was clearly very important, as
the conventional ’t Hooft anomaly-matching algorithm could not tell us which
type of the vacua were the correct ones. We found indeed that both the recent
generalized anomaly study (Sec. 2.2 and Sec. 4) and strong anomaly
consideration (Sec. 5.2 \- Sec. 5.5) seem to favor the dynamical Higgs phase,
in all these models.
Of course, consideration of the strong-anomaly effective action is relevant
also in other models. For illustration we discuss below a few models studied
in Sec. 3.
#### 5.6.1 $SU(6)$ model with a single fermion in a self-adjoint
antisymmetric representation
Consider an $SU(6)$ model with a single left-handed fermion in the
representation,
${\underline{20}}\,=\,\yng(1,1,1)\,$ (5.59)
which was studied in [59, 69] and reviewed in Sec. 3.1.1 above. The lessons
learned from the the gauging of the 1-form ${\mathbbm{Z}}_{3}^{\rm c}$
symmetry have been that the nonanomalous ${\mathbbm{Z}}_{6}^{\psi}\,$ symmetry
must break spontaneously as
${\mathbbm{Z}}_{6}^{\psi}\longrightarrow{\mathbbm{Z}}_{2}^{\psi}\;,$ (5.60)
implying a three-fold vacuum degeneracy[59, 69]. This could either be because
of a four-fermion condensate [59]
$\langle\psi\psi\psi\psi\rangle\sim\Lambda^{6}\neq
0\;,\qquad\langle\psi\psi\rangle=0\;,$ (5.61)
or due to a gauge-symmetry breaking bi-fermion condensate [69]
$\langle\psi\psi\rangle\sim\Lambda^{3}\neq 0\;,$ (5.62)
with $\psi\psi$ in the adjoint representation of $SU(6)$. Both scenarios are
consistent.
Let us see if considerations on the strong-anomaly can clarify which scenario
is actually realized. A particularly simple representation of the strong-
anomaly is
$\frac{i}{2}q(x)\log\psi\psi\psi\psi\psi\psi+{\rm h.c.}\;.$ (5.63)
Based on our viewpoint that the argument of the logarithmic function acquires
a nonvanishing VEV, the assumption of four-fermion condensate (5.61) appears
to leads to a difficulty. In constrast, the assumption of bi-fermion
condensates (5.62) looks perfectly consistent, with
$\langle\psi\psi\psi\psi\psi\psi\rangle\sim\langle\psi\psi\rangle^{i}_{j}\langle\psi\psi\rangle^{j}_{k}\langle\psi\psi\rangle^{k}_{i}\neq
0\;.$ (5.64)
#### 5.6.2 Adjoint QCD with $N_{\rm c}=N_{\rm f}=2$
It is interesting to apply the same logics also to the adjoint QCD, previously
analized from the point of view of generalized symmetries and their anomalies.
In particular let us focus on the $N_{\rm c}=2$, $N_{\rm f}=2$ case, because
of the renewed interest in this particular model, raised by the work of Anber
and Poppitz [56].
The conventional thinking holds that a gauge invariant bi-fermion condensate
$\langle\lambda\lambda\rangle\neq 0$ (5.65)
forms, breaking the flavor symmetry as $SU(2)_{\rm f}\to SO(2)_{\rm f}$,
leading to $2$ NG bosons, and reducing the discrete ${\mathbbm{Z}}_{8}$
symmetry to ${\mathbbm{Z}}_{2}$ resulting four degenerate vacua. The Anber and
Poppitz’s proposal [56] is that, instead, the system might develop a four-
fermion condensates but not bifermion condensate:
$\langle\lambda\lambda\lambda\lambda\rangle\neq
0\;,\qquad\langle\lambda\lambda\rangle=0\;.$ (5.66)
The discrete $\mathbbm{Z}_{8}$ symmetry is now broken to its $\mathbbm{Z}_{4}$
subgroup (therefore there are only two degenerate vacua). Massless baryons
$\sim\lambda\lambda\lambda$ (5.67)
(necessarily a doublet of $SU(2)_{\rm f}$) matches the UV-IR Witten anomaly of
$SU(2)_{f}$.
As said above, the two possibilities are both consistent with the generalized
’t Hooft anomaly matching, therefore an indication from the strong-anomaly
effective action would be very welcome.
The analogue of (5.4), (5.46) and (5.63), is in this case,
$\frac{i}{2}q(x)\log\lambda\lambda\ldots\lambda+{\rm h.c.}\;.$ (5.68)
with eight $\lambda$’s inside the argument of the logarithmic function.
Therefore, in contrast to what we saw in the preceding model Sec. 5.6.1, our
strong-anomaly algorithm does not seem to be able to discriminate the two
dynamical possibilities, (5.65), or (5.66).
Before concluding this section, we note that in the case with $N_{\rm f}=1$,
arbitrary $N_{\rm c}$, the adjoint QCD becomes ${\cal N}=1$ supersymmetric
Yang-Mills theory. The strong-anomaly effective action (5.68) with $2N_{\rm
f}N_{\rm c}=2N_{\rm c}$ $\lambda$’s, reduces precisely to the Veneziano-
Yankielowicz effective potential [35], implying
$\langle\lambda\lambda\rangle\neq 0$. In this case the assumption of the bi-
fermion condensate, $\langle\lambda\lambda\rangle\neq 0$, the breaking of the
discrete symmetry ${\mathbbm{Z}}_{2N_{\rm c}}\to{\mathbbm{Z}}_{2}$, and the
resulting $N_{\rm c}$ fold degeneracy of the vacua (Witten’s index), are
generally accepted as a well-established fact.
## 6 Summary and discussion
We have reviewed in this article the first applications of generalized
anomalies and discussed what the consequent stronger anomaly matching
conditions tell us about various chiral gauge theories based on $SU(N)$ gauge
group. Our discussion was divided in two class of models. In the first, the
system has a 1-form ${\mathbbm{Z}}_{k}$ symmetry ($k$ being a divisor of $N$)
under which the matter fermions do not transform. The treatment in this case
is relatively straightforward: certain discrete symmetries, respected by
instantons, are often found to be further made anomalous, due to the
fractional ’t Hooft fluxes accompanying the gauging of the discrete 1-form
symmetries. The discussion of Sec. 3 has illustrated that their consequences
depend nontrivially on the types of matter fermions present, and an
interesting and rich variety of predictions on the possible condensates,
symmetry breaking patterns and phases, have been found.
A second group of models (BY and GG models) have a color-flavor locked
${\mathbbm{Z}}_{N}$ 1-form symmetry, in which matter fermions transform
together with the $SU(N)$ gauge field. A careful analysis of the global
properties of the symmetry group is needed before actually introducing the
gauging of this discrete 1-form symmetry. This has been worked out in detail
in all of the generalized Bars-Yankielowicz and Georgi-Glashow models, and the
results of the analysis reviewed in Sec. 4. A surprising implication is that,
at least for even $N$ theories, color-flavor locked
${\mathbbm{Z}}_{2}-U(1)-{\mathbbm{Z}}_{N}$ 1-form symmetry does not allow its
gauging which is a new kind of ’t Hooft anomaly, and that this is consistent
with the low-energy system being in dynamical Higgs phase characterized by
certain bifermion condensates.
After the examination of the results from the generalized anomalies (involving
the 1-form symmetries and gauging of some discrete center symmetries), we
changed the topics, and turned to the very recent observation [96] about the
strong-anomaly associated effects. This is the idea discussed in the context
of the so-called $U(1)_{A}$ problem in QCD many years ago, but for some reason
it was almost never applied to the discussion of the physics of strongly-
coupled chiral gauge theories. It is found that the requirement that the
massless degrees of freedom of the hypothesized infrared phase should be able
to describe appropriately the strong anomaly gives a rather clear indication
on the physics of BY and GG models: the structure of the strong-anomaly
effective action favors the dynamical Higgs vacua, against the confining,
fully flavor symmetric vacua, in agreement with the implications from the
generalized anomaly matching algorithm, explored in the first part of this
review.
The fact that both mixed anomalies [70, 71] and the strong-anomaly effective
action [96] imply dynamical Higgs phase in chiral BY and GG models is
certainly not accidental. Both arise by taking properly the strong chiral
$U(1)$ anomalies into account.
These discussions, rather unexpectedly, brought us to note certain analogies
and contrasts between the strong-interaction dynamics of vector-like and
chiral gauge theories [96]. Let us now compare the standard QCD with $N_{\rm
f}$ light flavors of quarks and antiquarks, and the $\psi\eta$, $\chi\eta$
models as well as more general Bars-Yankielowicz and Georgi-Glasow models.
In many senses, the bifermion condensates such as $U=\psi\eta$ in the
$\psi\eta$ model (and $\chi\eta$, $\chi\chi$ condensates in the $\chi\eta$
model), can be regarded as a perfect analogue of the quark condensate
$U={\bar{\psi}}_{R}\psi_{L}$ in QCD. All of these composite scalars enter the
strong-anomaly effective action in a similar way, as
${\hat{L}}=\frac{i}{2}q(x)\log\det U/U^{\dagger}\;,\qquad
q(x)=\frac{g^{2}}{32\pi^{2}}F_{\mu\nu}^{a}{\tilde{F}}^{a,\mu\nu}\;.$ (6.1)
(See Sec. 5 for more careful discussions.) And in all cases this implies
condensation of $\langle U\rangle\propto{\mathbf{1}}$, i.e., the color-flavor-
locked Higgs phase in the $\psi\eta$ or $\chi\eta$ models on the one hand, and
the chiral-symmetry broken vacuum in QCD., on the other.
Another fact pointing to a similarity between massless QCD and BY and GG
models is the following. In general Bars-Yankielowicz models (with $p$ pairs
of additional matter fermions), we saw that there are two natural bifermion
condensate channels:
$\displaystyle\psi\Big{(}\raisebox{-3.0pt}{\yng(2)}\Big{)}\,\eta\Big{(}\bar{\raisebox{-3.0pt}{\yng(1)}}\Big{)}\qquad\
$ $\displaystyle{\rm forming}\qquad\raisebox{-3.0pt}{\yng(1)}\;,$
$\displaystyle{\rm and}$
$\displaystyle\quad\xi\Big{(}{\raisebox{-3.0pt}{\yng(1)}}\Big{)}\,\eta\Big{(}\bar{\raisebox{-3.0pt}{\yng(1)}}\Big{)}\qquad\
$ $\displaystyle{\rm forming}\qquad(\cdot)\;:$ (6.2)
the gluon-exchange strengths in the two channels are, respectively,
proportional to
$-\frac{(N+2)(N-1)}{N}$ and $-\frac{N^{2}-1}{N}$ . The $\psi\eta$ channel is
slightly more attractive; the strength is however identical in the large $N$
limit. Note also that $\xi\eta$ has the same quantum numbers as
${\bar{\psi}}_{R}\psi_{L}$ in QCD. Similarly for the comparison between the
condensates, $\langle\chi\eta\rangle$ and $\langle\xi\eta\rangle$ in the
Georgi-Glashow models. These considerations, based on rather naïve MAC [4]
idea and thus are not rigorous, nevertheless give supports to the idea that
the quark condensates in QCD and the bifermions condensates in the chiral
gauge theories under study in this review, are really on a very similar
footing.
Of course, there are significant differences, or contrast, in the vectorlike
and chiral gauge theories. The quark condensate
$\langle{\bar{\psi}}_{R}\psi_{L}\rangle$ is a color singlet, $SU(N_{\rm
f})_{L}\times SU(N_{\rm f})_{R}$ flavor matrix. $\langle\psi\eta\rangle$ is
instead in a color-flavor bifundamental form, which means that it breaks the
color completely, and reduces (partially or totally) the unbroken flavor
symmetry. The most important difference however is the existence of colored NG
bosons in the $\psi\eta$ (or in the $\chi\eta$) models. It means that these
are coupled linearly to the gauge boson fields, making them massive. These
processes are absent in QCD, as all NG bosons are color singlets. It is in
this sense that one talks about confinement phase in QCD, in spite of the fact
that the inter-quark confining strings can be broken by the spontaneous quark-
pair production from the vacuum.
The mass spectra are also qualitatively different in QCD and in the chiral
gauge theories discussed here. One is the presence of certain degenerate
massive vector bosons (the color-flavor locked $SU(N)_{\rm cf}$ symmetry)
found in the chiral gauge theories in the Higgs phase. But especially the
massless spectrum exhibits striking differences. In all chiral gauge theories
studied here, it contains in general both a number of composite fermions
(baryons) as well as some composite scalars (pions), a feature certainly not
shared by massless QCD. In other words, the way the chiral symmetries of the
theory are realized in the IR, is notably different, in vector-like and chiral
gauge theories.
It is possible to see a closer analogy - from a formal point of view - between
the vector-like theories and chiral theories, if one considers color
superconductivity phase in the high-density region of QCD [100, 101]. The
dynamics of QCD in that phase is believed to be such that some colored di-
quark condensates
$\langle\psi_{L}\psi_{L}\rangle\neq
0\;,\qquad\langle\psi_{R}\psi_{R}\rangle\neq 0\;.$ (6.3)
form. In particular, in the case with $N_{\rm f}=3$ flavors these are
condensates of color-flavor diagonal form, showing some similarity to
$\langle\psi\eta\rangle$ or $\langle\chi\eta\rangle$ in the chiral theories
discussed here. Of course, the details of the dynamics will be quite
different.
Summarizing, the implications of new, mixed anomalies and the associated
stricter anomaly-matching constraints reviewed in this article, and the
consideration of the strong-anomaly effective actions, together, seem to allow
us to get a clearer picture of the infrared dynamics of many strongly-coupled
chiral gauge theories than before. It is to be seen whether some of these
developments will turn out to be useful in a future effort to construct a
realistic theory of Nature beyond the standard Glashow-Weinberg-Salam-QCD
model of the fundamental interactions.
## Acknowledgments
The work is supported by the INFN special research initiative grant, ”GAST”
(Gauge and String Theories).
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|
On the Nature of Spatial Universes
in 3D Lorentzian Quantum Gravity
J. Brunekreef and R. Loll
Institute for Mathematics, Astrophysics and Particle Physics, Radboud
University
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands.
email<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract
Three-dimensional Lorentzian quantum gravity, expressed as the continuum limit
of a nonperturbative sum over spacetimes, is tantalizingly close to being
amenable to analytical methods, and some of its properties have been described
in terms of effective matrix and other models. To gain a more detailed
understanding of three-dimensional quantum gravity, we perform a numerical
investigation of the nature of spatial hypersurfaces in three-dimensional
Causal Dynamical Triangulations (CDT). We measure and analyze several quantum
observables, the entropy exponent, the local and global Hausdorff dimensions,
and the quantum Ricci curvature of the spatial slices, and try to match them
with known continuum properties of systems of two-dimensional quantum
geometry. Above the first-order phase transition of CDT quantum gravity, we
find strong evidence that the spatial dynamics lies in the same universality
class as two-dimensional Euclidean (Liouville) quantum gravity. Below the
transition, the behaviour of the spatial slices does not match that of any
known quantum gravity model. This may indicate the existence of a new type of
two-dimensional quantum system, induced by the more complex nature of the
embedding three-dimensional quantum geometry.
###### Contents
1. 1 Introduction
2. 2 Three-dimensional CDT quantum gravity
3. 3 Implementation
4. 4 Geometric observables on spatial hypersurfaces
1. 4.1 Vertex order
2. 4.2 Entropy exponent
3. 4.3 Hausdorff dimension
1. 4.3.1 Local Hausdorff dimension
2. 4.3.2 Global Hausdorff dimension
3. 4.3.3 Discussion of results
4. 4.4 Curvature profile
5. 5 Summary and conclusion
6. A Estimating the entropy exponent
7. B Determining confidence intervals for best fit parameters
1. B.1 Best fit parameter estimation
2. B.2 Confidence intervals
3. B.3 Potential caveats
## 1 Introduction
A complete theory of quantum gravity may offer insights into how the spacetime
we observe and inhabit can emerge from first principles. The nonperturbative
gravitational path integral is a promising route toward such a theory,
formulated within a purely quantum field-theoretic setting [1]. If one is
interested in concrete Planckian or near-Planckian results in the full, four-
dimensional theory, like information on the spectra of diffeomorphism-
invariant observables, Causal Dynamical Triangulations or CDT quantum gravity
[2, 3] is arguably the path integral approach that is furthest developed.
Recall that the continuum path integral for pure gravity is given by
$Z=\int\limits_{{\cal G}(M)}\\!\\!{\cal D}[g]\,{\rm e}^{\,iS^{\rm
EH}[g]},\;\;\;\;\;\;S^{\rm EH}[g]=\frac{1}{16\pi G_{\rm
N}}\,\int\limits_{M}d^{4}x\,\sqrt{-\det(g)}\,(R-2\Lambda),$ (1)
where ${\cal G}(M)$ denotes the space of diffeomorphism-equivalence classes
$[g]$ of Lorentzian metrics $g_{\mu\nu}(x)$ on the manifold $M$, and $S^{\rm
EH}$ is the Einstein-Hilbert action. In the CDT set-up this formal expression
is given a precise meaning, namely, as the continuum limit of a regularized
version of (1), with ${\cal G}(M)$ approximated by a space of piecewise flat
Lorentzian spacetimes. Although the primary physical interest is in spacetime
dimension $D\\!=\\!4$, the CDT path integral has also been studied in two and
three dimensions.
Hallmarks of this strictly nonperturbative approach are (i) the presence of a
well-defined analytic continuation or “Wick rotation”, mapping the regularized
path integral to a real partition function, which enables its analytical
evaluation in $D\\!=\\!2$ [4] and numerical evaluation in $D\\!=\\!2$, 3 and 4
[5, 6], (ii) its formulation on a space of geometries, avoiding the need to
gauge-fix the diffeomorphism symmetry and isolate its physical degrees of
freedom, (iii) following the logic of critical phenomena, a high degree of
uniqueness and universality if a continuum limit can be shown to exist, (iv) a
nonperturbative cure of the conformal divergence, which by default renders
Euclidean path integrals in $D\\!\geq\\!3$ ill defined [7], and (v) unitarity,
in the form of reflection positivity of the regularized path integral, with
respect to a notion of discrete proper time [6, 2].
In terms of results in $D\\!=\\!4$, in addition to the presence of second-
order phase transitions [8, 9, 10], necessary for the existence of a continuum
limit, an important finding of CDT is the emergence of an extended four-
dimensional universe [11, 12]. With the standard choice
$M\\!=\\!S^{1}\\!\times\\!S^{3}$ for the topology, in terms of the quantum
observables measured so far (spectral and Hausdorff dimensions [13, 14], shape
of the universe, including quantum fluctuations [15, 16], average Ricci
curvature [17]), its behaviour on sufficiently coarse-grained scales is
compatible with that of a de Sitter universe. This is remarkable because it
represents nontrivial evidence of a classical limit, one of the high hurdles
to clear for any nonperturbative and manifestly background-independent
approach to quantum gravity.
After Wick rotation, the gravitational path integrals of CDT become partition
functions of statistical systems, whose elementary geometric building blocks
(flat $D$-dimensional simplices, see Sec. 2 for further details) are assembled
into piecewise flat manifolds $T$ – the triangulations – each one contributing
with a Boltzmann weight $\exp(-S^{\rm EH}[T])$.111Note that $S^{\rm EH}[T]$ is
the so-called bare action of the regularized theory, depending on bare
coupling constants, which in the continuum limit will typically undergo
renormalization. There are a couple of reasons why such seemingly simple
ingredients can give rise to interesting continuum theories of quantum gravity
and quantum geometry. On the one hand, there is the highly nontrivial
combinatorics of how the simplicial building blocks can be glued together to
yield distinct curved spacetimes $T$. Especially in dimension $D\\!\geq\\!3$,
this reflects the complexities of local geometry and curvature, already
familiar from the classical theory. On the other hand, there is a complicated
interplay between “energy” (the bare action) and “entropy” (the number of
distinct triangulations for a given value of the bare action), which depends
on the values of the bare coupling constants, i.e. the point in phase space at
which the path integral is evaluated.
An enormous amount has been learned about such nonperturbative systems of
geometry over the last 35 years, beginning with the Euclidean analogue and
precursor of the Lorentzian CDT theory, based on Euclidean dynamical
triangulations or dynamical triangulations (DT) for short [18, 19, 20]. A
crucial role in the exploration of these systems has been played by Monte
Carlo methods, which are employed to numerically evaluate the path integral
and expectation values of observables by importance sampling [21, 22]. This is
also true in dimension $D\\!=\\!2$, where in addition a variety of
nonperturbative analytical solution techniques are available, e.g.
combinatorial, matrix model and transfer matrix methods [23, 4, 24], leading
to compatible results.
Monte Carlo simulations should be seen as numerical experiments, providing
tests and feedback for the construction of the theory. For full quantum
gravity, the quantitative information on the nonperturbative sector obtained
from numerical analysis is extremely valuable, since it cannot currently be
substituted by anything else. Although we do not know in detail what a theory
of quantum gravity will eventually look like, it seems unlikely that it will
come in closed analytic form. A potential scenario would be akin to QCD, where
we manage to extract nonperturbative information about the theory’s spectrum
(of suitable quantum-geometric observables) with ever greater accuracy, using
a background-independent analogue of lattice gauge theory such as (C)DT.
Despite its conventional, quantum field-theoretic setting and the absence of
any exotic ingredients, this type of lattice gravity has already uncovered
unexpected features of strongly quantum-fluctuating geometry, like the
dynamical dimensional reduction of spacetime near the Planck scale [13], which
is conjectured to be universal [25].
The focus of the present work will be the Lorentzian CDT path integral in
three spacetime dimensions [26]. More specifically, as a stepping stone
towards a more detailed geometric understanding of this quantum gravity model,
we will investigate the geometry of its two-dimensional spatial hypersurfaces.
A key question is whether in a continuum limit the behaviour of these surfaces
falls into one of the known universality classes [27] of nonperturbative
quantum gravity in two dimensions, or whether there is evidence for a
different type of quantum dynamics. The two universality classes in question
are that of (the scaling limit of) two-dimensional DT [18, 20], which also
contains Liouville quantum gravity, and that of two-dimensional CDT quantum
gravity [4, 28].
Our study will be numerical in nature, but – depending on the outcome – may
well provide input for further analytical work, which could be technically
feasible because of the effective two-dimensional character of the spatial
slices. Note that we do not claim that there is a direct physical
interpretation of the properties of these spatial geometries from a three-
dimensional point of view (inasmuch as a lower-dimensional toy model of
quantum gravity can be called “physical” at all). Although in our set-up a
spatial slice at constant proper time is an invariantly defined concept222It
is defined as the set of all points at a given proper-time distance to a given
initial spatial surface or an initial singularity, in the spirit of similar
constructions in the continuum [29]. Note also that the proper-time slicing is
not related to any gauge-fixing, since the CDT set-up is manifestly
diffeomorphism-invariant (see e.g. [3] for a detailed discussion)., it is not
clear to what extent its properties can be thought of as “observable”, because
of the highly nonlocal construction of the hypersurfaces and because of their
singular nature (“moments in time”) from the point of view of the quantum
theory. To obtain true quantum observables in a three-dimensional, spacetime
sense would presumably require some smearing in the time direction.
Nevertheless, our measurements within the slices of constant time are
perfectly well defined operationally and give us a quantitative handle on the
influence of the three-dimensional quantum geometry in which the spatial
slices are embedded, as we will demonstrate. We will start by investigating
the distribution of the vertex order in the slices, which counts the number of
spatial edges meeting at a vertex. This quantity is not per se related to a
continuum observable, but can be compared with known exact results for the
ensembles of two-dimensional DT and CDT geometries. The core of the paper
consists of measuring and analyzing the following quantum observables: (i) the
entropy exponent $\gamma$, also known as the string susceptibility, which
determines the subexponential growth of the partition function at fixed two-
volume $A$, as a function of $A$; (ii) the Hausdorff dimension $d_{H}$,
obtained by comparing volumes with their linear extension, where we
distinguish between a local and a global variant; (iii) the so-called
curvature profile $\mathcal{R}(\delta)$ of the spatial slices, measuring the
average quantum Ricci curvature [30] of the surfaces as a function of a linear
coarse-graining scale $\delta$. We find convincing evidence that the effective
dynamics of the spatial slices in the so-called degenerate phase of three-
dimensional CDT quantum gravity is described by two-dimensional DT quantum
gravity. However, we do not find a match with any known two-dimensional system
of quantum geometry in the so-called de Sitter phase, where the dynamics of
the hypersurfaces is much richer due to the nontrivial influence of the
embedding three-geometry. Further research is needed to determine the
continuum nature of the effective spatial dynamics in this phase.
The remainder of the paper is structured as follows. In the next section, we
recall the main ingredients of CDT quantum gravity in $D\\!=\\!3$ and review
previous research on the subject, and what it has revealed about its phase
structure and physical characteristics. In Sec. 3, we discuss the numerical
implementation of the three-dimensional CDT path integral in terms of Markov
chain Monte Carlo methods. Sec. 4 contains a detailed description of the
properties of the spatial slices that we have studied numerically. We present
the results of our measurements, and describe the overall picture that emerges
from them. In Sec. 5 we summarize and discuss our findings. A couple of
technical discussions have been relegated to appendices, to improve the
readability of the main part of the paper.
## 2 Three-dimensional CDT quantum gravity
Quantum gravity in three spacetime dimensions [31] provides an interesting
test case for the full gravitational path integral. Although the pure gravity
theory does not have any local propagating degrees of freedom, the path
integral has the same functional form in terms of the three-dimensional metric
as its four-dimensional counterpart (1), and therefore looks equally ill-
behaved with regard to its behaviour under renormalization. How to reconcile
the difficulties of solving this metric path integral with the “topological”
nature of three-dimensional gravity333More precisely, the physical degrees of
freedom of three-dimensional gravity are global modes of the metric, described
by Teichmüller parameters, which are present when the genus of the spatial
slices is larger than or equal to 1. The present work uses spherical slices,
without such parameters., which leads to considerable simplifications in a
first-order, Chern-Simons formulation, without any quantum field-theoretic
divergences, is only partially understood (see [32] for a discussion).
The CDT formulation has thrown some light on the nonperturbative aspects of
this question, uncovering both similarities and differences between the three-
dimensional and the physical, four-dimensional theory [26, 33, 34]. For a
better understanding of the issues involved and to set the stage for the main
part of the paper, let us briefly recall the set-up in three dimensions. After
applying the Wick rotation mentioned in the previous section, the regularized
CDT path integral in $D\\!=\\!3$ takes the form of a partition function
$Z=\sum\limits_{\text{triang.}\,T}\frac{1}{C_{T}}\,{\rm e}^{-S^{\rm
EH}[T]},\;\;\;\;\;\;\;S^{\rm EH}[T]=-k_{0}N_{0}(T)+k_{3}N_{3}(T),$ (2)
where $S^{\rm EH}[T]$ denotes the Regge form of the Einstein-Hilbert action on
the piecewise flat triangulation $T$, $N_{0}(T)$ and $N_{3}(T)$ are the
numbers of vertices (“zero-simplices”) and tetrahedra (“three-simplices”), and
$C_{T}$ is the order of the automorphism group of $T$. The coupling $k_{0}$ is
proportional to the inverse bare Newton constant and $k_{3}$ depends linearly
on the bare cosmological constant (see [26] for details). The sum is taken
over simplicial manifolds of a given, fixed topology, which in our case will
be $S^{1}\\!\times\\!S^{2}$, a periodically identified time interval times a
two-sphere.
The triangulated configurations $T$ of the Lorentzian CDT path integral have a
discrete product structure, representing a simplicial version of global
hyperbolicity, and are assembled from flat, Minkowskian tetrahedra [2, 3]. A
given spacetime geometry can be thought of as a sequence of two-dimensional
curved, spacelike triangulations, labeled by an integer proper time
$t=1,2,3,\dots,t_{\rm tot}$, and made of equilateral triangles. The spacetime
volume between each pair of adjacent constant-time slices is completely filled
in with tetrahedra, resulting in a “sandwich” of simplicial three-dimensional
spacetime with topology $[0,1]\times S^{2}$. The tetrahedral edges linking
neighbouring spatial slices are _timelike_ (and all of equal length), while
the edges lying within a spatial slice are of course _spacelike_ (and also of
equal length). We can therefore classify the building blocks according to
their constituent vertices. A tetrahedron of type $(p,q)$ is defined as having
$p$ vertices in slice $t$ and $q$ vertices in slice $t\\!+\\!1$, giving rise
to the types (1,3), (2,2) and (3,1), as illustrated by Fig. 1. Note that up to
time reversal a (1,3)- and a (3,1)-tetrahedron are geometrically identical.
Since the analytic continuation only affects the edge length assignments and
not the topology of the triangulation, this characterization of the tetrahedra
continues to be meaningful after the Wick rotation.
Figure 1: Three types of tetrahedral building blocks of three-dimensional CDT:
type (1,3) (left), type (2,2) (centre), and type (3,1) (right). Note that the
two-dimensional triangulations at times $t$ and $t\\!+\\!1$ are not drawn
isometrically; they are in general curved surfaces.
The key findings of the original, mostly numerical investigation of three-
dimensional CDT quantum gravity on $S^{1}\\!\times\\!S^{2}$ inside the range
$k_{0}\\!\in\\![3,7]$ were as follows [26, 33, 34]. After fine-tuning the bare
“cosmological” constant $k_{3}$ to its critical value from inside the region
of convergence444This region exists because the number of three-dimensional
CDT configurations is exponentially bounded as a function of the discrete
volume $N_{3}$ [35]. of the partition function Z, a two-phase structure was
found, consisting of what we shall call a _degenerate phase_ for $k_{0}\\!\geq
k_{0}^{\rm c}$ and a _de Sitter phase_ for $k_{0}\\!\leq k_{0}^{\rm c}$. These
two phases are very reminiscent of corresponding phases in four-dimensional
CDT quantum gravity [2] with regard to their volume profiles, i.e. the
behaviour of their spatial volume $V_{2}$ as a function of the proper time
$t$. In the degenerate phase, $V_{2}(t)$ oscillates wildly, indicating that
spacetime disintegrates into a sequence of uncorrelated two-dimensional
geometries (see also Fig. 4 below). By contrast, in the de Sitter phase a
nontrivial “blob” forms when the time extension is chosen sufficiently large,
whose shape matches that of a Euclidean de Sitter space (with $\langle
V_{2}(t)\rangle\propto\cos^{2}(\mathit{c}\,t)$), analogous to what has been
observed in $D\\!=\\!4$.555Volume profiles for nonperiodic boundary conditions
in time, with random, fixed two-spheres of various sizes as spatial boundaries
have been considered in [36, 37]. This constitutes nontrivial evidence that a
well-defined and macroscopically three-dimensional ground state of
geometry666in the sense of minimizing the effective Euclidean action governing
the nonperturbative quantum dynamics exists nonperturbatively. However, unlike
in four dimensions the transition at the critical point $k_{0}^{c}$ appears to
be a first- and not a second-order phase transition, and no fine-tuning of the
inverse gravitational coupling $k_{0}$ is needed to obtain a continuum limit.
This is in line with the expectation that no higher-order transitions are
present, due to the absence of propagating degrees of freedom.
Following these results, three-dimensional CDT quantum gravity has been
studied from various perspectives. Considerable effort has been focused on the
transfer matrix associated with a single time step $\Delta t\\!=\\!1$, which
captures the amplitude of going from one spatial two-geometry to an adjacent
one. More precisely, one usually considers a simpler, reduced transfer matrix,
whose in- and out-states are labelled by the spatial two-volume (and possibly
Teichmüller parameters). Given our knowledge about the physical degrees of
freedom of three-dimensional gravity, these are the parameters that are
_expected_ to be the only relevant ones in the continuum limit. The sandwich
geometries contributing to the transfer matrix are closer to two-dimensional
quantities and therefore potentially more amenable to an analytic treatment.
In this spirit, a variant of the model was introduced in [38], in which the
(1,3)- and (3,1)-building blocks are substituted by (1,4)- and (4,1)-pyramids,
something that is not expected to affect the universal properties of the
model. The motivation for considering this variant is that taking a midsection
of a sandwich geometry at half-integer time yields a quadrangulation, whose
dual graph is a configuration described by a Hermitian two-matrix model with
$ABAB$-interaction, for which analytical results are available. However, the
bicoloured graph configurations generated by the matrix model form a much
larger class than those coming from CDT sandwich geometries, and correspond to
geometries that in general violate the simplicial manifold conditions of the
two-dimensional slices and of the interpolating three-dimensional piecewise
flat geometries in specific ways. Three of these four conditions (following
the enumeration in [38]) are considered mild, in the sense that violating them
is conjectured not to affect the universality class of the CDT model. As
corroborating evidence, [38] cites new numerical simulations of the CDT model
in $D\\!=\\!3$, where these conditions are relaxed, but which nevertheless
reproduce the results found in the de Sitter phase of the earlier work that
used strict simplicial manifolds [26]. Interestingly, they note that the
degenerate phase completely disappears in the simulations of this generalized
variant of CDT quantum gravity, and conjecture that the presence of this phase
constitutes a discretization artefact. Although our present work does not
directly address these various conjectures (and works with simplicial
manifolds only), our results suggest that there may be more scope for
different universality classes in three dimensions than has been considered up
to now, and that it may be fruitful to re-examine the influence of regularity
conditions on continuum results in greater detail. (Of course, not all
universality classes may be associated with interesting models of quantum
gravity.) A similar sentiment was expressed in [39], which gives a precise
characterization of the bicoloured two-dimensional cell complexes associated
with midsections of CDT geometries with spherical and disc-like spatial
slices.
The configurations described by the $ABAB$-matrix model violate also a fourth
regularity condition [38], which is associated with a considerable enlargement
of the space of three-geometries. It allows for the appearance of spatial
wormholes and a new phase, not present in standard CDT quantum gravity, where
these wormholes are abundant. Whether this phase is interesting from a physics
point of view remains to be understood. The association of CDT quantum gravity
with the $ABAB$-matrix model was also used to analyze the behaviour of the
bare coupling constants of the former under renormalization [40]. An
asymmetric version of the matrix model was studied in [41], motivated by the
search for a Hamiltonian associated with the reduced transfer matrix. Without
invoking matrix models, a continuum Hamiltonian of this kind was derived for
the first time in [42], for spatial slices of cylinder topology, albeit for a
sub-ensemble of CDT configurations with certain ordering restrictions. The
effective action for the two-volumes of spatial slices with toroidal topology
was investigated in [43], and the dynamics of its Teichmüller parameters in
[44, 45]. The phase structure of a one-dimensional balls-in-boxes model, meant
to capture the effective dynamics of the two-volume of CDT quantum gravity,
was analyzed in [46], and shown to reproduce certain features of CDT as well
as Hořava-Lifshitz-inspired gravity models (see also [47]). The spectral
dimension of the CDT model in the de Sitter phase was measured and found to be
compatible with the classical value of 3 on large scales and to exhibit a
dynamical dimensional reduction to a value compatible with 2 on short scales,
similar to what happens in CDT for $D\\!=\\!4$ [48]. Lastly, a generalized
model of CDT quantum gravity with causally well-behaved configurations, but
without a preferred proper-time slicing was defined and investigated
numerically, and found to reproduce the volume profile of a de Sitter space
[49, 50].
## 3 Implementation
Expectation values of geometric observables $\mathcal{O}$ in CDT quantum
gravity are computed as
$\langle\mathcal{O}\rangle=\frac{1}{Z}\,\sum_{T}\frac{1}{C_{T}}\,\mathcal{O}[T]\,{\rm
e}^{-S^{\textrm{EH}}[T]},$ (3)
where $Z$ is the partition function defined in (2). As already mentioned, the
focus of our present work is a set of observables pertaining to the two-
dimensional spatial triangulations of constant integer time $t$ of the three-
dimensional CDT configurations, which will be the subject of Sec. 4. Since it
is not known how to compute $Z$ analytically in three dimensions, we will
compute statistical estimates of the expectation value of an observable by
sampling the CDT ensemble through Monte Carlo simulations.777Our
implementation code can be found at [51]. In these simulations, we construct a
random walk in the ensemble of CDT geometries by performing local updates
(“moves”) on a triangulation. The basic set of moves we used is shown in Fig.
2, see also [26]. If we impose so-called detailed balance [52] on the updating
procedure, by accepting or rejecting such moves with an appropriate
probability, this random walk corresponds to a sample of the ensemble where
geometries appear with a relative rate according to their Boltzmann weight.
Since subsequent geometries in a random walk are almost identical, we must
perform a large number of local moves on a given geometry to obtain a new and
sufficiently independent one. This procedure is iterated to obtain a sequence
of independent triangulations, and an estimate of the expectation value of an
observable is computed as the weighted average (3) over this sequence. To
study the continuum properties of observables, the number of building blocks
should be taken to infinity. Since this is impossible in practice, due to the
finiteness of our computational resources, we use finite-size scaling methods
[22] to estimate the behaviour of the system in the continuum limit. For more
details on computer simulations of three-dimensional CDT we refer the
interested reader to [33].
Figure 2: Three basic local moves in 3D CDT quantum gravity together with
their inverses, and their effect on spatial slices. Spacelike edges are drawn
in blue and timelike ones in red. Top left: subdivision of a spatial triangle
into three; top right: flip of a spatial edge. The move at the bottom does not
affect the spatial triangulations (time inverse not shown).
For a given value of the gravitational coupling $k_{0}$, the cosmological
coupling $k_{3}$ is always tuned to its $k_{0}$-dependent pseudocritical
value888which in the limit $N_{3}\\!\rightarrow\\!\infty$ would become the
critical value $k_{3}^{c}(k_{0})$, which means that we are investigating a
one-dimensional phase space parametrized by $k_{0}$. The location of the
critical point $k_{0}^{c}$ along this line, associated with the first-order
transition mentioned in Sec. 2, is not a universal quantity. For example, it
depends on the regularity conditions imposed on the ensemble, and the time
extension $t_{\rm tot}$ of the geometries [26]. In our analysis of the spatial
slices we will use standard CDT simplicial manifolds999The effects of relaxing
the local manifold constraints have been investigated further in [53], where
it was found that the order of the phase transition is likely unchanged,
although the location of the critical point shifts to a smaller value as the
restrictions are loosened. and $t_{\rm tot}\\!=\\!3$ with periodic boundary
conditions in time, for which we have found $k_{0}^{c}\\!\approx\\!6.24$.
Figure 3: Expectation value of the order parameter $O_{2}\\!=\\!N_{22}/N_{31}$
as a function of the bare coupling $k_{0}$, exhibiting a first-order phase
transition at $k_{0}^{c}\\!\approx\\!6.24$. To the left of the transition is
the de Sitter phase, and to its right the degenerate phase. Our measurements
on spatial slices are taken for the $k_{0}$-values $0.0$, $5.0$ and $8.0$, as
indicated.
A convenient order parameter to locate the phase transition is the ratio
$O_{2}(T)=\frac{N_{22}(T)}{N_{31}(T)},$ (4)
where $N_{22}(T)$ and $N_{31}(T)$ denote the numbers of (2,2)- and
(3,1)-simplices of the triangulation $T$ respectively. Its expectation value
$\left\langle O_{2}\right\rangle$ is nonvanishing for small $k_{0}$ and drops
to zero rapidly as the transition point $k_{0}^{c}$ is approached, beyond
which it remains zero for all values of $k_{0}\\!>\\!k_{0}^{c}$ we have
investigated. The measured values of $\left\langle O_{2}\right\rangle$ as a
function of $k_{0}$ are shown in Fig. 3, obtained in a system with
$N_{3}\\!=\\!64.000$ and $t_{\rm tot}=3$. The regions to the left and right of
the transition correspond to the de Sitter and degenerate phases introduced
earlier. Snapshots of typical volume profiles $V_{2}(t)$, counting the number
of triangles in the spatial slice at time $t$, are depicted in Fig. 4 for a
system with $N_{31}\\!=\\!16.000$ and $t_{\rm tot}\\!=\\!32$. The volumes of
neighbouring slices in the degenerate phase are largely uncorrelated, while
they tend to align in the de Sitter phase. When taking an ensemble average of
the latter with the “centres of volume” aligned, the expectation value
$\langle V_{2}(t)\rangle$ matches that of a three-dimensional Euclidean de
Sitter universe in a proper-time parametrization [26].
Figure 4: Volume profiles $V_{2}(t)$ of typical configurations appearing in
the de Sitter phase (left) and the degenerate phase (right) of the three-
dimensional CDT model on $S^{1}\\!\times\\!S^{2}$.
We will perform measurements of the spatial slices for three different
$k_{0}$-values, two in the de Sitter phase at $k_{0}\\!=\\!0.0$ and $5.0$ and
one in the degenerate phase at $k_{0}\\!=\\!8.0$, cf. Fig. 3. The volumes
$N_{31}$ of the systems we investigate take values in the range
$[1.000,96.000]$. In choosing these particular values of $k_{0}$, we are
staying away from the direct vicinity of the first-order transition at
$k_{0}^{c}\\!\approx\\!6.24$, to avoid that the system jumps between the two
phases during the simulation. The two points chosen in the de Sitter phase are
spaced well apart, while taking into account that the Monte Carlo algorithm
becomes increasingly inefficient as $k_{0}$ is lowered. It may be worth
pointing out that the expression for the discretized, bare Einstein-Hilbert
action in (2) is highly non-unique, and that the value $k_{0}\\!=\\!0.0$ is
therefore in no way physically distinguished.
The observed decoupling of neighbouring slices provides a strong argument for
an effective slice behaviour that is characteristic for the universality class
of Euclidean dynamical triangulations, something our data in Sec. 4 below will
confirm. As a cross check that the system’s behaviour stays the same
throughout the degenerate phase, we have performed a short series of
measurements for all observables (except the curvature profile) at the much
higher value of $k_{0}\\!=\\!15.0$, which found no differences compared with
$k_{0}\\!=\\!8.0$. Little is known about the slice behaviour in the de Sitter
phase; a preliminary investigation of the Hausdorff dimension for small slice
volumes $V_{2}(t)\sim 1.000$ was made in [26], resulting in the estimate
$d_{H}\\!=\\!3.4\pm 0.4$. Furthermore, a measure of homogeneity for the
spatial slices was formulated and implemented in [54], but with inconclusive
results.
A final technical issue to be discussed before turning to a description of the
measurement results is that of volume-fixing. As usual, we perform all
measurements at fixed spacetime volume; more precisely, since the Monte Carlo
moves are not volume-preserving, the simulations are run in the vicinity of a
target volume. The latter can be stated in terms of the total volume $N_{3}$
or in terms of $N_{31}$, which is what we will do below. Both prescriptions
are essentially equivalent, since in the case of periodic boundary conditions
in time we have the identity $N_{3}\\!=\\!2N_{31}+N_{22}$, and since for fixed
$k_{0}$ the ratio between $N_{22}$ and $N_{31}$ is approximately constant, and
can be read off the graph in Fig. 3. Note also that $N_{31}$ is equal to the
total number of _spatial_ triangles in the triangulation, i.e. the sum over
all $t$ of $V_{2}(t)$. The approximate volume-fixing is implemented by adding
a term
$S_{\textrm{fix}}[T]=\epsilon\,\big{(}N_{31}(T)-\tilde{N}_{31}\big{)}^{2}$ (5)
to the bare action, where $\tilde{N}_{31}$ denotes the desired target volume
and the value of the small, positive parameter $\epsilon$ determines the
typical size of the fluctuations of $N_{31}$ around $\tilde{N}_{31}$. In the
simulations performed for this work, we generally set $\epsilon$ to values on
the order of $10^{-5}$.
In addition, since we are interested in the intrinsic properties of the
spatial slices and in extracting their continuum behaviour from finite-size
scaling, we must collect measurement data at different, fixed _slice_ volumes.
Instead of adding further volume-fixing terms for the individual slices to the
action, which would run the risk of introducing an unwanted bias, we let the
individual slices fluctuate freely, subject only to the total volume
constraint (5), but take data only when a slice hits a precise desired value
$\tilde{V}_{2}$. More concretely, in between two measurements we first perform
a fixed number of attempted moves101010These attempted moves will be accepted
or rejected according to the detailed balance condition mentioned earlier.,
collectively referred to as a _sweep_. Different observables can have
different autocorrelation times (measured in Monte Carlo steps) and therefore
require different sweep sizes. A typical sweep size is taken to be on the
order of 1.000 times the target volume $\tilde{N}_{31}$. After completion of a
sweep, we continue performing local updates until one of the slices hits the
target two-volume $\tilde{V}_{2}$. We then perform a measurement of the
observable under consideration on this slice, and subsequently start a new
sweep.
The choice of the target three-volume $\tilde{N}_{31}$ that maximizes the
probability of encountering slices with target two-volume $\tilde{V}_{2}$
depends on the phase. In the de Sitter phase and for the small time extension
$t_{\rm tot}$ we have used, the total volume spreads roughly evenly over the
available slices and an appropriate choice is $\tilde{N}_{31}\\!=\\!t_{\rm
tot}\cdot\tilde{V}_{2}$. In the degenerate phase, the volume tends to
concentrate on one of the slices, and a good choice is
$\tilde{N}_{31}\\!=\\!\tilde{N}_{2}+m$, where for $t_{\rm tot}\\!=\\!3$ and
$\tilde{V}_{2}\\!>\\!1.000$ we found that $m\\!=\\!100$ is a convenient
choice.
To maximize the volume of the spatial slices, we work with $t_{\rm
tot}\\!=\\!3$, the minimal number allowed by our simulation code. Both this
choice and our choice of periodic boundary in time can in principle have an
influence on the behaviour of observables, even if our measurements are
confined to individual slices. Investigations of the transfer matrix in four-
dimensional CDT quantum gravity have indicated that such a set-up can be
appropriate, at least for selected observables [55]. As an extra check, we
have performed a few measurements on systems with larger $t_{\rm
tot}\\!\lesssim\\!32$, and found the same behaviour for the slice geometries.
This is reassuring, but not a substitute for a more systematic investigation
of the influence of these global choices, which goes beyond the scope of our
present work. This proviso should be kept in mind when interpreting the
outcomes of our research, which will be presented next.
## 4 Geometric observables on spatial hypersurfaces
The main objective of our work is a detailed measurement of the geometric
properties of the two-dimensional spatial hypersurfaces of constant integer
proper time $t$ in three-dimensional CDT quantum gravity, both in the de
Sitter and the degenerate phase. We will compare our findings with known
results for nonperturbative models of two-dimensional quantum gravity. There
are two pure-gravity systems in $D\\!=\\!2$ available for reference, Euclidean
DT and Lorentzian CDT quantum gravity. However, especially in the de Sitter
phase there are no stringent reasons why the slice geometries should match
these known systems, since they are part of a larger, three-dimensional
geometry. For example, the ambient geometry may induce extrinsic curvature
terms on the spatial slices, which are not present in intrinsically two-
dimensional situations. The following subsections will deal in turn with the
four quantities we have studied on the spatial slices: the vertex order, the
entropy exponent, the Hausdorff dimension and the curvature profile.
### 4.1 Vertex order
We first examined the distribution of the vertex order, which counts the
number $q(v)$ of spacelike edges meeting at a given vertex $v$. For the
simplicial manifolds we use in the simulations, the possible vertex orders are
$q(v)\\!=\\!3,4,5,\dots$. As already mentioned earlier, the distribution of
the $q(v)$ is not strictly speaking an observable. It is a non-universal
property of the discrete lattice, which depends on the details of the lattice
discretization and does not have an obvious continuum counterpart. For
example, using quadrangulations instead of triangulations leads to the same
continuum theory of two-dimensional Euclidean DT quantum gravity [56], but the
two models have different distributions of vertex orders. We have studied this
quantity nevertheless, since it is known analytically for both the DT and CDT
ensembles and gives us a first idea of whether and how our system changes as a
function of the bare coupling $k_{0}$.
The normalized probability distribution $P(q)$ for the vertex order in two-
dimensional DT quantum gravity in the thermodynamic limit has been determined
analytically as [57]
$P_{\textrm{DT}}(q)=16\left(\frac{3}{16}\right)^{q}\frac{(q-2)(2q-2)!}{q!(q-1)!},\quad\quad
q\geq 2,$ (6)
with a large-$q$ behaviour $\sim\\!\left(\frac{3}{4}\right)^{q}$. The
corresponding probability distribution for two-dimensional CDT quantum gravity
is given by [58]
$P_{\textrm{CDT}}(q)=\frac{q-3}{2^{q-2}},\quad\quad q\geq 4,$ (7)
with a fall-off behaviour $\sim\\!\left(\frac{1}{2}\right)^{q}$ for large $q$.
Both distributions are shown in Fig. 5.
Figure 5: Probability distribution of the vertex order $q$ for the ensembles
of two-dimensional Euclidean DT and Lorentzian CDT quantum gravity, in the
limit of infinitely many triangles.
We took measurements at the three chosen phase space points $k_{0}=0.0,5.0$
and $8.0$, with a target volume $\tilde{V}_{2}\\!=\\!4.000$ for the spatial
slices, corresponding to a target three-volume $\tilde{N}_{31}\\!=\\!12.000$
in the de Sitter phase, and $\tilde{N}_{31}\\!=\\!4.400$ in the degenerate
phase. The sweep size was set to $100\cdot\tilde{N}_{31}$ in each case. For
each value of $k_{0}$, we collected measurements of $P(q)$ for 100k different
slices, by recording for each slice the full set of vertex orders $q(v)$ for
all vertices and normalizing the resulting histogram. We then approximated the
eigenvalue $\langle P(q)\rangle$ by taking the ensemble average over this data
set according to the prescription (3). The results of the measurements for
$q\\!\in\\![3,20]$ are shown in Fig. 6, and are clearly distinct for the three
$k_{0}$-values. While the distribution for $k_{0}\\!=\\!8.0$ in the degenerate
phase is a very good match for the analytical result, the distributions for
$k_{0}\\!=\\!0.0$ and $5.0$ in the de Sitter phase are not good matches and
are also different from each other. This is confirmed when plotting the
measurements on a logarithmic scale, taking into account a much larger range
of $q\\!\leq\\!180$, as depicted in Fig. 7, which also includes the known
distributions for Euclidean DT and CDT as thin straight lines.
Figure 6: Measured probability distribution of the vertex order $q$ on spatial
slices of volume $V_{2}\\!=\\!4.000$ in three-dimensional CDT, for
$k_{0}\\!=\\!0.0$, $5.0$ and $8.0$ and $q\\!\leq\\!20$. For comparison, the
line connecting the exact values for DT from Fig. 5 are included.
We see that within measurement accuracy, the distribution of vertex orders in
the degenerate phase of the model is indistinguishable from (6), a result that
is also compatible with the numerical simulations of pure Euclidean DT quantum
gravity performed in [57]. By contrast, the distributions obtained in the de
Sitter phase exhibit a very different behaviour, at least for large $q$. High-
order vertices are relatively speaking more probable, and the data cannot be
fitted to a single exponential over the $q$-range we have explored. Moreover,
unlike what happens in the degenerate phase, the distributions within the de
Sitter phase depend on the value of $k_{0}$.
Figure 7: Measured probability distribution of the vertex order $q$
(logarithmic scale) on spatial slices of volume $V_{2}\\!=\\!4.000$ in three-
dimensional CDT, for $k_{0}\\!=\\!0.0$, $5.0$ and $8.0$. The unbroken straight
line is the analytical result (6) for two-dimensional DT, and the dashed
straight line the analytical result (7) for two-dimensional CDT.
To obtain a more detailed picture of the dependence on $k_{0}$, we performed a
series of shorter simulation runs at additional points in the de Sitter phase.
When entering the de Sitter phase from the degenerate phase by crossing the
phase transition at $k_{0}^{c}$, the vertex order distribution jumps
discontinuously to a shape similar to that for $k_{0}\\!=\\!5.0$. As $k_{0}$
is decreased further inside this phase, the distribution changes shape in a
continuous way; the distributions we measured for points in the interval
$0.0\\!<\\!k_{0}\\!<\\!5.0$ interpolate in a straightforward manner between
the ones for $k_{0}=0.0$ and $k_{0}=5.0$ shown in Figs. 6 and 7.
To summarize, the measurements of the vertex order distribution in the
degenerate phase produce an excellent match with the known one for DT quantum
gravity. By contrast, the distributions found in the de Sitter phase do not
match those of the standard DT or CDT ensembles in $D\\!=\\!2$. As mentioned
earlier, this does not necessarily mean that the slice geometries in the de
Sitter phase do not lie in either of the associated universality classes, but
it is a first indication that they may not. By looking at genuine observables
next, we will be able to make more definite statements about the universal
geometric properties of the slice geometries.
### 4.2 Entropy exponent
An important parameter characterizing two-dimensional systems of random
geometry is the entropy exponent $\gamma$, which contains information about
the behaviour of the partition function at fixed two-volume $N_{2}$ (number of
triangles), in the limit as $N_{2}$ becomes large.111111We use $N_{2}$ to
denote two-volumes in two-dimensional models of quantum gravity and $V_{2}$ to
denote the two-volume of a spatial slice in three-dimensional quantum gravity.
Recall that the path integral of DT quantum gravity in $D\\!=\\!2$ with bare
cosmological constant $\lambda$ can be written as the infinite sum
$Z(\lambda)=\sum_{N_{2}}Z(N_{2})\,{\rm e}^{-\lambda N_{2}},$ (8)
which is the (discrete) Laplace transform of the partition function $Z(N_{2})$
for fixed volume. For large $N_{2}$, $Z(N_{2})$ behaves like
$Z(N_{2})\sim{\rm
e}^{\lambda^{c}N_{2}}N_{2}^{\gamma-3}\left(1+{\mathcal{O}}(1/N_{2})\right),$
(9)
whose leading exponential growth is governed by a (non-universal) critical
cosmological constant $\lambda^{c}\\!>\\!0$ and whose subleading power-law
behaviour defines the universal entropy exponent $\gamma$, which for DT
quantum gravity is given by $\gamma\\!=\\!-1/2$. The asymptotic functional
form (9) continues to hold when conformal matter of central charge $c\\!<\\!1$
is added to the Euclidean quantum gravity model, giving rise to the entropy
exponents [59]
$\gamma=\frac{1}{12}\left(c-1-\sqrt{(25-c)(1-c)}\right),$ (10)
with $c\\!=\\!0$ corresponding to the pure-gravity case. Two-dimensional CDT
quantum gravity, which is not described by formula (10), is characterized by
$\gamma\\!=\\!1/2$ [4]. Computer simulations have demonstrated that adding
matter with $c\\!=\\!4$ to the CDT system induces a phase transition in the
geometry [60], but the corresponding entropy exponent is not known.
According to [61], the distribution of so-called baby universes in two-
dimensional Euclidean quantum gravity – parts of a geometry that are connected
to the larger bulk geometry via a thin neck – depends in a simple way on the
entropy exponent $\gamma$. This insight was used subsequently to formulate a
prescription of how to extract $\gamma$ by measuring the distribution of
minimal-neck baby universes (“minbus”) in the DT ensemble with the help of
Monte Carlo simulations [62]. A minbu is a simply connected subset of disk
topology of a two-dimensional triangulation $T$, which is connected to the
rest of $T$ along a loop consisting of three edges, which is the minimal
circumference of a neck allowed by the simplicial manifold conditions.
As shown in [61, 62], it follows from relation (9) that the average number
$\bar{n}_{N_{2}}(B)$ of minbus of volume $B$ (counting the number of triangles
in the minbu) in a spherical triangulation of volume $N_{2}$ for sufficiently
large $B$, $N_{2}$ behaves like
$\bar{n}_{N_{2}}(B)\sim(N_{2}-B)^{\gamma-2}B^{\gamma-2}.$ (11)
By measuring the distribution of minbus across a range of volumes $B$ for
fixed $N_{2}$ in a DT ensemble and fitting the results to the function (11),
the expected results $\gamma\\!=\\!-1/2$ for pure gravity and
$\gamma\\!=\\!-1/3$ for gravity coupled to Ising spins ($c\\!=\\!1/2$) were
reproduced within measuring accuracy [62].
We have carried out a similar analysis on the spatial slices of triangulations
generated by Monte Carlo simulations of three-dimensional CDT quantum gravity.
There is no obvious reason why (9) should hold for some “effective” fixed-
volume partition function for the two-volume $N_{2}\\!=\\!V_{2}$ of a single
spatial slice in this three-dimensional system. However, if the number of
(2,2)-simplices drops essentially to zero _and_ neighbouring slices decouple,
as is the case in the degenerate phase, the three-dimensional partition
function at fixed volume will depend only on $N_{31}$ (equal to the total two-
volume), which makes it plausible that (9) holds on individual spatial slices,
with $N_{2}\\!=\\!V_{2}$. In the de Sitter phase, if the spatial geometries
can be described in terms of two-dimensional DT quantum gravity, the minbu
method will presumably also lead to $\gamma\\!=\\!-1/2$.
We measured the distribution $\bar{n}_{V_{2}}(B)$ of minbu sizes $B$ for
target slice volumes $\tilde{V}_{2}\\!=\\!1.000$ and $2.000$ at the three
phase space points $k_{0}=0.0,5.0$ and $8.0$. The sweep size was set to
$10^{4}\\!\cdot\\!\tilde{V}_{2}$ for measurements in the degenerate phase, and
$10^{5}\\!\cdot\\!\tilde{V}_{2}$ in the de Sitter phase. We used longer sweeps
in the de Sitter phase because the observed autocorrelations were much larger,
especially at $k_{0}\\!=\\!0.0$, where the algorithm is much less efficient.
We collected on the order of $5\\!\cdot\\!10^{4}$ minbu size histograms for
$\tilde{V}_{2}\\!=\\!1.000$ and $1.5\\!\cdot\\!10^{5}$ histograms for
$\tilde{V}_{2}\\!=\\!2.000$ in the de Sitter phase, and $4\cdot 10^{5}$
histograms for $\tilde{V}_{2}\\!=\\!1.000$ and $2\cdot 10^{5}$ histograms for
$\tilde{V}_{2}\\!=\\!2.000$ in the degenerate phase. The resulting expectation
values $\langle\bar{n}_{V_{2}}\rangle$ of the minbu size distribution as a
function of the normalized ratio $B/V_{2}\in[0,1/2]$ are shown in Fig. 8,
together with best fits of the form (11) for specific values of $\gamma$. The
best fits were determined following the procedure used in [62], and involved a
subleading correction term to the power law $B^{\gamma-2}$, as is described in
more detail in Appendix A below.
Figure 8: Expectation values of the distribution $\bar{n}_{V_{2}}$ of minbu
sizes for spatial slices of volume $V_{2}\\!=\\!1.000$ and $2.000$ in three-
dimensional CDT configurations, in the degenerate phase ($k_{0}\\!=\\!8.0$,
top) and the de Sitter phase ($k_{0}\\!=\\!0.0$, bottom left, and
$k_{0}\\!=\\!5.0$, bottom right), on a log-log scale. The continuous lines are
best fits of the form (11) for specific values of $\gamma$. Error bars are
smaller than the dot size.
In the degenerate phase (Fig. 8, top), the choice $\gamma\\!=\\!-1/2$ fits the
data extremely well throughout the entire range of $B/V_{2}$, with the
exception of the smallest minbu sizes. Our results coincide within error bars
with those reported in Table 1 of [62]. This confirms that the spatial slices
in the degenerate phase exhibit behaviour consistent with DT quantum gravity
in two dimensions.
In the de Sitter phase, there is no $\gamma$-value that leads to a good fit
over the full range of $B/V_{2}$, even when we disregard the region of small
minbu size $B$, where (11) is known to be inaccurate. The plots for
$k\\!=\\!0.0$ and $k_{0}\\!=\\!5.0$ (Fig. 8, bottom) illustrate the optimum of
what can be achieved, namely, a fit that works reasonably well for an
intermediate range of $B/V_{2}$, in this case, a fit corresponding to
$\gamma\\!=\\!-1$. However, this clearly does not fit the data for large
minbus near $B/V_{2}\\!=\\!1/2$, especially not for the smaller value of
$k_{0}$, and the discrepancy seems to get worse with increasing volume
$V_{2}$.
We conclude that the minbu distribution in the de Sitter phase does not follow
the functional form of the right-hand side of relation (11), at least not for
the slice volumes we have considered (and which seem to be sufficient in the
degenerate phase). A possible explanation is that the “effective” partition
function $Z(V_{2})$, which is obtained from the three-dimensional CDT
partition function $Z(N_{3})$ for fixed three-volume by integrating out all
degrees of freedom except for a single slice volume $V_{2}$, does not have the
asymptotic form (9). This would imply that it is not in the universality class
of a DT model with central charge $c\\!<\\!1$ or of CDT quantum gravity. A
more subtle scenario would be that $Z(V_{2})$ does behave according to (9)
(and perhaps does correspond to a known gravity model in $D\\!=\\!2$), but
that the derivation of the minbu distribution (11) is invalidated by the
presence of correlations between the bulk and a baby universe that exist
because of the embedding of the spatial slice in a three-dimensional
simplicial geometry. Such correlations are not present in the degenerate phase
because of the absence of (2,2)-simplices. In the following two subsections,
we will look at observables which also characterize the intrinsic geometry of
the spatial slices, but whose determination is less subtle than that of the
entropy exponent.
### 4.3 Hausdorff dimension
The Hausdorff dimension is a notion of fractal dimension that can be used to
characterize a quantum geometry in an invariant manner. Broadly speaking, it
is extracted by comparing volumes with their characteristic linear size,
measured in terms of a geodesic distance. There have been many studies of the
Hausdorff dimension in the context of DT and CDT quantum gravity, including
extensive investigations in two-dimensional Euclidean DT models with and
without matter (see, for example, [63, 64, 20, 65] and references therein).
Following [63], we will investigate a local and a global (“cosmological”)
variant of this observable on the spatial slices. From analytical
considerations, it is known that in two-dimensional Euclidean DT quantum
gravity without matter both types of Hausdorff dimension are equal to four
(i.e. different from the topological dimension of the triangular building
blocks), while for two-dimensional CDT quantum gravity they are both equal to
two [4, 66].
When measuring the Hausdorff dimension numerically, one can use either the
link distance or the dual link distance as a discrete implementation of the
geodesic distance. In known systems of pure gravity in $D\\!=\\!2$, they lead
to equivalent notions of geodesic distance in the continuum limit, but for
finite lattice sizes, one particular choice may be more convenient. This is
true for our investigation below, where we will use the dual link distance,
which is defined between dual vertices (equivalently, centres of triangles)
and given by the length of the shortest path along dual links between the
vertices.
When comparing our results with previous measurements of the Hausdorff
dimension in the context of two-dimensional Euclidean DT [67, 64], one must
take into account that the latter employed a larger ensemble of geometries.
This generalized ensemble allows for local gluings of the equilateral
triangles that violate the strict simplicial manifold conditions121212two
triangles cannot share more than one edge, any two vertices cannot be
connected by more than one edge. When characterizing the triangulations in
terms of their dual, trivalent graphs, the generalization consists in allowing
for tadpole and self-energy insertions. It has been demonstrated to reduce
finite-size effects [68], and is justified by the fact that the model on the
enlarged ensemble can be shown to lie in the same universality class (see e.g.
[69] and references therein). Since no analogous result is available in three-
dimensional CDT quantum gravity, it is prudent to use only simplicial
manifolds, which implies that the spatial slices are simplicial manifolds too.
This may affect the quality of our results, compared with the earlier, purely
two-dimensional investigations. Note also that nonlocal minbu surgery moves
were used in [64] to complement the standard local Monte Carlo moves and
reduce autocorrelation times, something we cannot easily implement on two-
dimensional embedded triangulations.
#### 4.3.1 Local Hausdorff dimension
A key quantity in determining the local Hausdorff dimension $d_{h}$ of a two-
dimensional triangulation $T$ is the shell volume $S(r)$, which in our
implementation counts the number of dual vertices (equivalently, triangles) at
dual link distance $r$ from a given dual vertex. The corresponding observable
is the quantity $\bar{S}(r)$, obtained by averaging $S(r)$ over all dual
vertices of $T$. The reason for using the dual link distance is that shells
with respect to the link distance quickly cover a large fraction of the
geometry as $r$ grows. This implies that the average shell volumes
$\bar{S}(r)$ cover only a small range of radii $r$ before dropping to zero,
yielding too few data points to make reliable estimates of either Hausdorff
dimension. The local Hausdorff dimension is extracted from the expectation
value $\bar{S}(r)$ in the ensemble at fixed two-volume $N_{2}$ according to
$\langle\bar{S}(r)\rangle_{N_{2}}\sim r^{d_{h}-1},$ (12)
for small $r$. In other words, $d_{h}$ captures the initial, volume-
independent power-law growth of small geodesic spherical shells, where $r$
must be sufficiently large to avoid dominance by discretization artefacts and
sufficiently small to avoid significant corrections to the simple power law
behaviour (12), if such a behaviour is indeed present.
We have measured the expectation values of average shell volumes as a function
of the dual link distance $r$, at slice volumes $V_{2}\\!=\\!16k$ and $32k$
and at the three chosen phase space points $k_{0}$, which is all
straightforward. The local Hausdorff dimension $d_{h}$ was extracted by
fitting the measured data to the functional form
$\langle\bar{S}(r)\rangle_{V_{2}}=c\cdot(r+a)^{d_{h}-1},$ (13)
where – following [64] – we have introduced an offset $a$ to account for
short-distance discretization artefacts, and $c$ is a multiplicative
parameter. The dependence of the Hausdorff dimension on the chosen fitting
range $r\\!\in\\![r_{\textrm{min}},r_{\textrm{max}}]$ will be analyzed in more
detail below.
Figure 9: Expectation values $\langle\bar{S}(r)\rangle_{V_{2}}$ of average
shell volumes in the degenerate phase $(k_{0}\\!=\\!8.0)$ for slice volumes
$V_{2}\\!=\\!16k,32k$ (left), and for all three phase space points
$k_{0}\\!=\\!0.0$, $5.0$ and $8.0$ for slice volume $V_{2}\\!=\\!32k$ (right).
Error bars are smaller than dot size.
The measured expectation values of the average shell volume are shown in Fig.
9 for $0\\!\leq\\!r\\!\leq\\!40$. The plot on the left, describing the
behaviour of the system in the degenerate phase, illustrates the difference
between the data for slice volumes $16k$ and $32k$. The data points for the
smaller volume start deviating from the common behaviour around
$r\\!\approx\\!20$, indicating that a power-law fit becomes inadequate beyond
this point. The plot on the right illustrates the dependence of the initial
slope on the value of $k_{0}$. Note that although the curve for
$k_{0}\\!=\\!0.0$ lies in between the two other curves, the corresponding
Hausdorff dimension obtained from fitting to the functional form (13) comes
out lower than that for $k_{0}\\!=\\!5.0$, see below.
Since the criteria for fixing the fitting range
$r\\!\in\\![r_{\textrm{min}},r_{\textrm{max}}]$ for eq. (13) are only
approximate, it is important to understand which choice is most appropriate
and how stable the results for $d_{h}$ are when the range is varied. Some
earlier work used the range $r\\!\in\\![5,15]$ in terms of the dual link
distance, and for a system of volume $N_{2}\\!=\\!64k$ [70].
To investigate the influence of the fitting range in a systematic way, we have
performed fits for a set of ranges
$r\\!\in\\![r_{\textrm{min}},r_{\textrm{min}}\\!+\\!w]$ of varying width $w$,
and with $r_{\textrm{min}}\\!\in\\![5,14]$. The resulting best fit values for
the local Hausdorff dimension $d_{h}$ as a function of $r_{\textrm{min}}$ are
shown in Fig. 10, for two different widths $w\\!=\\!10$ and $12$.
Figure 10: Best fit values for $d_{h}$ from fitting (13) to the measured
expectation values $\langle\bar{S}(r)\rangle_{V_{2}}$ in the range
$r\\!\in\\![r_{\textrm{min}},r_{\textrm{min}}\\!+\\!w]$ for spatial slices in
the degenerate phase ($k_{0}\\!=\\!8.0$, slice volumes $V_{2}\\!=\\!16k,32k$)
and in the de Sitter phase ($k_{0}\\!=\\!0.0,5.0$, slice volume
$V_{2}\\!=\\!32k$). The error bars correspond to the 95% confidence intervals
of a $\chi^{2}$-test.
Starting our analysis in the degenerate phase, we observe a good stability of
the value of $d_{h}$ when $r_{\textrm{min}}$ is increased away from its
lowest, “canonical” cutoff value of 5, which indicates that the region
$r_{\textrm{min}}\\!\gtrsim\\!5$ is not affected by short-distance artefacts
and that data in the corresponding interval
$r\\!\in\\![r_{\textrm{min}},r_{\textrm{min}}\\!+\\!w]$ are well approximated
by a pure power law. Shifting the fitting range to start beyond
$r_{\textrm{min}}\\!=\\!7$ changes the extracted best $d_{h}$, mildly for the
larger volume $V_{2}\\!=\\!32k$ and more strongly for $V_{2}\\!=\\!16k$.
Taking into account Fig. 9, left, this indicates that one is leaving the
region where the functional form (13) is an appropriate fit. Lastly, setting
$w\\!=\\!12$ instead of 10 reduces the error bars without appreciably changing
$d_{h}$, and is therefore preferable. To conclude, the optimal choice of range
among the possibilities we have investigated at $k_{0}\\!=\\!8.0$ appears to
be $r\\!\in\\![5,17]$. The associated local Hausdorff dimension for
$V_{2}\\!=\\!32k$ is given by
$d_{h}=3.31(4),\quad\quad\textrm{degenerate phase }(k_{0}=8.0),$ (14)
obtained from fitting to (13), with fit parameters $a\\!=\\!4.0(2)$ and
$c\\!=\\!0.08(1)$. To illustrate the excellent quality of the fit, Fig. 11
shows the measured data together with the best-fit curve and the fits at the
edges of the 95% confidence interval, which basically fall on top of each
other (see Appendix B for details on how to compute such confidence
intervals). The fits for the data at $k_{0}\\!=\\!5.0$ and $k_{0}\\!=\\!0.0$
are of similar quality. We postpone a discussion of the compatibility of this
result with the analytical value $d_{h}\\!=\\!4$ for DT quantum gravity to
later, after having investigated the global Hausdorff dimension.
Figure 11: Expectation values $\langle\bar{S}(r)\rangle_{V_{2}}$ of average
shell volumes in the degenerate phase ($k_{0}\\!=\\!8.0$) and slice volume
$V_{2}\\!=\\!32k$. The curves are plots of the fit function (13) for three
different sets of fit parameters: the optimal fit which minimizes $\chi^{2}$,
and the two fits at the boundaries of the 95% confidence interval. The fit is
performed to the data points in the range $5\leq r\leq 17$, as indicated by
the unshaded region.
Turning next to a discussion of the de Sitter phase, Fig. 10 shows the best
fit values for $d_{h}$ we extracted from the shell volume data. They are
consistently smaller than those in the degenerate phase, and within the de
Sitter phase decrease further with decreasing $k_{0}$. We observe only a
shorter range of values $r_{\textrm{min}}\\!\gtrsim\\!5$ where the Hausdorff
dimension is reasonably stable. Examining the corresponding curves in Fig. 9,
right, there does not seem to be a very extended initial-growth regime, before
the curves straighten out to become approximately linear. It is possible that
finite-size effects affect the data points at the upper end of the chosen
ranges $[r_{\textrm{min}},r_{\textrm{max}}]$ and cause the observed deviations
from a simple power-law behaviour. Alternatively, this behaviour may be a
bona-fide feature of the embedded slices. The error bars for larger
$r_{\textrm{min}}$ are significantly larger than in the degenerate phase,
especially for $k_{0}\\!=\\!0.0$. At least in part, this appears to be due to
a statistical uncertainty of the measured shell volumes for increasing $r$. As
before, the error bars are smaller for $w=12$ than for $w=10$. For
$k_{0}\\!=\\!5.0$, the local Hausdorff dimension seems to decrease slightly
with growing $r_{\textrm{min}}$, but in view of the width of the 95%
confidence interval this may not have any significance.
From a best fit in the interval $r\\!\in\\![5,17]$, we find for the local
Hausdorff dimension
$\displaystyle d_{h}$ $\displaystyle=2.91(5),\quad\quad\textrm{de Sitter phase
}(k_{0}=0.0),$ (15) $\displaystyle d_{h}$
$\displaystyle=3.10(4),\quad\quad\textrm{de Sitter phase }(k_{0}=5.0),$ (16)
for the fit parameters $a\\!=\\!2.5(3)$, $c\\!=\\!0.26(5)$ and
$a\\!=\\!3.9(3)$, $c\\!=\\!0.11(2)$ respectively. For the time being, we take
note of these results and refer to Sec. 4.3.3 below for a summary and
attempted interpretation of all Hausdorff dimension measurements.
#### 4.3.2 Global Hausdorff dimension
The global Hausdorff dimension of a two-dimensional triangulation describes
its behaviour as a whole and can again be characterized by the average volume
$\bar{S}_{N_{2}}(r)$ of spherical shells of radius $r$, where we have added an
explicit subscript $N_{2}$ to emphasis that the total volume of the
triangulation will now play an important role. Given an ensemble of geometries
of volume $N_{2}$, a global Hausdorff dimension $d_{H}$ can be extracted if in
the limit of large $N_{2}$ the eigenvalue of the distribution of shell volumes
over the entire $r$-range can be described by the functional form
$\langle\bar{S}_{N_{2}}(r)\rangle=N_{2}^{1-1/d_{H}}\mathcal{F}(x),\quad\quad
x=\frac{r}{N_{2}^{1/d_{H}}},$ (17)
where $\mathcal{F}$ is a universal function that depends on the rescaled
geodesic distance $x$. This is known to be the case for DT quantum gravity in
two dimensions, where $\mathcal{F}$ has been computed explicitly [64], but
there is no guarantee that the scaling law (17) holds for general systems of
geometries in $D\\!=\\!2$. Even when a global Hausdorff dimension $d_{H}$ can
be assigned in this manner, it need not be equal to the local Hausdorff
dimension $d_{h}$ [64, 71].
We will attempt to extract a global Hausdorff dimension for the spatial slices
in three-dimensional CDT by performing a finite-size scaling analysis where we
collect data for $\langle\bar{S}_{V_{2}}(r)\rangle$ for the full range of
radii $r$ and several slice sizes $V_{2}$, and then try to rescale the
resulting distributions according to (17). If we can find a single value
$d_{H}$ such that the rescaled distributions fall on top of each other for all
volumes $V_{2}$, we define this $d_{H}$ to be the global Hausdorff dimension
of the system.
Following [64], we will work with the normalized shell volume distributions
$n_{V_{2}}(r)\\!:=\\!\langle\bar{S}_{V_{2}}(r)\rangle/V_{2}$, for which the
scaling law (17) assumes the form
$n_{V_{2}}(r)=V_{2}^{-1/d_{H}}\mathcal{F}(x),\quad\quad
x=\frac{r}{V_{2}^{1/d_{H}}}.$ (18)
Note that the measured distributions $n_{V_{2}}(r)$ are functions of a
discrete variable $r\in\mathbb{N}_{0}$. To perform a smooth rescaling, we
first construct continuous functions that interpolate between these discrete
values, which by slight abuse of notation we continue to call $n_{V_{2}}(r)$.
Following the methodology of [65], we then rescale, for each system volume
$V_{2}$ separately, the corresponding distribution $n_{V_{2}}(r)$ such that it
maximally overlaps with the normalized distribution $n_{V_{\textrm{max}}}(r)$
for the largest slice volume $V_{\textrm{max}}$ in the simulation, which we
are using as a reference distribution. We denote these rescaled distance
profiles by $\tilde{n}_{V_{2}}(\tilde{r})$, where $\tilde{r}$ is a rescaled
length variable. They take the form
$\tilde{n}_{V_{2}}(\tilde{r}_{V_{2}})=\left(\frac{V_{2}}{V_{\textrm{max}}}\right)^{1/d}n_{V_{2}}(\tilde{r}),\quad\quad\tilde{r}_{V_{2}}=\left(\frac{V_{2}}{V_{\textrm{max}}}\right)^{1/d}(r+a)-a,$
(19)
where the two fit parameters are a rescaling dimension $d$ and a
phenomenological shift $a$ [64, 65] that corrects for discretization effects
at small $r$, similar to the prescription (13) we used for the local Hausdorff
dimension.
Figure 12: Distributions $\tilde{n}_{V_{2}}(\tilde{r}_{V_{2}})$ of shell
volumes rescaled with the averaged parameters $(\bar{d},\bar{a})$ in the
degenerate phase ($k_{0}\\!=\\!8.0$, top) and the de Sitter phase
($k_{0}\\!=\\!5.0$, bottom left, and $k_{0}\\!=\\!0.0$, bottom right).
Note that $\tilde{r}_{V_{\textrm{max}}}$ is equal to the original discrete
length parameter $r$, so we can use them interchangeably, and that
$\tilde{n}_{V_{\textrm{max}}}\\!=\\!n_{V_{\textrm{max}}}$. For each system
size $V_{2}\\!<\\!V_{\textrm{max}}$ we determine the corresponding fit
parameters $a$, $d$ from the condition that the sum of the squared differences
$\left(\tilde{n}_{V_{2}}(\tilde{r}_{V_{2}})-\tilde{n}_{V_{\textrm{max}}}(r)\right)^{2}$
should be minimized, where $\tilde{r}_{V_{2}}$ depends implicitly on the
discrete parameter $r$. We perform the fit in the range of integers $r$ where
$\tilde{n}_{V_{\textrm{max}}}(r)>\frac{1}{5}\,\textrm{max}_{r}\,\tilde{n}_{V_{\textrm{max}}}(r)$,
which means we disregard contributions from the tails of the distribution,
where discretization effects are more likely to be present. If over a large
range of $V_{2}$ the rescaled distributions overlap to a common curve or are
reasonably close to doing so, we take the mean $\bar{d}$ of all the rescaling
dimensions $d$ and define this to be the global Hausdorff dimension,
$d_{H}\\!:=\\!\bar{d}$. We also average the shift parameters to obtain one
optimal shift $\bar{a}$ for the system. If we do not find sufficient overlap,
the method fails and we cannot assign a global Hausdorff dimension.
We measured the expectation value of the shell volume distribution for eight
different slice volumes $V_{2}\\!\in\\![1.000,32.000]$ in the degenerate
phase, and for seven slice volumes $V_{2}\\!\in\\![1.000,8.000]$ in the de
Sitter phase. Again, the autocorrelation times are much larger in the de
Sitter phase, and the resulting uncertainties especially large near the peaks
of the distributions. Since the location and height of these peaks are
important features for finding the appropriate rescaling dimension required
for a collapse, large uncertainties in this region imply large error bars for
the best fit parameters. This led to our choice for the smaller volume range
in the de Sitter phase.
Figure 13: Values for $d$ extracted by comparing measured distributions of
shell volumes at a given volume $V_{2}$ with those at the top volume, for
$k_{0}\\!=\\!0.0$, $5.0$ and $8.0$, as explained in the text.
For each of the three phase space points, Fig. 12 shows a collection of curves
for a range of volumes $V_{2}$, which have been rescaled according to (19),
using the averaged pair $(\bar{d},\bar{a})$ for all of them. The topmost
graph, for $k_{0}\\!=\\!8.0$, makes a convincing case for the presence of
finite-size scaling in the degenerate phase. Using $\bar{d}\\!=\\!3.30$ and
$\bar{a}\\!=\\!2.64$ as the joint rescaling parameters leads to a curve
collapse of good, although not perfect quality for slice volumes up to $32k$.
This is not the case for the rescaled curves in the de Sitter phase (Fig. 12,
bottom), which were obtained for the averaged fit parameters
$\bar{d}\\!=\\!2.68$ and $\bar{a}\\!=\\!-0.46$ at $k_{0}\\!=\\!5.0$ and
$\bar{d}\\!=\\!3.02$ and $\bar{a}\\!=\\!2.08$ at $k_{0}\\!=\\!0.0$. Although
the slice volumes span a significantly smaller range than in the degenerate
phase, our rescaled data do not support the presence of finite-size scaling in
this phase. In addition to the mismatch among the curves, we also note that
the distributions in the two phases look different as a function of the
rescaled radius $x\\!=\\!r/V_{2}^{1/\bar{d}}$. While in the degenerate phase
its range extends to around 6.7, and the peak is located near 2.4, the slice
geometries in the de Sitter phase have a smaller linear extension, with $x$
reaching on the order of 4 (5.7) and the peak located near 1.4 (2.0) for
$k_{0}\\!=\\!5.0$ (0.0). One could be tempted to disregard the lack of overlap
between the curves for different volumes and simply _define_ the global
Hausdorff dimension to be equal to the average $\bar{d}\\!=\\!2.68$ for
$k_{0}\\!=\\!5.0$ and $\bar{d}\\!=\\!3.02$ for $k_{0}\\!=\\!0.0$. However,
Fig. 13 shows that this would be misguided, since the values for $d$ exhibit a
strong dependence on the volume in the range we have investigated. Especially
the curve for $k_{0}\\!=\\!0.0$ shows a steep rise, with little indication of
asymptoting to a constant value, very different from the curve for the
degenerate phase, which we have included for comparison. Note also that the
$d$-values extracted from measurements in the de Sitter phase are not in any
obvious way related to the values we found for the local Hausdorff dimension,
namely, $d_{h}\\!=\\!3.10(4)$ for $k_{0}\\!=\\!5.0$ and $d_{h}\\!=\\!2.91(5)$
for $k_{0}\\!=\\!0.0$. This appears to be yet another indication that the
behaviour of the shell distributions is not governed by a single scale and
that the scaling hypothesis (17) is simply not valid in the de Sitter phase.
Based on these observations, we are unable to associate a global Hausdorff
dimension with the phase space points in the de Sitter phase. By contrast,
finite-size scaling is observed in the degenerate phase, and the associated
global Hausdorff dimension is given by
$d_{H}=3.30(2),\quad\quad\textrm{degenerate phase }(k_{0}=8.0),$ (20)
which within error margins coincides with the local Hausdorff dimension
$d_{h}\\!=\\!3.31(4)$ of eq. (14) we determined in the previous subsection.
#### 4.3.3 Discussion of results
Let us consider first the results we obtained in the degenerate phase. We
found clear evidence for the presence of finite-size scaling when analyzing
the global Hausdorff dimension, and mutually consistent values for both
Hausdorff dimensions, with $d_{h}\\!=\\!3.31(4)$ for the local and
$d_{H}\\!=\\!3.30(2)$ for the global variant. Despite the large discrepancy
with the analytical value $d_{h}\\!=\\!d_{H}\\!=\\!4$ for two-dimensional DT
quantum gravity, we nevertheless believe that the observed Hausdorff dimension
of the spatial slices is compatible with this model of quantum gravity. This
interpretation is supported by known difficulties in numerically extracting
the Hausdorff dimension in two-dimensional systems of random geometry (see
[72] and references therein), with a tendency of the Hausdorff dimension
measurements to underestimate its true value.131313More precisely, this is
true for DT models; numerical measurements of the Hausdorff dimension for two-
dimensional CDT found $d_{H}\\!\approx\\!2$ [58] and $d_{H}\\!=\\!2.2(2)$
[72]. In previous works, these difficulties have motivated the use of a more
general geometric ensemble, the introduction of additional minbu moves and of
a phenomenological shift parameter, as well as refined fitting techniques [64,
67, 65]. As mentioned earlier, several of these improvements are unfortunately
not directly applicable in our case, because our two-dimensional geometries
are parts of larger, three-dimensional triangulations.
Earlier numerical results for pure DT quantum gravity whose derivation most
closely resembles our treatment are the finite-size scalings obtained for
$N_{2}\\!\leq\\!32k$ from collapsing curves for the shell volume distributions
in terms of the dual link distance in [67]. Depending on the fitting method,
they yielded the values $d_{H}\\!=\\!3.150(31)$ and $d_{H}\\!=\\!3.411(89)$.
Unlike ours, this work used a generalized ensemble, but the final results for
the Hausdorff dimension are broadly in line with our findings. Another aspect
well illustrated by [67] is the increase of the measured Hausdorff dimension
with the system volume, something we also observed in the degenerate phase
(Fig. 10).
Our results in the de Sitter phase are less clear-cut, but point to a system
that is in a different universality class from DT quantum gravity. The values
we determined for the local Hausdorff dimensions, $d_{h}\\!=\\!3.10(4)$ for
$k_{0}\\!=\\!5.0$ and $d_{h}\\!=\\!2.91(5)$ for $k_{0}\\!=\\!0.0$, are even
further removed from 4, but this might conceivably still be due to some even
more serious underestimate than in the degenerate phase. More significant is
the absence of finite-size scaling and the ensuing impossibility to associate
a consistent global Hausdorff dimension to the system. The most likely
explanation is the absence of a single scale governing the dynamics, which
would imply a different universality class from that of the degenerate phase.
Discarding an interpretation of the de Sitter phase in terms of DT quantum
gravity leaves us without any obvious alternative candidate theory to explain
these results. There are a number of two-dimensional CDT models with matter
coupling that have a local and/or global Hausdorff dimension of or near 3,
including CDT with eight Ising spins [60], several massless scalar fields
[73], or restricted hard dimers [74]. Also two pure-gravity models of so-
called locally causal dynamical triangulations, generalizing the strict
slicing of two-dimensional CDT, were found to have Hausdorff dimensions near 3
[72]. There is nothing obvious that would link one of these models to the
Euclidean slices we are dealing with here, but we cannot exclude this
possibility either without a further investigation, which however would take
us beyond the scope of the present work.
### 4.4 Curvature profile
The last quantity we will measure on the spatial hypersurfaces is a curvature
observable. It is based on the quantum Ricci curvature, a generalized notion
of Ricci curvature introduced in [30]. The quantum Ricci curvature depends on
a neighbourhood of linear size $\delta$ of a point $x$ and has the
interpretation of a coarse-grained Ricci curvature associated with the scale
$\delta$. It is defined on a range of metric spaces of Riemannian signature,
including nonsmooth ones, and its introduction was motivated by the search for
a notion of (renormalized) curvature suitable for nonperturbative quantum
gravity. It can also be defined on classical, smooth Riemannian spaces, where
in the limit $\delta\\!\rightarrow\\!0$ it reproduces the standard notion of
Ricci curvature [30, 75]. The simplest way to turn this (quasi-)local notion
of curvature into a quantum observable depending on the scale $\delta$, dubbed
the _curvature profile_ [76], is by averaging it over all points of a given
metric space and in each point over all directions, leading to a coarse-
grained, averaged notion of a Ricci scalar. The quantum Ricci curvature and
associated curvature profiles have been studied for a wide range of smooth and
piecewise flat spaces [75, 76] and used to characterize the curvature
properties of DT quantum gravity in $D\\!=\\!2$ [75], CDT quantum gravity in
$D\\!=\\!2$ [77], and full, four-dimensional CDT quantum gravity, producing
further evidence for the de Sitter nature of the quantum geometry in four
dimensions [17]. In the following, we will recall briefly the ingredients and
construction of the curvature profile, and refer to the literature [30, 76,
77] for more detailed discussions.
The main ingredient in determining the quantum Ricci curvature is the
measurement of the _average sphere distance_
$\bar{d}(S_{p}^{\delta},S_{p^{\prime}}^{\delta})$ of two overlapping geodesic
spheres $S_{p}^{\delta}$, $S_{p^{\prime}}^{\delta}$, each of radius $\delta$,
whose centres $p$ and $p^{\prime}$ are also a geodesic distance $\delta$
apart. To compute the ($\delta$-dependent) quantity $\bar{d}$, we average over
the distance between all pairs of points $(q,q^{\prime})\in
S_{p}^{\delta}\times S_{p^{\prime}}^{\delta}$ on the two spheres. The geometry
of the situation is depicted in Fig. 14 for the two-dimensional case, where
the spheres are given by circles. When applying the prescription on a smooth
Riemannian manifold, one extracts the Ricci curvature $Ric(v,v)$ at $p$ in the
limit $\delta\\!\rightarrow\\!0$, where $v$ is the tangent vector at $p$ in
the direction of $p^{\prime}$ [30, 75]. However, we will implement the
prescription on piecewise flat triangulations and for non-infinitesimal
$\delta$, where distances are defined in terms of an integer-valued link
distance or dual link distance, and the limit $\delta\\!\rightarrow\\!0$ is
neither well defined nor physically interesting because of short-distance
discretization artefacts.
Figure 14: The average sphere distance $\bar{d}$ of two geodesic circles
$S_{p}^{\delta}$ and $S_{p^{\prime}}^{\delta}$, whose centres $p$ and
$p^{\prime}$ are a distance $\delta$ apart, is obtained by averaging over the
distance $d(q,q^{\prime})$ of all point pairs $(q,q^{\prime})$ along the two
circles.
In what follows, we will use the link distance $d(q,q^{\prime})$ between pairs
$q,q^{\prime}$ of vertices, to allow for a direct comparison with previous
measurements of the quantum Ricci curvature in two-dimensional DT quantum
gravity, which also used the link distance [75], on the ensemble of regular,
simplicial manifolds. In the triangulated setting, a geodesic “sphere”
$S_{p}^{\delta}$ centred at the vertex $p$ is defined as the set of vertices
at link distance $\delta$ from $p$, $N_{0}(S_{p}^{\delta})$ counts the number
of vertices in this set, and the average sphere distance takes the form of a
normalized double sum
$\bar{d}(S^{\delta}_{p},S^{\delta}_{p^{\prime}})=\frac{1}{N_{0}(S^{\delta}_{p})}\frac{1}{N_{0}(S^{\delta}_{p^{\prime}})}\sum_{q\in
S^{\delta}_{p}}\sum_{q^{\prime}\in
S^{\delta}_{p^{\prime}}}d(q,q^{\prime}),\;\;\;\;\;d(p,p^{\prime})=\delta.$
(21)
We have used inverted commas since the vertices of a point set
$S_{p}^{\delta}$ will in general not form a genuine sphere: it is not required
that the vertices form a sequence of nearest neighbours which can be joined
pairwise by $N_{0}(S_{p}^{\delta})$ edges, resulting in a unique one-
dimensional simplicial submanifold of the topology of a circle. Rather, such a
procedure generally yields multiple circles, and results in self-intersections
and -overlaps.
The quantum Ricci curvature $K_{q}(p,p^{\prime})$ associated with the point
pair $(p,p^{\prime})$ is defined in terms of the normalized average sphere
distance as
$\bar{d}(S_{p}^{\delta},S_{p^{\prime}}^{\delta})/\delta=:c_{q}\,(1-K_{q}(p,p^{\prime})),\quad\quad\delta=d(p,p^{\prime}).$
(22)
The factor $c_{q}$ is a positive constant which describes the
$\delta$-independent part of the average sphere distance. In the continuum, it
can be defined by the limit $c_{q}\\!:=\\!\lim_{\delta\to 0}\bar{d}/\delta$
and depends only on the dimension of the manifold. The function
$K_{q}(p,p^{\prime})$ captures the non-trivial dependence of the average
sphere distance on the direction of the vector $\overline{pp^{\prime}}$ and
the scale $\delta$. The most straightforward way to construct a genuine,
diffeomorphism-invariant curvature observable is by taking an average
$\bar{d}_{\rm av}$ of the average sphere distance $\bar{d}$ of eq. (22) over
all pairs $(p,p^{\prime})$ of centre points at a fixed distance
$\delta=d(p,p^{\prime})$ in the triangulation $T$,
$\bar{d}_{\rm av}(\delta):=\frac{1}{{\cal N}_{\delta}}\sum_{p\in
T}\sum_{p^{\prime}\in
T}\bar{d}(S_{p}^{\delta},S_{p^{\prime}}^{\delta})\,\delta_{K}(d(p,p^{\prime}),\delta).$
(23)
The symbol $\delta_{K}$ denotes a discrete Kronecker delta, implementing the
distance constraint on $p,p^{\prime}$, and the normalization ${\cal
N}_{\delta}$ is defined by the double sum
${\cal N}_{\delta}=\sum_{p\in T}\sum_{p^{\prime}\in
T}\delta_{K}(d(p,p^{\prime}),\delta).$ (24)
The so-called curvature profile, introduced in [76], is given by the quotient
$\bar{d}_{\rm av}(\delta)/\delta$. Since the double sum (23) includes an
average over all directions, it allows us to extract a scale-dependent quantum
Ricci scalar $K_{\rm av}(\delta)$ from the curvature profile via
$\bar{d}_{\rm av}(\delta)/\delta=:c_{\rm av}(1-K_{\rm av}(\delta)).$ (25)
Since in the simplicial setting the factor $c_{\textrm{av}}$ cannot be fixed
through a limit $\delta\\!\to\\!0$, it is set to the expectation value of the
normalized average sphere distance for the minimal value of $\delta$ such that
discretization artefacts are no longer dominant. In [75], this value was taken
to be $\delta\\!=\\!5$.
Figure 15: Expectation value $\langle\bar{d}_{\rm av}/\delta\rangle$ of the
normalized average sphere distance, measured in two-dimensional DT quantum
gravity (blue squares) [75], and for the spatial slices of three-dimensional
CDT quantum gravity in the degenerate phase (yellow dots), both for volume
$V_{2}\\!=\\!60k$. (Error bars are smaller than dot size.)
Continuing our investigation of three-dimensional CDT quantum gravity, we
measured the expectation value $\langle\bar{d}_{\rm av}/\delta\rangle$ of the
curvature profile of its spatial hypersurfaces at the phase space points
$k_{0}\\!=\\!0.0$, 5.0 and 8.0. We used slice volumes in the range
$V_{2}\\!\in\\![4k,60k]$ in the degenerate phase and $V_{2}\\!\in\\![4k,20k]$
in the de Sitter phase. For the volumes $V_{2}\\!=\\!20,30,40$ and $60k$, we
compared the curvature profiles in the degenerate phase at $k_{0}\\!=\\!8.0$
to those of two-dimensional DT quantum gravity [75], and in each case found
agreement within statistical error bars 141414We thank N. Klitgaard for making
the original data available to us.. For illustration, the results for
$V_{2}\\!=\\!60k$ are shown in Fig. 15. There is an excellent match of our
present data, taken for $\delta\\!\in\\![1,25]$, with those of DT quantum
gravity in the range where they overlap, except at $\delta\\!=\\!1$. The fact
that this agreement extends even into most of the region of discretization
artefacts at small $\delta$ provides additional evidence that the spatial
hypersurfaces in the degenerate phase behave like two-dimensional DT
geometries.
The monotonically decreasing curvature profiles we found in both the
degenerate and the de Sitter phase clearly indicate the presence of positive
curvature, as can be seen from eq. (25). Motivated by the fact that curvature
profiles of two-dimensional DT quantum gravity can best be fitted to those of
a five-dimensional continuum sphere with some effective curvature radius
$\rho_{\textrm{eff}}$ [75], we tried to do the same for our data. As expected
from the match of the curvature profiles, our results for the effective
curvature radii in the degenerate phase are close to the values listed in
Table 1 of [75]. Note that a rough estimate for the onset of finite-size
effects in measuring $\langle\bar{d}_{\rm av}/\delta\rangle$ on a sphere of
curvature radius $\rho$ is $\delta\\!\approx\rho$, where the extension
$3\delta$ of the double circle of Fig. 14 is approximately equal to $\pi\rho$,
half of the circumference of the sphere. This is in good agreement with the
findings in [75].
Consistent with this argument, we found that for $V_{2}\\!=\\!60k$, a fitting
range $\delta\\!\in\\![5,15]$ is appropriate, while for the smaller slice
volume $V_{2}\\!=\\!20k$, which is the maximal size available for measurements
in the de Sitter phase, the smaller range $\delta\\!\in\\![5,10]$ should be
used. The measured curvature profile for $V_{2}\\!=\\!20k$ at
$k_{0}\\!=\\!5.0$ in the de Sitter phase is shown in Fig. 16, where for
comparison we present it alongside the result for the same slice volume at
$k_{0}\\!=\\!8.0$ in the degenerate phase. The continuous lines are best fits
to a 5D continuum sphere. Following [75], an additive shift was used such that
the data point at $\delta\\!=\\!5$ always lies on the continuum curve. Because
of the small fitting range we cannot and do not claim that the data taken at
this (or even smaller) volume represent convincing evidence for the curvature
behaviour of a sphere, and the effective curvature radii extracted
($\rho_{\textrm{eff}}\\!=\\!13.5$ for $k_{0}\\!=\\!8.0$,
$\rho_{\textrm{eff}}\\!=\\!11.1$ for $k_{0}\\!=\\!5.0$) should be taken with a
large grain of salt.151515The data at $k_{0}\\!=\\!0.0$ are somewhat similar
to those at $k_{0}\\!=\\!5.0$, but their quality is even worse, and we do not
show them here. In the degenerate phase we can say a bit more, as we have
seen, since we can go up to volume $V_{2}\\!=\\!60k$ and essentially reproduce
the results of DT quantum gravity. With regard to the de Sitter phase, we can
at this stage only conclude that the curvature profiles are not in
contradiction with those of a 5D continuum sphere, but it is clear that the
quality of our results is not sufficient to definitely say that it is a
sphere, let alone determine its dimension and curvature radius reliably. This
would require us to probe much larger systems, which especially in the de
Sitter phase was not feasible in our set-up.
Figure 16: Comparing measured curvature profiles in the range
$\delta\\!\in\\![5,10]$ with those of five-dimensional continuum spheres, for
slice volume $V_{2}\\!=\\!20k$. Left: fit to a sphere with
$\rho_{\textrm{eff}}\\!=\\!13.5$ in the degenerate phase ($k_{0}\\!=\\!8.0$).
Right: fit to a sphere with $\rho_{\textrm{eff}}\\!=\\!11.1$ in the de Sitter
phase ($k_{0}\\!=\\!5.0$).
## 5 Summary and conclusion
We set out to gain a more detailed understanding of the properties of three-
dimensional CDT quantum gravity by studying the intrinsic geometric properties
of its spatial slices at integer proper time. We worked with the “classic”
ensemble of three-dimensional simplicial manifolds, which implies that the
spatial hypermanifolds under consideration also satisfy manifold conditions,
and can be characterized by dual trivalent graphs without tadpoles or self-
energy insertions. The original work on this quantum gravity model found two
distinct phases on either side of a first-order transition as a function of
the bare inverse gravitational coupling $k_{0}$ [26]: a de Sitter phase of
extended geometry for $k_{0}\\!<\\!k_{0}^{c}$ and a degenerate phase for
$k_{0}\\!>\\!k_{0}^{c}$, characterized by a strongly fluctuating volume
profile, the almost complete absence of (2,2)-tetrahedra and an approximate
decoupling of nearby spatial slices.
We investigated the intrinsic geometric properties of the spatial slices in
both phases, well away from the critical coupling
$k_{0}^{c}\\!\approx\\!6.24$, at $k_{0}\\!=\\!0.0$ and $5.0$ in the de Sitter
and at $k_{0}\\!=\\!8.0$ in the degenerate phase. The quantities considered
were the expectation values of the average coordination number of vertices in
the slices, of the entropy exponent extracted from the distribution of
minimal-neck baby universes, of the local and global Hausdorff dimension, and
of the curvature profile, obtained by averaging the quantum Ricci curvature.
They are for the most part well studied in two-dimensional DT and CDT quantum
gravity, the primary systems of reference for our results on the spatial
slices. One aim of our investigation, motivated by the observed decoupling
behaviour of the spatial slices, was to verify that the behaviour of the
slices in the degenerate phase lies in the same universality class as DT
quantum gravity in $D\\!=\\!2$. What happens in the de Sitter phase, and to
what extent the embedding three-dimensional geometry influences the effective
dynamics of the hypersurfaces in this phase is much less clear a priori.
Summarizing our results, we found convincing evidence that the behaviour of
the spatial slices in the degenerate phase is indeed compatible with that of
two-dimensional DT quantum gravity. The measured distribution of the vertex
order follows the analytical prediction almost perfectly (Fig. 6). The same is
true for the distribution of (sufficiently large) minbu sizes (Fig. 8, top),
yielding an entropy exponent $\gamma\\!=\\!-1/2$, the known value for pure
Euclidean gravity in $D\\!=\\!2$. Measurement of the local and global
Hausdorff dimension yielded mutually compatible results, with
$d_{h}\\!=\\!3.31(4)$ and $d_{H}\\!=\\!3.30(2)$, exhibiting finite-size
scaling for the latter. We argued that the discrepancy between the measured
values and the analytically known value $d_{h}\\!\equiv\\!d_{H}\\!=\\!4$ is in
line with expectations for a simplicial manifold ensemble and the relatively
small volumes under consideration here. Finally, the measured curvature
profile matched very well that of a previous investigation of DT quantum
gravity in $D\\!=\\!2$ (Fig. 15), at least up to the slice volume
$V_{2}\\!=\\!60k$ we could investigate.
By contrast, in the de Sitter phase we could not establish a corresponding
overall match of the behaviour of the measured observables with that of any
known quantum gravity model in two dimensions. In particular, we did not find
any evidence that the spatial slices behave according to DT quantum gravity in
$D\\!=\\!2$, which one may argue is the most natural hypothesis, given the
absence in the slices of a preferred direction or a time-space asymmetry,
which characterizes two-dimensional CDT configurations. We also found that the
behaviour within the de Sitter phase depends on the value of the bare coupling
constant $k_{0}$. Since it was tangential to our main focus, we did not study
the nature of this $k_{0}$-dependence more closely, which may be a worthwhile
project in itself. Within the limited range of couplings
$k_{0}\\!\in\\![3.0,6.0]$, earlier work found some evidence that results
inside the de Sitter phase can be mapped onto each other by a
$k_{0}$-dependent rescaling of the time- and space-like length units [26]. It
would be interesting to understand whether this also extends to the value
$k_{0}\\!=\\!0$ we have been using, or even to negative $k_{0}$.
Returning to the specifics of our results, the vertex order in the de Sitter
phase was found to obey a very different distribution from that in the
degenerate phase, with large coordination numbers being more prevalent (Fig.
7). We also saw that the distribution for $k_{0}\\!=\\!5.0$ is even further
removed from that for $k_{0}\\!=\\!8.0$ than the distribution for
$k_{0}\\!=\\!0.0$. The method of determining the entropy exponent $\gamma$
from the distribution of minbu sizes does not appear to be applicable in the
de Sitter phase, which we conjectured to be due to the presence of
correlations in the three-dimensional embedding triangulations. Although this
does not necessarily disprove a DT-like behaviour of the spatial slices, it
does not present any evidence in favour of it either. The measured local
Hausdorff dimensions, given by $d_{h}\\!=\\!3.10(4)$ for $k_{0}\\!=\\!5.0$ and
$d_{h}\\!=\\!2.91(5)$ for $k_{0}\\!=\\!0.0$ are significantly smaller than the
value found in the degenerate phase, and point to a different continuum limit
than that of two-dimensional DT quantum gravity. Of course, it could be the
case that the de Sitter phase is subject to much larger discretization
artefacts, because of the genuinely three-dimensional nature of the underlying
geometries, and that one needs to go to larger slice volumes to get a better
approximation of continuum behaviour. However, even taking this possibility
into account, a yet stronger indication that the slices do not exhibit DT-like
behaviour comes from the absence of finite-size scaling at fixed $k_{0}$ to
extract a global Hausdorff dimension, from which we deduced that the slice
dynamics is likely governed by more than just one scale. Finally, the
measurement of the curvature profiles showed the presence of a positive
average quantum Ricci scalar. Matching with a continuum sphere was in
principle possible, but should at this stage be regarded as inconclusive,
since it was based on only a handful of measurement points, which could be
compatible with other curvature profiles also. It therefore cannot serve as
evidence that the system is equivalent to two-dimensional DT quantum gravity.
Having largely dismissed an interpretation of the spatial slices in the de
Sitter phase in terms of two-dimensional DT or CDT quantum gravity does not
leave any obvious alternatives to associate them with (the universality
classes of) other known systems of random geometry. DT gravity coupled to
matter with a conformal charge $c\\!<\\!1$ is disfavoured, because the
apparent absence of finite-size scaling for the global Hausdorff dimension of
the spatial slices contradicts the scale-invariance of these systems. On the
one hand, the quality of our data is far removed from the precision
measurements of the Hausdorff dimension of such systems [65], and we cannot
entirely exclude that finite-size scaling will appear at much larger volumes
than the ones we could probe here. On the other hand, we would urge caution
when comparing to ensembles with less stringent regularity conditions, like
those used in [65]: even if the relaxation of simplicial manifold conditions
does not change the universality class in specific two-dimensional models,
this may not hold in general in three-dimensional quantum gravity models, and
may depend on the details of the regularity conditions. This is part of a more
general question, namely, are there natural larger ensembles of three-
dimensional triangulations which contain the simplicial manifold ensemble of
CDT, but lie in the same universality class? Conversely, are there strictly
smaller ensembles contained in that of standard CDT quantum gravity, which
still belong to the same universality class? Larger ensembles may facilitate
numerical simulations and lead to faster convergence, while smaller ensembles
may be easier to enumerate and handle analytically (see [78] for a recent
example in three-dimensional DT quantum gravity). Of course, there may be more
than one physically interesting universality class associated with three-
dimensional Lorentzian random geometries, like the wormhole phase described by
the ABAB-matrix model [38] already mentioned in the introduction.
Returning to ensembles of simplicial manifolds, our results in the de Sitter
phase may indicate the existence of another, new model of two-dimensional
quantum geometry, where the embedding three-dimensional geometry induces some
effective dynamics on the spatial slices, presumably through $k_{0}$-dependent
extrinsic curvature contributions. Further research is needed to understand
whether such an induced model exists and whether it can in turn be interpreted
as a two-dimensional quantum field theory with properties like locality and
unitarity, as is the case in the degenerate phase. Contrasting our relatively
straightforward verification of the DT nature of the spatial dynamics in the
degenerate phase with the difficulties we encountered when investigating the
de Sitter phase highlights the fact that quantum geometry in three dimensions
– here by leaving its imprint on the hypersurfaces – is significantly more
complex and complicated than quantum geometry in two dimensions. Much remains
to be done to illuminate its mathematical and physical properties.
Acknowledgments. This work was partly supported by a Projectruimte grant of
the Foundation for Fundamental Research on Matter (FOM, now defunct),
financially supported by the Netherlands Organisation for Scientific Research
(NWO). J.B. would like to thank Wouter van Amsterdam for useful comments on
Appendix B.
## Appendix A Estimating the entropy exponent
This appendix describes the procedure used in Sec. 2.1 of [62] to determine
best fit values for the entropy exponent $\gamma$ with associated error bars,
together with the results of this analysis applied to our data. One introduces
a subleading correction to the right-hand side of (11) by replacing
$B^{\gamma-2}\to
B^{\gamma-2}\left(1+\frac{c}{B}+O\left(\frac{1}{B^{2}}\right)\right),$ (26)
which allows for a better fit in the regime of small $B$, without
significantly affecting the behaviour of the function at intermediate and
large $B$. One then takes the logarithm on both sides of (11) to obtain
$\log(\bar{n}_{N_{2}}(B))=a+(\gamma-2)\log\left(B(N_{2}-B)\right)+\frac{c}{B},$
(27)
where the best fit parameters $a$, $\gamma$ and $c$ should now be determined
from the observed baby universe distributions shown in Fig. 8.161616Note that
the argument in the logarithm on the right-hand side of this expression
differs from eq. (2.2) in [62] by a multiplicative factor $N_{2}$, which is
absorbed by the fit parameter $a$. The goodness of fit is defined through the
$\chi^{2}$-statistic, described in greater detail in App. B below.
As mentioned before, the prediction (11) is only expected to hold for
sufficiently large baby universes, where discretization effects are
negligible. We therefore introduce a lower cut-off $B_{0}$ on $B$ on the data
before extracting the best fit parameters. The resulting values of the entropy
exponent $\gamma$ as a function of the cut-off $B_{0}$ are shown in Fig. 17
for $N_{2}\\!\equiv V_{2}\\!=\\!1.000$, both with and without the correction
term $c/B$ in (27). Our figure resembles Fig. 2 of reference [62] extremely
closely, including the magnitude of the error bars. The authors of [62]
subsequently obtain an estimate for $\gamma$ by fitting an exponential of the
form
$\gamma(B_{0})=\gamma-c_{1}\,e^{-c_{2}\,B_{0}}.$ (28)
The resulting values for $\gamma$ shown in Table 1 of [62] are (within
statistical error) identical to the ones we found from our data, using the
same method.
Figure 17: Fitted values of the entropy exponent $\gamma$ in the degenerate
phase for slice volume $N_{2}\\!=\\!1.000$ and different lower cut-offs
$B_{0}$. We show the best fit values with and without the correction term
appearing in (27).
## Appendix B Determining confidence intervals for best fit parameters
While researching the literature on numerical estimates of quantities like the
Hausdorff dimension and the entropy exponent, we found that previous work is
often not very explicit about the methodology used to determine the
uncertainty in the best fit parameters, if such margins of error are provided
at all. Since our goal was to investigate whether the behaviour of the spatial
slices is consistent with that of known models of two-dimensional random
geometry, we considered it important to attach a degree of confidence to the
results obtained. This led us to a more detailed study of the statistical
methods available for performing such an analysis. Here we summarize our
findings, in the hope that others may find them useful. We also comment on the
assumptions required to make use of these methods, and to what extent they
apply in the present context. Our main aim is to motivate the choices made in
computing the confidence intervals; we do not aim for full mathematical or
statistical rigour.
The generic starting point consists of a set of $N$ data points
$y_{n}\in\mathbb{R}$ measured at positions $\bm{x}_{n}$, to which we want to
fit a function $f(\bm{x},\bm{\theta})$ with parameters $\bm{\theta}$. Firstly,
we require a measure of the goodness-of-fit which allows us to find the set of
best fit parameters $\bm{\theta}_{\textrm{min}}$ that minimizes this measure.
Secondly, since the measurements are inherently noisy, we are interested in
specifying a range for the fit parameters in which we expect to find the
“true” values with a certain degree of confidence. In the method of least
squares, the goodness-of-fit is defined through the sum of squares of the
residuals, and the optimal fit is obtained when this sum is minimized.
However, the standard unweighted sum of least squares assumes equal variances
on all the data points, which typically is not the case for the measurements
we perform in lattice quantum gravity. In what follows, we will show how the
$\chi^{2}$-distribution can be used in the context of a linear regression
model to define a goodness-of-fit and corresponding confidence intervals for
the parameters $\bm{\theta}$. The models we fit in the main text of this work
do not fall into this class of linear models, so we subsequently discuss how
the analysis is affected when the condition of linearity is relaxed.
### B.1 Best fit parameter estimation
A linear model takes the form
$f\left(\bm{x};\bm{\theta}\right)=\sum_{i=1}^{p}\theta_{i}f_{i}(\bm{x}),$ (29)
where the $f_{i}$ are functions of the independent variables. These functions
are allowed to be nonlinear in the $x_{i}$ — the linearity condition applies
to the fit parameters $\theta_{i}$ only. Suppose the “correct” model has fit
parameters $\bm{\theta}_{0}$. If the errors $\sigma_{n}$ on the $N$
measurement outcomes are Gaussian, we can consider the $y_{n}$ to be normally
distributed random variables with mean $f(\bm{x}_{n};\bm{\theta}_{0})$ and
standard deviation $\sigma_{n}$. We can turn these into $N$ standard normals
$Z_{n}=\frac{y_{n}-f(\bm{x}_{n};\bm{\theta}_{0})}{\sigma_{n}}$. Let us define
the _$\chi^{2}$ -statistic_ of a choice of fit parameters $\bm{\theta}$ as the
sum of the squares of the $Z_{n}$,
$\chi^{2}\left(\bm{\theta}\right)=\sum_{n=1}^{N}\left(\frac{y_{n}-f\left(\bm{x}_{n};\bm{\theta}\right)}{\sigma_{n}\,}\right)^{2}.$
(30)
The reason we call this the $\chi^{2}$-statistic is that the sum of $k$
independent standard normals follows a so-called $\chi^{2}$-distribution with
$k$ degrees of freedom. Such a distribution has expectation value $k$ and
variance $2k$. The quantity $\chi^{2}\left(\bm{\theta}\right)$ should be
minimized to find a _maximum likelihood estimator_ for $\bm{\theta}_{0}$,
corresponding to the set of best fit parameters.
In the current situation, where we are trying to fit a model to the data
points, we must take into account that the individual $y_{n}$ are explicitly
_not_ independent — after all, we have hypothesized the existence of a model
$f(\bm{x};\bm{\theta})$ that can predict the outcomes of our measurements. The
sum of $N$ squared standard normal distributions therefore follows a
$\chi^{2}$-distribution with a certain number $k\\!<\\!N$ of degrees of
freedom. For a linear model without priors (i.e. restrictions on the fit
parameters), we have $k\\!=\\!N-p$, where $p$ is the number of fit parameters.
Therefore, when fitting a linear model $f(\bm{x};\bm{\theta})$ with $p$ fit
parameters to $N$ data points $y_{n}$ with Gaussian errors $\sigma_{n}$, we
expect the $\chi^{2}$-statistic to follow a $\chi^{2}$-distribution with
$k=N-p$ degrees of freedom. As mentioned earlier, the best fit parameters
$\bm{\theta}_{\textrm{min}}$ are determined by finding the minimum possible
value $\chi^{2}_{\textrm{min}}$ of the $\chi^{2}$-statistic. The fit is
considered good when $\chi^{2}_{\textrm{min}}\\!\approx\\!k$, since this is
the expectation value of the corresponding $\chi^{2}$-distribution.
Significantly larger values of $\chi^{2}_{\textrm{min}}$ indicate that no
reasonable fit could be found (e.g. our assumptions about the model may be
wrong), whereas a $\chi^{2}_{\textrm{min}}$ significantly lower than $k$ could
mean we are overfitting the data or overestimating the measurement errors
$\sigma_{n}$.
### B.2 Confidence intervals
With the best fit parameters $\bm{\theta}_{\textrm{min}}$ at our disposal, we
can now turn to determining _confidence intervals_ on these parameters. After
all, slightly varying the parameters around their best fit values should
produce approximately equal $\chi^{2}$-statistics. Moreover, the measurement
data we are using to compute $\chi^{2}$ is inherently noisy, which implies
that we have merely obtained an estimate of the “true” best fits. To specify a
range in which we expect to find the true values with a certain degree of
confidence, we can use the properties of the $\chi^{2}$-distribution.
Confidence intervals (CIs) are computed at a certain _confidence level_ ,
specified by a percentage (a common choice is the 95% CI). Alternatively, we
can specify a significance level $\alpha$, corresponding to a
$(1\\!-\\!\alpha)\%$ confidence level. Given $\alpha$, we can determine the
_critical value_ $\chi^{2}_{\textrm{crit},\alpha}$ of the $\chi^{2}$-statistic
for our fitted model containing $p$ parameters by solving
$P(\chi^{2}\\!\leq\\!\chi^{2}_{\textrm{crit},\alpha})\\!=\\!(1-\\!\alpha)$,
where $P(\chi^{2}\\!\leq\\!x)$ is the cumulative distribution function for a
$\chi^{2}$-distribution with $k\\!=\\!p$ degrees of freedom.171717When
estimating the best fit entropy exponent and local Hausdorff dimension, we
were only interested in one out the $p$ fit parameters. In this case, we
should match to a $\chi^{2}$-distribution with one degree of freedom. We can
determine $\chi^{2}_{\textrm{crit},\alpha}$ by the use of quantile functions
or lookup tables. The $(1\\!-\\!\alpha)\%$ confidence interval for the fit
parameters $\bm{\theta}$ is then defined [79] as the region for which
$\chi^{2}(\bm{\theta})-\chi^{2}_{\textrm{min}}<\chi^{2}_{\textrm{crit},\alpha}.$
(31)
Typically, this region is an ellipsoid in $\bm{\theta}$-space, which can be
determined numerically by performing a grid search around
$\bm{\theta}_{\textrm{min}}$. As an example, when determining the 95%
confidence intervals for a linear model with two parameters, we find
$\chi^{2}_{\textrm{crit},0.05}\\!\approx\\!5.991$, and the joint confidence
intervals of the two fit parameters are the region in $\mathbb{R}^{2}$ for
which (31) holds.
### B.3 Potential caveats
We have used the procedure just described to determine the confidence
intervals for the best fit parameters in the main text of this work. However,
as pointed out earlier, the analysis rests on several assumptions that do not
necessarily apply to our models and measurements. An important prerequisite
for using the $\chi^{2}$-distribution is that the measurement errors are
Gaussian, otherwise the sum (30) is not a sum of squares of standard normals.
We often found a slight degree of skewness in the distribution of our
measurement results, potentially invalidating the use of $\chi^{2}$-methods.
However, the skewness factors were always near zero, so that we may still
consider the computed bounds of the confidence intervals to be good
approximations to their true values.
A second potential issue is that the models we fit in our work are not linear.
Both for the minbu sizes and the microscopic Hausdorff dimension, one of the
fit parameters appears in the exponent of an independent variable. Although
the best fit parameters for such models can still be obtained by minimizing
(30), determining the correct number of degrees of freedom is known to be
difficult [80]. This means that we do not know the proper expectation value of
(30), and therefore do not have a reference point to compare our
$\chi^{2}_{\textrm{min}}$ to. However, not knowing the true number of degrees
of freedom has more serious consequences for computing confidence intervals.
The number $p$ of fit parameters in our models is small, $p\\!<\\!4$, and
choosing a different number of degrees of freedom near zero has a strong
effect on the resulting $\chi^{2}_{\textrm{crit},\alpha}$. This can
significantly alter the width of the corresponding confidence interval. For
our purposes, we do not consider this to be a major issue. The main goal of
our analysis was to check whether our results are consistent with previously
known results from the literature, requiring an order-of-magnitude estimate of
the confidence interval. This order of magnitude is not affected if we over-
or underestimate the degree-of-freedom count by a few units. Furthermore,
since all confidence intervals in this work were obtained by using the same
methods, any comparison among our confidence intervals is likely to still be
meaningful.
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|
# Detection of 49 Weak Dispersed Radio Pulses in a Parkes Observation of the
X-ray Pulsar PSR J0537$-$6910
Fronefield Crawford Department of Physics and Astronomy, Franklin and Marshall
College, P.O. Box 3003, Lancaster, PA 17604, USA
###### Abstract
I conducted a new search for dispersed radio pulses from the X-ray pulsar PSR
J0537$-$6910 in the Large Magellanic Cloud in a long (11.6 hr) archival 1.4
GHz Parkes search observation. I searched dispersion measures (DMs) between 0
and 10000 pc cm-3 and detected 49 pulses with a signal-to-noise ratio (S/N)
greater than 7 at a wide range of DMs using the HEIMDALL and FETCH pulse
detection and classification packages. All of the pulses were weak, with none
having a S/N above 8.5. There was a significant excess of pulses observed in
the DM range of the known pulsar population in the LMC, suggesting that these
pulses may originate from LMC pulsars. Three repeat pulses, each having widths
$\lesssim 1$ ms, were detected in a single DM trial of 103.412 pc cm-3, which
is in the LMC DM range. This is unlikely to occur by chance in a single DM
trial in this search at the (marginally significant) 4.3$\sigma$ level. It
remains unclear whether any of the detected pulses in the sample are from PSR
J0537$-$6910 itself.
Pulsars (1306); Radio Transient Sources (2008)
††facilities: Parkes††software: HEIMDALL (Barsdell, 2012; Barsdell et al.,
2012), FETCH (Agarwal et al., 2020)
## 1 Introduction and Background
PSR J0537$-$6910 is a young rotation-powered X-ray pulsar associated with the
supernova remnant (SNR) N157B in the Large Magellanic Cloud (LMC). It was
first discovered as a 16-ms pulsed X-ray source by Marshall et al. (1998), and
it is the most rapidly rotating unrecycled pulsar currently known. Despite
previous searches for both periodic radio emission and single radio pulses
with the Parkes 64-m radio telescope (“Murriyang”) (Crawford et al., 1998,
2005), PSR J0537$-$6910 has not yet been detected as a radio emitter.
PSR J0537$-$6910 is a particularly good candidate to search for giant radio
pulses. Cognard et al. (1996) suggested a relationship between the giant pulse
emission mechanism and the magnetic field strength at the light cylinder
radius, defined as the equatorial radius at which co-rotation with the pulsar
would equal the speed of light. This magnetic field strength can be computed
from the spin parameters of the pulsar according to $B_{lc}=3\times
10^{8}\dot{P}^{1/2}P^{-5/2}$ G, where $P$ and $\dot{P}$ are the pulsar period
in seconds and its time derivative, respectively.
Along with two other young pulsars that are known emitters of giant radio
pulses, the Crab pulsar (Staelin & Reifenstein, 1968; Lundgren et al., 1995)
and PSR B0540$-$69, which also resides in the LMC (Johnston & Romani, 2003),
PSR J0537$-$6910 has a large light-cylinder magnetic field strength. It is
noteworthy that five millisecond pulsars (MSPs) with much different properties
than these young pulsars but with large light-cylinder magnetic field values
have also been observed to emit giant radio pulses (Cognard et al., 1996;
Romani & Johnston, 2001; Johnston & Romani, 2003; Joshi et al., 2004; Knight
et al., 2005, 2006). PSR J0537$-$6910 has a light-cylinder field strength that
is more than twice as large as either of the next two highest light-cylinder-
field pulsars (the Crab pulsar and the MSP PSR B1937+21), making it a good
target to test this hypothesis.
Table 1 presents a current listing of the pulsars with the largest light-
cylinder field strengths, with the observed giant radio pulse emitters
identified. This is also illustrated in Fig. 1, where pulsars with
$B_{lc}>10^{5}$ G are plotted as a function of the spin period (see also Table
1 of both Crawford et al. (2005) and McLaughlin & Cordes (2003), and Fig. 4 of
Cognard et al. (1996); however, these references do not have the more recently
discovered pulsars shown in Table 1 and Fig. 1).
## 2 Radio Search History
Prior searches for both periodic and single-pulse radio emissions from PSR
J0537$-$6910 were conducted (unsuccessfully) using Parkes at several
frequencies. McLaughlin & Cordes (2003) searched for the pulsar in a single
0.5 hr pointing at 435 MHz, while Crawford et al. (1998) searched two 4 hr
pointings at 660 MHz and a single 6-hr pointing at 1374 MHz. Subsequently,
Crawford et al. (2005) observed the pulsar with a longer 11.6 hr integration
at a center frequency of 1390 MHz. This observation had a 256 MHz bandwidth
split into 512 channels and was sampled at 80 $\mu$s. In all of these
searches, no dispersion measures (DMs) above 300 pc cm3 were searched. This
encompassed the DMs of the known LMC pulsar population (which currently ranges
from 45 to 273 pc cm-3; see, e.g., Hisano et al. 2022). No convincing
astrophysical signals were seen in any of these prior searches. This last (and
longest) observation is the one I have searched again with newer software
packages over a wider range of DMs.
## 3 Data Analysis
I reprocessed this single long Parkes observation with the HEIMDALL pulse
detection package (Barsdell,
2012)111https://sourceforge.net/projects/heimdall-astro at trial DMs ranging
from 0 to 10000 pc cm-3 in order to search for single-pulse events, including
fast radio bursts (FRBs; Lorimer et al. 2007). A total of 1011 DM trials were
produced and searched by HEIMDALL. Boxcar-matched filtering windows of $2^{n}$
samples, with $n$ ranging from 0 to 9, were applied to each dedispersed time
series to maintain maximum sensitivity to pulses with widths up to $\sim 41$
ms. This is significantly larger than the widths expected for any pulses from
PSR J0537$-$6910 given its 16 ms spin period.
All of the pulses detected by HEIMDALL were then analyzed by
FETCH.222https://github.com/devanshkv/fetch FETCH is a pulse classifier that
assigns a probability of being real to each detected pulse based on its
morphology and characteristics (Agarwal et al., 2020). FETCH rated each
detected pulse using its Model A (see Table 4 of Agarwal et al. 2020) and
assigned a probability of being a real, astrophysical pulse of between 0 and
1. I also searched the data for periodicities at DMs ranging from 0 to to 5000
using PRESTO tools (Ransom, 2001) in case other pulsars were present in the
same beam. No promising periodicity signals were detected.
## 4 Results and Discussion
A total of 49 single pulses were detected with a signal-to-noise ratio (S/N)
above 7 which had a probability assigned by FETCH that was greater than 0.5.
All of these pulses were subsequently checked visually to ensure they were not
obvious radio frequency interference (RFI) signals that had been misidentified
by FETCH. All of the detected pulses were weak, with none having a S/N above
8.5. This corresponds to a fluence threshold of 0.6 Jy ms (for a putative
pulse width of $W=1$ ms; this sensitivity limit scales as $\sqrt{W}$). Table 2
lists these 49 pulse detections with their characteristics.
### 4.1 Possible False Detections
As a check to see if these pulses might have been artificially generated by
the software during the detection and classification process, I repeated the
search over the same range of negative DMs (0 to $-10000$ pc cm-3) with the
same S/N threshold of 7 in order to see whether a significant number of
spurious candidates would be produced. This exercise produced only one
artificial negative-DM candidate that was classified as real (with S/N = 7.0).
This is in contrast to the results of Perera et al. (2022) and Hisano et al.
(2022) who each conducted a similar test on two different large-scale pulsar
surveys and found more artificial, low-S/N candidates at negative DMs than
corresponding positive-DM candidates. This indicated that their sample of
detected low-S/N candidates might be largely artificial.
Nimmo et al. (2023) reported several cases in which weak pulses detected from
the repeating FRB 20200120E were not classified as real by FETCH. These pulses
were ultimately determined to be real owing to their DM proximity to the FRB.
In this case, real, weak pulses had been classified as RFI and missed (false
negatives), but not the reverse (no false positives were reported). Thus, this
same misclassification issue would not have produced spurious, false-positive
detections in this sample of 49 pulses (although it could possibly have
resulted in missing some real pulses).
For comparison, in 36.6 hr of targeted observations of several SNRs that used
a similar 1.4 GHz Parkes observing setup and the same analysis procedure used
here, no pulses (spurious or otherwise) were detected and classified as real,
apart from four pulses from a known bright pulsar in the vicinity (Crawford,
2023).
Another indication that the pulses could be artificial is if their measured
pulse widths are smaller than the corresponding dispersive smearing time
within the finite frequency channels at that DM. Such narrow intrinsic pulses
would be expected to be broadened by dispersion if the pulses were real. This
dispersion smearing scales linearly with DM and is determined by
$\tau=(202/f_{c})^{3}\,\Delta f\,{\rm DM}$, where $\tau$ is the smearing in
ms, $f_{c}$ and $\Delta f$ are the center observing frequency and the channel
width in MHz (1390 and 0.5, respectively), and DM is in pc cm-3. Fig. 2 shows
the measured pulse widths for the sample of 49 pulses from Table 2 plotted as
a function of DM, with the expected width from dispersive broadening also
plotted. As seen in the figure, none of the detected pulses lie below the
minimum (dispersive) width, lending further support to the notion that the
pulses are not obviously artificial or otherwise generated by the software.
Note that although scattering contributions to the broadening are not
considered here, the Galactic contribution to scattering along this line of
sight is negligible (Cordes & Lazio, 2002).
### 4.2 Pulse Detection Rate
The average detection rate in this single observation was one pulse detected
and classified as real above S/N of 7 for every 14 minutes of observing time.
This rate is high compared to a similar search conducted of a large-scale
survey of the LMC with Parkes using HEIMDALL and FETCH (Hisano et al., 2022).
That survey used a similar (though not identical) observing system that had a
comparable raw sensitivity. They searched 702 beams totaling 1677 hr for
single pulses out to a DM of 10000 pc cm-3, and they used a similar maximum
boxcar width (33 ms). A total of 229 pulses were found in that survey with a
S/N above 7, a DM above 50 pc cm-3, and a FETCH probability of being real
above 0.9 (this included nine pulses detected from the giant pulse emitter PSR
B0540$-$69). When using these same cutoffs and filtering criteria for the set
of detections reported here, 33 of the 49 original pulses are retained.
However, these were detected over the much smaller 11.6 hr of integration
time. The pulse detection rate per unit of observing time is therefore $\sim
20$ times larger than the large-scale LMC survey analysis of Hisano et al.
(2022). Some of this difference may be attributable to the fact that the lone
observation analyzed here targeted the 30 Doradus star formation region, where
more pulsars may be present than on average in the LMC. However, this large
difference remains difficult to reconcile completely if the pulses detected
here are indeed real and coming from the LMC. This discrepancy becomes even
more pronounced if some fraction of the weak pulses detected by Hisano et al.
(2022) are not actually real (see the discussion above).
As outlined in Crawford et al. (2005), if the Crab pulsar were located at the
distance of the LMC, it would emit a giant pulse every 20 minutes that would
be detectable with this observing system. This would lead to several dozen
such detections in this single observation. Crawford et al. (2005) also
indicate that a giant pulse from PSR B0540$-$69 in the LMC should be
detectable every 0.5 hr with such an observing setup. This is broadly
consistent with the detection of nine pulses from PSR B0540$-$69 (four of
which were above S/N of 9) in a single 2.4-hr Parkes survey beam of the LMC
which covered the location of that pulsar (Hisano et al. 2022; see their Table
1). The sample reported here clearly does not include any pulses from a single
source (such as PSR J0537$-$6910) with this brightness or frequency of
occurrence.
### 4.3 DM Distribution of Detected Pulses
The largest DM of any currently known radio pulsar in the LMC is 273 pc cm-3
(Ridley et al., 2013), but the remainder of the known LMC population has DMs
that lie between 45 and 147 pc cm-3 (Manchester et al., 2006; Ridley et al.,
2013). Fig. 3 shows our 49 detected pulses with S/N plotted against DM. The
majority of the detected pulses (34 out of 49) fall within the observed DM
range of the currently known LMC pulsar population. However, only 26% of the
DM trials in the search were in this range, and so we would expect only 13
events to occur here by chance. Given this, the likelihood of detecting 34 or
more pulses by chance in the DM trials in this range is less than $10^{-6}$,
suggesting that the observed excess is real. Therefore, it is possible that
many of these pulses are from as-yet-unidentified pulsars in the LMC.
The wide distribution of DMs of the detected pulses indicates that they cannot
all be coming from the same object. Some may be coming from other, as-yet-
unidentified LMC pulsars (possible, given the excess of pulses seen in the LMC
DM range). The few pulses with very large DMs (4 of the 49 had DM $>800$ pc
cm-3; see Table 2 and Fig. 3) could be FRBs originating from well beyond the
LMC.
### 4.4 Repeat Pulses
Three pulses were detected in a single DM trial (DM = 103.412 pc cm-3). These
three pulses are shown in Fig. 4 and are identified in Table 2. The likelihood
of three or more pulses occurring by chance in a single DM trial can be
estimated using the total number of pulses detected (49) and the total number
of DM trials in the search (1011). Following the analysis outlined in Section
4.2 of Paine et al. (2024) for a similar likelihood estimate for a single-
pulse search of M82, I determined that this is unlikely to occur by chance at
the (marginally significant) 4.3$\sigma$ level. No other DM trial had any
repeat pulses (see Table 2).
This DM value is well within the observed DM range for LMC pulsars, and all
three pulses had comparable widths of between 0.3 and 1.0 ms, as determined
from the HEIMDALL detections (see Table 2). Note that the dispersive smearing
within frequency channels at this DM is 0.15 ms. This is significantly smaller
than the measured pulse widths (see also Fig. 2), indicating that these
measured widths are largely intrinsic to the pulsar. This also suggests that
these are not micropulses or nanoshots like those seen in giant pulses from
the Crab pulsar (Hankins & Eilek, 2007; Hankins et al., 2016).
It is possible that these three pulses could all be coming from the same
pulsar in the LMC. The pulse widths are much smaller than the 16 ms period of
PSR J0537$-$6910, so they could be coming from PSR J0537$-$6910 specifically.
However, the time separations between the three pulses are not close to an
integer number of pulse periods from PSR J0537$-$6910\. To determine this, the
topocentric period and its uncertainty were determined for PSR J0537$-$6910
from the ATNF catalog parameters (Manchester et al.,
2005)333https://www.atnf.csiro.au/research/pulsar/psrcat/ using PRESTO tools.
The topocentric period uncertainty was combined with the measured half-widths
of the pulses to obtain an uncertainty in the time of each pulse separation.
In all three cases, the uncertainty (which was 2%-3% of the pulse period of
PSR J0537$-$6910) was much less than the remainder when the pulse time
difference was divided by the pulsar period (these remainders were 41%, 18%,
and 77%). This disfavors the conclusion that the pulses are from PSR
J0537$-$6910\. However, given the large and prolific timing glitches seen for
the pulsar (e.g., Ho et al. 2020), this may not be definitive. I also
dedispersed the raw data at this DM and folded the resulting time series using
the ephemeris of PSR J0537$-$6910\. No signal was detected in this fold.
## 5 Conclusions
In a new analysis of an archival Parkes search observation of the LMC X-ray
pulsar PSR J0537$-$6910, I detected 49 dispersed single radio pulses with a
S/N greater than 7. All 49 pulses had a FETCH likelihood of being real that
was greater than 0.5. None of the 49 detected pulses had a S/N above 8.5,
corresponding to a fluence threshold of 0.6 Jy ms (for a putative 1 ms pulse
width). A significant excess of the detected pulses (34 out of 49) occurred
within the DM range of the known LMC pulsar population, suggesting that some
of these pulses may be from as-yet-unidentified LMC pulsars or possibly from
PSR J0537$-$6910 itself. Three pulses having widths $\lesssim 1$ ms were
detected in a single DM trial (DM = 103.412 pc cm-3). This is unlikely to
occur by chance at the 4.3$\sigma$ level. This DM value is within the observed
range for LMC pulsars, suggesting that the pulses may originate from a pulsar
in the LMC. Future observations with more sensitive, next-generation
facilities may be useful for determining whether any of the pulses detected in
the sample are from PSR J0537$-$6910.
The Parkes radio telescope is part of the Australia Telescope National
Facility (grid.421683.a), which is funded by the Australian Government for
operation as a National Facility managed by CSIRO. This work was supported in
part by National Science Foundation (NSF) Physics Frontiers Center award Nos.
1430284 and 2020265, and used the Franklin and Marshall College compute
cluster, which was funded through NSF grant 1925192.
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Table 1: Cataloged Pulsars with the Largest Light-Cylinder Magnetic Field
Strengths
PSR | $B_{lc}$ | Type | Giant Pulse
---|---|---|---
| ($10^{5}$ G) | | References
J0537$-$6910 | 20.7 | younga |
B1937+21 | 10.2 | MSP | Cognard et al. (1996)
B0531+21 (Crab) | 9.6 | young | Staelin & Reifenstein (1968)
J1402+13 | 7.8 | MSPa |
B1821$-$24A | 7.4 | MSP | Romani & Johnston (2001)
J0058$-$7218 | 7.3 | younga |
J1748$-$2446ak | 5.8 | MSP |
J1701$-$3006F | 5.6 | MSP |
J1737$-$0314A | 5.3 | MSP |
J1555$-$2908 | 4.7 | MSP |
J1835$-$3259B | 4.4 | MSP |
B1957+20 | 3.8 | MSP | Joshi et al. (2004)
B0540$-$69 | 3.6 | young | Johnston & Romani (2003)
J1400$-$6325 | 3.5 | young |
J0218+4232 | 3.2 | MSP | Knight et al. (2006)
Note. — aNot a radio pulsar. Data were taken from the ATNF Pulsar Catalog
(version 1.70). The top fifteen pulsars ranked by $B_{lc}$ are listed. Bold
entries indicate pulsars observed to emit giant radio pulses. Note that one
other pulsar, PSR B1820$-$30A, is an MSP that emits observable giant radio
pulses (Knight et al., 2005), but it is not listed here since it ranks 23rd on
this list. See also Fig. 1.
Table 2: List of Detected Single Pulses Ordered by DM
Pulse | Time of | DM Trial | S/N | FETCH | Pulse Width
---|---|---|---|---|---
Number | Pulse (s) | (pc cm-3) | | Likelihood | (ms)
1 | 6676.2417(3) | 22.161 | 7.5 | 99.997% | 0.6
2 | 2442.0319(6) | 35.742 | 7.5 | 99.985% | 1.3
3 | 8397.3183(2) | 38.948 | 7.4 | 99.841% | 0.3
4 | 5369.5697(2) | 44.762 | 7.8 | 99.515% | 0.3
5 | 20223.2706(2) | 50.421 | 7.3 | 71.246% | 0.4
6 | 40026.1995(1) | 56.835 | 7.3 | 99.988% | 0.2
7 | 3497.6510(1) | 59.695 | 7.1 | 99.438% | 0.2
8* | 5924.5415(5) | 103.412 | 7.2 | 99.970% | 1.0
9* | 6764.7114(2) | 103.412 | 7.2 | 98.097% | 0.3
10* | 40538.7033(2) | 103.412 | 7.0 | 62.850% | 0.4
11 | 26744.4141(3) | 109.731 | 7.1 | 99.983% | 0.6
12 | 5523.0110(2) | 111.903 | 7.0 | 99.886% | 0.5
13 | 2083.5429(2) | 117.103 | 7.0 | 99.990% | 0.3
14 | 24753.1893(2) | 118.623 | 7.6 | 52.817% | 0.4
15 | 11974.3024(6) | 120.160 | 7.1 | 99.945% | 1.3
16 | 8762.3732(3) | 122.494 | 7.6 | 99.989% | 0.6
17 | 18975.3036(5) | 124.071 | 7.0 | 77.577% | 1.0
18 | 22657.4292(8) | 128.902 | 7.6 | 66.141% | 1.7
19 | 5913.4912(8) | 139.035 | 8.4 | 99.758% | 1.5
20 | 8012.5940(8) | 143.452 | 7.2 | 99.981% | 1.6
21 | 27263.3490(8) | 153.597 | 7.8 | 96.984% | 1.5
22 | 36148.5644(10) | 155.507 | 7.4 | 99.468% | 2.0
23 | 1460.5734(8) | 161.362 | 7.2 | 99.995% | 1.7
24 | 12319.5048(7) | 167.412 | 7.4 | 99.998% | 1.4
25 | 26833.7275(12) | 169.473 | 7.2 | 89.701% | 2.5
26 | 18547.8670(9) | 172.608 | 8.2 | 98.323% | 1.8
27 | 9213.3246(5) | 174.726 | 7.0 | 96.529% | 1.0
28 | 14323.1635(13) | 177.947 | 7.3 | 59.221% | 2.6
29 | 15232.4035(6) | 179.033 | 7.5 | 94.064% | 1.1
30 | 26228.9132(15) | 185.673 | 7.6 | 99.998% | 3.0
31 | 40129.3022(3) | 189.077 | 7.4 | 90.696% | 0.6
32 | 34427.6883(2) | 194.876 | 7.1 | 79.459% | 0.3
33 | 38263.8594(9) | 199.633 | 7.5 | 93.066% | 1.8
34 | 21103.5138(10) | 206.969 | 7.4 | 99.992% | 1.9
35 | 6778.7168(5) | 210.730 | 7.0 | 61.309% | 1.0
36 | 20002.8843(3) | 222.397 | 7.4 | 88.357% | 0.6
37 | 1891.0116(2) | 230.508 | 7.3 | 97.514% | 0.5
38 | 22474.6054(4) | 236.069 | 7.1 | 99.995% | 0.8
39 | 32339.2015(2) | 311.730 | 7.4 | 85.183% | 0.5
40 | 5607.8609(6) | 315.418 | 7.2 | 94.921% | 1.1
41 | 12581.6178(15) | 365.243 | 7.1 | 99.765% | 3.0
42 | 7558.9811(13) | 401.071 | 7.3 | 94.033% | 2.6
43 | 39081.7106(21) | 403.421 | 7.7 | 99.410% | 4.2
44 | 12254.4407(46) | 405.784 | 7.0 | 52.500% | 9.1
45 | 33980.0101(6) | 536.747 | 7.3 | 99.978% | 1.3
46 | 18378.7853(37) | 810.519 | 8.5 | 99.847% | 7.4
47 | 16925.0990(32) | 839.202 | 7.3 | 99.502% | 6.4
48 | 21693.3493(138) | 1844.050 | 7.1 | 91.929% | 27.5
49 | 26754.9458(128) | 3463.760 | 7.6 | 99.987% | 25.6
Note. — The time of the pulse is relative to the start of the integration at
MJD 52888.61267361. The figure in parentheses represents the uncertainty in
the last digit of the time of the pulse as determined by HEIMDALL. The pulse
widths were also measured from the HEIMDALL detections. Pulses 8, 9, and 10
(indicated with asterisks) are the three pulses that appeared in a single DM
trial. See also Fig. 4.
Figure 1: Light-cylinder magnetic field strength vs. spin period for pulsars
in the ATNF pulsar catalog (version 1.70) (Manchester et al., 2005) that have
$B_{lc}>10^{5}$ G. PSR J0537$-$6910 has the largest $B_{lc}$ by a factor of
two and is indicated by the open circle. The seven pulsars observed to emit
giant radio pulses are indicated by open stars; the two pulsars that have the
next highest values of $B_{lc}$ after PSR J0537$-$6910 are labeled (PSR
B1937+21 and the Crab pulsar). The population shown here can be divided into
MSPs on the left and young pulsars on the right. See also Table 1. Figure 2:
Pulse width vs. DM for 49 detected pulses. Also plotted is a dashed line
indicating where the pulse width equals the dispersive smearing within the
frequency channels. Real, astrophysical pulses would not be expected to lie
below this line. Figure 3: Pulse S/N vs. DM for 49 single-pulse detections
with S/N above 7 which had a FETCH-assigned probability of being real greater
than 0.5. There is an excess of pulses in the observed DM range of the known
LMC pulsar population (45 to 273 pc cm-3, indicated by the dashed vertical
lines), suggesting that some of these pulses could be from LMC sources.
Whether PSR J0537$-$6910 is the source of any of these pulses remains
speculative.
Figure 4: Three weak single pulse detections occurring in a single DM trial of
103.412 pc cm-3. The pulses are shown in the order given in Table 2, which
provides further details. The top panel in each case shows flux vs. time, with
the pulse centered at time zero. The middle panel shows frequency vs. time
after dedispersion has been applied. A broadband signal (straight vertical
line) would be expected in this panel for a typical pulse from a pulsar. The
bottom panel shows DM vs. time. A localized signal appearing at a non-zero DM
would be expected for an astrophysical signal. The likelihood of three or more
of the 49 detected pulses appearing in a single DM trial by chance in this
search is small (unlikely at the 4.3$\sigma$ level). All three pulses have
comparable widths and could be coming from the same pulsar in the LMC, though
probably not from PSR J0537$-$6910 (see the discussion in the main text).
|
18215 LABEL:LastPageJan. 11, 2021Jun. 07, 2022 biblatexPatching footnotes
failed
# Quotients, inductive types,
& quotient inductive types ,
Marcelo P. Fiore Department of Computer Science and Technology
University of Cambridge
United Kingdom {marcelo.fiore, andrew.pitts<EMAIL_ADDRESS>,
Andrew M. Pitts and S. C. Steenkamp(🖂)
###### Abstract.
This paper introduces an expressive class of indexed quotient-inductive types,
called QWI types, within the framework of constructive type theory. They are
initial algebras for indexed families of equational theories with possibly
infinitary operators and equations. We prove that QWI types can be derived
from quotient types and inductive types in the type theory of toposes with
natural number object and universes, provided those universes satisfy the
Weakly Initial Set of Covers (WISC) axiom. We do so by constructing QWI types
as colimits of a family of approximations to them defined by well-founded
recursion over a suitable notion of size, whose definition involves the WISC
axiom. We developed the proof and checked it using the Agda theorem prover.
###### Key words and phrases:
dependent type theory, higher inductive types, quotient types, well-founded
relation, weakly initial set of covers, topos theory
## 1\. Introduction
Inductive types are an essential feature of many type theories and the proof
assistants and programming languages that are based upon them. Homotopy Type
Theory [Uni13] introduces a powerful extension to the notion of inductive
type: higher inductive types (HITs). To define an ordinary inductive type one
declares the ways in which its elements are constructed, and these
constructions may depend inductively on smaller elements. To define a HIT one
not only declares element constructors, but also declares equality
constructors valued in identity types (possibly iterated ones), specifying how
the element constructors (and possibly the equality constructors) are related.
In the language of homotopy theory, these correspond to points, paths,
surfaces, and so on.
In this paper, rather than using HoTT, we work in the simpler setting of
Extensional Type Theory [Mar82], where any propositional equality is reflected
as a definitional equality. However, our results also hold for an intensional
type theory with the uniqueness of identity proofs axiom (UIP), as
demonstrated by our Agda development which can be found at [FPS21]. In either
case identity types are trivial in dimensions higher than one, so that any two
proofs of identity are equal. Nevertheless, as [AK16] point out, HITs are
still useful in such a one-dimensional setting. They introduced the term
_quotient inductive type_ (QIT) for this truncated form of HIT.
One specific advantage of QITs over inductive types is that they provide a
natural way to study universal algebra in type theory, since general algebraic
theories cannot be represented in an equation-free way just using inductive
notions [MM16]. As an example, the free commutative monoid on carrier $X$ is
given by the quotient inductive type of finite multisets111Other presentations
of multisets are available, aside from the two presentations given here.
Incidentally, the algebra of monoids has an equation-free presentation [Kel80,
section 23], [Kel92] – the algebra of Lists – and this is why no equation for
associativity is needed in either definition of multisets., $\mathsf{Bag}\,X$,
with constructors:
$\displaystyle{}[]$ $\displaystyle:\mathsf{Bag}\,X$ (1)
$\displaystyle\\_{\mathrel{::}}\\_$
$\displaystyle:X\rightarrow\mathsf{Bag}\,X\rightarrow\mathsf{Bag}\,X$
$\displaystyle\mathsf{swap}$
$\displaystyle:\prod_{x,y:X}\prod_{\mathit{zs}:\mathsf{Bag}\,X}x\mathrel{::}y\mathrel{::}\mathit{zs}=y\mathrel{::}x\mathrel{::}\mathit{zs}$
This can be seen as the element constructors for lists ($[]$ and
$\mathrel{::}$), along with an equality constructor $\mathsf{swap}$ that
transposes two elements and thereby allows all reorderings of a list to be
identified. Notice that the $\mathsf{swap}$ constructor lands in the identity
type on $\mathsf{Bag}\,X$ (just written as $\\_{=}\\_$), making it an
_equation_ constructor, compared to the _element_ constructors, $[]$ and
$\\_{\mathrel{::}}\\_$, which are familiar from ordinary inductive type
definitions. There is no need to include a constructor for the congruence
property of $\\_{\mathrel{::}}\\_$ since that is automatic from the congruence
properties of identity types.
An alternative presentation of multisets, whose equations may be easier to
work with, is:
$\displaystyle{}[]$ $\displaystyle:\mathsf{Bag}^{\prime}\,X$ (2)
$\displaystyle\\_{\mathrel{::}}\\_$
$\displaystyle:X\rightarrow\mathsf{Bag}^{\prime}\,X\rightarrow\mathsf{Bag}^{\prime}\,X$
$\displaystyle\mathsf{comm}$
$\displaystyle:\prod_{x,y:X}\prod_{\mathit{as},\mathit{bs},\mathit{cs}:\mathsf{Bag}^{\prime}\,X}\mathit{as}=y\mathrel{::}\mathit{cs}\rightarrow
x\mathrel{::}\mathit{cs}=\mathit{bs}\rightarrow
x\mathrel{::}\mathit{as}=y\mathrel{::}\mathit{bs}$
Notice that the $\mathsf{comm}$ equality constructor is _conditional_ on two
equations, $\mathit{as}=y\mathrel{::}\mathit{cs}$ and
$x\mathrel{::}\mathit{cs}=\mathit{bs}$, in order to deduce the final equation
$x\mathrel{::}\mathit{as}=y\mathrel{::}\mathit{bs}$. The QWI-types introduced
in this paper are not able to encode such conditional QITs (at least, not
obviously so); see the discussion in the Conclusion.
Given the parameter $X$, the types $\mathsf{Bag}\,X$ and
$\mathsf{Bag}^{\prime}\,X$ are single quotient-inductive types, rather than
indexed families defined quotient-inductively. As an example of the latter,
consider the go-to example of an inductively defined family, the vector type
(length-indexed lists). This has a commutative variant, $\mathsf{AbVec}\,X\,n$
for $n:\mathbb{N}$, which makes it an example of an indexed QIT. It has
constructors:
$\displaystyle{}[]$ $\displaystyle:\mathsf{AbVec}\,X\,0$ (3)
$\displaystyle\\_{\mathrel{::}}\\_$
$\displaystyle:X\rightarrow\prod_{i:\mathbb{N}}\mathsf{AbVec}\,X\,i\rightarrow\mathsf{AbVec}\,X\,(i+1)$
$\displaystyle\mathsf{swap}$
$\displaystyle:\prod_{x,y:X}\prod_{i:\mathbb{N}}\prod_{\mathit{zs}:\mathsf{AbVec}\,X\,i}x\mathrel{::}y\mathrel{::}\mathit{zs}=y\mathrel{::}x\mathrel{::}\mathit{zs}$
Our final example of a QIT is unordered countably-branching trees over $X$,
called $\mathsf{\omega Tree}\,X$, with constructors:
$\displaystyle{}\mathsf{leaf}$ $\displaystyle:\mathsf{\omega Tree}\,X$ (4)
$\displaystyle\mathsf{node}$
$\displaystyle:X\rightarrow(\mathbb{N}\rightarrow\mathsf{\omega
Tree}\,X)\rightarrow\mathsf{\omega Tree}\,X$ $\displaystyle\mathsf{perm}$
$\displaystyle:\prod_{x:X}\prod_{b:\mathbb{N}\rightarrow\mathbb{N}}\prod_{b^{\prime}:\mathsf{isIso}\,b}\prod_{f:\mathbb{N}\rightarrow\mathsf{\omega
Tree}\,X}\mathsf{node}\,x\,f=\mathsf{node}\,x\,(f\circ b)$
where elements of $\mathsf{isIso}\,b$ witness that $b$ is a bijection. (As
with $\mathsf{AbVec}$, one could similarly consider a depth-indexed variant of
$\mathsf{\omega Tree}$.) The significance of this example is that the
$\mathsf{node}\,x\,\\_$ and $\mathsf{perm}\,x\,b\,b^{\prime}\,\\_$
constructors have infinite arity, making this an example of an _infinitary_
QIT. This type is one of the original motivations for considering QITs [AK16],
since they can enable constructive versions of structures that classically use
non-constructive choice principles, as we explain next.
The examples of QITs in (1)–(3) only involve element constructors of _finite_
arity. For example in (1), $[]$ is nullary and for each $x,y:X$,
$x\mathrel{::}\\_$ and $\mathsf{swap}\,x\,y\,\\_$ are unary. Consequently
$\mathsf{Bag}\,X$ is isomorphic to the type obtained from the ordinary
inductive type of finite lists over $X$ by quotienting by the congruence
generated by $\mathsf{swap}$. By contrast, (4) involves element and equality
constructors with countably infinite arity. So if one first forms the ordinary
inductive type of _ordered_ (or planar) countably-branching trees (by dropping
the equality constructor $\mathsf{perm}$ from the declaration) and then
quotients by a suitable relation to get the equalities specified by
$\mathsf{perm}$, one appears to need the axiom of countable choice to be able
to lift the $\mathsf{node}$ element constructor to the quotient [AK16, Section
2.2].
The construction of the Cauchy reals as a higher inductive-inductive type
[Uni13, Section 11.3] provides a similar, but more complicated example where
use of countable choice is avoided. So it seems that without some form of
choice principle, quotient inductive types are strictly more expressive than
the combination of ordinary inductive types with quotient types. Indeed,
[LS19, Section 9] turn an infinitary equational theory due to [Bla83, Section
9] into a higher-inductive type that cannot be proved to exist in ZF set
theory without the Axiom of Choice. In this paper we show that in fact a much
weaker and constructively acceptable form of choice is sufficient for
constructing indexed quotient inductive types. This is the _Weakly Initial Set
of Covers_ (WISC) axiom, originally called the “Type Theoretic Collection
Axiom” TTCAf by [Str05] and (in the context of constructive set theory) the
“Axiom of Multiple Choice” by [vdBM14]. WISC is constructively acceptable in
that it is known to hold in the internal logic of a wide range of toposes
[vdBM14] (for example, all Grothendieck and all realizability toposes over the
topos of classical sets), including those commonly used in the semantics of
Type Theory. Section 4 reviews WISC and our motivation for using it herein.
We make two contributions:
* •
First we define a class of indexed quotient inductive types called _QWI-types_
and give elimination and computation rules for them (Section 3). The usual
W-types of [Mar82] are inductive types giving the algebraic terms over a
possibly infinitary signature. One specifies a QWI-type by giving a family of
equations between such terms. So such types give initial algebras for
(families of) possibly infinitary algebraic theories. They can encode a very
wide range of examples of, possibly infinitary, indexed quotient inductive
types (Section 7). In classical set theory with the Axiom of Choice, QWI-types
can be constructed simply as quotients of the underlying Indexed W-type—hence
the name.
* •
Second, our main result (Section 6) is that with only WISC it is still
possible to construct QWI-types from inductive types and quotients in the
constructive type theory of toposes, but not simply by quotienting a W-type.
Instead, quotienting is interleaved with an inductive construction. To make
sense of this and prove that the resulting type has the required universal
property, we construct the type as the colimit of a family of approximations
to it defined by well-founded recursion over a suitable notion of _size_ (a
constructive version of what classically would be accomplished with a sequence
of transfinite ordinal length). Our notion of size is developed in Section 5.
Sizes are elements of a W-type equipped with the _plump_ [Tay96] well-founded
order; and WISC allows us to make the W-type big enough that a given
polynominal endofunctor preserves size-indexed colimits (Section 5.1). This
fact is then applied in Section 6 to form QWI-types as size-indexed colimits.
### Note
This paper is a greatly revised and expanded version of our earlier paper
[FPS20], which introduced QW-types (see Section 3.2 which is the non-indexed
version of Section 3.3). There we made use of _sized types_ [Abe12] as they
are implemented in the Agda theorem prover [Agd20]. Though the results of that
paper were formalised in Agda version 2.5.2, unfortunately, in the current
version of Agda, 2.6.2, sized types are logically unsound. In this paper, not
only do we consider the more expressive indexed form of QW-types, namely QWI-
types, we also put the construction of QW-types on a firm semantic footing by
avoiding Agda-style sized types (and positivity assumptions on quotient types,
see Section 6.1) in favour of a well-founded notion of size built from WISC.
The Agda formalization of the results in this paper is available at [FPS21].
## 2\. Type theory
The results in this paper are proved in a version of Extensional Type Theory
[Mar82] that is an internal language for toposes [Joh02] with natural numbers
object and universes [Str05a]. We use this type theory in an informal way, in
the style and notation of the HoTT Book [Uni13]. Thus dependent function types
are written $\prod_{x:A}B\,x$ and non-dependent ones as either $A\rightarrow
B$ or $B^{A}$. Dependent product types are written $\sum_{x:A}B\,x$ and non-
dependent ones as $A\times B$. Coproduct types are written $A+B$ (with
injections $\iota_{1}:A\rightarrow A+B$ and $\iota_{2}:B\rightarrow A+B$); and
finite types are written $\mathbb{0}$ (with no elements), $\mathbb{1}$ (with a
single element $0$), $\mathbb{2}$ (with two elements $0,1$), etc. The identity
type for $x:A$ and $y:A$ is $x=_{A}y$, or just $x=y$ when the type $A$ is
known from the context. When we need to refer to the judgement that $x$ and
$y$ are definitionally equal, we write $x\equiv y$. Unlike in the HoTT Book
[Uni13], the type $x=y$ is an extensional identity type and so it is inhabited
iff $x\equiv y$ holds. In particular, proofs of identity are unique when they
exist and so it makes sense to use McBride’s heterogeneous form of identity
type [McB99] wherever possible. Given $x:A$ and $y:B$, we denote the
heterogeneous identity type by $x\mathrel{{=}{=}}y$; thus this type is
inhabited iff $A\equiv B$ and $x\equiv y$.
The type theory has universes of types,
$\mathcal{U}_{0}:\mathcal{U}_{1}:\mathcal{U}_{2},\ldots$ which we write just
as $\mathcal{U}$ when the universe level is immaterial. We use Russell-style
universes (there is no syntactic distinction between an element
$A:\mathcal{U}$ and the corresponding type of its elements). Influenced by
Agda (see below) and unlike the HoTT Book [Uni13], we do not assume universes
are cumulative, but instead use Agda-style closure under $\prod$-types: if
$A:\mathcal{U}_{i}$ and $B:A\rightarrow\mathcal{U}_{j}$, then
$\prod_{x:A}B\,x:\mathcal{U}_{\max ij}$ (and similarly for $\sum$-types).
Since we only consider toposes that have a natural numbers object, we can
assume the universes are closed under forming not just W-types [MP00,
Proposition 3.6], but also all inductively defined indexed families of types
[GH04]. Indexed W-types are considered in more detail in the next section.
The lowest universe contains an impredicative universe $\mathsf{Prop}$ of
propositions, corresponding to the subobject classifier in a topos.
$\mathsf{Prop}$ contains the identity types, is closed under intuitionistic
connectives ($\wedge$, $\vee$, $\rightarrow$, $\leftrightarrow$, etc.) and
quantifiers ($\forall(x:A).\phi(x)$ and $\exists(x:A).\phi(x)$, for $A$ in any
universe), and satisfies propositional extensionality:
$\mathsf{propext}:\forall(p,q:\mathsf{Prop}).\,(p\leftrightarrow
q)\rightarrow{p=q}$ (5)
Being an extensional type theory, we also have function extensionality:
$\mathsf{funext}:\forall\left(f,g:\prod_{x:A}B\,x\right).\,(\forall(x:A).\,f\,x=g\,x)\rightarrow
f=g$ (6)
Note that like the universes $\mathcal{U}_{i}$, $\mathsf{Prop}$ is also a
Russell-style universe, in that we do not make a notational distinction
between a proposition $p:\mathsf{Prop}$ and the type of its proofs. Given
$A:\mathcal{U}$ and $\phi:A\rightarrow\mathsf{Prop}$, we regard the
comprehension type $\\{x:A\mid\phi\,x\\}$ in $\mathcal{U}$ as synonymous with
the dependent product $\sum_{x:A}\,\phi\,x$.
Toposes have coequalizers and effective epimorphisms [Joh02, A2.4].
Correspondingly the type theory contains quotient types, with notation as in
Figure 1 (recall that $\mathrel{{=}{=}}$ stands for heterogeneous identity).
These can be constructed in the usual way via equivalence classes, using
$\mathsf{Prop}$’s impredicative quantifiers and the fact that toposes satisfy
unique choice:
$\mathsf{uniquechoice}:(\exists(x:A).\forall(y:A).x=y)\rightarrow A$ (7)
Thus $\mathsf{uniquechoice}$ is a function mapping proofs that $A$ has exactly
one element to a name for that one element. From $\mathsf{uniquechoice}$ it
follows that given $A:\mathcal{U}$, $B:\mathcal{U}^{A}$ and
$\phi:\prod_{x:A}(B\,x\rightarrow\mathsf{Prop})$, from a proof of
$\forall(x:A)\exists!(y:B\,x).\;\phi\,x\,y$ we get there is a unique
$f:\prod_{x:A}B\,x$ such that $\forall(x:A).\;\phi\,x\,(f\,x)$.
• Given $A:\mathcal{U}$ and $R:A\rightarrow A\rightarrow\mathsf{Prop}$, we
have: $\displaystyle A/R:\mathcal{U}$ $\displaystyle[\\_]_{R}:A\rightarrow
A/R$
$\displaystyle\mathsf{qeq}_{R}:\forall(x,y:A)\mathbin{.}\,R\,x\,y\rightarrow[x]_{R}=[y]_{R}$
• Furthermore, given $B:A/R\rightarrow\mathcal{U}$, $f:\prod_{x:A}B([x]_{R})$,
and $p:\forall(x,y:A)\mathbin{.}\,R\,x\,y\rightarrow
f\,x\mathrel{{=}{=}}f\,y$, we have:
$\displaystyle\mathsf{qelim}_{R}\,B\,f\,p:\prod_{z:A/R}B\,z$
$\displaystyle\mathsf{qcomp}_{R}\,B\,f\,p:\forall(x:A)\mathbin{.}\,\mathsf{qelim}\,B\,f\,[x]_{R}=f\,x$
• Quotients of equivalence relations are effective: writing $\mathsf{ER}_{R}$
for the proposition that $R$ is reflexive, symmetric and transitive, we have:
$\displaystyle\mathsf{qeff}_{R}:\mathsf{ER}_{R}\rightarrow\forall(x,y:A)\mathbin{.}\,[x]_{R}=[y]_{R}\rightarrow
R\,x\,y$ Figure 1. Quotient types
### Agda development
We have developed and checked a version of the results in this paper using the
Agda theorem prover [Agd20]. In particular our Agda development gives the full
details of the construction of the indexed version of QW-types, whereas in the
paper, for readability, we only give the construction in the non-indexed case
(Section 6).
Being intensional and predicative, the type theory provided by Agda is weaker
than the one described above, but can be soundly interpreted in it. One has to
work harder to establish the results with Agda, since two expressions that are
definitionally equal in Extensional Type Theory may not be so in Agda and
hence one has to produce a term in a corresponding identity type to prove them
equal. On the other hand, when a candidate term is given, Agda can decide
whether or not it is a correct proof, because validity of the judgements of
the type theory it implements is decidable222though not always feasibly so, in
contrast with the situation for Extensional Type Theory [Hof95]. Our
development is also made easier by extensive use of the predicative universes
of proof-irrelevant propositions that feature in recent versions of Agda. Not
only are any two proofs of such propositions definitionally equal, but
inductively defined propositions (such as the well-founded ordering on sizes
used in Section 6) can be eliminated in proofs using dependent pattern
matching, which is a very great convenience. We still need these propositions
to satisfy the extensionality (5) and unique choice (7) properties, so we add
them as postulates in Agda. Also, the impredicative construction of quotient
types is not available, so we get quotients as in Figure 1 by postulating them
and using a user-declared rewrite to make their computation rule a
definitional equality. Although function extensionality (6) is not provable in
Agda’s core type theory, it becomes so once one has such quotient types. Our
Agda development is available at [FPS21].
## 3\. Indexed polynomial functors and equational systems
In this section we introduce a class of indexed quotient inductive types,
called _QWI-types_ , which are indexed families of free algebras for possibly
infinitary equational theories. To begin with we describe the simpler, non-
indexed case (_QW-types_) and then treat the general, indexed case in Section
3.3.
### 3.1. W-types
We recall some facts about types of well-founded trees, the W-types of
[Mar82]. We take _signatures_ (also known as _containers_ [AAG05]) to be
elements of the dependent product
$\textstyle\mathsf{Sig}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sum_{A:\mathcal{U}}(A\rightarrow\mathcal{U})$ (8)
So a signature is given by a pair $\mathrm{\Sigma}=(A,B)$ consisting of a type
$A:\mathcal{U}$ and a family of types $B:A\rightarrow\mathcal{U}$. Each such
signature determines a polynomial endofunctor [GH04, AAG05]
$S_{\mathrm{\Sigma}}:\mathcal{U}\rightarrow\mathcal{U}$ whose value at
$X:\mathcal{U}$ is the following dependent product
$\textstyle S_{\mathrm{\Sigma}}(X)\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sum_{a:A}(B\,a\rightarrow X)$ (9)
and whose action on a function $f:X\rightarrow Y$ in $\mathcal{U}$ is the
function $S_{\mathrm{\Sigma}}f:S_{\mathrm{\Sigma}}(X)\rightarrow
S_{\mathrm{\Sigma}}(Y)$ where
$S_{\mathrm{\Sigma}}f\,(a,b)\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}(a,f\circ b)\qquad(a:A,b:B\,a\rightarrow X)$ (10)
An _$S_{\mathrm{\Sigma}}$ -algebra_ is by definition an element of the
dependent product
$\textstyle\mathsf{Alg}_{\mathrm{\Sigma}}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sum_{X:\mathcal{U}}(S_{\mathrm{\Sigma}}(X)\rightarrow X)$ (11)
$S_{\mathrm{\Sigma}}$-algebra morphisms
$(X,\alpha)\rightarrow(X^{\prime},\alpha^{\prime})$ are given by functions
$h:X\rightarrow X^{\prime}$ together with an element of the type
$\mathsf{isHom}_{\alpha,\alpha^{\prime}}\,h\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\prod_{a:A}\prod_{b:B\,a\rightarrow X}(\alpha^{\prime}(a,h\circ
b)=h(\alpha(a,b)))$ (12)
Then the W-type
$\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{7.00009pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{5.00006pt}{}{$\mathrm{W}$}}}}}\slimits@_{\mathrm{\Sigma}}$
determined by a signature $\mathrm{\Sigma}$ is the underlying type of an
initial object in the evident category of $S_{\mathrm{\Sigma}}$-algebras. More
generally, [Dyb97] shows that the initial algebra of any non-nested, strictly
positive endofunctor on $\mathcal{U}$ is given by a W-type; and [AAG05] extend
this to the case with nested uses of W-types as part of their work on
containers. (These proofs take place in extensional type theory [Mar82], but
work just as well in intensional type theory with uniqueness of identity
proofs and function extensionality, which is what we use in Agda.)
More concretely, given a signature $\mathrm{\Sigma}=(A,B)$, if one thinks of
elements $a:A$ as names of operation symbols whose (not necessarily finite)
arity is given by the type $B\,a:\mathcal{U}$, then the elements of the W-type
$\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{7.00009pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{5.00006pt}{}{$\mathrm{W}$}}}}}\slimits@_{\mathrm{\Sigma}}$
represent the closed algebraic terms (i.e. well-founded trees) over the
signature. From this point of view it is natural to consider not only closed
terms solely built up from operations, but also open terms additionally built
up with variables drawn from some type $V$. As well as allowing operators of
possibly infinite arity, we also allow terms involving possibly infinitely
many variables (example (4) involves such terms). Categorically, the type
$T_{\mathrm{\Sigma}}(V)$ of such open terms is the free
$S_{\mathrm{\Sigma}}$-algebra on $V$ and is another W-type, for the signature
obtained from $\mathrm{\Sigma}$ by adding the elements of $V$ as nullary
operations. Nevertheless, it is convenient to give a direct inductive
definition: the value of
$T_{\mathrm{\Sigma}}:\mathcal{U}\rightarrow\mathcal{U}$ at some some
$V:\mathcal{U}$, is the inductively defined type with constructors
$\displaystyle\eta$ $\displaystyle:V\rightarrow T_{\mathrm{\Sigma}}\,V$ (13)
$\displaystyle\sigma$
$\displaystyle:S_{\mathrm{\Sigma}}(T_{\mathrm{\Sigma}}\,V)\rightarrow
T_{\mathrm{\Sigma}}\,V$
Given an $S_{\mathrm{\Sigma}}$-algebra
$(X,\alpha):\mathsf{Alg}_{\mathrm{\Sigma}}$ and a function $f:V\rightarrow X$,
the unique morphism of $S_{\mathrm{\Sigma}}$-algebras from the free
$S_{\mathrm{\Sigma}}$-algebra on $V$ to $(X,\alpha)$ that extends $f$,
$\\_\mathbin{\gg\\!=}f:(T_{\mathrm{\Sigma}}(V),\sigma)\rightarrow(X,\alpha)$,
has underlying function $T_{\mathrm{\Sigma}}(V)\rightarrow X$ mapping each
$t:T_{\mathrm{\Sigma}}(V)$ to the element $t\mathbin{\gg\\!=}f$ defined as
follows by recursion on the structure of $t$:
$\displaystyle\eta\,x\mathbin{\gg\\!=}f$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny def}}{=}}}f\,x$ (14)
$\displaystyle\sigma\,(a,b)\mathbin{\gg\\!=}f$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\alpha\,(a,\lambda\,x\mathbin{.}b\,x\mathbin{\gg\\!=}f)$
As the notation suggests, $\mathbin{\gg\\!=}$ is the Kleisli lifting operation
(“bind”) for a monad structure on $T_{\mathrm{\Sigma}}$. Indeed,
$T_{\mathrm{\Sigma}}$ is the free monad on the endofunctor
$S_{\mathrm{\Sigma}}$ and in particular the functorial action of
$T_{\mathrm{\Sigma}}$ on $f:V\rightarrow V^{\prime}$ yields the function
$T_{\mathrm{\Sigma}}f:T_{\mathrm{\Sigma}}(V)\rightarrow
T_{\mathrm{\Sigma}}(V^{\prime})$ given by
$T_{\mathrm{\Sigma}}f\,t\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}t\mathbin{\gg\\!=}(\eta\circ f)\qquad(t:T_{\mathrm{\Sigma}}(V))$
(15)
### 3.2. QW-types
The notion of _QW-type_ that we introduce in this section is obtained from
that of a W-type by considering not only the algebraic terms over a given
structure, but also equations between terms. To code equations we use a type-
theoretic rendering of a categorical notion of equational system introduced by
[FH09], referred to as _term equational system_ [FH09, Section 2] and as
_monadic equational system_ [Fio13, Section 5], here instantiated to the
special case of free monads on signatures.
###### Definition .
A _system of equations_ over a signature $\mathrm{\Sigma}:\mathsf{Sig}$ is
specified by:
* •
A type $E:\mathcal{U}$, whose elements $e:E$ can be thought of as names for
each equation.
* •
A function $V:E\rightarrow\mathcal{U}$. For each equation $e:E$, the type
$V\,e:\mathcal{U}$ represents the variables used in the equation $e$.
* •
For each equation $e:E$, two elements called $l\,e$ and $r\,e$ of type
$T_{\mathrm{\Sigma}}(V\,e)$, which is the free $S_{\mathrm{\Sigma}}$-algebra
on the variables $V\,e$. So $l\,e$ and $r\,e$ can be thought of as the
abstract syntax trees of terms with some leaves being free variables drawn
from $V\,e$.
So systems of equations are elements of the dependent product
$\mathsf{Syseq}_{\mathrm{\Sigma}}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sum_{E:\mathcal{U}}\sum_{V:E\rightarrow\mathcal{U}}\bigg{(}\prod_{e:E}T_{\mathrm{\Sigma}}(V\,e)\bigg{)}\times\bigg{(}\prod_{e:E}T_{\mathrm{\Sigma}}(V\,e)\bigg{)}$
(16)
An $S_{\mathrm{\Sigma}}$-algebra $\alpha:S_{\mathrm{\Sigma}}(X)\rightarrow X$
_satisfies_ the system of equations
$\varepsilon\equiv(E,V,l,r):\mathsf{Syseq}_{\mathrm{\Sigma}}$ if there is a
proof of
$\mathsf{Sat}_{\alpha,\varepsilon}(X)\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\forall(e:E).\forall(\rho:V\,e\rightarrow
X).\;(l\,e\mathbin{\gg\\!=}\rho)=(r\,e\mathbin{\gg\\!=}\rho)$ (17)
The category-theoretic view of QW-types is that they are simply
$S_{\mathrm{\Sigma}}$-algebras that are initial among those satisfying a given
system of equations:
###### Definition (QW-type).
A _QW-type_ for a signature $\mathrm{\Sigma}\equiv(A,B):\mathsf{Sig}$ and a
system of equations
$\varepsilon\equiv(E,V,l,r):\mathsf{Syseq}_{\mathrm{\Sigma}}$ is given by a
type $\mathsf{QW}:\mathcal{U}$ equipped with an $S_{\mathrm{\Sigma}}$-algebra
structure and a proof that it satisfies the equations. Thus there are
functions
$\displaystyle\mathsf{qwintro}$
$\displaystyle:S_{\mathrm{\Sigma}}(\mathsf{QW})\rightarrow\mathsf{QW}$ (18)
$\displaystyle\mathsf{qwequate}$
$\displaystyle:\mathsf{Sat}_{\mathsf{qwintro},\varepsilon}(\mathsf{QW})$ (19)
together with functions that witness that it is an initial such algebra:
$\displaystyle\mathsf{qwrec}$
$\displaystyle:\prod_{X:\mathcal{U}}\prod_{\alpha:S_{\mathrm{\Sigma}}\,X\rightarrow
X}\mathsf{Sat}_{\alpha,\varepsilon}(X)\rightarrow\mathsf{QW}\rightarrow X$
(20) $\displaystyle\mathsf{qwrechom}$
$\displaystyle:\prod_{X:\mathcal{U}}\prod_{\alpha:S_{\mathrm{\Sigma}}\,X\rightarrow
X}\prod_{p:\mathsf{Sat}_{\alpha,\varepsilon}(X)}\mathsf{isHom}\,(\mathsf{qwrec}\,X\,\alpha\,p)$
(21) $\displaystyle\mathsf{qwuniq}$
$\displaystyle:\prod_{X:\mathcal{U}}\prod_{\alpha:S_{\mathrm{\Sigma}}\,X\rightarrow
X}\prod_{p:\mathsf{Sat}_{\alpha,\varepsilon}(X)}\prod_{f:\mathsf{QW}\rightarrow
X}\mathsf{isHom}\,f\rightarrow\mathsf{qwrec}\,X\,\alpha\,p=f$ (22)
###### Remark (Free algebras).
Section 3.2 defines QW-types as initial algebras for systems of equations
$\varepsilon$ over signatures $\mathrm{\Sigma}$. More generally, the _free_
$(\mathrm{\Sigma},\varepsilon)$-algebra on a type $X:\mathcal{U}$ is a type
$F_{\mathrm{\Sigma},\varepsilon}\,X:\mathcal{U}$ equipped with an
$S_{\mathrm{\Sigma}}$-algebra structure
$S_{\mathrm{\Sigma}}(F_{\mathrm{\Sigma},\varepsilon}\,X)\rightarrow
F_{\mathrm{\Sigma},\varepsilon}\,X$ satisfying the system of equations
$\varepsilon$ and an inclusion of generators $\eta_{X}:X\rightarrow
F_{\mathrm{\Sigma},\varepsilon}\,X$ which is universal among such data.
Initial algebras are the $X=\mathbb{0}$ special case of free algebras. But
once one has them, one also has free algebras, by change of signature:
$F_{\mathrm{\Sigma},\varepsilon}\,X$ is the QW-type for the signature
$\mathrm{\Sigma}_{X}$ and system of equations $\varepsilon_{X}$ defined as
follows. If $\mathrm{\Sigma}=(A,B)$, then $\mathrm{\Sigma}_{X}=(X+A,B_{X})$
where $B_{X}:X+A\rightarrow\mathcal{U}$ satisfies
$B_{X}(\iota_{1}\,x)=\mathbb{0}$ and $B_{X}(\iota_{2}\,a)=B\,a$. And if
$\varepsilon=(E,V,l,r)$, then $\varepsilon_{X}=(E,V,l_{X},r_{X})$ where for
each $e:E$, $l_{X}\,e=(l\,e\mathbin{\gg\\!=}\eta)$ and
$r_{X}\,e=(r\,e\mathbin{\gg\\!=}\eta)$ (using the
$S_{\mathrm{\Sigma}}$-algebra structure $s$ on $T_{\mathrm{\Sigma}}(V\,e)$
given by $s(a,b)=\sigma(\iota_{2}\,a,b)$).
The definitions of $S_{\mathrm{\Sigma}}$ in (9) and
$\mathsf{Sat}_{\alpha,\varepsilon}$ in (17) suggest that QW-types are an
instance of the notion of quotient-inductive type [AK16], with
$\mathsf{qwintro}$ as the element constructor and $\mathsf{qwequate}$ as the
equality constructor. To show that QW-types are indeed quotient-inductive
types, they need to have the requisite dependently-typed elimination and
computation properties333In this paper we work with extensional type theory,
so the computation property, (26), is also a definitional equality. In our
Agda development we work in intensional type theory, so there we only
establish the computation property up to propositional equality; using the
terminology of [Shu18], those are _typal_ indexed quotient-inductive types.
for these elements and equality constructors. We show that these follow from
(20)–(22) and function extensionality. To state this proposition we need a
dependent version of the bind operation (14). For each “motive” $P$ and
induction step $p$,
$\displaystyle P$ $\displaystyle:\mathsf{QW}\rightarrow\mathcal{U}$ (23)
$\displaystyle p$
$\displaystyle:\prod_{a:A}\prod_{b:B\,a\rightarrow\mathsf{QW}}\bigg{(}\prod_{x:B\,a}P(b\,x)\bigg{)}\rightarrow
P(\mathsf{qwintro}(a,b))$
as well as a type $X:\mathcal{U}$, a substitution
$\rho:\left(X\rightarrow\sum_{x:\mathsf{QW}}P\,x\right)$, and term
$t:T_{\mathrm{\Sigma}}\,X$, we have an element
$\mathsf{lift}_{P,X}\,p\,\rho\,t:P\,(t\mathbin{\gg\\!=}(\pi_{1}\circ\rho))$
defined by recursion on the structure of $t$:
$\begin{array}[]{lcl}\mathsf{lift}_{P,X}\,p\,\rho\,(\eta\,x)&\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}&\pi_{2}\,(\rho\,x)\\\
\mathsf{lift}_{P,X}\,p\,\rho\,(\sigma\,(a,b))&\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}&p\,a\,(\lambda\,x\mathbin{.}b\,x\mathbin{\gg\\!=}(\pi_{1}\circ\rho))\,((\mathsf{lift}_{P,X}\,p\,\rho)\circ
b)\end{array}$ (24)
Note that the substitution $\rho$ gives terms in the “dependent telescope”
$\sum_{x:\mathsf{QW}}P\,x$, which will be written as $P^{\prime}$.
###### Proposition .
For a QWI-type as defined above, given $P$ and $p$ as in (23), and a term
$p_{\mathsf{resp}}$ of type
$\prod_{e:E}\prod_{\rho:V\,e\rightarrow\sum_{x:\mathsf{QW}}P\,x}\mathsf{lift}_{P,X}\,p\,\rho\,(l\,e)==\mathsf{lift}_{P,X}\,p\,\rho\,(r\,e)$
(25)
there are elimination and computation terms
$\displaystyle\mathsf{qwelim}$ $\displaystyle:\prod_{x:\mathsf{QW}}P\,x$ (26)
$\displaystyle\mathsf{qwcomp}$
$\displaystyle:\prod_{a:A}\prod_{b:B\,a\rightarrow\mathsf{QW}}\mathsf{qwelim}\,(\mathsf{qwintro}\,(a,b))=p\,a\,b\,(\mathsf{qwelim}\circ
b)$
(Note that (25) uses heterogeneous identity $\mathrel{{=}{=}}$, because
$\mathsf{lift}_{P,X}\,p\,\rho\,(l\,e)$ and
$\mathsf{lift}_{P,X}\,p\,\rho\,(r\,e)$ inhabit different types, namely
$P\,(l\,e\mathbin{\gg\\!=}(\pi_{1}\circ\rho))$ and
$P\,(r\,e\mathbin{\gg\\!=}(\pi_{1}\circ\rho))$ respectively.)
###### Proof.
To define the eliminator, we must first use the algebra map on the more
general type $P^{\prime}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sum_{x:\mathsf{QW}}P\,x$. The algebra map, $\mathsf{qwrec}$,
requires that this type $P^{\prime}$ has an algebra structure
$\beta:S_{\mathrm{\Sigma}}(P^{\prime})\rightarrow P^{\prime}$, which is given
by:
$\beta\,(a,b)\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}(\mathsf{qwintro}(a,\pi_{1}\circ b),p\,a\,(\pi_{1}\circ
b)\,(\pi_{2}\circ b))$ (27)
Moreover, we must show that the recursor satisfies the equations by giving a
proof $s:\mathsf{Sat}_{\beta,\varepsilon}(P^{\prime})$, which we do pointwise
on the two elements of the dependent product $P^{\prime}$. Taking projections
distributes (possibly dependently) over $\mathbin{\gg\\!=}$; so to construct
$s$ it suffices that, given any $e:E$ and $\rho:V\,e\rightarrow P^{\prime}$,
we have the two terms:
$\begin{array}[]{rcrcl}\mathsf{qwequate}\,e\,\rho&:&l\,e\mathbin{\gg\\!=}(\pi_{1}\circ\rho)&=&r\,e\mathbin{\gg\\!=}(\pi_{1}\circ\rho)\\\
p_{\mathsf{resp}}\,e\,\rho&:&\mathsf{lift}\,p\,\rho\,(l\,e)&==&\mathsf{lift}\,p\,\rho\,(r\,e)\end{array}$
(28)
Then the eliminator is defined by taking the second projection of this
recursor.
$\mathsf{qwelim}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\pi_{2}\circ\mathsf{qwrec}\,P^{\prime}\,\beta\,s$ (29)
Given an element $(a,b):S_{\mathrm{\Sigma}}(\mathsf{QW})$, we can prove that
applying the eliminator to it computes as expected, that is
$\mathsf{qwelim}\,(\mathsf{qwintro}\,(a,b))=p\,a\,b\,(\mathsf{qwelim}\circ
b)$, as follows, where we write $r^{\prime}$ for
$\mathsf{qwrec}\,P^{\prime}\,\beta\,s$:
$\displaystyle\mathsf{qwelim}\,(\mathsf{qwintro}\,(a,b))$ $\displaystyle=={}$
$\displaystyle\pi_{2}\,(r^{\prime}(\mathsf{qwintro}\,(a,b)))$ def. of
$\mathsf{qwelim}$ $\displaystyle=={}$
$\displaystyle\pi_{2}\,(\beta\,(a,r^{\prime}\circ b))$ using
$\mathsf{qwrechom}$ against $\mathsf{qwintro}\,(a,b)$ $\displaystyle=={}$
$\displaystyle p\,a\,(\pi_{1}\circ r^{\prime}\circ b)\,(\pi_{2}\circ
r^{\prime}\circ b)$ def. of $\beta$ $\displaystyle=={}$ $\displaystyle
p\,a\,b\,(\pi_{2}\circ r^{\prime}\circ b)$ $\displaystyle=={}$ $\displaystyle
p\,a\,b\,(\mathsf{qwelim}\circ b)$ def. of $\mathsf{qwelim}$
### 3.3. QWI-types
In this section we consider _indexed_ quotient inductive types specified by
families of (possibly infinitary) equational theories. As for ordinary
W-types, the indexed version of QW-types gives convenient expressive power;
see Section 7. To describe the indexed case we first introduce some notation
for indexed families of types and functions.
Given a type $I:\mathcal{U}$, when we map between two $I$-indexed types, say
$A:\mathcal{U}^{I}$ to $B:\mathcal{U}^{I}$, we will write
$A\rightharpoondown B\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\prod_{k:I}A_{k}\rightarrow B_{k}$ (30)
for the function type. The composition of $f:A\rightharpoondown B$ and
$g:B\rightharpoondown C$ will just be written as
$g\circ f\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\lambda i.\lambda
x.\,g_{i}(f_{i}\,x)$ (31)
We also often need to define an indexed family of types
$B_{i}\,a:\mathcal{U}^{I}$ which is dependent on some $i:I$ and element
$a:A_{i}$ for $A:\mathcal{U}^{I}$. The type of such families will be written
as
$A\nrightharpoondown\mathcal{U}^{I}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\prod_{i:I}(A_{i}\rightarrow\mathcal{U}^{I})$ (32)
We take our notion of an indexed signature for a WI-type (indexed W-type) to
be a particular case of indexed containers [AAG05], namely an element of the
type
$\sum_{I:\mathcal{U}}\sum_{A:\mathcal{U}^{I}}\prod_{i:I}(A_{i}\rightarrow\mathcal{U}^{I})$,
which with the above notational conventions we write as
$\mathsf{Sig}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sum_{I:\mathcal{U}}\sum_{A:\mathcal{U}^{I}}(A\nrightharpoondown\mathcal{U}^{I})$
(33)
We only need indexed containers whose source and target indexing types are the
same, since we only need to consider endofunctors on the category of
$I$-indexed families in $\mathcal{U}$ and their algebras. Thus an indexed
signature $\mathrm{\Sigma}\equiv(I,A,B)$ is a triple that can be thought of as
a set of indices (or sorts) $I$, an $I$-indexed family of operators $A_{i}$,
and a function mapping each operator $a:A_{i}$ to an $I$-indexed family of
arities in $\mathcal{U}^{I}$. For instance, the type of vectors over some type
$D$ has an indexed signature as follows (see Section 7.2):
* •
natural numbers as indices;
* •
for index $0$ a parameter-less operator for the empty vector, and for index
$n+1$ an operator parametrised by $D$ for cons; and
* •
for the operator indexed by $0$, a family of empty types for arities, and for
operators indexed by $n+1$, a family of arities which is the unit type just in
the $n$th index, and empty otherwise.
Each indexed signature $\mathrm{\Sigma}\equiv(I,A,B)$ determines a polynomial
endofunctor $S_{\mathrm{\Sigma}}:\mathcal{U}^{I}\rightarrow\mathcal{U}^{I}$,
which is defined at a family $X:\mathcal{U}^{I}$ by
$(S_{\mathrm{\Sigma}}(X))_{i}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sum_{a:A_{i}}(B_{i}\,a\rightharpoondown X)\quad\text{(for each
$i:I$)}$ (34)
An $S_{\mathrm{\Sigma}}$-_algebra_ is by definition an element of the
dependent product
$\mathsf{Alg}_{\mathrm{\Sigma}}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sum_{X:\mathcal{U}^{I}}(S_{\mathrm{\Sigma}}(X)\rightharpoondown X)$
(35)
(For example, when $\mathrm{\Sigma}$ is the signature for vectors over some
type $D$, then an algebra $(X,\alpha):\mathsf{Alg}_{\mathrm{\Sigma}}$ amounts
to $\alpha_{0}:\mathbb{1}\rightarrow X_{0}$ and $\alpha_{n+1}:D\times
X_{n}\rightarrow X_{n+1}$.)
$S_{\mathrm{\Sigma}}$-algebra morphisms
$(X,\alpha)\rightarrow(X^{\prime},\alpha^{\prime})$ are given by an
$I$-indexed family of functions $h:X\rightharpoondown X^{\prime}$ together
with a family of elements in the types
$(\mathsf{isHom}\,h)_{i}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\prod_{a:A_{i}}\prod_{b:B_{i}\,a\rightharpoondown
X}(\alpha^{\prime}_{i}\,(a,h\circ b)=h_{i}(\alpha_{i}(a,b)))$ (36)
The WI-type
$\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{7.00009pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{5.00006pt}{}{$\mathrm{W}$}}}}}\slimits@_{i:I,a:A_{i}}B_{i}\,a$,
or
$\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{7.00009pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{5.00006pt}{}{$\mathrm{W}$}}}}}\slimits@_{\mathrm{\Sigma}}$,
determined by $\mathrm{\Sigma}\equiv(I,A,B)$ is the underlying type of an
initial $S_{\mathrm{\Sigma}}$-algebra. The elements of
$\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{7.00009pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{5.00006pt}{}{$\mathrm{W}$}}}}}\slimits@_{\mathrm{\Sigma}}$
represent an $I$-indexed family of _closed_ algebraic terms (i.e. well-founded
trees) over the signature $\mathrm{\Sigma}$. Given a family
$V:\mathcal{U}^{I}$, the family $T_{\mathrm{\Sigma}}(V):\mathcal{U}^{I}$ of
_open_ terms with variables from $V$ is the inductively defined family with
constructors
$\displaystyle\eta$ $\displaystyle:V\rightharpoondown T_{\mathrm{\Sigma}}\,V$
(37) $\displaystyle\sigma$
$\displaystyle:S_{\mathrm{\Sigma}}(T_{\mathrm{\Sigma}}\,V)\rightharpoondown
T_{\mathrm{\Sigma}}\,V$
Given an $S_{\mathrm{\Sigma}}$-algebra
$(X,\alpha):\mathsf{Alg}_{\mathrm{\Sigma}}$ and an $I$-indexed family of
functions $f:V\rightharpoondown X$, the unique morphism of
$S_{\mathrm{\Sigma}}$-algebras from the $S_{\mathrm{\Sigma}}$-algebra
$(T_{\mathrm{\Sigma}}(V),\sigma)$ to $(X,\alpha)$ that extends $f$ has an
underlying family of functions $T_{\mathrm{\Sigma}}(V)\rightharpoondown X$
mapping each $t:(T_{\mathrm{\Sigma}}\,V)_{i}$ to the element
$t\mathbin{\gg\\!=}f$ defined analogously to (14).
Given an indexed signature $\mathrm{\Sigma}\equiv(I,A,B):\mathsf{Sig}$, a
_system of equations_ for it is an element of the dependent product
$\mathsf{Syseq}_{\mathrm{\Sigma}}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sum_{E:\mathcal{U}^{I}}\sum_{V:E\nrightharpoondown\mathcal{U}^{I}}\bigg{(}\prod_{i:I}\prod_{e:E_{i}}(T_{\mathrm{\Sigma}}(V_{i}\,e))_{i}\bigg{)}\times\bigg{(}\prod_{i:I}\prod_{e:E_{i}}(T_{\mathrm{\Sigma}}(V_{i}\,e))_{i}\bigg{)}$
(38)
An $S_{\mathrm{\Sigma}}$-algebra
$\alpha:S_{\mathrm{\Sigma}}(X)\rightharpoondown X$ satisfies the system of
equations $\varepsilon\equiv(E,V,l,r):\mathsf{Syseq}_{\mathrm{\Sigma}}$ if for
each $i:I$ there is a proof of
$(\mathsf{Sat}_{\alpha,\varepsilon}(X))_{i}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\forall(e:E_{i}).\forall(\rho:V_{i}\,e\rightharpoondown
X).\;((l_{i}e)\mathbin{\gg\\!=}\rho)=((r_{i}e)\mathbin{\gg\\!=}\rho)$ (39)
###### Definition (QWI-type).
A _QWI-type_ for an indexed signature
$\mathrm{\Sigma}\equiv(I,A,B):\mathsf{Sig}$ and a system of equations
$\varepsilon\equiv(E,V,l,r):\mathsf{Syseq}_{\mathrm{\Sigma}}$ is given by an
$I$-indexed family of types $\mathsf{QW}:\mathcal{U}^{I}$ equipped with an
$S_{\mathrm{\Sigma}}$-algebra structure and a proof that it satisfies the
equations
$\displaystyle\mathsf{qwintro}$
$\displaystyle:S_{\mathrm{\Sigma}}(\mathsf{QW})\rightharpoondown\mathsf{QW}$
(40) $\displaystyle\mathsf{qwequate}$
$\displaystyle:\mathsf{Sat}_{\mathsf{qwintro},\varepsilon}(\mathsf{QW})$ (41)
together with functions that witness that it is an initial such algebra:
$\displaystyle\mathsf{qwrec}$
$\displaystyle:\prod_{X:\mathcal{U}^{I}}\prod_{\alpha:S_{\mathrm{\Sigma}}(X)\rightharpoondown
X}\left(\mathsf{Sat}_{\alpha,\varepsilon}(X)\rightarrow\left(\mathsf{QW}\rightharpoondown
X\right)\right)$ (42) $\displaystyle\mathsf{qwrechom}$
$\displaystyle:\prod_{X:\mathcal{U}^{I}}\prod_{\alpha:S_{\mathrm{\Sigma}}(X)\rightharpoondown
X}\prod_{p:\mathsf{Sat}_{\alpha,\varepsilon}(X)}\mathsf{isHom}\,(\mathsf{qwrec}\,X\,\alpha\,p)$
(43) $\displaystyle\mathsf{qwuniq}$
$\displaystyle:\prod_{X:\mathcal{U}^{I}}\prod_{\alpha:S_{\mathrm{\Sigma}}(X)\rightharpoondown
X}\prod_{p:\mathsf{Sat}_{\alpha,\varepsilon}(X)}\prod_{f:\mathsf{QW}\rightharpoondown
X}\mathsf{isHom}\,f\rightarrow\mathsf{qwrec}\,X\,\alpha\,p=f$ (44)
## 4\. Weakly initial sets of covers
In Section 3 we defined a QWI-type in a topos to be an initial algebra for a
given (possibly infinitary) indexed signature and system of equations (Section
3.3). If one interprets these notions in the topos of sets in classical
Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), one regains the
usual notion from universal algebra of initial algebras for equational
theories that are multi-sorted [BL70] and infinitary [Lin66]. Thus in ZFC, the
QWI-type for a signature $\mathrm{\Sigma}=(I,A,B)$ and system of equations
$\varepsilon=(E,V,l,r)$ can be constructed by first forming the $I$-indexed
family
$\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{7.00009pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{5.00006pt}{}{$\mathrm{W}$}}}}}\slimits@_{\mathrm{\Sigma}}$
of sets of well-founded trees over $\mathrm{\Sigma}$ and then quotienting by
the congruence relation $\sim_{\varepsilon}$ on
$\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{7.00009pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{5.00006pt}{}{$\mathrm{W}$}}}}}\slimits@_{\mathrm{\Sigma}}$
generated by $\varepsilon$. The $I$-indexed family of quotient sets
$(\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{7.00009pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{5.00006pt}{}{$\mathrm{W}$}}}}}\slimits@_{\mathrm{\Sigma}})/{\sim_{\varepsilon}}$
yields the desired initial algebra for $(\mathrm{\Sigma},\varepsilon)$
provided the $S_{\mathrm{\Sigma}}$-algebra structure on
$\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{7.00009pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{5.00006pt}{}{$\mathrm{W}$}}}}}\slimits@_{\mathrm{\Sigma}}$
induces one on the quotient sets. It does so, because for each operator in the
signature, using the Axiom of Choice (AC) one can pick representatives of the
(possibly infinitely many) equivalence classes that are the arguments of the
operator, apply the interpretation of the operator in
$\mathop{\vphantom{\sum}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{10.00012pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{7.00009pt}{}{$\mathrm{W}$}}}}{\vbox{\hbox{\leavevmode\resizebox{5.00006pt}{}{$\mathrm{W}$}}}}}\slimits@_{\mathrm{\Sigma}}$
and then take the equivalence class of that. So the topos of sets in ZFC has
QWI-types.
Is this use of AC really necessary? [Bla83, Section 9] shows that if one drops
AC and just works in ZF, then provided a certain large cardinal axiom is
consistent with ZFC, it is consistent with ZF that there is an infinitary
equational theory with no initial algebra. He shows this by first exhibiting a
countably presented equational theory whose initial algebra has to be an
uncountable regular cardinal; and secondly appealing to the construction of
[Git80] of a model of ZF with no uncountable regular cardinals (assuming a
certain large cardinal axiom). [LS19] turn the infinitary equational theory of
Blass into a higher-inductive type that cannot be proved to exist in ZF (and
hence cannot be constructed in type theory just using pushouts and the natural
numbers). We show in Section 7.2 that this higher inductive type can be
presented as a QWI-type.
So one cannot hope to construct QWI-types using a type theory which is
interpretable in just ZF. However, the type theory in which we work (Section
2) already requires going beyond ZF to be able to give a classical set-
theoretic interpretation of its universes (by assuming the existence of enough
strongly inaccessible cardinals, for example). So the above considerations
about non-existence of initial algebras for infinitary equational theories in
ZF do not necessarily rule out the construction of QWI-types in other toposes
with natural numbers object and universes. In Section 6, we show that QWI
types exist in a topos if it satisfies a weak form of choice principle that we
call IWISC and which is known to hold for a wide range of toposes. IWISC is a
version for universes in the type theory in which we are working of a property
introduced by [vdBM14], who work in constructive set theory CZF and call it
the _Axiom of Multiple Choice_ (building upon previous work by [MP02]); it is
related to the _Type Theoretic Collection Axiom_ TTCAf of [Str05] (see Section
4).
###### Definition (Indexed WISC).
A _family_ in a universe $\mathcal{U}$ is just a pair $(A,B)$ where
$A:\mathcal{U}$ and $B:A\rightarrow\mathcal{U}$. A family $(C,E)$ in
$\mathcal{U}$ is a _wisc_ for $Y:\mathcal{U}$ if for all $X:\mathcal{U}$ and
surjective functions $q:X\rightarrow Y$ in $\mathcal{U}$, there exists $c:C$
and $f:E\,c\rightarrow X$ for which $q\circ f:E\,c\rightarrow Y$ is
surjective. A family $(C,E)$ is a wisc for a family $(A,B)$ if it is a wisc
for each type $B\,a$ in the family. We say that _the universe $\mathcal{U}$
satisfies IWISC_ if for all families $(A,B)$ in $\mathcal{U}$, there exists a
family in $\mathcal{U}$ which is a wisc for $(A,B)$.
Following [https://ncatlab.org/nlab/show/WISC], “wisc” stands for “weakly
initial set of covers”, with the terminology being justified as follows. A
_cover_ of $Y:\mathcal{U}$ is just a surjective function $q:X\rightarrow Y$
for some $X:\mathcal{U}$. If $(C,E)$ is a wisc for $Y$ as in the above
definition, then the family of all covers of the form $E\,c\rightarrow Y$ with
$c:C$ is weakly initial, in the sense that for every cover $q:X\rightarrow Y$
in $\mathcal{U}$, there is a cover in the family that factors through $q$.
Note that in classical set theory AC implies IWISC; this is because the Axiom
of Choice implies that covers split and hence any family $(A,B)$ is a wisc for
itself. IWISC is independent of ZF in classical logic [Kar14, Rob15]. The
results of [vdBM14] imply that in constructive logic it is preserved by many
ways of making new toposes from existing ones, in particular by the formation
of sheaf toposes and realizability toposes over a given topos. Thus it holds
in all the toposes commonly used for the semantics of Type Theory. It is in
this sense that IWISC is a constructively acceptable form of the axiom of
choice. ([Rob15] gives examples of toposes which fail to satisfy it.)
###### Remark .
IWISC is a property of a single universe, in contrast to Streicher’s type
theoretic formulation of a WISC axiom, TTCAf [Str05], which uses two universes
(and which can be shown to imply IWISC). With a single universe it seems
necessary to quantify over indexed families in the universe (hence our choice
of terminology), rather than just over types in the universe as in _loc. cit._
; see [Lev21, Section 5.1]. Note that IWISC asks that a wisc merely exists for
each family in $\mathcal{U}$. A stronger requirement on $\mathcal{U}$ would be
to ask for a function assigning a wisc to each family; and this is equivalent
to asking for a function mapping each $B:\mathcal{U}$ to a wisc for $B$. (For
note that, given a family $(A,B)$, if we have a function mapping each $a:A$ to
a wisc $(C_{a},E_{a})$ for $B\,a$, then the single family $(C,E)$ with
$C=\sum_{a:A}C_{a}$ and $E(a,c)=E_{a}\,c$ is a wisc for all the $B\,a$
simultaneously.) [Lev21] calls this property Global WISC. Although the topos
of sets in ZFC with Grothendieck universes satisfies Global WISC, it is not
known which other toposes do.
The following lemma gives a convenient reformulation of the wisc property from
Section 4 that we use in the next section.
###### Lemma .
Suppose that $A:\mathcal{U}$ has a wisc
$(C:\mathcal{U},E:C\rightarrow\mathcal{U})$. If $B:A\rightarrow\mathcal{U}$
and $\phi:\prod_{x:A}(B\,x\rightarrow\mathsf{Prop})$ satisfy
$\forall(x:A).\exists(y:B\,x).\;\phi\,x\,y$, then there exist $c:C$, a
surjection $p:E\,c\rightarrow A$ and a function $q:\prod_{z:E\,c}B(p\,z)$
satisfying $\forall(z:E\,c).\;\phi\,(p\,z)\,(q\,z)$.
###### Proof.
Consider the type $\mathrm{\Phi}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sum_{x:A}\sum_{y:B\,x}\,\phi\,x\,y$ in $\mathcal{U}$. By assumption
$\pi_{1}:\mathrm{\Phi}\rightarrow A$ is a surjection, so since $(C,E)$ is a
wisc for $A$, there exist $c:C$ and $e:E\,c\rightarrow\mathrm{\Phi}$ with
$\pi_{1}\circ e$ a surjection. So we can take
$p\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\pi_{1}\circ e$ and
$q\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\pi_{1}\circ\pi_{2}\circ e$;
and then $\pi_{2}(\pi_{2}(e\,z))$ is a proof that $\phi\,(p\,z)\,(q\,z)$
holds. ∎
## 5\. Size
[Swa18] uses IWISC to construct _W-types with reductions_ , which are a simple
special case of QWI-types (see Section 7.2). We will see that in fact IWISC
implies the existence of all QWI-types, but using a different approach from
that of [Swa18]. We show in this section that, starting with a given signature
$\mathrm{\Sigma}$ and system of equations $\varepsilon$ (Section 3.2), using
IWISC one can construct a well-founded type of “sizes” with the property that
colimits of diagrams indexed by such sizes are preserved when taking exponents
by the arity types of $\mathrm{\Sigma}$ and $\varepsilon$. This will enable us
to construct the QW-type for $(\mathrm{\Sigma},\varepsilon)$ as a colimit in
Section 6. We consider the general case of indexed signatures in Section 5.1;
and the construction of QWI-types as a sized-indexed colimit is detailed in
the accompanying Agda formalization [FPS21]. Size is here playing a role in
the constructive type theory of toposes that in classical set theory would be
played by ordinal numbers. The next definition gives the properties of size
that we need.
###### Definition (Size).
A type $\mathsf{Size}$ in a universe $\mathcal{U}$ will be called a _type of
sizes_ if it comes equipped with
* •
a binary relation
$\\_<\\_:\mathsf{Size}\rightarrow\mathsf{Size}\rightarrow\mathsf{Prop}$ which
is transitive
$\forall i,j,k.\;i<j\rightarrow j<k\rightarrow i<k$ (45)
and well-founded
$\forall(\phi:\mathsf{Size}\rightarrow\mathsf{Prop}).(\forall i.(\forall
j.\;{j<i}\rightarrow\phi\,j)\rightarrow\phi\,i)\rightarrow\forall i.\;\phi\,i$
(46)
* •
a distinguished element $0^{s}$
* •
a binary operation
$\\_\sqcup^{s}\\_:\mathsf{Size}\rightarrow\mathsf{Size}\rightarrow\mathsf{Size}$
which gives upper bounds with respect to $<$ for pairs of elements
$\forall i,j.\;i<i\sqcup^{s}j\;\wedge\;j<i\sqcup^{s}j$ (47)
Note that defining ${i}^{+}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}i\sqcup^{s}i$ we get a form of successor operation satisfying
$\forall i.\;i<{i}^{+}$ (48)
For the examples of types of sizes given in Section 5 this successor operation
preserves $<$, but we do not need that property for the results in this
paper.444In fact we do not need the distinguished element $0^{s}$ either, but
it does no harm to require it and then sizes are directed with respect to $<$
in the usual sense of having upper bounds for all finite subsets, including
the empty one.
Such a type of sizes has many properties of the classic notion of limit
ordinal, except that we do not require the order to be total ($\forall
i,j.\;i<j\vee i=j\vee j<i$); that would be too strong in a constructive
setting and indeed does not hold in the examples below. Nor do we have any
need for an extensionality property ($\forall i,j.\;(\forall
k.\;k<i\leftrightarrow k<j)\rightarrow i=j$). In the precursor to this paper
[FPS20] we made use of Agda’s built in type $\mathsf{Size}$ and its version of
sized types [Abe12]. This shares only some of the properties of the above
definition. In particular, it has a “stationary” size $\infty$ satisfying
$\infty<\infty$ and hence the relation $<$ for Agda’s notion of size is not
well-founded.555Unfortunately in the current version of Agda (2.6.2) it is
possible to prove well-foundedness of $<$ for its built in type of sizes
$\mathsf{Size}$, and this leads immediately to logical unsoundness; see
github.com/agda/agda/issues/3026. For this reason we avoid the use of Agda’s
sized types in the current Agda development accompanying this paper.
###### Notation (Bounded quantification over sizes).
We write $\prod_{j<i}\,\\_$ for $\prod_{j:(\downarrow i)}\,\\_$, where
$\downarrow i\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\\{j:\mathsf{Size}\mid j<i\\}$ (49)
is the type of sizes below $i:\mathsf{Size}$. Similarly for other binders
involving sizes, such as $\forall{j<i}.\;\\_$ and
$\lambda\,{j<i}\mathbin{.}\\_$.
In the next section we need the fact that well-founded recursion (50) can be
reduced to well-founded induction (46) by defining the graph of the recursive
function and then appealing to unique choice (7) and function extensionality
(6):
###### Proposition (Well-founded recursion).
For any family of types $A:\mathsf{Size}\rightarrow\mathcal{U}$ and function
$\alpha:\prod_{i}\left(\left(\prod_{j<i}\,A\,j\right)\rightarrow A\,i\right)$,
there is a unique function $\mathop{\mathsf{rec}}A\,\alpha:\prod_{i}\,A\,i$
satisfying
$\forall
i.\;\mathop{\mathsf{rec}}A\,\alpha\,i=\alpha\,i\,(\lambda\,{j<i}\mathbin{.}\mathop{\mathsf{rec}}A\,\alpha\,j)$
(50)
###### Proof.
Let $R:\prod_{i}(A\,i\rightarrow\mathsf{Prop})$ be the least666Instead of the
impredicative definition of $R$ (and $<$ and $\leq$ in Section 5) as the least
relation closed under some rules, in our Agda development such relations are
constructed as inductively defined datatypes in the universe $\mathsf{Prop}$,
allowing one to use dependent pattern-matching to simplify proofs about them.
family of relations satisfying
$\textstyle\forall
i.\forall(f:\prod_{j<i}\,A\,j).\;(\forall{j<i}.\;R_{j}(f\,j))\rightarrow
R_{i}(\alpha\,i\,f)$ (51)
The fact that $R$ is least with this property allows one to prove $\forall
i.\exists!(x:A_{i}).\;R_{i}\,x$ by applying (46) with
$\phi\,i\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\exists!\,(x:A_{i}).\;R_{i}\,x$. So by unique choice (7) there is
$r:\prod_{i}\,A\,i$ satisfying $\forall i.\;R_{i}(r\,i)$ and hence by (51)
also satisfying $\forall
i.\;R_{i}(\alpha\,i\,(\lambda\,{j<i}\mathbin{.}r\,j))$. Since for each $i$
there is a unique $x$ with $R_{i}\,x$, it follows that
$r\,i=\alpha\,i\,(\lambda\,{j<i}\mathbin{.}r\,j)$. So we can take
$\mathop{\mathsf{rec}}A\,\alpha$ to be $r$. If $r^{\prime}:\prod_{i}\,A\,i$
also satisfies $\forall
i.\,r^{\prime}i=\alpha\,i\,(\lambda\,{j<i}\mathbin{.}r^{\prime}j)$, then
$\forall i.\,r^{\prime}i=r\,i$ follows from another application of (46), so
that $r^{\prime}=r$ by function extensionality (6). ∎
###### Example (Plump order).
Let $\mathsf{Size}:\mathcal{U}$ be the W-type determined by some type
$\mathit{Op}:\mathcal{U}$ and family
$\mathit{Ar}:\mathit{Op}\rightarrow\mathcal{U}$ (see Section 3.1). Thus
$\mathsf{Size}$ is inductively defined with constructor
$\sigma:(\sum_{a:\mathit{Op}}(\mathit{Ar}\,a\rightarrow\mathsf{Size}))\rightarrow\mathsf{Size}$.
The _plump_ ordering [Tay96] on $\mathsf{Size}$ is given by the least
relations
${\\_<\\_},{\\_\leq\\_}:\mathsf{Size}\rightarrow\mathsf{Size}\rightarrow\mathsf{Prop}$
satisfying for all $a:\mathit{Op}$, $b:\mathit{Ar}\,a\rightarrow\mathsf{Size}$
and $i:\mathsf{Size}$
$\displaystyle(\forall(x:\mathit{Ar}\,a).\;b\,x<i)\rightarrow\sigma\,a\,b\leq
i$ (52) $\displaystyle(\exists(x:\mathit{Ar}\,a).\;i\leq b\,x)\rightarrow
i<\sigma\,a\,b$ (53)
It is not hard to see that the relation $<$ is transitive (45) and well-
founded (46). Furthermore one can prove that $\leq$ is reflexive and hence
from (53) we get
$\forall(x:\mathit{Ar}\,a).\;b\,x<\sigma\,a\,b$ (54)
So for each operation $a:\mathit{Op}$, every family
$b:\mathit{Ar}\,a\rightarrow\mathsf{Size}$ of elements of $\mathsf{Size}$
indexed by its arity $\mathit{Ar}\,a$ has an upper bound with respect to $<$,
namely $\sigma\,a\,b$. In particular, if the signature
$(\mathit{Op},\mathit{Ar})$ contains an operation $a_{2}:\mathit{Op}$ whose
arity is $\mathit{Ar}\,a_{2}=\mathbb{2}$, then an upper bound for any
$i,j:\mathsf{Size}$ with respect to $<$ is given by
$i\sqcup^{s}j\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sigma\,a_{2}\,b_{i,j}$ (55)
with $b_{i,j}:\mathbb{2}\rightarrow\mathsf{Size}$ defined by $b_{i,j}\,0=i$
and $b_{i,j}\,1=j$. So (47) is satisfied in this case. Similarly, if the
signature $(\mathit{Op},\mathit{Ar})$ contains an operation
$a_{0}:\mathit{Op}$ whose arity is $\mathit{Ar}\,a_{0}=\mathbb{0}$, then
$\mathsf{Size}$ contains a distinguished element
$0^{s}\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\sigma\,a_{0}\,b$ (56)
with $b:\mathbb{0}\rightarrow\mathsf{Size}$ uniquely determined. Therefore, we
have:
###### Lemma .
If the signature
$\mathit{Op}:\mathcal{U},\mathit{Ar}:\mathit{Op}\rightarrow\mathcal{U}$
contains a nullary operation ($a_{0}:\mathit{Op}$ with
$\mathit{Ar}\,a_{0}=\mathbb{0}$) and a binary operation ($a_{2}:\mathit{Op}$
with $\mathit{Ar}\,a_{2}=\mathbb{2}$), then the corresponding W-type
$\mathsf{Size}:\mathcal{U}$, equipped with the plump order $<$, is a type of
sizes in the sense of Section 5. Furthermore for any $a:\mathit{Op}$, every
$(\mathit{Ar}\,a)$-indexed family of sizes
$b:\mathit{Ar}\,a\rightarrow\mathsf{Size}$ has an upper bound with respect to
$<$ (namely $\sigma\,a\,b$). ∎
Given a signature $\mathrm{\Sigma}$ in a universe $\mathcal{U}$ satisfying
IWISC, the lemma allows us to prove the existence of a type of sizes
$\mathsf{Size}$ with enough upper bounds to be able to prove a cocontinuity
property of the polynomial endofunctor associated with $\mathrm{\Sigma}$
(Section 5.1 below); we apply this in Section 6 to construct QW-types for a
given signature and system of equations (and QWI-types for indexed signatures
and systems of equations as defined in Section 3.3). The cocontinuity property
has to do with colimits of $\mathsf{Size}$-indexed diagrams in $\mathcal{U}$.
To state it, we first recall some semi-standard777They are only _semi_
-standard, because $<$ is not reflexive (indeed is irreflexive, because of
well-foundedness), so that $(\mathsf{Size},<)$ is only a (thin) _semi_
-category. category-theoretic notions to do with diagrams and colimits.
### 5.1. Colimits of size-indexed diagrams
By definition, given a type of sizes $\mathsf{Size}$ in a universe
$\mathcal{U}$ (Section 5), a _$\mathsf{Size}$ -indexed diagram_ is given by a
family of types $D:\mathsf{Size}\rightarrow\mathcal{U}$ equipped with
functions $\delta_{i,j}:D_{i}\rightarrow D_{j}$ for all $i,j:\mathsf{Size}$
with $i<j$, satisfying
$\forall(i,j,k:\mathsf{Size}).\;i<j<k\rightarrow\delta_{i,k}=\delta_{j,k}\circ\delta_{i,j}$
(57)
As usual, the _colimit_ of such a diagram is a cocone of functions in
$\mathcal{U}$
$(\nu_{D})_{i}:D_{i}\rightarrow\mathop{\mathsf{colim}}D\qquad\forall(i,j:\mathsf{Size}).\;i<j\rightarrow(\nu_{D})_{i}=(\nu_{D})_{j}\circ\delta_{i,j}$
(58)
with the universal property that for any other cocone
$f_{i}:D_{i}\rightarrow C\qquad\forall(i,j:\mathsf{Size}).\;i<j\rightarrow
f_{i}=f_{j}\circ\delta_{i,j}$
there is a unique function $f:\mathop{\mathsf{colim}}D\rightarrow C$
satisfying $\forall i.\;f_{i}=f\circ(\nu_{D})_{i}$.
Colimits can be constructed using quotient types: we define a binary relation
$\\_{\sim}\\_$ on $\sum_{i}\,D_{i}$ by:
$(i,x)\sim(j,y)\;\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\;\exists(k:\mathsf{Size}).\;{i<k}\;\wedge\;{j<k}\;\wedge\;{\delta_{i,k}\,x=\delta_{j,k}\,y}$
(59)
This is an equivalence relation, because $(\mathsf{Size},{<})$ has property
(47). Quotienting by it yields
$\textstyle\mathop{\mathsf{colim}}D\;\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\;\left(\sum_{i}D_{i}\right)/{\sim}$ (60)
with the universal cocone functions $(\nu_{D})_{i}$ given by mapping each
$x:D_{i}$ to the equivalence class $[i,x]_{\sim}$.
###### Definition (Preservation of colimits by taking a power).
Given a $\mathsf{Size}$-indexed diagram $(D,\delta)$ in $\mathcal{U}$ and a
type $X:\mathcal{U}$, we get a _power_ diagram $(D^{X},\delta^{X})$ in
$\mathcal{U}$ with
$\displaystyle(D^{X})_{i}$ $\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}X\rightarrow D_{i}$ $\displaystyle(i:\mathsf{Size})$ (61)
$\displaystyle(\delta^{X})_{i,j}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\lambda\,(f:X\rightarrow D_{i})\mathbin{.}\delta_{i,j}\circ f$
$\displaystyle(i,j:\mathsf{Size})$
Post-composition with $(\nu_{D})_{i}$ gives a cocone under the diagram $D^{X}$
with vertex $X\rightarrow\mathop{\mathsf{colim}}D$ and by universality this
induces a function
$\displaystyle\kappa_{D,X}:\mathop{\mathsf{colim}}(D^{X})\rightarrow(X\rightarrow\mathop{\mathsf{colim}}D)$
(62) $\displaystyle\kappa_{D,X}\,[i,f]_{\sim}=(\nu_{D})_{i}\circ f$
One says that _taking a power by $X:\mathcal{U}$ preserves
$\mathsf{Size}$-indexed colimits_ if for all $\mathsf{Size}$-indexed diagrams
$(D,\delta)$ in $\mathcal{U}$ the function $\kappa_{D,X}$ is an isomorphism.
###### Theorem (Cocontinuity).
Suppose $\mathcal{U}$ is a universe satisfying IWISC in a topos with natural
numbers object. Given $A:\mathcal{U}$ and $B:A\rightarrow\mathcal{U}$, there
exists a type of sizes $\mathsf{Size}:\mathcal{U}$ with the property that for
all $a:A$, taking a power by the type $B\,a:\mathcal{U}$ preserves
$\mathsf{Size}$-indexed colimits.
###### Proof.
The proof is in three parts, (1)–(3). In part (1) we show how to find a
suitable type of sizes $\mathsf{Size}$ using Section 5; so $\mathsf{Size}$ is
a W-type for a suitable signature derived from $A:\mathcal{U}$ and
$B:A\rightarrow\mathcal{U}$. The signature contains arities arising from a
wisc whose existence is guaranteed by IWISC. In fact we need to consider not
only the covers in such a wisc, but also a wisc for the family of (subtypes
of) covers in this wisc, for the same reason that Swan uses “2-cover bases”
[Swa18, Section 3.2.2]. Next, we have to prove that (62) is an isomorphism
when $X:\mathcal{U}$ is $B\,a$, for any $a:A$; and for this it suffices to
prove that it is (2) injective and (3) surjective. For then we can apply
unique choice (7) to construct a two-sided inverse for $\kappa_{D,X}$. By
definition of $\sim$ (59), $\kappa_{D,X}$ is injective iff
$\forall(i,j:\mathsf{Size}).\forall(f:X\rightarrow
D_{i}).\forall(f^{\prime}:X\rightarrow D_{j}).\;(\nu_{D})_{i}\circ
f=(\nu_{D})_{j}\circ f^{\prime}\rightarrow{}\\\
\exists(k:\mathsf{Size}).\;{i<k}\;\wedge\;{j<k}\;\wedge\;{\delta_{i,k}\circ
f=\delta_{j,k}\circ f^{\prime}}$ (63)
and surjective iff
$\forall(f:X\rightarrow\mathop{\mathsf{colim}}D).\exists(i:\mathsf{Size}).\exists(f^{\prime}:X\rightarrow
D_{i}).\;(\nu_{D})_{i}\circ f^{\prime}=f$ (64)
1. (1)
_Construction of $\mathsf{Size}$_: Given $A:\mathcal{U}$ and
$B:A\rightarrow\mathcal{U}$, using IWISC let $(C,F)$ be a wisc for the family
$(A,B)$. For each $c:C$, $a:A$ and function $f:F\,c\rightarrow B\,a$ we can
form the kernel of $f$
$\textstyle\mathsf{Ker}f\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sum_{x,x^{\prime}:F\,c}(f\,x=f\,x^{\prime})$ (65)
Applying IWISC again, let $(C^{\prime},F^{\prime})$ be a wisc for this family
of kernels indexed by $(c,a,f):\sum_{(c,a):C\times A}(F\,c\rightarrow B\,a)$.
Finally, consider the signature with
$\mathit{Op}=\mathbb{1}+\mathbb{1}+A+C+C^{\prime}$ and
$\mathit{Ar}:\mathit{Op}\rightarrow\mathcal{U}$ the function mapping
$0:\mathbb{1}$ in the first summand of $\mathit{Op}$ to $\mathbb{0}$,
$0:\mathbb{1}$ in the second summand to $\mathbb{2}$, $a:A$ in the third
summand to $B\,a$, $c:C$ in the fourth summand to $F\,c$ and
$c^{\prime}:C^{\prime}$ in the fifth summand to $F^{\prime}c^{\prime}$. Then
as in Section 5, the W-type for the signature $(\mathit{Op},\mathit{Ar})$ is a
type of sizes. Call it $\mathsf{Size}$.
2. (2)
_Proof of ( 63)_: Recall that $X:\mathcal{U}$ is of the form $B\,a$ for some
$a:A$. Suppose we have $f:X\rightarrow D_{i}$ and $f^{\prime}:X\rightarrow
D_{j}$ satisfying $(\nu_{D})_{i}\circ f=(\nu_{D})_{j}\circ f^{\prime}$. So by
definition of $\mathop{\mathsf{colim}}D$ (60) we have
$\forall(x:X).\exists(k:\mathsf{Size}).\;i<k\;\wedge\;j<k\;\wedge\;\delta_{i,k}(f\,x)=\delta_{j,k}(f^{\prime}\,x)$
(66)
Using the version of the wisc property of $(C,F)$ from Section 4, there is
$c:C$, $p:F\,c\rightarrow X$ and $s:F\,c\rightarrow\mathsf{Size}$ with $p$
surjective and satisfying
$\forall(z:F\,c).\;i<s\,z\;\wedge\;j<s\,z\;\wedge\;\delta_{i,s\,z}(f(p\,z))=\delta_{j,s\,z}(f^{\prime}(p\,z))$
(67)
Since $F\,c$ is one of the arities in the signature of the W-type
$\mathsf{Size}$, by Section 5 we have that $s:F\,c\rightarrow\mathsf{Size}$
has an upper bound, i.e. there is $k:\mathsf{Size}$ with
$\forall(z:F\,c).\;s\,z<k$; and by (47) we can assume further that $i<k$ and
$j<k$. Furthermore, from (67) and (57) we get
$\forall(z:F\,c).\;\delta_{i,k}(f(p\,z))=\delta_{j,k}(f^{\prime}(p\,z))$; but
$p$ is surjective, so $\delta_{i,k}\circ f=\delta_{j,k}\circ f^{\prime}$, as
required for (63).
3. (3)
_Proof of ( 64)_: Suppose we have $f:X\rightarrow\mathop{\mathsf{colim}}D$.
Since the quotient function
$[\\_]_{\sim}:D_{i}\rightarrow\mathop{\mathsf{colim}}D$ is surjective, using
Section 4 again, there is $c:C$, $p:F\,c\rightarrow X$,
$s:F\,c\rightarrow\mathsf{Size}$ and $g:\prod_{z:F\,c}D_{s\,z}$ with $p$
surjective and satisfying $\forall(z:F\,c).\;f(p\,z)=[s\,z,g\,z]_{\sim}$. As
before, we have that $s:F\,c\rightarrow\mathsf{Size}$ has an upper bound, i.e.
there is $j:\mathsf{Size}$ with $\forall(z:F\,c).\;s\,z<j$. So we get a
function $g^{\prime}:F\,c\rightarrow D_{j}$ by defining
$g^{\prime}\,z\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\delta_{s\,z,j}(g\,z)$ and hence
$f(p\,z)=[s\,z,gz]_{\sim}=[j,\delta_{s\,z,j}(gz)]_{\sim}=(\nu_{D})_{j}(g^{\prime}z)$
(68)
Let $Y\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\mathsf{Ker}\,p$ be the
kernel of $p$ as in (65). Then for any $(z,z^{\prime},\\_):Y$, since
$p\,z=p\,z^{\prime}$, we have
$(\nu_{D})_{j}(g^{\prime}z)=f(p\,z)=f(p\,z^{\prime})=(\nu_{D})_{j}(g^{\prime}z^{\prime})$
and hence $\exists
k.\;j<k\wedge{\delta_{j,k}(g^{\prime}z)=\delta_{j,k}(g^{\prime}z^{\prime})}$.
So we can apply Section 4 again to deduce the existence of
$c^{\prime}:C^{\prime}$, $\langle
p_{1}^{\prime},p_{2}^{\prime},\\_\rangle:F^{\prime}c^{\prime}\rightarrow Y$
and $s^{\prime}:F^{\prime}c^{\prime}\rightarrow\mathsf{Size}$ with $\langle
p_{1}^{\prime},p_{2}^{\prime}\rangle$ surjective and satisfying
$\forall(z^{\prime}:F^{\prime}c^{\prime}).\;{j<s^{\prime}z^{\prime}}\;\wedge\;\delta_{j,s^{\prime}z^{\prime}}(g^{\prime}(p_{1}^{\prime}\,z^{\prime}))=\delta_{j,s^{\prime}z^{\prime}}(g^{\prime}(p_{2}^{\prime}\,z^{\prime}))$
(69)
Since $F^{\prime}c^{\prime}$ is one of the arities in the signature of the
W-type $\mathsf{Size}$, by Section 5 we have that the family of sizes
$s^{\prime}:F^{\prime}c^{\prime}\rightarrow\mathsf{Size}$ has an upper bound,
$i$ say; and by (47) we can assume $j<i$. Let
$g^{\prime\prime}:F\,c\rightarrow D_{i}$ be $\delta_{j,i}\circ g^{\prime}$.
Thus from (68) we have
$f(p\,z)=(\nu_{D})_{i}(g^{\prime\prime}z)$ (70)
and from (69) also $g^{\prime\prime}\circ p_{1}^{\prime}=g^{\prime\prime}\circ
p_{2}^{\prime}$. So altogether we have functions
$\textstyle{{F\,c^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\langle
p_{1}^{\prime},p_{2}^{\prime}\rangle}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\textstyle{{F\,c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime\prime}}$$\textstyle{D_{i}}$
(71)
with $g^{\prime\prime}\circ\pi_{1}\circ\langle
p_{1}^{\prime},p_{2}^{\prime}\rangle=g^{\prime\prime}\circ
p_{1}^{\prime}=g^{\prime\prime}\circ
p_{2}^{\prime}=g^{\prime\prime}\circ\pi_{2}\circ\langle
p_{1}^{\prime},p_{2}^{\prime}\rangle$. Since $\langle
p_{1}^{\prime},p_{2}^{\prime}\rangle$ is surjective, this implies
$g^{\prime\prime}\circ\pi_{1}=g^{\prime\prime}\circ\pi_{2}$. But since
$Y=\mathsf{Ker}\,p$, $p$ is the coequalizer of $\pi_{1}$ and $\pi_{2}$ (since
as we noted in Section 2, toposes have effective epimorphisms); therefore
there is a unique function $f^{\prime}:X\rightarrow D_{i}$ satisfying
$f^{\prime}\circ p=g^{\prime\prime}$. Since $(\nu_{D})_{i}\circ
f^{\prime}\circ p=(\nu_{D})_{i}\circ g^{\prime\prime}=f\circ p$ by (70) and
$p$ is surjective, we have $(\nu_{D})_{i}\circ f^{\prime}=f$, completing the
proof of (64) and hence of Section 5.1. ∎
###### Remark .
Note that the type of sizes constructed in the proof of the theorem has the
property that for any $a:A$, $\mathsf{Size}$ has upper bounds (with respect to
$<$) for families of sizes indexed by $B\,a$ (because by construction $B\,a$
is one of the arities in the signature of the W-type $\mathsf{Size}$ and so
Section 5 applies). Such upper bounds are not needed in the above proof. We
included them because they are needed below in the proof of Section 6 when
proving property (89).
###### Definition (Preservation of colimits by polynomial endofunctors).
If $(D,\delta)$ is a $\mathsf{Size}$-indexed diagram in $\mathcal{U}$, then we
get another such, $(S_{\mathrm{\Sigma}}\circ
D,S_{\mathrm{\Sigma}}\circ\delta)$, by composing with the polynominal
endofunctor $S_{\mathrm{\Sigma}}:\mathcal{U}\rightarrow\mathcal{U}$ associated
with the signature $\mathrm{\Sigma}$ as in (9):
$\displaystyle(S_{\mathrm{\Sigma}}\circ D)_{i}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}S_{\mathrm{\Sigma}}(D_{i})$ $\displaystyle(i:\mathsf{Size})$ (72)
$\displaystyle(S_{\mathrm{\Sigma}}\circ\delta)_{i,j}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}S_{\mathrm{\Sigma}}(\delta_{i,j})$
$\displaystyle(i,j:\mathsf{Size})$
Applying $S_{\mathrm{\Sigma}}$ to (58) gives a cocone under
$(S_{\mathrm{\Sigma}}\circ D,S_{\mathrm{\Sigma}}\circ\delta)$ with vertex
$S_{\mathrm{\Sigma}}(\mathop{\mathsf{colim}}D)$ and this induces a function in
$\mathcal{U}$,
$\kappa_{D,\mathrm{\Sigma}}:\mathop{\mathsf{colim}}(S_{\mathrm{\Sigma}}\circ
D)\rightarrow S_{\mathrm{\Sigma}}(\mathop{\mathsf{colim}}D)$ (73)
One says that _the polynomial endofunctor $S_{\mathrm{\Sigma}}$ preserves
$\mathsf{Size}$-indexed colimits_ if $\kappa_{D,\mathrm{\Sigma}}$ is an
isomorphism for all diagrams $(D,\delta)$.
###### Corollary (Cocontinuity of $S_{\mathrm{\Sigma}}$).
In a topos with natural numbers object, given a signature
$\mathrm{\Sigma}=(A:\mathcal{U},B:A\rightarrow\mathcal{U})$ in a universe
$\mathcal{U}$ satisfying IWISC, there exists a type of sizes $\mathsf{Size}$
with the property that the associated polynominal endofunctor
$S_{\mathrm{\Sigma}}:\mathcal{U}\rightarrow\mathcal{U}$ preserves
$\mathsf{Size}$-indexed colimits.
###### Proof.
We apply Section 5.1 to find a type of sizes, $\mathsf{Size}$ for the given
$A:\mathcal{U}$ and $B:A\rightarrow\mathcal{U}$. So for each diagram
$(D,\delta)$ in $\mathcal{U}$, we have properties (63) and (64) when $X$ is
$B\,a$, for any $a:A$. The function (73) satisfies for all $i:\mathsf{Size}$,
$a:A$ and $b:B\,a\rightarrow D_{i}$
$\kappa_{D,\mathrm{\Sigma}}[i,(a,b)]_{\sim}=(a,(\nu_{D})_{i}\circ b)$ (74)
We have that $\kappa_{D,\mathrm{\Sigma}}$ is injective, because if
$(a,(\nu_{D})_{i}\circ b)=(a^{\prime},(\nu_{D})_{j}\circ b^{\prime})$, then
$a=a^{\prime}$ and $(\nu_{D})_{i}\circ b=(\nu_{D})_{j}\circ b^{\prime}$; but
then from (63) with $X=B\,a=B\,a^{\prime}$, there is some $k$ with $i,j<k$ and
$\delta_{i,k}\circ b=\delta_{j,k}\circ b^{\prime}$, so that $[i,(a,b)]_{\sim}$
and $[j,(a^{\prime},b^{\prime})]_{\sim}$ are equal terms of
$\mathop{\mathsf{colim}}(S_{\mathrm{\Sigma}}\circ D)$.
Furthermore, $\kappa_{D,\mathrm{\Sigma}}$ is surjective because given
$(a,b):S_{\mathrm{\Sigma}}(\mathop{\mathsf{colim}}D)$, from (64) with
$X=B\,a$, there exist $i:\mathsf{Size}$ and $b^{\prime}:B\,a\rightarrow D_{i}$
with $(\nu_{D})_{i}\circ b^{\prime}=b$; so $[i,(a,b^{\prime})]_{\sim}$ in
$\mathop{\mathsf{colim}}(S_{\mathrm{\Sigma}}\circ D)$ is mapped by
$\kappa_{D,\mathrm{\Sigma}}$ to $(a,b)$.
Since $\kappa_{D,\mathrm{\Sigma}}$ is both injective and surjective, we can
apply unique choice (7) to conclude that it is an isomorphism. ∎
###### Remark (Cocontinuity for polynomial endofunctors on
$\mathcal{U}^{I}$).
There are indexed versions of Section 5.1 and Section 5.1. To state them for
an indexing type $I:\mathcal{U}$, we need to consider $\mathsf{Size}$-indexed
diagrams in $\mathcal{U}^{I}$ and their colimits. Since the latter are given
pointwise by colimits in $\mathcal{U}$, this makes what follows a simple
extension of the previous development.
Given a $\mathsf{Size}$-indexed diagram $(D,\delta)$ in $\mathcal{U}^{I}$ and
a family $X:\mathcal{U}^{I}$, the indexed version of (61) is a _power_ diagram
$(D^{X},\delta^{X})$ in $\mathcal{U}$ with
$\displaystyle(D^{X})_{i}$ $\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}X\rightharpoondown D_{i}$ $\displaystyle(i:\mathsf{Size})$ (75)
$\displaystyle(\delta^{X})_{i,j}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\lambda\,(f:X\rightharpoondown D_{i})\mathbin{.}\delta_{i,j}\circ f$
$\displaystyle(i,j:\mathsf{Size})$
Post-composition with $(\nu_{D})_{i}$ gives a cocone under the diagram $D^{X}$
with vertex $X\rightharpoondown\mathop{\mathsf{colim}}D$ and this induces a
function which is the indexed version of (62):
$\displaystyle\kappa_{D,X}:\mathop{\mathsf{colim}}(D^{X})\rightharpoondown(X\rightharpoondown\mathop{\mathsf{colim}}D)$
(76) $\displaystyle\kappa_{D,X}\,[i,f]_{\sim}=(\nu_{D})_{i}\circ f$
One says that _taking a power by $X:\mathcal{U}^{I}$ preserves
$\mathsf{Size}$-indexed colimits_ if for all $\mathsf{Size}$-indexed diagrams
$(D,\delta)$ in $\mathcal{U}^{I}$ the function $\kappa_{D,X}$ is an
isomorphism. Then we have:
> _If $\mathcal{U}$ is a universe satisfying IWISC in a topos with natural
> numbers object, then given $A,I:\mathcal{U}$ and
> $B:A\rightarrow\mathcal{U}^{I}$, there exists a type of sizes
> $\mathsf{Size}:\mathcal{U}$ with the property that for all $a:A$, taking a
> power by a family $B\,a:\mathcal{U}^{I}$ preserves $\mathsf{Size}$-indexed
> colimits._
The proof of this is similar to the proof of Section 5.1 and can be found in
the Agda development accompanying this paper [FPS21].
Given a $\mathsf{Size}$-indexed diagram $(D,\delta)$ in $\mathcal{U}^{I}$, and
an $I$-indexed signature $\mathrm{\Sigma}$ as in (34), then we get another
$\mathsf{Size}$-indexed diagram, $(S_{\mathrm{\Sigma}}\circ
D,S_{\mathrm{\Sigma}}\circ\delta)$, by composing with the polynominal
endofunctor $S_{\mathrm{\Sigma}}:\mathcal{U}^{I}\rightarrow\mathcal{U}^{I}$:
$\displaystyle(S_{\mathrm{\Sigma}}\circ D)_{i}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}S_{\mathrm{\Sigma}}(D_{i})$ $\displaystyle(i:\mathsf{Size})$ (77)
$\displaystyle(S_{\mathrm{\Sigma}}\circ\delta)_{i,j}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}S_{\mathrm{\Sigma}}(\delta_{i,j})$
$\displaystyle(i,j:\mathsf{Size})$
Applying $S_{\mathrm{\Sigma}}$ to the $I$-indexed version of (58) gives a
cocone under $(S_{\mathrm{\Sigma}}\circ D,S_{\mathrm{\Sigma}}\circ\delta)$
with vertex $S_{\mathrm{\Sigma}}(\mathop{\mathsf{colim}}D)$ and this induces a
family of functions in $\mathcal{U}^{I}$
$\kappa_{D,\mathrm{\Sigma}}:\mathop{\mathsf{colim}}(S_{\mathrm{\Sigma}}\circ
D)\rightharpoondown S_{\mathrm{\Sigma}}(\mathop{\mathsf{colim}}D)$ (78)
One says that _the polynomial endofunctor
$S_{\mathrm{\Sigma}}:\mathcal{U}^{I}\rightarrow\mathcal{U}^{I}$ preserves
$\mathsf{Size}$-indexed colimits_ if $\kappa_{D,\mathrm{\Sigma}}$ is a family
of isomorphisms, for all diagrams $(D,\delta)$. Then the indexed
generalization of Section 5.1 is:
> _In a topos with natural numbers object, given an indexed signature
> $\mathrm{\Sigma}=(I,A,B)$ in a universe $\mathcal{U}$ satisfying IWISC,
> there exists a type of sizes $\mathsf{Size}$ with the property that the
> associated polynominal endofunctor
> $S_{\mathrm{\Sigma}}:\mathcal{U}^{I}\rightarrow\mathcal{U}^{I}$ preserves
> $\mathsf{Size}$-indexed colimits._
This is proved as a corollary of the indexed version of Section 5.1 given
above, and can also be found in the accompanying Agda development [FPS21].
## 6\. Construction of QWI-types
We aim to prove the following theorem about existence of QWI-types (Section
3.3):
###### Theorem .
Suppose $\mathcal{U}$ is a universe satisfying IWISC in a topos with natural
numbers object. Then for every indexed signature and system of equations in
$\mathcal{U}$, there exists a QWI-type for it.
The proof follows from the cocontinuity results of the previous section (the
indexed versions of Section 5.1 and Section 5.1 given in Section 5.1). For
simplicity, here we only give the proof for QW-types (Section 3.2), that is,
for the non-indexed $I=\mathbb{1}$ case of signatures. The general case is
similar, but notationally more involved since there are indexes ranging both
over an index type and over sizes. The proof for the general, indexed case can
be found in our Agda development [FPS21].
_So in this section we fix a signature
$\mathrm{\Sigma}=(A:\mathcal{U},B:A\rightarrow\mathcal{U})$ and system of
equations
$\varepsilon=(E:\mathcal{U},V:E\rightarrow\mathcal{U},l,r:\prod_{e:E}T_{\mathrm{\Sigma}}(V\,e))$
over it in some universe $\mathcal{U}$ satisfying IWISC in a topos with
natural numbers object._
### 6.1. Motivating the construction
We noted at the start of Section 4 that QW-types can be constructed in the
category of ZFC sets as initial algebras for possibly infinitary equational
theories by first forming the W-type of terms of the theory and then
quotienting that by the congruence relation generated by the equations of the
theory. AC is used when constructing the algebra structure of the quotient,
because the signature’s arities may be infinite. To avoid this use of AC,
instead of forming all terms in one go and then quotienting, we consider
interleaving quotient formation with the construction of terms of free
algebras for equational systems (cf. the categorical construction by [FH09]).
$\begin{array}[]{l}\mathtt{mutual}\\\
\quad\mathtt{data}\;W:\mathcal{U}\;\mathtt{where}\\\ \quad\quad
q\tau:T_{\mathrm{\Sigma}}(W/{\sim})\rightarrow W\\\
\quad\mathtt{data}\;\\_\sim\\_:W\rightarrow
W\rightarrow\mathsf{Prop}\;\mathtt{where}\\\ \quad\quad
q\varepsilon:\forall(e:E).\forall(\rho:V\;e\rightarrow
W/{\sim}).\;q\tau(T_{\mathrm{\Sigma}}\,\rho\,(l\,e))\sim
q\tau(T_{\mathrm{\Sigma}}\,\rho\,(r\,e))\\\ \quad\quad
q\eta:\forall(t:T_{\mathrm{\Sigma}}(W/{\sim})).\;q\tau(\eta\,[q\tau\,t]_{\sim})\sim
q\tau\,t\\\ \quad\quad q\sigma:\forall(a:A).\forall(b:B\,a\rightarrow
T_{\mathrm{\Sigma}}(W/{\sim})).\;q\tau(\sigma(a,b))\sim
q\tau(\sigma(a,\eta\circ[\\_]_{\sim}\circ q\tau\circ b))\\\
\mathsf{QW}=W/{\sim}\end{array}$ Figure 2. First attempt at constructing QW-
types
Figure 2 gives the idea, using the constructors $\eta$ and $\sigma$ from (13)
and Agda-like notation for inductive definitions. We would like to construct
the QW-type for $(\mathrm{\Sigma},\varepsilon)$ as a quotient
$\mathsf{QW}\mathrel{\smash{\overset{\text{\tiny def}}{=}}}W/{\sim}$, but now
the type $W:\mathcal{U}$ and the relation $\\_\sim\\_:W\rightarrow
W\rightarrow\mathsf{Prop}$ are mutually inductively defined, with constructors
as indicated in the figure. Note that whereas the construction in ZFC uses AC
to get an $S_{\mathrm{\Sigma}}$-algebra structure for $\mathsf{QW}$, here we
get one trivially from the constructor
$q\tau:T_{\mathrm{\Sigma}}(W/{\sim})\rightarrow W$:
$S_{\mathrm{\Sigma}}(\mathsf{QW})\equiv
S_{\mathrm{\Sigma}}(W/{\sim})\xrightarrow{S_{\mathrm{\Sigma}}\,\eta}S_{\mathrm{\Sigma}}(T_{\mathrm{\Sigma}}(W/{\sim}))\xrightarrow{\sigma}T_{\mathrm{\Sigma}}(W/{\sim})\xrightarrow{q\tau}W\xrightarrow{[\\_]_{\sim}}W/{\sim}\equiv\mathsf{QW}$
(79)
Furthermore the property $q\varepsilon$ of $\sim$ in Figure 2 ensures that the
$S_{\mathrm{\Sigma}}$-algebra $W/{\sim}$ satisfies the equational system
$\varepsilon$. The use of $T_{\mathrm{\Sigma}}$ rather than
$S_{\mathrm{\Sigma}}$ in the domain of $q\tau$ seems necessary for this method
of construction to go through (once we have fixed up the problems mentioned in
the next paragraph); but it does mean that as well as $q\varepsilon$, we have
to impose the conditions $q\eta$ and $q\sigma$ to ensure that $W/{\sim}$ has a
$S_{\mathrm{\Sigma}}$-algebra structure, or equivalently, an algebra structure
for the monad $T_{\mathrm{\Sigma}}$.
Note that the domain of the constructor $q\tau$ combines
$T_{\mathrm{\Sigma}}(\\_)$ with $\\_/\\_$. While the first is unproblematic
for inductive definitions, the second is not: if one thinks of the semantics
of inductively defined types in terms of initial algebras of endofunctors, it
is not clear what endofunctor (in some class known to have initial algebras)
is involved here, given that both arguments to $\\_/\\_$ are being defined
simultaneously. Agda uses a notion of “strict positivity” as a conservative
approximation for such a class of functors; and one can instruct Agda to
regard quotienting as a strictly positive operation through the use of its
POLARITY declarations. If one does so, then a definition like the one in
Figure 2 is accepted by Agda. The semantic justification for regarding
quotients as strictly positive constructions needs further investigation. We
avoid the need for that here and replace the attempt to define $W$ and $\sim$
inductively by a size-indexed version that uses definition by well-founded
recursion over a type of sizes as in the previous section. This also avoids
another difficulty with Figure 2: even if one can define $W$ and $\sim$
inductively, one still has to verify that $W/{\sim}$ has the universal
property (20)–(22) required of a QW-type. In particular, there is an obvious
recursive definition of $\mathsf{qwrec}$ following the shape of the inductive
definition in Figure 2, but it is not at all clear why this recursive
definition is terminating (i.e. gives a well-defined, total function). Well-
founded recursion over sizes will solve this problem as well.
### 6.2. QW-type via sizes
We are given a signature
$\mathrm{\Sigma}=(A:\mathcal{U},B:A\rightarrow\mathcal{U})$ and system of
equations
$\varepsilon=(E:\mathcal{U},V:E\rightarrow\mathcal{U},l,r:\prod_{e:E}T_{\mathrm{\Sigma}}(V\,e))$.
Let $\mathsf{Size}:\mathcal{U}$ be the type of sizes whose existence is
guaranteed by Section 5.1 when in the theorem we take $A$ to be
$AE\mathrel{\smash{\overset{\text{\tiny def}}{=}}}A+E$ and $B$ to be the
function $BV:AE\rightarrow\mathcal{U}$ mapping $a:A$ to $B\,a$ and $e:E$ to
$V\,e$. It follows (as in the proof of Section 5.1) that $S_{\mathrm{\Sigma}}$
preserves $\mathsf{Size}$-indexed colimits.
###### Definition .
A _$\mathsf{Size}$ -indexed $\mathrm{\Sigma}$-structure_ in $\mathcal{U}$ is
specified by a family of types $D:\mathsf{Size}\rightarrow\mathcal{U}$
equipped with functions
$\tau_{j,i}:T_{\mathrm{\Sigma}}(D_{j})\rightarrow D_{i}\quad\text{for all
$i,j:\mathsf{Size}$ with $j<i$}$ (80)
Similarly, for each $i:\mathsf{Size}$, a _$(\downarrow i)$ -indexed
$\mathrm{\Sigma}$-structure_ is the same thing, except with $D$ only defined
on the subsemicategory $\downarrow i$ (49) rather than the whole of
$\mathsf{Size}$. Clearly, given a $\mathsf{Size}$-indexed
$\mathrm{\Sigma}$-structure $(D,\tau)$, for each $i:\mathsf{Size}$ we get a
$(\downarrow i)$-indexed one $(D\;(\downarrow i),\tau\;(\downarrow i))$ by
restriction.
If $i:\mathsf{Size}$ and $(D,\tau)$ is a $(\downarrow i)$-indexed
$\mathrm{\Sigma}$-structure, let $\Diamond_{i}D:\mathcal{U}$ be the quotient
type (see Figure 1)
$\Diamond_{i}D\;\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\;\left.\left(\sum_{j<i}T_{\mathrm{\Sigma}}\,D_{j}\right)\right/R_{i}$
(81)
with the relation
$R_{i}:(\sum_{j<i}T_{\mathrm{\Sigma}}\,D_{j})\rightarrow(\sum_{j<i}T_{\mathrm{\Sigma}}\,D_{j})\rightarrow\mathsf{Prop}$
defined by:
$R_{i}\,(j,t)\,(k,t^{\prime})\;\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\;{}\\\ \begin{aligned}
&&&({k=j}\;\wedge\;\exists(e:E).\exists(\rho:V\,e\rightarrow
D_{j}).\;{t=T_{\mathrm{\Sigma}}\,\rho\,(l\,e)}\;\wedge\;{t^{\prime}=T_{\mathrm{\Sigma}}\,\rho\,(r\,e)})\\\
&\vee&&({k<j}\;\wedge\;t=\eta(\tau_{k,j}\,t^{\prime}))\\\
&\vee&&({k<j}\;\wedge\;\exists(a:A).\exists(b:B\,a\rightarrow
T_{\mathrm{\Sigma}}D_{k}).\;{t=\sigma(a,b)}\;\wedge\;{t^{\prime}=\sigma(a,\eta\circ\tau_{k,j}\circ
b)})\end{aligned}$ (82)
(The three clauses in the above definition correspond to the three
constructors $q\varepsilon$, $q\eta$ and $q\sigma$ in Figure 2.) We will see
that the QW-type for $(\mathrm{\Sigma},\varepsilon)$ can be constructed from
$\mathsf{Size}$-indexed $\mathrm{\Sigma}$-structures $(D,\tau)$ satisfying the
following fixed-point property.
###### Definition .
A $\mathsf{Size}$-indexed $\mathrm{\Sigma}$-structure $(D,\tau)$ is a
_$\Diamond$ -fixed point_ if for all $i:\mathsf{Size}$
$D_{i}=\Diamond_{i}(D\;(\downarrow i))$ (83)
and for all $j<i$ and $t:T_{\mathrm{\Sigma}}D_{j}$
$\tau_{j,i}\,t=[j,t]_{R_{i}}$ (84)
Suppose that $(D,\tau)$ is a $\Diamond$-fixed point. Because of (83) and (84),
for $i,j:\mathsf{Size}$ with $i<j$ the functions
$\delta_{i,j}:D_{i}\rightarrow
D_{j}\qquad\delta_{i,j}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\tau_{i,j}\circ\eta$ (85)
satisfy
$\delta_{i,j}([k,t]_{R_{i}})=[k,t]_{R_{j}}\quad\text{(for all $k<i$ and
$t:T_{\mathrm{\Sigma}}\,D_{k}$)}$ (86)
In particular they satisfy composition (57) and so $(D,\delta)$ is a
$\mathsf{Size}$-indexed diagram in $\mathcal{U}$ whose colimit we can form as
in Section 5.1. We prove that this colimit
$\mathsf{QW}\;\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\;\mathop{\mathsf{colim}}D$ (87)
has the structure (18)–(22) of a QW-type for the signature
$(\mathrm{\Sigma},\varepsilon)$.
$\mathsf{qwintro}$:
Given how we chose $\mathsf{Size}$ at the start of this section, from the
(proof of) Section 5.1 we have that
$S_{\mathrm{\Sigma}}:\mathcal{U}\rightarrow\mathcal{U}$ preserves
$\mathsf{Size}$-indexed colimits. We claim that the functions
$S_{\mathrm{\Sigma}}(D_{i})\mathbin{\smash{\xrightarrow{S_{\mathrm{\Sigma}}\eta}}}S_{\mathrm{\Sigma}}(T_{\mathrm{\Sigma}}\,D_{i})\mathbin{\smash{\xrightarrow{\sigma}}}T_{\mathrm{\Sigma}}\,D_{i}\mathbin{\smash{\xrightarrow{\tau_{i,{i}^{+}}}}}D_{{i}^{+}}\mathbin{\smash{\xrightarrow{(\nu_{D})_{{i}^{+}}}}}\mathop{\mathsf{colim}}D$
(88)
form a cocone under the diagram $S_{\mathrm{\Sigma}}\circ D$ and hence induce
a function $\mathop{\mathsf{colim}}(S_{\mathrm{\Sigma}}\circ
D)\rightarrow\mathop{\mathsf{colim}}D$; then we obtain
$\mathsf{qwintro}:S_{\mathrm{\Sigma}}(\mathop{\mathsf{colim}}D)\rightarrow\mathop{\mathsf{colim}}D$
by composing this with the isomorphism
$S_{\mathrm{\Sigma}}(\mathop{\mathsf{colim}}D)\cong\mathop{\mathsf{colim}}(S_{\mathrm{\Sigma}}\circ
D)$ from Section 5.1. That the functions in (88) form a cocone for
$S_{\mathrm{\Sigma}}\circ D$ follows from the fact that the functions in (85)
satisfy for all sizes $i<j$
$\forall(t:T_{\mathrm{\Sigma}}(D_{i})).\exists(k:\mathsf{Size}).\;j<k\;\wedge\;\tau_{j,k}(T_{\mathrm{\Sigma}}\,\delta_{i,j}(t))=\tau_{i,k}(t)$
(89)
from which it follows that $(\nu_{D})_{{j}^{+}}\circ\tau_{j,{j}^{+}}\circ
T_{\mathrm{\Sigma}}\,\delta_{i,j}=(\nu_{D})_{{i}^{+}}\circ\tau_{i,{i}^{+}}$
and hence also the cocone property. Property (89) can be proved by induction
on the structure of $t:T_{\mathrm{\Sigma}}(D_{i})$, with the $t=\sigma(a,b)$
case of (13) proved using the fact that $\mathsf{Size}$ has $<$-upper bounds
for families of sizes indexed by $B\,a$ (Section 5.1).
$\mathsf{qwequate}$:
Given $e:E$ and $\rho:V\,e\rightarrow\mathop{\mathsf{colim}}D$, by Section 5.1
there exist $i:\mathsf{Size}$ and $\rho^{\prime}:V\,e\rightarrow D_{i}$ with
$\rho=(\nu_{D})_{i}\circ\rho^{\prime}$. By standard properties of the bind
operation $\mathbin{\gg\\!=}$ (14) (which depends implicitly on the
$S_{\mathrm{\Sigma}}$-algebra structure $\mathsf{qwintro}$ that we have just
constructed), we have
$\begin{array}[]{@{}r@{}c@{}r@{}c@{}l@{}r@{}}{}(l\,e\mathbin{\gg\\!=}\rho)&{}={}&(l\,e\mathbin{\gg\\!=}(\nu_{D})_{i}\circ\rho^{\prime})&{}={}&(T_{\mathrm{\Sigma}}\,\rho^{\prime}&(l\,e)\mathbin{\gg\\!=}(\nu_{D})_{i})\\\
(r\,e\mathbin{\gg\\!=}\rho)&{}={}&(r\,e\mathbin{\gg\\!=}(\nu_{D})_{i}\circ\rho^{\prime})&{}={}&(T_{\mathrm{\Sigma}}\,\rho^{\prime}&(r\,e)\mathbin{\gg\\!=}(\nu_{D})_{i})\end{array}$
(90)
From the definition of $\mathsf{qwintro}$ and (84) it follows that for any
$t:T_{\mathrm{\Sigma}}\,D_{i}$, there is a proof of
$(t\mathbin{\gg\\!=}(\nu_{D})_{i})=(\nu_{D})_{{i}^{+}}(\tau_{i,{i}^{+}}\,t)$
(91)
So from above we have that
$(l\,e\mathbin{\gg\\!=}\rho)\;=\;(\nu_{D})_{{i}^{+}}(\tau_{i,{i}^{+}}(T_{\mathrm{\Sigma}}\rho^{\prime}(l\,e)))$
and
$(r\,e\mathbin{\gg\\!=}\rho)\;=\;(\nu_{D})_{{i}^{+}}(\tau_{i,{i}^{+}}(T_{\mathrm{\Sigma}}\rho^{\prime}(r\,e)))$.
Since from (84) and the first clause in the definition of $R_{i}$ (82) we also
have
$\tau_{i,{i}^{+}}(T_{\mathrm{\Sigma}}\rho^{\prime}(l\,e))=\tau_{i,{i}^{+}}(T_{\mathrm{\Sigma}}\rho^{\prime}(r\,e))$,
it follows that there is a proof of
$(l\,e\mathbin{\gg\\!=}\rho)=(r\,e\mathbin{\gg\\!=}\rho)$.
$\mathsf{qwrec}$:
Given an $S_{\mathrm{\Sigma}}$-algebra
$(X,\alpha):\sum_{X:\mathcal{U}}(S_{\mathrm{\Sigma}}\,X\rightarrow X)$
satisfying the system of equations $\varepsilon$, the function
$\mathsf{qwrec}:\mathop{\mathsf{colim}}D\rightarrow X$ is induced by a cocone
of functions $r:\prod_{i}(D_{i}\rightarrow X)$ under the diagram $D$, defined
by well-founded recursion (Section 5). More precisely, a strengthened
“recursion hypothesis” is needed: instead of $\prod_{i}(D_{i}\rightarrow X)$
we use $\prod_{i}\,F_{i}$ where
$F_{i}\;\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\;\\{f:D_{i}\rightarrow
X\mid\forall
j<i.\forall(t:T_{\mathrm{\Sigma}}\,D_{j}).\;(t\mathbin{\gg\\!=}(f\circ\delta_{j,i}))=f([j,t]_{R_{i}})\\}$
(92)
(The definition uses $\alpha$ implicitly in $\mathbin{\gg\\!=}$ and relies on
the fact (83) that $D_{i}=\Diamond_{i}(D\;(\downarrow i))$.) For each
$i:\mathsf{Size}$, if we have $r_{j}:F_{j}$ for all $j<i$, then we get a
function $r_{i}:F_{i}$ well-defined by
$r_{i}([j,t]_{R_{i}})\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}t\mathbin{\gg\\!=}r_{j}\quad\text{for all $j<i$ and
$t:T_{\mathrm{\Sigma}}\,D_{j}$}$ (93)
(The defining property of $F_{i}$ is needed to see that the right-hand side of
this definition respects the relation $R_{i}$.) Hence by well-founded
recursion (Section 5) we get an element $r:\prod_{i}\,F_{i}$. One can prove
$\forall i.\forall j<i.\;r_{j}=r_{i}\circ\delta_{j,i}$ by well-founded
induction (46); so $r$ is a cocone and induces a function
$\mathsf{qwrec}:\mathop{\mathsf{colim}}D\rightarrow X$.
$\mathsf{qwrechom}$:
We noted at the start of this section that by choice of $\mathsf{Size}$, the
functor $S_{\mathrm{\Sigma}}$ preserves $\mathsf{Size}$-indexed colimits. So
to prove that
$\textstyle{{S_{\mathrm{\Sigma}}(\mathop{\mathsf{colim}}D)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{qwintro}}$$\scriptstyle{S_{\mathrm{\Sigma}}\mathsf{qwrec}}$$\textstyle{{\mathop{\mathsf{colim}}D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{qwrec}}$$\textstyle{{S_{\mathrm{\Sigma}}\,X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{X}$
(94)
commutes, by the definitions of $\mathsf{qwintro}$ and $\mathsf{qwrec}$, it
suffices to prove that
$\textstyle{{S_{\mathrm{\Sigma}}\,D_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{i,{i}^{+}}\circ\sigma\circ
S_{\mathrm{\Sigma}}\eta}$$\scriptstyle{S_{\mathrm{\Sigma}}\,r_{i}}$$\textstyle{{D_{{i}^{+}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{{i}^{+}}}$$\textstyle{{S_{\mathrm{\Sigma}}\,X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{X}$
(95)
does for each $i:\mathsf{Size}$. But each $(a,b):S_{\mathrm{\Sigma}}\,D_{i}$
is mapped by $\tau_{i,{i}^{+}}\circ\sigma\circ S_{\mathrm{\Sigma}}\eta$ to
$[i,\sigma(a,\eta\circ b)]_{R_{{i}^{+}}}$ (using the fact that
$D_{{i}^{+}}=\Diamond_{{i}^{+}}(D\;(\downarrow{i}^{+}))$); and by (93), that
is mapped by $r_{{i}^{+}}$ to $\sigma(a,\eta\circ b)\mathbin{\gg\\!=}r_{i}$,
which is indeed equal to $\alpha(S_{\mathrm{\Sigma}}\,r_{i}(a,b))$ by
definition of $\mathbin{\gg\\!=}$ (14) and the action of $S_{\mathrm{\Sigma}}$
on functions (10).
$\mathsf{qwuniq}$:
If $h:\mathop{\mathsf{colim}}D\rightarrow X$ is a morphism of
$S_{\mathrm{\Sigma}}$-algebras, then one can prove by well-founded induction
for $<$ that $\forall i.\;h\circ(\nu_{D})_{i}=r_{i}$ holds: for if we have
$h\circ(\nu_{D})_{j}=r_{j}$ for all $j<i$, then for any $[j,t]_{R_{i}}$ in
$D_{i}=\Diamond_{i}(D\;(\downarrow i))$
$\begin{array}[]{r@{}c@{}l@{\hspace{2em}}l}{}r_{i}([j,t]_{R_{i}})&{}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}{}&t\mathbin{\gg\\!=}r_{j}\hfil\hskip
20.00003pt&\text{\eqref{eq:qwrec}}\\\ &=&t\mathbin{\gg\\!=}(h\circ(\nu
D)_{j})\hfil\hskip 20.00003pt&\text{by induction hypothesis}\\\
&=&h(t\mathbin{\gg\\!=}(\nu_{D})_{j})\hfil\hskip 20.00003pt&\text{since $h$ is
a morphism of $S_{\mathrm{\Sigma}}$-algebras}\\\
&=&h((\nu_{D})_{{j}^{+}}[j,t]_{R_{{j}^{+}}})\hfil\hskip
20.00003pt&\text{by\leavevmode\nobreak\ \eqref{eq:qwequate}
and\leavevmode\nobreak\ \eqref{eq:fixed2}}\\\
&=&h((\nu_{D})_{i}[j,t]_{R_{i}})\hfil\hskip
20.00003pt&\parbox[c]{186.45341pt}{\raggedright{}since by\leavevmode\nobreak\
\eqref{eq:fixed-diag},
$\delta_{{j}^{+},{j}^{+}\sqcup^{s}i}([j,t]_{R_{{j}^{+}}})=[j,t]_{R_{{j}^{+}\sqcup^{s}i}}=\delta_{i,{j}^{+}\sqcup^{s}i}([j,t]_{R_{i}})$.\@add@raggedright}\end{array}$
So by well-founded induction, $h\circ(\nu_{D})_{i}=r_{i}$ holds for all
$i:\mathsf{Size}$, and hence by definition of $\mathsf{qwrec}$ and the
uniqueness part of the universal property of colimits we have
$h=\mathsf{qwrec}$.
Thus we have proved:
###### Proposition .
Let $\mathsf{Size}$ be the type of sizes defined at the start of this section,
whose existence is guaranteed by Section 5.1. If the $\mathsf{Size}$-indexed
$\mathrm{\Sigma}$-structure $(D,\tau)$ is a $\Diamond$-fixed point, then
$\mathop{\mathsf{colim}}D$ has the structure of a QW-type for the signature
$(\mathrm{\Sigma},\varepsilon)$. ∎
Now we can complete the proof of the main theorem:
###### Proof of non-indexed version of Section 6.
In view of Section 6.2, it suffices to construct a $\mathsf{Size}$-indexed
$\mathrm{\Sigma}$-structure which is a $\Diamond$-fixed point in the sense of
Section 6.2.
For each $i:\mathsf{Size}$, say that a $(\downarrow i)$-indexed
$\mathrm{\Sigma}$-structure (Section 6.2) is an _upto- $i$_ $\Diamond$-fixed
point if
$\forall j<i.\;D_{j}=\Diamond_{j}(D\;(\downarrow j))\;\wedge\;\forall
k<j.\forall t.\;\tau_{k,j}\,t\mathrel{{=}{=}}[k,t]_{R_{j}}$ (96)
(cf. (83) and (84)). Note that:
1. (A)
_Given $j<i$, any upto-$i$ $\Diamond$-fixed point restricts to an upto-$j$
$\Diamond$-fixed point._
2. (B)
_For all $i$, any two upto-$i$ $\Diamond$-fixed points are equal_ (proof by
well-founded induction (46)).
Using these two facts, it follows by well-founded recursion (Section 5) that
there is an upto-$i$ $\Diamond$-fixed point for all $i:\mathsf{Size}$. For if
$(D^{(j)},\tau^{(j)})$ is an upto-$j$ $\Diamond$-fixed point for all $j<i$,
then we get $D^{(i)}:{\downarrow i}\rightarrow\mathcal{U}$ by defining for
each $j<i$
$(D^{(i)})_{j}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\Diamond_{j}(D^{(j)})$ (97)
If $k<j<i$, then by (A) we have that $D^{(j)}(\downarrow k)$ is an upto-$k$
$\Diamond$-fixed point and hence by (B) that $D^{(j)}(\downarrow k)=D^{(k)}$.
So together with (96) this gives:
$(D^{(i)})_{k}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\Diamond_{k}(D^{(k)})=\Diamond_{k}(D^{(j)}(\downarrow
k))=(D^{(j)})_{k}$ (98)
Hence we can define
$(\tau^{(i)})_{k,j}:T_{\mathrm{\Sigma}}((D^{(i)})_{k})\rightarrow(D^{(i)})_{j}$
by $(\tau^{(i)})_{k,j}\,t\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}[k,t]_{R_{j}}$ and this makes $(D^{(i)},\tau^{(i)})$ into a
$(\downarrow i)$-indexed $\mathrm{\Sigma}$-structure which by construction is
an upto-$i$ $\Diamond$-fixed point. Thus by well-founded recursion (Section 5)
we have an upto-$i$ $\Diamond$-fixed point $D^{(i)}$ for all
$i:\mathsf{Size}$.
Let $D:\mathsf{Size}\rightarrow\mathcal{U}$ be given by
$D_{i}\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\Diamond_{i}(D^{(i)})$.
If $j<i$, then by (A) and (B) we have $D^{(i)}(\downarrow j)=D^{(j)}$ and
together with (96) this gives:
$D_{j}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\Diamond_{j}(D^{(j)})=\Diamond_{j}(D^{(i)}(\downarrow
j))=(D^{(i)})_{j}$ (99)
So we can define $\tau_{j,i}:T_{\mathrm{\Sigma}}\,D_{j}\rightarrow D_{i}$ by
$\tau_{j,i}\,t\mathrel{\smash{\overset{\text{\tiny def}}{=}}}[j,t]_{R_{i}}$.
This makes $(D,\tau)$ into a $\mathrm{\Sigma}$-structure which by construction
is a $\Diamond$-fixed point. ∎
## 7\. Encoding QITs as QWI-types
The general notion of indexed quotient inductive type (QIT) was discussed by
example in the Introduction. We wish to show that a wide variety of QITs can
be expressed as QWI-types, namely those which do not use conditional equality
constructors (as in the example in (2)). We first introduce a schema for such
QITs that combines desirable features of ones that occur in the literature. As
in the previous section, for simplicity sake we will confine our attention to
the non-indexed case of QITs and QW-types.
### 7.1. General QIT schemas and encodings
[BGvdW17] present a schema for infinitary QITs that do not support conditional
path equations. Constructors are defined by arbitrary polynomial endofunctors
built up using (non-dependent) products and sums, which means in particular
that parameters and arguments can occur in any order; however, they require
constructors to be in uncurried form. They also construct a model of simple
1-cell complexes and other non-recursive QITs.
[DM18, Sections 3.1 and 3.2] present a schema for finitary QITs that does
allow curried constructors (and also supports _conditional_ path equations),
but requires all parameters to appear before all arguments. This contrasts
with the more convenient schema for regular inductive types in Agda, which
allows parameters and arguments in any order.
[KKA19] define an encoding of finitary quotient inductive-inductive (QIIT)
types, called the _theory of signatures_ (ToS), which is a restriction of the
_theory of codes_ for HIITs [KK18]. In the ToS a QI(I)T is encoded as a
context of a small internal type theory. This encoding is not quite as
convenient as the schema for regular inductive types in Agda, but, when using
named variables, it is much closer to Agda than QWI-types are. They also
construct a model for finitary QIITs (and [KK20] reduce the problem of
modelling infinitary QIITs to just one “universal” QIIT – a generalised ToS).
Building on the first two schemas mentioned above, we provide a schema for
infinitary, non-conditional QITs combining the arbitrarily ordered parameters
and arguments of the first [BGvdW17] with the curried constructors of the
second [DM18].
In the following we fix the context $\mathrm{\Gamma}$ in which the QIT, $Q$,
($Q\notin\mathrm{\Gamma}$) is defined, whereas $\mathrm{\Delta}$, $A$, $B$,
$H$, $K$, $a$, $b$, $x$, $y$, etc. are metavariables. When deriving an element
or equality constructor, $\mathrm{\Delta}$ will always be equal to
$\mathrm{\Gamma}$ or an extension of it. A type _strictly-positive in $Q$_ is
made up with $\prod$-types and $\sum$-types and can use $Q$ and constants or
variables in the context, provided that $Q$ never occurs on the LHS of a
$\prod$-type, and that the RHS of a $\sum$-type never depends on $Q$, even if
it occurs on the LHS. $\mathrm{\Delta}\vdash K\mathsf{StrPos}$
$\mathrm{\Delta}$ $\vdash$ B : $\mathcal{U}$ 1[Param]$\mathrm{\Delta}$
$\vdash$ B $\mathsf{StrPos}$ $\mathrm{\Delta}$ $\vdash$ B : $\mathcal{U}$
$\mathrm{\Delta}$, b : B $\vdash$ K $\mathsf{StrPos}$
2[StrPosFun]$\mathrm{\Delta}$ $\vdash$ $\prod$_b : B K $\mathsf{StrPos}$ .
0[IndArg]$\mathrm{\Delta}$ $\vdash$ Q $\mathsf{StrPos}$ $\mathrm{\Delta}$
$\vdash$ A $\mathsf{StrPos}$ $\mathrm{\Delta}$, a :
A${[\nicefrac{{{\mathbb{1}}}}{{{Q}}}\\!]}$ $\vdash$ K $\mathsf{StrPos}$
2[Prod]$\mathrm{\Delta}$ $\vdash$ $\sum$_a : A K’ $\mathsf{StrPos}$ . Where
$K^{\prime}$ is found from $K$ by recursively replacing each sub-term of type
$\mathbb{1}$, $A\rightarrow\mathbb{1}$, etc. with the corresponding unique
term $0$, $!$, etc. _Note: We can see that these rules give strictly-positive
types because: (1) at this point $\mathrm{\Delta}\vdash Q:\mathcal{U}$ is not
derivable since $Q\notin\mathrm{\Gamma}$ and no rule for $\mathsf{StrPos}$
adds $Q$ to the context, so $Q$ can never appear in the LHS of a $\prod$-type.
Also (2) the Prod rule ensures that abstracting with $\sum$ cannot result in
dependencies of $Q$ in the codomain. _ The type of an element constructor is
an (iterated) $\prod$-type with codomain $Q$, with zero or more strictly-
positive arguments. $\mathrm{\Delta}\vdash H\mathsf{ElConstr}$
$\mathrm{\Delta}$, Q : $\mathcal{U}$ $\vdash$ H $\mathsf{ElType}$
1[ElConstr]$\mathrm{\Delta}$ $\vdash$ H $\mathsf{ElConstr}$ .
0[ElCoDom]$\mathrm{\Delta}$ $\vdash$ Q $\mathsf{ElType}$
$\mathrm{\Delta}$${[\nicefrac{{{\mathbb{1}}}}{{{Q}}}\\!]}$ $\vdash$ K
$\mathsf{StrPos}$ $\mathrm{\Delta}$, a : K’ $\vdash$ H $\mathsf{ElType}$
2[ElArg]$\mathrm{\Delta}$ $\vdash$ $\prod$_a : K’ H $\mathsf{ElType}$ . Where
$\mathrm{\Delta}{[\nicefrac{{{\mathbb{1}}}}{{{Q}}}\\!]}$ is the context that
results after replacing occurrences of $Q$ in $\mathrm{\Delta}$ by the type
$\mathbb{1}$. _Note: Deriving
$\mathrm{\Gamma}\vdash\prod_{a:K^{\prime}}Q\mathsf{ElConstr}$ via ElArg for
some strictly-positive $K$ does not imply
$\mathrm{\Gamma}\vdash\prod_{a:K^{\prime}}Q:\mathcal{U}$ since
$\mathrm{\Gamma}\nvdash Q:\mathcal{U}$ (since $Q\notin\mathrm{\Gamma}$), and
also $\mathrm{\Gamma}\nvdash K^{\prime}:\mathcal{U}$ in general since
$K^{\prime}$ may contain $Q$. This distinction also applies to the following
judgements. _ The type of an equality constructor is an (iterated)
$\prod$-type with codomain $x=_{Q}y$, with zero or more strictly-positive
arguments. Note that each of the element constructors are added to the context
and can be used to derive $x:Q$ and $y:Q$. $\mathrm{\Delta}\vdash
K\mathsf{EqConstr}$ $\mathrm{\Delta}$, Q : $\mathcal{U}$, c1 : C1, $\cdots$,
cm : Cm $\vdash$ K : $\mathsf{EqType}$ 1[EqConstr]$\mathrm{\Delta}$ $\vdash$ K
$\mathsf{EqConstr}$ $\mathrm{\Delta}$ $\vdash$ x : Q $\mathrm{\Delta}$
$\vdash$ y : Q 2[EqCoDom]$\mathrm{\Delta}$ $\vdash$ x = y $\mathsf{EqType}$
$\mathrm{\Delta}$${[\nicefrac{{{\mathbb{1}}}}{{{Q}}}\\!]}$ $\vdash$ K
$\mathsf{StrPos}$ $\mathrm{\Delta}$, a : K’ $\vdash$ H $\mathsf{EqType}$
2[EqArg]$\mathrm{\Delta}$ $\vdash$ $\prod$_a : K’ H $\mathsf{EqType}$ Figure
3. Rules for QIT element and equality constructors.
The following definition should be read as an extension of a formalisation of
the type theory described in Section 2 in terms of typing contexts
($\mathrm{\Gamma},\mathrm{\Delta},\ldots$) and various judgements-in-context
(such as $\mathrm{\Gamma}\vdash a:A$, $\mathrm{\Gamma}\vdash a=a^{\prime}:A$,
etc.); see the HoTT Book [Uni13, Appendix].
###### Definition .
A (_non-conditional_) _QIT_ , $Q$, in a context $\mathrm{\Gamma}$,
$Q\notin\mathrm{\Gamma}$ is specified by a list of element constructors
${c_{1}:C_{1},}\ldots,{c_{m}:C_{m}}$, followed by a list of equality
constructors ${d_{1}:D_{1},}\ldots,{d_{n}:D_{n}}$, where the $C_{i}$s and
$D_{j}$s are derived as $\mathrm{\Gamma}\vdash C_{i}\,\mathsf{ElConstr}$ and
$\mathrm{\Gamma}\vdash D_{j}\,\mathsf{EqConstr}$ respectively, according to
the rules in Figure 3.
Given a list of element and equality constructors built up according to these
rules, the newly-defined QIT, $Q$, has formation rule
$\prooftree\infer 0{\mathrm{\Gamma}\vdash Q:\mathcal{U}}$
for the specific context $\mathrm{\Gamma}$ in which it was defined. And it has
an introduction rule for each element constructor and each equality
constructor:
$\prooftree\infer 0{\mathrm{\Gamma}\vdash
c_{1}:C_{1}}\quad\cdots\quad\prooftree\infer 0{\mathrm{\Gamma}\vdash
c_{m}:C_{m}}\qquad\qquad\prooftree\infer 0{\mathrm{\Gamma}\vdash
d_{1}:D_{1}}\quad\cdots\quad\prooftree\infer 0{\mathrm{\Gamma}\vdash
d_{n}:D_{n}}$
We show how to derive elimination and computation rules for $Q$ from these
formation and introduction rules (compare with [BGvdW17, Definition 10]) after
first giving an example of how the schema is instantiated.
###### Example .
Consider multisets as in example (1) with two element constructors $[]$ and
$\\_\mathrel{::}\\_$, and one equality constructor $\mathsf{swap}$. Given the
context containing parameter $X:\mathcal{U}$, we define the QIT
$\cdot,X\mathbin{:}\mathcal{U}\vdash\mathsf{Bag}\mathbin{:}\mathcal{U}_{1}$ by
providing those constructors along with their types such that they are derived
by the rules in Figure 3:
Let $\mathrm{\Gamma}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}(\cdot,X:\mathcal{U})$ and
$\mathrm{\Delta}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}(\mathrm{\Gamma},\mathsf{Bag}:\mathcal{U})=(\cdot,X:\mathcal{U},\mathsf{Bag}:\mathcal{U})$;
so that, by definition,
$\mathrm{\Delta}{[\nicefrac{{{\mathbb{1}}}}{{{\mathsf{Bag}}}}\\!]}=\mathrm{\Gamma}$.
* •
$[]$
0[ElCoDom]$\mathrm{\Delta}$ $\vdash$ $\mathsf{Bag}$$\mathsf{ElType}$
1[ElConstr]$\mathrm{\Gamma}$ $\vdash$ $\mathsf{Bag}$$\mathsf{ElConstr}$
* •
$\\_\mathrel{::}\\_$
0$\mathrm{\Gamma}$ $\vdash$ X : $\mathcal{U}$ 1[Param]$\mathrm{\Gamma}$
$\vdash$ X $\mathsf{StrPos}$ 0[IndArg]$\mathrm{\Gamma}$, a : X $\vdash$
$\mathsf{Bag}$$\mathsf{StrPos}$ 0[ElCoDom]$\mathrm{\Delta}$, a : X, b :
$\mathsf{Bag}$$\vdash$ $\mathsf{Bag}$$\mathsf{ElType}$
2[ElArg]$\mathrm{\Delta}$, a : X $\vdash$ ($\mathsf{Bag}$$\rightarrow$
$\mathsf{Bag}$) $\mathsf{ElType}$ 2[ElArg]$\mathrm{\Delta}$ $\vdash$ (X
$\rightarrow$ $\mathsf{Bag}$$\rightarrow$ $\mathsf{Bag}$) $\mathsf{ElType}$
1[ElConstr]$\mathrm{\Gamma}$ $\vdash$ (X $\rightarrow$
$\mathsf{Bag}$$\rightarrow$ $\mathsf{Bag}$) $\mathsf{ElConstr}$
Let
$\displaystyle{}\mathrm{\Theta}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\mathrm{\Delta},[]:\mathsf{Bag},\\_\mathrel{::}\\_:X\rightarrow\mathsf{Bag}\rightarrow\mathsf{Bag}$
$\displaystyle\mathrm{\Xi}$ $\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\mathrm{\Theta}{[\nicefrac{{{\mathbb{1}}}}{{{\mathsf{Bag}}}}\\!]}\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\mathrm{\Gamma},[]:\mathbb{1},\\_\mathrel{::}\\_:X\rightarrow\mathbb{1}\rightarrow\mathbb{1}$
$\displaystyle\mathrm{\Phi}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\mathrm{\Theta},x\;y:X,\mathit{zs}:\mathsf{Bag}$
$\displaystyle\mathit{eqn}$ $\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}x\mathrel{::}y\mathrel{::}\mathit{zs}=y\mathrel{::}x\mathrel{::}\mathit{zs}$
* •
$\mathsf{swap}$
0$\mathrm{\Xi}$ $\vdash$ X : $\mathcal{U}$ 1$\mathrm{\Xi}$ $\vdash$ X
$\mathsf{StrPos}$ 0$\mathrm{\Xi}$, x : X $\vdash$ X : $\mathcal{U}$
1$\mathrm{\Xi}$, x : X $\vdash$ X $\mathsf{StrPos}$ 0$\mathrm{\Xi}$, x y : X
$\vdash$ $\mathsf{Bag}$$\mathsf{StrPos}$ $\vdots$ 1$\mathrm{\Phi}$ $\vdash$ x
$\mathrel{::}$ y $\mathrel{::}$ $\mathit{zs}$: $\mathsf{Bag}$ $\vdots$
1$\mathrm{\Phi}$ $\vdash$ y $\mathrel{::}$ x $\mathrel{::}$ $\mathit{zs}$:
$\mathsf{Bag}$ 2[EqCoDom] $\mathrm{\Phi}$ $\vdash$ eqn$\mathsf{EqType}$
2[EqArg] $\mathrm{\Theta}$, x y : X $\vdash$ $\prod$_$\mathit{zs}$:
$\mathsf{Bag}$ eqn$\mathsf{EqType}$ 2[EqArg] $\mathrm{\Theta}$, x : X $\vdash$
$\prod$_y : X $\prod$_$\mathit{zs}$: $\mathsf{Bag}$ eqn$\mathsf{EqType}$
2[EqArg] $\mathrm{\Theta}$ $\vdash$ $\prod$_x y : X $\prod$_$\mathit{zs}$:
$\mathsf{Bag}$ eqn$\mathsf{EqType}$ 1[EqConstr]$\mathrm{\Gamma}$ $\vdash$
$\prod$x y : X $\prod$_$\mathit{zs}$: $\mathsf{Bag}$ eqn$\mathsf{EqConstr}$
###### Definition (Elimination and computation).
The _arguments_ of a constructor $C_{j}$, written $\mathsf{Arg}(C_{j})$, is
the list of $K^{\prime}$ for all strictly-positive types $K$ introduced with
the rule ElArg.
Given a QIT defined by constructors $c_{1},\ldots,c_{m},d_{1},\ldots,d_{n}$ as
above, the _underlying inductive type_ $\lfloor Q\rfloor$ is the inductive
type defined by only the element constructors $c_{1},\ldots,c_{m}$, ignoring
the equalities; then the _underlying W-type_ is the W-type that encodes this
inductive type. (Recall that every strictly-positive description of a
polynomial endofunctor gives rise to a W-type with the same initial algebra;
see [Dyb97], and for the more general indexed and nested case see [AGHMM15].)
Define _leaf application_ on a strictly-positive term $a:A$ (that is, a term
with a strictly-positive type) for a function $f:\prod_{q:Q}R\,q$ into some
type $R$, written $f\mathbin{\mathdollar}a$, by induction on the
$\mathsf{StrPos}$ structure of $A$:
IndArg $\displaystyle f\mathbin{\mathdollar}x$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny def}}{=}}}f\,x$ (100) Param
$\displaystyle f\mathbin{\mathdollar}b$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny def}}{=}}}b$ Prod
$\displaystyle f\mathbin{\mathdollar}(a,b)$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}(f\mathbin{\mathdollar}a,f\mathbin{\mathdollar}b)$ StrPosFun
$\displaystyle f\mathbin{\mathdollar}g$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}(f\mathbin{\mathdollar}\\_)\circ g$
In order to define the elimination and computation rules we must first define
the induction step. First, given a “motive” $\mathrm{\Gamma}\vdash
P:Q\rightarrow\mathcal{U}$, define the type
$P^{\prime}\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\sum_{x:Q}P\,x$. Now
given a strictly-positive argument $A$, define the type $A^{\prime}$ (it will
be (one of) the induction hypotheses) by induction on the structure of $A$,
replacing each occurrence of $Q$ (IndArg) with $P^{\prime}$. Terms
$a^{\prime}:A^{\prime}$ are also strictly-positive and admit leaf application
for functions of type $\prod_{p^{\prime}:P^{\prime}}S\,p^{\prime}$ for some
type $S$.
To find the induction step
$\savestack{\tmpbox}{\stretchto{\scaleto{\leavevmode\hbox
to24.80237pt{\hss\hbox
to13.35547pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{\leavevmode\hbox
to38.15784pt{\hskip 38.15784pt\hbox
to9.53928pt{\hss\leavevmode\hbox{\set@color\leavevmode\hbox
to47.69711pt{\hss\hbox
to0.35pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{
\scalerel*[\widthof{\small$C_{j}\;C_{j}$}]{\kern 0.1pt\mathchar
12382\relax\kern 0.1pt}{\rule{0.0pt}{505.89pt}}}}}}}\hss}\hskip
47.69711pt}}}\hss}}}}}}\hss}\hskip
24.80237pt}}{}}{2.4ex}}\stackon[-6.9pt]{C_{j}}{\tmpbox}$ for each element
constructor $c_{j}:C_{j}$, replace each of the strictly-positive arguments
$\prod_{a_{1}:A_{1}}\,\cdots\,\prod_{a_{p}:A_{p}}$ with
$\prod_{a_{1}:A_{1}^{\prime}}\,\cdots\,\prod_{a_{p}:A_{p}^{\prime}}$ , as
defined above, and replace the target $Q$ of the constructor with
$P\,(c_{j}\,(\pi_{1}\mathbin{\mathdollar}a_{1})\,\cdots\,(\pi_{1}\mathbin{\mathdollar}a_{p}))$.
The induction cases for each of the element constructors are then
$h_{1}:\savestack{\tmpbox}{\stretchto{\scaleto{\leavevmode\hbox
to24.96461pt{\hss\hbox
to13.44284pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{\leavevmode\hbox
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to9.60167pt{\hss\leavevmode\hbox{\set@color\leavevmode\hbox
to48.00912pt{\hss\hbox
to0.35pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{
\scalerel*[\widthof{\small$C_{1}\;C_{1}$}]{\kern 0.1pt\mathchar
12382\relax\kern 0.1pt}{\rule{0.0pt}{505.89pt}}}}}}}\hss}\hskip
48.00912pt}}}\hss}}}}}}\hss}\hskip
24.96461pt}}{}}{2.4ex}}\stackon[-6.9pt]{C_{1}}{\tmpbox},\ldots,h_{m}:\savestack{\tmpbox}{\stretchto{\scaleto{\leavevmode\hbox
to26.946pt{\hss\hbox
to14.50975pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{\leavevmode\hbox
to41.45575pt{\hskip 41.45575pt\hbox
to10.36374pt{\hss\leavevmode\hbox{\set@color\leavevmode\hbox
to51.81949pt{\hss\hbox
to0.35pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{
\scalerel*[\widthof{\small$C_{m}\;C_{m}$}]{\kern 0.1pt\mathchar
12382\relax\kern 0.1pt}{\rule{0.0pt}{505.89pt}}}}}}}\hss}\hskip
51.81949pt}}}\hss}}}}}}\hss}\hskip
26.946pt}}{}}{2.4ex}}\stackon[-6.9pt]{C_{m}}{\tmpbox}$, and these provide an
eliminator $\mathsf{elim}\,h_{1}\,\cdots\,h_{m}$ for the underlying inductive
type $\lfloor Q\rfloor$.
Given $h_{1},\ldots,h_{m}$, define
$\savestack{\tmpbox}{\stretchto{\scaleto{\leavevmode\hbox
to25.88763pt{\hss\hbox
to13.93985pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{\leavevmode\hbox
to39.82748pt{\hskip 39.82748pt\hbox
to9.95667pt{\hss\leavevmode\hbox{\set@color\leavevmode\hbox
to49.78415pt{\hss\hbox
to0.35pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{
\scalerel*[\widthof{\small$D_{k}\;D_{k}$}]{\kern 0.1pt\mathchar
12382\relax\kern 0.1pt}{\rule{0.0pt}{505.89pt}}}}}}}\hss}\hskip
49.78415pt}}}\hss}}}}}}\hss}\hskip
25.88763pt}}{}}{2.4ex}}\stackon[-6.9pt]{D_{k}}{\tmpbox}$, for each equality
constructor $d_{k}:D_{k}$, in the same way. Replace the homogeneous equality
type with a heterogeneous equality type and replace the endpoints $l,r$ of the
equality with $\savestack{\tmpbox}{\stretchto{\scaleto{\leavevmode\hbox
to17.96158pt{\hss\hbox
to9.67186pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{\leavevmode\hbox
to27.63344pt{\hskip 27.63344pt\hbox
to6.90822pt{\hss\leavevmode\hbox{\set@color\leavevmode\hbox
to34.54166pt{\hss\hbox
to0.35pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{
\scalerel*[\widthof{\small$l\;l$}]{\kern 0.1pt\mathchar 12382\relax\kern
0.1pt}{\rule{0.0pt}{505.89pt}}}}}}}\hss}\hskip
34.54166pt}}}\hss}}}}}}\hss}\hskip
17.96158pt}}{}}{2.4ex}}\stackon[-6.9pt]{l}{\tmpbox},\savestack{\tmpbox}{\stretchto{\scaleto{\leavevmode\hbox
to19.4674pt{\hss\hbox
to10.48271pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{\leavevmode\hbox
to29.95012pt{\hskip 29.95012pt\hbox
to7.48738pt{\hss\leavevmode\hbox{\set@color\leavevmode\hbox
to37.4375pt{\hss\hbox
to0.35pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{
\scalerel*[\widthof{\small$r\;r$}]{\kern 0.1pt\mathchar 12382\relax\kern
0.1pt}{\rule{0.0pt}{505.89pt}}}}}}}\hss}\hskip
37.4375pt}}}\hss}}}}}}\hss}\hskip
19.4674pt}}{}}{2.4ex}}\stackon[-6.9pt]{r}{\tmpbox}$ inductively defined by
cases depending on if the abstracted variable is a constructor or not:
$\displaystyle{}\savestack{\tmpbox}{\stretchto{\scaleto{\leavevmode\hbox
to21.4937pt{\hss\hbox
to11.57384pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{\leavevmode\hbox
to33.06754pt{\hskip 33.06754pt\hbox
to8.26672pt{\hss\leavevmode\hbox{\set@color\leavevmode\hbox
to41.33426pt{\hss\hbox
to0.35pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{
\scalerel*[\widthof{\small$c_{j}\;c_{j}$}]{\kern 0.1pt\mathchar
12382\relax\kern 0.1pt}{\rule{0.0pt}{505.89pt}}}}}}}\hss}\hskip
41.33426pt}}}\hss}}}}}}\hss}\hskip
21.4937pt}}{}}{2.4ex}}\stackon[-6.9pt]{c_{j}}{\tmpbox}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny def}}{=}}}h_{j}$ (101)
$\displaystyle\savestack{\tmpbox}{\stretchto{\scaleto{\leavevmode\hbox
to20.33406pt{\hss\hbox
to10.94939pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{\leavevmode\hbox
to31.28345pt{\hskip 31.28345pt\hbox
to7.82071pt{\hss\leavevmode\hbox{\set@color\leavevmode\hbox
to39.10416pt{\hss\hbox
to0.35pt{\leavevmode\hbox{\set@color\resizebox{}{}{{\leavevmode\hbox{\set@color{
\scalerel*[\widthof{\small$x\;x$}]{\kern 0.1pt\mathchar 12382\relax\kern
0.1pt}{\rule{0.0pt}{505.89pt}}}}}}}\hss}\hskip
39.10416pt}}}\hss}}}}}}\hss}\hskip
20.33406pt}}{}}{2.4ex}}\stackon[-6.9pt]{x}{\tmpbox}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny def}}{=}}}x:P^{\prime}_{i}$
The elimination rule is then:
$\mathrm{\Gamma}$ $\vdash$ P : Q $\rightarrow$ $\mathcal{U}$ [no rule]1
$\mathrm{\Gamma}$ $\vdash$ h1 : *[$C_{1}\;C_{1}$]^ 2.4ex[-6.9pt]C1 $\ldots$
$\mathrm{\Gamma}$ $\vdash$ hm : *[$C_{m}\;C_{m}$]^ 2.4ex[-6.9pt]Cm [no rule]1
$\mathrm{\Gamma}$ $\vdash$ p1 : *[$D_{1}\;D_{1}$]^ 2.4ex[-6.9pt]D1 $\ldots$
$\mathrm{\Gamma}$ $\vdash$ p_nm :
*[$D_{\mathrlap{n}\phantom{m}}\;D_{\mathrlap{n}\phantom{m}}$]^
2.4ex[-6.9pt]D_nm 1[QITelim]$\mathsf{qitelim}$ h1 $\cdots$ hm p1 $\cdots$ pn :
$\prod$_x : Q P x
Finally the computation rules are, for each element constructor $c_{i}:C_{i}$,
_same hypotheses as QITelim_ $\mathrm{\Gamma}$ $\vdash$ a1,$\ldots$,ap :
$\mathsf{Arg}$(Ci) 2[QITcomp] $\mathsf{qitelim}$ h1 $\cdots$ hm p1 $\cdots$ pn
(ci a1 $\cdots$ ap) [no rule]1 =_P (ci a1 $\cdots$ ap) hi (a1 ,
($\mathsf{qitelim}$ $\cdots$) $\mathbin{\mathdollar}$a1) $\cdots$ (ap ,
($\mathsf{qitelim}$ $\cdots$) $\mathbin{\mathdollar}$ap)
###### Example .
The elimination rule for $X:\mathcal{U}\vdash\mathsf{Bag}$ is:
$\mathrm{\Gamma}$ $\vdash$ P : $\mathsf{Bag}$$\rightarrow$ $\mathcal{U}$
$\mathrm{\Gamma}$ $\vdash$ nil: P [] $\mathrm{\Gamma}$ $\vdash$ cons:
$\prod$_x : X $\prod$_$\mathit{xs}$’ : ($\sum$_$\mathit{xs}$: $\mathsf{Bag}$ P
$\mathit{xs}$) P (x $\mathrel{::}$ ($\pi$1 $\mathit{xs}$’)) [no rule]1
$\mathrm{\Gamma}$ $\vdash$ resp: $\prod$_x y : X $\prod$_$\mathit{zs}$’ :
($\sum$_$\mathit{zs}$: $\mathsf{Bag}$ P $\mathit{zs}$) cons x (y
$\mathrel{::}$ ($\pi$1 $\mathit{zs}$’), cons y $\mathit{zs}$’) == cons y (x
$\mathrel{::}$ ($\pi$1 $\mathit{zs}$’), cons x $\mathit{zs}$’) 1Bagelim nil
cons resp: $\prod$_x : $\mathsf{Bag}$ P x
And it computes:
$\displaystyle\mathsf{Bagelim}\,\mathit{nil}\,\mathit{cons}\,\mathit{resp}\,[]$
$\displaystyle=_{P\,[]}$ $\displaystyle\mathit{nil}$
$\displaystyle\mathsf{Bagelim}\,\mathit{nil}\,\mathit{cons}\,\mathit{resp}\,(x\mathrel{::}\mathit{xs})$
$\displaystyle=_{P\,(x\mathrel{::}\mathit{xs})}$
$\displaystyle\mathit{cons}\,x\,(\mathit{xs},\mathsf{Bagelim}\,\mathit{nil}\,\mathit{cons}\,\mathit{resp}\,\mathit{xs})$
### 7.2. From QIT to QWI-type
We claim that any QIT $Q$ in the sense of Section 7.1 can be constructed as
the QW-type for a signature and equational system derived from the declaration
of the QIT; and that the same is true for the indexed version of Section 7.1
and QWI-types. That is, QWI-types are universal for non-conditional QITs in
the same sense that W-types are for inductive types in a sufficiently
extensional type theory.
We saw above that the data $c_{1}:C_{1},\ldots,c_{m}:C_{m}$ in Section 7.1
gives rise to an underlying inductive type $\lfloor Q\rfloor$ and hence to a
W-type [AGHMM15], with signature $\mathrm{\Sigma}=(A,B)$. The parameters and
arguments of the equality constructors $d_{1}:D_{1},\ldots,d_{n}:D_{n}$ are
also encoded in the same way by a signature $(E,V)$. Then the endpoints of the
equality constructors can be encoded by the $l$ and $r$ arguments of an
equational system $\varepsilon=(E,V,l,r)$ in the sense of Section 3.2; the
encoding follows the structure of $\mathsf{EqType}$ judgements in Figure 3,
using the $\eta$ constructor of $T_{\mathrm{\Sigma}}$ (13) for variables and
the $\sigma$ constructor for $c_{1},\cdots,c_{m}$ introduced by EqConstr in
$\mathrm{\Theta}$. We illustrate the encoding by example, beginning with the
three examples from the Introduction.
###### Example (Finite multisets).
The element constructors of the QIT, $\mathsf{Bag}\,X$, of finite multisets
over $X:\mathcal{U}$ in (1) are encoded exactly as the W-type for List over
$X$: we take $A:\mathcal{U}$ to be $\mathbb{1}+X$, where $\iota_{1}\,0$
corresponds to $[]$ and $\iota_{2}\,x$ corresponds to $x\mathrel{::}\\_$ for
each $x:X$. The arity of $[]$ is zero, and the arity of each
$x\mathrel{::}\\_$ is one; so we take $B:A\rightarrow\mathcal{U}$ to be the
function mapping $\iota_{1}\,0$ to $\mathbb{0}$ and each $\iota_{2}\,x$ to
$\mathbb{1}$. The $\mathsf{swap}$ equality constructor is parametrised by
elements of $E\mathrel{\smash{\overset{\text{\tiny def}}{=}}}X\times X$ and
for each $(x,y):E$, $\mathsf{swap}\,(x,y)$ yields an equation involving a
single free variable (called $\mathit{zs}:\mathsf{Bag}\,X$ in (1)); so we
define $V\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\lambda\,\\_\mathbin{.}\mathbb{1}:E\rightarrow\mathcal{U}$. Each
side of the equation named by $\mathsf{swap}\,(x,y)$ is coded by an element of
$T_{\mathrm{\Sigma}}\,(V\,(x,y))=T_{\mathrm{\Sigma}}\,\mathbb{1}$. Recalling
the definition of $T_{\mathrm{\Sigma}}$ from Section 3.1, the single free
variable, $\mathit{zs}$, corresponds to
$\eta\,0:T_{\mathrm{\Sigma}}\,\mathbb{1}$. Then the left-hand side of the
equation, $x\mathrel{::}y\mathrel{::}zs$, is encoded as
$\sigma\,(\iota_{2}\,x,(\lambda\,\\_\mathbin{.}\sigma\,(\iota_{2}\,y,(\lambda\,\\_\mathbin{.}\eta\,0))))$,
and similarly the right-hand side, $y\mathrel{::}x\mathrel{::}zs$, is encoded
as
$\sigma\,(\iota_{2}\,y,(\lambda\,\\_\mathbin{.}\sigma\,(\iota_{2}\,x,(\lambda\,\\_\mathbin{.}\eta\,0))))$.
So altogether, the encoding of $\mathsf{Bag}\,X$ as a QW-type uses the non-
indexed signature $\mathrm{\Sigma}=(A,B)$ and equational system
$\varepsilon=(E,V,l,r)$, where:
$\displaystyle A$ $\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\mathbb{1}+X$ $\displaystyle E$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny def}}{=}}}X\times X$
$\displaystyle B(\iota_{1}\,0)$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\mathbb{0}$
$\displaystyle V(x,y)$ $\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\mathbb{1}$ $\displaystyle B(\iota_{2}\,x)$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\mathbb{1}$
$\displaystyle l(x,y)$ $\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sigma(\iota_{2}\,x,(\lambda\,\\_\mathbin{.}\,\sigma(\iota_{2}\,y,(\lambda\,\\_\mathbin{.}\,\eta\,0))))$
$\displaystyle r(x,y)$ $\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\sigma(\iota_{2}\,y,(\lambda\,\\_\mathbin{.}\,\sigma(\iota_{2}\,x,(\lambda\,\\_\mathbin{.}\,\eta\,0))))$
###### Example (Length-indexed multisets).
The QWI-type encoding the QIT $\mathsf{AbVec}\,X$ of length-indexed multisets
in (3) is an indexed version of the previous example, using the index type
$I\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\mathbb{N}$. The indexed
signature $\mathrm{\Sigma}=(\mathbb{N},A,B)$ has $A:\mathcal{U}^{\mathbb{N}}$
and $B:\prod_{i:\mathbb{N}}(A_{i}\rightarrow\mathcal{U}^{\mathbb{N}})$ given
by:
$\displaystyle A_{0}$ $\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\mathbb{1}$ $\displaystyle A_{i+1}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny def}}{=}}}X$ $\displaystyle
B_{0}\,0\,j$ $\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\mathbb{0}$ $\displaystyle B_{i+1}\,x\,j$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny def}}{=}}}(i=j)$
The indexed system of equations $\varepsilon=(\mathbb{N},E,V,l,r)$ over
$\mathrm{\Sigma}$ has $E:\mathcal{U}^{\mathbb{N}}$,
$V:\prod_{i:\mathbb{N}}(E_{i}\rightarrow\mathcal{U}^{\mathbb{N}})$ and
$l,r:\prod_{i:\mathbb{N}}\prod_{e:E_{i}}(T_{\mathrm{\Sigma}}(V_{i}\,e))_{i}$
given by:
$\displaystyle E_{0}$ $\displaystyle\mathrel{\smash{\overset{\text{\tiny
def}}{=}}}\mathbb{0}$ $\displaystyle E_{1}$
$\displaystyle\mathrel{\smash{\overset{\text{\tiny def}}{=}}}\mathbb{0}$
|
# A Clinical Dataset for the Evaluation of Motion Planners in Medical
Applications
Inbar Fried1,2, Jason A. Akulian3, and Ron Alterovitz1 This research was
supported by the U.S. National Institutes of Health (NIH) under awards
R01EB024864 and F30CA265234, and by the National Science Foundation (NSF)
under awards 2008475 and 2038855.The authors acknowledge the National Cancer
Institute and the Foundation for the National Institutes of Health, and their
critical role in the creation of the free publicly available LIDC/IDRI
Database used in this study.The MR brain images from healthy volunteers used
in this paper were collected and made available by the CASILab at The
University of North Carolina at Chapel Hill and were distributed by the MIDAS
Data Server at Kitware, Inc.1 I. Fried and R. Alterovitz are with the
Department of Computer Science, University of North Carolina at Chapel Hill,
Chapel Hill, NC 27599, USA. {ifried01<EMAIL_ADDRESS>I. Fried is also with
the Medical Scientist Training Program, University of North Carolina School of
Medicine, Chapel Hill, NC, 27599, USA.3J. A. Akulian is with the Division of
Pulmonary Diseases and Critical Care Medicine at the University of North
Carolina at Chapel Hill, NC 27599, USA<EMAIL_ADDRESS>
###### Abstract
The prospect of using autonomous robots to enhance the capabilities of
physicians and enable novel procedures has led to considerable efforts in
developing medical robots and incorporating autonomous capabilities. Motion
planning is a core component for any such system working in an environment
that demands near perfect levels of safety, reliability, and precision.
Despite the extensive and promising work that has gone into developing motion
planners for medical robots, a standardized and clinically-meaningful way to
compare existing algorithms and evaluate novel planners and robots is not well
established. We present the Medical Motion Planning Dataset (Med-MPD), a
publicly-available dataset of real clinical scenarios in various organs for
the purpose of evaluating motion planners for minimally-invasive medical
robots. Our goal is that this dataset serve as a first step towards creating a
larger robust medical motion planning benchmark framework, advance research
into medical motion planners, and lift some of the burden of generating
medical evaluation data.
## I Introduction
Automation of medical robots for clinical procedures or subtasks is
increasingly being shown to be feasible. Achieving autonomy in interventional
medical procedures has a lot of potential benefits for patient care and
hospital efficiency. Much like teleoperated medical robots, such as the da
Vinci (Intuitive Surgical Inc., Sunnyvale, CA), can compensate for physician
fatigue and hand instability, autonomous medical robotics can further improve
and standardize patient care by accounting for inter- and intra-physician
variability while also focusing the physician’s time on sub-tasks that require
their expertise. However, beyond the technical challenges that exist in
hardware and software, a critical, if not the most important, challenge in
these systems is making them safe and reliable. To address these challenges
and still benefit from the advantages of automation, integrating motion
planning into medical robots to ensure safe motions is essential.
One class of medical robots that has been studied extensively over the past
couple decades has been medical continuum robots, which include, for example,
concentric tube robots and steerable needles [1]. Many mechanical designs have
been proposed for these devices, but at their core, medical continuum robots
can follow curvilinear trajectories in 3D, allowing them to curve around
obstacles and access regions of the anatomy that are otherwise inaccessible
when using straight rigid tools. The potential benefit of these devices has
been proposed in numerous organs and for various medical procedures. The
complex kinematics of these devices in conjunction with the precision required
for safe medical procedures make manual operation of these devices unintuitive
and impractical. To overcome this challenge, autonomous robots have been
proposed that actuate the medical continuum robot following a planned
trajectory.
Despite the numerous motion planners that have been proposed for medical
continuum robots, to the best of our knowledge, a benchmarking dataset to
evaluate the performance of these algorithms does not exist. The lack of a
shared benchmarking resource has lead each research group to generate their
own testing data, which is often a time intensive effort. Since the motion
planners have been tested in various organs, in different anatomical models of
those organs, and likely with different obstacle resolutions, it is difficult
to properly assess the benefits and drawbacks of each proposed motion planning
approach and to compare motion planners. To help evaluate the benefits of
robot automation in medicine, it is important to have benchmarks that can
robustly and equitably evaluate the performance of algorithms in clinically
relevant scenarios.
In this work, we propose Med-MPD, a medical benchmarking dataset consisting of
real clinical motion planning environments for assessing motion planners for
medical continuum robots and related minimally-invasive medical robots. The
data includes benchmark scenarios defined by the relevant anatomy and the
clinical problem in the lungs, liver, and brain. We make Med-MPD publicly
available at https://github.com/UNC-Robotics/Med-MPD.
## II Related Work
There are several robotics datasets and benchmarking suites that have been
developed specifically to allow robust evaluation of motion planners [2, 3, 4,
5] (and citations within). These works focus on non-medical robots. There have
also been several medical robotics datasets that have been published, but
these are mostly focused on computer vision problems like tool or anatomy
segmentation, physician training or assessment, and object manipulation, but
not on motion planning [6, 7, 8, 9]. Several works have proposed simulators
for medical robots [10, 11, 12], but there is no set benchmarking dataset with
which to compare different motion planning algorithms.
A variety of algorithms for medical continuum robots have been proposed that
encompass various organs and clinical applications. Within the broad class of
medical continuum robots, there has been substantial work in developing motion
planners for steerable needles in the lungs [13, 14, 15], liver [16, 17],
prostate [18, 19, 20, 21], and brain [22, 23, 24]. It is difficult to
effectively compare these motion planning algorithms since they span different
organs and different instances of these organs.
## III Med-MPD
Med-MPD contains anatomical environments and specifications of clinically
relevant scenarios in the lungs, liver, and brain. These three organs have
received considerable research attention from the continuum medical robots
motion planning community, especially for steerable needles. The description
of each environment, the clinical motivation, and several relevant evaluation
criteria are presented below. Although each organ has various pathologies each
defined by a different clinical objective, the planning problem we consider is
target reach. This objective encompasses many clinical procedures, including
biopsy, ablation, and drug delivery. Future iterations of the data could be
adapted to evaluate motion planners for scenarios where the objective is
different, such as manipulation at the target.
We represent the environments as three-dimensional binary maps that indicate
the presence or absence of obstacles at each corresponding voxel location in
the original medical image. This environmental representation is used by many
of the medical robot motion planners referenced above. We also provide a
collection of clinically-motivated start poses in each environment, along with
target points that correspond to true clinical targets.
### III-A Lungs
It is estimated that roughly one million pulmonary nodules are discovered
every year in the United States. In order to get a definitive diagnosis for
these nodules, a tissue biopsy is required. There are several methods to reach
lung nodules, but the least invasive and safest approach is via bronchoscopy
where a physician navigates a bronchoscope through the airways and inserts a
needle into the lung tissue towards the target. Since physicians currently use
straight rigid tools to perform the biopsy which are limited in reach and
access, there have been efforts to use flexible steerable needles to overcome
some of the existing challenges and increase the number of patients for which
bronchoscopy can be used [25]. Delivery of the robot can be done through the
working channel of a bronchoscope.
At a high level, the anatomy of the lungs consists of the airways, major blood
vessels, and the pleura (lung boundary) (see Figure 1). The remaining space
inside the lung (known as the parenchyma) is composed of functional tissue and
is the location where lung nodules that may be suspicious for cancer often
present. We present 5 motion planning scenarios in the lungs that reflect real
clinical scenarios of patients with lung nodules. The data is part of the Lung
Image Database Consortium and Image Database Resource Initiative (LIDC-IDRI)
image collection [26, 27] from The Cancer Imaging Archive (TCIA) [28]. Each
environment consists of obstacles including the airways, major blood vessels,
lung fissures, and pleura, and segmented nodules in the parenchyma. The
fissures and nodules were manually segmented while all other objects were
automatically segmented [29]. The start poses correspond to areas along the
airway wall that are accessible with a bronchoscope through which a medical
robot can be passed. An example plan for a flexible steerable needle is shown
in Figure 1.
Figure 1: A representative lung environment consisting of blood vessels, lung
fissures, bronchial tree, and pleural boundary. A nodule (target) is shown on
the right along with a sample planned trajectory. The steerable needle starts
at a valid point along the airway wall and can travel through the space within
the pleural boundary that is not occupied by obstacles.
### III-B Liver
Currently, percutaneous liver biopsies, where a physician inserts a needle
through the abdominal wall and into the liver, are most commonly performed
with straight rigid tools. The mechanical constraint of existing devices make
it hard to reach posterior sites that are obstructed by critical anatomy.
Additionally, when multiple targets exist, a physician will need to re-insert
the needle for each target. Medical devices such as steerable needles that are
able to curve around obstacles and reach multiple sites from a single point-
of-entry can alleviate some of these clinical challenges.
Similar to the lungs, the liver motion planning environment consists of major
blood vessels and the organ boundary. The remaining space within the liver
(also referred to as the parenchyma) is traversable. We present 5 motion
planning environments in the liver in patients with hepatocellular carcinoma
with segmentations of relevant obstacles. The data is derived from the
Hepatocellular Carcinoma Transarterial Chemoembolization Segmentation (HCC-
TACE-Seg) dataset [30, 31] from TCIA [28]. A sample liver planning environment
is shown in Figure 2.
Figure 2: A liver environment from the dataset showing the segmented hepatic
arteries, hepatic veins, portal vein, liver boundary, nodule (target), and a
sample trajectory. The three boxes on the bottom show the view in the CT
slices (transverse planes).
### III-C Brain
The brain is one of the most complex organs in the body, with nearly every
portion of tissue critical to some physiologic function. From a planning
perspective, while obvious obstacles exist such as blood vessels and
ventricles, there are many other regions of the brain and properties of the
tissue that are important to consider. For example, the directionality of
white matter fibers, which can be analyzed via tractography, can play an
important role in evaluating trajectories through the brain. Given the density
of critical regions in the brain and their fragility, medical robots and
motion planning algorithms that consider and account for these constraints can
have a large impact in this domain. We include 5 motion planning environments
in the brain where the targets are the globi pallidi for deep brain
stimulation. We consider blood vessels and ventricles as traditional
obstacles, whereas all other segmentations of brain regions can be assigned a
cost since some subset of them must be traversed. White matter fiber tracts
are not currently included in the data. The data is part of the Healthy MR
Database [32]. Blood vessels were manually segmented and all other structures
were segmented using FastSurfer [33]. A sample environment is depicted in
Figure 3.
Figure 3: A brain environment from the dataset showing the segmented blood
vessels, lateral ventricles (without the temporal horn), globi pallidi, brain
boundary, a sample trajectory, and various segmented regions of the brain
(left: multiple colors).
### III-D Evaluation Criteria
In order to evaluate and compare the performance of different motion planning
algorithms, we describe several criteria that are relevant to many medical
applications. This is a general and non-exhaustive list of relevant criteria,
and in many cases, each organ and clinical application has domain-specific
considerations that would be valuable to use as evaluation metrics.
The following metrics can be reported for a single clinical target or as a
statistic across a collection of clinical targets in the data.
* •
Path Length: the length of the collision-free motion plan from the start pose
to the target.
* •
Computation Time: the amount of computation time that the motion planner took
to find a kinematically-feasible collision-free plan from the start pose to
the target prior to the procedure.
* •
Replanning Time: the amount of time that the motion planner took to find a
kinematically-feasible collision-free plan from its current intraoperative
pose to the target following a random deviation event.
* •
Obstacle Clearance Statistics: the minimum, mean, and median of the Euclidean
distances between every pose along the motion plan to its nearest obstacle.
* •
Procedure Success (clinical targets): the percentage of clinical targets that
the motion planner was able to successfully plan to.
* •
Coverage of the Anatomy (random targets): the percentage of random goals that
the motion planner was able to successfully plan to. This is an approximation
for the motion planner’s ability to generalize to any target.
## IV Discussion
In this work, we proposed Med-MPD, a new medical benchmarking dataset for
motion planners for medical continuum robots and related minimally invasive
medical robots. At its current state, Med-MPD is a stand-alone collection of
clinically-relevant motion planning scenarios. It is our hope to extend the
benchmarking suite in the various ways described throughout this paper, as
well as to integrate it directly into an existing motion planning framework.
We also hope to expand the provided start poses to include start regions from
which motion planning can begin. This would introduce an interesting and
medically relevant problem where the choice of start pose is itself a motion
planning challenge that can be optimized as part of the procedure. We also
hope to introduce uncertainty into the data by implementing methods that allow
for target and obstacle deviation during a medical procedure. Uncertainty is
highly likely to have an impact during a procedure because of the deformable
nature of organ tissue. Ideally, the environmental uncertainty would be
incorporated into a simulator that would consider a robot’s model and
differential constraints to enable more realistic evaluations. Since the data
is not directly tied to medical continuum robots, the clinical environments
can be used to evaluate other existing and novel medical robots. It is our
intention that this dataset be used to advance research in motion planning for
autonomous medical robots towards the ultimate goal of leveraging these
systems to improve patient care.
## References
* [1] R. J. Webster III and B. A. Jones, “Design and kinematic modeling of constant curvature continuum robots: A review,” _The International Journal of Robotics Research_ , vol. 29, no. 13, pp. 1661–1683, 2010.
* [2] C. Chamzas, C. Quintero-Pena, Z. Kingston, A. Orthey, D. Rakita, M. Gleicher, M. Toussaint, and L. E. Kavraki, “Motionbenchmaker: A tool to generate and benchmark motion planning datasets,” _IEEE Robotics and Automation Letters_ , vol. 7, no. 2, pp. 882–889, 2021.
* [3] M. Moll, I. A. Sucan, and L. E. Kavraki, “Benchmarking motion planning algorithms: An extensible infrastructure for analysis and visualization,” _IEEE Robotics & Automation Magazine_, vol. 22, no. 3, pp. 96–102, 2015\.
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# Inflation and reheating in quadratic metric-affine gravity with derivative
couplings
Ioannis D. Gialamas 0000-0002-2957-5276 , Theodoros Katsoulas
0000-0003-4103-7937 and Kyriakos Tamvakis 0009-0007-7953-9816
###### Abstract
Within the framework of metric-affine theories of gravity, where both the
metric and connection are treated as independent variables, we consider
actions quadratic in the Ricci scalar curvature coupled non-minimally to a
scalar field through derivative couplings. Our analysis delves into the
inflationary predictions, revealing their consistency with the latest
observational constraints across a wide range of parameters. This
compatibility permits adjustments such as an increase in the spectral index
and a reduction in the tensor-to-scalar ratio. While we do not propose a
specific reheating mechanism, our analysis demonstrates that within the
quadratic model of inflation, the maximum reheating temperature can reach
$\sim 3\times 10^{15}\;\mathrm{GeV}$.
## 1 Introduction
Cosmological inflation [1, 2, 3, 4] offers a natural explanation of how
quantum fluctuations of gravitational and matter fields can be promoted to the
cosmological perturbations [5, 6, 7, 8, 9, 10] that are the origin of large
scale structure of the Universe. The standard way of realizing inflation is
through the vacuum energy of a scalar degree of freedom (the inflaton), either
introduced as a fundamental field or as part of gravity itself. In both cases
non-minimal couplings of the inflaton to curvature are expected to be present
as corrections to the classical general relativity (GR) action arising from
the quantum interactions of gravitating matter fields. Such are couplings of
the scalar field to the Ricci scalar curvature $\sim f(\phi){R}$ [11] or
derivative couplings to the Ricci tensor
$\sim(\partial_{\mu}\phi\partial_{\nu}\phi)R^{\mu\nu}$ [12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Quadratic terms of
the curvature invariants [31] are also expected to modify the classical
action, a particular case being that of a quadratic Ricci scalar term $\sim
R^{2}$ giving rise to the Starobinsky model of inflation [32].
The standard metric formulation of GR, where the connection is given by the
Levi-Civita relation in terms of the metric, is known to be equivalent to the
so-called metric-affine formulation, where the connection is an independent
variable. Nevertheless, this equivalence ceases to be true if we depart from
the Einstein-Hilbert form of the action and consider modifications such as
non-minimal couplings of scalar fields to the curvature or quadratic curvature
terms. In the general framework of metric-affine theories of gravity, while
general covariance is preserved, the connection is promoted to an independent
variable independent of the metric. The difference of the independent
connection and the Levi-Civita one is the so-called distortion tensor, which
in cases that it can be integrated out, leads to a resulting effective metric
theory with or without additional dynamical degrees of freedom. A
characteristic example is the case of the metric-affine (Palatini) version of
the ${\cal{R}}^{2}$ model [33] which, in contrast to the analogous metric
model, does not predict any propagating scalar mode, although, when coupled to
a scalar field, gives rise to a characteristic inflationary plateau
independently of the scalar self interaction potential. This general feature
of metric-affine ${\cal{R}}^{2}$ models [33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76] is maintained
in the presence of non-minimal $f(\phi){\cal{R}}$ couplings [77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99,
100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114,
115, 116, 117, 118, 119, 120]. Nevertheless, the effect of derivative
couplings $\sim(\partial\phi)^{2}{\cal{R}}$ or
$\sim(\partial_{\mu}\phi\partial_{\nu}\phi)\mathcal{R}^{\mu\nu}$ on this
behaviour [121, 122], being an open issue, is the main focus of the present
paper. As working examples we consider the case of a scalar field with a
quadratic potential and the case of a scalar field with a quartic self
interaction. In both models the inclusion of the derivative couplings can
increase the value of the spectral index. In the absence of the
$\mathcal{R}^{2}$ term the tensor-to-scalar ratio, $r$, is reduced but the
reduction is not enough to render the models compatible with the observational
data. The largest reduction of $r$ comes from the inclusion of the
$\mathcal{R}^{2}$ term [36, 38]. The effect of the derivative couplings in
combination with the $\mathcal{R}^{2}$ term can rescue both models aligning
their inflationary predictions more closely with the latest observational
constraints. We also analyze the issue of reheating in the simpler case of the
quadratic model and, without adopting any specific reheating mechanism, we
find that the maximum reheating temperature $T_{\rm ins}$, is of order $\sim
3\times 10^{15}\;\mathrm{GeV}$ , with small deviations for varying the
derivative coupling parameter $\tilde{\alpha}$ and small values of the
${\cal{R}}^{2}$-parameter $\beta$. For large $\beta$ the maximum reheating
temperature is independent of $\tilde{\alpha}$ and behaves as $\beta^{-1/4}$.
The outline of the paper is as follows. In section 2 we set up the theoretical
framework of metric-affine gravity with derivative couplings and quadratic in
curvature terms. Section 3 discusses the inflationary predictions of the
quadratic and quartic models. The reheating temperature is computed in section
4. Finally, we summarize and conclude in section 5.
Throughout the paper, we adopt the mostly plus signature for the metric and we
use natural units, setting the reduced Planck mass $M_{\rm P}$ to one.
## 2 The model
Metric-affine theories of gravity treat the metric tensor $g_{\mu\nu}$ and the
connection $\tensor{\Gamma}{{}^{\lambda}_{\mu}{}_{\nu}}$ as independent
variables. The independent connection can be decomposed as
$\tensor{\Gamma}{{}^{\lambda}_{\mu}{}_{\nu}}=\\{\tensor{}{{}^{\lambda}_{\mu}{}_{\nu}}\\}+\tensor{\mathcal{C}}{{}_{\mu}^{\lambda}{}_{\nu}}\,,$
(2.1)
where $\\{\tensor{}{{}^{\lambda}_{\mu}{}_{\nu}}\\}$ is the usual Levi-Civita
tensor and $\tensor{\mathcal{C}}{{}^{\lambda}_{\mu}{}_{\nu}}$ is the so-called
distortion tensor. The Riemann tensor is given by
$\displaystyle\tensor{\mathcal{R}}{{}^{\alpha}_{\beta}{}_{\gamma}{}_{\delta}}$
$\displaystyle=\partial_{\gamma}\tensor{\Gamma}{{}^{\alpha}_{\delta}{}_{\beta}}-\partial_{\delta}\tensor{\Gamma}{{}^{\alpha}_{\gamma}{}_{\beta}}+\tensor{\Gamma}{{}^{\alpha}_{\gamma}{}_{\mu}}\tensor{\Gamma}{{}^{\mu}_{\delta}{}_{\beta}}-\tensor{\Gamma}{{}^{\alpha}_{\delta}{}_{\mu}}\tensor{\Gamma}{{}^{\mu}_{\gamma}{}_{\beta}}$
$\displaystyle=\tensor{R}{{}^{\alpha}_{\beta}{}_{\gamma}{}_{\delta}}+\nabla_{\gamma}\tensor{\mathcal{C}}{{}_{\delta}^{\alpha}{}_{\beta}}-\nabla_{\delta}\tensor{\mathcal{C}}{{}_{\gamma}^{\alpha}{}_{\beta}}+\tensor{\mathcal{C}}{{}_{\gamma}^{\alpha}{}_{\mu}}\tensor{\mathcal{C}}{{}_{\delta}^{\mu}{}_{\beta}}-\tensor{\mathcal{C}}{{}_{\delta}^{\alpha}{}_{\mu}}\tensor{\mathcal{C}}{{}_{\gamma}^{\mu}{}_{\beta}}\,,$
(2.2)
where $\tensor{R}{{}^{\alpha}_{\beta}{}_{\gamma}{}_{\delta}}$ is the metric
Riemann tensor constructed from the metric and $\nabla$ is the covariant
derivative in terms of the Levi-Civita connection. The only symmetry of
$\tensor{\mathcal{R}}{{}^{\alpha}_{\beta}{}_{\gamma}{}_{\delta}}$ is the
antisymmetry under the interchange of the last two indices. As a result, there
are three non-zero contractions given by
$\tensor{\mathcal{R}}{{}_{\mu}{}_{\nu}}=\tensor{\mathcal{R}}{{}^{\rho}_{\mu}{}_{\rho}{}_{\nu}}\,,\qquad\tensor{\widehat{\mathcal{R}}}{{}^{\mu}_{\nu}}=g^{\alpha\beta}\tensor{\mathcal{R}}{{}^{\mu}_{\alpha}{}_{\beta}{}_{\nu}}\,,\qquad\tensor{\tilde{\mathcal{R}}}{{}_{\mu}{}_{\nu}}=\tensor{\mathcal{R}}{{}^{\alpha}_{\alpha}{}_{\mu}{}_{\nu}}\,,$
(2.3)
called the Ricci, co-Ricci, and homothetic curvature tensor, respectively.
There is a single Ricci scalar determined through an additional contraction of
either the Ricci tensor or the co-Ricci tensor, expressed as follows:
$\mathcal{R}=g^{\mu\nu}\tensor{\mathcal{R}}{{}_{\mu}{}_{\nu}}=-\tensor{\widehat{\mathcal{R}}}{{}^{\mu}_{\mu}}\,.$
(2.4)
We consider the following metric-affine action of a scalar field $\phi$
coupled non-minimally to the Ricci scalar as well as the Ricci tensors through
derivative couplings:
$\displaystyle\mathcal{S}=\int{\rm
d}^{4}x\sqrt{-g}\bigg{(}\frac{1}{2}(f(\phi)+\alpha_{1}X)\mathcal{R}-\frac{1}{2}K(\phi)X+\alpha_{2}\mathcal{R}^{\mu\nu}X_{\mu\nu}+\alpha_{3}\tensor{\widehat{\mathcal{R}}}{{}^{\mu}{}^{\nu}}X_{\mu\nu}+\frac{\beta}{4}\mathcal{R}^{2}-V(\phi)\bigg{)}\,,$
(2.5)
where
$X_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi\qquad\text{and}\qquad
X=g^{\mu\nu}X_{\mu\nu}\,.$ (2.6)
There is no coupling of the homothetic curvature Ricci tensor
$\tensor{\tilde{\mathcal{R}}}{{}_{\mu}{}_{\nu}}$ due to the antisymmetry of
its indices. Note that in the metric case (i.e.
$\tensor{\mathcal{C}}{{}_{\mu}^{\lambda}{}_{\nu}}=0$) there is no separate
$a_{3}$ coupling since $\widehat{R}_{\mu\nu}=-R_{\mu\nu}$. The functions
$F(\phi)$, $K(\phi)$, and $V(\phi)$ represent nonminimal couplings, non-
canonical kinetic terms, and the potential term of the scalar field,
respectively. As we have remarked in the introduction quadratic terms of the
curvature are bound to be generated from quantum corrections due to
gravitating matter fields. Nevertheless, general quadratic terms of the
Riemann and Ricci tensors [123, 124, 55] are known to be associated with
unphysical degrees of freedom [125, 126, 127, 128, 129, 130], in contrast to
quadratic terms of the Ricci scalar which is renowned for its reliable
inflationary predictions [36, 38] both in the metric as well as the metric-
affine case. Therefore, in the action (2.5), we have also incorporated a
quadratic Ricci scalar term. Note that the $\mathcal{R}^{2}$ term is not the
only possible safe quadratic term that can be added to the action. A quadratic
term constructed from the Holst invariant
$\tensor{\epsilon}{{}_{\alpha}^{\beta}{}^{\gamma}{}^{\delta}}\tensor{\mathcal{R}}{{}^{\alpha}_{\beta}{}_{\gamma}{}_{\delta}}$,
can also be incorporated. The implications of such terms in inflation have
been investigated in studies such as [64, 67, 74]. Nevertheless, in the
present study we concentrate on the presence of $\mathcal{R}^{2}$.
The $\mathcal{R}^{2}$ term can be written in terms of the auxiliary scalar
field $\chi$ as $\mathcal{R}^{2}=2\chi\mathcal{R}-\chi^{2}$, so the action
(2.5) takes the form
$\displaystyle\mathcal{S}=\int{\rm
d}^{4}x\sqrt{-g}\bigg{(}\frac{1}{2}(\mathcal{F}(\phi,\chi)+\alpha_{1}X)\mathcal{R}-\frac{1}{2}K(\phi)X+\alpha_{2}\mathcal{R}^{\mu\nu}X_{\mu\nu}+\alpha_{3}\tensor{\widehat{\mathcal{R}}}{{}^{\mu}{}^{\nu}}X_{\mu\nu}-U(\phi,\chi)\bigg{)}\,,$
(2.7)
with
$\mathcal{F}(\phi,\chi)=f(\phi)+\beta\chi\qquad\text{and}\qquad
U(\phi,\chi)=V(\phi)+\frac{\beta}{4}\chi^{2}\,.$ (2.8)
To delve into the dynamics of inflation within the theory, it becomes
necessary to rephrase the action in what is known as the Einstein frame. While
actions constructed solely from the Ricci scalar can be transformed via a Weyl
rescaling (or conformal transformation), since our action (2.5) involves
derivative couplings of the scalar field to the Ricci tensors, we need to
employ a broader set of transformations, namely the so-called disformal
transformations. These transformations are defined as (we have closely
followed the notation of [122])
$g_{\mu\nu}=\gamma_{1}(\phi,\tilde{X})\tilde{g}_{\mu\nu}+\gamma_{2}(\phi,\tilde{X})X_{\mu\nu}\,,$
(2.9)
where $\tilde{X}=\tilde{g}^{\mu\nu}X_{\mu\nu}$. The inverse transformation is
$\tilde{g}_{\mu\nu}=\tilde{\gamma}_{1}(\phi,X)g_{\mu\nu}+\tilde{\gamma}_{2}(\phi,X)X_{\mu\nu}\,,$
(2.10)
with $\tilde{\gamma}_{1}=1/\gamma_{1}$ and
$\tilde{\gamma}_{2}=-\gamma_{2}/\gamma_{1}$, while the determinants are
related by the equation
$g=\tilde{g}\gamma_{1}^{3}(\phi,\tilde{X})(\gamma_{1}(\phi,\tilde{X})+\gamma_{2}(\phi,\tilde{X})\tilde{X})$.
Using the relations for the inverse metrics $g^{\mu\nu}$ and
$\tilde{g}^{\mu\nu}$ we obtain also that
$X=\frac{\tilde{X}}{\gamma_{1}(\phi,\tilde{X})+\gamma_{2}(\phi,\tilde{X})\tilde{X}}\qquad\text{and}\qquad\tilde{X}=\frac{X}{\tilde{\gamma}_{1}(\phi,X)+\tilde{\gamma}_{2}(\phi,X)X}\,.$
(2.11)
Following [122] we may replace the co-Ricci tensor with the so-called average
Ricci tensor, defined as
$\overline{\mathcal{R}}_{\mu\nu}=(\mathcal{R}_{\mu\nu}+\widehat{\mathcal{R}}_{\mu\nu})/2$,
which vanishes if the connection is metric compatible. In general metric-
affine theories have non-zero torsion
$T_{\mu\lambda\nu}=2\mathcal{C}_{[\mu|\lambda|\nu]}$ (see [131] for a recent
review on gravitational theories with torsion) and non-metricity
$\nabla_{\rho}g_{\mu\nu}=-2\mathcal{C}_{(\mu\nu)\rho}$. In the Einstein-Cartan
gravity framework, non-metricity is assumed to be zero, whereas in Palatini
gravity, the torsion is required to vanish. In the subsequent discussion, we
will initially maintain the coupling to the average Ricci tensor. However,
ultimately, we will omit it, focusing our analysis solely on the Einstein-
Cartan case.
By substituting the average Ricci tensor into equation (2.7) and applying the
disformal transformation, we derive
$\displaystyle\mathcal{S}=\int{\rm d}^{4}x\sqrt{-g}\bigg{[}$ $\displaystyle
F_{1}(\phi,X,\chi)\frac{\mathcal{R}}{2}+F_{2}(\phi,X)\overline{\mathcal{R}}^{\mu\nu}X_{\mu\nu}+F_{3}(\phi,X,\chi)\mathcal{R}^{\mu\nu}X_{\mu\nu}$
$\displaystyle-F_{4}(\phi,X)X-F_{5}(\phi,X,\chi)U(\phi,\chi)\bigg{]}\,.$
(2.12)
The functions $F_{i}$ are given by
$\displaystyle F_{1}(\phi,X,\chi)$ $\displaystyle=(1+\gamma
X)^{1/2}\left(\gamma_{1}\mathcal{F}(\phi,\chi)+\frac{\alpha_{1}X}{1+\gamma
X}\right)\,,$ (2.13a) $\displaystyle F_{2}(\phi,X)$
$\displaystyle=\alpha_{3}(1+\gamma X)^{-1/2}\,,$ (2.13b) $\displaystyle
F_{3}(\phi,X,\chi)$ $\displaystyle=\frac{1}{2}(1+\gamma
X)^{-1/2}\left(-\gamma_{2}\mathcal{F}(\phi,\chi)+\frac{\alpha_{2}-\alpha_{3}-(\alpha_{1}+\alpha_{3})\gamma
X}{(1+\gamma X)}\right)\,,$ (2.13c) $\displaystyle F_{4}(\phi,X)$
$\displaystyle=\frac{1}{2}\gamma_{1}(1+\gamma X)^{-1/2}K(\phi)\,,$ (2.13d)
$\displaystyle F_{5}(\phi,X,\chi)$ $\displaystyle=\gamma_{1}^{2}(1+\gamma
X)^{1/2}\,,$ (2.13e)
where, for brevity, we have omitted the tildes from the rescaled quantities
and the arguments from the functions $\gamma$ and $\gamma_{1}$. Our action (2)
aligns with the one presented in [122], with the sole distinction being the
replacement given by equation (2.8). Note however that the presence of the
auxiliary scalar $\chi$ will turn out to have important effects on the
inflationary behaviour.
If we assume that $\alpha_{3}=0$ (i.e. the Einstein-Cartan case), to derive
the action in the EF, we essentially need to solve the system of equations
$F_{1}(\phi,X,\chi)=1$ and $F_{3}(\phi,X,\chi)=0$. Solving this system is
inherently challenging. However, we can approximate the solutions by assuming
that in the slow-roll approximation, the higher-order kinetic terms are
negligible (i.e. $X\ll 1$), particularly during inflation [43], as well as
during reheating [53]. An approximate solution under this assumption is111Note
that the invertibility of the disformal transformation requires
$\gamma_{1}>0,\,\,\,\,\gamma_{2}\geq
0,\,\,\,\,\gamma_{1}+\tilde{X}\gamma_{2}>0,\,\,\newline
\tilde{\gamma}_{1}-X\partial\tilde{\gamma}_{1}/\partial
X-X^{2}\partial\tilde{\gamma}_{2}/\partial X\neq 0$. Using the approximate
solution (2.14) we see that the above system is equivalent to $\alpha_{2}>0$,
$\alpha_{1}+\alpha_{2}<0$ and $|X|<1/(-\alpha_{1}+\alpha_{2}/2)$.
Additionally, we can infer that the combination $4\alpha_{1}+\alpha_{2}$
involved in the inflationary dynamics is negative, as required.
$\gamma\simeq\alpha_{2}-\frac{\alpha_{2}^{2}}{2}X+\frac{5\alpha_{2}^{3}}{8}X^{2}\,,\quad\gamma_{1}\simeq\frac{1}{\mathcal{F}(\phi,\chi)}\left(1-(\alpha_{1}+\alpha_{2}/2)X+(\alpha_{1}\alpha_{2}+5\alpha_{2}^{2}/8)X^{2}\right)\,,$
(2.14)
where we kept terms up to $\mathcal{O}(X^{2})$. Substituting the solution back
to the action we obtain
$\displaystyle\mathcal{S}=\int{\rm d}^{4}x\sqrt{-g}\bigg{[}$
$\displaystyle\frac{R}{2}-\frac{K(\phi)X}{2\mathcal{F}(\phi,\chi)}(1-(\alpha_{1}+\alpha_{2})X)$
$\displaystyle-\frac{U(\phi,\chi)}{\mathcal{F}^{2}(\phi,\chi)}\left(1-(2\alpha_{1}+\alpha_{2}/2)X+(\alpha_{1}^{2}+2\alpha_{1}\alpha_{2}+5\alpha_{2}^{2}/8)X^{2}\right)\bigg{]}\,,$
(2.15)
where now the Ricci scalar $R$ is the one constructed by the Levi-Civita
connection. Note that for $\alpha_{1}=\alpha_{2}=0$ the above action is
reduced to the known Palatini-$\mathcal{R}^{2}$ action [36, 38]. The
subsequent step involves varying the action with respect to the auxiliary
field $\chi$. Upon doing so, the solution $\chi=\chi(\phi,X)$ must be re-
expanded in powers of the kinetic term $X$ and then substituted back into the
action. The resulting effective action will represent the final metric action
of the scalar field $\phi$, featuring modified potential and kinetic terms.
The variation gives
$\frac{\delta\mathcal{S}}{\delta\chi}=0\quad\Rightarrow\quad\chi=\frac{4V(\phi)+A(\phi)X+B(\phi)X^{2}}{f(\phi)+C(\phi)X+D(\phi)X^{2}}\,,$
(2.16)
with
$\displaystyle A(\phi)=$ $\displaystyle
K(\phi)f(\phi)-4V(\phi)(2\alpha_{1}+\alpha_{2}/2)\,,$ (2.17a) $\displaystyle
B(\phi)=$ $\displaystyle
4V(\phi)(\alpha_{1}^{2}+2\alpha_{1}\alpha_{2}+5\alpha_{2}^{2}/8)-(\alpha_{1}+\alpha_{2})K(\phi)f(\phi)\,,$
(2.17b) $\displaystyle C(\phi)=$ $\displaystyle-\beta
K(\phi)-f(\phi)(2\alpha_{1}+\alpha_{2}/2)\,,$ (2.17c) $\displaystyle D(\phi)=$
$\displaystyle\beta(\alpha_{1}+\alpha_{2})K(\phi)+(\alpha_{1}^{2}+2\alpha_{1}\alpha_{2}+5\alpha_{2}^{2}/8)f(\phi)\,.$
(2.17d)
Expanding in powers of $X$ we obtain222In the minimal case $f(\phi)=K(\phi)=1$
the auxiliary field reads $\chi\simeq 4V(\phi)+(1+4\beta V(\phi))X+(1+4\beta
V(\phi))(\beta+\alpha_{1}-\alpha_{2}/2)X^{2}+\mathcal{O}(X^{3})\,.$ (2.18)
$\chi\simeq\frac{4V(\phi)}{f(\phi)}+XK(\phi)\left(1+4\beta\frac{V(\phi)}{f^{2}(\phi)}\right)+X^{2}\frac{K(\phi)}{2f(\phi)}\left(1+4\beta\frac{V(\phi)}{f^{2}(\phi)}\right)\left(\,(2\alpha_{1}-\alpha_{2})f(\phi)+2\beta
K(\phi)\right)\,.$ (2.19)
Substituting the eq. (2.19) back to the action (2) and re-expanded in powers
of $X$ the $\chi$-dependent part of the first line gives
${\cal{L}}_{1}\simeq-X\frac{K(\phi)}{2f(\phi)\left(1+4\beta\frac{V(\phi)}{f^{2}(\phi)}\right)}+X^{2}\frac{K(\phi)}{2f^{2}(\phi)\left(1+4\beta\frac{V(\phi)}{f^{2}(\phi)}\right)}\left((\alpha_{1}+\alpha_{2})f(\phi)+\beta
K(\phi)\right)\,,$ (2.20)
while the second line reads
${\cal{L}}_{2}\simeq\frac{V(\phi)}{f^{2}(\phi)+4\beta
V(\phi)}-\frac{X}{2}\frac{(4\alpha_{1}+\alpha_{2})V(\phi)}{f^{2}(\phi)+4\beta
V(\phi)}+\frac{X^{2}}{8}\frac{\left(2\beta
K^{2}(\phi)+(8\alpha_{1}^{2}+5\alpha_{2}^{2}+16\alpha_{1}\alpha_{2})V(\phi)\right)}{f^{2}(\phi)+4\beta
V(\phi)}\,.$ (2.21)
Having done all the groundwork, we can substitute the last expansions in the
action (2) to obtain
$\mathcal{S}=\int{\rm
d}^{4}x\sqrt{-g}\left(\frac{R}{2}-\bar{K}(\phi)\frac{X}{2}-\bar{U}(\phi)+\mathcal{O}(X^{2})\right)\,,$
(2.22)
with
$\bar{K}(\phi)=\frac{-\tilde{\alpha}V(\phi)+f(\phi)K(\phi)}{\left(f^{2}(\phi)+4\beta
V(\phi)\right)}\qquad\text{and}\qquad\bar{U}(\phi)=\frac{V(\phi)}{f^{2}(\phi)+4\beta
V(\phi)}\,,$ (2.23)
where we have defined $4\alpha_{1}+\alpha_{2}=\tilde{\alpha}$. Although we
have retained the functions $f(\phi)$ and $K(\phi)$ for generality in this
discussion, moving forward, we will simplify the analysis by considering the
minimal case where $f(\phi)=K(\phi)=1$. Note that, achieving canonical form
for the kinetic term is possible through the field redefinition ${\rm
d}\phi_{c}=\sqrt{\bar{K}(\phi)}{\rm d}\phi$. Consequently, the canonically
normalized inflaton $\phi_{c}$ can, in principle, be expressed as a function
of $\phi$. However, it is not obligatory to employ a canonical field for
determining inflationary parameters. By directly working with $\phi$, this
impediment can be bypassed.
## 3 Inflation
Regarding cosmological observables and assuming the slow-roll approximation,
we initiate the discussion by introducing the scalar ($\mathcal{P}_{\zeta}$)
and tensor ($\mathcal{P}_{T}$) power spectra, crucial elements in inflationary
cosmology. By selecting an arbitrary pivot scale $k_{\star}$ that exited the
horizon, the expressions for the scalar and tensor power spectra take the
form:
$\mathcal{P}_{\zeta}(k)=A_{s}\left(\frac{k}{k_{\star}}\right)^{n_{s}-1},\quad
A_{s}\simeq\frac{1}{24\pi^{2}}\frac{\bar{U}(\phi_{\star})}{\epsilon_{\bar{U}}(\phi_{\star})}\qquad\text{and}\qquad\mathcal{P}_{T}(k)\simeq\frac{2\bar{U}(\phi_{\star})}{3\pi^{2}}\left(\frac{k}{k_{\star}}\right)^{n_{t}}\,,$
(3.1)
where $A_{s}$ is the amplitude of the power spectrum of scalar perturbations.
The scalar ($n_{s}$) and tensor ($n_{t}$) spectral indices given by
$n_{s}-1=\frac{{\rm d}\ln\mathcal{P}_{\zeta}(k)}{{\rm d}\ln
k}\simeq-6\epsilon_{\bar{U}}+2\eta_{\bar{U}}\qquad\text{and}\qquad
n_{t}=\frac{{\rm d}\ln\mathcal{P}_{T}(k)}{{\rm d}\ln k}\,,$ (3.2)
characterize the scale-dependence of the power spectra (3.1). In the equations
above we have used the potential slow-roll parameters
$\epsilon_{\bar{U}}=\frac{1}{2\bar{K}(\phi)}\left(\frac{\bar{U}^{\prime}(\phi)}{\bar{U}(\phi)}\right)^{2}\qquad\text{and}\qquad\eta_{\bar{U}}=\frac{\left(\bar{K}^{-1/2}(\phi)\bar{U}^{\prime}(\phi)\right)^{\prime}}{\bar{K}^{1/2}(\phi)\bar{U}(\phi)}\,.$
(3.3)
In these equations primes denote derivatives with respect the scalar field,
while both slow-roll parameters are small $(\ll 1)$ during inflation and one
of them approaches unity near its end.
The tensor-to-scalar ratio is defined as
$r=\frac{\mathcal{P}_{T}(k)}{\mathcal{P}_{\zeta}(k)}\simeq
16\epsilon_{\bar{U}},$ (3.4)
while the duration of inflation is measured by the number of $e$-folds
$N_{\star}=\int^{\phi_{\star}}_{\phi_{\rm
end}}\bar{K}(\phi)\frac{\bar{U}(\phi)}{\bar{U}^{\prime}(\phi)}{\rm d}\phi\,.$
(3.5)
The inflationary predictions are significantly constrained by observations of
the cosmic microwave background (CMB), as demonstrated in [132, 133]. The most
recent combination of Planck, BICEP/Keck, and BAO data has established the
following limits on the observable values at the pivot scale
$k_{\star}=0.05\,{\rm Mpc}^{-1}$:
$A_{s}=(2.10\pm 0.03)\times 10^{-9},\qquad n_{s}=0.9649\pm
0.0042\quad(1\sigma\mbox{ region}),\qquad r<0.036\,.$ (3.6)
Subsequently, we examine particular models and delve into their predictions.
We focus on an intriguing category of models where the potential, $V$, takes
the form of a monomial in the field $\phi$, i.e. $V\sim\phi^{n}$, with $n$
even integer. More precisely we will study the quadratic $(n=2)$ and quartic
$(n=4)$ models of inflation.
### 3.1 The quadratic model
We first consider the simple case of the minimally coupled ($f(\phi)=1$)
quadratic model with a potential
$V(\phi)=\frac{m^{2}}{2}\phi^{2}\,,$ (3.7)
where $m$ is a parameter of dimension ${\rm mass}^{2}$. Provided that the
$\mathcal{O}(X^{2})$ terns are small, the first slow-roll parameters (3.3) are
given by
$\epsilon_{\bar{U}}=\frac{4}{\phi^{2}(2-\tilde{\alpha}m^{2}\phi^{2})(1+2\beta
m^{2}\phi^{2})}\,,\qquad\eta_{\bar{U}}=\frac{8\left(1+\beta
m^{2}\phi^{2}(3\tilde{\alpha}m^{2}\phi^{2}-4)\right)}{\phi^{2}(2-\tilde{\alpha}m^{2}\phi^{2})^{2}(1+2\beta
m^{2}\phi^{2})}\,.$ (3.8)
The number of $e-$folds (3.5), left to the end of on inflation are
$N_{\star}=\frac{1}{16}(\phi_{\rm
end}^{2}-\phi_{\star}^{2})\left(\tilde{\alpha}m^{2}(\phi_{\rm
end}^{2}+\phi_{\star}^{2})-4\right)\simeq\frac{1}{16}\phi_{\star}^{2}(4-\tilde{\alpha}m^{2}\phi_{\star}^{2})\,,$
(3.9)
where the second equality holds for $\phi_{\rm end}^{2}\ll\phi_{\star}^{2}$.
The above equation (without neglecting $\phi_{\rm end}$ is a quadratic
equation for $\phi_{\star}^{2}$. The sole solution that restores the correct
$\tilde{\alpha}=0$ limit is
$\phi_{\star}^{2}=\frac{2-\sqrt{\tilde{\alpha}^{2}m^{4}\phi_{\rm
end}^{4}-4\tilde{\alpha}m^{2}\phi_{\rm
end}^{2}+4-16\tilde{\alpha}m^{2}N_{\star}}}{\tilde{\alpha}m^{2}}\xrightarrow{\tilde{a}\rightarrow
0}\phi_{\rm end}^{2}+4N_{\star}\,.$ (3.10)
The field value at the end of inflation is defined by the condition
$\epsilon_{\bar{U}}(\phi_{\rm end})=1\Rightarrow$
$2\tilde{\alpha}\beta m^{4}\phi_{\rm end}^{6}+(\tilde{\alpha}m^{2}-4\beta
m^{2})\phi_{\rm end}^{4}-2\phi_{\rm end}^{2}+4=0\,.$ (3.11)
In [43], it has been shown that for $\tilde{\alpha}=0$ the field value at the
end of inflation is bounded from above, namely $\phi_{\rm end}^{2}<2$. So in
this case, the condition $\phi_{\rm end}^{2}\ll\phi_{\star}^{2}$ is indeed
true. In our scenario we have seen numerically that such a bound for the
$\phi_{\rm end}^{2}$ also exists, thereby ensuring that the approximation
$\phi_{\rm end}^{2}\ll\phi_{\star}^{2}$ holds true.
Figure 1: Left: The mass parameter $m^{2}$ given by eq. (3.14) as function of
the parameter $\tilde{\alpha}$. Right: The spectral index, given by eq.
(3.16), as function of the parameter $\tilde{\alpha}$ for $\beta=0$. The
shaded regions represent the permissible parameter space at confidence levels
of $68\%$ (dark blue) and $95\%$ (light blue), as derived from the most recent
combination of Planck, BICEP/Keck, and BAO data [132, 133].
In what follows, we safely omit the term $\phi_{\rm end}$. Under this
approximation the field value at the horizon crossing is given by
$\phi_{\star}^{2}\simeq\frac{2-2\sqrt{1-4\tilde{\alpha}m^{2}N_{\star}}}{\tilde{\alpha}m^{2}}\,.$
(3.12)
Using $\phi_{\star}$, given above, $A_{s}$ of eq. (3.1) is written in terms of
$N_{\star}$ as
$A_{s}\simeq\frac{\sqrt{1-4\tilde{\alpha}m^{2}N_{\star}}\left(1-\sqrt{1-4\tilde{\alpha}m^{2}N_{\star}}\right)^{2}}{24\pi^{2}\tilde{\alpha}^{2}m^{2}}\,,$
(3.13)
which under this approximation does not depend on the parameter $\beta$. The
above equation can be used in order to fix one of the parameters. If we solve
with respect to $m^{2}$, the only solution that recovers the correct
$\tilde{\alpha}=0$ limit, namely $m^{2}\simeq 6\pi^{2}A_{s}/N_{\star}^{2}$, is
$m^{2}=\frac{3\pi^{2}A_{S}}{4\mathring{A}N_{\star}^{3}}\left[N_{\star}^{2}+4\mathring{A}N_{\star}-\mathring{A}^{2}-(N_{\star}-\mathring{A})\left(N_{\star}^{2}-6\mathring{A}N_{\star}+\mathring{A}^{2}\right)^{1/2}\right]\,,\quad\mathring{A}=6\pi^{2}A_{s}\tilde{\alpha}\,.$
(3.14)
The impact of the parameter $\tilde{\alpha}$ on $m^{2}$ is visible once the
parameter $\mathring{A}$ reaches a comparable or larger magnitude than
$N_{\star}$.333This can be easily observed if we expand around
$\mathring{A}\sim 0$. The mass for small $|\mathring{A}|$ is given by
$m^{2}\simeq\frac{6\pi^{2}A_{s}}{N_{\star}^{2}}\left[1+\left(\frac{\mathring{A}}{N_{\star}}\right)^{2}+4\left(\frac{\mathring{A}}{N_{\star}}\right)^{3}+\mathcal{O}(\mathring{A}^{4}/N_{\star}^{4})\right]\,.$
(3.15) For $N_{\star}\sim 50-60$ this occurs for $|\tilde{\alpha}|\gtrsim
10^{8}$.
The left panel of Fig. 1 illustrates the dependence of the parameter $m^{2}$
on the parameter $\tilde{\alpha}$ according to the aforementioned equation. It
is evident that for “small” values of $|\tilde{\alpha}|$ there is no
discernible effect on the value of the parameter $m^{2}$. However, for
$|\tilde{\alpha}|\gg 10^{8}$ it exhibits a linear growth as
$m^{2}\simeq-\frac{9\pi^{4}A_{s}^{2}\tilde{\alpha}}{N_{\star}^{3}}\left[1+\mathcal{O}(N_{\star}/\mathring{A})\right]$.
The spectral index (3.2) is given by
$n_{s}\simeq
1-\frac{1+\sqrt{1-4\tilde{\alpha}m^{2}N_{\star}}-6\tilde{\alpha}m^{2}N_{\star}}{N_{\star}(1-4\tilde{\alpha}m^{2}N_{\star})}\simeq\begin{cases}\displaystyle
1-\frac{2}{N_{\star}}\,,&\text{if }\,\,|\mathring{A}|/N_{\star}\ll
1\\\\[8.5359pt] \displaystyle 1-\frac{3}{2N_{\star}}\,,&\text{if
}\,\,|\mathring{A}|/N_{\star}\gg 1\,.\end{cases}$ (3.16)
As in the $\tilde{\alpha}=0$ case [36], the spectral index is
$\beta$-independent to leading order in the slow-roll parameters as well as
the scalar power spectrum and the number of $e$-folds. Therefore, for small
$|\mathring{A}|$, always compared to $N_{\star}$, the prediction aligns with
that of the simple quadratic model of inflation. As $|\mathring{A}|$ increases
the spectral index also increases eventually reaching the asymptotic value
$n_{s}\simeq 1-3/(2N_{\star})$. As is also depicted in the right panel of Fig.
1 the asymptotic region for large $|\mathring{A}|$ is marginally outside of
the $2\sigma$ observational bounds, namely $N_{\star}$ is forced to be
$\lesssim 56$.
Finally, the tensor-to-scalar ratio (3.4) is given by
$r\simeq\frac{16\tilde{\alpha}m^{2}}{\sqrt{1-4\tilde{\alpha}m^{2}N_{\star}}(\sqrt{1-4\tilde{\alpha}m^{2}N_{\star}}-1)\left[4\beta(\sqrt{1-4\tilde{\alpha}m^{2}N_{\star}}-1)-\tilde{\alpha}\right]}\,,$
(3.17)
while its limiting cases are
$r\simeq\begin{cases}\displaystyle\frac{8}{N_{\star}+48\pi^{2}A_{s}\beta}\,,&\text{if
}\,\,|\mathring{A}|/N_{\star}\ll 1\\\\[8.5359pt]
\displaystyle\frac{4}{N_{\star}+24\pi^{2}A_{s}\beta}\,,&\text{if
}\,\,|\mathring{A}|/N_{\star}\gg 1\,.\end{cases}$ (3.18)
As shown in Fig. 2, the decrease in the tensor-to-scalar ratio for large
$|\mathring{A}|$ alone is insufficient to bring the value of $r$ within the
observational limit of $r<0.036$. As highlighted in the right panel, the
introduction of a substantial $\beta$ parameter ($\beta\gtrsim 10^{8}$)
becomes necessary.
Figure 2: The tensor-to-scalar ratio, given by eq (3.17), as function of the
parameter $\tilde{\alpha}$ for $\beta=0$ (left) and $\beta=10^{9}$ (right).
The latest observational data excludes the shaded region in the left panel,
since $r_{0.05}<0.036$ at $95\%$ confidence [133].
### 3.2 The quartic model
As a second model worth studying, is the quartic model
$V(\phi)=\frac{\lambda}{4}\phi^{4}\,.$ (3.19)
Here, the dimensionless coupling parameter $\lambda$ is referred to as the
quartic coupling. However, the inflationary predictions of this model do not
align with observations, whether considering the pure $\phi^{4}$ model or with
the inclusion of the $\mathcal{R}^{2}$ term. In the former case, both the
tensor-to-scalar ratio and the spectral index deviate from the established
bounds. In the latter case, the inclusion of the $\mathcal{R}^{2}$ term
succeeds in reducing the value of $r$, while $n_{s}$ remains around $\sim
0.94-0.96$ for $N_{\star}=50-60$. Subsequently, we will explore how derivative
couplings may enhance the value of $n_{s}$ to bring it within the
observationally allowed range444It is worth noting that the quartic model can
be rescued if one assumes a non-minimal coupling of the form
$\sim\xi\phi^{2}\mathcal{R}$ both in the metric [11] and the Palatini [77]
formulation. Nevertheless, here we shall consider the minimally coupled case
of $f(\phi)=1$ to isolate the effect of a non-minimal derivative coupling on
predictions..
As in the quadratic model, the first slow-roll parameters, are easily computed
to be
$\epsilon_{\bar{U}}=\frac{32}{\phi^{2}(4-\tilde{\alpha}\lambda\phi^{4})(1+\beta\lambda\phi^{4})}\,,\qquad\eta_{\bar{U}}=\frac{16\left[12(1-\beta\lambda\phi^{4})+\tilde{\alpha}\lambda\phi^{4}(5\beta\lambda\phi^{4}-1)\right]}{\phi^{2}(4-\tilde{\alpha}\lambda\phi^{4})^{2}(1+\beta\lambda\phi^{4})}\,,$
(3.20)
while the number of $e-$folds left to the end of inflation are
$N_{\star}=\frac{1}{96}\left[12(\phi_{\star}^{2}-\phi_{\rm
end}^{2})+\tilde{\alpha}\lambda(\phi_{\rm
end}^{6}-\phi_{\star}^{6})\right]\,.$ (3.21)
Using again the approximation $\phi_{\rm end}^{2}\ll\phi_{\star}^{2}$ we
obtain that the field value at the horizon crossing is
$\phi_{\star}^{2}\simeq\frac{-2(\tilde{\alpha}\lambda+Y^{2})}{\tilde{\alpha}\lambda
Y}\,,\qquad\text{with}\qquad
Y=\left(6\tilde{\alpha}^{2}\lambda^{2}N_{\star}^{2}+\sqrt{\tilde{\alpha}^{3}\lambda^{3}(36\tilde{\alpha}\lambda
N_{\star}^{2}-1)}\right)^{1/3}\,.$ (3.22)
Figure 3: Left: The quartic coupling $\lambda$ as function of the parameter
$\tilde{\alpha}$. Right: The spectral index, given by eq. (3.2), as function
of the parameter $\tilde{\alpha}$ for $\beta=0$. The shaded regions represent
the permissible parameter space at confidence levels of $68\%$ (dark blue) and
$95\%$ (light blue), as derived from the most recent combination of Planck,
BICEP/Keck, and BAO data [132, 133].
In contrast to the quadratic model, the intricate expression of
$\phi_{\star}^{2}$ prevents us from presenting a concise formula for the
quartic coupling $\lambda$, akin to what was done for the mass parameter in
equation (3.14). However, we can provide approximate limiting expressions
using the fact that the value of the quartic coupling $\lambda$ is fixed by
the observed value of the amplitude of the scalar power spectrum, $A_{s}\simeq
2.1\times 10^{-9}$ at the pivot scale $k_{\star}=0.05\,{\rm Mpc}^{-1}$. As
depicted in the left panel of Fig. 3, for $|\tilde{\alpha}|\ll 10^{8}$, we
recover the familiar expression for the quartic model, specifically
$\lambda\approx 3\pi^{2}A_{s}/(2N_{\star}^{3})$. On the other hand, for
$|\tilde{\alpha}|\gg 10^{8}$, the quartic coupling increases as
$\lambda\approx 32\pi^{6}A_{s}^{3}|\tilde{\alpha}|^{2}/(9N_{\star}^{5})$.
Furthermore, since an analytic expression for the quartic coupling $\lambda$
is unavailable, the spectral index is expressed as a function of it, given by:
$\displaystyle n_{s}\simeq$
$\displaystyle-\tilde{\alpha}\lambda\big{[}-\tilde{\alpha}\lambda
Y^{2}+\tilde{\alpha}^{2}\lambda^{2}(1+N_{\star}(48N_{\star}Y^{2}+2Y-1))+12\tilde{\alpha}^{3}\lambda^{3}N_{\star}^{2}(3N_{\star}-5)$
$\displaystyle+(1-8N_{\star}Y)Y(Y^{3}-6\tilde{\alpha}^{2}\lambda^{2}N_{\star}^{2})\big{]}/(Y^{3}-6\tilde{\alpha}^{2}\lambda^{2}N_{\star}^{2})^{2}\,.$
(3.23)
The limiting cases are given by
$n_{s}\simeq\begin{cases}\displaystyle 1-\frac{3}{N_{\star}}\,,&\text{if
}\,\,|\tilde{\alpha}|\ll 10^{8}\\\\[8.5359pt] \displaystyle
1-\frac{5}{3N_{\star}}\,,&\text{if }\,\,|\tilde{\alpha}|\gg
10^{8}\,.\end{cases}$ (3.24)
The $|\tilde{\alpha}|\ll 10^{8}$ limit, lies outside of the observational
bounds, since it coincides with the pure $\phi^{4}$ model. In the
$|\tilde{\alpha}|\gg 10^{8}$ limit, the spectral index takes the form
$n_{s}\simeq 1-5/(3N_{\star})\simeq 0.967-0.972$ for $N_{\star}=50-60$ (see
the right panel of Fig. 3). These values fall well within observational
bounds, marking a significant achievement as the derivative couplings, in
conjunction with $\mathcal{R}^{2}$, rescue the quartic model of inflation.
Figure 4: The tensor-to-scalar ratio, given by eq (3.25), as function of the
parameter $\tilde{\alpha}$ for $\beta=0$ (left) and $\beta=10^{9}$ (right).
The latest observational data excludes the shaded region in the left panel,
since $r_{0.05}<0.036$ at $95\%$ confidence [133].
As previously indicated, the derivative couplings alone are insufficient to
rescue this model. The tensor-to-scalar ratio is given by
$r\simeq
64\tilde{\alpha}^{4}\lambda^{3}Y^{5}\left[(Y^{2}+\tilde{\alpha}\lambda)(Y^{4}+\tilde{\alpha}\lambda
Y^{2}+\tilde{\alpha}^{2}\lambda^{2})(4\beta
Y^{4}+\tilde{\alpha}\lambda(\tilde{\alpha}+8\beta)Y^{2}+4\tilde{\alpha}^{2}\lambda^{2}\beta)\right]^{-1}\,,$
(3.25)
with the limits being
$r\simeq\begin{cases}\displaystyle\frac{16}{N_{\star}+96\pi^{2}A_{s}\beta}\,,&\text{if
}\,\,|\tilde{\alpha}|\ll 10^{8}\\\\[8.5359pt]
\displaystyle\frac{16/3}{N_{\star}+32\pi^{2}A_{s}\beta}\,,&\text{if
}\,\,|\tilde{\alpha}|\gg 10^{8}\,.\end{cases}$ (3.26)
As depicted in the left panel of Fig. 4, the tensor-to-scalar ratio is reduced
by approximately $67\%$, which is insufficient for agreement with the
observational bound $r<0.036$. The inclusion of $\mathcal{R}^{2}$ is
inevitable, and a value of $\beta>10^{8}$ (see the right panel of Fig. 4) is
adequate to relocate the tensor-to-scalar ratio within the allowed region.
## 4 Reheating
As an illustrative example within this section, we will compute the reheating
temperature, for the quadratic model discussed in Section 3.1, giving analytic
approximations in regions of the parameter space.
The number of $e$-folds during the reheating era is given by
$N_{\rm reh}\equiv\ln\left[\frac{a_{\rm reh}}{a_{\rm
end}}\right]=-\frac{1}{3(1+w)}\ln\left[\frac{\rho_{\rm reh}}{\rho_{\rm
end}}\right]\,,$ (4.1)
where the subscripts “reh” and “end” in the cosmic scale factor $a$ and the
energy densities $\rho$ indicate that these quantities are evaluated at the
end of the reheating period and inflation, respectively. The equation of state
parameter $w$ is a free parameter, taking the value of $-1/3$ during inflation
and $1/3$ during the radiation-dominated phase. In terms of the number of
$e$-folds during the reheating era, the reheating temperature is given by
$T_{\rm reh}=T_{\rm ins}\exp\left(-\frac{3(1+w)N_{\rm reh}}{4}\right)\,,\qquad
T_{\rm ins}\equiv\left(\frac{30}{\pi^{2}}\frac{\rho_{\rm
end}}{g_{\star}(T_{\rm reh})}\right)^{1/4}\,,$ (4.2)
where $T_{\rm ins}$ is the maximum temperature defined as the instantaneous
reheating temperature obtained for $a_{\rm reh}=a_{\rm end}$
($w=1/3$)555Regarding Palatini inflationary models, the authors of [43, 47,
54, 109, 66, 134] have investigated the inflationary predictions across a
range of values for the equation of state parameter, spanning from $-1/3$ to
$1$. The resulting reheating temperature, exhibits significant variability,
ranging from relatively low values near those during BBN, $\sim\mathrm{MeV}$,
up to much higher values of around $\sim 10^{16}\,\mathrm{GeV}$.. It’s worth
noting that $g_{\star}(T_{\text{reh}})$ represents the effective degrees of
freedom of entropy density, which is approximately $106.75$ under the
assumption of Standard Model particle content and temperatures around $\sim
1\;\mathrm{TeV}$ or higher.
To estimate the instantaneous reheating temperature, we only require the value
of the energy density at the end of inflation, which can be approximated
as666This equality holds precisely when there are no higher-order kinetic
terms. References [43, 66] have considered the inclusion of higher-order
kinetic terms, revealing that they yield only a negligible correction, as
demonstrated therein. $\rho_{\rm end}\simeq\frac{3}{2}\bar{U}(\phi_{\rm
end})\,.$ The field value at the end of inflation is determined by solving
equation (3.11). Since, this is a cubic equation for $\phi_{\rm end}^{2}$ its
solution and consequently the $T_{\rm ins}$ are too complicated to be
presented, although analytic expression for the unique positive solution does
exist. Therefore, our objective is to provide analytic expressions for the
limits $|\mathring{A}|/N_{\star}\ll 1$ and $|\mathring{A}|/N_{\star}\gg 1$, as
described in the preceding section, as well as for the cases $\beta\ll 10^{9}$
and $\beta\gg 10^{9}$.
Figure 5: In the $\beta$, $|\tilde{\alpha}|$ plane we display the
instantaneous reheating temperature, as given by eq. (4.2), for
$N_{\star}=55$.
For small values of $\beta$, i.e. $\beta\ll 10^{9}$ the instantaneous
reheating temperature can be approximated as
$T_{\rm ins}\simeq\begin{cases}\displaystyle
0.38\times\left(\frac{1-\sqrt{1-4\mathring{A}/N_{\star}^{2}}}{\tilde{\alpha}}\right)^{1/4}\stackrel{{\scriptstyle|\tilde{\alpha}|\rightarrow
0}}{{\simeq}}2.8\times
10^{15}\left(\frac{55}{N_{\star}}\right)^{1/2}\;\mathrm{GeV}\,,&\text{if
}\,\,|\mathring{A}|/N_{\star}\ll 1\\\\[8.5359pt] \displaystyle
0.38\times\left(\frac{1-\sqrt{1+\mathring{A}^{2}/N_{\star}^{3}}}{\tilde{\alpha}}\right)^{1/4}\stackrel{{\scriptstyle|\tilde{\alpha}|\rightarrow\infty}}{{\simeq}}3.9\times
10^{15}\left(\frac{55}{N_{\star}}\right)^{3/8}\;\mathrm{GeV}\,,&\text{if
}\,\,|\mathring{A}|/N_{\star}\gg 1\,.\end{cases}$ (4.3)
For large values of $\beta$, i.e. $\beta\gg 10^{9}$ in both $\tilde{\alpha}$
regimes we have that
$T_{\rm ins}\simeq 7.8\times 10^{17}\beta^{-1/4}\;\mathrm{GeV}\,.$ (4.4)
Note that we have reinstated the units by multiplying with $M_{\rm Pl}\simeq
2.4\times 10^{18}\;\mathrm{GeV}$.
In Fig. 5 we display the instantaneous reheating temperature, as given by eq.
(4.2), for $N_{\star}=55$. The light and dark blue regions represent the
limits outlined by equation (4.3), with the white-dotted contour indicating
the highest temperature $\sim 3.9\times 10^{15}\;\mathrm{GeV}$. As $\beta$
increases, the approximation (4.3) becomes more accurate, with the temperature
decreasing as $\beta^{-1/4}$ in aggreement with [43, 66]. In conclusion, as
indicated by the figure and the approximate expressions, the maximum
instantaneous reheating temperature is attained for small values of the
parameter $\beta$, typically ranging from approximately $(2.8-3.9)\times
10^{15}\;\mathrm{GeV}$ (for $N_{\star}=55$), as $\tilde{a}$ increases.
Conversely, for large values of the parameter $\beta$, it diminishes as
$\beta^{-1/4}$, regardless of the value of $\tilde{\alpha}$.
## 5 Summary
In the present paper we considered theories of a scalar field, coupled to
gravity through non-minimal couplings to curvature that include derivatives of
the field, in the general framework of metric-affine theories incorporating
quadratic Ricci scalar curvature terms. We employed disformal transformations
of the metric to transform the action into the Einstein frame, focusing on the
Einstein-Cartan case that allows derivative couplings to the Ricci-tensor and
the Ricci scalar. We proceeded to study the inflationary predictions in the
simple cases of two models, namely that of a quadratic potential and that of a
quartic self-interacting scalar. We find that in both models the effect of the
derivative couplings in combination with the quadratic Ricci curvature term
leads to predictions aligned with the latest observational data. Derivative
couplings tend in general to increase the spectral index, while a reduction to
the tensor-to-scalar ration comes mostly from the quadratic Ricci scalar term.
We have also studied reheating in the simpler case of the quadratic model that
allows for approximate analytic treatment and found a maximum reheating
temperature of $\mathcal{O}(3\times 10^{15})\;\mathrm{GeV}$ for small values
of the quadratic Ricci scalar parameter.
## Acknowledgments
The work of IDG was supported by the Estonian Research Council grants
MOBJD1202, RVTT3, RVTT7, and by the CoE program TK202 “Fundamental Universe”.
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|
# Piloting Diversity and Inclusion Workshops in Artificial Intelligence and
Robotics for Children
A. Badillo-Perez, D. Badillo-Perez, D. Coyotzi-Molina, D. Cruz, R. Montenegro,
L. Vazquez and M. Xochicale air4children: Artificial Intelligence and
Robotics for Children
Xicohtzinco, México
<EMAIL_ADDRESS>
###### Abstract
In this paper, we present preliminary work from a pilot workshop that aimed to
promote diversity and inclusion for fundamentals of Artificial Intelligence
and Robotics for Children (air4children) in the context of developing
countries. Considering the scarcity of funding and the little to none
availability of specialised professionals to teach AI and robotics in
developing countries, we present resources based on free open-source hardware
and software, open educational resources, and alternative education programs.
That said, the contribution of this work is the pilot workshop of four lessons
that promote diversity and inclusion on teaching AI and Robotics for children
to a small gender-balanced sample of 14 children of an average age of 7.64
years old. We conclude that participant, instructors, coordinators and parents
engaged well in the pilot workshop noting the various challenges of having the
right resources for the workshops in developing countries and posing future
work. The resources to reproduce this work are available at
https://github.com/air4children/hri2022.
###### Index Terms:
Open Educational Resources, Educational robots, Child-Robot Interaction
## I Introduction
Accessible and affordable technology in conjunction with open educational
resources can promote equal opportunities for childhood education [1].
However, teaching state-of-the-art technologies such as Artificial
Intelligence and Robotics, AIR, is a current challenge for low-income and
often politically or culturally marginalized countries. Additionally, creating
the right environment to promote inclusivity and diversity to teach AIR has
been little investigated. Astobiza et al. 2019, for instance, reported the
need of collaborations between industry and a multidisciplinary group of
researchers to address concerns on the paradigm of inclusivity in robotics
[2]. In that sense, Astobiza et al. suggested that inclusive robotics should
be based on two points: ”1) they should be easy to use, and 2) they must
contribute to making accessibility easier in distinct environments” [2].
Peixoto et al. in 2018 reported the use of robots as tool to promote diversity
leading to improve competences in communication, teamwork, leadership, problem
solving, resilience and entrepreneurship [3, 4]. Recently, Pannier et al. in
2020 pointed out the challenges of increasing the participation of women and
underrepresented minorities in the areas of Mechatronics and Robotics
Engineering as well as the creation of community of educators to promote
diversity and inclusion [5]. Similarly, Pannier et al. mentioned that the
prevalence of free and open-source software and hardware made mechatronics and
robotics more accessible to a diverse group of population. Pannier et al. also
touched on the importance of offering workshops to different range of
underrepresented students leading to inspire other programs and to create
outreach activities for students, trainers and workshops [5]. In March 2021,
we introduced air4children, Artificial Intelligence and Robotics for Children,
as a way (a) to address aspects for inclusion accessibility, equity and
fairness and (b) to create affordable child-centred materials in AI and
Robotics (AIR) in developing countries [6]. That said, in this work, we are
addressing the challenges of piloting and organising workshops in the context
of communities in developing countries where little to none is known about the
demographics, education levels and socio-economical factors that impact on
teaching AIR. For instance, considering the town of Xicohtzinco, Tlaxcala
México as our case study, where Xicohtzinco has a total population of 14,197
(6762 males and 7435 females) [7] and 19 schools including seven kindergartens
(3 public and 4 private), seven primaries (4 public and 3 private), four
secondaries (2 public and 2 private) and one public hight-school [8]. However,
neither the census [7] nor the education information site [8] provide further
information about teaching technological subjects in AI and Robotics. That
said, we hypothesised that piloting workshops of air4chidren in a town such as
Xicohtzinco might led us to have better understanding of the needs and
challenges of promoting diversity and inclusion considering state-of-the-art
technologies with open education resources.
This short paper is organised as follows: Section II presents resources to
promote diversity and inclusion in AI and Robotics for children. Section III
presents the design of workshops for children from 6 to 8 years old. Section
IV presents outcomes of a four lessons pilot workshop for 14 children
including the engagement of instructors and coordinators. We present results
of the workshops and finalise it with conclusions, limitations and future
work.
## II Resources to promote Diversity and Inclusion in AI and Robotics for
children
### II-A Free and open-source software, open-source hardware and open
educational resources
In March 2021, we presented examples to create educational resources aimed to
be ”affordable, educational and fun”, such examples are (a) Otto DIY – an
educational open source robot and (b) JetBot platform – open source
educational robot to create new AI projects [6]. Similarly, considering Open
Educational Resources (OER) which aim to provide ”teaching and learning
materials that are available without access fees” seems to be a right
direction to afford innovation through OER-enabled pedagogy [9]. However,
Wiley et al. in 2014 contrasted positives and negatives of OERS where for
instance of the benefits of OERs is to make course development process quicker
and easier but also highlighting the challenges of making OER material for
people easier to find but with the challenge of making financially self-
sustainable programs among many other difficulties [10]. Hence, in this work,
we consider Otto Humanoid as a good option because of its affordability with a
cost of 200 EUROS, the block diagram programming interface, the multiple
sensors and actuators (servos and LCD matrix display) aligned with open-source
software and hardware and OER principles [11].
### II-B Ensuring education and Inclusive Learning
Recently, Opertti et al. in 2021 discussed ideas in the forum ”Ensuring
education and Inclusive Learning for Educational Recovery 2021” [12]. Such
ideas to ensure and inclusive learning are summarised as: (1) personalisation
of education including the recognition of specific learning expectations and
needs, (2) designing inclusive, emphatic and participatory curriculum for an
plural and open participation of a diversity of actors and institutions, (3)
appropriation of technology as a community resource to strength ties between
students, educators, families and communities, (4) empowering knowledge,
learning, collaboration, trust and listening among peers, and (5) the
visualisation of schools as lifelong learning spaces. Therefore, such ideas
would help to promote diversity and inclusion of teaching AI and Robotics to
children.
### II-C Alternative education programs with new technologies
Alternative education programs such as Montessori, Waldorf and Regio Emilia
considers children as active authors of their own development [13, 14]. In the
last 5 years such programs are starting to include topics on AI, robotics and
computational thinking into their curriculum [15, 16]. For instance, Aljabreen
pointed out the adoptions of new technologies and how early child education is
re-conceptualised [16]. Elkin et al. in 2014 explored the how robots can be
used in the Montessori curriculum [15]. Similarly Elkin et al. posed the
question on the revision of new curriculums that include technology should not
deviate from the main purpose of the Montessori classroom [15]. Drigas and
Gkeka in 2016 reviewed the application of information and communication
technologies in the Montessori Method, mentioning the Manipulatives, as
objects to develop motor skills or understand mathematical abstractions, are
based on cultural areas, language, mathematics and sensoria but little to none
on technological areas [17]. Drigas and Gkeka reviewed Montessori materials of
the 21st century where interactive systems with sounds and lights, touch
application to enhance visual literacy or the development of computational
thinking and constructions of the physical world [17]. These indicate that the
incorporation of such manipulatives with the use of robotics might led to
reach scenarios to explore motor skill development, visualisation and
computational thinking. Recently, Scippo and Ardolino reported a longitudinal
study of the use of computational thinking in five years participants of
primary school in a Montessori school [18] Scippo and Ardolino pointed out the
importance of alignment of the Montessori material with the computational
thinking activities. That said, previous authors stated various challenges on
the incorporation of new technologies into their curriculum posing more
questions on creating curriculums that should be more accessible to a diverse
group of population as it has been done in other areas such as the case of
open educational resources.
## III Designing Diversity and Inclusion Workshops
To design diversity and inclusive workshops we considered: (a) free and open-
source software, open-source hardware and open educational resources, (b)
Ensuring education and Inclusive Learning, and (c) alternative education
programs with new technologies. That said, with the combination of such
resources, we proposed a four lesson workshop with three-fold aims: (a) to
promote diversity of and inclusion to children to teach AI and Robotics with
recreational and engaging activities, (a) to encourage children to discover
and increase their interest in AI and Robotics with open source hardware and
software, and (c) to develop the Montessori concept of ’concrete to abstract’
to make abstract concepts clearer with hands-on learning materials.
Figure 1: Curriculum of the pilot workshop with four lessons. Lesson 01
introduce the course (L01), lesson 02 provides the basics of anatomy (L02),
lesson 03 covers algorithms (L03), and lesson 04 wrap-up and showcase the
project of children (L04). The arrows in the figure illustrate the connection
from the final to initial part of each lesson and how all the lessons where
connected to the final section of the workshop.
#### Lesson 01: Breaking the ice and motivations
The educational goal for this lesson was to develop the children’s curiosity
about AI and Robotics while emphasizing the importance of interpersonal
connections that will evolve into a collaboration work for the following
lessons. That said, this lesson started with a recreational activity where
each student and teacher introduced themselves with name, favorite food and a
superpower or ability that we would like to have and that was related to
robots. This lesson also covers basic concepts and examples of AI and Robotics
in different fields and daily life, how the brain works and how the human
senses and body parts relates to the way a robot is built and how it works and
perform activities.
#### Lesson 02: Human senses and coding my first robot
The main purpose of this lesson was to understand fundamentals of Robotics.
Children began to work with more abstract concepts, developing problem-solving
skills as well as cooperatively working relationships. The first activity,
outside of the classroom, was a true or false game, were the teacher told a
sentence about AI and Robotics, and the children jumped in front of a rope if
the sentence was true, or if the sentence was false, the kids jumped behind of
the rope. In the second activity, the instructor explain about the human
senses and their relationship with inputs and outputs. After that, instructors
explain examples of sequences and codes. In the last activity, participants
were asked to sort out tangram in a group in which a leader of the group
provide instructions to the team-mates as an analogy of coding a robot with
algorithms.
#### Lesson 03: Playing with reaction-action activities
The educational goal of this lesson is to cover the concept of the effect of
causes and consequences with daily life examples and the computational
thinking of robots. Hence, this lesson start with a match game consisting of
figures and shadow’s figures where participants develop their comparing skills
to match similar or different robots. This lesson covered points on how robot
works with sensors, processors, actuators and programming. The “find the
effect” activity was also introduced where participants have to relate
pictures of cause and consequences, for example “the cause is the rain and the
consequently is a rainbow”. Afterwards, we worked with the Otto humanoids in
which we programmed the sensor presence for that robot moves, dances and that
emit texts with the 8x8 matrix.
#### Lesson 04: Develop your own AIR
The four lesson aimed to summaries what was covered in the previous lessons
emphasising the relationship of the human body anatomy (brain, neurons and
body parts) with humanoid robots (computer, sensors and actuators). This
lesson covered real-word application of AI and Robotics including medicine,
spacial robotics and smart cities. Three projects were prepared to be
introduced to each team in which every participant have a role. Each team
prepare a short speech of their application using AI and Robotics.
See figure 1 that illustrates four lessons of the workshops.
## IV Piloting Diversity and Inclusion Workshops
To pilot the four lesson workshop, we invited 14 participants (6 female and 8
male) with range of age from 6 to 11 years old (average age of 7.64) (Figure
2). For the workshop, three instructors with three years of experience in
teaching and two coordinators with ten years of teaching experience
volunteered to deliver four lessons of 90 minutes in the workshop (as shown in
the proposed curriculum Fig 1). During the initial three lessons of the
workshop, children incorporated the gained knowledge to relate fundamental
human body anatomy (brain, neurons, body and senses) to robot parts
(microcontroller, motors, sensors) based on open source hardware and software.
In the final lesson, children showcased their final work promoting a sense of
achievement in the children working not only with their mind but also with
their social emotional well-being. In all lessons, instructors encouraged and
engaged every participant for individual and group activities.
Figure 2: Instructors demonstrating fundamentals of AI and Robotics (A, and
B). Children engaging with classmates, robots and instructors (B, and C).
We however noted that each lessons was originally planned to be 90 minutes of
length and we did not consider breaks nor the participant’s energy levels to
which in the second to four lesson a 15 minutes break was incorporated.
Additionally, we piloted surveys to (a) children with ten questions about
their understanding and feelings towards different type of robots and (b) to
parents with 30 questions about their understanding of AI and robotics and how
parents were aware of technological advances in AI and Robotics. Although the
aim of the surveys was not to be reported but only to understand how
participants and parents feel about being surveyed and how the logistics of
surveys would be followed with more participants. That said, we noticed that
participants require support as few participants were not familiar with
reading surveys to which the content of 10 questions was spread into five
questions into two sessions. On other hand, parents felt that surveys were
lengthy, taking more than 60 minutes, and we also realised that a paper-based
survey require more work as scans and transcripts are required.
## V Conclusions, limitations and future work
In this paper, we posed the challenges of promoting Diversity and Inclusion to
teach ”Artificial Intelligence and Robotics for Children” in developing
countries, considering resources of open-source hardware and software and
principles of Montessori education. We think that the goals of each lessons
were intended to remain beyond the learning of a single concept but to
contribute to develop skills of inclusion and diverstiy that children can take
and apply to other areas in their life. That said, for the pilot of the
workshop, we considered a small sample of 14 children of an average age of
7.64 years old from the town Xicohtzinco Tlaxcala México. The workshops were
free of cost as a way encourage participation and inclusion of anyone. During
the pilot workshop, children were enthusiastic about learning the fundamentals
for AIR by coding, designing and playing with open-sourced robots. The
instructors embraced the different set of skills each child had by working in
small groups and supporting the students during all the activities. However,
we noted that grouping children of four participants with one instructor was
not creating an engaging experience as each group has only one robot and one
computer and the space and number of participants was leaving sometimes one
participant outside of the reachable robot-computer setup.
In terms of limitations, the pilot surveys only helped us to identify the
gender and age of the participants and no other insights such as needs of the
target group were considered. Similarly, this work did no consider metrics to
quantify the impact of the workshop but to identify the needs of the workshop
that might be addressed in future work. The workshops were free of cost but no
sustainable model was considered for this pilot experiment.
As a future work, we are planing to run another pilot in late 2022 or early
2023 with more lessons and perhaps more participants considering the addition
of a study design and to run a pre-survey to identify the needs of
participants of the workshops. For the curriculum of the workshops, we are
planning to improve the activities to be more engaging, diverse and inclusive
and to provide further evidence on how alternative education programs (e.g.
Montessori, Waldorf, Regio Emilia [13]; and ”synthesis program” [19]) with new
technologies might lead to potential new avenues of inclusivity and diversity.
## Acknowledgment
To Marta Pérez and Donato Badillo for their support in organising the pilot of
the workshops. To Rocio Montenegro for her contributions with the design of
the Montessori curriculum for the workshops. To Donato Badillo Peréz, Antonio
Badillo Peréz and Diego Coyotzi Molina for volunteering as instructors of the
workshops. To Leticia Vázquez for her support with the logistics and feedback
to improve the workshops. To Adriana Pérez Fortis for her contributions and
discussion to prepare draft pilot surveys for the parents and children. To
Elías Méndez Zapata for his support and feedback on the hardware design of the
robot. To Dago Cruz for his feedback and discussions on the design of the
workshops. To Angel Mandujano, Elva Corona and others who have contributed
with feedback and support to keep AIR4children project alive
## Contributions
Antonio Badillo-Peréz: Contributing to design and write up of lesson 02.
Donato Badillo-Perézo: Contributing to design and write up of lesson 01. Diego
Coyotzi-Molina: Contributing to design and write up of lesson 03. Dago Cruz:
Contributing to proofreading, edition and feedback. Rocio Montenegro:
Contributing to the write up of designing and piloting workshops. Leticia
Vázquez: Write up and refinement of the conclusions. Miguel Xochicale:
Contributing to create the open source and reproducible workflow, drafting,
write-up, edition, and submission of the paper.
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|
# Learning Lie Group Symmetry Transformations with Neural Networks
Alex Gabel Victoria Klein Riccardo Valperga Jeroen S. W. Lamb Kevin
Webster Rick Quax Efstratios Gavves
###### Abstract
The problem of detecting and quantifying the presence of symmetries in
datasets is useful for model selection, generative modeling, and data
analysis, amongst others. While existing methods for hard-coding
transformations in neural networks require prior knowledge of the symmetries
of the task at hand, this work focuses on discovering and characterizing
unknown symmetries present in the dataset, namely, Lie group symmetry
transformations beyond the traditional ones usually considered in the field
(rotation, scaling, and translation). Specifically, we consider a scenario in
which a dataset has been transformed by a one-parameter subgroup of
transformations with different parameter values for each data point. Our goal
is to characterize the transformation group and the distribution of the
parameter values. The results showcase the effectiveness of the approach in
both these settings.
Machine Learning, ICML
## 1 Introduction
It has been shown that restricting the hypothesis space of functions that
neural networks are able to approximate using known properties of data
improves performance in a variety of tasks (Worrall & Welling, 2019; Cohen et
al., 2018; Weiler et al., 2018; Zaheer et al., 2017; Cohen & Welling, 2016).
The field of Deep Learning has produced a prolific amount of work in this
direction, providing practical parameterizations of function spaces with the
desired properties that are also universal approximators of the target
functions (Yarotsky, 2022). In physics and, more specifically, time-series
forecasting of dynamical systems, symmetries are ubiquitous and laws of motion
are often symmetric with respect to various transformations such as rotations
and translations, while transformations that preserve solutions of equations
of motions are in one way or another associated with conserved quantities
(Noether, 1918). In computer vision, successful neural network architectures
are often invariant with respect to transformations that preserve the
perceived object identity as well as all pattern information, such as
translation, rotation and scaling. Many of these transformations are smooth
and differentiable, and thus belong to the family of Lie groups, which is the
class of symmetries we deal with in this work.
Figure 1: The distribution of transformations in a toy dataset that correspond
to the Lie groups of rotation and (isotropic) scaling, given in terms of the
parameters degree and scaling factor respectively; crucially, these groups are
differentiable and can be (locally) decomposed into one-parameter subgroups.
Although methods that hard-code transformations are capable of state-of-the-
art performance in various tasks, they all require prior knowledge about
symmetries in order to restrict the function space of a neural network. A
broad class of, a priori unknown, transformations come into play in the
context of modelling dynamical systems and in applications to physics. On the
other hand, in vision tasks, identity-preserving transformations are often
known beforehand. Despite this, these transformations are expressed
differently by different datasets. As a result, algorithms for not only
_discovering_ unknown symmetries but also _quantifying_ the presence of
specific transformations in a given dataset, may play a crucial role in
informing model selection for scientific discovery or computer vision, by
identifying and describing physical systems through their symmetries and
selecting models that are invariant or equivariant with respect to only those
symmetries that are _actually_ present in the dataset under consideration.
In this work, we address the problem of qualitatively and quantitatively
detecting the presence of symmetries with respect to one-parameter subgroups
within a given dataset (see Figure 1). In particular, let $\phi(t)$ be a one
parameter subgroup of transformations. We consider the scenario in which a
dataset $\\{x_{i}\\}_{i=1}^{N}$ has been acted on by $\phi(t)$, with a
_different_ value of the parameter $t$ for every point $x_{i}$. Our goal is to
characterise the group of transformations $\phi(t)$, as well as the
_distribution_ from which the parameters $t$ have been sampled. We propose two
models: a naive approach that successfully manages to identify the underlying
one-parameter subgroup, and an autoencoder model that learns transformations
of a one-parameter subgroup in the latent space and is capable of extracting
the overall shape of the $t$-distributions. The cost of the latter is that the
one-parameter subgroup in the latent space is not necessarily identical to
that in pixel space. The work is structured as follows: Section 2 introduces
some basic tools from Lie group theory; Section 3 outlines the method; Section
5 provides an overview of the existing methods that are related to our own;
and lastly, results are shown in Section 4.
## 2 Background
The theoretical underpinnings of symmetries or invariance can be described
using group theory (Fulton & Harris, 1991). In particular, we present the
necessary theory of one-parameter subgroups (Olver, 1993) on which our method
is based, following the logic of Oliveri (2010).
### 2.1 One-parameter subgroups
We focus on learning invariances with respect to one-parameter subgroups of a
Lie group $G$, which offer a natural way to describe continuous symmetries or
invariances of functions on vector spaces.
###### Definition 2.1.
A one-parameter subgroup of $G$ is a differentiable homomorphism
$\phi:\mathbb{R}\to G$, more precisely, such that $\phi(t+s)=\phi(t)\phi(s)$
for all $t,s\in\mathbb{R}$.
Let the action of $\phi$ on the vector space $X\subset\mathbb{R}^{n}$ be a
transformation $T:X\times\mathbb{R}\to X$ that is continuous in $x\in X$ and
$t\in\mathbb{R}$. Because of continuity, for sufficiently small $t$ and some
fixed $x\in X$, the action is given by
$T(x,t)\approx x+tA(x)\;\text{where}\;A(x):=\frac{\partial T(x,t)}{\partial
t}\Bigg{|}_{t=0}.$ (1)
Note that this is equivalent to taking a first-order Taylor expansion in $t$
around $t=0$.
### 2.2 Generators
In general, we can use $A(x)$ in (1) to construct what is known as the
generator of a one-parameter subgroup $\phi$ of a Lie group $G$, that in turn
will characterise an ordinary differential equation, the solution to which
coincides with the action $T$ on $X$.
Let $C^{\infty}(X)$ be the space of smooth functions from $X$ to $X$. The
generator of $\phi$ is defined as a linear differential operator
$L:C^{\infty}(X)\to C^{\infty}(X)$ such that
$L=\sum_{i=0}^{n}(A(x))_{i}\frac{\partial}{\partial x_{i}}$ (2)
describing the vector field of the infinitesimal increment $A(x)t$ in (1),
where $\partial/\partial x_{i}$ are the unit vectors of $X$ in the coordinate
directions for $i=1,\ldots,n$. It can be shown (Olver, 1993) that, for a fixed
$x\in X$, that $T(x,t)$ is the solution to the ordinary differential equation
$\frac{dT(x,t)}{dt}=LT(x,t)\quad\text{where}\quad T(x,0)=x.$ (3)
The solution to (3) is the exponential $T(x,t)=e^{tL}x$ where
$e^{tL}:=\sum_{k=0}^{\infty}\frac{(tL)^{k}}{k!},$ (4)
where $L^{k}$ is the operator $L$ applied $k$ times iteratively.
For a one-parameter subgroup $\phi$ of a matrix Lie group $G\subset
GL(n,\mathbb{R})$ and a fixed $x\in X$, it can be shown (Olver, 1993) that
there exists a unique matrix $A\in\mathbb{R}^{n\times n}$ such that $A(x)=Ax$.
This is a more restrictive approach as groups such as translations cannot be
written as a matrix multiplication.
Figure 2: Model architecture.
## 3 Method
As in Rao & Ruderman (1998); Sanborn et al. (2022); Dehmamy et al. (2021) the
semi-supervised symmetry detection setting that we consider consists of
learning the generator $L$ of a one-parameter subgroup $\phi$ from pairs of
observations of the form
$\left\\{\left(x_{i},\bar{x}=T(x_{i},t_{i})\right)\right\\}_{i=1}^{N}$, where
$N$ is the number of observations and each $t_{i}\in\mathbb{R}$ is drawn from
some unknown distribution $p(t)$. Not only do we attempt to learn the
generator $L$, but also the unknown distribution $p(t)$ of the parameters
$\\{t_{i}\\}_{i=1}^{N}$.
### 3.1 Parametrisation of the generator
Deciding how to parametrise $L$ has an effect on the structure of the model
and ultimately on what one-parameter subgroups we are able to learn. For
simplicity, consider one-parameter subgroups acting on
$X\subset\mathbb{R}^{2}$, although this operator can be defined for higher-
dimensional vector spaces. The generator $L$ of $\phi$ is given as in Eq. (2)
and we parametrise $A(x,y)$ as a linear operator in the basis $\\{1,x,y\\}$
with a coefficient matrix $A=\alpha\in\mathbb{R}^{2\times 3}$, giving
$\displaystyle\begin{split}L^{\alpha}&:=(\alpha_{11}+\alpha_{12}x+\alpha_{23}y)\frac{\partial}{\partial
x}\\\ &+(\alpha_{12}+\alpha_{22}x+\alpha_{23}y)\frac{\partial}{\partial
y}.\end{split}$ (5)
In this particular basis, for different values of $\alpha$, the generator
$L^{\alpha}$ is able to express one-parameter sub-groups of the affine group.
This includes the “traditional” symmetries that are usually considered
(translation, rotation, and isotropic scaling) and all other affine
transformations111Alternatively, the constant terms can be thought of as the
drift terms (i.e. translation) and the four others can be arranged into a
diffusion matrix.. This can be generalized to any functional form of the
generator by augmenting the basis accordingly.
### 3.2 Discretisation and interpolation
The generator $L^{\alpha}$ is constructed as an operator that acts on a
function $f:\mathbb{R}^{2}\to\mathbb{R}$, given, in practice, by
$I\in\mathbb{R}^{n\times n}$ such that $I_{ij}=f(i,j)$ are evaluations of $f$
on a regularly-sampled $n\times n$ grid $M$ of points
$M_{ij}=(i,j)\in\mathbb{R}^{2}$. We then vectorise $I$, obtaining a point in a
vector space $\tilde{I}\in\mathbb{R}^{n^{2}}$ such that
$\tilde{I}_{i+j}:=I_{ij}$ and construct the matrix operator
$L^{\alpha}\in\mathbb{R}^{n^{2}\times n^{2}}$ as
$\displaystyle\begin{split}L^{\alpha}&:=(\alpha_{11}+\alpha_{12}X_{x}+\alpha_{13}X_{y})\frac{\partial}{\partial
X_{x}}\\\
&+(\alpha_{21}+\alpha_{22}X_{x}+\alpha_{23}X_{y})\frac{\partial}{\partial
X_{y}},\end{split}$ (6)
acting on $\tilde{I}$, where $X_{x}\in\mathbb{R}^{n^{2}\times n^{2}}$ and
$X_{y}\in\mathbb{R}^{n^{2}\times n^{2}}$ are such that $(X_{x})_{ij}:=i$ and
$(X_{y})_{ij}:=j$, while ${\partial}/{\partial X_{x}}$ and
${\partial}/{\partial X_{y}}$ are also matrix operators in
$\mathbb{R}^{n^{2}\times n^{2}}$. The exponential in (4) and the action $T$
coincides with the matrix exponential.
In order to define ${\partial}/{\partial X_{x}}$ and ${\partial}/{\partial
X_{y}}$ as operators that transform by infinitesimal amounts at discrete
locations, we require an interpolation function. The Shannon-Whittaker theorem
(Marks, 2012) states that any square-integrable, piecewise continuous function
that is band-limited in the frequency domain can be reconstructed from its
discrete samples if they are sufficiently close and equally spaced. For sake
of interpolations, we will also assume that the function is periodic.
##### Interpolation: 1D
In the case where $M$ is a discrete set of $n$ points in 1D, we have that
$I(i+n)=I(i)$ for all $i=1,\ldots,n$ samples. Shannon-Whittaker interpolation
reconstructs the signal for all $x\in\mathbb{R}$ as
$\begin{split}&I(x)=\sum_{i=0}^{n-1}I(i)Q(x-i),\quad\text{where}\\\
&Q(x)=\frac{1}{n}\left[1+2\sum_{p=1}^{n/2-1}\cos\left(\frac{2\pi
px}{n}\right)\right]\end{split}$ (7)
Differentiating $Q$ with respect to $x$ and evaluating it at every $x_{i}\in
M$ gives an analytic expression for a vector field in $\mathbb{R}^{n}$,
describing continuous changes in $x$ at all $n$ points (Rao & Ruderman, 1998).
This is precisely what ${\partial}/{\partial x}$ or ${\partial}/{\partial y}$
in (5) are.
##### Interpolation: 2D
In the case where $M$ is a grid of $n\times n$ points in 2D, we construct the
$n\times n$ matrices of the partial derivatives of $Q$ with respect to $x$ and
$y$, analogously to the 1D case, stacking them to construct the ${n^{2}\times
n^{2}}$ block diagonal matrices ${\partial}/{\partial X_{x}}$ and
${\partial}/{\partial X_{y}}$. It is worth noting that alternative
interpolation techniques can be used to obtain the operators and the method
does not depend on any specific one.
Two different architectures, the main model and the latent model, are proposed
to learn $L^{\alpha}$ and, in doing so, the action $T$.
#### 3.2.1 Naive model
The coefficients $\alpha$ of $L^{\alpha}$ are approximated by fixed
coefficients that are shared across the dataset, while the parameter $t_{i}$
is approximated by $\hat{t}_{i}$ that depends on the input pair
$(x_{i},\bar{x}_{i})$. We learn
1. 1.
the coefficients $\alpha\in\mathbb{R}^{2\times 3}$ of the generator
$L^{\alpha}$ and
2. 2.
the parameters $\theta$ of an MLP $f_{\theta}$ that returns
$f_{\theta}(x_{i},\bar{x}_{i})=:\hat{t}_{i}$ as a function of every input
pair,
such that the solution to (3) for $L^{\alpha}$ is approximated by
$\hat{T}(x_{i},\bar{x}_{i}):=e^{f_{\theta}(x_{i},\bar{x}_{i})L^{\alpha}}\,x_{i}.$
(8)
The model objective is then given by the reconstruction loss
$\mathcal{L}_{T}(x_{i},\bar{x}_{i})=||\hat{T}_{\phi}(x_{i},\bar{x}_{i})-\bar{x}_{i}||^{2}.$
(9)
#### 3.2.2 Latent model
While the model described above will prove to work sufficiently well for
learning the coefficients $\alpha$ of $L^{\alpha}$, the matrix exponential
function in $\hat{T}$ in (8) can be costly to compute and difficult to
optimise in high dimensions; consider that the cost of the matrix exponential
in a single forward pass is roughly $O(n^{3})$ using the algorithm of Al-Mohy
& Higham (2010).
As a result, a different version of the model is proposed that incorporates an
autoencoder for reducing dimension. The concept remains the same, but $x_{i}$
is now mapped to some latent space $Z\subset\mathbb{R}^{n_{Z}}$ for $n_{Z}\ll
n$, such that the exponential is taken in a significantly lower dimension.
This is done by an encoder $h_{\psi}:X\to Z$ and a decoder $d_{\psi}:Z\to X$
such that $z_{i}=h_{\psi}(x_{i})$ and $x_{i}\approx d_{\psi}(z_{i})$.
We learn
1. 1.
the parameters $\psi$ of an MLP autoencoder,
2. 2.
the coefficients $\tilde{\alpha}\in\mathbb{R}^{2\times 3}$ of the generator
$L^{\tilde{\alpha}}$ for a one-parameter subgroup $\phi_{Z}$ acting on the
latent space $Z$,
3. 3.
the parameters $\theta$ of an MLP $f_{\theta}$ that returns
$f_{\theta}(x_{i},\bar{x}_{i})=:\hat{t}_{i}$ as a function of every original
input pair $(x_{i},\bar{x}_{i})$,
such that the solution to (3) for $L^{\alpha}$, the generator in the original
space, is approximated by
$\hat{T}^{Z}(x_{i},\bar{x}_{i})=d_{\psi}(e^{f_{\theta}(x_{i},\bar{x}_{i})L^{\tilde{\alpha}}}h_{\psi}(x_{i})).$
(10)
It is important to note that enforcing good reconstruction of the autoencoder
alone does not enforce the commutativity of the diagram in Figure 3. To make
it commutative, we use an objective that is a weighted sum of multiple terms.
A simple reconstruction term for the autoencoder on each input example
$\mathcal{L}_{R}(x_{i}):=||d_{\psi}(h_{\psi}(x_{i}))-x_{i}||^{2},$ (11)
a transformation-reconstruction term in the original space
$\mathcal{L}^{X}_{T}(x_{i},\bar{x}_{i}):=||\hat{T}^{Z}_{\phi}(x_{i},\bar{x}_{i})-\bar{x}_{i}||^{2},$
(12)
a transformation-reconstruction term in the latent space
$\mathcal{L}^{Z}_{T}(x_{i},\bar{x}_{i}):=||e^{f_{\theta}(x_{i},\bar{x}_{i})L^{\tilde{\alpha}}}h_{\psi}(x_{i})-h_{\psi}(\bar{x}_{i})||^{2},$
(13)
and a Lasso term on the generator coefficients $\tilde{\alpha}$. The overall
loss of the latent model is
$\displaystyle\begin{split}\mathcal{L}(x_{i},\bar{x}_{i})&=\lambda_{R}(\mathcal{L}_{R}(x_{i})+\mathcal{L}_{R}(\bar{x}_{i}))\\\
&+\,\lambda_{X}\mathcal{L}^{X}_{T}(x_{i},\bar{x}_{i})+\lambda_{Z}\mathcal{L}^{Z}_{T}(x_{i},\bar{x}_{i})\\\
&+\lambda_{L}||\mathbf{\tilde{\alpha}}||^{2},\end{split}$ (14)
where $\lambda_{R},\lambda_{X},\lambda_{Z},\lambda_{L}\in\mathbb{R}$ are
treated as hyperparameters.
$X$$X$$Z$$Z$$T(\cdot,t)$$e^{f_{\theta}(\cdot,\cdot)L^{\tilde{\alpha}}}$$h_{\psi}$$d_{\psi}$
Figure 3: The commuting diagram enforced by the objective function in the
latent model: $T(x,t)\approx
d_{\psi}(e^{f_{\theta}(x_{i},\bar{x}_{i})L^{\tilde{\alpha}}}h_{\psi}(x))$.
##### Recovering the group
It is important to note that the one-parameter subgroup corresponding to the
generator $L^{\tilde{\alpha}}$ and the generator $L^{\alpha}$ are _not_
necessarily the same; $L^{\tilde{\alpha}}$ is the generator corresponding to
some action on $X$ of a one-parameter subgroup $\phi$, while $L^{\alpha}$ is a
different generator corresponding to some action on $Z$ of a different one-
parameter subgroup $\phi_{Z}$.
### 3.3 Uniqueness
For both the naive model in Section 3.2.1 and the latent model in 3.2.2, the
approximations $\hat{t}_{i}$ for the values of the parameters $t_{i}$ require
interpretation. Both models parameterise $\hat{T}$ or $\hat{T}^{Z}$ with the
products $\hat{t}_{i}L^{\alpha}$ or $\hat{t}_{i}L^{\tilde{\alpha}}$
respectively, where $\hat{t}_{i}=f_{\theta}(x_{i},\bar{x}_{i})$. While both
the values of $\hat{t}_{i}L^{\alpha}$ and $\hat{t}_{i}L^{\tilde{\alpha}}$ are
unique for a given action on $X$ and $Z$ respectively, their decomposition is
only unique up to a constant. Therefore, $L^{\alpha}$ or $L^{\tilde{\alpha}}$
and $\hat{t}$ approximate the generators and the parameter respectively up to
a constant. Consequently, the one-parameter subgroup $\phi$ can only be
deduced by the values of the individual coefficients in $\alpha$ _relative to
one another_ , as opposed to in absolute, likewise for $\phi_{Z}$ and
$\tilde{\alpha}$ . We therefore recover a scaled approximation for the
distribution of $\hat{t}_{i}$.
### 3.4 The most general setting
Suppose we are given a labelled dataset
$\mathcal{D}=\left\\{(x_{i},c_{i})\right\\}_{i=1}^{N}$ and a one-parameter
subgroup $\phi$. Then we call $\mathcal{D}$ symmetric or invariant with
respect to $\phi$ if the action of $\phi$ preserves the object identity of the
data points, where by object identity we mean any property of the data that we
might be interested in. For example, in the case of MNIST handwritten digits,
rigid transformations preserve their labels 222With the exception of the
number ’9’ that, if rotated 180 degrees becomes a ’6’. and therefore, can be
considered symmetries of the dataset. Now suppose that every $x_{i}$ in
$\mathcal{D}$ is acted on with a one-parameter subgroup $\phi_{t}$ to get
$T\mathcal{D}=\left\\{(T(x_{i},t_{i}),c_{i})\right\\}_{i=1}^{N}$. The most
general, fully unsupervised symmetry detection setting consists of learning
$\phi$, and characterize the distribution of the parameter $t$ from just
$\bar{\mathcal{D}}$. The idea is that, under the assumption that points with
the same label are sufficiently similar for the subgroup transformation to
account for the important difference333Keeping MNIST hand-written digits as
our paradigmatic example, digits with the same label differ by small
transformations that account for handwriting style differences., we can use
labels to group data points, and compare those data points using methods such
as the one presented in this paper. We leave the fully unsupervised symmetry
detection setting for future work although we will emphasize that the proposed
method can, in principle, be used in such setting without substantial changes
to the architecture.
## 4 Experiments
### 4.1 Experiment setting
In practice, we experiment with a dataset of MNIST digits transformed with
either 2D rotations or translations in one direction. To test the method’s
ability to learn distributions of these transformations, for each one-
parameter subgroup $\phi\in\\{SO(2),T(2)\\}$ we construct a dataset
$\left\\{x_{i},T(x_{i},t_{i})\right\\}_{i=1}^{N}$ by sampling the parameters
$t_{i}\in\mathbb{R}$ from various _multimodal_ distributions.
As in (Rao & Ruderman, 1998), the dataset is composed of signals
$I:M\longrightarrow\mathbb{R}$ regularly-sampled from a discrete grid of
$n^{2}$ points $(x,y)\in\mathbb{R}^{2}$ for $n=28$. The signals $I$ are
vectorised into points in $\mathbb{R}^{784}$ as described in Section 3.2. The
implementation of the naive model is available here.
### 4.2 Main model experiments
The naive model architecture outlined in 3.2.1 consists of a fully-connected,
3-layer MLP for $f_{\theta}$ that was trained jointly with the coefficients
$\alpha_{ij}$ using Adam (Kingma & Ba, 2014) with a learning rate of 0.001.
Given the disproportionate number of trainable parameters in $f_{\theta}$ and
the 6 coefficients in $\alpha$, updating $\alpha_{ij}$ roughly 10 times for
every update of $\theta$ in $f_{\theta}$ was found to be beneficial during
training.
##### Coefficients
Figure 4 shows the evolution of $\alpha_{ij}$ during training. It can be seen
that after a few hundred steps, the coefficients $\alpha_{ij}$ that do not
correspond to the infinitesimal generator of the symmetry expressed by the
dataset drop to zero, while those that do, settle to values compatible with
those of the ground truth generator $L$.
(a) Rotation
(b) Translation in $x$
Figure 4: Training evolution of the coefficients $\alpha$ defining the
generator $L^{\alpha}$ of the one-parameter subgroup, that are shown to
converge to the ground-truth non-zero coefficients $\alpha$ for rotated
($-\alpha_{22}=\alpha_{13}=1$ and $0$ otherwise) and translated
($\alpha_{11}=1$ and $0$ otherwise) MNIST.
### 4.3 Latent model experiments
The latent model outlined in 3.2.2 consists of a fully-connected, 3-layer MLP
$f_{\theta}$, as in (8), to approximate $\hat{t}$, and two fully-connected,
3-layer MLPs with decreasing/increasing hidden dimensions for the encoder
$h_{\psi}$ and $d_{\psi}$. We set the latent space to $n_{Z}=25$. Similar to
the naive model experiment above, $f_{\theta}$ was trained jointly with the
coefficients $\alpha_{ij}$ using Adam (Kingma & Ba, 2014) with learning rate
0.001.
##### Parameters
After every epoch (roughly 500 steps), the outputs of $\hat{t}=f_{\theta}$
were collected in a histogram to show $p(\hat{t})$. Figure 5 shows how the
distribution of $\hat{t}$ changes during training and how multimodal
distributions are clearly recovered, showing the same number of modes as the
ground truth distribution from which the transformations were sampled.
(a) Unimodal distribution
(b) Bimodal distribution
(c) 3-mode distribution
(d) 5-mode distribution
Figure 5: Training evolution of the distributions $p(\hat{t})$ of the learned
parameters $\hat{t}$ computed by $f_{\theta}$ for the validation set. The
figure shows that $p(\hat{t})$ resembles the original multi-modal
distributions $p(t)$ of the transformations expressed by the dataset.
## 5 Related Work
Symmetries in Neural Networks Numerous studies have tackled the challenges
associated with designing neural network layers and/or models that are
equivariant with respect to specific transformations (Finzi et al., 2021).
These transformations include continuous symmetries such as scaling (Worrall &
Welling, 2019), rotation on spheres (Cohen et al., 2018), local gauge
transformations (Cohen et al., 2019) and general E(2) transformations on the
Euclidean plane (Weiler & Cesa, 2019), as well as discrete transformations
like permutations of sets (Zaheer et al., 2017) and reversing symmetries
(Valperga et al., 2022). Another line of research focuses on establishing
theoretical principles and practical techniques for constructing general
group-equivariant neural networks. Research in such areas show improved
performances on tasks related to symmetries, but nonetheless require prior
knowledge about the symmetries themselves.
Symmetry Detection Symmetry detection aims to discover symmetries from
observations, a learning task that is of great importance in of itself.
Detecting symmetries in data not only lends itself to more efficient and
effective machine learning models but also in discovering fundamental laws
that govern data, a long-standing area of interest in the physical sciences.
Learned symmetries can then be incorporated after training in equivariant
models or used for data augmentation for downstream tasks. In physics and
dynamical systems, the task of understanding and discovering symmetries is a
crucial one; in classical mechanics and more generally Hamiltonian dynamics,
continuous symmetries of the Hamiltonian are of great significance since they
are associated, through Noether’s theorem (Noether, 1918), to conservation
laws such as conservation of angular momentum or conservation of charge.
The first work on learning symmetries of one-parameter subgroups from
observations were Rao & Ruderman (1998) and Miao & Rao (2007), which outline
MAP-inference methods for learning infinitesimally small transformations.
Sohl-Dickstein et al. (2010) propose a transformation-specific smoothing
operation of the transformation space to overcome the issue of a highly non-
convex reconstruction objective that includes an exponential map. These
methods are close to ours in that we also make use of the exponential map to
obtain group elements from their Lie algebra. Despite this, Sohl-Dickstein et
al. (2010) do not consider the task of characterizing the distribution of the
parameter of the subgroup nor do they consider the whole of pixel-space, using
small patches instead. Cohen & Welling (2014) focus on disentangling and
learning the distributions of multiple compact “toroidal” one-parameter
subgroups in the data.
Neural Symmetry Detection A completely different approach to symmetry
discovery is that of Sanborn et al. (2022), who’s model uses a group invariant
function known as the bispectrum to learn group-equivariant and group-
invariant maps from observations. Benton et al. (2020) consider a task similar
to ours, attempting to learn groups with respect-to-which the data is
invariant, however, the objective places constraints directly on the network
parameters as well as the distribution of transformation parameters with which
the data is augmented. Alternatively, Dehmamy et al. (2021) require knowledge
of the specific transformation parameter for each input pair (differing by
that transformation), unlike our model, where no knowledge of the one-
parameter group is used in order to find the distribution of the
transformation parameter.
Latent Transformations Learning transformations of a one-parameter subgroup in
latent space (whether that subgroup be identical to the one in pixel space or
not) has been accomplished by Keurti et al. (2023) and Zhu et al. (2021).
Nevertheless, other works either presuppose local structure in the data by
using CNNs instead of fullly-connected networks or focus on disentangling
interpretable features instead of directly learning generators that can be
used as an inductive bias for a new model.
In contrary to the other works mentioned above, we propose a promising
framework in which we can simultaneously
* •
perform symmetry detection in pixel-space, without assuming any inductive
biases are present in the data a priori,
* •
parametrize the generator such that non-compact groups (e.g. translation) can
be naturally incorporated,
* •
and learn both the generator and the parameter distributions.
## 6 Discussion
In this work we proposed a framework for learning one-parameter subgroups of
Lie group symmetries from observations. Our method uses a neural network to
predict the one-parameter of every transformation that has been applied to
datapoints, and the coefficients of a linear combination of pre-specified
generators. We show that our method can learn the correct generators for a
variety of transformations as well as characterize the distribution of the
parameter that has been used for transforming the dataset.
While the goal of learning both the coefficients of the generator and the
distribution of the transformation parameter has not been accomplished by only
one model in this work, modifying our existing framework to do so is a
priority for future work. In addition, the proposed method lends itself well
to being composed to form multiple layers, which can then be applied to
datasets that express multiple symmetries. By doing so, ideally, each layer
would learn one individual symmetry. We leave this study, and the more
general, fully unsupervised setting described in 3.4, for future work.
## Acknowledgements
This publication is based on work partially supported by the EPSRC Centre for
Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and
Simulation (EP/S023925/1) and the Dorris Chen Award granted by the Department
of Mathematics, Imperial College London.
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|
11institutetext: Department of Electronic Engineering, The Chinese University
of Hong Kong, Shatin, Hong Kong SAR
11email<EMAIL_ADDRESS>
# DiffRect: Latent Diffusion Label Rectification for Semi-supervised Medical
Image Segmentation
Xinyu Liu Wuyang Li Yixuan Yuan${}^{\text{{\char 0\relax}}}$
###### Abstract
Semi-supervised medical image segmentation aims to leverage limited annotated
data and rich unlabeled data to perform accurate segmentation. However,
existing semi-supervised methods are highly dependent on the quality of self-
generated pseudo labels, which are prone to incorrect supervision and
confirmation bias. Meanwhile, they are insufficient in capturing the label
distributions in latent space and suffer from limited generalization to
unlabeled data. To address these issues, we propose a Latent Diffusion Label
Rectification Model (DiffRect) for semi-supervised medical image segmentation.
DiffRect first utilizes a Label Context Calibration Module (LCC) to calibrate
the biased relationship between classes by learning the category-wise
correlation in pseudo labels, then apply Latent Feature Rectification Module
(LFR) on the latent space to formulate and align the pseudo label
distributions of different levels via latent diffusion. It utilizes a
denoising network to learn the coarse to fine and fine to precise consecutive
distribution transportations. We evaluate DiffRect on three public datasets:
ACDC, MS-CMRSEG 2019, and Decathlon Prostate. Experimental results demonstrate
the effectiveness of DiffRect, e.g. it achieves 82.40% Dice score on ACDC with
only 1% labeled scan available, outperforms the previous state-of-the-art by
4.60% in Dice, and even rivals fully supervised performance. Code is released
at https://github.com/CUHK-AIM-Group/DiffRect.
###### Keywords:
Semi-supervised Medical Image Segmentation Diffusion Models Label
Rectification.
## 1 Introduction
Medical image segmentation is crucial for clinical applications but often
requires large amounts of pixel-wise or voxel-wise labeled data, which is
tedious and time-consuming to obtain [17, 19, 18, 1, 28]. Such a heavy
annotation cost has motivated the community to develop semi-supervised
learning methods [10, 20, 14, 38]. Existing semi-supervised image segmentation
methods can be generally categorized into self-training and consistency
regularization. For self-training methods [1, 31, 4, 7, 34, 15, 40, 23], they
generate pseudo labels for unlabeled images, then use the pseudo-labeled
images in conjunction with labeled images to update the segmentation model
iteratively. This paradigm could effectively incorporate unlabeled data by
minimizing their entropy. For consistency regularization methods [5, 27, 21,
30, 35, 9, 22, 37, 33], they are designed based on the assumption that
perturbations should not change the predictions of the model, and have
achieved more promising performance recently. Perturbations are applied on the
input or the network level, and the models are enforced to achieve an
invariance of predictions.
Despite the progress, the semi-supervised medical image segmentation remains
challenging due to the following factors. (1) Reliance Risk: Existing methods
typically rely on self-generated pseudo labels to optimize the model [30, 37,
11, 35], which is ill-posed since errors in pseudo labels are preserved during
iterative optimization. The overfitting to incorrect supervision could lead to
severe confirmation bias [16] and considerable performance degradation.
Besides, they do not fully utilize the category-wise correlation in the pseudo
labels, and the label quality is sensitive to the perturbation design and
network structure. (2) Distribution Misalignment: Most methods only apply
consistency regularization and auxiliary supervision at the output mask level
to encourage the model to produce consistent mask predictions between
different perturbations [27, 5]. However, these approaches are insufficient in
capturing the semantics in the latent space and tend to overlook the
underlying label distributions, resulting in limited generalization to
unlabeled data.
To address the reliance risk issue, we first propose a Label Context
Calibration Module (LCC). Different from methods that directly use the self-
generated pseudo labels, LCC calibrates the biased semantic context, i.e., the
relationships between different semantic categories, and reduce the errors in
the pseudo labels. It starts with a semantic coloring scheme that encodes the
one-hot pseudo labels and ground truth masks into the visual space, and
subsequently feeds them into a semantic context embedding block to adjust the
features of the pseudo labels in the latent space. Notably, LCC introduces
explicit calibration guidance by encoding the dice score between the pseudo
labels and the ground truth, thereby providing more reliable calibration
directions for model optimization.
To tackle the distribution misalignment problem, some previous works have
proposed to model data distributions with VAE [41] or GAN [42]. However, their
adversarial training scheme could suffer from mode collapse and conflict
between generation and segmentation tasks, resulting in suboptimal
performance. Different from them, the denoising diffusion probabilistic model
(DDPM) is a new class of generative models trained using variational inference
[8, 24, 12, 13], which alleviates the above problem by formulating the complex
data distribution with probabilistic models. Therefore, we design a Latent
Feature Rectification Module (LFR), which models the consecutive refinement
between different latent distributions with a generative latent DDPM [25]. LFR
leverages the power of DDPM to learn the latent structure of the semantic
labels. Specifically, it first applies Gaussian noise on fine-grained label
features with a diffusion schedule, then uses the coarse-grained label
features as conditions to recover the clean feature. With the denoising
process, the consecutive transportations of coarse to fine and fine to precise
distributions of the pseudo labels are formulated and aligned, and the pseudo
labels are progressively rectified for better supervision. Based on LCC and
LFR, we construct a semi-supervised medical image segmentation framework named
Latent Diffusion Label Rectification Model (DiffRect). Extensive experimental
results show that our method outperforms prior methods by significant margins.
## 2 Methodology
### 2.1 Preliminary: Conditional DDPM
DDPM is a class of latent variable generative model that learns a data
distribution by denoising noisy images [8]. The forward process diffuses the
data samples with pre-defined noise schedules. Concretely, given a clean data
$z^{0}$, sampling of $z^{t}$ is expressed in a closed form:
$q(z^{t}\|z^{0})=\mathcal{N}(z^{t};\sqrt{\overline{\alpha}_{t}}z^{0},(1-\overline{\alpha}_{t})\mathbf{I}),$
(1)
where $\overline{\alpha}_{t}$ is the noise schedule variable [24, 8]. During
the reverse process, we are given an optional condition $\rho$ [6], and each
step is expressed as a Gaussian transition with learned mean
$\boldsymbol{\mu}_{\epsilon}$ and variance $\sigma_{\epsilon}$ from the
denoising model $\epsilon$:
$p\left(z^{t-1}\mid
z^{t},\rho\right):=\mathcal{N}\left(z^{t-1};\boldsymbol{\mu}_{\epsilon}\left(z^{t},t,\rho\right),\sigma_{\epsilon}\left(z^{t},t,\rho\right)\mathbf{I}\right).$
(2)
By decomposing the above equation, we have:
$z^{t-1}\leftarrow\frac{1}{\sqrt{\alpha_{t}}}(z^{t}-\frac{1-\alpha_{t}}{\sqrt{1-\overline{\alpha}_{t}}}\epsilon(z^{t},t,\rho))+\sigma_{\epsilon}\eta,$
(3)
where $\eta\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ is a sampled noise that
ensures each step is stochastic. In this work, we extend the conditional DDPM
to the latent space of pseudo labels, and model the distribution
transportations for label rectification.
### 2.2 Label Context Calibration Module (LCC)
Existing semi-supervised training schemes that rely extensively on self-
generated pseudo labels are often ill-posed, where errors in low-quality
pseudo labels accumulate and degrade performance. To address this issue, we
introduce LCC that effectively captures and calibrates the semantic context
within the visual space, thereby mitigating the impact of noisy labels. As in
Fig. 1(a), given the one-hot pseudo labels $y_{s},y_{w}\in\mathbb{R}^{H\times
W\times C}$ with height $H$ and width $W$ from the segmentation network, we
encode them to semantic pseudo labels $m_{s}$ and $m_{w}$ with dimensions of
$\mathbb{R}^{H\times W\times 3}$, using a proposed semantic coloring scheme
(SCS).
Concretely, for a dataset that contains $C$ different classes, we build a
color set $M_{C}$ that is composed of $C$ RGB colors, and each color is
represented by a tuple of three values within the range $[0,255]$. We maximize
the color difference between each encoded category to avoid semantic
confusion. Therefore, it can be represented by a functional mapping $f:C\to
M_{C}$, which is defined as:
$m_{(h,w)}=f(y_{(h,w)}),\quad{\forall}h\in[1,2,...,H],w\in[1,2,...,W],$
(4)
where $m$ is the semantic pseudo label in the visual space, and $m_{(h,w)}$
represents the mapped RGB color of the pixel at location $(h,w)$ in $m$. The
$y_{(h,w)}$ represents the class of the corresponding pixel in one-hot mask
$y$. The semantic coloring scheme can effectively incorporate color
information into the segmentation task, which enables the model to exploit
additional cues with the rich semantics from colors, and improves the
discrimination ability [32, 3] as well as the interpretability of the model.
Figure 1: Overall framework of DiffRect. (a) Label Context Calibration Module
(LCC). (b) Latent Feature Rectification Module (LFR). (c) Segmentation
Network.
To perform context calibration with the semantic labels, we design a semantic
context embedding block $\mathbf{B}_{sem}$, which embeds the pseudo labels to
the latent features $z_{s},z_{w},z_{l}$ with the dimensions of
$\mathbb{R}^{\frac{H}{16}\times\frac{W}{16}\times 256}$. Specially, additional
calibration guidance (CG) $\tau^{u}$ for unlabeled data and $\tau^{l}$ for
labeled data are also encoded into the block using the sinusoidal embeddings
[8, 29],
$\displaystyle\\{z_{s},z_{w}\\}$
$\displaystyle=\mathbf{B}_{\text{sem}}(m_{s},m_{w}\|\tau^{u})\quad\text{for
unlabeled data},$ (5) $\displaystyle\\{z_{w},z_{l}\\}$
$\displaystyle=\mathbf{B}_{\text{sem}}(m_{w},m_{l}\|\tau^{l})\quad\text{for
labeled data},$
where the $\tau^{u}$ and $\tau^{l}$ values for unlabeled and labeled data are
computed using the dice coefficient between the one-hot segmentation masks of
different qualities, which is denoted as follows:
$\tau^{u}=\text{Dice}(y_{s},y_{w}),\quad\tau^{l}=\text{Dice}(y_{w},y_{l}).$
(6)
By using the dice coefficient as the calibration guidance factor, the model
can simultaneously measure the quality of pseudo labels and integrate this
information into the learning process. It enables the model to better capture
the semantic context and refine the pseudo labels for both unlabeled and
labeled data.
### 2.3 Latent Feature Rectification Module (LFR)
To address the distribution misalignment issue between the pseudo labels with
different levels of quality, we propose a Latent Feature Rectification Module
(LFR), which is illustrated in Fig. 1(b).
Concretely, LFR applies a latent diffusion process to model the transportation
of label quality distributions. For each unlabeled data $I_{u}$, the strongly
and weakly semantic context embedding $z_{s}$ and $z_{w}$ are first obtained
with LCC. We then construct a diffusion process from $z_{w}$ to the diffused
noisy feature $z^{T}_{w}$ with $T$ timestamps as follows:
$\displaystyle z^{T}_{w}$
$\displaystyle=\sqrt{\alpha_{T}}z^{T-1}_{w}+\sqrt{1-\alpha_{T}}\eta^{T-1}$ (7)
$\displaystyle=\cdot\cdot\cdot=\sqrt{\overline{\alpha}_{T}}z_{w}+\sqrt{1-\overline{\alpha}_{T}}\eta,$
where $\alpha_{T}$ and $\overline{\alpha}_{T}$ are the schedule variables in
the diffusion forward process, (e.g., cosine [24]), and
$\overline{\alpha}_{T}=\prod^{T}_{i=1}\alpha_{i}$. The $\eta^{t}$ is the
corresponding noise sampled from Gaussian distribution at the $t$-th step.
Then, we train a denoising U-Net $\epsilon$ to learn to reverse this process.
Since the individual reverse diffusion process is unconditioned, we add
$z_{s}$ as the conditional input and also feed it into the denoising model.
Therefore, the model is encouraged to learn the distribution transportation
from coarse-grained masks $p(z_{s})$ (strong pseudo labels) to the latent
distributions of fine-grained masks $p(z_{w})$ (weak pseudo labels), where we
denote it as a strong-to-weak transportation (S2W). The reverse diffusion is
formulated as the following Markov chain:
$\begin{aligned}
&p_{\epsilon}\left(z_{w}^{0:T}\right):=p\left(z_{w}^{T}\right)\prod_{t=1}^{T}p_{\epsilon}\left(z_{w}^{t-1}\mid
z_{w}^{t},z_{s}\right),\quad
z_{w}^{T}\sim\mathcal{N}(\mathbf{0},\mathbf{I})\\\
&p_{\epsilon}\left(z_{w}^{t-1}\mid
z_{w}^{t},z_{s}\right):=\mathcal{N}\left(z_{w}^{t-1};\boldsymbol{\mu}_{\epsilon}\left(z_{w}^{t},t,z_{s}\right),\sigma_{\epsilon}\left(z_{w}^{t},t,z_{s}\right)\mathbf{I}\right),\end{aligned}$
(8)
where $\boldsymbol{\mu}$ and $\sigma$ are the predicted data mean and variance
from the denoising U-Net model. For the training with unlabeled input, the
latent loss for optimization can be expressed as follows:
$\mathcal{L_{\text{Lat-U}}}=E_{z_{w},t}\left[\left\|z_{w}-r_{w}\right\|_{2}\right],$
(9)
where $r_{w}=\epsilon\left(z_{w}^{T},z_{s},t\right)$, which is the
reconstructed version of the weakly semantic context embedding $z_{w}$. The
objective minimizes the $\ell_{2}$ distance between the clean and denoised
feature and encourages the model to learn the distribution transportation from
a coarse pseudo label to a fine pseudo label.
Similarly, we can obtain the weak semantic context embedding of labeled data
$z_{w}$ and the ground truth $z_{l}$. We then learn the reverse process that
recovers $z_{l}$ based on the $T$-timestamp diffused noisy feature
$z_{l}^{T}$, with the $z_{w}$ as condition:
$\begin{aligned}
&p_{\epsilon}\left(z_{l}^{0:T}\right):=p\left(z_{l}^{T}\right)\prod_{t=1}^{T}p_{\epsilon}\left(z_{l}^{t-1}\mid
z_{l}^{t},z_{w}\right),\quad
z_{l}^{T}\sim\mathcal{N}(\mathbf{0},\mathbf{I})\\\
&p_{\epsilon}\left(z_{l}^{t-1}\mid
z_{l}^{t},z_{w}\right):=\mathcal{N}\left(z_{l}^{t-1};\boldsymbol{\mu}_{\epsilon}\left(z_{l}^{t},t,z_{w}\right),\sigma_{\epsilon}\left(z_{l}^{t},t,z_{w}\right)\mathbf{I}\right),\end{aligned}$
(10)
and the training objective for the reconstructed feature
$r_{l}=\epsilon\left(z_{l}^{T},z_{w},t\right)$ is:
$\mathcal{L_{\text{Lat-L}}}=E_{z_{l},t}\left[\left\|z_{l}-r_{l}\right\|_{2}\right].$
(11)
With the above latent diffusion process, the continual distribution
transportations from fine-grained mask distributions $p(z_{w})$ (weak pseudo
labels) to precise mask distributions $p(z_{l})$ (ground truth) are also
formulated in the latent space, which is denoted as the weak-to-groud truth
transportation (W2G). The denoising U-Net is hence capable to achieve latent
feature rectification.
Afterwards, the weak pseudo labels of unlabeled data are fed into the
denoising U-Net for obtaining the rectified features with progressive
denoising. Specifically, we randomly sample a Gaussian noise
$r_{l}^{T}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ as the input of the
denoising U-Net, which simulates the $T$-timestamp noisy feature of the
rectified pseudo label $y_{r}$. The rectified feature $r_{l}$ is generated via
a progressive reverse diffusion process, with the weak pseudo label features
$z_{w}$ as condition. Mathematically, a single denoising from step $t$ to
$t-1$ is formulated as:
$r_{l}^{t-1}\leftarrow\frac{1}{\sqrt{\alpha_{t}}}(r_{l}^{t}-\frac{1-\alpha_{t}}{\sqrt{1-\overline{\alpha}_{t}}}\epsilon(r_{l}^{t},t,z_{w}))+\sigma_{\epsilon}\eta,$
(12)
where $\eta\sim\mathcal{N}(\mathbf{0},\mathbf{I})$ which ensures each step is
stochastic as in DDPM [8]. The rectified label is obtained with an upsampling
of the feature $r_{l}$ to the input resolution $y_{r}=Upsample(r_{l})$, which
is utilized as a better and more precise supervision signal for the
segmentation model.
### 2.4 Loss Function
The training of the DiffRect frameworks includes two parts: (1) the
optimization of segmentation U-Net $\theta$ (with Seg Loss) and (2) the joint
optimization of the rectification components $\mathbf{B}_{sem}$ and $\epsilon$
(with Diff Loss). The overall loss is:
$\mathcal{L_{\text{DiffRect}}}=\underbrace{{\mathcal{L^{\text{Seg}}_{\text{Semi}}}+\mathcal{L_{\text{Rect}}}}}_{\text{Seg
Loss}}+\underbrace{{\mathcal{L^{\text{Lat}}_{\text{Semi}}}+\lambda_{1}\mathcal{L_{\text{Lat-U}}}+\lambda_{2}\mathcal{L_{\text{Lat-L}}}}}_{\text{Diff
Loss}},$
(13)
where $\mathcal{L^{\text{Seg}}_{\text{Semi}}}$ and
$\mathcal{L^{\text{Lat}}_{\text{Semi}}}$ are the semi-supervised losses for
segmentation as in [27]. The $\lambda_{1}$ and $\lambda_{2}$ are trade-off
factors to balance the contribution of each term. $\mathcal{L_{\text{Rect}}}$
is the rectified supervision loss between $y_{w}$ and the rectified pseudo
label $y_{r}$, where the summation of cross-entropy and Dice score are used:
$\mathcal{L_{\text{Rect}}}=\text{CE}(y_{w},y_{r})+\text{Dice}(y_{w},y_{r}).$
(14)
During inference, the input is directly fed into segmentation network in Fig.
1(c) to produce the segmentation result, thus no extra inference cost is
required.
Table 1: Segmentation results on the ACDC validation and test sets.
Method | Labeled | ACDC Validation Set | ACDC Test Set
---|---|---|---
Ratio | Dice$\uparrow$ | Jac$\uparrow$ | HD95$\downarrow$ | ASD$\downarrow$ | Dice$\uparrow$ | Jac$\uparrow$ | HD95$\downarrow$ | ASD$\downarrow$
UAMT [39] | 1% | 42.28 | 32.21 | 40.74 | 18.58 | 43.86 | 33.36 | 38.60 | 18.33
FixMatch [27] | 69.67 | 58.34 | 37.92 | 14.41 | 60.80 | 49.14 | 36.81 | 14.75
CPS [5] | 56.70 | 44.31 | 24.97 | 10.48 | 52.28 | 41.68 | 20.38 | 7.35
ICT [30] | 43.03 | 30.58 | 34.92 | 15.23 | 42.91 | 32.81 | 25.42 | 10.80
MCNetV2 [35] | 57.49 | 43.29 | 31.31 | 10.97 | 49.92 | 39.16 | 24.64 | 8.47
INCL [43] | 77.80 | 66.13 | 11.69 | 3.22 | 67.01 | 56.22 | 13.43 | 3.35
DiffRect (Ours) | 82.40 | 71.96 | 10.04 | 2.90 | 71.85 | 61.53 | 5.79 | 2.12
UAMT [39] | 5% | 72.71 | 60.89 | 21.48 | 7.15 | 69.93 | 58.45 | 17.01 | 5.25
FixMatch [27] | 83.12 | 73.59 | 9.86 | 2.61 | 74.68 | 64.12 | 11.18 | 2.93
CPS [5] | 75.24 | 64.67 | 10.93 | 2.98 | 74.67 | 63.51 | 9.37 | 2.55
ICT [30] | 74.20 | 62.90 | 17.01 | 4.32 | 73.10 | 60.69 | 11.92 | 3.70
MCNetV2 [35] | 78.96 | 68.15 | 12.13 | 3.91 | 75.86 | 65.20 | 9.85 | 2.88
INCL [43] | 85.43 | 75.76 | 6.37 | 1.37 | 80.64 | 70.78 | 5.29 | 1.42
DiffRect (Ours) | 86.95 | 78.08 | 4.07 | 1.23 | 82.46 | 71.76 | 7.18 | 1.94
UAMT [39] | 10% | 85.14 | 75.90 | 6.25 | 1.80 | 86.23 | 76.72 | 9.40 | 2.56
FixMatch [27] | 88.31 | 79.97 | 7.35 | 1.79 | 87.96 | 79.37 | 5.43 | 1.59
CPS [5] | 84.63 | 75.20 | 7.57 | 2.27 | 85.61 | 75.76 | 9.29 | 3.00
ICT [30] | 85.15 | 76.05 | 4.27 | 1.46 | 86.77 | 77.43 | 8.01 | 2.16
MCNetV2 [35] | 85.97 | 77.21 | 7.55 | 2.11 | 88.75 | 80.28 | 6.16 | 1.64
INCL [43] | 88.28 | 80.09 | 1.67 | 0.49 | 88.68 | 80.27 | 4.34 | 1.13
DiffRect (Ours) | 90.18 | 82.72 | 1.38 | 0.48 | 89.27 | 81.13 | 3.85 | 1.00
Supervised [26] | 100% | 91.48 | 84.87 | 1.12 | 0.34 | 91.65 | 84.95 | 1.14 | 0.50
Table 2: Segmentation results on MS-CMRSEG 2019 with 20% data labeled.
Method | Dice $\uparrow$ | Jac$\uparrow$ | HD95$\downarrow$ | ASD$\downarrow$
---|---|---|---|---
UAMT [39] | 84.27 | 73.69 | 12.15 | 4.18
FixMatch [27] | 84.31 | 73.57 | 17.79 | 4.81
CPS [5] | 83.66 | 73.03 | 15.01 | 4.30
ICT [30] | 83.66 | 73.06 | 17.24 | 4.85
MCNetV2 [35] | 83.93 | 73.45 | 13.10 | 3.39
INCL [43] | 84.33 | 73.92 | 9.95 | 2.61
DiffRect | 86.78 | 77.13 | 6.39 | 1.85
Supervised [26] | 88.19 | 79.28 | 4.21 | 1.32
Table 3: Segmentation results on Decathlon Prostate with 10% data labeled.
Method | Dice$\uparrow$ | Jac$\uparrow$ | HD95$\downarrow$ | ASD$\downarrow$
---|---|---|---|---
UAMT [39] | 40.91 | 29.13 | 28.32 | 10.45
FixMatch [27] | 54.70 | 41.07 | 16.82 | 5.24
CPS [5] | 43.51 | 31.18 | 26.93 | 8.31
ICT [30] | 39.91 | 28.95 | 24.73 | 7.59
MCNetV2 [35] | 40.58 | 28.77 | 21.29 | 7.11
INCL [43] | 55.67 | 41.91 | 31.09 | 15.78
DiffRect | 62.23 | 48.64 | 10.36 | 3.41
Supervised [26] | 73.81 | 61.25 | 7.28 | 1.94
## 3 Experiments
### 3.1 Experimental Setup
We examine all methods with identical settings for fair comparison, and
trained on a NVIDIA 4090 GPU for 30k iterations. For the $\mathbf{B}_{sem}$
which downsamples the input to $\frac{H}{16}\times\frac{W}{16}$, we use two
$3\times 3$ convolution layers followed by BN and LeakyReLU before the
2$\times$ downsample in each stage, and repeat for four stages. The Denoising
U-Net $\epsilon$ down and upsamples the input by 4$\times$, which also uses
two $3\times 3$ convolution layers per stage. The multi-scale image feature is
embedded into the model via concatenation as in [36]. For the weak
perturbation, we apply random flipping and rotation. For the strong
perturbation, we apply random Gaussian blur and additional random image
adjustments, including contrast, sharpness, and brightness enhancement. For
ACDC, we test the 1%, 5%, and 10% labeling regimes following [20]. For MS-
CMRSEG 2019, 20% labeling regime is tested, while 10% labeled data is used in
Decathlon Prostate.
### 3.2 Comparison with State-of-the-art Methods
We validate the effectiveness of the proposed approach on the ACDC dataset [2]
in Tab. 1. Our method shows superior results under all labeling regimes.
Compared with MCNetV2 [35], our method possesses superior capability with
increments of 24.91%, 7.99%, 4.21% in Dice, 28.67%, 9.93%, 5.51% in Jaccard on
the validation set with 1%, 5%, and 10% scans available. DiffRect displays
better segmentation performance even when the labeled samples are extremely
scarce (e.g. 82.40% Dice with 1% scans available), suggesting it can model the
transportation of the pseudo label distributions precisely and produce refined
masks. Results in MS-CMRSEG 2019 are shown in Tab. 3. DiffRect shows
consistent performance gain on all metrics, with 86.78% in Dice, 77.13% in
Jaccard, 6.39mm in HD95, and 1.85mm in ASD, outperforming the state-of-the-art
method INCL [43] by 2.45% Dice, 3.21% Jaccard, 3.56mm HD95, and 0.76mm in ASD,
respectively. On Decathlon Prostate in Tab. 3, DiffRect remains showing
compelling results, demonstrating its capability in various modalities.
Table 4: Ablation study of the proposed modules.
Method | w/o | Dice$\uparrow$ | Jac$\uparrow$ | HD95$\downarrow$ | ASD$\downarrow$
---|---|---|---|---|---
Baseline | - | 69.67 | 58.34 | 37.92 | 14.41
+LCC | SCS | 73.83 | 61.83 | 29.49 | 11.71
| CG | 76.12 | 64.69 | 26.24 | 8.31
| - | 78.28 | 66.97 | 20.46 | 5.60
+LCC | S2W | 79.97 | 69.31 | 14.07 | 4.91
& LFR | W2G | 78.57 | 66.38 | 21.07 | 5.91
| - | 82.40 | 71.96 | 10.04 | 2.90
Table 5: Ablation study of different calibration guidance choices in LCC.
Choice | Dice$\uparrow$ | Jac$\uparrow$ | HD95$\downarrow$ | ASD$\downarrow$
---|---|---|---|---
Dice | 82.40 | 71.96 | 10.04 | 2.90
Jaccard | 82.37 | 71.82 | 11.33 | 2.87
Fixed | 80.34 | 69.67 | 14.97 | 4.47
Random | 80.60 | 69.99 | 13.15 | 3.75
Both | 81.67 | 71.45 | 10.28 | 2.47
### 3.3 Further Analysis
Ablation study of the proposed modules. We evaluate the effect of individual
modules in DiffRect in Tab. 5. Adopting LCC achieves 78.28% Dice and 66.97%
Jaccard, with 8.61% and 8.63% gains compared with the Fixmatch baseline [27].
Removing the semantic coloring scheme (SGS) shows a large performance drop
(73.83% Dice and 61.83% Jaccard), showing the importance of exploiting the
semantics in the visual domain. No calibration guidance (CG) causes 2.16% Dice
drop due to the impact of noisy calibration directions. Adding LFR improves
Dice by 4.12% and 10.42mm in HD95. Removing the strong to weak transportation
(S2W) shows a 2.43% Dice drop while removing the weak to ground truth (W2G)
causes a severe Dice drop to 78.57%. The results demonstrate the necessity of
each sub-component.
Different Calibration Guidance Choices. To analyze the effectiveness and the
optimal choice of calibration guidance, experiments were conducted to compare
the performance of models trained with different calibration guidance in Tab.
5, including Dice score, Jaccard score, Fixed (using a fixed value 0.5),
Random (using a random sampled value within 0$\sim$1), and Both (using the
summation of Dice and Jaccard). It is shown that Dice, Jaccard, and Both have
similar performance, and outperform the fixed and random strategies, which
validates the reliable directions provided for optimization.
## 4 Conclusion
In this paper, we identify the reliance risk and distribution misalignment
issues in semi-supervised medical image segmentation, and propose DiffRect, a
diffusion-based framework for this task. It comprises two modules: the LCC
aims to calibrate the biased relationship between classes in pseudo labels by
learning category-wise correlation, and the LFR models the consecutive
transportations between coarse to fine and fine to precise distributions of
the pseudo labels accurately with latent diffusion. Extensive experiments on
three datasets demonstrate that DiffRect outperforms existing methods by
remarkable margins.
#### 4.0.1 Acknowledgements
This work was supported by Hong Kong Research Grants Council (RGC) General
Research Fund 14204321.
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|
# RTFormer: Re-parameter TSBN Spiking Transformer
1st Hongzhi Wang School of Software Technology
Zhejiang University
Ningbo, China
<EMAIL_ADDRESS>2nd Xiubo Liang *Corresponding author School of Software
Technology
Zhejiang University
Ningbo, China
<EMAIL_ADDRESS>3rd Mengjian Li Research Center for Data Hub and Security
Zhejiang Lab
Hangzhou, China
<EMAIL_ADDRESS>4th Tao Zhang School of Software Technology
Zhejiang University
Ningbo, China
<EMAIL_ADDRESS>
###### Abstract
The Spiking Neural Networks (SNNs), renowned for their bio-inspired
operational mechanism and energy efficiency, mirror the human brain’s neural
activity. Yet, SNNs face challenges in balancing energy efficiency with the
computational demands of advanced tasks. Our research introduces the RTFormer,
a novel architecture that embeds Re-parameterized Temporal Sliding Batch
Normalization (TSBN) within the Spiking Transformer framework. This innovation
optimizes energy usage during inference while ensuring robust computational
performance. The crux of RTFormer lies in its integration of re-parameterized
convolutions and TSBN, achieving an equilibrium between computational prowess
and energy conservation. Our experimental results highlight its effectiveness,
with RTFormer achieving notable accuracy on standard datasets like ImageNet
(80.54%), CIFAR-10 (96.27%), and CIFAR-100 (81.37%), and excelling in
neuromorphic datasets such as CIFAR10-DVS (83.6%) and DVS128 (98.61%). These
achievements illustrate RTFormer’s versatility and establish its potential in
the realm of energy-efficient neural computing.
###### Index Terms:
SNNs, LIF, Transformer, Normalization
## I Introduction
Inspired by the human brain, deep Artificial Neural Networks (ANNs) have
garnered significant success, particularly in areas such as computer vision
[1, 2] and nature language processing [3, 4, 5, 6] . However, these
accomplishments come at a considerable computational cost. ANNs consume
approximately 12 times [7] more energy than the human brain, rendering high-
energy models tough to deploy onto resource-constrained appliances, for
example, smartphones and IoT devices. Utilising the brain’s efficient
computational paradigm to create low-energy neural networks on such platforms
holds significant value.
Why SNN? Spiking Neural Networks (SNNs) present themselves as paragons of
energy efficiency in the computational realm. While structurally echoing the
design of traditional ANNs, SNNs diverge in their unique handling of data
through discrete binary events. The zero denotes a dormant state, whereas one
signals the firing of a neuron, a spike that conveys information. This binary
data processing leads to sparse activations within the network, ensuring that
energy is expended only when necessary. Such efficiency is not merely
incidental but a core feature of SNNs, enabling them to operate with a
fraction of the power required by their ANN counterparts and thus answering
the call for sustainable and energy-conscious computing.
How can we design a structure in which inference is more energy efficient?
Addressing the question of how to craft a more energy-efficient structure for
inference leads us to the inception of the Spatial-Temporal Core. The Spatial
Core streamlines the convolutional process by employing structurally
reparameterized convolutions, significantly reducing the computational burden
during inference without compromising the integrity of learned features.
Simultaneously, the Temporal Core introduces the concept of Temporal Sliding
Batch Normalization (TSBN), which tailors the batch normalization process to
the temporal aspects of data, ensuring that the network remains responsive to
the temporal dynamics inherent in real-world scenarios. Together, these cores
form a robust framework that not only excels in energy efficiency but also
maintains high fidelity in data processing, making it an ideal candidate for
deployment in energy-constrained environments like neuromorphic hardware.
What’s the meaning of intaking the ST-Core? The Spatial-Temporal Core is not
just a technological innovation; it’s a conceptual shift towards creating
neural networks that operate with a level of energy efficiency akin to the
human brain. By drawing inspiration from nature’s most sophisticated computing
machine, we aim to bridge the gap between the computational prowess of deep
learning models and the energy limitations of the devices they run on. This
synergy of spatial efficiency and temporal precision paves the way for the
next generation of neural network models that are both powerful and
sustainable, ready for deployment in the increasingly connected and mobile
world we live in.
Our contributions are summarized below:
* •
We introduce the Spatial-Temporal Core, a harmonious fusion of structurally
reparameterized convolutions (Spatial) and dynamic temporal batch
normalization (Temporal) , crafted to deliver enhanced spatial efficiency and
temporal adaptability, thereby elevating neuromorphic computing to new heights
of processing excellence.
* •
We introduce the TSBN, an ingenious mechanism that aligns batch normalization
with the temporal dimension of data, allowing for precise, context-sensitive
normalization across sequential inputs, thereby bolstering the temporal
coherence and predictive performance of neural networks.
* •
Extensive experiments confirming the proposed architecture’s superiority over
state-of-the-art SNNs on neuromorphic and non-neuromorphic datasets,
underlining its practical significance in advancing Spatial-Temporal data
processing.
## II Related Works
### II-A SNN Learning Methods
Spiking Neural Networks (SNNs) are heralded as the third generation of neural
network models due to their biological fidelity, intrinsic event-driven
computation, and energy efficiency on neuromorphic platforms [8]. These
characteristics have catalyzed a surge in SNN research, positioning them as
formidable contenders to their Artificial Neural Networks (ANNs) counterparts.
The fundamental divergence between SNNs and ANNs stems from the employment of
spiking neurons as the fundamental computational unit, which facilitates
biological interpretability and adeptness in processing temporal information
[9, 10, 11, 12, 13]. ANNs, with their robust gradient backpropagation training
frameworks, contrast with SNNs, which predominantly utilize two training
paradigms: ANN-to-SNN conversion and direct training with surrogate gradients.
The conversion approach [14, 15, 16] entails substituting a pre-trained ANN’s
ReLU layers with spiking neurons, necessitating fine-tuning of hyperparameters
to retain accuracy. Nevertheless, this technique is limited by extended
conversion time-steps and the architectural rigidity of the source ANN. To
circumvent these limitations, [9] employs surrogate gradients to facilitate
direct SNN training, yielding high accuracy within minimal temporal intervals.
Methodologies have facilitated breakthroughs across domains, with Spiking-Yolo
[17] and EMS-yolo [18] paving the way in object detection, Spiking-UNet [19]
advancing semantic segmentation, SpikingGPT [20], and SpikingBert [21]
emerging in language modeling, and SpikingGAN [22] introducing generative
capabilities. For graph-based learning, SpikingGCN [23] and SpikingGAT [24]
have shown promise. The advent of neuromorphic chips such as TrueNorth [25],
Loihi [26], and Tianjic [27] further underscores the potential of SNNs to
become prevalent in near-term computational ecosystems.
### II-B Transformer Architecture In SNNs
Thanks to the well-established and effective network architectures in ANNs,
SNNs can utilise them to construct high-performance models, such as [28, 29,
7, 30]. The attention mechanism, currently the most efficient method in ANNs,
has also been integrated into SNNs, including the implementation of the
Transformer, its most classic network architecture. Spikformer [31] is the
initial directly-trained Transformer within SNNs. It adopts a new spike-form
self-attention named Spiking Self Attention (SSA). However, the current
configuration of the Spikformer, which includes residual [32] connections,
still involves non-spike computation. Therefore, the Spike-Driven [33]
Transformer presents novel structures for preserving the spike computation.
The issue of non-spike computation is resolved by Spike-Driven Transformer
through introducing Spike-Driven Self-Attention (SDSA). The integration of
attention mechanisms within SNNs has facilitated the adaptation of Transformer
architectures to the SNN paradigm. However, previous implementations have not
adequately addressed constraints encountered during inference. To this end, we
propose RTFormer, which leverages a reparameterization strategy to ensure that
SNNs benefit from reduced parameter complexity during inference, thereby
enhancing deployment efficiency.
### II-C BatchNormalization In SNNs
In the realm of SNNs, the integration of Batch Normalization (BN) techniques
has been pivotal in mitigating challenges associated with training dynamics,
such as the vanishing or exploding gradient problem. One innovative approach,
termed Batch Normalization Through Time (BNTT), was proposed by [34]. BNTT
uniquely calculates BN statistics and parameters independently at each time-
step, enhancing the network’s adaptability to instantaneous changes. However,
this method may not fully account for the temporal correlations present in
input spike sequences, potentially overlooking crucial sequential information.
To address this limitation, [35] introduced the threshold-dependent Batch
Normalization (tdBN) methodology. This technique consolidates BN statistics
and parameters across the temporal dimension, thus maintaining the
conventional BN’s benefits while accommodating the temporal structure inherent
in SNNs. By aggregating data temporally, tdBN circumvents the instability
often encountered in gradients during SNN training. Further expanding upon
these developments, the Temporal Effects Batch Normalization (TEBN) method,
conceived by [36], has refined the approach by merging data along the temporal
axis for shared BN statistics. TEBN then introduces temporal dynamics into the
BN process by applying distinct scaling weights. This method captures the
essential temporal dynamics, thereby providing a more nuanced normalization
process that aligns with the temporal nature of SNNs. We introdece the TSBN to
selectively leverage the accumulated pre-synaptic inputs in the temporal
domain, consistent with the properties of LIF neurons.
## III Method
We introduce the RTFormer, a novel fusion of the Transformer architecture with
the Re-parameter and Temporal Sliding Batch Normalization(TSBN). This section
will begin by providing a concise overview of spike neurons’ working
principles, followed by an in-depth exploration of the Spatial-Temporal Core
and the Spiking Guided Attention (SGA) module. Finally, we will discuss the
energy consumption aspect.
### III-A Preliminaries
In SNNs, spike neurons control the release of spikes based on a threshold. In
this paper we use LIF [37] neurons, which work in the following way:
$\displaystyle
U[t]=V[t-1]+\frac{1}{k_{\tau}}\left(X[t]-(V[t-1]-V_{reset})\right)$ (1)
$\displaystyle S[t]=\mathcal{H}(U[t]-V_{th})$ (2) $\displaystyle
V[t]=U[t]~{}(1-S[t])+V_{reset}S[t]$ (3)
where $k_{\tau}$, $V_{th}$, and $V_{reset}$ represent the decay factor, firing
threshold, and reset membrane potential, respectively, which are pre-set to
default values. The notation $X[t]$ refers to the input at time step $t$,
while $U[t]$ denotes the membrane potential. The function
$\mathcal{H}$($\cdot$) represents the Heaviside step function. The spike
output, denoted by $S[t]$, is calculated based on the membrane potential and
the threshold. Additionally, $V[t]$ and $V[t-1]$ signify the temporal output
at time t.
In our study, we employ the spikingjelly [38], utilizing default values for
$k_{\tau}$, $V_{th}$, and $V_{reset}$, specifically set to 2.0, 1.0, and 0,
respectively.
Figure 1: Illustration of the structural reparameterization and simplification
from a complex multi-branch system to a streamlined model. The top dashed box
represents the parameters of the TSBN incorporated into Conv3, and the bottom
dashed box represents the parameters of the TSBN incorporated into the
trainable threshold.
### III-B Spatial-Temporal Core
In the exploration of SNNs, our innovative design, termed the ”Spatial-
Temporal Core,” shown in Fig.1, addresses the nuanced realms of spatial and
temporal processing. This sophisticated framework divides the architecture
into two synergistic components: the ”Spatial Core” and the ”Temporal Core.”
Each core is uniquely adapted to manage distinct aspects of data processing -
the former concentrating on spatial features and the latter on temporal
dynamics.
Spatial Core. The ”Spatial Core” echoes the principles of structural
reparameterization, akin to approaches seen in Artificial Neural Networks
(ANNs). Here, we utilize depthwise convolutions (DW-Conv) that are
strategically reparameterized to reduce the complexity of the model during
inference. This innovative arrangement entails parallel DW-Conv layers with
diverse kernel sizes, notably 1x1 and 3x3, enhancing spatial feature
extraction efficiency. The transformation from four consecutive 3x3 layers to
a trio of ”Spatial Core” units signifies a leap in spatial detail capture, all
while maintaining a lean model structure.
In the innovative architecture depicted in Fig.1, the STCore is structured
with five concurrent branches, each contributing unique convolutional
parameters to the network’s composite function. Notably, one of these
branches—the identity branch—functions effectively as a 1 × 1 convolution,
utilizing an identity matrix as its kernel. This integration ensures that each
branch’s convolutional characteristics are distinctly represented. The fusion
of these diverse convolutional influences is succinctly captured in Eq.4:
$y=\sum_{i=1}^{n}TSBN_{i}(x*W^{(i)},\mu_{i},\sigma_{i},\gamma_{i},\beta_{i})$
(4)
Here, the index $i$ extends over $n$ distinct branches, with this work
specifically focusing on $n=5$. The variable $x$ symbolizes the input, while
$y$ represents the resultant output. The operator $"*"$ denotes the
convolution operation, and $(i)$ signifies the convolutional kernelassociated
with the $i_{th}$ branch. The parameters $\mu,\sigma,\gamma,\beta$ correspond
to the accumulated mean, standard deviation, as well as the learned scaling
factor and bias, respectively, derived from the BN layer that follows each
convolutional operation. These parameters are crucial for the TSBN process,
which refines the data at each branch, ensuring a harmonized and effective
integration of the temporal dynamics into the network’s overall learning
process.
Temporal Core. Conversely, the ”Temporal Core” integrates Temporal Sliding
Batch Normalization (TSBN) with the adjustable thresholds of spiking neurons.
This integration is a pivotal aspect of our approach, aligning with the innate
dynamics of temporal information processing inherent in SNNs. By incorporating
the TSBN parameters directly into the neurons’ thresholding mechanism (denoted
as $V_{th}$), our model gains a robustness in handling temporal sequences,
which is essential for cognitive functions.
In practice, a sliding window mechanism judiciously controls the extent of
Batch Normalization across time, allowing for refined data processing at each
stage. Diverging from traditional BN methods like tdBN and BNTT, our focus is
on data closer to the current timestep, ensuring a more context-sensitive
normalization approach. This methodology enhances the network’s capability to
adeptly respond to dynamic temporal shifts, as captured in Eq.5.
$y_{[t-w:t]}=\gamma(t)\frac{x_{[t-w:t]}-\mu_{[t-w:t]}}{\sqrt{\sigma_{[t-w:t]}^{2}+\epsilon}}+\beta(t)$
(5)
Here, $x_{[t-w:t]}$ and $y_{[t-w:t]}$ signify the input and output,
respectively, spanning a temporal window of width $w$. The variable $t$
specifies the current timestep, anchoring the window’s position within the
sequence. $\gamma(t)$ and $\beta(t)$ serve as the scale and shift factors at
each specific timestep $t$. $\mu_{[t-w:t]}$ and $\sigma_{[t-w:t]}$ are the
statistical core of this equation, representing the mean and variance computed
over the inputs within the sliding window from $t-w$ to $t$, $\epsilon$ is a
small number to avoid dividing zero .
Moreover, we fold the TSBN into the neuron’s threshold , enabling an efficient
integration of temporal normalization into the spiking mechanism of the
neurons. This folding process is encapsulated in Eq.7, transforming $V_{th}$
into a trainable parameter , which dynamically adapts to the temporal
normalization.
${s}^{(t)}=\begin{cases}1&\text{if
}\gamma(t)\frac{x_{[t-w:t]}-\mu_{[t-w:t]}}{\sqrt{\sigma_{[t-w:t]}^{2}+\epsilon}}+\beta(t)>V_{\rm
th}\\\ 0&\text{otherwise}\end{cases}.$ (6)
${\tilde{V}}_{\rm th}=\frac{(V_{\rm
th}-{\beta}(t)){\sqrt{{\sigma}_{[t-w:t]}^{2}}}}{{\gamma}(t)}+{\mu}_{[t-w:t]}$
(7)
Here, $V_{th}$ is the threshold of spiking neuron, ${\tilde{V}}_{\rm th}$ is
its trainable counterpart in this work. $s^{(t)}$ is the spiking output matrix
at timestep t.
The ”Spatial-Temporal Core” stands as a hallmark of innovation in SNN
architecture. Its ability to fluidly navigate both spatial and temporal
dimensions positions it as a vital development towards emulating the
sophisticated computational capabilities of the human brain. This structure
not only ensures model efficiency but also accentuates the biological
resemblance and pulse-like nature of SNNs, underscoring its potential for
real-world applications where both performance and biologically-inspired
functionality are paramount.
TABLE I: Experiments on ImageNet. ’T’ denotes the timestep. The architecture abbreviations ’S-V,’ ’S-R,’ and ’S-T’ correspond to Spiking VGG, Spiking Resnet, and Spiking Transformer, respectively. Dataset | Methods | Type | Architecture | T | Param(M) | Acc(%) | Energy(mJ)
---|---|---|---|---|---|---|---
ImageNet | RMP [39] | ANN2SNN | S-V-16 | 4096 | 138.4 | 73.09 | 49.86
Calibration [40] | ANN2SNN | S-V-16 | 2048 | 138.4 | 75.32 | 25.98
SEW-ResNet [29] | SNN training | S-R-152 | 4 | 60.19 | 69.26 | 12.89
MS-ResNet [30] | SNN training | S-R-104 | 4 | 77.28 | 76.02 | 10.19
Att-MS-ResNet [7] | SNN training | S-R-104 | 4 | 78.37 | 77.08 | 7.3
tdBN [35] | SNN training | S-R-34 | 6 | 21.79 | 63.72 | 6.39
TEBN [36] | SNN training | S-R-34 | 4 | 21.79 | 64.29 | 7.05
MPBN [41] | SNN training | S-R-34 | 4 | 21.79 | 64.71 | 6.56
spikformer [31] | SNN training | S-T-8-768 | 4 | 66.34 | 74.81 | 21.47
spike-driven [33] | SNN training | S-T-8-768* | 4 | 66.34 | 77.07 | 6.09
This work | SNN training | S-T-8-768 | 4 | 58.86 | 80.54 | 5.59
### III-C Analyse Of Energy Consumption
In ANNs, computational demands largely stem from Floating-point Operations
(FLOPs), predominantly due to Multiply and Accumulate (MAC) operations. SNNs,
however, primarily rely on Accumulate (AC) operations, which reduces the need
for MAC operations. This shift not only cuts down on FLOPs but also aligns
with SNNs’ energy-efficient ethos by reducing power usage.
Yet, MAC operations remain a factor in the initial stages of data processing,
where raw images are converted into spike-encoded formats. To gauge energy
use, an assessment of MAC and AC operations throughout the network’s
computational processes is essential.
The total energy consumption ($E$) can be expressed as follows:
$\displaystyle E=E_{MAC}\times FL_{conv}+E_{AC}\times$ (8)
$\displaystyle(FL_{STCore}+FL_{SGA}+FL_{STMLP})$
Here, $E_{MAC}$ and $E_{AC}$ represent the energy consumption associated with
MAC and AC operations, respectively. Experimental measurements have determined
$E_{MAC}$ to be approximately 4.6 picojoules (pJ), and $E_{AC}$ to be
approximately 0.9 picojoules (pJ), based on testing conducted on 45nm
technology [42], $FL_{\cdot}$ denotes the float number of the specific layer.
Calculating energy consumption offers an accurate measure of the computational
and energy efficiency enhancements delivered by our framework, considering the
interplay of both MAC and AC operations across the processing pipeline.
## IV Experiments
Our experimental evaluation encompasses both non-neuromorphic datasets like
CIFAR10, CIFAR100, and ImageNet, as well as neuromorphic datasets such as
CIFAR10-DVS and DVS128 Gesture. The visualizations of these results are
presented in Fig.2, with ImageNet findings detailed in Tab.I. Results for
other datasets are compiled in Tab.II, while the outcomes of our ablation
studies are summarized in Tab.III. Additionally, visual representations of
these ablation studies can be found in Fig.3.
### IV-A Non-neuromorphic Datasets Classification
#### IV-A1 ImageNet
Dataset Description. The ImageNet dataset, a cornerstone in the field of
computer vision, consists of approximately 1.3 million images spanning 1,000
classes for training, alongside 50,000 validation images.
Figure 2: Display of a comparative visualization across four columns for a
series of images. Visualization comprises original images, Grad-CAM
representations, Attention Maps , and Spiking Fire Rate (SFR) maps.
RTFormer’s performance. As shown in Table I, the RTFormer model with TSBN and
structurally reparameterized DW-conv achieves remarkable accuracy.
Specifically, the Spikformer-8-768 model, equipped with 58.86M parameters,
attains a top-1 accuracy of 80.54%, which is a notable enhancement over
previous SNN models like SEW-ResNet and MS-ResNet. This leap in performance is
also accompanied by a reduction in energy consumption, underlining the
efficiency of the model’s transformer architecture and optimized components.
Comparision with BN methods in SNNs. RTFormer significantly surpasses the
tdBN method, achieving a higher accuracy rate of 80.54% compared to 64.29%
obtained by tdBN. This stark difference in performance indicates the
effectiveness of the architectural and BN improvements. Moreover, the energy
consumption of ”This work” is lower (5.59mJ) compared to tdBN (7.05mJ),
highlighting improved efficiency. Similarly, RTFormer showcases superior
accuracy over TEBN, with a 13.87% increase in top-1 accuracy. RTFormer
achieves an accuracy rate that is 13.45% higher than that of the MPBN method.
The energy savings are also notable, with RTFormer consuming less energy,
thereby presenting a strong case for the enhancements brought by the TSBN and
Re-parameter Transformer architecture.
Comparision with Transformers in SNN. The Spikformer shows competitive
performance, achieving a 74.81% accuracy rate. However, RTFormer outperforms
Spikformer by 3.35%, which is a significant margin in the realm of deep
learning models. The Spike-Driven Transformer model, another variant of the
Spiking Transformer, achieves a 77.07% accuracy rate, which is commendable.
Nonetheless, RTFormer has a slight edge with an accuracy of 78.16%. The energy
efficiency of our architecture is notably better, with an energy consumption
of 5.59mJ compared to the spike-driven model’s 6.09mJ, showcasing that the
model’s improvements do not come at the cost of increased energy usage.
TABLE II: Experiments on both Static and DVS datasets. In this table, ’Ts’ refers to the Timestep for Static datasets, while ’Td’ indicates the Timestep for Neuromorphic (DVS) datasets. The architecture abbreviations ’S-V,’ ’S-R,’ and ’S-T’ correspond to Spiking VGG, Spiking Resnet, and Spiking Transformer, respectively. Asterisked results (*) represent outcomes from our implementations of these methods. Detailed hyperparameters used in these experiments are meticulously documented in the appendix for reference. Methods | Architecture | Param(M) | Ts | Acc | Td | Acc
---|---|---|---|---|---|---
cifar10 | cifar100 | cifar10-dvs | dvs 128
RMP [39] | S-V-16 | 138.4 | 4096 | 93.63 | 70.93 | - | - | -
Calibration [40] | S-V-16 | 138.4 | 2048 | 95.79 | 77.87 | - | - | -
SEW-ResNet [29] | S-R-21 | 21.79 | 4 | 95.34* | 78.32* | 16 | 74.4 | 97.9
MS-ResNet [30] | S-R-18 | 11.69 | 4 | 94.79* | 78.15* | 16 | 75.56 | 97.54*
Att-MS-ResNet [7] | S-R-18 | 11.87 | 4 | 95.07* | 77.89* | 20 | 77.35* | 98.23
tdBN [35] | S-R-19 | 12.63 | 4 | 92.92 | 70.86 | 16 | 67.8 | 96.9
TEBN [36] | S-R-19 | 12.63 | 4 | 94.7 | 76.13 | 10 | 83.3* | 97.95*
MPBN [41] | S-R-19 | 12.63 | 2 | 96.05* | 79.51 | 10 | 74.4 | 98.26*
Spikformer [31] | S-T-4-384 | 9.32 | 4 | 95.19 | 77.86 | 16 | 80.9 | 98.3
Spike-Driven [33] | S-T-4-384 | 9.32 | 4 | 95.6 | 78.4 | 16 | 80 | 97.9*
This work | S-T-4-384 | 7.93 | 4 | 96.27 | 81.37 | 16 | 83.6 | 98.61
2 | 96.12 | 80.9 | 10 | 82.9 | 98.61
#### IV-A2 CIFAR
Dataset Description. The CIFAR-10 dataset is a well-known collection of
60,000 32x32 color images split into 10 classes, with each class represented
by 6,000 images. The CIFAR-100 dataset, while similar in structure to
CIFAR-10, offers a more challenging task with its 100 classes, each comprising
600 images, also totaling 60,000. Both datasets, developed by the Canadian
Institute for Advanced Research, serve as fundamental benchmarks for machine
learning and computer vision, facilitating the development and validation of
innovative image classification models.
Comparision with previous works. Firstly, as shown in Tab.II when benchmarked
against previous SNN models like RMP and others, RTFormer demonstrates
superior performance. While traditional SNNs like RMP and Calibration have
laid the groundwork in the field, RTFormer capitalizes on their foundation and
pushes the boundaries further. For instance, on CIFAR-100, RTFormer achieves
an accuracy of 81.37%, which is a substantial improvement over the 70.93% and
77.87% accuracy rates achieved by RMP and Calibration, respectively. This leap
in performance can be attributed to RTFormer’s more sophisticated temporal
dynamics capturing capabilities and its optimized training methodologies.
Comparision with BN methods in SNN. Secondly, in comparison with other SNN
models employing various BN techniques, RTFormer stands out for its effective
use of TSBN. This technique provides RTFormer with an edge, allowing for
better normalization of the neuron’s output across different timesteps, which
is crucial for datasets with a high degree of intra-class variability like
CIFAR-100. The improvement in normalization contributes to more stable and
faster convergence during training, as evidenced by the higher accuracy rates
when compared to the tdBN, TEBN, and MPBN methods.
Comparision with Transformers in SNN. Thirdly, against the backdrop of Spike
Transformer architectures, RTFormer’s refined approach shines through.
RTFormer’s innovative BN approach, coupled with structurally reparameterized
depthwise convolutions (DWconv), significantly boosts its performance. While
Spikformer and spike-driven models have shown the viability of Transformer
architectures in SNNs, RTFormer optimizes these designs, achieving an
impressive 96.27% on CIFAR-10 and 81.37% on CIFAR-100. This represents not
only an improvement over the aforementioned models but also highlights the
RTFormer’s architectural benefits, which are particularly advantageous for
complex and nuanced datasets like CIFAR-100.
In conclusion, RTFormer, with its strategic modifications and enhancements,
stands as a testament to the potential of SNNs, particularly in processing
complex visual data, and sets a new benchmark for accuracy and efficiency in
the field.
Figure 3: The figure presents two line graphs, where the blue line represents
the baseline model, and the green line indicates the performance after
incorporating TSBN. The graph on the left illustrates the results obtained on
the CIFAR-10 dataset, while the right graph showcases the outcomes on the
CIFAR10-DVS dataset.
### IV-B Neuromorphic Datasets Classification
Dataset Description. The CIFAR10-DVS dataset is a neuromorphic version of the
well-known CIFAR-10 dataset, converted using a Dynamic Vision Sensor (DVS). It
presents everyday objects in a format compatible with neuromorphic vision
systems, capturing temporal changes in pixel intensity. DVS128 Gesture, on the
other hand, is a gesture recognition dataset specifically designed for
neuromorphic processing. It comprises hand gesture data from 29 individuals
under various lighting conditions, captured through a DVS camera, making it
ideal for developing and testing gesture recognition models on SNNs and
neuromorphic hardware.
As shown in Tab.II, RTFormer exhibits unique advantages in Neuromorphic
datasets, such as CIFAR10-DVS and DVS128 Gesture, which capitalize on the
intrinsic features of Spiking Neural Networks (SNNs) and the architectural
innovations specific to RTFormer.
TABLE III: Ablation Experiment for TSBN. In the ”Architecture” column of the table, the abbreviation ”S” stands for ”Spiking,” ”R” represents ”ResNet,” and ”T” denotes ”Transformer.” Dataset | Method | Architecture | Acc.(%)
---|---|---|---
CIFAR10 | Baseline | S-R-19 | 95.28%
w/ TSBN | S-R-19 | 96.04%
Baseline[33] | S-T-4-384 | 95.60%
w/ TSBN | S-T-4-384 | 96.27%
CIFAR100 | Baseline | S-R-19 | 74.52%
w/ TSBN | S-R-19 | 79.37%
Baseline[31] | S-T-4-384 | 80.90%
w/ TSBN | S-T-4-384 | 83.60%
Compared to Previous Research. RTFormer outshines prior studies, notably in
neuromorphic datasets like CIFAR10-DVS and DVS128 Gesture. It demonstrates
superior accuracy, a clear advancement over earlier methods like RMP and
Calibration, which do not present results for these DVS datasets.
Against Other BN Methods in SNNs. When compared to other batch normalization
techniques such as tdBN, TEBN, and MPBN, RTFormer exhibits noteworthy
improvements in accuracy on neuromorphic datasets. This suggests that its
approach to integrating batch normalization is more effective for handling the
dynamic nature of these datasets.
Versus Other Spiking Transformer Architectures. RTFormer also excels in
comparison to other spiking transformer architectures like Spikformer and
Spike-Driven. It achieves higher accuracy rates, highlighting its
effectiveness in processing the temporally rich data characteristic of
neuromorphic datasets, thereby underscoring its advanced capability in
handling spatiotemporal data complexities.
### IV-C Ablation Study
To verify the effectiveness of the TSBN, a lot of ablative studies using
different architecture were conducted on the CIFAR10 and CIFAR10-DVS datasets.
Table III clearly demonstrates that the integration of Temporal Sliding Batch
Normalization (TSBN) leads to an enhancement in accuracy, regardless of
whether the underlying backbone is a Spiking ResNet or a Spiking Transformer.
This improvement is consistent across both Static and Neuromorphic datasets, a
fact that is also visually evident in the accompanying Figure 3.
### IV-D Performance Insights of RTFormer
Architectural Synergy with Neuromorphic Data. Neuromorphic datasets
inherently contain temporal information that traditional static datasets lack,
and RTFormer is adept at leveraging this. The model’s architecture, influenced
by the Transformer design, is inherently suited for handling sequences, making
it exceptionally well-aligned with the time-sensitive data in neuromorphic
datasets. The RTFormer uses Spiking Transformer blocks tailored to process the
spatio-temporal dynamics present in such data, enabling it to capture the
nuanced temporal patterns that are pivotal for recognition tasks in
neuromorphic vision.
Effectiveness on Static Data. The RTFormer’s effectiveness in processing
static datasets can be attributed to its novel integration of re-parameterized
convolutions. The spatial core’s re-parameterized convolutions adeptly capture
complex spatial patterns in static data, while the TSBN, even in a non-
temporal context, provides adaptive normalization that enhances the network’s
ability to generalize from training to unseen data. This combination not only
boosts computational efficiency but also ensures a high level of accuracy,
making RTFormer a versatile tool in both dynamic and static data environments.
Efficient Temporal Encoding. The RTFormer’s use of Temporal Sliding Batch
Normalization (TSBN) is particularly beneficial for neuromorphic datasets.
This specialized BN method ensures that RTFormer’s neurons maintain an optimal
firing rate, preventing both the vanishing and exploding gradient problems
that are common in SNNs. This allows the RTFormer to efficiently encode
temporal information, a critical aspect when dealing with datasets like
CIFAR10-DVS, where each pixel’s intensity changes over time are encoded into
spike trains.
Robust to Diverse Visual Stimuli. RTFormer’s robustness to diverse visual
stimuli, stemming from its Transformer roots, is evident in its neuromorphic
dataset performance. The attention mechanisms allow the model to focus on the
most salient features within the spike trains, enhancing its ability to
discern between different gestures and visual patterns with high accuracy.
## V Conlcusion
To ensure efficient inference on spiking chips, all convolutional operations
within our model have been reparameterized structurally. In conjunction, we
have refined the subsequent Batch Normalization (BN) technique to align with
the characteristics of Leaky Integrate-and-Fire (LIF) neurons, resulting in
the introduction of Temporal Sliding Batch Normalization (TSBN). By embedding
TSBN into the transformer architecture, we have crafted the RTFormer, which
achieves unprecedented results on both static and neuromorphic datasets,
setting new benchmarks in performance.
## Acknowledgment
This work was partly supported by ”Pioneer” and ”Leading Goose” R&D Program of
Zhejiang (2023C01045) and Ningbo Leading Talents Training Project.
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|
# Using LLMs to discover emerging coded antisemitic hate-speech in extremist
social media
††thanks: Research supported by American University’s Signature Research
Initiative program. We thank Jacob Levine for his inspiration in the initial
steps of this project.
Dhanush Kikkisetti, Raza Ul Mustafa, Wendy Melillo, Roberto Corizzo, Zois
Boukouvalas, Jeff Gill, Nathalie Japkowicz American University, 4400
Massachusetts Ave NW, Washington, DC 20016, USA
<EMAIL_ADDRESS>
###### Abstract
Online hate speech proliferation has created a difficult problem for social
media platforms. A particular challenge relates to the use of coded language
by groups interested in both creating a sense of belonging for its users and
evading detection. Coded language evolves quickly and its use varies over
time. This paper proposes a methodology for detecting emerging coded hate-
laden terminology. The methodology is tested in the context of online
antisemitic discourse. The approach considers posts scraped from social media
platforms, often used by extremist users. The posts are scraped using seed
expressions related to previously known discourse of hatred towards Jews. The
method begins by identifying the expressions most representative of each post
and calculating their frequency in the whole corpus. It filters out
grammatically incoherent expressions as well as previously encountered ones so
as to focus on emergent well-formed terminology. This is followed by an
assessment of semantic similarity to known antisemitic terminology using a
fine-tuned large language model, and subsequent filtering out of the
expressions that are too distant from known expressions of hatred. Emergent
antisemitic expressions containing terms clearly relating to Jewish topics are
then removed to return only coded expressions of hatred.
###### Index Terms:
hate speech, coded antisemitic terminology
## I Introduction
Online hate speech detection111Warning: Some of the paper’s content may be
disturbing to the reader. is a complex problem for social media platforms. A
particular challenge, not much discussed in the literature, relates to the use
of coded language. The following post illustrates the issue in the context of
antisemitic hate speech:
> “Nope Globalist want us intertwined and run by the elites, Globalist don’t
> lay tariffs on their friends you stupid fu****”. [posted on Dec. 31, 2022,
> on the Disqus platform ]
According to the American Jewish Committee (AJC) Translate Hate
Glossary222https://www.ajc.org/translatehate/globalist, a globalist, in its
unbiased definition, is “a person who advocates the interpretation or planning
of economic and foreign policy in relation to events and developments
throughout the world”. According to this definition, the term is rather
flattering. Indeed, that is the way it is intended in the Hyatt hotel’s
welcoming message to its club members seen in Figure 1333Photo by one of the
authors on 11/3/23 at a Hyatt Texas property.. In that commercial context a
globalist refers to someone “who gets it!” and should feel good about it! The
term does not, in any way, refer to Jews.
Figure 1: Non antisemitic use of the term ”globalist”
Yet, the AJC Translate Hate Glossary argues that the term has an antisemitic
connotation when it “is used to promote the antisemitic conspiracy that Jewish
people do not have allegiance to their countries of origin, like the United
States, but to some worldwide order—like a global economy or international
political system—that will enhance their control over the world’s banks,
governments, and media”. In the above post, it is clear that the antisemitic
connotation is implied. From this post, we surmise that
1. 1.
The globalists (a.k.a., the Jews) are distinct from “us”, presumably, the good
American citizens;
2. 2.
They control “our” fate to be run by the elites (a subset of these
Jews)444“Elite” appears in the AJC Glossary in the context of “Cosmpolitan
Elite”: ““Cosmopolitan” and “elite” are terms that have separately incited
antisemites across the political spectrum. Based on stereotypes of Jewish
wealth and insularity, Jews have been accused of being part of an elite class
for centuries.”;
3. 3.
They help each other by not imposing the same tariffs on each other as those
they impose on “us”.
The above post, thus has a double meaning. To a recipient who is unaware of
its antisemitic connotation, some category of people, the globalists, do not
seem to behave very nicely. Yet, to an informed audience, it is a very pointed
post that reiterates old Nazi and Soviet anitisemitic propaganda555c.f.
“Globalist” and “Cosmopolitan Elite” in the AJC Glossary. and propagates it
further. Furthermore, on social media platforms, it does so without setting
off any serious alerts since, except for the “stupid fu****” mention, which
could raise a flag, no offensive terms are used.
Though the usefulness and importance of catching such subtle posts and their
impact on society beyond the small extremist groups they are primarily
intended for are important subjects that we debate elsewhere, this paper is
concerned with the automatic discovery of “coded” terms similar to globalists
and cosmopolitan elite which carry both a “regular” and an antisemitic
connotation depending on the context in which they are used. Such an automated
process is necessary due to the fact that coded terminology evolves rapidly
online and fixed glossaries such as the AJC glossary become quickly outdated.
In addition, due to the large volume of posts appearing on social media, human
monitoring cannot be performed without the assistance of automated tools
pointing them in the right direction. The purpose of our approach is just
that: to create an automated monitoring tool to assist human monitors by
suggesting emerging, potentially coded, antisemitic terminology, along with
the posts that use that terminology.
Though the topic of hate speech is, unfortunately, quite vast, this study
focuses on antisemitism. The choice of a particular category of hate speech
comes from our belief that we can perform a more thorough analysis of the
problem by remaining focused. Antisemitism was selected because of the
reported increase in antisemitic incidents in the months preceding the
beginning of this study, in 2022. The lessons learned from this particular
study will apply to other categories of hatred including hatred against Black,
Muslim, Asian, and LGPTQ+ populations amongst others.
The main contribution of this paper is a methodology for the novel problem of
extracting emerging coded hate-laden terminology (antisemitism, in this paper)
from extremist social posts, along with a practical pipeline to demonstrate
its effectiveness. The methodology is based on the hypothesis that coded
antisemitic terminology begets coded antisemitic terminology. In other words,
those who use coded terminology to remain under the radar of social media
monitors will, when not able to express new ideas with existing coded terms,
derive or invent new ones. Based on this hypothesis, we harvest terminology
used in similar contexts as known coded antisemitic terminology and propose it
as potential emerging antisemitic coded terminology to human monitors, along
with the context in which that terminology occurs. We propose four different
versions of our pipeline and validate them using a quantitative approach. The
most advanced version is also evaluated qualitatively. We conclude with a
discussion of our approach’s practical utility.
The remainder of the paper is structured as follows: Section II presents
background and related work. In Section III, we discuss data preparation
matters. Next, the methodology and pipeline for extracting coded terminology
is introduced in detail in Section IV. This is followed by a presentation and
discussion of the results in Section V. Finally, Section VI concludes the
paper and discusses future work.
## II Background and related work
With the advent of the internet and social media, technology has increased the
speed at which language evolves. Propaganda in the form of hate speech now
travels the world at such a fast pace that it is beyond human capacity to keep
up with. Harmful words take on new meanings in both direct and coded ways,
inciting hatred in the minds of those only too willing to believe them as they
reinforce and justify preexisting prejudices.
### II-A Machine Learning Methods for Hate Speech Detection
In recent times, there has been a notable rise in hate crimes across the
United States.666https://bjs.ojp.gov/library/publications/hate-crime-recorded-
law-enforcement-2010-2019 While establishing a clear relationship between hate
crimes and online content is not straightforward, a report by the US
Department of Justice points to the simultaneous purchases of Facebook ads
containing dividing content and hate crime. These two reports777(1)
https://bit.ly/2xeeF5h; (2)
https://www.ojp.gov/pdffiles1/nij/grants/304532.pdf thus suggest that hate
speech should not be considered harmless, and coming up with methods to curb
it is an important goal.
Previous work aims to detect hate speech from social media using various
Machine Learning (ML) methods as documented by a number of surveys written in
the last six years [1, 2, 3, 4]. One of the most recent surveys shows that
while up to 2016, fewer than 10 papers were published on the topic each year,
since then, there has been a huge increase in interest in the topic with over
150 papers published in 2020, the last year for which their survey had
complete information [4]. Hate speech detection has been attempted using a
wide variety of techniques and applied to many different problems. Founta et
al.[5], for example, used Recurrent Neural Networks (RNN) to classify racism
and sexism. Serrà et al. [6] showed that character level based Long Short-Term
Memory networks (LSTMs) for abusive language detection could be useful.
Similarly, Convolutional Neural Networks (CNNs) have also been shown to be
successful in hate speech detection and classification [7]. More recently,
large language models have been used for these tasks like in the work of [8]
who propose different fine-tuned and non-fined-tuned variations of pre-trained
models such as BERT, RoBERTa, ALBERT, etc. on offensive language detection.
Most of these studies, however, consider hate speech as a whole and,
typically, do not distinguish the community towards which it is directed. We
feel that this generalized approach is too broad and decided, instead, to use
a divide-and-conquer approach by focusing on particular communities
separately. Our first attempt focused on the Jewish community and the problem
of antisemitic speech in social media.
### II-B Antisemitism in Social Media and its Detection
Antisemitism specifically targets Jewish individuals or the Jewish community
[9]. In [10], authors use the outcomes of two surveys from EU and ADL to
assess how the level of antisemitism relates to the perception of antisemitism
by the Jewish community in eight different EU countries. A recent survey finds
that 20% of American Jewish adults have experienced an act of antisemitism,
such as an attack either online or on social media.888https://bit.ly/41FV6ei
In another study, the authors address the challenges of quantifying and
measuring online antisemitism. It raises the question of whether the number of
antisemitic messages is increasing proportionally to other content or if the
share of antisemitic content is rising. Additionally, the paper aims to
determine the extent of online Jew-hatred beyond well-known websites, forums,
and closed social media groups [11].999These studies preceded 10/7/23 when the
situation worsened drastically.
A few studies have attempted to combat online antisemitism in a way similar to
the way in which generalized hate speech has been countered in the works
discussed in the previous section. In [12], for example, the authors prepared
a data set that includes both social posts and associated images, when
available. They labeled the entries as antisemitic or not, and if antisemitic,
indicated the kind of antisemitism: political, economic, religious or racial.
They used a bimodal deep learning approach for classifying the data into these
categories. [13] considers a subset of the text-only part of this datasetin an
attempt to classify antisemitic posts using a less computationally-intensive
approach. Focusing on the class imbalance problem in the data while taking
advantage of OpenAI’s GPT technology, they compared GPT-based resampling
techniques against other traditional kinds. Very recently, [14] proposed a new
data set for antisemitism detection in social media posts that uses a strict
annotating process. The data set is so recent, however, that it has not yet
been used for classification or the results obtained on such efforts have not
yet been published. There are other projects that consider the detection of
online antisemitism using AI approaches as well. In particular, the project
entitled “Decoding Antisemitism”101010https://decoding-antisemitism.eu/ calls
itself an “AI-driven Study on Hate Speech and Imagery Online”, and already
produced five published reports on the subject. The project specifically aims
at linking national or international events reported in the traditional media
to antisemitic online social media discussions.
### II-C Alternatives to automated hate speech detection
In [15], the authors question whether the way in which hate speech detection
has been handled by the machine learning community is the way forward, or
whether hate speech detection is a lot more complex than previously assumed by
the researchers who labeled data sets and applied classifiers to them.
Furthermore, the authors note that some hateful content may occur without the
use of well-known slurs and that on top of it all, the nature of hate speech
is constantly evolving.
In contrast to previous studies, our work takes these observations into
consideration and focuses on identifying emerging, potentially coded terms
related to antisemitism using NLP methods. There has been a lack of rigorous
research in finding emerging antisemitic coded terms that can lead to the
detection of hate speech and, perhaps, subsequently, to the prevention of hate
crimes. This paper aims to bridge this gap and provide an approach for the
detection of emerging antisemitic, sometimes coded, terminology used on
extremist social media platforms.
## III Data Preparation
This study is part of a large multi-disciplinary project sponsored by our
institution which, simultaneously, collects and analyzes the use of coded
language to express antisemitic sentiment in lightly moderated social media
platforms typically preferred by individuals with extremist tendencies and
studies the migration of this language from these extremist platforms to the
general population. The overall project includes a data team, a population
team, and a software team which collaborate closely and work in parallel. The
pipeline illustrating our proposed methodology is shown in Figure 2.
Figure 2: Emergent Coded Antisemitic Terminology Extraction Pipeline
### III-A Dataset
The project is constantly evolving, though for this study, we considered the
first delivery of the data curated by the data team in June 2023. The data
team’s objectives concerning this study was to analyze the usage of
antisemitic terms. We describe the data gathering and cleanup methodologies
summarized by the 3 leftmost components in Figure 2.
#### III-A1 Data Scraping and Labeling
To build the corpus, the data team analyzed antisemitic social media posts
from various extremist social media platforms including Discuss, Telegram,
Minds, and GETTR. It used antisemitic expressions obtained from the previously
mentioned American Jewish Committee (AJC) Translate Hate Glossary as well as
the Southern Poverty Law Center (SPLC) to collect social media posts. This
collection effort was facilitated by Pyrra111111https://www.pyrratech.com/, a
private software company that allows its users to scrape posts from alt-social
media platforms according to a list of seed terms. The data team considered
the 46 seed expressions available from the AJC Glossary at the time as well as
the term “Cultural Marxism”, discussed in a SPLC
article121212https://www.splcenter.org/fighting-hate/intelligence-
report/2003/cultural-marxism-catching and chose 16 of them to make the process
tractable. It analyzed the 659 retrieved posts related to these seed
expressions to determine whether the post was antisemitic or not.131313A copy
of the coding statement is available upon request. The 16 terms used in the
subset were selected based on their potential to reveal posts that had
emerging new antisemitic terms in them. The list of seed words used is: Cabal,
Cosmopolitan Elite, Cultural Marxism, Deicide, The Goyim Know, Holocough,
Jewish Capitalist, Jewish Communist, Jew Down, Jewish Lobby, New World Order,
Not the Real Jews, Rothschild, Soros, Zionist, and Zionist Occupied
Government. Since the distribution of posts with respect to each seed
expression is not ideal, though the software team used all the posts retrieved
from the 16 seed expressions, it used only the seed expressions with at least
5 posts related to them to conduct its analysis. The terms dropped from the
list according to this criterion are Jew Down and Cosmopolitan elite, leaving
us with 14 seed words for the remainder of the study.
#### III-A2 Preprocessing
Text preprocessing is a critical step in Natural Language Processing (NLP). It
involves transforming raw text data into a format that can be easily analyzed
by machine learning algorithms. The preprocessing steps usually used involve
several techniques, such as tokenization, stop word removal, stemming, and
lemmatization [16]. During the first phase of cleaning the corpus, we removed
the urls and lower-cased all the posts to normalize them. This initial
procedure was followed by stop words removal. Then we lemmatized the text to
get a single root form for each word prior to passing it on to the coded
antisemitic terms extraction process, which will be discussed in the next
section. Bigrams and trigrams were formed by running two- and three- word
windows through all the posts.141414We also considered unigrams but were not
able to filter them effectively using our current methodology. Their treatment
was left for future work. It was important to filter out badly-formed
expressions obtained through that approach. In particular, we decided to
include bigrams and trigrams that only contain nouns, proper nouns,
adjectives, and verbs, since others were judged less relevant to our
quest.Since the emphasis of this study is on the novel proposed extraction
process discussed next, we did not experiment with the various pre-processing
techniques suggested in the literature on hate speech for social media posts
[17]. It was left for future work.
## IV Coded Antisemitic Terms Extraction Approach
As previously mentioned, the purpose of this study is the extraction of
emerging coded antisemitic terms. In order to carry out this goal, we designed
a method for operationalizing each term of that expression. That
operationalization and the linking of its resulting components into a
functional system constitute the main contribution of this work. The purpose
of this section is to discuss the process. To begin with, we consider each
word in the emerging coded antisemitic terms expression and give it the
specific meaning shown below.
* •
Terms: the extracted expressions are limited to grammatically consistent
bigrams and trigrams; they have to be relevant enough to the documents in
which they appear and appear frequently enough in the corpus.
* •
Antisemitic: the candidate expressions have to be semantically related to
antisemitic discourse.
* •
Coded: antisemitic expressions that contain terms relating to obvious Jewish
concepts are removed.
* •
Emerging: already known coded antisemitic expressions are removed in order to
concentrate on new terminology.
These operations are divided into two phases. In Phase 1, we address the
extraction of emerging coded terminology without worrying about its semantic
relation to antisemitism. In Phase 2, we address semantics using large
language models. Phase 1 is represented by the “Important Terms Extraction”
component in Figure 2. Phase 2 is represented by the combination of the LLM
Generation, Similarity Scoring, Antisemitic Terminology Extraction, and
Monitoring components in Figure 2. Both phases of the pipeline are implemented
using two approaches: a standard solution and an advanced solution. We
subsequently test all four combinations, yielding a baseline approach composed
of two standard solutions, two hybrid approaches composed of one standard and
one advanced solution, and one advanced approach composed of two advanced
solutions.
### IV-A Phase 1: Emerging Coded Trending Terms Extraction
For the first part of Phase 1, the extraction of trending terms, we explore
the use of off-the-shelf NLP tools for our standard solution and then propose
our advanced solution that combines tf-idf and frequency. Once the trending
terms are extracted, we propose a strategy to remove non-emerging and non-
coded terms from the list of extracted terms. This strategy is applied to both
the standard and advanced solutions.
#### IV-A1 Standard Solution: Trending Terms Extraction using Concordance and
Collocation tools
In this first attempt at trending terms extraction, we use traditional NLP
techniques to extract bi-grams and tri-grams using concordance and collocation
algorithms from the NLTK Toolkit[18]. Concordance is a technique that provides
a comprehensive view of how a given term appears in a corpus. Using this
approach, we use the 14 seed terms from Section III-A1 for analyzing patterns
and gaining insights into language usage. For each occurrence of a seed term,
this approach provides the surrounding words context. We use default settings
for the extraction of context. Next, using collocation, we find the most
frequent bi-grams and tri-grams in the collected contexts. Collocation is a
technique that finds a meaningful combination of words from a corpus that are
semantically coherent. Different statistical measures can be used to detect
collocations including frequency, pointwise mutual information (PMI), and log-
likelihood ratio (LLR) among others. We use frequency, here since that is the
measure also used in the advanced approach In the future, we plan to
experiment with other statistical measures for both approaches. The standard
approach yielded 126 trending terms.
#### IV-A2 Advanced Solution: Trending Terms Extraction using TF-IDF and
Frequency
Our proposed advanced approach is presented in Algorithm 1 which uses TF-IDF
feature-weighting [19] and frequency to extract trending terms. In a nutshell,
this was done by selecting all the terms that obtained a TF-IDF value greater
than a self-set threshold, listing these terms in decreasing order of
frequency, and selecting the top 200 terms from the list.151515We assume that
at least 200 terms had a TF-IDF value larger than the self-set threshold. When
the same term appeared in several documents, the highest TF-IDF value it
received was retained Algorithms 1 shows the approach that was followed in
detail. The algorithm is explained line by line next.
Algorithm 1 Trending terms extraction
1:Initialize Trending_terms $\triangleright$ Stores top 200 trending terms
2:Initialize $D_{s}$ $\triangleright$ Stores terms’ highest TF-IDF scores (s)
3:Initialize $D_{f}$ $\triangleright$ Stores terms’ values and frequencies
4:Set T$\triangleright$ Stores all the vocabulary terms (value).
5:Set F $\triangleright$ Stores the frequency of each term in the corpus.
6:Calculate the TF-IDF scores for each term in each document and store them in
matrix $\mathbf{W}\in\mathbb{R}^{d\times v}$, where $d$ denotes the number of
documents and $v$ denotes the vocabulary size.
7:for each term $t$ in ${\bf T}$ do
8: for each row in $\mathbf{W}$ do$\triangleright$ Each row is a document
9: $D_{s}[t]\leftarrow max(D_{s}[t],W[row,t])$ $\triangleright$ Finds t’s
highest TF-IDF score, $s$, across all documents
10: end for
11:end for
12:$\delta\leftarrow Average(D_{s})$ $\triangleright$ $\delta$ is the average
of all $s$’s
13:i = 1
14:for each t in $D_{s}$ do
15: if $D_{s}[t]\geq\delta$ then
16: $D_{f}[i]\leftarrow(T[t],F[t])$ $\triangleright$ If t’s highest TF-IDF
score is larger than threshold $\delta$, store t’s value and frequency in
$D_{f}$
17: i = i + 1
18: end if
19:end for
20:Sort $D_{f}$ in descending order of frequency
21:for $i=1,2,\ldots,200$ do
22: $Trending\\_terms\leftarrow D_{f}[i][T]$ $\triangleright$ Store the most
frequent terms in Trending_terms (drop the frequencies)
23:end for
The algorithm begins by initializing the Trending_terms list which is the list
that will return the 200 bigrams and trigrams (terms) that received the
highest combination of TF-IDF and Frequency scores. Next, $D_{s}$ and $D_{f}$
are also initialized. $D_{s}$ will be used to store all the terms’ highest TF-
IDF scores, whereas $D_{f}$ will store all the terms and their associated
frequencies. Since terms are subsequently referred to according to their
indices, $T$, which is set next, serves as the reference vector that
associates an index with the actual value of the term (i.e., the actual bigram
or trigram). Next, the frequency of each term in the corpus is calculated and
saved in vector F. The TF-IDF values obtained for each unique term and each
document are then calculated and placed in the matrix $\mathbf{W}$ of size d x
v where d represents the number of documents whereas, v is the number of
terms.
On lines 7-11, we take each term in matrix $\mathbf{W}$ and find the highest
score across all the rows (documents) of the matrix and store it in $D_{s}$.
To remove the less relevant terms, we compute the average of all the values in
$D_{s}$ and use this value, $\delta$, as a threshold. This allows us to
consider only the terms with TF-IDF values greater than the average value of
all the scores in $D_{s}$. $\delta$ is calculated on line 12. On lines 13-19,
we check if the terms’ TF-IDF value is greater than $\delta$. If so, we save
the terms’ values and their frequencies in $D_{f}$. Finally, on line 20, we
sort the terms in $D_{f}$ in descending order of their frequency values and
select the top 200 terms, storing them in Trending_terms.
#### IV-A3 Removal Strategy: Redundant, Non-Emergent and Non-Coded terms
Removal
Once the list of most trending terms have been extracted using either the
standard or advanced solution, three categories of terms are removed from it.
First, as we consider expressions that are both bigrams and tri-grams, there
is a possibility of encountering bigrams within trigrams. Such redundant
bigrams are removed from the list of expressions. Next, we remove the terms
that have occurred earlier. For now, this corresponds to the original list of
16 seed words used to retrieve the posts. In the future, this list will grow
as we intend to use the system continuously, using newly discovered terms of
interest as new seed terms. Lastly, the terms that are considered non-coded
are removed. These correspond to terms that contain words that obviously
pertain to Jewish themes. The list of words currently used includes jew,
jewish, kike, and zionist. Expressions that include these words either as
stand-alone words or embedded within other words are removed. After the
removal phase is applied, we are left with 52 and 94 trending terms for the
standard and advanced trending term extraction solutions, respectively.
### IV-B Phase 2: Embeddings and Comparisons
Though the bigrams and trigrams extracted in the previous section are known to
be trending, their semantics are unknown and, in particular, there is no
information as to whether or not these terms are antisemitic. To find out
which of these trending expressions are antisemitic, we compare the context in
which they are used to the context in which the known antisemitic expressions
are used. If a trending term appears in contexts similar to those in which
seed expressions occur, it will be deemed antisemitic. Otherwise, it will be
discarded as non-antisemitic. To compute embeddings for the trending and seed
terms, we begin by fine-tuning BERT. Since BERT was not specifically trained
on instances of hate speech or antisemitism, we fined-tuned it with additional
data collected using the same seed expressions as before (since time elapsed
between the original collection and the new collection, more posts were
available for this exercise). This fine-tuned version of BERT is then used to
generate contextual embeddings for both the trending terms discovered in the
last section and the seed terms used to extract posts. We present the details
of BERT’s fine-tuning followed by the standard and advanced embedding
solutions we implemented.
#### IV-B1 Fine-tuning the BERT model
The generalized BERT model does not possess domain-specific vocabulary, thus
it is not capable of handling coded hate speech such as antisemitism. Indeed,
when such out-of-vocabulary terms occur, they get broken down into smaller
tokens for which embeddings are generated. These are treated as rare tokens,
yielding unsatisfactory results. To avoid this issue, we fine-tune the BERT
model using an additional 56K posts extracted using the same seed words as
before on Pyrra. We, thus, extend BERT’s vocabulary from 30k to 55k tokens,
and fine-tune it using the Masked Language Modeling (MLM) approach. MLM is a
pre-training approach that masks a few tokens. The model is subsequently
trained to predict the masked tokens from the words that surround
them.161616https://huggingface.co/learn/nlp-course/chapter7/3
#### IV-B2 Comparing Trending Terms to Seed Terms
To differentiate between antisemitic and non-antisemitic terms during Phase 2,
we compare the trending terms’ embeddings to the seed terms’ embeddings using
Cosine Similarity. We generate two types of embeddings following i) the
standard pre-truncate embedding method and ii) the advanced post-truncate
embedding method.
Figure 3: Pre-truncate embedding approach for a window of size 5.
In pre-truncate embedding, we truncate the post containing the term to be
embedded prior to embedding it. In post-truncate embedding, we embed the
entire post containing the term of interest, and truncate the resulting
embedding afterwards.171717Since we cannot embed posts exceeding 512 tokens,
we turned large posts into multiple ones.
##### Standard Solution: Pre-truncate embeddings
In this approach, we consider context windows of 5 to 14 words, where the size
of the windows refers to the twin windows located before and after the term
being embedded, respectively. We show an example of windows of size 5 in
Figure 3. Since the same term may be found in more than one post, we
concatenate all the embeddings extracted from fine-tuned BERT using the same
window size and take their average. Embedding, here, refers to the pooled
layer obtained from the 12 layers of the BERT architecture. We follow the same
procedure for all the trending terms we extracted and the 14 seed words
retained in Section III-A1.
Next, we determine the trending terms antisemitic nature using Algorithm 2.
After some initialisations on lines 1-4, $S[tt]$, the “similarity to
antisemitism” value for trending term $tt$, is computed as follows: $tt$’s
embedding is compared to each of the 14 seed terms (the $st$’s)’s embeddings
using Cosine Similarity as described on lines 6-8. On line 9, the 14 resulting
measurements are averaged and assigned to $S[tt]$ The process is repeated for
each trending term (lines 5-10) and the median of all the $S[tt]$’s, $\gamma$,
is calculated on line 11. $\gamma$ is then used as our threshold for potential
antisemitism on lines 12-18: if $S[tt]$ for trending term $tt$ is greater than
$\gamma$, $tt$ will be given the partial label “potentially antisemitic”
($TT\\_PL\\_w[tt]$ = 1). Otherwise, it will be given the partial label
“probably not antisemitic” ($TT\\_PL\\_w[tt]$ = 0). (We used the median as it
offered more flexibility than the mean.) Algorithm 2 is repeated 10 times,
once for each window size $w$ considered. This yields 10 partial labels
$TT\\_PL\\_w[tt]$, $w=1\dots 10$ for each term $tt$, and the final labeling
for $tt$ is “antisemitic” if $m$ out of the 10 partial labels are “potentially
antisemitic”. It is “not antisemitic”, otherwise. The optimal value of $m$ was
7 for the pre-truncate case.
Algorithm 2 Comparing semantic similarity–window size $w$
1:$Embeddings\\_tt\leftarrow\\{et\\_1,et\\_2\dots,et\\_n\\}$$\triangleright$ n
pre- or post- truncate trending terms embeddings at window size $w$
2:$Embeddings\\_st\leftarrow\\{es\\_1,es\\_2\dots,es\\_14\\}$$\triangleright$
14 pre- or post- truncate seed words embeddings at window size $w$
3:Initialize TT_PL_w. TT_PL_w will store the n trending terms & predicted
antisemitic label for window size $w$.
4:Initialize $S$. $S$ will store the average semantic score for each trending
term at window size $w$.
5:for each tt in ${\bf Embeddings\\_tt}$ do
6: for each st in ${\bf Embeddings\\_st}$ do
7: $tt\\_scores[tt]\leftarrow Sim(et\\_tt,es\\_st)$ $\triangleright$ Cosine
Sim
8: end for
9: $S[tt]\leftarrow Average({tt\\_scores[tt]}$) $\triangleright$ Average all
the 14 semantic scores between tt and all the st’s
10:end for
11:$\gamma\leftarrow Median(S)$ $\triangleright$ $\gamma$ is the median of all
the scores
12:for each tt in ${\bf S}$ do
13: if $S[tt]>\gamma$ then$\triangleright$ check if score greater than
$\gamma$
14: $TT\\_PL\\_w[tt]\leftarrow 1$$\triangleright$ if score greater than
$\gamma$
15: else
16: $TT\\_PL\\_w[tt]\leftarrow 0$$\triangleright$ if score less than $\gamma$
17: end if
18:end for
Figure 4: Post-truncate embedding approach for a window of size 5.
##### Advanced Solution: Post-truncate embeddings
In this approach, we begin by embedding each complete post using fine-tuned
BERT. The approach is illustrated in Figure 4 for the 18-word post AND THE
EVIL LYING DEEP STATE CABAL SATANIC SCUM BAGS ALL NEED TO BE ROUNDED UP AND
EXECUTED. This yields an $18$ x $12$ x $768$ tensor representing the total
number of words in the post, the total number of encoding layers, and their
dimension. This embedding can be thought of as a word embeddings lookup table
that provides complete context for each post.181818We assume that each word in
the post has a token id in Bert’s vocabulary. Once this embedding is
constructed, we follow the same procedure described in Section IV-B2 except
for the fact that we now extract word-level contextual embeddings from the
lookup table (see Figure 4). The advantage of this approach over the previous
one is that it builds more informed embeddings given its use of a complete
rather than partial context. Please note that there are three additional
differences between the standard and the advanced approach: in the standard
approach, we used context window sizes between 5 and 14 while in the advanced
approach, we used context window sizes between 1 and 10. That is because a
window of 1 word does not convey much information in the standard approach
whereas it does in the advanced approach. As a result, we started at size 5 in
the standard approach and 1 in the advanced approach and used 10 different
window sizes in each case. Furthermore, in the advanced approach, the
embeddings are generated by averaging the final encoder layer of BERT rather
than using the pooled layer since that yielded better results. Finally, the
optimal value for $m$ in the advanced approach was 9 rather than 7.
## V Results and Discussion
The purpose of our study was to design a methodology for extracting emerging
coded antisemitic terminology from online posts appearing on social media
platforms often used by extremist groups. We proposed a pipeline to implement
this methodology and instantiated this pipeline with standard and advanced
components. The difficult part of our evaluation is the assessment of whether
the approach yields a significant number of terms and whether these terms can,
indeed in some contexts, have an antisemitic connotation. In order to answer
these questions, we created a gold standard and tested our results according
to it.
### V-A Construction of a gold standard:
The gold standard we created uses two complementary methodologies. One for the
terms already familiar to the community that fights antisemitism, and the
other, for the terms unknown or not yet catalogued by that community.191919In
this paper, we created a prototype system based on the seed words provided to
us by the data team. These seed words are only a small subset of the already
known coded antisemitic terms. As a result, some of the emergent terms
discovered by our system are emergent vis-a-vis the system’s knowledge but not
vis-a-vis the broader current knowledge. Discovering terms known to the
community but not known by the system constitutes a useful proof of concept.
The discovery of terms not currently known by the community constitutes an
added demonstration of the worth of the approach.
Known Terms For the first category, we simply compiled a general glossary from
three existing sources: the Institute for Curriculum Services’ Glossary
spanning the history of European Antisemitism, which we took in its entirety;
the American Jewish Committee “Translate Hate” glossary which we also used in
its entirety (prior to its recent expansion from 46 to 70 terms) and portions
of the Glossary of Terms and Acronyms constructed by the R2Pris project on
Radicalization and violent extremism. Since this last source encompassed
hatred of different types, for this specific study, we restricted ourselves to
the terms whose composition or definition included a known antisemitic term
(e.g., nazi, Aryan, anti-semitic, Ku Klux Klan, SS, Swastika, Fascism, White
Supremacist, etc.).202020 The sources we used can be found at the following
websites: https://bit.ly/45kEtYB; https://bit.ly/3MIjKpt; and
http://www.r2pris.org/glossary.html
New Terms The new terms are the terms that do not appear in the glossaries
just mentioned and that need to be manually verified through an internet
search. We used the following systematic procedure to assign ground labels to
new terms:
* •
Each extracted term not found in the glossary compilation was searched for on
Google in two ways: the term alone or together with “+ antisemitism” added to
the search.
* •
The documents retrieved on the first page of the Google browser for both
searches were examined for references to antisemitism.
* •
If, based on this analysis, the term was found to be associated with
antisemitism (e.g., “deep state” was found to be associated with a conspiracy
theory against the Jews), it was coded as antisemitic in our gold standard
database. If, on the other hand, the term did not carry any clear meaning
(e.g., “late 20th”) or was not associated with antisemitism (e.g., “new york
city”), it was coded as not antisemitic in our gold standard database.
Qualitative evaluation We conducted two types of qualitative evaluation. The
first one simply consisted of observing the terms extracted by the approach to
assess whether they made sense when taken out of context. The second one can
be thought of as a sanity check. For terms extracted and labeled as either
antisemitic or not, we went back to the the posts from which the term was
extracted to assess whether, within the context of the post, it was used in an
antisemitic way or not. Though we do not use these qualitative assessments in
our quantitative evaluation, we show examples of the different situations that
arose in terms of agreement or disagreement between our system and our gold
standard.
### V-B Results
TABLE I: Accuracy, Precision, Recall and F-score using the four versions of our pipeline. Model+Embedding | Approach Type | Accuracy | Precision | Recall | F-score
---|---|---|---|---|---
colloc-pretrunc | standard | 0.74 | 0.34 | 1 | 0.51
colloc-posttrunc | hybrid | 0.76 | 0.36 | 1 | 0.53
tfidf-pretrunc | hybrid | 0.67 | 0.47 | 0.55 | 0.51
tfidf-posttrunc | advanced | 0.80 | 0.63 | 0.83 | 0.72
Quantitative Results: We tested four different versions of our proposed
pipeline, by combining the standard and advanced solutions proposed for
trending term extraction with the standard and advanced solutions proposed for
term embedding. These combinations resulted in one standard, two hybrid, and
one advanced implementation. Table I lists the results obtained by concordance
+ collocation followed by pre-truncation embedding (colloc-pretrunc) or post-
truncation embedding (colloc-posttrunc); and those obtained by tfidf +
frequency followed by pre-truncation embedding (tfidf-pretrunc) or post-
truncation embedding (tfidf-posttrunc). The results were obtained using our
gold standard labels. The approach using the two advanced components stands
out as the absolute winner: tfidf-posttrunc, although the results for all four
methods, including tfidf-posttrunc, show a higher level of recall than
precision. Future work will attempt to improve all these metrics scores, with
a focus on precision so as not to unduly label terms as antisemitic when they
are, in fact, benign. When comparing the numbers in Table I, it is important
to note that the number and type of terms retrieved differ between the two
term extraction processes, colloc and tfidf. While colloc extracted 52 terms
of which only 7 were truly antisemitic, tfidf extracted 94 of which 29 were
truly antisemitic. The recall of 1 obtained by the two colloc-based methods,
thus means that both pretrunc and posttrunc were able to identify these 7
antisemitic terms. Their low level of precision, however, suggests that they
are too liberal in their labeling of terms as antisemitic.
TABLE II: List of trending terms that are predicted antisemitic by the most advanced version of the pipeline. False Positives | | Known Terms | | New Terms | | Neutral |
---|---|---|---|---|---|---|---
plain sight | german people | white genocide | interest groups | FEMA camps | color revolution | end game | world war
new york city | big part | nostra aetate | federal reserve | central bank | critical race theory | western civilization | democrat party
Qualitative Results: Our qualitative evaluation was applied to the version of
our pipeline that obtained the best results: tfidf-posttrunc, i.e., the
advanced version. Table II shows some of the terms extracted by that version.
The terms in red correspond to terms incorrectly classified as antisemitic
with no good explanation; those in black are correctly classified as
antisemitic as they correspond to our Known Terms; those in blue were verified
to be antisemitic as they correspond to our New Terms; and those in purple
were incorrectly classified as antisemitic, although the context in which they
arise is clearly antisemitic. As discussed below, we call these terms Neutral
(in an antisemitic context).
Sanity Check: In Table III, we show a few sample posts containing the
following trending antisemitic terms discovered by tfifd-posttrunc: Interest
groups, White Genocide, Deep state. Each of these terms had an entry in the
antisemitic glossary compilation described earlier. For instance, White
Genocide, refers to a conspiracy theory rooted in white supremacist ideology,
claiming that there is an intentional effort by Jews to destroy the white race
through immigration, mixed-racial marriage, LGBTQ+ identification, etc.
TABLE III: Posts on social media with automatically labeled antisemitic coded terms as per the most advanced version of the pipeline. Coded Term | Status | Website | Post
---|---|---|---
Interest groups | Known Term | Minds | the united states government is controlled by interest groups that are only seeking to enlarge their own power. the us government does not represent the will of the citizenry, and condemning it is not a condemnation on the principles of freedom, democracy, etc.the usa is being set up to fail.the rootless cosmopolitan elite have been constructing elaborate safehouses for decades in preparation for this.
Deep State | Known Term | 4chan | US deep state MIGApede detected. The real deep state is the Jewish lobby.
White Genocide | Known Term | Truth Social | Rotten Eggs - Dr. Reiner Fuellmich and Whitney Webb! Vatican Pro-Abortion- False Prophet Francis Owned By New World Order! Jacob’s Trouble = White Genocide! Pandemic Of The Double Dosed.Inflation Spiking, More Lockdowns, The Worst Is Yet To Come!
End Game/FEMA camps | Neutral/New Term | Truth Social | FEMA is not a good thing! FEMA camps are concentration camps. FEMA camps are the end game of the New World Order
Big Part | False Positive | 4chan | all turds need to be deported from the West. turds are brown MENA sunni muslim garbage. they are a big part of the non-white invasion. many of the turkish Iraqi and syrian immigrants who rape women and girls are actually ethnic turds. turds are also zionists and turdistan is a base for israeli ops. imagine sympathizing with these zio-muslim invaders.
Table III also shows an instance of a new term —FEMA camps. This corresponds
to a conspiracy theory where FEMA is believed to plan the incarceration and
possible execution of US citizens in favor of the establishment of a New World
Order, one of our seed words which often refers to the establishment of a new
form of government controlled by a Jewish elite.On the other hand, during the
process of extracting coded antisemitic terms, some terms were labeled as
antisemitic despite the fact that they do not appear in our gold standard. In
certain cases, that represents an outright mistake like in the case of Big
Part in Table III where the context is certainly racist, but not specifically
antisemitic, though antisemitism is part of the post, but in other situations,
a case could be made for the antisemitic label. For example, our approach
predicts End game as a coded antisemitic term, even though we did not find any
reason for it in our glossary or internet search. A look at the post in which
the term appeared (Table III) helps us understand how the antisemitic context
of the post that includes the terms “concentration camps” and ”new world
order” led the system to mislabel it. We conclude that, in such cases, our
approach is extracting the right term according to the context, but the term
should be considered Neutral (in an antisemitic context) rather than
antisemitic.
### V-C Discussion
Though we assume that our approach could still be refined, we note that the
results obtained by the most successful version of our system are encouraging,
suggesting the viability of our hypothesis that emergent coded terms could be
discovered automatically using distance measures in embedding spaces. The
sanity checks suggest that the terms identified by our approach are, usually,
warranted as the context shown in the posts attests to the antisemitic nature
of the way in which the identified terms are used. These checks also point to
the errors made by the system and will help us improve our results. We also
believe that our approach could have important practical uses. After being
vetted by a human team, the emergent coded terminology it discovers could be
input to the moderating algorithms used by social media platforms to discover
problematic discourse or users currently avoiding discovery.
## VI Conclusion
This paper proposes an approach for detecting the emergence of new antisemitic
coded terminology which offers a valuable resource in combating online
antisemitism and contributes to the ongoing efforts to create safer and more
inclusive online spaces. We achieve an accuracy of 80% and F-Score of 72% in
extracting antisemitic terms using this approach which relies on NLP
techniques including POS tagging, TF-IDF, and Fined-tuned large language
models such as BERT. In the future, we intend to refine our semantic
similarity technique by exploring other deep learning and large language model
approaches and their various parameter combinations. Similarly, we will
experiment with different types of text pre-processing approaches to deal
specifically with hate-speech and social media text. This will be done in the
context of a lifelong-learning setting where the trending terms discovered
will be used as input to the data scraping component in the following
iteration. We also intend to create a more user-friendly version that will be
convenient for people working in this space. Finally, our goal is to extend
this study to hateful terminology against other minority groups.
## References
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* [2] P. Fortuna and S. Nunes, “A survey on automatic detection of hate speech in text,” ACM Computing Surveys (CSUR), vol. 51, pp. 1 – 30, 2018.
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* [9] M. Schwarz-Friesel and J. Reinharz, Inside the antisemitic mind: the language of Jew-Hatred in contemporary Germany. Brandeis University Press, 2017.
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* [11] G. Jikeli, D. Cavar, and D. Miehling, “Annotating antisemitic online content. towards an applicable definition of antisemitism,” arXiv preprint arXiv:1910.01214, 2019.
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|
# Are Large Language Models a Threat to Digital Public Goods? Evidence from
Activity on Stack Overflow
Maria del Rio-Chanona 1,2, Nadzeya Laurentsyeva 3, and Johannes Wachs 4,5,1,∗
1 Complexity Science Hub, Vienna.
2 Harvard Kennedy School
3 Faculty of Economics, LMU Munich
4 Corvinus University of Budapest
5 Centre for Economic and Regional Studies, Hungary Direct correspondence to
<EMAIL_ADDRESS>
(2023-07-14)
###### Abstract
Large language models like ChatGPT efficiently provide users with information
about various topics, presenting a potential substitute for searching the web
and asking people for help online. But since users interact privately with the
model, these models may drastically reduce the amount of publicly available
human-generated data and knowledge resources. This substitution can present a
significant problem in securing training data for future models. In this work,
we investigate how the release of ChatGPT changed human-generated open data on
the web by analyzing the activity on Stack Overflow, the leading online Q&A
platform for computer programming. We find that relative to its Russian and
Chinese counterparts, where access to ChatGPT is limited, and to similar
forums for mathematics, where ChatGPT is less capable, activity on Stack
Overflow significantly decreased. A difference-in-differences model estimates
a 16% decrease in weekly posts on Stack Overflow. This effect increases in
magnitude over time, and is larger for posts related to the most widely used
programming languages. Posts made after ChatGPT get similar voting scores than
before, suggesting that ChatGPT is not merely displacing duplicate or low-
quality content. These results suggest that more users are adopting large
language models to answer questions and they are better substitutes for Stack
Overflow for languages for which they have more training data. Using models
like ChatGPT may be more efficient for solving certain programming problems,
but its widespread adoption and the resulting shift away from public exchange
on the web will limit the open data people and models can learn from in the
future.
## 1 Introduction
Over the last thirty years, humans have constructed a vast library of
information on the web. Using powerful search engines anyone with an internet
connection can access valuable information from online knowledge repositories
like Wikipedia, Stack Overflow, and Reddit. New content and discussions posted
online are quickly integrated into this ever-growing ecosystem, becoming
digital public goods used by people all around the world to learn new
technologies and solve their problems (Hess & Ostrom, 2003; Henzinger &
Lawrence, 2004; Lemmerich et al., 2019; Piccardi et al., 2021).
More recently, these public goods have been used to train artificial
intelligence (AI) systems, in particular, large language models (LLMs)
(Vaswani et al., 2017). For example, the LLM ChatGPT (OpenAI, 2023) answers
user questions by summarizing the information contained in these repositories.
The remarkable effectiveness of ChatGPT is reflected in its quick adoption
(Teubner et al., 2023) and application across diverse fields including
auditing (Gu et al., 2023), astronomy (Smith & Geach, 2023), medicine (Kanjee
et al., 2023), and chemistry (Guo et al., 2023). Randomized control trials
show that using LLMs significantly boosts productivity in computer
programming, professional writing, and customer support tasks (Peng et al.,
2023; Noy & Zhang, 2023; Brynjolfsson et al., 2023). Indeed, the widely
reported successes of LLMs like ChatGPT suggest that we will observe a
significant change in how people search for, create and share information
online.
Ironically, if LLMs like ChatGPT present substitute traditional ways of
searching and interrogating the web, then they will displace the very human
behavior that generated their original training data. User interactions with
ChatGPT are the exclusive property of OpenAI, its creator. Only OpenAI will be
able to learn from the information contained in these interactions. As people
begin to use LLMs instead of online knowledge repositories to find
information, contributions to these repositories will likely decrease,
diminishing the quantity and quality of these digital public goods. While such
a shift would have significant social and economic implications, we have
little evidence on whether people are actually substituting their consumption
and creation of digital public goods with ChatGPT.
The aim of this paper is to evaluate the impact of LLMs on the generation of
open data on question-and-answer (Q&A) platforms. Since LLMs perform
relatively well on software programming tasks (Peng et al., 2023), we study
Stack Overflow, the largest online Q&A platform for software development and
programming. We present three results. First, we examine whether the release
of ChatGPT has decreased the volume of posts, i.e. questions and answers,
posted on the platform. We measure the overall effect of ChatGPT’s release on
Stack Overflow activity using a difference-in-differences model. We compare
the weekly posting activity on Stack Overflow against that of four comparable
Q&A platforms. These counterfactual platforms are less likely to be affected
by ChatGPT either because their users are less able to access ChatGPT or
because ChatGPT performs poorly in questions discussed on those platforms. We
find that posting activity on Stack Overflow decreased by about $16\%$
following the release of ChatGPT, increasing over time to around $25\%$ within
six months.
Second, we investigate whether ChatGPT is simply displacing simpler or lower
quality posts on Stack Overflow. To do so, we use data on up- and downvotes,
simple forms of social feedback provided by other users to rate posts. We
observe no change in the votes posts receive on Stack Overflow since the
release of ChatGPT. This finding suggests that ChatGPT is displacing a wide
variety of Stack Overflow posts, including high-quality content.
Third, we study the heterogeneity of the impact of ChatGPT across different
programming languages discussed on Stack Overflow. We test for these
heterogeneities using an event study design. We observe that posting activity
in some languages like Python and Javascript has decreased significantly more
than the global site average. Using data on programming language popularity on
GitHub, we find that the most widely used languages tend to have larger
relative declines in posting activity.
Our analysis points to several significant implications for the sustainability
of the current AI ecosystem. The first is that the decreased production of
open data will limit the training of future models (Villalobos et al., 2022).
LLM-generated content itself is an ineffective substitute for training data
generated by humans for the purpose of training new models (Gudibande et al.,
2023; Shumailov et al., 2023; Alemohammad et al., 2023). One analogy is that
training an LLM on LLM-generated content is like making a photocopy of a
photocopy, providing successively less satisfying results (Chiang, 2023). And
while human feedback to LLMs may facilitate continued learning, such feedback
remains private information. This suggests a second issue: ChatGPT’s initial
advantage can compound if it effectively learns from its interactions with
users while simultaneously crowding out the generation of new open data
(Arthur, 1989). More broadly, a shift from open data to a more closed web will
likely have significant second-order impacts on the digital economy and how we
access and share information.
The rest of the paper is organized as follows. We introduce our empirical set-
up, including the data and models used in our analysis, in Section 2. Section
3 presents our results. In Section 4, we discuss their implications. We argue
that our findings of a significant decline in activity on Stack Overflow
following the release of ChatGPT have important implications for the training
of future language models, competition in the artificial intelligence sector,
the provision of digital public goods, and how humans seek and share
information.
## 2 Data and Methods
### 2.1 Stack Exchange and Segmentfault data
To understand the effect ChatGPT can have on digital public goods, we compare
the change in Stack Overflow’s activity with the activity on a set of similar
platforms. These platforms are similar to Stack Overflow in that they are
technical Q&A platforms, but are less prone to substitution by ChatGPT given
their focus or target group. Specifically, we focus on the Stack Exchange
platforms Mathematics and Math Overflow and on the Russian-language version of
Stack Overflow. We also examine a Chinese-language Q&A platform on computer
programming called Segmentfault.
Mathematics and Math Overflow focus on university- and research-level
mathematics questions respectively. We consider these sites to be less
susceptible to replacement by ChatGPT given that, during our study’s period of
observation, the free-tier version of ChatGPT performed poorly (0-20th
percentile) on advanced high-school mathematics exams (OpenAI, 2023), and was
therefore unlikely to serve as a suitable alternative to these platforms.
The Russian Stack Overflow and the Chinese Segmentfault have the same scope as
Stack Overflow, but target users located in Russia and China, respectively. We
consider these platforms to be less affected by ChatGPT given that ChatGPT is
officially unavailable in the Russian Federation, Belarus, Russian-occupied
Ukrainian territory, and the People’s Republic of China. Although people in
these places can and do access ChatGPT via VPNs (Kreitmeir & Raschky, 2023),
such barriers still represent a hurdle to widespread fast adoption.
We extract all posts (questions or answers) on Stack Overflow, Mathematics,
Math Overflow, and Russian Stack Overflow from their launch to early June 2023
using https://archive.org/details/stackexchange. We scraped the data from
Segmentfault directly. Our dataset comprises 58 million posts on Stack
Overflow, over 900 thousand posts for the Russian-language version of Stack
Overflow, 3.5 million posts on Mathematics Stack Exchange, 300 thousand posts
for Math Overflow, and about 300 thousand for Segmentfault. We focus our
analysis on data from January 2019 to June 2023, noting that our findings are
robust to alternative time windows.
For each post, our dataset includes the number of votes (up – positive
feedback, or down – negative feedback) the post received, the author (user),
and whether the post is a question or an answer. Furthermore, each post can
have up to 5 tags – predefined labels that summarize the content of the post,
for instance, an associated programming language. For more details on the data
used, we refer the reader to section 5. From this point forward, we will refer
to Mathematics, Math Overflow, Russian Stack Overflow, and Segmentfault, along
with their corresponding posts, as the counterfactual platforms and posts.
### 2.2 Models
#### Difference-in-differences
We estimate the effect of ChatGPT for posting activity on Stack Overflow using
a difference-in-differences method with four counterfactual platforms. We
aggregate posting data at platform- and week-level and fit a regression model
using ordinary least squares (OLS):
$IHS(Posts_{p,t})=\alpha_{p}+\lambda_{t}+\beta\times Treated_{p,t}+\sum_{p\in
P}\theta_{p}t+\epsilon_{p,t}$ (1)
where $Posts_{p,t}$ is the number of posts on platform $p$ in a week $t$,
which we transform using the inverse-hyperbolic sine function (IHS) (Burbidge
et al., 1988).111We prefer this transformation because then the coefficient of
interest can be roughly interpreted as a percent change in posting activity.
The IHS behaves similarly to a natural log transformation for positive values
but remains defined for zeroes. Our estimates are qualitatively similar to
using log transformation, standardization or raw data. $\alpha_{p}$ are
platform fixed effects, $\lambda_{t}$ are time (week) fixed effects,
$\theta_{p}$ are platform-specific linear time trends, and $\epsilon_{p,t}$ is
the error term.
The coefficient of interest is $\beta$, which captures the estimated effect of
ChatGPT on posting activity on Stack Overflow relative to the less affected
platforms: $Treated$ equals one for weeks after the release of ChatGPT
(starting November 27, 2022) when the platform $p$ is Stack Overflow and zero
otherwise. We report robust standard errors clustered at the monthly level.
To check the dynamics of the effect and to examine pretrends, we employ a
similar specification but instead of $\beta\times Treated_{p,t}$, we use
$\sum_{t}\beta_{t}\times I(week=t)\times I(platform=StackOverlow)$. We
standardize the effects to 0 in the week before the public release of ChatGPT
by dropping the indicator for that week from the regression. Separate
coefficients for 25 weeks following the release of ChatGPT show how the
effects of ChatGPT realized over time. Separate coefficients for the first 100
weeks before the release allow us to verify that posts on Stack Overflow had
evolved similarly to the activity on counterfactual platforms prior to the
introduction of ChatGPT.
The advantage of the difference-in-differences method compared to a simple
event study with Stack Overflow data only is that we estimate ChatGPT effects
net of possible weekly shocks that are common across the technical Q&A
platforms. For the interpretation of the coefficient, we note that we estimate
relative change in posting activity on Stack Overflow compared to activity on
other platforms before vs. after the release of ChatGPT.
#### Event Study
When analyzing the effect of ChatGPT on activity across programming languages,
we can no longer compare data from Stack Overflow with the counterfactual
platforms. This is because the tags annotating posts are different between
Stack Exchange platforms. Therefore, we study ChatGPT’s heterogeneous effects
using an event-study specification. For each programming language $i$
(identified by a tag), we model the standardized number of posts in a week $t$
on Stack Overflow by fitting a simple linear time trend with seasonal effects:
$\overline{Posts}_{i,t}=\beta_{0}+\beta_{1}(t)+\beta_{2}(ChatGPT)+\beta_{3}(t\times
ChatGPT)+\eta+\epsilon_{i,t}$ (2)
where $t$ is the linear time trend and $\eta$ are seasonal (month of year)
fixed effects. $ChatGPT$ equals one if the week $t$ is after the release of
ChatGPT and zero otherwise. Coefficient $\beta_{2}$ captures the change in the
intercept and coefficient $\beta_{3}$ reflects the change in the slope of the
time trend following the release of ChatGPT. In the tag-level analysis, we
standardize the dependent variable in order to be better able to compare
effects across programming languages with different numbers of posts.222We
standardize the number of posts within each tag by subtracting the mean and
dividing by the standard deviation. Both statistics are calculated before the
release of ChatGPT. We report HAC standard errors.
## 3 Results
Figure 1: A) Time series of weekly posts to Stack Overflow since early 2016.
The number of weekly posts decreases at a rate of about 7,000 posts each year
from 2016 to 2022. In the six months after the release of ChatGPT, the weekly
posting rate decreases by around 20,000 posts. B) Comparing posts to Stack
Overflow, its Russian- and Chinese-language counterparts, and mathematics Q&A
platforms since early 2022. Post counts are standardized by the average and
standard deviation of post counts within each platform prior to the release of
ChatGPT. Posting activity on Stack Overflow falls significantly more relative
to activity on other platforms.
### 3.1 Decrease in posting activity
Figure 1A, shows the evolution of activity on Stack Overflow from January 2016
to June 2023. Up to 2022 there was a gradual decrease in activity from roughly
110,000 to 60,000 posts per week, that is roughly 7,000k posts less per week
each year. However, after the release of ChatGPT (November 30th, 2022) posting
activity decreased sharply, with the weekly average falling from around 60,000
posts to 40,000 within six months. Compared to the pre-ChatGPT trend, this
decrease represents more than five years worth of deceleration in just half a
year.
The decrease in activity on Stack Overflow is larger than for similar
platforms for which we expect ChatGPT to be a less viable substitute. Figure
1B shows the standardized posting activity on Stack Overflow, the Russian- and
Chinese-language counterparts of Stack Overflow, and two mathematics Q&A
platforms. We standardize posting activity by the average and standard
deviation of post counts within each platform prior to the release of ChatGPT.
Figure 1B shows that Stack Overflow activity deviates markedly from activity
on the other platforms after the release of ChatGPT. The plot visualizes the
standardized posting activity within each platform since early 2022. Smoothed
weekly activity varies between plus and minus two standard deviations for all
platforms for most of 2022. Events, such as the Chinese New Year and other
holidays and the start of the Russian invasion of Ukraine, are visible.
Following the release of ChatGPT, we observe a significant decline in activity
on Stack Overflow.
We report the estimated effect of our difference-in-differences model in Table
1 and visualize the weekly estimates of the relative change in the Stack
Overflow activity in Figure 2. Table 1 indicates that ChatGPT decreased
posting activity on Stack Overflow by 15.6% ($1-e^{-0.17}$). These results are
robust to changes in the controls and starting point of the data time series.
We also tested for heterogeneity in subsets of the data: considering only
questions (rather than counting both questions and answers) and posts on
weekdays. In both subsets our estimates did not deviate significantly from the
main result: we estimate a 12% relative decrease in questions and 14% relative
decrease in posts on weekdays.
| (1) | (2) | (3)
---|---|---|---
| Number of posts | Number of questions | Weekday posts
Stack Overflow $\times$ Post-GPT | -0.170** | -0.112+ | -0.149*
| (0.0607) | (0.0619) | (0.0636)
Observations | 1,150 | 1,150 | 1,150
R-squared | 0.995 | 0.994 | 0.993
R-squared within | 0.290 | 0.315 | 0.232
Outcome mean | 16363 | 7273 | 13191
Outcome std. dev. | 29088 | 12661 | 23685
Table 1: Results of a difference-in-differences model, estimating the change
in activity observed weekly on Stack Overflow following the release of
ChatGPT, relative to activity on four other platforms less likely to have been
impacted. All regressions comprise platform fixed effects, week fixed effects,
and platform-specific linear time-trends. The standard-error of the estimate
clustered on month is reported in parentheses. Significance codes: ***:
$p<0.001$, **: $p<0.01$, *: $p<0.05$, +: $p<0.1$.
Figure 2 shows that the impact of ChatGPT is increasing over time and is by
the end of our study greater in magnitude than the average post-ChatGPT effect
estimated in Table 1. By the end of April 2023, the estimated effect
stabilizes at around 25%. Interestingly, ChatGPT use, in general, peaked
around this time.333https://www.similarweb.com/blog/insights/ai-news/chatgpt-
bard/
Figure 2: Difference-in-differences analysis for posting activities. The
dashed line marks November 30, 2023 the release date of ChatGPT. Eight weeks
after its introduction, we observe a steady decline in the activity of Stack
Overflow. The plotted coefficients correspond to the interaction between a
weekly dummy and posting on Stack Overflow. The coefficients are normalized to
that in the week before the release of ChatGPT. The reported confidence
intervals are at 95%. The regression comprises platform fixed effects, week
fixed effects, and platform-specific linear time-trends.
#### Voting activity
A decrease in overall activity on Stack Overflow does not necessarily signify
a problem; it could indicate a beneficial shift toward fewer but higher
quality posts, as less valued or simplistic questions may be outsourced to
ChatGPT. We investigate this possibility using data on voting activity but
observe no significant change in the typical appreciation of posts after
ChatGPT’s release.
The time series of upvotes and downvotes, which we use as a proxy for the
overall quality of posts, remain stable across the release of ChatGPT.
Specifically, Figure 3 reports the average number of upvotes and downvotes
that posts from a given week receive within five weeks of their creation.
Upvotes are shown in grey and downvotes in blue; neither series changes
significantly. Indeed the relative stability of voting behavior suggests that
the quality of posts on Stack Overflow has not meaningfully changed after the
introduction of ChatGPT.
Figure 3: The time series of upvotes and downvotes accruing to posts within
five weeks of their appearance. We observe no significant change since the
release of ChatGPT. The horizontal axis indicates the week of the post.
### 3.2 Heterogeneities across tags
Studying posts about different programming languages on Stack Overflow, we
find significant heterogeneities in the impact of ChatGPT on posting behavior
across languages.
In Facet A of Figure 4, we plot the estimated effects (slope changes in the
linear time trend after the introduction of ChatGPT) for those 69 tags that we
connected to a programming language on GitHub. We estimate a negative effect
of ChatGPT for most tags, but the estimates range between a 0.25 standard
deviation decrease in slope (i.e. change per week following the ChatGPT
release) to a 0.03 standard deviation increase. We observe that some of the
widely used languages like Python and Javascript are the most impacted by
ChatGPT. Interestingly, the model estimates that posts about CUDA have
increased (though not significantly) after ChatGPT was released. CUDA is an
application programming interface created by Nvidia, a graphics card
manufacturer, that facilitates the use of graphics cards for computational
tasks, in particular for machine learning and artificial intelligence. This
exception again demonstrates the impact of ChatGPT on the world of computer
programming: people are increasingly interested in software relating to
artificial intelligence.
Figure 4: A) The event study estimates of the effect of ChatGPT’s release on
activity on a selection of tags on Stack Overflow. We report HAC-corrected 95%
confidence intervals. B) The relationship between estimated effects and salary
data from the Stack Overflow developer survey. We find no significant
relationship. C) The relationship between the number of GitHub repositories
using a tag and the estimated effect of ChatGPT on that tag. In both B) and C)
we plot a linear fit with bootstrapped 95% confidence intervals. The dashed
line in B) indicates that the correlation is not significant
In Facet B, we compare the estimated impact of ChatGPT on different languages
against salary data of developers using those languages. We source salary data
from the 2022 Stack Overflow developer survey, focusing on US-based developers
and calculating medians of reported salaries. We observe no clear relationship
between the estimated labor market value of a specific language and changes in
posting behavior in that language post-ChatGPT.
To better understand the relationship between the size of the user base of a
programming language and how it is impacted by ChatGPT, we compare our
estimates with data from GitHub, the largest online platform for collaborative
software development. Among other sources, ChatGPT was trained on data from
GitHub. Because training data was collected up to September 2021, we use data
on language use on GitHub up to June 2021. In Facet C of Figure 4, we
visualize the relationship between the number of GitHub repositories (coding
projects) in a specific language and the estimated impact of ChatGPT on that
language. We observe that languages with more GitHub repositories tend to be
more significantly impacted by the release of ChatGPT in terms of associated
activity on Stack Overflow (Pearson’s $\rho=-0.45,p<.001$).
## 4 Discussion
The rate at which people have adopted ChatGPT is one of the fastest in the
history of technology (Teubner et al., 2023). It is essential that we better
understand what activities this new technology displaces and what second-order
effects this substitution may have (Schumpeter, 1942; Aghion & Howitt, 1992).
This paper shows that after the introduction of ChatGPT there was a sharp
decrease in human content creation on Stack Overflow. We compare the decrease
in activity on Stack Overflow with other Stack Exchange platforms where
current LLMs are less likely to be used. Using a difference-in-differences
model, we find about $16\%$ relative decrease in posting activity on Stack
Overflow, with a larger effect in later months. We observed no large change in
social feedback on posts, measured using votes, following ChatGPT’s release,
suggesting that average post quality has not changed. Posting activity related
to more popular programming languages decreased more on average than that for
more niche languages. These results suggest that users partially substituted
Stack Overflow with ChatGPT. Consequently, the wide adoption of LLMs can
decrease the provision of digital public goods, in particular, the open data
previously generated by interactions on the web.
Our results and data have some shortcomings that point to open questions about
the use and impact of LLMs. First, while we can present strong evidence that
ChatGPT decreased the posting activity in Stack Overflow, we can only
partially assess quality of posting activity using data on upvotes and
downvotes. Users may be posting more challenging questions, ones that LLMs
cannot (yet) address, to Stack Overflow. Future work should examine whether
continued activity on Stack Overflow is more complex or sophisticated on
average than posts from prior to ChatGPT release. Similarly, ChatGPT may have
reduced the volume of duplicate questions about simple topics, though this is
unlikely to impact our main results as duplicates are estimated to account for
only $3\%$ of posts (Correa & Sureka, 2013), and we do not observe significant
changes in voting outcomes.
A second limitation of our work is that we cannot observe the extent to which
Russian- and Chinese-language users of the corresponding Q&A platforms are
actually hindered from accessing ChatGPT; indeed recent work has shown a spike
in VPN and Tor activity following the blocking of ChatGPT in Italy (Kreitmeir
& Raschky, 2023). Given the potential economic importance of ChatGPT and
similar LLMs, it is anyway essential that we better understand how such bans
and blocks impact the accessibility of these tools (Gaessler & Piezunka,
2023). Finally, we do not address the issue that ChatGPT may be used to
generate Stack Overflow content. Stack Overflow policy effectively banned
posts authored by ChatGPT within a week of its release. In any case, a
significant amount of ChatGPT activity on Stack Overflow would mean that our
measures underestimate the effect of ChatGPT.
Despite these shortcomings, our results have important implications for the
future of digital public goods. Before the introduction of ChatGPT, more
human-generated content was posted to Stack Overflow, forming a collective
digital public good due to their non-rivalrous and non-exclusionary nature –
anyone with internet access can view, absorb, and extend this information,
without diminishing the value of the knowledge. Now, this information is
rather fed into privately owned LLMs like ChatGPT. This represents a
significant and trending shift of knowledge from the public domain to the
private ones.
This observed substitution effect poses several issues for the future of
artificial intelligence in general. The first is that if language models crowd
out open data creation, they will be limiting their own future training data
and effectiveness. The second is that owners of the current leading models
have exclusive access to user inputs and feedback, which, with a relatively
smaller pool of open data, gives them a significant advantage against new
competitors in training future models. Third, the decline of public resources
on the web would reverse progress made by the web toward democratizing access
to knowledge and information. Finally, the consolidation of humans searching
for information around one or a few language models could narrow our
explorations and focus our attention on mainstream topics. We briefly
elaborate on these points, then conclude with a wider appeal for more research
on the political economy of open data and AI, and how we can incentivize
continued contributions to digital public goods.
#### Training future models
Our findings suggest that the widespread adoption of ChatGPT may make it
difficult to train few iterations (Taleb, 2012). Though researchers have
already expressed concerns about running out of data for training AI models
(Villalobos et al., 2022), our results show that the use of LLMs can slow down
the creation of new data. Given the growing evidence that data generated by
LLMs cannot effectively train new LLMs (Gudibande et al., 2023; Shumailov et
al., 2023; Alemohammad et al., 2023), modelers face the real problem of
running out of useful data. If ChatGPT truly is a ‘‘blurry JPEG’’ of the web
(Chiang, 2023), then, in the long run, it cannot effectively replace its most
important input: data derived from human activity. The proliferation of LLMs
has already impacted other forms of data creation: many Amazon Mechanical Turk
workers now generate content (i.e. respond to surveys, evaluate texts) using
ChatGPT (Veselovsky et al., 2023).
#### Competition in the artificial intelligence sector
A firm’s early advantage in technological innovation often leads to
significant market share (Arthur, 1989). In our case, ChatGPT is
simultaneously decreasing the amount of open training data that competitors
could use to build competing models, while capturing a valuable private source
of user data. There is also a growing concentration in tech driven by a shift
from companies going public to acquisitions (Ederer & Pellegrino, 2023) –
indeed OpenAI is partially owned by Microsoft. These forces may lead to a
compounding advantage for OpenAI. Though firms have long used the massive
amounts of open data created by users of platforms like Wikipedia, Stack
Overflow, GitHub, OpenStreetMap or Reddit to create products and capture value
(Henzinger & Lawrence, 2004; Vincent et al., 2018; Vincent & Hecht, 2021),
these products have not generally replaced those platforms.
#### Lost economic value
Digital public goods generate value in many ways besides feeding LLMs and
other algorithms. For instance, Wikipedia is an important source of
information worldwide, but in developing countries, readers are more often
motivated by intrinsic learning goals and tend to read articles in greater
detail (Lemmerich et al., 2019). Unequal access to artificial intelligence may
also compound inequalities in growth and innovation between countries
(Gaessler & Piezunka, 2023).
Digital public goods also provide direct value to the many websites that
extract data from open data to complement their core services with extra
information (Piccardi et al., 2021). For instance, there is substantial
interdependence between sites like Wikipedia, Reddit, and Stack Overflow and
the search engines that use them to enrich responses to user queries via
infoboxes (McMahon et al., 2017; Vincent & Hecht, 2021).
Contributors to digital public goods like Stack Overflow or Open Source
Software (OSS) often enjoy indirect benefits (Lerner & Tirole, 2002). For
instance, while OSS itself provides significant value in the global economy
(Greenstein & Nagle, 2014), OSS contributions are valuable signals of a firm’s
capabilities to investors (Conti et al., 2021). Individual contributions to
Stack Overflow are used to signal ability on the labor market (Xu et al.,
2020). Any general tendency of ChatGPT to crowd out contributions to digital
public goods, may limit these valuable signals that reduce economic frictions.
On the other hand, such signaling activity may serve as a powerful incentive
to keep people contributing.
#### Narrowing of information seeking
The substitution effect we report likely has important second-order effects on
how people search for information and their exposure to new ideas. LLMs likely
favor well-established perspectives and due to their efficiency decrease the
need for users to forage for information. These features of LLMs may reinforce
a trend observed earlier in the context of the web. Specifically, internet
search engines are thought to have pushed science toward consensus and
narrower topics by improving efficiency of information search and improving
the visibility of mainstream information (Evans, 2008). LLMs may also
disincentivize the use of new or niche tools because they most amplify our
productivity with those tools for which it has much training data. For
instance, ChatGPT may not be able to help users of a new programming language
that is has not seen many examples of. Given that LLMs are poised to change
how we do research (Grossmann et al., 2023) and present a strong competitor to
search engines (Xu et al., 2023), we need to understand what LLM efficiency
implies for our contact with diverse sources of information and incentives to
try new things.
More generally, models like ChatGPT are going to generate political and
economic winners and losers like many previous breakthrough technologies.
While early evidence shows that these models enhance productivity especially
among new and inexperienced workers (Noy & Zhang, 2023; Brynjolfsson et al.,
2023), there are other ways in which they may contribute to inequality between
people and firms (Rock, 2019), for instance via potential negative side
effects of automation (Acemoglu & Restrepo, 2019; Eloundou et al., 2023). Our
results suggest that the economics of data creation and ownership will become
more salient: as data becomes more valuable, there will be growing interest in
how creators of data can capture some of that value (Li et al., 2023). These
multi-faceted aspects of the impact of LLMs suggest that the political economy
of data and AI will be especially important in the next years (Lehdonvirta,
2022; Johnson & Acemoglu, 2023).
In this context, our work highlights the specific issue that valuable digital
public goods may be under-produced as a result of the proliferation of AI. A
natural follow-up question is how we can incentivize the creation of such
goods. While unemployment shocks are known to increase the provision of
digital public goods (Kummer et al., 2020), it would be an unsatisfying
solution to suggest that people put out of work by automation will fill this
gap. In the case of platforms like Stack Overflow, active users are often
motivated by social feedback and gamification (Anderson et al., 2012), but the
continual onboarding of new users is what keeps these platforms relevant in
the long run (Danescu-Niculescu-Mizil et al., 2013). For the sake of a
sustainable open web and an AI ecosystem that draws on its data, we should
think about how to keep people exchanging information and knowledge online.
## 5 Appendix
### Data
#### Stack Exchange platform sites
The raw dataset obtained from https://archive.org/details/stackexchange
contains nearly all posting activity on the question and answer platforms
hosted on the Stack Exchange network from its launch in 2008 to early June
2023. These include Stack Overflow, its Russian language version, and Math
Overflow and Math Stack Exchange. Stack Overflow is the largest online Q&A
platform for topics relating to computer programming and software development.
It provides a community-curated discussion of issues programmers face
(Anderson et al., 2012). Questions have multiple answers, and users debate the
relative merits of solutions and alternatives in comments. A track record on
Stack Overflow has value on the labor market as a signal of an individual’s
skills (Xu et al., 2020).
The data contains over 58 million posts, including both questions and answers.
Posts are linked to their posting users, from which we infer poster previous
activity and can identify posts made by new users. Questions are annotated
with tags indicating the topic of the post including programming languages
used. Users can give posts upvotes or downvotes, providing posting users with
social feedback and reputation points. The Russian language version of Stack
Overflow (over 900 thousand posts) and the mathematics-oriented platforms Math
Stack Exchange (over 3.5 million posts) and Math Overflow (over 300 thousand
posts) have identically structured data dumps hosted in the same location.
Registered users can upvotes and downvote posts made on Stack Exchange
platforms. These votes provide a valuable signal of the value of posts
Anderson et al. (2012); Mamykina et al. (2011). They are the primary way users
earn reputation points and status on Stack Exchange platforms. Votes also
influence the ranking of posts in user feeds and search engine results,
facilitating information filtering. Downvotes are used to moderate. The Stack
Exchange data dump contains data on every vote cast, including the
corresponding post, the date the vote was made, and whether it was an upvote
or downvote.
#### Segmentfault
Segmentfault is a Chinese language platform with a Q&A platform for developers
that has many similarities with the Stack Exchange sites. Users post questions
on programming language topics and other users post answers. Questions are
tagged by relevant languages and technologies, and there are similar
gamification elements on the platform. We scraped data on all posts as of
early June 2023, gathering over 300 thousand in total.
#### Selection of tags
Stack Overflow posts are annotated by tags which describe the concepts and
technologies used in the post. For example, many tags indicate programming
languages, webframeworks, database technologies, or programming concepts like
functions or algorithms. Stack Overflow reconciles tags referring to the same
things via a centralized synonym dictionary. We selected the 1,000 most used
tags up to early June 2023, and focused on those 69 which could be directly
linked to language statistics reported by GitHub, described next.
#### GitHub data on programming language use
We use data from the June 2021 GHTorrent data dump (Gousios & Spinellis, 2012)
as a proxy measure for the amount of open data available for each programming
language. The dataset reports which languages are used in each project or
repository on GitHub. We simply count the number of repositories mentioning
each language. We then link the languages with tags on Stack Overflow. As an
alternative, we count the number of commits, elemental code contributions to
repositories, to each repository, hence language. In the main paper we
visualize the estimated effects of ChatGPT on specific tags that we can link
to GitHub languages. We exclude some tags which refer to file formats or plain
text, specifically: yaml, json, text, svg, markdown, and xml.
### Data and Code availability
Data and code to reproduce our analyses will be made available in a subsequent
draft. The Stack Overflow data dump is available here:
https://archive.org/details/stackexchange.
### Acknowledgments
We thank Frank Neffke, Gergő Tóth, Christoffer Koch, Sándor Juhász, Martin
Allen, Manran Zhu, Karl Wachs, László Czaller, and Helene Strandt for helpful
comments and discussions.
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|
# GraVAC: Adaptive Compression for Communication-Efficient Distributed DL
Training
Sahil Tyagi Indiana University Bloomington, USA
<EMAIL_ADDRESS>Martin Swany Indiana University Bloomington, USA
<EMAIL_ADDRESS>
###### Abstract
Distributed data-parallel (DDP) training improves overall application
throughput as multiple devices train on a subset of data and aggregate updates
to produce a globally shared model. The periodic synchronization at each
iteration incurs considerable overhead, exacerbated by the increasing size and
complexity of state-of-the-art neural networks. Although many gradient
compression techniques propose to reduce communication cost, the ideal
compression factor that leads to maximum speedup or minimum data exchange
remains an open-ended problem since it varies with the quality of compression,
model size and structure, hardware, network topology and bandwidth. We propose
_GraVAC_ , a framework to dynamically adjust compression factor throughout
training by evaluating model progress and assessing gradient information loss
associated with compression. _GraVAC_ works in an online, black-box manner
without any prior assumptions about a model or its hyperparameters, while
achieving the same or better accuracy than dense SGD (i.e., no compression) in
the same number of iterations/epochs. As opposed to using a static compression
factor, _GraVAC_ reduces end-to-end training time for ResNet101, VGG16 and
LSTM by 4.32$\times$, 1.95$\times$ and 6.67$\times$ respectively. Compared to
other adaptive schemes, our framework provides 1.94$\times$ to 5.63$\times$
overall speedup.
###### Index Terms:
deep learning, data-parallel training, gradient compression, sparsification,
adaptive systems
## I Introduction
Deep Learning (DL) is a supervised machine learning approach that optimizes a
loss function over a non-convex surface by comparing model predictions with
ground truth. Each training iteration in DL involves forward and backward
pass, i.e., generate predictions from input data, assess loss, compute
gradients and update model parameters via optimization method like gradient
descent. Training is an iterative process, typically involving multiple passes
over the entire dataset where each pass is called an _epoch_. DL is also
heavily influenced by certain _hyperparameters_ that affect training speed,
quality, or both. Commonly used hyperparameters are learning rate, momentum,
batch size, weight decay, epochs, activation function, etc.
Distributed data-parallel (DDP) methods further scale training across multiple
nodes that train a globally shared model with I.I.D. data (independent and
identically distributed) by periodically aggregating locally computed
gradients at the end of each iteration. The compute requirements to train DL
models doubles every 3.5 months [1], while the compute gains in chip design
for ML accelerators and bandwidth gains in telecommunications networks double
every 24 and 18 months [2, 3]. Thus, the infrastructure required to train
state-of-the-art models tends to fall behind their compute and networking
demands. Since upgrading network stack in the cloud, datacenter and HPC
clusters can be infrequent as compared to appending new accelerators in pre-
existing systems, gradient communication tends to be the major bottleneck in
distributed training [4].
Different compression techniques have been proposed in recent years to
mitigate this synchronization overhead. However, the optimal compression
factor (CF) that minimizes data exchange or end-to-end training time depends
on the model itself (i.e., its size, structure and depth), available network
bandwidth and the compression overhead itself. Unlike traditional HPC and
distributed computing applications that only measure parallel efficiency, DDP
training has an additional statistical efficiency associated with it. Although
the amount of computation performed on each iteration is the same, some
iterations tend to be more crucial than others towards the overall learning of
the model. Updates are especially sensitive in early stages and to
hyperparameters like learning rate schedule, momentum and weight decay [5]. It
would thus be intuitive to compare information loss in gradients on account of
compression, and use a lower CF when considerably more information is lost and
a higher CF when most information is preserved under compression. We can
subsequently increase compression as training continues and gradients
saturate, and decrease it back during the aforementioned critical stages.
We take into account the parallel and statistical efficiency aspect of
gradient compression in this work: a high CF improves overall throughput
(i.e., number of samples processed per second) by reducing communication cost,
but increases information loss in the gradients resulting in either slower or
insignificant updates. The two metrics in DDP compression are pareto-related
as one improves at the detriment of the other. We propose GraVAC: {Gra}dient
{V}ariance-based {A}daptive {C}ompression to dynamically adjust CF by
comparing information loss from compression with that of the original
gradients computed in backpropagation. _GraVAC_ evaluates different CFs in a
given search space and determines the CF that best balances parallel and
statistical efficiency in DDP training with compression. We validate our
approach over a variety of DL models and directly compare with static CF on
compressors like Top-$\mathit{k}$ [6], Deep Gradient Compression or DGC [7],
Redsync [9] and Random-$\mathit{k}$ [6].
## II Background and related work
DDP training can be implemented either via MPI-based collectives (AllReduce)
[10, 11, 12] or using one or more centralized parameter servers (PS) [13] to
accumulate and distribute model updates among workers.
### II-A Scaling Efficiency of DDP Training
DL training is an iterative process that involves parameter updates at each
step via gradient descent (GD) [14]. Full GD uses entire training data at
every step, making the whole process slow and compute-intensive, while
Stochastic GD processes a single sample and does not vectorize multiple
samples on fast accelerators. Mini-batch GD is the optimal middle ground
between Full and Stochastic GD where _b_ samples are randomly sampled from
I.I.D. data. Eqn. (1) describes the update rule in mini-batch GD where
parameters $\mathit{w}$ at $(\mathit{i}+1)$-th iteration on $\mathit{N}$
workers minimize loss function $\mathit{\mathcal{L}(\cdot)}$ on input samples
$\mathit{x_{j}}$ of size $\mathit{b}$ from distribution $\mathcal{X}_{j}$ with
learning rate $\mathit{\eta}$. With weak scaling, we can increase the amount
of per-iteration work by adding more workers and keeping per-worker batch-size
$\mathit{b}$ the same.
$w_{i+1}=w_{i}-\eta\dfrac{1}{N}\sum_{n=1}^{n=N}{\dfrac{1}{|b|}\sum_{j\in
b}\dfrac{\partial}{\partial w_{i}}\mathcal{L}(x_{(j,n)},w_{i})}$ (1)
The _ideal_ throughput of a distributed application $\mathit{T_{N}}$ executed
across $N$ workers is $N$ times the throughput of a single worker
$\mathit{T_{1}}$. The deviation is measured via “scaling efficiency“ in Eqn.
2a. Assuming negligible IO overhead, iteration time in dense SGD is bounded by
computation and communication time (Eqn. (2b)). It may be possible to overlap
communication with computation, but only partially since the latter is
comparatively much lower on modern GPUs and TPUs. Model communication has been
shown to be an order of hundreds or even thousands of magnitudes higher than
gradient computation. Thus, frequent synchronization ($t_{sync}$) is the
bottleneck that halts linear scaling in DDP. Table 1 describes the size,
density and convergence target of ResNet101 [15], VGG16 [16] and LSTM [17]
with dense SGD communication. Latency is further exacerbated on constrained
networks with limited bandwidth as large volumes of data is exchanged by
multiple workers simultaneously.
$\eta_{scaling}=T_{N}/N\cdot T_{1}$ (2a) $t_{iter}\approx
t_{compute}+t_{sync}$ (2b)
TABLE 1: DL model description Model | Layers | Size (MB) | Dataset | Test target
---|---|---|---|---
ResNet101 | 101 | 170 | CIFAR10 | 80% Top-1
LSTM | 2 | 252 | PTB | 22.0 PPL
VGG16 | 16 | 528 | CIFAR100 | 90% Top-5
(a) Scaling efficiency in DDP
(b) Initial gradient sensitivity
Figure 1: Communication overhead and early critical period in DDP training.
For a DL model with a total of $M$ parameters, the time cost based on the
$\alpha$-$\beta$ communication model (where $\alpha$ is the latency and
$\beta$ is the inverse of bandwidth) for tree-based allreduce is $(2\alpha
logN+2MlogN\beta)$ [18]. For ring-based allreduce, this becomes
$2(N-1)\alpha+2M\beta(N-1)/N$. Hence, communication cost increases as more
workers are added to the mix in distributed training. Fig. 1a shows how
overall throughput deviates from the ideal as cluster-size increases. The
scaling efficiency is also influenced by the message size, i.e., total
gradients/parameters to be communicated. In dense SGD, we observed scaling to
be affected by the tensor-size distributions across the layers of a model as
well. For e.g., LSTM has a better $\eta_{scaling}$ than ResNet101 despite
being a larger model. This is because parameters in LSTM are spread across
just 2 layers, compared to 101 in ResNet101.
### II-B Gradient Variance in Deep Learning
Prior work has demonstrated that gradient information can help measure the
statistical efficiency of distributed training [19, 20]. There is a strong
correlation between changes in the eigen values of second-order hessian [21]
and first-order gradients (i.e., variance). [22, 23] explores how gradients
behave in early stages of DL training and during certain critical periods,
influenced by hyperparameters like learning rate schedule, gradient clipping
and type of SGD used (e.g., zero, first or second-order moments). Fig. 1b
attests those findings where we plot variance over the starting iterations and
notice how drastically the gradients change and saturate over training.
### II-C Gradient Compression
Many lossy compression techniques have been proposed for DDP and federated
learning in recent years. Lossy compression incurs a fundamental trade-off
between data-size and information loss; one can either reduce message size by
losing more information, or preserve data quality by keeping majority of the
original bits intact. In the context of DDP, higher CF reduces communication
time at the cost of accuracy degradation or more steps/epochs required for the
same convergence. CF measures the size of original gradients to the size of
compressed tensors. E.g., compressing 10% gradients gives CF of
10$\mathsf{x}$, while 1% gives 100$\mathsf{x}$. Lossy compression can be
broadly classified into _quantization_ , _sparsification_ or _low-rank
approximations_.
The bit-width of single-precision (32-bit) floats is reduced in gradient
quantization. Techniques like automatic mixed precision (AMP) [24] reduces
gradients to half-precision, resulting in 2$\mathsf{x}$ CF. QSGD [25] balances
the trade-off between accuracy and quantization precision. 1-bit SGD [26]
reduces 32-bit floats to 1-bit and propagates quantization error via error-
feedback. Sparsification methods communicate only a fraction of the gradient
values along with their indices and set everything else to 0. Top-k
sparisifies by extracting the top k% values while Random-k does so randomly
with negligible compression overhead. DGC discards gradients below a certain
threshold along with using momentum correction and gradient clipping. Methods
like Redsync [40] combine quantization and sparsification, but the estimation
quality is not accurate [27]. Approaches like PowerSGD [28] and Pufferfish
[29] achieve compression via low-rank updates. The former can be viewed as
adding regularization in DL, while the latter performs low-rank factorization
on fully connected, convolutional and LSTM layers.
_What should be the ideal CF in Compression-based DDP?_
The ideal CF is one that reduces communication time without trimming too much
gradients which can be detrimental to final model. Compression has its own
associated costs depending on the target CF and computational complexity of
the mechanism itself. These factors affect both the parallel efficiency of
distributed training as well as statistical inefficiency due to information
loss from compression. Fig. 2 aptly demonstrates this where the CF that gives
maximum speedup varies for each model and compression technique employed. The
models are trained to Table 1 targets. ResNet101 on Top-k achieves most
speedup at 100$\mathsf{x}$, while VGG16 and LSTM peak at CFs 1000$\mathsf{x}$
and 10$\mathsf{x}$ respectively. On the other hand, ResNet101 fails to
converge for any CF with Random-k compression. VGG16 and LSTM converged with
10$\mathsf{x}$ and failed with other CFs. Although a typical ML practitioner
may not necessarily need to think about a plethora of compression methods,
choosing the right CF with any compressor and DL model that minimizes training
time, or even converges successfully, presents a non-trivial challenge.
(a) Top-k
(b) Random-k
Figure 2: CF with maximal speedup (to reach Table 1 targets) varies for each
model and compression technique used. The results are normalized by
10$\mathsf{x}$ CF while a speedup of 0.0 implies convergence failure.
Dynamic compression mechanisms like AdaQS [30] perform quantization using
gradient mean to standard deviation ratio (MSDR). Systems like Accordion [31]
and ScaDLES [32] switch between low and high compression based on critical
regime identification. We tackle the ideal CF exploration problem in _GraVAC_
in a gradient-driven manner by comparing variance of prior and post-
compression gradients. For clarity, prior-compression gradients refer to the
original tensors computed in backward pass. By measuring the information lost
in compression, we dynamically adjust CF over each iteration. Starting with a
low CF initially, we gradually increase compression as training progresses. On
encountering senstive or critical regions, _GraVAC_ switches to a lower CF
that least degrades convergence.
## III Design and implementation
In this section, we first describe the trade-off between parallel and
statistical efficiency of DDP training _with_ compression. Then we describe
the metrics “compression gain“ and “compression throughput“ to combine the
two, and explain _GraVAC_ ’s adaptive compression algorithm.
### III-A Parallel Efficiency of Gradient Compression
The end goal of gradient compression is to improve DDP scaling efficiency.
Application scaling is governed by the DDP mechanism (ring-based, tree-based
allreduce or parameter servers), communication library used (MPI, NCCL [11],
Gloo [10] or RPC) and available bandwidth. Keeping the latter and network
infrastructure aside, speedup in any DL model depends on the target CF,
quality of estimation and compression overhead. The overall iteration time in
Eqn. 2b is adjusted for compression as
$\mathit{t_{iter}^{(\mathit{c})}}\approx\mathit{t_{compute}}+\mathit{t_{sync}^{(\mathit{c})}}+\mathit{t_{compress}^{(\mathit{c})}}+\mathit{t_{decompress}^{(\mathit{c})}}$
where it takes $\mathit{t_{compress}^{(\mathit{c})}}$ time to reduce gradients
to CF $\mathit{c}$ such that it reduces communication time to
$\mathit{t_{sync}^{(c)}}$. $\mathit{t_{decompress}^{(\mathit{c})}}$ is the
time taken to reconstruct the compressed gradients to the same dimension as
the original gradients. A viable compressor must have its compression time
considerably lower than synchronization time.
The parallel efficiency of a distributed application suffers with more workers
due to higher synchronization costs. Improving the network bandwidth
alleviates this to only a certain extent. [4] investigates how DDP throughput
improves marginally with higher bandwidth. They observed that ResNet50 peaks
to 75% scale-out on a 25 Gbps network and remains the same even for 100 Gbps.
Its because network transport implementation of current DL frameworks cannot
fully utilize the available network bandwidth. Thus, even though cloud
providers like GCP provide anywhere from 10-32 Gbps bandwidth depending on the
machine type and VM size, they may not be utilized to their full potential.
Fig. 3 shows how the throughput increases and communication overhead reduces
with compression. The results are relative to CF 10$\mathsf{x}$ for each
model. We perform layerwise DGC compression over a 32 GPU cluster. System
throughput is determined only by compression overhead and communication time
as the compute time in backpropagation stays the same across all CFs. Based on
the compressor used, compression latency may vary with target CF. For e.g., it
decreases with larger CF as Top-k uses max-heap and sorts the top k% elements
in $O(N+\mathit{k}\log{}\mathit{k})$ time. Throughput for ResNet101 and VGG16
saturates at 500$\mathsf{x}$ and does not improve thereafter, while LSTM
saturates at 1000$\mathsf{x}$ (Fig. 3a). Communication savings also diminish
at higher CFs due to small message size and network saturation (Fig. 3b).
Thus, the highest CF may not necessarily correspond to the largest throughput.
(a) Relative throughput
(b) Relative communication
Figure 3: Throughput and communication speedup for layerwise DGC compression,
normalized by 10$\mathsf{x}$ CF.
### III-B Statistical Inefficiency of Gradient Compression
Gradient compression mechanisms rely on _error-feedback_ [35, 36] which
essentially acts as delayed updates, as commonly noted in asynchronous
training. The gradients ineligible for compression in the current iteration
are not discarded, but added to residual gradients which in turn are added to
gradients computed in the next iteration. Residual gradients and error-
feedback helps preserve important features and is critical to convergence [6,
7, 8]. Applying compression without error-feedback has been shown to achieve
lower accuracy in deep learning models [35]. At the same time, residual
gradients can sometimes degrade generalization performance due to stale
updates.
DDP training with very high CFs can negatively impact training time,
convergence quality, or both if the compressed gradients are too sparse or
quantized to update the model in any significant way. _It is thus crucial to
have an indicator that quantifies information loss between compressed and the
original gradients._ We do so by comparing variance between the original and
compressed tensors on every iteration and see how it relates to actual model
convergence. Denoting the original gradients as _BC_ (Before-Compression) and
compressed tensors as _AC_ (After-Compression), we compare BC and AC tensors
in two separate configurations with CFs 10$\mathsf{x}$ and 1000$\mathsf{x}$ in
Fig. 4, 5 and 6. We compare the convergence curves for the two CFs with _Dense
SGD_ (i.e., no compression) to see how much accuracy degrades with
compression.
_AC_ 10$\mathsf{x}$ is nearly identical to its _BC_ counterpart in ResNet101
(Fig. 4a) while there is considerably more information loss in between _BC_
and _AC_ 1000$\mathsf{x}$ (Fig. 4b). This translates to their convergence
curves in Fig. 4c as well where 10$\mathsf{x}$ and dense SGD runs follow a
similar convergence trajectory while 1000$\mathsf{x}$ achieves considerably
lower accuracy for the same iterations.
(a) 10$\mathsf{x}$ CF
(b) 1000$\mathsf{x}$ CF
(c) Convergence curve
(d) Compression gain
Figure 4: ResNet101: Prior and Post-Compression gradients, test accuracy and
compression gain for CFs 10$\mathsf{x}$ and 1000$\mathsf{x}$.
VGG16 follows a similar trend with 10$\mathsf{x}$ CF. The _BC_ and _AC_
gradient variance (Fig. 5a) is nearly identical and so are the convergence
curves for 10$\mathsf{x}$ and Dense SGD (Fig. 5c). We notice a slight
deviation between _BC_ and _AC_ at 1000$\mathsf{x}$ initially in Fig. 5b,
which correlates to slow convergence in the early iterations for
1000$\mathsf{x}$ in Fig. 5c. As the deviation _BC_ and _AC_ decreases, we see
both CFs converge to the same accuracy as Dense SGD in the same iterations.
(a) 10$\mathsf{x}$ CF
(b) 1000$\mathsf{x}$ CF
(c) Convergence curve
(d) Compression gain
Figure 5: VGG16: Prior and Post-Compression gradients, test accuracy and
compression gain for CFs 10$\mathsf{x}$ and 1000$\mathsf{x}$.
The _AC_ 10$\mathsf{x}$ and 1000$\mathsf{x}$ gradients lie on similar scales
as _BC_ in LSTM, although the higher CF has slightly higher variance (Fig. 6a
and 5b). As seen from Fig. 5c, Dense SGD has the least perplexity (thus,
better model quality), followed by 10$\mathsf{x}$ and 1000$\mathsf{x}$ CFs.
(a) 10$\mathsf{x}$ CF
(b) 1000$\mathsf{x}$ CF
(c) Convergence curve
(d) Compression gain
Figure 6: LSTM: Prior and Post-Compression gradients, test perplexity (lower
is better) and compression gain for CFs 10$\mathsf{x}$ and 1000$\mathsf{x}$.
To compare the information loss between the original and gradients compressed
to CF _c_ , we define a simplistic metric called _Compression gain_. As part
of error feedback, we update the gradients such that
$\mathit{g_{ef}^{(i)}}=\mathit{g_{0}^{(i)}}+\;\mathsf{residual\\_gradients}^{(i-1)}$
for $i\geq 1$. Here, $\mathit{g_{0}^{(i)}}$ are the original gradients
calculated via backpropagation at iteration $i$, while
$\mathsf{residual\\_gradients}^{(i-1)}$ are left-overs from the last iteration
$(i-1)$ and before, which are added back as part of error-feedback to produce
$\mathit{g_{ef}^{(i)}}$ for the current iteration. With compression operator
$\mathcal{C}$, gradients are compressed as
$\mathit{g_{c}^{(i)}}=\mathcal{C}[\mathit{g_{ef}^{(i)}}]$. Compression gain is
then measured as the ratio of expected variance of compressed gradients
$\mathit{g_{c}^{(i)}}$ and the original gradients modified with error-
feedback, i.e., $\mathit{g_{ef}^{(i)}}$:
$\mathsf{Compression\
gain}=\frac{\mathbb{E}[||g_{c}^{(i)}||^{2}]}{\mathbb{E}[||g_{ef}^{(i)}||^{2}]}$
In prior work, gradient noise has been well studied in deep learning
literature pertaining to divergence between locally-computed and aggregated
gradients in DDP [20, 37, 38]. These works use gradient information to tweak
the global batch-size in DDP to optimize job completion time or allocate
optimal resources for a job. Instead of looking at local and global gradients,
_GraVAC_ ’s novelty comes from evaluating the noise between the original and
compressed tensors. The gradients computed over each iteration can be noisy.
Thus, we keep a moving average of the respective variances of the original and
compressed gradients. The computation and memory footprint of this approach is
low since the window-size in moving average is finite and only a single-
precision floating point is stored for every iteration. Compression gain is
bounded between $\\{0,1]$ such that it is low when $\mathcal{C}$ trims too
much information. As models keep training, gradients saturate and higher
compression becomes more viable in later stages of training. Hence,
compression gain increases over training as compressed tensors become more
aligned with the original gradients.
We plot compression gains for the three models when training with fixed CF
10$\mathsf{x}$ and 1000$\mathsf{x}$ respectively, shown in Fig. 4d, 5d and 6d.
In each model, 10$\mathsf{x}$ has higher compression gain than
1000$\mathsf{x}$ since more information is preserved in the smaller CF. _It
should also be apparent that Dense SGD training has a constant gain of 1.0._
For all models, convergence curve of 10$\mathsf{x}$ follows a similar
trajectory as Dense SGD. Correspondigly, the compression gain of
10$\mathsf{x}$ stays close to 1.0 throughout. In ResNet101, gain of
1000$\mathsf{x}$ is low initially and grows in an oscillating manner, although
still lower than gains of 10$\mathsf{x}$ and Dense SGD. The low gains in the
first 1000 iterations of CF 1000$\mathsf{x}$ correlates to the considerable
gap between _BC_ and _AC_ gradients in Fig. 4b and lower accuracy in Fig. 4c.
VGG16 is more robust to higher CFs (Fig. 5c), as also seen from the high
compression gains of CF 1000$\mathsf{x}$ in Fig. 5d. For LSTM, compression
gain for 10$\mathsf{x}$ stays close to 1.0 and between 0.8-0.9 for
1000$\mathsf{x}$. The proximity of the two CFs to Dense SGD’s gain of 1.0 is
equivalent to their perplexity curves in Fig. 6c. From these results we see
how compression gain serves as a viable indicator of the statistical
efficiency of DDP with compression.
### III-C Combining System Throughput and Compression Gain
As described earlier in II-C as well as Fig. 2, choosing a high CF
unintuitively does not necessarily improve training time and may even degrade
final model quality. Thus, to account for both the parallel and statistical
efficiency DDP training _with_ gradient compression, we combine _system
throughput_ ($\text{T}_{system}$) and _compression gain_ into a single metric
called _Compression Throughput_ :
$\text{T}_{compression}=\;\text{T}_{system}\;\times\;\mathsf{Compression\
gain}$
If CF is high, system throughput would be high as well but compression gain
would relatively be lower, decreasing the resulting $\text{T}_{compression}$.
On the other hand, compression gain will be high for a low CF, but system
throughput will be lower due to relatively higher communication overhead.
_With Compression Throughput, we capture this pareto-relationship between the
parallel (system throughput) and statistical efficiency (compression gain) of
gradient compression in DDP._
1 Input: $\theta_{min}$, $\theta_{max}$, $\epsilon$, $\theta_{s}$, $\omega$,
$\mathsf{window}$, compressor $\mathcal{C}$
2 $w_{o}:$ initial model state, N: total nodes, b: per-worker batch-size,
residual = 0; $\text{T}_{sys},\text{T}_{compress}$ = empty()
3 Train for _i = 1,2,3… $\triangleright$ training iterations_
4 $g_{o}^{(i)},t_{o}=\nabla f(x^{(i)},w_{i})$ $\triangleright$ backpropagation
5 $g_{o}^{(i)}=g_{o}^{(i)}+$ residual $\triangleright$ error-feedback
6 $g_{min}^{(i)},t_{min}=\mathcal{C}(g_{o}^{(i)},\theta_{min})$
$\triangleright$ compress to CF $\theta_{min}$
7
$\delta_{min}=\text{EWMA}$($\frac{||g_{min}^{(i)}||^{2}}{||g_{o}^{(i)}||^{2}}$)
$\triangleright$ $\theta_{min}$ compression gain
8 $g_{c}^{(i)},t_{c}^{(i)}=\mathcal{C}(g_{min}^{(i)},\theta_{s})$
$\triangleright$ compress to CF ($\theta_{s}\cdot\theta_{min}$)
9 $\delta_{c}=\text{EWMA}$($\frac{||g_{c}^{(i)}||^{2}}{||g_{o}^{(i)}||^{2}}$)
$\triangleright$ gain for CF ($\theta_{s}\cdot\theta_{min}$)
10 $t_{compress}=t_{min}+t_{c}$ $\triangleright$ total compression time
11 if _$\delta_{c}\geq\epsilon:$_
12 $\tilde{g}^{(i)}$, $t_{s}$ = Aggregate($g_{c}^{(i)}$) $\triangleright$
synchronize $g_{c}^{(i)}$
13 residual = $g_{o}^{(i)}-g_{c}^{(i)}$ $\triangleright$ update residual
14 $t_{iter}$ = $t_{o}$ \+ $t_{compress}$ \+ $t_{s}$ $\triangleright$
iteration time
15 UpdateStep(_$\theta_{s}\cdot\theta_{min},\delta_{c},t_{iter}$_)
16 else if _$\delta_{c} <\epsilon\;\text{and}\;\delta_{min}\geq\epsilon:$_
17 $\tilde{g}^{(i)}$, $t_{s}$ = Aggregate($g_{min}^{(i)}$) $\triangleright$
synchronize $g_{min}^{(i)}$
18 residual = $g_{o}^{(i)}-g_{min}^{(i)}$ $\triangleright$ update residuals
19 $t_{iter}$ = $t_{o}$ \+ $t_{compress}$ \+ $t_{s}$ $\triangleright$
iteration time
20 UpdateStep(_$\theta_{min},\delta_{min},t_{iter}$_)
21 else
22 $\tilde{g}^{(i)}$, $t_{s}$ = Aggregate($g_{o}^{(i)}$) $\triangleright$
synchronize $g_{o}^{(i)}$
23 residual = 0 $\triangleright$ no residual gradients
24 $t_{iter}$ = $t_{o}$ \+ $t_{s}$ $\triangleright$ iteration time
25 UpdateStep(_$1,1,t_{iter}$_)
26 $w_{i+1}=w_{i}-\eta\cdot\tilde{g}^{(i)}$ $\triangleright$ apply SGD update
27 $\theta_{s}$ = CheckGraVAC(_i, $\theta_{s},\delta_{min},\delta_{c}$_)
28procedure _UpdateStep(_$\theta,\delta,t_{iter}$_)_:
29 $\text{T}_{sys}$ = $\text{N}\cdot\text{b}/t_{iter}$ $\triangleright$ system
throughput
30 $\text{T}_{compress}[\theta]=\text{T}_{sys}\cdot\delta$ $\triangleright$
compression throughput
31procedure _CheckGraVAC(_i, $\theta_{s},\delta_{min},\delta_{c}$_)_:
32 if _i % $\mathsf{window}==0:$_
33 $\theta_{s}=\textsf{ScalingPolicy}(\theta_{s})$ $\triangleright$
compression scale-up
34 if _$\omega\geq\frac{|\delta_{min}-\delta_{c}|}{\delta_{min}}:$_
35 $\theta_{min}=\theta_{s}\cdot\theta_{min}$ $\triangleright$ scale-up
minimum CF
36 ct = sort($\text{T}_{compress}.\text{values}()$) $\triangleright$
$\text{T}_{compress}$ vals
37 if _$|\frac{\text{ct}[-1]\;-\;\text{ct}[-2]}{\text{ct}[-2]}|\leq\omega:$_
38 $\theta_{ideal}=\text{T}_{compress}.get(\text{ct}[-2])$ $\triangleright$
ideal CF
39 return $\theta_{ideal}/\theta_{min}$ $\triangleright$ gives optimal
$\theta_{s}$
40
41 else
42 return $\theta_{s}$ $\triangleright$ else use old scaling factor
Algorithm 1 _GraVAC_ ’s Adaptive Compression
We build _GraVAC_ as a modular extension on top of PyTorch’s [33] DDP module
[34] using Python in about 3000 lines of code. A base
$\mathsf{GravacOptimizer}$ wraps common SGD optimizers implemented in PyTorch
by extending the base $\mathsf{torch.optim.Optimizer}$ class. The optimizer
takes an additional $\mathsf{Compressor}$ object that specifies the type of
compression technique used. We implement four pre-existing techniques as
compressor classes in this paper: Top-k, DGC, Redsync and Random-k.
Compression for the appropriate CF and its gain is computed before the
optimizer $\mathsf{step}$ function which applies the aggregated gradient
updates on model parameters.
_GraVAC Algorithm:_ Alg. 1 describes _GraVAC_ ’s approach of using compressor
$\mathcal{C}$ to scale CFs in the exploration space
[$\theta_{min},\theta_{max}$], where each candidate CF is evaluated for
$\mathsf{window}$ steps and incremented in step-size of $\theta_{s}$ w.r.t.
$\theta_{min}$. For e.g., scaling from CF 10$\mathsf{x}$ to CF 20$\mathsf{x}$
means $\theta_{s}=20/10=2\mathsf{x}$. The threshold $\epsilon$ denotes the
minimum compression gain required for any CF to be eligible for communication
in _GraVAC_ , while threshold $\omega$ is used to measure saturation in
compression throughputs and for scaling up $\theta_{min}$. We explain this in
the following sections in more detail. For every iteration, we compute
gradients $g_{o}^{(i)}$ with model parameters $w_{i}$ on training sample
$x^{(i)}$ in time $t_{o}$ (line 4). To incorporate error-feedback,residual
holds the leftover gradients not communicated from previous iterations. The
shape and memory size of tensors in residual is the same as gradients itself.
As shown in line 5, we add residual gradients to the gradients computed in the
current iteration. In the first stage, we compress original gradients using
$\mathcal{C}$ to compressed gradients $g_{min}^{(i)}$ corresponding to minimum
CF $\theta_{min}$ (line 6). We then compute the compression gain corresponding
to $\theta_{min}$ (line 7), and smoothen out the inter-iteration gain through
exponential weighted moving average (EWMA) smoothing. In our evaluation, we
set the EWMA smoothing factor to $N$/100, where $N$ is the number of
participating workers. We evaluate the next candidate CF by stepping up the
previous $\theta_{min}$ and further compressing the already compressed
gradients $g_{min}^{(i)}$ by stepsize $\theta_{s}$ (line 8). Thus, candidate
CF evaluated in this case is $\theta_{s}\cdot\theta_{min}$. This is done as
part of our multi-level compression strategy to avoid compressing the large,
original tensors $g_{o}^{(i)}$ twice. We measure the time savings of our
multi-level approach in section IV-C.
Next, we compute the gradients and compression gain of candidate CF
$\theta_{s}\cdot\theta_{min}$ (line 8-9), and denote the total compression
time $t_{compress}$ as the sum of time to compress original gradients to
$g_{min}^{(i)}$ (line 6) and the time to further compress $g_{min}^{(i)}$ to
$g_{c}^{(i)}$ (line 8). Based on the compression gains obtained and threshold
$\epsilon$, we choose the appropriate gradients to call the collective
operation on. If the gain of our candidate CF meets $\epsilon$ (line 11), we
go ahead and communicate compressed gradients $g_{c}^{(i)}$ among workers. We
update the residual gradients in accord with $g_{c}^{(i)}$ as well (line 13),
calculate the total iteration time (line 14) and update the system as well as
compression throughput for CF $\theta_{s}\cdot\theta_{min}$ via
$\mathsf{UpdateStep}$ function. $\text{T}_{compress}$ is a dictionary or a
hashmap that stores compression throughput of each candidate CF, min-max CF as
well as dense SGD setting (i.e., CF 1$\mathsf{x}$).
If the gain of $g_{c}^{(i)}$ does not meet the threshold, but gain
$\delta_{min}$ of $\theta_{min}$ does (line 16), we instead synchronize
compressed gradients $g_{min}^{(i)}$ corresponding to $\theta_{min}$. In a
similar fashion as before, we update the residuals, this time with
$g_{min}^{(i)}$ instead of $g_{c}^{(i)}$ (line 18), compute iteration time and
assess compression throughput. _It is important to remember that
synchronization overhead to communicate $g_{min}^{(i)}$ is more than
$g_{c}^{(i)}$ due to the former’s lower CF. The trade-off we make in GraVAC is
to incur higher communication latency for more accurate representation of the
original gradients (measured by compression gain) and vice-versa._
If both $\theta_{min}$ and currently evaluated CF do not meet the set
threshold, we incur maximum communication latency by transmitting the original
gradients via dense SGD (line 22). In this case, residual gradients are set to
0 and no compression overhead is included as part of iteration time and
computing system/compression throughput. The CF and compression gain are both
1, as set in the $\mathsf{UpdateStep}$ function at line 25.
Following SGD update (line 26), we evaluate _GraVAC_ to assess the performance
of CFs evaluated so far. This happens at a frequency determined by
$\mathsf{window}$. Here, we adjust $\theta_{s}$ by a certain factor to scale
up compression, determined by the chosen ScalingPolicy. The scaling policy
tunes compression only until the upper bound $\theta_{max}$. We explore two
scaling policies in this paper that we describe in detail under section IV-B.
After scaling $\theta_{s}$, we also assess if the minimum CF, i.e.,
$\theta_{min}$ can be scaled up as well. The intuition is that as training
progresses, model gradually starts converging as well and we can use higher
compression even for the minimum CF later on. In addition to candidate CFs, we
thus scale up the minimum CF as well. The transition is made if the current
gain $\delta_{c}$ is within $\omega$% of the gain of previous $\theta_{min}$
(line 34). Once enough CFs are evaluated, we look at the two largest
compression throughputs (line 36) and fetch the corresponding CF if they are
within the bounds of $\omega$. We do this as it means the compression
throughput has saturated and thus, we pick the lower CF as $\theta_{ideal}$
(line 38) and send the appropriate step-size (line 39). If the threshold
$\omega$ is not met, we use $\theta_{s}$ as is.
_When does compression scale-up?_ As seen from Alg. 1, the compression scale-
up happens during _GraVAC_ ’s evaluation phase where we scale the step-size
$\theta_{s}$ in accordance with a specific scaling policy. At the same time,
we escalate the minimum CF $\theta_{min}$ to currently evaluated CF if the two
compression gains are within $\omega$% of each other.
_When does compression scale-down?_ Compression scale-down is determined by
$\epsilon$ (shown via conditional statements lines 11-25). If current CF loses
considerably more information in compressed gradients $g_{c}^{(i)}$, we use
the lower CF $\theta_{min}$. If the latter fails to meet $\epsilon$ as well,
we send uncompressed gradients $g_{o}^{(i)}$ as a last resort.
## IV Evaluation
### IV-A Cluster Setup and Training Hyperparameters
We evaluate _GraVAC_ on a 32 GPU setup on the Google Cloud Platform (GCP)
across 8 VMs. Each VM is a $\mathsf{n1}$-$\mathsf{standard}$-$\mathsf{8}$
machine type with 8 vCPUs, 30 GB system memory and 4 NVIDIA V100 GPUs with 16
GB VRAM each. The machines are configured with PyTorch 1.10.1, CUDA 11.3, CUDA
driver 465.19.01 and NCCL 2.10.3.
We evaluate the three models described in Table 1. ResNet101 is trained with
per-worker batch size 32, momentum 0.9, weight decay 0.0001 and SGD optimizer
with initial learning rate (lr) 0.1 decayed by a factor of 10 at 9K and 14K
iterations respectively. VGG16 is also trained with per-worker batch-size 32,
weight decay 0.0005, momentum 0.9 and SGD with fixed lr 0.1. Lastly, LSTM is
measured with test perplexity (i.e., exponential of test loss) with per-worker
batch-size 20, momentum 0.9, weight decay 0.0001 and SGD with fixed lr 0.1.
The model is initialized with 1500 embedding dimensions and 2 hidden layers
with 35 bptt steps.
We evaluate _GraVAC_ with different scaling policies and look at their
convergence curves (i.e. test accuracy/perplexity vs. iterations), average
compression throughput of candidate CFs and kernel density estimates (KDE) of
training iterations using different CFs over the course of training. KDE gives
the distribution over the iterations for all CFs and plotted on the log-scale
with smoothing bandwidth of $0.1$ passed to the gaussian KDE.
### IV-B _GraVAC_ ’s Adaptive Compression Policies
In this section, we look at how _GraVAC_ achieves optimal CF for a given
$\theta_{min}$, $\theta_{max}$, $\epsilon$, $\mathsf{window}$, $\omega$ and
$\mathsf{stepsize}$. To see how a model converges and communication costs vary
by evaluating different candidate CFs in the search space, we employ an
_Exponential_ policy that upscales CFs aggressively, and a relatively smoother
_Geometric_ scaling policy that scales CFs as a geometric progression.
#### IV-B1 Exponential scaling policy
In this policy, we implement the ScalingPolicy function from Alg. 1 such that
CFs are scaled up in exponents of 2 w.r.t the first initialized
$\theta_{min}$. On top of DGC, we set $\theta_{min}$ and $\theta_{max}$ to
10$\mathsf{x}$ and 1000$\mathsf{x}$, $\mathsf{window}$=500 and $\omega$=1%. So
we scale up by factors of $2^{1}$, $2^{2}$, $2^{4}$, $2^{8}$ w.r.t
10$\mathsf{x}$ up until 1000$\mathsf{x}$. The candidate CFs thus evaluated in
this policy are 10$\mathsf{x}$, 20$\mathsf{x}$, 40$\mathsf{x}$,
160$\mathsf{x}$ and 1000$\mathsf{x}$. We run _GraVAC_ on two configuration
with different thresholds on compression gain, $\epsilon$ = 0.7 and 0.9. The
lower $\epsilon$ relaxes the constraint on the gain for higher CFs to be
eligible for communication, thus achieving higher compression. A large
$\epsilon$ (i.e., close to 1) allows for compression only if the compressed
tensors are highly representative of the original gradients. First, we compare
these two thresholds with Dense SGD as the latter demonstrates the ideal
convergence scenario. Then, we compare _GraVAC_ with different compression
techniques on static CFs and look at final model accuracy, communication
savings and overall speedup.
_ResNet101:_ Fig. 7 shows how _GraVAC_ achieves the same convergence as dense
SGD in the same number of iterations. The low and high $\epsilon$ reduce
overall communication volume by 163$\times$ and 19$\times$ over dense SGD. _We
measure communication volume as the ratio of cumulative single-precision
floats exchanged among workers in GraVAC relative to dense SGD._ Training
cycle is slightly more volatile with compression, as seen from the accuracy
drop due to lr decay at around 9000-th iteration. The drop is more apparent
for $\epsilon$ = 0.7 as we continue to train with higher CFs on account of the
lower threshold. Comparatively, $\epsilon$ = 0.9 is more robust to
hyperparameter tuning like lr decay as we tend to train with a lower CF due to
higher threshold. This is corroborated from Fig. 7b which shows distribution
of training iterations over the CFs. We equally train with 10$\mathsf{x}$ and
1000$\mathsf{x}$ for $\epsilon$ = 0.9, while we mostly train with
1000$\mathsf{x}$ for $\epsilon$ of 0.7. For the compression throughputs of
$\epsilon$ = 0.9 in Fig. 7c, it might seem counterintuitive at first that
although $T_{compression}$ is maximum for 1000$\mathsf{x}$ and minimum for
10$\mathsf{x}$, we still evenly train with the two CFs. This is on account of
the high threshold and because $\theta_{min}$ did not scale up and remained at
10$\mathsf{x}$ for ResNet101. Thus, whenever the compression gain of any
candidate CF did not meet the threshold, we synchronized gradients compressed
at 10$\mathsf{x}$. For $\epsilon$ of 0.7, compression throughput was maximum
for 1000$\mathsf{x}$ and we trained at this CF for most iterations as the
corresponding gain easily met that threshold.
(a) Test accuracy
(b) Iteration density
(c) $T_{compression}$
Figure 7: ResNet101:_GraVAC_ with $\epsilon$ = [0.7, 0.9] and Dense SGD.
_VGG16:_ Like ResNet101, VGG16 also converges to the same accuracy as dense
SGD within the same iterations, where $\epsilon$ = 0.7 and 0.9 reduce
communication volume by 80$\times$ and 13.5$\times$ over dense SGD (Fig. 9).
Although $T_{compression}$ is maximum at 1000$\mathsf{x}$ for $\epsilon$ =
0.9, the corresponding gain was _not_ as high to meet the threshold. Because
of this, we switch back to $\theta_{min}$ and thus train with 10$\mathsf{x}$
for majority iterations as seen from the kernel density estimates in Fig. 8b.
However, when $\epsilon$ was lower, we were able to find 40$\mathsf{x}$ CF to
meet that threshold. $T_{compression}$ corresponding to this CF was second
largest in our exploration space. As candidate CFs are evaluated over the
iterations, the model gradually converges and as a result, compression gain
improves even further on larger CFs as training progresses. Ultimately, we
arrive on $\theta_{ideal}$ = 1000$\mathsf{x}$ corresponding to the maximum
compression throughput (Fig. 8c).
(a) Test accuracy
(b) Iteration density
(c) $T_{compression}$
Figure 8: VGG16: _GraVAC_ with $\epsilon$ = [0.7, 0.9] and Dense SGD.
_LSTM:_ Like the models before, _GraVAC_ with either $\epsilon$ converged in
the same iterations as dense SGD training, while reducing the communication
volume by 279$\times$ and 289$\times$ for $\epsilon$ of 0.9 and 0.7
respectively. Given the dataset, model and training hyperparameters, we
already saw from Fig. 6d that compression gain for LSTM was high for both
10$\mathsf{x}$ and 1000$\mathsf{x}$. We observed a similar trend here as
compression gain corresponding to 1000$\mathsf{x}$ easily satisfied both
thresholds and thus, we train with the largest available CF for most
iterations (Fig. 9b). Correspondingly, the compression throughput is maximum
at this CF as well.
(a) Test accuracy
(b) Iteration density
(c) $T_{compression}$
Figure 9: LSTM: _GraVAC_ with $\epsilon$ = [0.7, 0.9] and Dense SGD. TABLE 2: _GraVAC_ ’s model quality and speedup over static CFs Model | Compression | Acc./Ppl | Diff. | Speedup
---|---|---|---|---
ResNet101 | Top-k 10$\mathsf{x}$ | 80.14% | +0.14% | 1$\times$
Top-k 1000$\mathsf{x}$ | 76.4% | $-$3.6% | 3.02$\times$
DGC 10$\mathsf{x}$ | 80.4% | +0.4% | 1.23$\times$
DGC 1000$\mathsf{x}$ | 78.6% | $-$1.4% | 5.19$\times$
Redsync 10$\mathsf{x}$ | 79.4% | $-$0.6% | 1.2$\times$
Redsync 1000$\mathsf{x}$ | 77.4% | $-$2.6% | 6.94$\times$
Random-k 10$\mathsf{x}$ | - | - | -
Random-k 1000$\mathsf{x}$ | - | - | -
_GraVAC_ | 80.2% | +0.2% | 4.32$\times$
VGG16 | Top-k 10$\mathsf{x}$ | 91.2% | +1.2% | 1$\times$
Top-k 1000$\mathsf{x}$ | 90.68% | +0.68% | 3.22$\times$
DGC 10$\mathsf{x}$ | 90.8% | +0.8% | 0.935$\times$
DGC 1000$\mathsf{x}$ | 90.4% | +0.4% | 3.35$\times$
Redsync 10$\mathsf{x}$ | 90.45% | +0.45% | 0.99$\times$
Redsync 1000$\mathsf{x}$ | 90.3% | +0.3% | 3.6$\times$
Random-k 10$\mathsf{x}$ | 87.8% | $-$2.2% | 0.7$\times$
Random-k 1000$\mathsf{x}$ | - | - | -
_GraVAC_ | 90.48% | +0.48% | 1.95$\times$
LSTM | Top-k 10$\mathsf{x}$ | 22.0 | +0.0 | 1$\times$
Top-k 1000$\mathsf{x}$ | 26.78 | $-$4.78 | 3.36$\times$
DGC 10$\mathsf{x}$ | 21.67 | +0.33 | 1.23$\times$
DGC 1000$\mathsf{x}$ | 25.14 | $-$3.14 | 6.25$\times$
Redsync 10$\mathsf{x}$ | 21.65 | +0.35 | 1.17$\times$
Redsync 1000$\mathsf{x}$ | 24.24 | $-$2.24 | 6.9$\times$
Random-k 10$\mathsf{x}$ | 24.15 | $-$2.15 | 1.3$\times$
Random-k 1000$\mathsf{x}$ | - | - | -
_GraVAC_ | 21.25 | +0.75 | 6.67$\times$
Further, we compare _GraVAC_ with static CFs running on different compression
techniques. In particular, we train our models with Top-k, DGC, Redsync and
Random-k at CFs 10$\mathsf{x}$ and 1000$\mathsf{x}$. We run each compression
technique to report the final accuracy/perplexity until it does not improve
any further, difference in convergence compared to dense SGD baseline from
Table 1, and relative training speedup over Top-k 10$\mathsf{x}$ for each
model. The results are tabulated in Table 2. We do not consider dense SGD
training in this comparison since we already established previously how
_GraVAC_ is able to achieve the same convergence in the same iterations, and
other compression techniques have already been compared to dense SGD in prior
works. For ResNet101, 1000$\mathsf{x}$ CF on Redsync, DGC and Top-k have
considerably high speedups than 10$\mathsf{x}$ Top-k. However, these methods
at 1000$\mathsf{x}$ CF achieve considerably less accuracy than Top-k at
10$\mathsf{x}$. At 1000$\mathsf{x}$, Top-k, DGC and Redsync do not improve
beyond 76.4%, 78.6% and 77.4% top-1 test accuracy. Random-k faild to converge
at either CF and accuracy did not improve beyond 20% . Because of _GraVAC_ ’s
adaptive scheme, we converge to 80.2% accuracy while still reducing training
time by 4.32$\times$.
For VGG16, we previously observed that the model is already quite robust to
high compression (Fig. 5). We see that again here for Top-k, DGC and Redsync
at 1000$\mathsf{x}$ cross 90% accuracy with 3.22, 3.35 and 3.6$\times$ speedup
over Top-k 10$\mathsf{x}$. Random-k at 10$\mathsf{x}$ also converged, albeit
to a lower 87.8% accuracy and slower convergence. Since _GraVAC_ attains
90.48% test accuracy with 1.95$\times$ training speedup, other compression
schemes were more optimal in this case simply because they used high CFs.
In LSTM, _GraVAC_ obtains the least perplexity of 21.25 while still providing
maximum speedup of 6.67$\times$ over Top-k 10$\mathsf{x}$. Random-k
10$\mathsf{x}$ converged to 24.15 perplexity and did not improve further,
while Random-k 1000$\mathsf{x}$ failed here again. Of all the configurations,
only Top-k, DGC and Redsync at 10$\mathsf{x}$ CF and _GraVAC_ achieved better
perplexity than dense SGD.
Thus, we see how _GraVAC_ is able to train models like ResNet101 and LSTM to
high accuracy/perplexity and still reduce training time significantly. Static
compression schemes achieve high accuracy at low CF at the cost of high
communication overhead, thus providing lower speedup. Large CFs considerably
reduce communication, but the final model quality is not at par with _GraVAC_.
On the flip side, some over-parameterized models like VGG16 can be robust to
compression and still converge successfully at high static CFs.
#### IV-B2 Geometric scaling policy
We also propose a relatively smoother compression policy where ScalingPolicy
increments CFs as a geometric progression with common ratio 2. We deploy
_GraVAC_ with Redsync on ResNet101 and set $\theta_{min}$ = 10$\mathsf{x}$,
$\theta_{max}$ = 2000$\mathsf{x}$, $\epsilon$ = 0.7, $\mathsf{window}$ = 2000
steps and $\omega$ = 1%. Thus, candidate CFs are 10$\mathsf{x}$,
20$\mathsf{x}$, 40$\mathsf{x}$, 80$\mathsf{x}$, 160$\mathsf{x}$,
320$\mathsf{x}$, 640$\mathsf{x}$, 1280$\mathsf{x}$ and 2000$\mathsf{x}$. Fig.
10a shows the accuracy curve over the iterations. Compared to dense SGD (Fig.
7a), _GraVAC_ with geometric scaling converged _while reducing communication
volume by 76 $\times$_. In contrast to exponential scaling, convergence is
relatively slower because we evaluate each candidate CF for a larger
$\mathsf{window}$ size. As a result, gradients get even smaller as _GraVAC_
gradually arrives at larger CFs and compression gain increases beyond
$\epsilon$. Thus, we see similar iteration densities from CF 10$\mathsf{x}$ to
640$\mathsf{x}$ (Fig. 10b). After the first 7 CFs are evaluated over 2000
steps each, we mostly train with CF 1280$\mathsf{x}$ from 16K iterations
onward (because 8 $\times$ 2000 = 16000). We did not scale to 2000$\mathsf{x}$
in our evaluation since compression throughput for 1280$\mathsf{x}$ and
2000$\mathsf{x}$ was 1029.9 and 1035.4, which falls within $\omega$’s bound of
1%. _This case highlights the effectiveness of GraVAC such that it does not
scale the CF beyond a point when it stop improving the parallel or statistical
efficiency of gradient compression_. In this case, _GraVAC_ does not compress
beyond 1280$\mathsf{x}$ as it corresponds to the maximum compression
throughput (and at a lower CF of 1280$\mathsf{x}$ compared to
2000$\mathsf{x}$).
(a) Convergence
(b) $T_{compression}$ and KDE
Figure 10: ResNet101: _GraVAC_ with Geometric scaling policy.
### IV-C Gains of Multi-level Compression in _GraVAC_
Alg. 1 explains how at each iteration, _GraVAC_ scales compression from
initial $\theta_{min}$ to current CF being evaluated (i.e., $\theta_{c}$), up
to the maximum allowed $\theta_{max}$. Thus, compressing the original
gradients (computed over backward pass) twice; i.e., once over $\theta_{min}$
and then again on $\theta_{c}$ can incur significant overhead, especially on
larger models. The latency of a compressor may vary with the size of the
tensor to compress as well as the target CF. To reduce the cumulative overhead
of compressing original tensors multiple times, we apply a multi-level
compression scheme as follows: given a compressor $\mathcal{C}$ and tensor
$\mathcal{X}$ to be compressed to CFs $\theta_{1}$ and $\theta_{2}$ such that
$\theta_{2}>\theta_{1}$, rather than compressing each CF on $\mathcal{X}$ as:
$\mathcal{X}_{1}=\mathcal{C}(\theta_{1},\mathcal{X})\;\text{and}\;\mathcal{X}_{2}=\mathcal{C}(\theta_{2},\mathcal{X})$
to produce compressed tensors where
$|\mathcal{X}_{2}|<|\mathcal{X}_{1}|<|\mathcal{X}|$. In _GraVAC_ , we first
compute $\mathcal{X}_{1}$ and then compress this tensor to
$\theta_{2}^{{}^{\prime}}$ to produce $\mathcal{X}_{2}^{{}^{\prime}}$:
$\mathcal{X}_{1}=\mathcal{C}(\theta_{1},\mathcal{X})\;\Longrightarrow\mathcal{X}_{2}^{{}^{\prime}}=\mathcal{C}(\theta_{2}^{{}^{\prime}},\mathcal{X}_{1})\;:\;\theta_{2}^{{}^{\prime}}=\frac{\theta_{2}}{\theta_{1}}$
The resulting tensor $\mathcal{X}_{2}^{{}^{\prime}}$ is such that
$\mathcal{X}_{2}^{{}^{\prime}}=\mathcal{X}_{2}$ for
$\theta_{2}^{{}^{\prime}}=\theta_{2}/\theta_{1}$. The appeal of doing so is
that the second compression operation is applied on a smaller tensor
$\mathcal{X}_{1}$ instead of $\mathcal{X}$ again. We tabulate the savings of
multi-level compression in Table 3. Let’s consider a scaling case of _GraVAC_
where $\theta_{min}=10\mathsf{x}$ and current CF evaluated is
1000$\mathsf{x}$. Then multilevel _GraVAC_ first compresses to 10$\mathsf{x}$
and then further compresses the reduced tensors to 100$\mathsf{x}$, i.e.,
$\theta_{1}=10\mathsf{x}$ and $\theta_{2}^{{}^{\prime}}=100\mathsf{x}$ so that
$\theta_{2}=1000\mathsf{x}$. In direct approach, we first compress original
gradients to 10$\mathsf{x}$, then compress the original gradients again to
1000$\mathsf{x}$. From our results, we see that multi-level compression is at
least 1.1$\times$ and up to 1.83$\times$ faster than directly compressing the
original tensors twice.
TABLE 3: _GraVAC_ ’s mulit-level (MTL) compression speedup Model | Method | Direct (ms) | MTL (ms) | Speedup
---|---|---|---|---
ResNet101 | Top-k | 606 | 332 | 1.83$\times$
DGC | 90 | 59 | 1.52$\times$
Redsync | 33 | 29.8 | 1.1$\times$
Random-k | 23 | 14 | 1.64$\times$
VGG16 | Top-k | 181 | 121 | 1.49$\times$
DGC | 122 | 95.5 | 1.27$\times$
Redsync | 101.4 | 87.7 | 1.16$\times$
Random-k | 41.6 | 31 | 1.34$\times$
LSTM | Top-k | 200 | 126 | 1.59$\times$
DGC | 88 | 63 | 1.4$\times$
Redsync | 69.4 | 46.4 | 1.5$\times$
Random-k | 56.4 | 37.4 | 1.5$\times$
### IV-D Comparing _GraVAC_ with Prior Art
In this section, we compare _GraVAC_ with another adaptive scheme called
Accordion [31]. For the three models, we use bounds of Rank-1 and Rank-4 for
compression in Accordion, as described in [31] and compare with _GraVAC_ in
terms of communication and time savings (i.e., training speedup) to achieve
the same test accuracy/perplexity. The savings are normalized by Accordion’s
performance for each respective model, shown in Table 4. For ResNet101,
_GraVAC_ reduces total communication volume by 44.5$\times$ and reduces
training time by 1.94$\times$ over Accordion. _GraVAC_ speeds up training by
5.63$\times$ over Accordion for communication-heavy models like VGG16. In LSTM
training, _GraVAC_ converges twice as fast by reducing communication volume up
to 104.2$\times$.
Accordion is based on detecting critical regions during training, i.e., when
inter-iteration gradients computed in backward pass change significantly and
cross a certain user-defined threshold. Accordion switches between 2
compression factors such that it uses the low CF in critical regions and the
higher CF otherwise. On the other hand, _GraVAC_ looks at information loss on
account of compression (i.e., statistical efficiency) and not just relative
gradient change in sensitive regions of training. That is, _GraVAC_ looks at
intra-iterations gradients as well (between original and gradients compressed
at different CFs). Additionally, _GraVAC_ scales compression across a wider
range and carefully inspects intermediary CFs as potential compression
candidates. Thus, we obtain higher speedups when training with _GraVAC_.
TABLE 4: _GraVAC_ vs. Accordion: Communication and Time savings Model | Method | Floats sent | Comm. sav. | Time sav.
---|---|---|---|---
ResNet101 | Accordion | 4.17 $\times 10^{11}$ | 1$\times$ | 1$\times$
_GraVAC_ | $\mathbf{9.38\times 10^{9}}$ | $\mathbf{44.5\times}$ | $\mathbf{1.94\times}$
VGG16 | Accordion | 3.83 $\times 10^{11}$ | 1$\times$ | 1$\times$
_GraVAC_ | $\mathbf{1.7\times 10^{10}}$ | $\mathbf{22.4\times}$ | $\mathbf{5.63\times}$
LSTM | Accordion | 4.2 $\times 10^{11}$ | 1$\times$ | 1$\times$
_GraVAC_ | $\mathbf{4\times 10^{9}}$ | $\mathbf{104.2\times}$ | $\mathbf{2.06\times}$
#### IV-D1 _GraVAC_ vs. Accordion on Random-k Compression
We previously saw in Fig. 2b and Table 2 that ResNet101 failed to converge at
any CF with Random-k compression. In this section, we present a special case
of using Random-k under the hood with both _GraVAC_ and Accordion. Although
the compression quality of Random-k is lower compared to other compressors, we
present this as a special case to demonstrate how _GraVAC_ is more dynamic and
operates at a finer granularity. We launch _GraVAC_ with Random-k on
$\theta_{min}$ = 1.5$\mathsf{x}$, $\theta_{max}$ = 1000$\mathsf{x}$,
$\mathsf{window}$ = 2000 and $\epsilon$ = 0.7. The CFs are scaled up via
_geometric scaling policy_. Accordion was also deployed with the same min-max
bounds on CF as _GraVAC_ , i.e., low CF = 1.5$\mathsf{x}$ and high CF =
1000$\mathsf{x}$. The convergence curves comparing _GraVAC_ and Accordion are
shown in Fig. 11a. Unlike static 10$\mathsf{x}$ Random-k compression (Fig. 2b)
that failed to converge, we were able to achieve to 78% top-1 test accuracy
for ResNet101 with _GraVAC_. The CFs used for training by _GraVAC_ were
1.5$\mathsf{x}$, 3$\mathsf{x}$, 6$\mathsf{x}$, 12$\mathsf{x}$, 24$\mathsf{x}$
and 48$\mathsf{x}$. All candidate CFs beyond this were ignored as they did not
meet the required threshold of $\epsilon$. CF 12$\mathsf{x}$ has the highest
density, implying most iterations used this CF for training (Fig. 11b).
Correspondingly, compression throughput is maximum for this CF as well.
Compared to dense SGD, we reduced overall communication volume by 18$\times$.
As for Accordion on Random-k, we see in Fig. 11a that training saturates at
20% accuracy. This is because Accordion does _not_ consider the efficacy of
the compression technique itself, and only switches between a low and high CF
if the uncompressed, inter-iteration gradients change beyond a certain
measure. With a low CF 1.5$\mathsf{x}$, information loss in Random-k was too
high to update ResNet101 in a meaningful way.
(a) Model convergence
(b) _GraVAC_ $T_{comp.}$ and KDE
Figure 11: _GraVAC_ and Accordion on Random-k compression.
## V Conclusion
Gradient noise has previously been used as a scalability indicator for batch
and cluster-size scaling in deep learning [39, 20, 19, 37, 38]. Adaptive
compression schemes like Accordion [31] switch between two compression levels
based on when the inter-iteration gradients change by some margin. _GraVAC_ ’s
key insight is to tweak compression factor over the course of training while
balancing the pareto-relationship between parallel and statistical efficiency
in gradient compression. We use “compression gain“ to measure information loss
on account of compression and choose a CF appropriately. In our evaluation, we
see that _GraVAC_ converges 1.95 to 6.67$\times$ faster than choosing a static
CF, while converging in the same number of iterations as dense SGD. Compared
to Accordion, we observed up to 5.63$\times$ reduction in end-to-end training
time.
One should be mindful when training models with _GraVAC_ as it introduces
parameters like compression threshold ($\epsilon$) and $\mathsf{window}$ size
that may affect overall training performance. Setting too small a
$\mathsf{window}$ size may result in poor convergence as all the candidate CFs
may be exhausted while the model is still in early training stages and
gradients are still volatile. As for $\epsilon$, choosing a very small
threshold may enable high compression but may lead to model degradation by
allowing high CF gradients from the beginning that will not update the model
in a significant way.
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# Complete separation of variables in the geodesic Hamilton–Jacobi equation
M. O. Katanaev
Steklov Mathematical Institute,
119991, Moscow, ul. Gubkina, 8 E-mail<EMAIL_ADDRESS>
###### Abstract
We consider a (pseudo)Riemannian manifold of arbitrary dimension. The
Hamilton–Jacobi equation for geodesic Hamiltonian admits complete separation
of variables for some (separable) metrics in some (separable) coordinate
systems. Separable metrics are very important in mathematics and physics. The
Stäckel problem is: “Which metrics admit complete separation of variables in
the geodesic Hamilton–Jacobi equation?” This problem was solved for inverse
metrics with nonzero diagonal elements, in particular, for positive definite
Riemannian metrics, long ago. However the question is open for indefinite
inverse metrics having zeroes on diagonals. We propose the solution. Separable
metrics are divided into equivalence classes characterised by the number of
Killing vector fields, quadratic indecomposable conservation laws for
geodesics, and the number of coisotropic coordinates. The paper contains
detailed proofs, sometimes new, of previous results as well as new cases. As
an example, we list all canonical separable metrics in each equivalence class
in two, three, and four dimensions. Thus the Stäckel problem is completely
solved for metrics of any signature in any number of dimensions.
## 1 Introduction
Many important and interesting problems in mathematical physics are related to
analysis of geodesics on a (pseudo)Riemannian manifold. In its turn,
integration of geodesic equations often reduces to solution of the
corresponding Hamilton–Jacobi equation. Models admitting complete separation
of variables in the Hamilton–Jacobi equations are of particular interest
because, in this case, there are $n$ independent conservation laws in
envolution for $n$ dimensional Hamiltonian system, and geodesic equations are
integrated in quadratures. Therefore finding metrics which admit complete
separation of variables is of great importance. This problem is interesting
both for Riemannian (positive definite) metrics and metrics of Lorentz
signature, the latter case being important in gravity models.
In 1891, Stäckel raised the question: “Which metrics admit complete separation
of variables in the respective Hamilton–Jacobi equation?” [1, 2, 3, 4], and
gave the answer in the case of diagonal (orthogonal) metrics in the presence
of only quadratic indecomposable conservation laws. These metrics are called
Stäckel or separable. The problem attracted much interest of mathematicians
and physicists and became classical. It is clear that if the metric has enough
symmetry then it may admit $n$ envolutive conservation laws. At the same time
some Stäckel metrics admit complete separation of variables even without any
symmetry.
Many interesting and important results for orthogonal separable metrics were
obtained in papers [5, 6, 7, 8], but we focus our attention on nondiagonal
metrics.
The separating action functions were found for nondiagonal metrics of
arbitrary signature under the assumption that all diagonal elements differ
from zero in [9, 10, 11]. This is always true for Riemannian positive definite
metrics. The corresponding separable metrics were derived in [12, 13]. These
results were culminated in [14] (see also [15]) were necessary and sufficient
conditions for complete separation of variables were found for more general
Hamiltonians depending on time and containing the term linear in momenta and
potential, again under the assumption that all diagonal elements of separable
metrics differ from zero.
A different technique was used in [16, 17] (see also [18]) for obtaining
separable metrics including the case when the diagonal inverse metric
components include zeroes (the corresponding coordinate lines were called
there “essential coordinates of type I” and we call them “coisotropic
coordinates” for brevity). However the full list of separating action
functions and conservation laws in Hamiltonian formulation was not derived.
This general problem was also attacked in [19, 20] and many important examples
especially with coisotropic coordinates were considered in [21, 22]. In the
present paper, we solve this problem using another technique which allows us
not only to derive separable metrics but, in addition, the full set of
separating action functions and conservation laws in the Hamiltonian
formulation.
The coordinate free formulation of separability criteria in general case was
proposed in [23].
It turns out that Stäckel metrics and complete multiplicative separation of
variables in the respective Laplace–Beltrami, Helmholtz, and Schrödinger
equations are closely related. Namely, complete multiplicative separation of
variables in the latter equations provides sufficient conditions for complete
additive separation of variables in the Hamilton–Jacobi equation [12, 13].
This observation increases the importance of separable metrics.
Separable metrics of Lorentzian signature are of great importance in gravity
models. Usually, one assumes existence of a large symmetry group for metric to
reduce the number of independent components which allows to obtain exact
solutions of Einstein’s equations in many cases. As a rule, such solutions
admit complete separation of variables in the geodesic Hamilton–Jacobi
equation. This approach can be inverted. One may assume complete separability
of metric which also reduces significantly the number of independent metric
components yielding a hope of obtaining exact solutions of gravity equations.
Such solutions are very attractive because the respective geodesic equations
are Liouville integrable, and this helps us to understand global structure of
respective space-times. This idea was successfully implemented in [24] for
separable metrics admitting two commutative Killing vector fields and two
indecomposable quadratic conservation laws. A large class of exact solutions
was described in this way including Schwarzschild, Reissner–Nordström, Kerr,
and many others.
It turns out that all nonzero components of separable metrics are given by
fixed functions of the set of arbitrary functions of single coordinates. This
means that vacuum Einstein’s equations reduce to a system of nonlinear
ordinary differential equations. This feature enhances the hope of obtaining
exact solutions.
Complete separation of variables for the geodesic Hamilton–Jacobi equation
occurs in the presence of Killing vectors and Killing second rank tensors. The
importance of second rank Killing tensors for linear Hamiltonian systems is
discussed in [25]. Some topological properties of complete separation of
variables for Killing tensors are discussed in [26].
In the present paper, we propose complete solution of the Stäckel problem for
metrics of any signature in any dimensions including cases of zero diagonal
inverse metric components. Our proofs in many cases differ from others. At the
end of the paper, we give complete lists of canonical separable metrics in
two, three, and four dimensions.
## 2 Separation of variables
We consider $n$-dimensional topologically trivial manifold
${\mathbb{M}}\approx{\mathbb{R}}^{n}$ covered by a global coordinate system
$x^{\alpha}$, $\alpha=1,\dotsc,n$. Let there be a geodesic Hamiltonian
(function on the phase space $(x,p)\in{\mathbb{T}}^{*}({\mathbb{M}})$)
$H_{0}(x,p):=\frac{1}{2}g^{\alpha\beta}(x)p_{\alpha}p_{\beta},$ (1)
where $g^{\alpha\beta}(x)$ is the inverse metric on ${\mathbb{M}}$ and
$p_{\alpha}$ are momenta. It is well known that it yields Hamiltonian
equations for geodesics on a (pseudo)Riemannian manifold $({\mathbb{M}},g)$.
All functions are assumed to be sufficiently smooth, and we shall not mention
this in what follows. Moreover we often say simply “metric” instead of
“inverse metric”.
If metric is positive definite then Hamiltonian (1) describes motion of a
point particle on Riemannian manifold $({\mathbb{M}},g)$. For Lorentzian
signature metric, Hamiltonian (1) describes worldlines of point particles on
space-time $({\mathbb{M}},g)$. Both cases are of considerable interest from
mathematical and physical points of view, and many properties of such
mechanical systems do not depend on the signature of the metric. We shall
consider metrics of arbitrary signature, transition to positive definite
metric being usually trivial.
The Hamilton–Jacobi equation for the truncated action function (characteristic
Hamilton function) $W(x)$ is
$H_{0}\left(x,\frac{\partial W}{\partial
x^{\alpha}}\right)=\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}W\partial_{\beta}W=E,\qquad
E={\sf\,const}.$ (2)
###### Definition.
A solution of Hamilton–Jacobi equation (2) $W(x,c)$ depending on $n$
independent parameters (integration constants)
$(c_{a})\in{\mathbb{V}}\subset{\mathbb{R}}^{n}$, $a=1,\dotsc,n$, such that
$\det\frac{\partial^{2}W}{\partial x^{\alpha}\partial c_{a}}\neq 0,$ (3)
is called complete integral. ∎
It is not a general solution of Eq.(2) which has functional arbitrariness.
However any solution of the Hamilton–Jacobi equation can be obtained from a
complete integral by variation of parameters (see, e.g., [27], Ch. IX, §3).
Therefore, if complete integral is known, then the problem may be considered
as solved.
Note that there are infinitely many complete integrals of the Hamilton–Jacobi
equation, if they exist. Therefore our aim is not to find all complete
integrals but only one. The domain $c\in{\mathbb{V}}$ depends on the metric
and therefore is not specified. It should be found in every particular case,
and requires some investigation. In general, constant $E(c)$ (energy) in the
Hamilton–Jacobi equation depends on parameters. In particular, it can be
considered as one of them.
It is clear that any solution of the Hamilton–Jacobi equation is defined up to
addition of arbitrary constant. This constant cannot be chosen as one of the
parameters $c$ because condition (3) is violated. Therefore all additive
integration constants of $W$ will be dropped as inessential.
We use Latin indices $a,b,\dotsc$ for enumeration of independent parameters
$c$ though they run the same values as the Greek ones. This is done to stress
important difference: tensor components with Greek indices are transformed
under coordinate transformation whereas with Latin indices are not. For
example, functions $\partial^{a}W:=\partial W/\partial c_{a}$ are scalars
while partial derivatives $\partial_{\alpha}W:=\partial W/\partial x^{\alpha}$
are components of covector.
###### Definition.
Coordinates $x^{\alpha}$, if they exist, are called separable, if
Hamilton–Jacobi equation (2) admits additive separation of variables in this
coordinate system, i.e. the action function is given by the sum
$W=\sum_{\alpha=1}^{n}W_{\alpha}(x^{\alpha},c)$ (4)
where every summand $W_{\alpha}$ is a function of only one coordinate
$x^{\alpha}$ and parameters $c_{a}$, taking values in some domain
${\mathbb{V}}\subset{\mathbb{R}}^{n}$. We require
$\det\frac{\partial^{2}W}{\partial x^{\alpha}\partial
c_{a}}=\det\frac{\partial^{2}W_{\alpha}}{\partial x^{\alpha}\partial
c_{a}}\neq 0$ (5)
for all $x$ and $c$. Metric in separable coordinate system is called
separable. Functions $W_{\alpha}$ in the sum (4) are called separating. ∎
Individual summand in the action function (4) may depend only on some part of
parameters, but the whole action function depends on all $n$ parameters.
For brevity, we use notation
$\partial^{a}W:=\frac{\partial W}{\partial c_{a}},\qquad
W^{\prime}_{\alpha}(x^{\alpha},c):=\partial_{\alpha}W_{\alpha}(x^{\alpha},c).$
Then requirement (5) is written as
$\det(\partial^{a}W^{\prime}_{\alpha})\neq 0.$ (6)
The problem of complete separation of variables in Hamilton–Jacobi equation
(2) is as follows. We have to describe all metrics $g^{\alpha\beta}(x)$,
depending only on coordinates $x$, for which there is a constant $E(c)\neq 0$
and functions $W^{\prime}_{\alpha}(x^{\alpha},c)$, depending on only one
coordinate $x^{\alpha}$ and parameters $c$ such that
$\det\partial^{a}W^{\prime}_{\alpha}\neq 0$ and equation
$g^{\alpha\beta}W^{\prime}_{\alpha}W^{\prime}_{\beta}=2E$ (7)
holds. During solution of this problem we find admissible form of separating
functions $W^{\prime}_{\alpha}$ which, as we shall see, define independent
envolutive conservation laws in the corresponding Hamiltonian systems. Thus,
we have to solve functional (not differential) equation (7) with respect to
$g^{\alpha\beta}(x)$ and $W^{\prime}_{\alpha}(x^{\alpha},c)$, which are
supposed sufficiently smooth both on $x$ and $c$.
All functions $W_{\alpha}$ is the sum (4) are scalars with respect to
coordinate transformations on ${\mathbb{M}}$ though they have the coordinate
index $\alpha$. Covector is defined by partial derivatives
$W^{\prime}_{\alpha}:=\partial_{\alpha}W_{\alpha}$. If the Hamilton–Jacobi
equation admits complete separation of variables, then the corresponding
Hamiltonian equations are Liouville integrable, the solution being given in
quadratures. Requirement (6) means that $n$ functions $W_{\alpha}$ are
functionally independent and can be chosen as new coordinates. These are the
action-angle coordinates in the corresponding Hamiltonian formulation.
Consider an important example of mechanical system in which metric has no
Killing vectors but admit complete separation of variables in the
Hamilton–Jacobi equation.
###### Example 2.1 (The Liouville system).
Consider a conformally flat metric
$g_{\alpha\beta}=\Phi^{2}\eta_{\alpha\beta}\quad\Leftrightarrow\quad
g^{\alpha\beta}:=\frac{1}{\Phi^{2}}\eta^{\alpha\beta},\qquad\Phi^{2}:=\phi_{1}+\dotsc+\phi_{n}>0,$
(8)
where every function $\phi_{\alpha}(x^{\alpha})$ (no summation) depends on
single coordinate, and $\eta_{\alpha\beta}$ denotes (pseudo)Euclidean metric
of arbitrary signature. This metric defines the Hamiltonian
$H:=\frac{\eta^{\alpha\beta}p_{\alpha}p_{\beta}}{2(\phi_{1}+\dotsc+\phi_{n})}+U(x)=\frac{1}{\Phi^{2}}\left(\frac{1}{2}\eta^{\alpha\beta}p_{\alpha}p_{\beta}+\Theta^{2}\right).$
(9)
Here we added potential energy $U$ of special type for generality
$U:=\frac{\Theta^{2}}{\Phi^{2}}=\frac{\theta_{1}(x^{1})+\dotsc+\theta_{n}(x^{n})}{\phi_{1}(x^{1})+\dotsc+\phi_{n}(x^{n})},$
every function $\theta_{\alpha}(x^{\alpha})$ being depended on single
coordinate. Hamiltonian equations of motion are
$\begin{split}\dot{x}^{\alpha}=&\frac{\eta^{\alpha\beta}p_{\beta}}{\Phi^{2}},\\\
\hphantom{\qquad{\sf\,ns\,}(\alpha)}\dot{p}_{\alpha}=&\frac{\eta^{\beta\gamma}p_{\beta}p_{\gamma}}{2\Phi^{4}}\partial_{\alpha}\phi_{\alpha}+\frac{\Theta^{2}\partial_{\alpha}\phi_{\alpha}-\Phi^{2}\partial_{\alpha}\theta_{\alpha}}{\Phi^{4}}=\frac{H}{\Phi^{2}}\partial_{\alpha}\phi_{\alpha}-\frac{\partial_{\alpha}\theta_{\alpha}}{\Phi^{2}},\qquad{\sf\,ns\,}(\alpha),\end{split}$
were the dot denotes differentiation with respect to the evolution parameter
$\tau\in{\mathbb{R}}$ (time), and summation on lower indices $\alpha$ is
absent. Here and in what follows, we denote this circumstance by special
symbol ${\sf\,ns\,}(\alpha)$ on the right of equations for brevity. Summation
over other repeated indices is performed as usual. The corresponding
Lagrangian is
$L:=p_{\alpha}\dot{x}^{\alpha}-H=\frac{\phi_{1}+\dotsc+\phi_{n}}{2}\eta_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}-\frac{\theta_{1}+\dotsc+\theta_{n}}{\phi_{1}+\dotsc+\phi_{n}}.$
Sure, energy $E$ for every trajectory is conserved,
$E:=H\big{|}_{\text{trajectory}}={\sf\,const}.$ (10)
Multiply Hamilton–Jacobi equation (7) by the conformal factor:
$\eta^{\alpha\beta}W^{\prime}_{\alpha}W^{\prime}_{\beta}+2\Theta^{2}=2E\Phi^{2}.$
Separation of variables for the Liouville system happens in the way
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}\eta^{\alpha\alpha}W^{\prime
2}_{\alpha}+2\theta_{\alpha}-2E\phi_{\alpha}=c_{\alpha},\qquad
c_{\alpha}\in{\mathbb{R}},\quad\forall\alpha,\qquad\qquad{\sf\,ns\,}(\alpha).$
(11)
The action function has the form
$W=\sum_{\alpha=1}^{n}W_{\alpha},\qquad
W_{\alpha}:=\int\\!\\!dx^{\alpha}\sqrt{\eta^{\alpha\alpha}\big{(}c_{\alpha}+2E\phi_{\alpha}-2\theta_{\alpha}\big{)}}.$
Certainly, the radicand is assumed to be positive. This complete integral
satisfies the Hamilton–Jacobi equation multiplied by the conformal factor:
$\delta^{\alpha\beta}\partial_{\alpha}W\partial_{\beta}W=2E\Phi^{2},$ (12)
where multipliers $\eta^{\alpha\alpha}$ are included in the action function
$W$.
In the Hamiltonian formalism, we have $n$ quadratic conservation laws
$\eta^{\alpha\alpha}p_{\alpha}^{2}+2\theta_{\alpha}-2E\phi_{\alpha}=c_{\alpha}.$
(13)
This can be easily checked by straightforward computation:
$\dot{c}_{\alpha}=2\eta^{\alpha\alpha}p_{\alpha}\dot{p}_{\alpha}+2\dot{\theta}_{\alpha}-2E\dot{\phi}_{\alpha}=\eta^{\alpha\alpha}p_{\alpha}\frac{2H\partial_{\alpha}\phi_{\alpha}}{\Phi^{2}}-2E\dot{x}^{\alpha}\partial_{\alpha}\phi_{\alpha}=\eta^{\alpha\alpha}p_{\alpha}\frac{2(H-E)}{\Phi^{2}}\partial_{\alpha}\phi_{\alpha}=0,$
because $H=E$ along every trajectory. For nontrivial functions $\theta$ and
$\phi$ these conservation laws are indecomposable, parameters satisfying the
condition $c_{1}+\dotsc+c_{n}=0$. Indeed, sum all conservation laws (13)
taking into account the definition of Hamiltonian (9):
$c_{1}+\dotsc+c_{n}=(2H-2E)(\phi_{1}+\dotsc+\phi_{n})=0.$ (14)
This is the sole relation between parameters. It shows that only $n-1$
conservation laws among (11) are independent. The last independent
conservation law is also quadratic
$\frac{1}{\Phi^{2}}\big{(}\eta^{\alpha\alpha}p_{\alpha}^{2}+2\Theta\big{)}=2E.$
(15)
It depends on all momenta and coordinates. The complete set of independent
parameters is, for example, $c_{1},\dotsc,c_{n-1},2E$.
Thus, all canonical pairs of variables are separated. The particular feature
of this separation is that conformally flat metric (8) has no Killing vector
for nonconstant functions $\phi_{\alpha}(x^{\alpha})$. Besides, variables were
separated only after multiplication of the Hamilton–Jacobi equation by the
conformal factor. Separation of variables for the Liouville system does not
depend on the metric signature. ∎
## 3 Separation of variables and Hamiltonian formalism
In this section, we show what happens in Hamiltonian formalism for complete
separation of variables in the Hamilton–Jacobi equation. Remind that we are
considering Hamiltonian (1) yielding equations of motion
$\begin{split}\dot{x}^{\alpha}=&[x^{\alpha},H_{0}]=g^{\alpha\beta}p_{\beta},\\\
\dot{p}_{\alpha}=&[p_{\alpha},H_{0}]=\frac{1}{2}\partial_{\alpha}g_{\beta\gamma}p^{\beta}p^{\gamma}.\end{split}$
(16)
Suppose that relations
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}G_{\alpha}(x^{\alpha},p_{\alpha},c):=p_{\alpha}-W^{\prime}_{\alpha}(x^{\alpha},c)=0,\qquad\forall\alpha,\qquad\qquad{\sf\,ns\,}(\alpha),$
(17)
hold, where $(W^{\prime}_{\alpha})$ is an arbitrary solution of the
Hamilton–Jacobi equation (7) depending on $n$ independent parameters. Due to
inequality (6), these equations can be locally solved with respect to $c$:
$c_{a}=F_{a}(x,p),$ (18)
where $F_{a}$ are some functions on canonical variables $x,p$ only.
###### Theorem 3.1.
Relations (18) hold along every trajectory of the Hamiltonian system. These
first integrals of Hamiltonian equations are in involution.
###### Proof.
Compute
$\dot{c}_{a}=\frac{\partial F_{a}}{\partial
x^{\alpha}}\dot{x}^{\alpha}+\frac{\partial F_{a}}{\partial
p_{\alpha}}\dot{p}_{\alpha}.$ (19)
To find partial derivatives of functions $F_{a}$, we do the following. Take
the differential of Eq.(17):
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}dp_{\alpha}-\partial_{\alpha}W^{\prime}_{\alpha}dx^{\alpha}-\partial^{a}W^{\prime}_{\alpha}dc_{a}=0,\qquad\qquad{\sf\,ns\,}(\alpha).$
These equations can be solved with respect to $dc_{a}$:
$dc_{a}=\sum_{\alpha=1}^{n}\left(\partial^{a}W^{\prime}_{\alpha}\right)^{-1}\big{(}dp_{\alpha}-\partial_{\alpha}W^{\prime}_{\alpha}dx^{\alpha}\big{)},$
(20)
because of inequality (6). On the other hand, the differential of Eq.(18)
yields
$dc_{a}=\frac{\partial F_{a}}{\partial x^{\alpha}}dx^{\alpha}+\frac{\partial
F_{a}}{\partial p_{\alpha}}dp_{\alpha}.$
Comparing Eqs.(19) and (20), we conclude that
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}\frac{\partial F_{a}}{\partial
x^{\alpha}}=-(\partial^{a}W^{\prime}_{\alpha})^{-1}\partial_{\alpha}W^{\prime}_{\alpha},\qquad\frac{\partial
F_{a}}{\partial
p_{\alpha}}=(\partial^{a}W^{\prime}_{\alpha})^{-1},\qquad\qquad{\sf\,ns\,}(\alpha).$
Now Eq.(19) yields:
$\dot{c}_{a}=\sum_{\alpha=1}^{n}(\partial^{a}W^{\prime}_{\alpha})^{-1}\left(-\partial_{\alpha}W^{\prime}_{\alpha}p_{\alpha}+\frac{1}{2}\partial_{\alpha}g_{\beta\gamma}p^{\beta}p^{\gamma}\right),$
where Hamiltonian equations (16) are used for exclusion of derivatives with
respect to evolution parameter $\tau$.
On the other hand, differentiate Hamilton–Jacobi equation (7) with respect to
$x^{\gamma}$:
$\partial_{\gamma}g^{\alpha\beta}W^{\prime}_{\alpha}W^{\prime}_{\beta}+2g^{\alpha\beta}W^{\prime}_{\alpha}\partial_{\gamma}W^{\prime}_{\beta}=-g^{\alpha\delta}g^{\beta\epsilon}\partial_{\gamma}g_{\delta\epsilon}W^{\prime}_{\alpha}W^{\prime}_{\beta}+2W^{\prime\alpha}\partial_{\gamma}W^{\prime}_{\alpha}=\\\
=-\partial_{\gamma}g_{\delta\epsilon}W^{\prime\delta}W^{\prime\epsilon}+2W^{\prime\alpha}\partial_{\gamma}W^{\prime}_{\alpha}=-\partial_{\gamma}g_{\delta\epsilon}p^{\delta}p^{\epsilon}+2p^{\gamma}\partial_{\gamma}W^{\prime}_{\gamma}=0,$
where Eq.(17) is used. Therefore $\dot{c}_{a}=0$, and the first statement of
the theorem is proved.
Now compute the Poisson bracket of two conservation laws with indices $a\neq
b$
$[F_{a},F_{b}]=\frac{\partial F_{a}}{\partial x^{\alpha}}\frac{\partial
F_{b}}{\partial p_{\alpha}}-\frac{\partial F_{a}}{\partial
p_{\alpha}}\frac{\partial F_{b}}{\partial x^{\alpha}}=\\\
=-\sum_{\alpha=1}^{n}\Big{(}(\partial^{a}W^{\prime}_{\alpha})^{-1}\partial_{\alpha}W^{\prime}_{\alpha}(\partial^{b}W^{\prime}_{\alpha})^{-1}-(\partial^{a}W^{\prime}_{\alpha})^{-1}(\partial^{b}W^{\prime}_{\alpha})^{-1}\partial_{\alpha}W^{\prime}_{\alpha}\Big{)}=0.$
This proves that conservation laws are in involution. ∎
We see that, if variables are completely separated in the Hamilton–Jacobi
equation, then the corresponding Hamiltonian system admits $n$ independent
conservation laws (18) which are in involution, and, consequently, it is
Liouville integrable. This statement is proved for the geodesic Hamiltonian
$H_{0}$. Moreover, it is valid in a general case.
If summands $W^{\prime}_{\alpha}(x^{\alpha},c)$ are found, then conservation
laws (18) are obtained by solution of Eqs. (17) with respect to parameters in
explicit form. Luckily, this can be always done for geodesic Hamiltonian.
If a complete integral of the Hamilton–Jacobi equation is known, then one can
always go to the action-angle coordinates at least implicitly.
###### Theorem 3.2.
Let us perform the canonical transformation $(x,p)\mapsto(X,P)$ with
generating function $S_{2}(x,P):=W(x,P)$, where momenta are substituted
instead of parameters: $(c_{a})\mapsto(P_{a})$. Then Hamiltonian equations for
new variables take the form
$\begin{split}\dot{X}^{a}=&\frac{\partial H}{\partial P_{a}}=\frac{\partial
H}{\partial p_{\alpha}}\partial^{a}\partial_{\alpha}W,\\\
\dot{P}_{a}=&0,\qquad\Rightarrow\qquad P_{a}=c_{a}={\sf\,const},\end{split}$
(21)
where the substitution $x=x(X,P)$ and $p=p(X,P)$ is performed on the right
hand side.
###### Proof.
If the generating function of the canonical transformation depends on old
coordinates and new momenta, then old momenta and new coordinates are given by
formulae (see, e.g., [28])
$p_{\alpha}=\partial_{\alpha}W,\qquad X^{a}=\partial^{a}W.$ (22)
The last equality can be solved with respect to old coordinates at least
locally due to inequality (3), and we find $x=x(X,P)$. Substitution of this
solution in the first equality defines $p=p(X,P)$.
Take the differential of Eqs. (22):
$\begin{split}dp_{\alpha}=&\partial^{2}_{\alpha\beta}Wdx^{\alpha}+\partial^{a}\partial_{\alpha}WdP_{a},\\\
dX^{a}=&\partial^{a}\partial_{\alpha}Wdx^{\alpha}+\partial^{a}\partial^{b}WdP_{b}.\end{split}$
(23)
This defines differentials
$\begin{split}dx^{\alpha}=&(\partial^{a}\partial_{\alpha}W)^{-1}(dX^{a}-\partial^{a}\partial^{b}WdP_{b}),\\\
dp_{\alpha}=&\partial^{2}_{\alpha\beta}W(\partial^{a}\partial_{\alpha}W)^{-1}dX^{a}+\big{[}\partial^{a}\partial_{\alpha}W-\partial^{2}_{\alpha\beta}W(\partial^{b}\partial_{\beta}W)^{-1}\partial^{a}\partial^{b}W\big{]}dP_{a}.\end{split}$
(24)
Consequently, partial derivatives are
$\begin{split}\frac{\partial x^{\alpha}}{\partial
X^{a}}=&(\partial^{a}\partial_{\alpha}W)^{-1},\\\ \frac{\partial
x^{\alpha}}{\partial
P_{a}}=&-(\partial^{b}\partial_{\alpha}W)^{-1}\partial^{a}\partial^{b}W,\\\
\frac{\partial p_{\alpha}}{\partial
X^{a}}=&\partial^{2}_{\alpha\beta}W(\partial^{a}\partial_{\beta}W)^{-1},\\\
\frac{\partial p_{\alpha}}{\partial
P_{a}}=&\partial^{a}\partial_{\alpha}W-\partial^{2}_{\alpha\beta}W(\partial^{b}\partial_{\beta}W)^{-1}\partial^{a}\partial^{b}W.\end{split}$
(25)
On the other hand the Hamilton–Jacobi equation implies
$H\left(x,\frac{\partial W(x,P)}{\partial x}\right)=2E(P).$
Differentiate this equality by $x^{\alpha}$:
$\frac{\partial H}{\partial x^{\alpha}}+\frac{\partial H}{\partial
p_{\beta}}\frac{\partial p_{\beta}}{\partial x^{\alpha}}=\frac{\partial
H}{\partial x^{\alpha}}+\frac{\partial H}{\partial
p_{\beta}}\partial_{\alpha\beta}W=0.$ (26)
Now we see that equalities
$\frac{\partial H}{\partial X^{a}}\equiv 0,\qquad\frac{\partial H}{\partial
P_{a}}=\frac{\partial H}{\partial p_{\alpha}}\partial^{a}\partial_{\alpha}W,$
hold where $H=H\big{(}x(X,P),p(X,P)\big{)}$. ∎
Separable coordinates, when they exist, are not uniquely defined, and there is
a functional arbitrariness related to coordinate transformations. To isolate
it, we give
###### Definition.
Two separable coordinate systems $x$ and $X$ on ${\mathbb{M}}$ are equivalent,
if there is a canonical transformation $(x,p)\mapsto(X,P)$ of respective
Hamiltonian systems, such that new coordinates $X(x)$ depend only on old ones
but not on momenta $p$. Moreover, separable coordinate systems are equivalent
in case when parameters are related by nondegenerate transformation
$c\mapsto\tilde{c}(c)$ which do not involve coordinates. Manifold
${\mathbb{M}}$ with metric $g$ which admits separable coordinates for Eq. (2)
in some neighbourhood of each point related by this equivalence relation in
overlapping domains is called Stäckel. ∎
Right hand sides of conservation laws (18) are scalars under canonical
transformations:
$F(x,p)\mapsto\tilde{F}_{a}(X,P)=F_{a}\big{(}x(X),p(X,P)\big{)}.$
Conservation laws are transformed into conservation laws and their involution
survives under arbitrary canonical transformation because the latter preserves
the Poisson bracket. We shall use canonical transformations in what follows to
bring separable metrics to the most simple (canonical) forms and eliminate
inessential functions.
There may occur inequivalent separable coordinate systems for one and the same
metric. In next sections, we obtain necessary and sufficient conditions for
existence of separable coordinate systems and present explicit form of
canonical separable metrics in each class of equivalent separable metrics.
Before solution of the Stäckel problem, we consider the simple instructive
example.
###### Example 3.1.
Consider Euclidean space ${\mathbb{R}}^{n}$ with metric $\delta^{ab}$,
$a,b=1,\dotsc,n$ in Cartesian coordinates $y^{a}$. The Hamilton–Jacobi
equation
$\delta^{ab}W^{\prime}_{a}W^{\prime}_{b}=2E$
admits complete separation of variables
$W^{\prime}_{a}=c_{a}={\sf\,const},$
the energy $2E=c^{a}c_{a}$ being the dependent parameter. The action function
is
$W=y^{a}c_{a},$ (27)
where we dropped inessential additive integration constant. It depends on $n$
independent parameters $c_{1},\dotsc,c_{n}$. Consequently, action function
(27) is the complete integral of the Hamilton–Jacobi equation, and the problem
is solved. Complete separation of variables (27) implies $n$ conservation laws
in the Hamiltonian formulation:
$p_{a}=c_{a},$
which means conservation of all momenta. We see that Cartesian coordinate
system in Euclidean space is separable. Vector fields $\partial_{a}$ are
Killing vector fields. At the same time most of curvilinear coordinates in
${\mathbb{R}}^{n}$ are not separable.
Indeed, let us go to curvilinear coordinates $y^{a}\mapsto x^{\alpha}(y)$,
$\alpha=1,\dotsc,n$. Then inverse functions $y^{a}(x)$ define vielbein (Jacobi
matrices) and functions $W^{\prime}_{\alpha}(x,c)$ are
$W^{\prime}_{\alpha}=e_{\alpha}{}^{a}c_{a},\qquad
e_{\alpha}{}^{a}:=\partial_{\alpha}y^{a},$
because they are covector components. It is necessary and sufficient for new
coordinates to be separable that vielbein components
$e_{\alpha}{}^{a}(x^{\alpha})$ depend only on one coordinate $x^{\alpha}$. It
means that transition functions must be specific
$y^{a}=\sum_{\alpha=1}^{n}k_{\alpha}{}^{a}(x^{\alpha}),$
where all elements of each row of the matrix $k_{\alpha}{}^{a}$ depend only on
one coordinate $x^{\alpha}$. Then the vielbein is
$e_{\alpha}{}^{a}(x^{\alpha})=\partial_{\alpha}k_{\alpha}{}^{a}=:k^{\prime}_{\alpha}{}^{a}$
with the sole condition $\det k^{\prime}_{\alpha}{}^{a}\neq 0$. Now we compute
metric $g^{\alpha\beta}$, and variables in the Hamilton–Jacobi equation are
separated:
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}W=\sum_{\alpha=1}^{n}W_{\alpha},\qquad
W_{\alpha}:=\int\\!\\!dx^{\alpha}e_{\alpha}{}^{a}(x^{\alpha})c_{a},\qquad\qquad{\sf\,ns\,}(\alpha).$
In the Hamiltonian formulation, we have $n$ independent conservation laws:
$p_{\alpha}e^{\alpha}{}_{a}(x)=c_{a}.$
Note that the inverse vielbein components $e^{\alpha}{}_{a}(x)$, in general,
depend on all coordinates.
After the inverse transformation $x^{\alpha}\mapsto y^{a}$, the conservation
laws take simple form $p_{a}=c_{a}$ which correspond to $n$ commuting Killing
vectors $\partial_{a}$ (translations along Cartesian coordinates).
All coordinate systems obtained in this way and only they (this will be proved
in section 4) are equivalent. To see this, we make canonical transformation
$(x^{\alpha},p_{\alpha})\mapsto(y^{a},p_{a})$ with generating function
depending on old coordinates and new momenta
$S_{2}(x^{\alpha},p_{a}):=\sum_{\alpha=1}^{n}k_{\alpha}{}^{a}(x^{\alpha})p_{a}.$
Then
$p_{\alpha}=\frac{\partial S_{2}}{\partial
x^{\alpha}}=k^{\prime}_{\alpha}{}^{a}p_{a}=e_{\alpha}{}^{a}p_{a},\qquad
y^{a}=\frac{\partial S_{2}}{\partial
p_{a}}=\sum_{\alpha=1}^{n}k_{\alpha}{}^{a},$ (28)
and we are led to previous formulae. Thus, if we solve functional equation
(7), then its general solution contains many arbitrary functions. In our case,
there are $n^{2}$ inessential functions $k_{\alpha}{}^{a}(x^{\alpha})$
describing transformations between equivalent separable coordinate systems. We
see that the problem can be essentially simplified by making canonical
transformations. Therefore it is sufficient to choose the most simple
separable metric in each class of equivalent metrics which is called
canonical. In the present case, it is the Euclidean metric, all other
equivalent separable metrics are related to it by suitable canonical
transformation. We note that transformation of momenta (28) is linear.
Therefore linear and quadratic conservation laws are transformed into linear
and quadratic ones, respectively.
At the same time, separation of variables in Euclidean space may happen in a
different way. Consider Euclidean plane in polar coordinates
$(y^{a})=(r,\varphi)\in{\mathbb{R}}^{2}$. In Cartesian coordinates, we have
two linear conservation laws
$p_{1}=c_{1},\qquad p_{2}=c_{2},$
corresponding to two commuting Killing vectors $\partial_{1}$ and
$\partial_{2}$. In polar coordinates,
$x:=y^{1}=r\cos\varphi,\qquad y:=y^{2}=r\sin\varphi.$
metric and its inverse are
$g_{\alpha\beta}=\begin{pmatrix}1&0\\\ 0&r^{2}\end{pmatrix},\qquad
g^{\alpha\beta}=\begin{pmatrix}1&0\\\ 0&\frac{1}{r^{2}}\end{pmatrix}.$
The Hamilton–Jacobi equation is
$W^{\prime 2}_{r}+\frac{1}{r^{2}}W^{\prime
2}_{\varphi}=\delta^{ab}c_{a}c_{b}=2E.$
Variables in this equation are separated as follows
$W^{\prime}_{\varphi}=c_{\varphi},\qquad W^{\prime
2}_{r}+\frac{c_{\varphi}^{2}}{r^{2}}=2E.$ (29)
The first equality corresponds to rotational invariance (Killing vector
$\partial_{\varphi}$), and the second is the conservation of energy. The
action function is now
$W=\int\\!\\!dr\sqrt{2E-\frac{c_{\varphi}^{2}}{r^{2}}}+\varphi c_{\varphi}.$
It is the complete integral of the Hamilton–Jacobi equation. If $E>0$, then
this expression makes sense only for $r^{2}>c_{\varphi}^{2}/2E$.
In the Hamiltonian formulation, we have two conservation laws
$p_{\varphi}=c_{\varphi},\qquad p_{r}^{2}+\frac{c_{\varphi}^{2}}{r^{2}}=2E,$
These are conservation of angular momentum and energy. Thus variables are also
separated in polar coordinates, but now we have one linear and one
indecomposable quadratic conservation laws.
Though variables are separated both in Cartesian and polar coordinates, these
coordinate systems are not equivalent in the sense of our definition. Indeed,
the transformation of coordinate does exist but it is not canonical because
conservation laws do not transform into conservation laws.
In polar and Cartesian coordinates, parameters $(E,c_{\varphi})$ and
$(c_{1},c_{2})$ constitute full sets of parameters, respectively, for the
complete action functions. We have $2E=c_{1}^{2}+c_{2}^{2}$, but parameter
$c_{\varphi}$ cannot be expressed entirely through $c_{1}$ and $c_{2}$.
Really, momenta in polar and Cartesian coordinates are related by
transformation
$p_{r}=\cos\varphi\,p_{1}+\sin\varphi\,p_{2},\qquad
p_{\varphi}=-r\sin\varphi\,p_{1}+r\cos\varphi\,p_{2}.$
It implies expression for the angular parameter
$c_{\varphi}=-r\sin\varphi\,c_{1}+r\cos\varphi\,c_{2}=-yc_{1}+xc_{2}.$
That is transformation of parameters $(c_{1},c_{2})\mapsto(E,c_{\varphi},x,y)$
necessarily include coordinates. We see that transformation between full sets
of independent parameters may include coordinates, and separation of variables
does depend on our choice. ∎
Considered examples teach us the following. First, a general solution of the
Stäckel problem contains many functions parameterizing transformations between
equivalent separable coordinate systems. In configuration space we have
ordinary coordinate transformations which do not alter geometric invariants,
e.g. scalar curvature or squares of curvature tensor. These functions are
inessential, and can be eliminated by suitable canonical transformation. Note
that these transformations do not exhaust the whole group of diffeomorphisms.
Second, transformation of one full set of parameters to the other
$(c_{a})\mapsto(\tilde{c}_{a},x)$ may include coordinates. In this case,
separation of variables holds differently, and corresponding separable
coordinates are not equivalent.
## 4 Linear conservation laws
Example 3.1 shows that separation of variables and conservation laws depend on
the choice of complete set of parameters. The right hand side of the
Hamilton–Jacobi equation (7) depends on one constant $E$ which therefore is
distinguished. In this section, we choose “symmetric” set of independent
parameters which does not include $E$. This is done as follows.
Fix a point ${\textsc{p}}\in{\mathbb{M}}$ and introduce parameters
$c_{a}:=W^{\prime}_{\alpha=a}\big{|}_{\textsc{p}}.$
Here we change indices into Latin ones, $\alpha\mapsto a$ to stress that the
set of parameters $(c_{a})$ after coordinate transformations is multiplied by
the inverse Jacobi matrix at a fixed point p. The Hamilton–Jacobi equation at
this point is
$g^{ab}c_{a}c_{b}=2E,\qquad
g^{ab}:=g^{\alpha\beta}\big{|}_{{\textsc{p}},\alpha=a,\beta=b},$
where subscript p means restriction of metric to this point. That is $E(c)$
becomes the dependent parameter. Now we choose the collection $(c_{a})$,
$a=1,\dotsc,n$, as the complete set of independent parameters. Moreover metric
$g^{ab}$ at point p can be transformed into diagonal form with $\pm 1$ on the
diagonal depending on the signature of the metric by suitable linear
coordinate transformation. Denote this matrix by
$\eta^{ab}:={\sf\,diag\,}(\underbrace{1,\dotsc,1,}_{r}\underbrace{-1,\dotsc,-1}_{s}),\qquad
r+s=n,$ (30)
where the pair $(r,s)$ designates the signature of the metric. Then the
Hamilton–Jacobi equation takes the “symmetrical” form
$g^{\alpha\beta}W^{\prime}_{\alpha}W^{\prime}_{\beta}=\eta^{ab}c_{a}c_{b},$
(31)
where each function $W^{\prime}_{\alpha}(x^{\alpha},c)$ depends on single
coordinate $x^{\alpha}$ and, probably, on the full set of parameters $c$. The
left hand side of this equation is a geometric invariant. Therefore we can
regard metric components $g^{\alpha\beta}$ and functions $W^{\prime}_{\alpha}$
as transforming by usual tensor transformation rules under coordinate changes
while all quantities with Latin indices remain unchanged.
The geometric meaning of this set of parameters is as follows. One and only
one geodesic goes through each point p in every direction. Therefore the set
of parameters $(x_{\textsc{p}},c)$ yields the Cauchy data and uniquely defines
geodesic line in some neighborhood of point p. In the Hamiltonian formulation,
the set $(c_{a})$ represent momenta at a fixed point p.
Now we derive necessary and sufficient conditions for complete separation of
variables for “symmetric” choice of parameters in Hamilton–Jacobi equation
(31). There is a global vielbein $e^{\alpha}{}_{a}(x)$ on every topologically
trivial manifold ${\mathbb{M}}\approx{\mathbb{R}}^{n}$:
$g^{\alpha\beta}=e^{\alpha}{}_{a}e^{\beta}{}_{b}\eta^{ab},$
which is defined up to local ${\mathbb{O}}(r,s)$ (pseudo)rotations which
include coordinate reflections. The inverse vielbein is denoted by
$e_{\alpha}{}^{a}$ ($e_{\alpha}{}^{b}e^{\beta}{}_{b}=\delta_{\alpha}^{\beta}$,
$e_{\alpha}{}^{a}e^{\alpha}{}_{b}=\delta^{a}_{b}$). Then Eq. (31) is rewritten
as
$\eta^{ab}W^{\prime}_{a}W^{\prime}_{b}=\eta^{ab}c_{a}c_{b},\qquad
W^{\prime}_{a}:=e^{\alpha}{}_{a}W^{\prime}_{\alpha}.$
It implies that covectors $W^{\prime}_{a}$ and $c_{a}$ are necessarily related
by some (pseudo)rotational matrix $S\in{\mathbb{O}}(r,s)$:
$W^{\prime}_{a}(x)=S_{a}{}^{b}(x)c_{b}.$
multiply this expression by inverse vielbein $e_{\alpha}{}^{a}$ and obtain
$W^{\prime}_{\alpha}=\tilde{e}_{\alpha}{}^{a}c_{a},\qquad\text{where}\qquad\tilde{e}_{\alpha}{}^{a}:=e_{\alpha}{}^{b}S_{b}{}^{a}.$
For complete separation of variables in the Hamilton–Jacobi equation, it is
necessary and sufficient that the right hand side of this relation depends
solely on $x^{\alpha}$. Drop the tilde sign and formulate the result.
###### Theorem 4.1.
Variables in Hamilton–Jacobi equation (31) with ”symmetric“ set of independent
parameters $(c_{a})$ are completely separated if and only if there exists the
vielbein $e_{\alpha}{}^{a}(x^{\alpha})$ such that its components with fixed
$\alpha$ depend solely on one coordinate $x^{\alpha}$ for all values of index
$a$. Then every term in action function (4) is a primitive
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}W_{\alpha}(x_{\alpha},c):=\int\\!\\!dx^{\alpha}e_{\alpha}{}^{a}(x^{\alpha})c_{a},\qquad\qquad{\sf\,ns\,}(\alpha).$
(32)
In the Hamiltonian formulation, there are $n$ linear in momenta conservation
laws
$e^{\alpha}{}_{a}(x)p_{\alpha}=c_{a}$ (33)
for all $a$.
###### Corollary.
Covariant components of the separable metric for Hamilton–Jacobi equation (31)
$g_{\alpha\beta}(x^{\alpha},x^{\beta})=e_{\alpha}{}^{a}(x^{\alpha})e_{\beta}{}^{b}(x^{\beta})\eta_{ab}$
depend only on two coordinates. ∎
The proof is evident. Sure, components of the inverse metric
$g^{\alpha\beta}(x)$ and vielbein $e^{\alpha}{}_{a}(x)$ depend on all
coordinates in general. That is conservation laws (33) are linear in momenta
but depend on all coordinates.
###### Example 4.1.
Complicate the problem considered in example 3.1. Suppose that rotationally
symmetric metric is given in polar coordinates on a plane
$(r,\varphi)\in{\mathbb{R}}^{2}$
$g_{\alpha\beta}=\begin{pmatrix}f^{2}&0\\\ 0&r^{2}\end{pmatrix},\qquad
g^{\alpha\beta}=\begin{pmatrix}\frac{1}{f^{2}}&0\\\
0&\frac{1}{r^{2}}\end{pmatrix},$ (34)
where $f(r)>0$ is some function of radius only. Then the Hamilton–Jacobi
equation in polar coordinates is
$\frac{W^{\prime 2}_{r}}{f^{2}}+\frac{W^{\prime 2}_{\varphi}}{r^{2}}=2E,$
and variables are completely separated:
$W^{\prime}_{\varphi}=c_{\varphi},\qquad\frac{W^{\prime
2}_{r}}{f^{2}}+\frac{c_{\varphi}^{2}}{r^{2}}=2E.$
Parameters $c_{\varphi}$ and $E$, as before, correspond to conservation of the
angular momentum and the energy.
Let us look what happens in Cartesian coordinates $(x,y)\in{\mathbb{R}}^{2}$.
In polar coordinates vielbein can be chosen in diagonal form
$e_{\alpha}{}^{a}=\begin{pmatrix}f&0\\\ 0&r\end{pmatrix},\qquad
e^{\alpha}{}_{a}=\begin{pmatrix}\frac{1}{f}&0\\\ 0&\frac{1}{r}\end{pmatrix}.$
In Cartesian coordinates it is written as
$e_{x}{}^{a}=\left(\frac{x}{r}f,\,-\frac{y}{r}\right),\qquad
e_{y}{}^{a}=\left(\frac{y}{r}f,\,\frac{x}{r}\right).$
Vielbein components $e_{x}{}^{a}$ explicitly depend on $y$, and variables are
not separated. According to theorem 4.1, separable coordinates exist if there
is the vielbein
$\tilde{e}_{\alpha}{}^{a}=e_{\alpha}{}^{b}S_{b}{}^{a},\qquad
S_{b}{}^{a}=\begin{pmatrix}\cos\omega&-\sin\omega\\\
\sin\omega&~{}~{}\cos\omega\end{pmatrix}\in{\mathbb{S}}{\mathbb{O}}(2),$
where $\omega(x,y)$ is some rotational angle, such that vielbein components
$\tilde{e}_{x}{}^{a}(x)$ and $\tilde{e}_{y}{}^{a}(y)$ depend only on $x$ and
$y$, respectively.
###### Proposition 4.1.
If $f^{2}\equiv\\!\\!\\!\\!\\!\\!/\>1$, then there is no rotational matrix $S$
such that components of the first row of vielbein $\tilde{e}_{x}{}^{a}$ depend
solely on $x$ for all $a=1,2$, and components of the second row
$\tilde{e}_{y}{}^{a}$ depend solely on $y$.
###### Proof.
Two components of vielbein in Cartesian coordinates after the rotation are
$\begin{split}\tilde{e}_{x}{}^{1}=&~{}~{}\frac{x}{r}f\cos\omega-\frac{y}{r}\sin\omega,\\\
\tilde{e}_{x}{}^{2}=&-\frac{x}{r}f\sin\omega-\frac{y}{r}\cos\omega.\end{split}$
Their independence on $y$ is given by equations
$\displaystyle\partial_{y}\tilde{e}_{x}{}^{1}=$
$\displaystyle-\frac{xy}{r^{3}}f\cos\omega+\frac{xy}{r^{2}}f^{\prime}\cos\omega-\frac{x}{r}f\sin\omega\partial_{y}\omega-\frac{x^{2}}{r^{3}}\sin\omega-\frac{y}{r}\cos\omega\partial_{y}\omega=0,$
(35) $\displaystyle\partial_{y}\tilde{e}_{x}{}^{2}=$
$\displaystyle~{}~{}\frac{xy}{r^{3}}f\sin\omega-\frac{xy}{r^{2}}f^{\prime}\sin\omega-\frac{x}{r}f\cos\omega\partial_{y}\omega-\frac{x^{2}}{r^{3}}\cos\omega+\frac{y}{r}\sin\omega\partial_{y}\omega=0,$
(36)
where $f^{\prime}:=df/dr$. Take the linear combination of theses equations
$(\ref{anskhg})\sin\omega+(\ref{anbswe})\cos\omega=-\frac{x}{r}f\partial_{y}\omega-\frac{x^{2}}{r^{3}}=0\qquad\Rightarrow\qquad
f\partial_{y}\omega=-\frac{x}{r^{2}}$
and substitute in Eq. (35). Finally, we obtain equation for $f$
$rff^{\prime}-f^{2}+1=0.$
Its general solution is
$f^{2}=1+Cr^{2},\qquad C={\sf\,const}.$
Now we write the independence of the second row of the vielbein on $x$,
$\partial_{x}\tilde{e}_{y}{}^{a}=0$, and make the same calculations as for
Eqs. (35) and (36). Then we obtain
$f\partial_{x}\omega=\frac{y}{r^{2}}$
with the same function $f$. Consequently, the rotation angle is defined by the
system of equation
$\partial_{x}\omega=\frac{y}{r^{2}f},\qquad\partial_{y}\omega=-\frac{x}{r^{2}f}.$
It is easily checked that integrability conditions are not fulfilled for
$f^{2}\neq 1$, and therefore function $\omega(x,y)$ does not exist. Therefore
variables for metric (34) are separated in Cartesian coordinates only for
$f^{2}\equiv 1$. ∎
Thus variables in the Hamilton–Jacobi equation for metric (34) are completely
separated in polar coordinates but not in Cartesian. This is a natural result
because metric (34) is not translationally invariant. ∎
The moral of this example is that if the Hamilton–Jacobi equation is not
separable for ”symmetric“ choice of parameters, then it does not mean that
separability does not happen for other choices of independent parameters.
Consequently, theorem 4.1 does not exhaust all possibilities of variable
separations.
Conservation laws (33) can be significantly simplified by canonical
transformation $(x^{\alpha},p_{\alpha})\mapsto(X^{a},P_{a})$ with generating
function depending on old coordinates $(x^{\alpha})$ and new momenta $(P_{a})$
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}S_{2}:=\sum_{\alpha=1}^{n}I_{\alpha},\qquad
I_{\alpha}:=\int\\!\\!dx^{\alpha}\,e_{\alpha}{}^{a}(x^{\alpha})P_{a},\qquad\qquad{\sf\,ns\,}(\alpha),$
(37)
where $I_{\alpha}$ are primitives. Then
$p_{\alpha}=\frac{\partial S_{2}}{\partial
x^{\alpha}}=e_{\alpha}{}^{a}P_{a},\qquad X^{a}=\frac{\partial S_{2}}{\partial
P_{a}}=\sum_{\alpha=1}^{\textsc{n}}\int\\!\\!dx^{\alpha}e_{\alpha}{}^{a}.$
This is exactly the coordinate transformation
$(x^{\alpha})\mapsto\big{(}X^{a}(x)\big{)}$. Indeed, the Jacobi matrix is
$\partial_{\alpha}X^{a}=e_{\alpha}{}^{a},$
and momenta components are transformed as covariant vector. In new
coordinates, the Hamiltonian is
$H_{0}=\frac{1}{2}\eta^{ab}P_{a}P_{b}.$
It implies $n$ independent linear conservation laws
$P_{a}=c_{a}$ (38)
(all momenta are conserved). Since metric components are constant in these
coordinates then there is the maximal number $n$ of linearly independent
commuting Killing vectors $\partial_{a}$ on ${\mathbb{M}}$. In this way, we
proved
###### Theorem 4.2.
Variables in the Hamilton–Jacobi equation (31) with the “symmetric” choice of
parameters are completely separable if and only if there exist $n$ linearly
independent at each point commuting Killing vector fields on
$({\mathbb{M}},g)$. Then there is a coordinate system in which metric
components are constant, and conservation laws have form (38).
This result is not surprising. Indeed, if vielbein components
$e_{\alpha}{}^{a}(x^{\alpha})$ depend on single coordinate $x^{\alpha}$ for
all $a$, then curvature tensor vanishes. These calculations are most easily
performed in Cartan variables. The nonholonomicity components are identically
zero
$c_{\alpha\beta}{}^{a}:=-\partial_{\alpha}e_{\beta}{}^{a}+\partial_{\beta}e_{\alpha}{}^{a}\equiv
0.$
Therefore the corresponding ${\mathbb{S}}{\mathbb{O}}(r,s)$ connection is also
zero, and, consequently, curvature vanishes. Therefore manifold ${\mathbb{M}}$
is locally (pseudo)Euclidean. This is enough for complete separation of
variables in Cartesian coordinates. Thus, if complete separation of variables
occurs for “symmetrical” choice of independent parameters, then manifold
${\mathbb{M}}$ is necessarily locally flat.
## 5 Quadratic conservation laws and coisotropic coordinates
Now we consider another possibility of complete variables separation of
variables in the Hamilton–Jacobi equation. Remember that our aim is not to
find all complete integrals, which are infinitely many, but to find at least
one of them for a given type of separable metric. Therefore our strategy is
the following.
1. 1.
Describe all possible types of separable metrics.
2. 2.
Choose independent parameters.
3. 3.
Specify the class of separating functions in each case.
4. 4.
Solve the functional Hamilton–Jacobi equation.
5. 5.
Choose the canonical separable metric in every equivalence class, find the
complete integral and the full set of conservation laws.
6. 6.
Prove that any additional conservation law is functionally dependent on the
full set of previously obtained conservation laws at least locally.
First, we introduce notation, choose parameters, and prove the simple lemma
which is important for further analysis. Suppose that we have exactly
$0\leq{\textsc{n}}\leq n$ commuting Killing vector fields and not more, and
variables are nevertheless completely separated. We assume that variables
corresponding to Killing vectors are separated, and they are the first n
coordinates. There are only two possibilities left: diagonal components of
separable metrics can differ from zero or vanish, which is possible only for
indefinite metrics. Assume that the number of nonzero diagonal metric
components is equal to $0\leq{\textsc{m}}\leq n$, and they precede zero
components. We shall see in what follows that the inequality
$n-{\textsc{n}}-{\textsc{m}}\leq{\textsc{n}}\qquad\Leftrightarrow\qquad
2{\textsc{n}}+{\textsc{m}}\leq n$ (39)
must hold, otherwise the metric becomes degenerate.
A curve $x^{\alpha}(\tau)$, $\tau\in{\mathbb{R}}$ is called isotropic if the
tangent vector $dx^{\alpha}$ is null, i.g. $g_{\alpha\alpha}\equiv 0$. By
analogy, we call the coordinate line coisotropic if $g^{\alpha\alpha}\equiv
0$. Sure, coisotropic lines exist only on pseudoriemannian manifolds. Thus the
last $n-{\textsc{n}}-{\textsc{m}}$ coordinates are coisotropic. We shall see
in what follows that the nonzero diagonal metric components correspond to
indecomposable quadratic conservation laws.
Introduce notation. Divide all coordinates on three groups
$(x^{\alpha},y^{\mu},z^{\varphi})\in{\mathbb{M}}$, where indices from the
beginning, middle, and end of the Greek alphabet take the following values:
$\displaystyle\alpha,\beta,\dotsc=$ $\displaystyle 1,\dotsc,{\textsc{n}}$
$\displaystyle\text{(commuting Killing vectors)},$ (40)
$\displaystyle\mu,\nu,\dotsc=$
$\displaystyle{\textsc{n}}+1,\dotsc,{\textsc{n}}+{\textsc{m}}$
$\displaystyle(\text{quadratic conservation laws},~{}g^{\mu\mu}\neq 0),$
$\displaystyle\varphi,\phi,\dotsc=$
$\displaystyle{\textsc{n}}+{\textsc{m}}+1,\dotsc,n$
$\displaystyle(\text{coisotropic coordinates},~{}g^{\varphi\varphi}\equiv 0).$
In this way we described all possible types of separable metrics.
Suppose that all variables corresponding to Killing vector fields are
separated, and the first n coordinates are chosen cyclic. Then the
corresponding separating functions and conservation laws are
$W^{\prime}_{\alpha}=c_{\alpha}\qquad\text{and}\qquad p_{\alpha}=c_{\alpha}.$
All parameters are also divided into three groups $(c,d,a)$. Parameters $c$
are already introduced and parameters $d$ and $a$ are defined as follows. In a
fixed point ${\textsc{p}}\in{\mathbb{M}}$, we set
$\displaystyle d_{ii}:=W^{\prime 2}_{\mu}\big{|}_{{\textsc{p}},\mu=i},$
$\displaystyle i={\textsc{n}}+1,\dotsc,{\textsc{n}}+{\textsc{m}},$ (41)
$\displaystyle a_{r}:=W^{\prime}_{\varphi}\big{|}_{{\textsc{p}},\varphi=r},$
$\displaystyle r={\textsc{n}}+{\textsc{m}}+1,\dotsc,n.$
which is always possible. For convenience, we consider parameters $d_{ii}$ as
diagonal elements of matrix $d=(d_{ij})$, $d_{ij}=0$ for $i\neq j$ (to
preserve the number of independent parameters). Sure, all diagonal elements
are positive, $d_{ii}>0$. Parameters $d$ for quadratic conservation laws and
$a$ for coisotropic coordinates are enumerated by Latin indices from the
middle and end of the alphabet, respectively.
At a fixed point p, the Hamilton–Jacobi equation becomes
$g^{\alpha\beta}_{\textsc{p}}c_{\alpha}c_{\beta}+2g^{\alpha
i}_{\textsc{p}}c_{\alpha}\sqrt{d_{ii}}+2g^{\alpha
r}_{\textsc{p}}c_{\alpha}a_{r}+g^{ij}_{\textsc{p}}\sqrt{d_{ii}d_{jj}}+2g^{ir}_{\textsc{p}}\sqrt{d_{ii}}\,a_{r}+g^{rs}_{\textsc{p}}a_{r}a_{s}=2E,$
where $g^{rr}=0$. Therefore parameters $(c,d,a)$ and $E$ are simply related.
Choose the independent set of parameters in the following ways.
Case 1. Energy $E$ corresponds to the coisotropic coordinate. Redefine
$a_{n}:=2E$. Then the set of independent parameters is
$(c_{\alpha},d_{ij},a_{{\textsc{n}}+{\textsc{m}}+1},\dotsc,a_{n-1\,n-1},a_{n}:=2E).$
(42)
Case 2. Energy $E$ corresponds to the indecomposable quadratic conservation
law. Redefine $d_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}:=2E$.
Then the set of independent parameters is
$(c_{\alpha},d_{{\textsc{n}}+1\,{\textsc{n}}+1},\dotsc,d_{{\textsc{n}}+{\textsc{m}}-1\,{\textsc{n}}+{\textsc{m}}-1},d_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}:=2E,a_{r}).$
(43)
This is always possible because redefinition of parameters $a_{n}$ and
$d_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}$ leads to
multiplication of separating functions $W^{\prime}_{n}$ and
$W^{\prime}_{{\textsc{n}}+{\textsc{m}}}$ on some nonzero constants.
We are looking for solutions of the Hamilton–Jacobi equation for separating
functions $W^{\prime 2}_{\mu}$ and $W^{\prime}_{\varphi}$ within the class
functions linear in parameters $d$ and $a$:
$\displaystyle W^{\prime 2}_{\mu}:=$ $\displaystyle
b_{\mu\mu}{}^{ij}(y^{\mu},c)d_{ij}+b_{\mu\mu}{}^{r}(y^{\mu},c)a_{r}+k_{\mu\mu}(y^{\mu},c)>0,$
(44) $\displaystyle W^{\prime}_{\varphi}:=$ $\displaystyle
b_{\varphi}{}^{ij}(z^{\varphi},c)d_{ij}+b_{\varphi}{}^{r}(z^{\varphi},c)a_{r}+l_{\varphi}(z^{\varphi},c),$
(45)
where $b_{\mu\mu}{}^{ii}(y^{\mu},c)$, $b_{\mu\mu}{}^{r}(y^{\mu},c)$,
$b_{\varphi}{}^{ii}(z^{\varphi},c)$, and $b_{\varphi}{}^{r}(z^{\varphi},c)$
are some functions of single coordinate and, possibly, the first group of
parameters $c$. We suppose that matrices $b_{\mu\mu}{}^{ii}$ whose elements
are enumerated by pairs of indices $\mu\mu$ and $ii$, and $b_{\varphi}{}^{r}$
are nondegenerate. This choice is justified by the result: the functional
Hamilton–Jacobi equation has solution within this class of separating
functions, and our aim is to find at least one solution. Then the
Hamilton–Jacobi equation becomes
$g^{\alpha\beta}c_{\alpha}c_{\beta}+2g^{\alpha\mu}c_{\alpha}W^{\prime}_{\mu}+2g^{\alpha\varphi}c_{\alpha}W^{\prime}_{\varphi}+g^{\mu\nu}W^{\prime}_{\mu}W^{\prime}_{\nu}+2g^{\mu\varphi}W^{\prime}_{\mu}W^{\prime}_{\varphi}+g^{\varphi\phi}W^{\prime}_{\varphi}W^{\prime}_{\phi}=2E.$
(46)
This equality must hold for all values of coordinates and parameters. For
terms quadratic in $a$, we have
$g^{\varphi\phi}b_{\varphi}{}^{r}b_{\phi}{}^{s}a_{r}a_{s}\equiv
0\qquad\Rightarrow\qquad g^{\varphi\phi}\equiv 0,$
because matrix $b_{\varphi}{}^{r}$ is nondegenerate. That is the whole metric
block for coisotropic coordinates must be equal to zero:
$g^{\varphi\phi}\equiv 0$ for all $\varphi$ and $\phi$.
There are irrational function on $d$ on the left hand side of Eq. (46) which
cannot be cancelled. Therefore the equalities $g^{\alpha\mu}\equiv 0$ and
$g^{\mu\varphi}\equiv 0$ must hold. In addition block $g^{\mu\nu}$ must be
diagonal. Thus we have proved
###### Lemma 5.1.
If independent parameters are chosen as in Eqs. (42) or (43) and separating
functions have form (44), (45), then separable metric has block form
$g^{**}=\begin{pmatrix}g^{\alpha\beta}(y,z)&0&g^{\alpha\phi}(y,z)\\\
0&g^{\mu\nu}(y,z)&0\\\ g^{\varphi\beta}(y,z)&0&0\end{pmatrix},$ (47)
where block $g^{\mu\nu}$ is diagonal, and the star takes all values:
$*:=(\alpha,\mu,\varphi)$.
In this way, three items of our strategy are met.
We shall see in what follows that variables are separated differently
depending on which group of parameters contains energy $E$: either $a_{n}=2E$
(case 1), or $d_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}=2E$
(case 2). Therefore each separable metric belongs to one of the equivalence
class $[{\textsc{n}},{\textsc{m}},n-{\textsc{n}}-{\textsc{m}}]_{1,2}$, where
the index shows the location of $E$, if it is considered as independent
parameter, i.e. if ${\textsc{m}}\neq 0$ and/or
$n-{\textsc{n}}-{\textsc{m}}\neq 0$. All Riemannian positive definite metrics
belong to classes $[{\textsc{n}},n-{\textsc{n}},0]_{2}$.
One and the same metric in different separable coordinates can be the member
of different classes. For example, the Euclidean metric on a plane in the
Cartesian and polar coordinates belong to classes $[2,0,0]$ and $[1,1,0]_{2}$,
respectively (see example 3.1).
In the Hamiltonian formalism, the complete variables separation leads to $n$
independent conservation laws in involution. For their derivation in explicit
form, we have to replace $W^{\prime}_{\mu}\mapsto p_{\mu}$,
$W^{\prime}_{r}\mapsto p_{r}$ and solve equalities (44), (45) with respect to
parameters $d_{ii}$ and $a_{r}$. This introduces some restrictions on matrix
$b$ which will be put later.
Now we solve functional Hamilton–Jacobi equation with respect to separating
functions and metric components for different values of m and n.
### 5.1 Quadratic conservation laws
Suppose that variables in the Hamilton–Jacobi equation are completely
separated, but Killing vectors and coisotropic coordinates are absent. Then
${\textsc{n}}=0$, ${\textsc{m}}=n$, and the Hamilton–Jacobi equation is
$g^{\mu\nu}W^{\prime}_{\mu}W^{\prime}_{\nu}=2E,$ (48)
where separating functions $W^{\prime}_{\mu}(y^{\mu},d)$ depend on single
coordinate $y^{\mu}$ and, in a general case, on all independent parameters
$d:=(d_{ij})={\sf\,diag\,}(d_{11},\dotsc,d_{n-1\,n-1},d_{nn}:=2E).$ (49)
Due to lemma 5.1 the separable metric must be diagonal. Separating functions
have the form (44)
$W^{\prime 2}_{\mu}(y^{\mu},d)=b_{\mu\mu}{}^{ij}(y^{\mu})d_{ij}>0,$ (50)
where $b_{\mu\mu}{}^{ii}(y^{\mu})$ is some invertible $(n\times n)$-matrix,
whose rows depend on single coordinate, and the inequalities restrict the form
of matrix $b$ for given parameters $d$. Without loss of generality, we assume
$b_{\mu\nu}{}^{ij}\big{|}_{i\neq j}\equiv 0,\qquad
b_{\mu\nu}{}^{ij}\big{|}_{\mu\neq\nu}\equiv 0,$ (51)
because the parameter matrix $d_{ij}$ and metric $g^{\mu\nu}$ are diagonal.
Differentiate the Hamilton–Jacobi equation (48) with respect to all
parameters. As the result, we obtain the system of $n$ linear equations for
$n$ diagonal metric components:
$\begin{split}\sum_{\mu=1}^{n}g^{\mu\mu}W^{\prime}_{\mu}\frac{\partial
W^{\prime}_{\mu}}{\partial d_{ii}}=&0,\qquad i=1,\dotsc,n-1,\\\
\sum_{\mu=1}^{n}g^{\mu\mu}W^{\prime}_{\mu}\frac{\partial
W^{\prime}_{\mu}}{\partial E}=&1.\end{split}$ (52)
The determinant of this system of equations on metric components differ from
zero:
$\hphantom{\qquad{\sf\,ns\,}(\alpha)}\det\left(W^{\prime}_{\mu}\frac{\partial
W^{\prime}_{\mu}}{\partial d_{ii}}\right)=W^{\prime}_{1}\dotsc
W^{\prime}_{n}\det\left(\frac{\partial W^{\prime}_{\mu}}{\partial
d_{ii}}\right)\neq 0,\qquad{\sf\,ns\,}(\mu),$
where matrix indices are enumerated by pairs $\mu\mu$ and $ii$ due to
condition (3) which takes the form
$\det\left(\frac{\partial W^{\prime}_{\mu}}{\partial d_{ii}}\right)\neq 0.$
(53)
in our case. Consequently, the system of equations (52) has unique solution.
To find metric components, we substitute Eqs. (50) in the left hand side of
the Hamilton–Jacobi equation (48):
$g^{\mu\nu}b_{\mu\nu}{}^{ij}d_{ij}=2E.$ (54)
Since $d_{nn}=2E$, it implies that diagonal metric components are
$g^{\mu\mu}=b_{nn}{}^{\mu\mu},$ (55)
where $b_{nn}{}^{\mu\mu}(y)$ is the last row of the inverse matrix
$b_{ii}{}^{\mu\mu}$:
$\sum_{i=1}^{n}b_{\nu\nu}{}^{ii}b_{ii}{}^{\mu\mu}=b_{\nu\nu}{}^{ij}b_{ij}{}^{\mu\mu}=\delta^{\mu}_{\nu},\qquad\sum_{\mu=1}^{n}b_{ii}{}^{\mu\mu}b_{\mu\mu}{}^{jj}=b_{ii}{}^{\mu\nu}b_{\mu\nu}{}^{jj}=\delta^{j}_{i},$
whose elements depend on all coordinates in general. In addition, we must
require that all elements of the last row of matrix (55) differ from zero,
otherwise the metric becomes degenerate.
Thus the problem is solved, and the complete integral of the Hamilton–Jacobi
equation is
$W(y,d)=\sum_{\mu=1}^{n}W_{\mu}(y^{\mu},d),$
where the right hand side contains primitives
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}W_{\mu}(y^{\mu},d):=\int\\!\\!dy^{\mu}\sqrt{b_{\mu\mu}{}^{ij}d_{ij}},\qquad\qquad{\sf\,ns\,}(\mu),\quad\forall\mu.$
Integration constants in the last equalities are inessential because the
action function is defined up to a constant.
Equalities (50) imply $n$ quadratic conservation laws
$b_{ii}{}^{\mu\nu}p_{\mu}p_{\nu}=d_{ii}$ (56)
in the Hamiltonian formulation. In general, the left hand side depends on all
coordinates and momenta. These conservation laws correspond to Killing tensors
of second rank.
Rewrite Eq. (56) in the form
$p_{\mu}^{2}=b_{\mu\mu}{}^{ij}d_{ij}.$
It implies that quadratic conservation laws are indecomposable if and only if
each row of matrix $b_{\mu\mu}{}^{ii}$ contains at least two nonzero and not
proportional elements. Otherwise, the square root can be taken, and linear
conservation law appears, which contradicts the assumption on the absence of
Killing vectors.
Thus we have proved
###### Theorem 5.1.
If all diagonal inverse metric components differ from zero, $g^{\mu\mu}\neq
0$, and Killing vector fields and coisotropic components are absent, then
there exist such independent parameters (49) and separating functions (50),
that all conservation laws are quadratic, and the Hamilton–Jacobi equation
admits complete separation of variables if and only if when inverse metric
$g^{\mu\nu}$ is diagonal with components (55), where $b_{ii}{}^{\mu\mu}$ is
the inverse matrix to an arbitrary nondegenerate matrix
$b_{\mu\mu}{}^{ii}(y^{\mu})$, whose rows depend on single coordinate $y^{\mu}$
and contain at least two nonzero not proportional elements. In addition,
inequalities $b_{nn}{}^{\mu\mu}\neq 0$ must hold for all values of index
$\mu$, and arbitrary functions $b_{\mu\mu}{}^{ii}$ must be chosen in such a
way that the system of equations for separating functions $W^{\prime}_{\mu}$
(50) have real solutions for given values of parameters $d$.
Consequently, diagonal metric components are parameterized by $n^{2}$
arbitrary functions on single coordinate $b_{\mu\mu}{}^{ii}(y^{\mu})$,
satisfying three conditions: $\det b_{\mu\mu}{}^{ii}\neq 0$,
$b_{nn}{}^{\mu\mu}\neq 0$ for all $\mu$, and each row of matrix
$b_{\mu\mu}{}^{ii}$ must contain at least two nonzero not proportional
elements. Note that components of the inverse matrix $b_{ii}{}^{\mu\mu}(y)$
depend on all coordinates in general.
In fact, we prove the following. The right hand sides of Eqs. (50) contain
arbitrary functions of single coordinates linear and homogeneous in $d$.
Therefore the separable metric is found for arbitrary indecomposable quadratic
conservation laws. All restrictions on arbitrary functions in the conservation
laws follow from nondegeneracy of the separable metric, indecomposability of
the quadratic Killing tensors, and existence of solutions for separating
functions $W^{\prime}_{\mu}$ in Eqs. (50).
In the Hamiltonian formulation, we have $n$ involutive quadratic conservation
laws (56). Consequently, the Hamiltonian system is Liouville integrable.
###### Example 5.1.
Consider two dimensional Euclidean space $(y^{1},y^{2})\in{\mathbb{R}}^{2}$
and choose matrix $b$ in a general form
$b_{\mu\mu}{}^{ii}:=\begin{pmatrix}\phi_{11}(y^{1})&\phi_{12}(y^{1})\\\
\phi_{21}(y^{2})&\phi_{22}(y^{2})\end{pmatrix}\qquad\Rightarrow\qquad
b_{ii}{}^{\mu\mu}:=\frac{1}{\det
b}\begin{pmatrix}~{}~{}\phi_{22}&-\phi_{12}\\\
-\phi_{21}&~{}~{}\phi_{11}\end{pmatrix}$ (57)
and assume that
$\det b=\phi_{11}\phi_{22}-\phi_{12}\phi_{21}\neq 0.$ (58)
If all matrix elements differ from zero, then conditions of theorem 5.1 hold:
elements of the first and second rows depend on $y^{1}$ and $y^{2}$,
respectively, and each row contains two nonzero elements assumed to be not
proportional. Separable metric corresponding to matrix $b$ us parameterized by
four functions of single coordinates:
$g^{\mu\nu}=\frac{1}{\det b}\begin{pmatrix}-\phi_{21}&0\\\
~{}~{}0&\phi_{11}\end{pmatrix}.$ (59)
Now we consider particular cases. Let $\phi_{11}\equiv 1$ and
$\phi_{21}\equiv-1$. Then
$b_{\mu\mu}{}^{ii}:=\begin{pmatrix}~{}~{}1&\phi_{12}(y^{1})\\\
-1&\phi_{22}(y^{2})\end{pmatrix}\qquad\Rightarrow\qquad
b_{ii}{}^{\mu\mu}:=\frac{1}{\phi_{12}+\phi_{22}}\begin{pmatrix}\phi_{22}&-\phi_{12}\\\
1&1\end{pmatrix}.$
The respective inverse metric is conformally Euclidean:
$g^{\mu\nu}=\frac{1}{\phi_{12}+\phi_{22}}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}.$ (60)
The Hamilton–Jacobi equation becomes
$W^{\prime 2}_{1}+W^{\prime 2}_{2}=2E(\phi_{12}+\phi_{22}).$
Complete separation of variables yields
$W^{\prime 2}_{1}=d_{11}+2E\phi_{12},\qquad W^{\prime
2}_{2}=-d_{11}+2E\phi_{22},$
where $d_{11}$ and $E$ are two independent parameters. The conservation laws
are quadratic
$\frac{1}{\phi_{12}+\phi_{22}}\big{(}\phi_{22}p_{1}^{2}-\phi_{12}p_{2}^{2}\big{)}=d_{11},\qquad\frac{1}{\phi_{12}+\phi_{22}}\big{(}p_{1}^{2}+p_{2}^{2}\big{)}=2E.$
(61)
In general, they are indecomposable for $d_{11}\neq 0$, $E>0$ and nontrivial
functions $\phi_{12}$, $\phi_{22}$. The parameters domain for $d_{11}$, $E$
and acceptable form of arbitrary functions are defined by the following
inequalities:
$d_{11}+2E\phi_{12}>0,\qquad-d_{11}+2E\phi_{22}>0.$
If, for example, $d_{11}>0$ and $E>0$, then
$\phi_{12}>-\frac{d_{11}}{2E},\qquad\phi_{22}>\frac{d_{11}}{2E}.$
Consequently this example is the particular two dimensional case of the
Liouville system for Riemannian metric considered in example 2.1.
Now put $\phi_{11}\equiv-1$ and $\phi_{21}\equiv-1$. Then
$b_{\mu\mu}{}^{ii}:=\begin{pmatrix}-1&\phi_{12}(y^{1})\\\
-1&\phi_{22}(y^{2})\end{pmatrix}\qquad\Rightarrow\qquad
b_{ii}{}^{\mu\mu}:=\frac{1}{\phi_{12}-\phi_{22}}\begin{pmatrix}\phi_{22}&-\phi_{12}\\\
1&-1\end{pmatrix}.$
These matrices imply conformally Lorentzian metric
$g^{\mu\nu}=\frac{1}{\phi_{12}-\phi_{22}}\begin{pmatrix}1&~{}~{}0\\\
0&-1\end{pmatrix}.$ (62)
The respective Hamilton–Jacobi equation is
$W^{\prime 2}_{1}-W^{\prime 2}_{2}=2E(\phi_{12}-\phi_{22}),$
and variables are separated:
$W^{\prime 2}_{1}=-d_{11}+2E\phi_{12},\qquad W^{\prime
2}_{2}=-d_{11}+2E\phi_{22}.$
They are indecomposable for $d_{11}\neq 0$, $E\neq 0$, and nonconstant
functions $\phi_{12}$, $\phi_{22}$. The conservation laws are quadratic:
$\frac{1}{\phi_{12}-\phi_{22}}\big{(}\phi_{22}p_{1}^{2}-\phi_{12}p_{2}^{2}\big{)}=d_{11},\qquad\frac{1}{\phi_{12}-\phi_{22}}\big{(}p_{1}^{2}-p_{2}^{2}\big{)}=2E.$
(63)
The parameter domain of definition and the form of arbitrary functions are
defined by inequalities:
$-d_{11}+2E\phi_{12}>0,\qquad-d_{11}+2E\phi_{22}>0.$
If $E>0$, then there are two possibilities when $\phi_{12}-\phi_{22}\neq 0$:
$\phi_{12}>\phi_{22}>\frac{d_{11}}{2E},\qquad\phi_{22}>\phi_{12}>\frac{d_{11}}{2E},$
for $g^{11}>0$ and $g^{11}<0$, respectively. This is also two-dimensional
Liouville system considered in example 2.1 but for Lorentzian signature
metric.
Let $\phi_{12}\equiv 0$ and $\phi_{21}\equiv-1$. Then
$b_{\mu\mu}{}^{ii}:=\begin{pmatrix}~{}~{}\phi_{11}(x)&0\\\
-1&\phi_{22}(y)\end{pmatrix}\qquad\Rightarrow\qquad
b_{ii}{}^{\mu\mu}:=\frac{1}{\phi_{11}\phi_{22}}\begin{pmatrix}\phi_{22}&0\\\
1&\phi_{11}\end{pmatrix}.$
The conditions of theorem 5.1 are not fulfilled because the first row of
matrix $b_{\mu\mu}{}^{ii}$ contains only one nonzero element. The respective
inverse metric is
$g^{\mu\nu}=\frac{1}{\phi_{11}\phi_{22}}\begin{pmatrix}1&0\\\
0&\phi_{11}\end{pmatrix}.$ (64)
The Hamilton–Jacobi equation for this metric
$W^{\prime 2}_{1}+\phi_{11}W^{\prime 2}_{2}=2E\phi_{11}\phi_{22},$
admits complete separation of variables:
$W^{\prime 2}_{1}=\phi_{11}d_{11},\qquad W^{\prime
2}_{2}=-d_{11}+2E\phi_{22}.$
The conservation laws are quadratic:
$\frac{1}{\phi_{11}\phi_{22}}p_{1}^{2}=d_{11},\qquad\frac{1}{\phi_{11}\phi_{22}}\big{(}p_{1}^{2}+\phi_{11}p_{2}^{2}\big{)}=2E.$
(65)
The parameter domain of definition and admissible form of arbitrary functions
are defined by inequalities:
$\phi_{11}d_{11}>0,\qquad-d_{11}+2E\phi_{22}>0.$
For $d_{11}>0$ and $E>0$, for example, they restrict arbitrary functions:
$\phi_{11}>0,\qquad\phi_{22}>\frac{d_{11}}{2E}.$
In this case the first conservation law (65) is decomposable: it is the square
of linear conservation law $p_{1}/\sqrt{\phi_{11}\phi_{22}}=\sqrt{d_{11}}$.
Consequently, we have in fact one linear and one quadratic conservation law. ∎
Now we simplify the form of matrix $b_{\mu\mu}{}^{ii}$ using the canonical
transformation. It turns out that one of nonzero elements in each row of
matrix $b_{\mu\mu}{}^{ii}$ can be transformed to unity. For example, let
$b_{\mu\mu}{}^{ii}\neq 0$ for given $\mu$ and $i$. For definiteness we assume
that it is the diagonal element, $\mu=i$. Choose the generating function of
canonical transformation for one pair of canonically conjugate variables
$(y^{\mu},p_{\mu})\mapsto(Y^{i},P_{i})$ as
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}S_{2}(y,P):=\int\\!\\!dy^{\mu}\sqrt{|b_{\mu\mu}{}^{ii}|}\,P_{i},\qquad\qquad{\sf\,ns\,}(\mu,i).$
(66)
Note that it must be linear in $P$, because otherwise the coordinate
transformation $x\mapsto X(x)$ depends on momenta which contradicts the
equivalence relation. Then we get for fixed $\mu$:
$p_{\mu}=\frac{\partial S_{2}}{\partial
y^{\mu}}=\sqrt{|b_{\mu\mu}{}^{ii}|}\,P_{i},\qquad Y^{i}=\frac{\partial
S_{2}}{\partial
P_{i}}=\int\\!\\!dy^{\mu}\sqrt{|b_{\mu\mu}{}^{ii}|},\qquad\qquad{\sf\,ns\,}(\mu,i),$
and Eq. (50) becomes
$|b_{\mu\mu}{}^{ii}|\,\tilde{W}^{\prime
2}_{i}=b_{\mu\mu}{}^{ii}\,d_{ii}+\sum_{j\neq i}b_{\mu\mu}{}^{jj}\,d_{jj}$
for transformed separating function $\tilde{W}_{i}$. Dividing it by
$|b_{\mu\mu}{}^{ii}|$, we obtain the quadratic equality
$\tilde{W}^{\prime 2}_{i}=\pm d_{ii}+\sum_{j\neq
i}\tilde{b}_{ii}{}^{jj}\,d_{jj}$ (67)
with some new functions $\tilde{b}_{ii}{}^{jj}(Y^{i})$. Similar transformation
can be performed for each row. So, without loss of generality, one of nonzero
elements in each row can be set to unity because signs $\pm 1$ can be
attributed to parameters $d$. It means that canonical separable metric is
parameterized by $n^{2}-n$ functions of single coordinates.
In fact, this statement is evident. Indeed, Eq. (50) depends only on single
coordinate, and one of nonzero elements can be transformed to $\pm 1$ by
coordinate transformation $y^{\mu}\mapsto\tilde{y}^{\mu}(y^{\mu})$.
### 5.2 Linear and quadratic conservation laws
Assume that metric admits exactly $1\leq{\textsc{n}}<n$ and not more commuting
Killing vector fields, and coisotropic coordinates are absent,
$n-{\textsc{n}}-{\textsc{m}}=0$. There is one linear conservation law for each
Killing vector. Then we need additional $n-{\textsc{n}}$ independent
involutive conservation laws to provide complete integrability. These
conservation laws are quadratic as was shown in the preceeding section. Now we
have two groups of coordinates $(x^{\alpha},y^{\mu})\in{\mathbb{M}}$ and two
groups of independent parameters $c$ and $d$ (43). Separable metric (47) in
this case is block diagonal
$g^{**}=\begin{pmatrix}g^{\alpha\beta}&0\\\ 0&g^{\mu\nu}\end{pmatrix},$ (68)
where the lower block $g^{\mu\nu}$ is diagonal.
Functions $k$ in equality (44) must be quadratic in $c$ as the consequence of
the Hamilton–Jacobi equation. Therefore separating functions are
$W^{\prime
2}_{\mu}=b_{\mu\mu}{}^{ij}d_{ij}+k^{\alpha\beta}_{\mu\mu}c_{\alpha}c_{\beta}>0,$
(69)
where $b_{\mu\mu}{}^{ii}(y^{\mu})$ is some nondegenerate matrix whose elements
in each row depend on single coordinate corresponding to the number of the
row, and functions $k^{\alpha\beta}_{\mu\mu}(y^{\mu})$ are arbitrary and
symmetric in indices $\alpha$ and $\beta$ but can be nondiagonal in them.
Their indices can be written one under another because they will never be
lowered or raised.
There is the new property. We shall see that the term
$k^{\alpha\beta}_{\mu\mu}c_{\alpha}c_{\beta}$ in Eq. (69) must not vanish for
nondegenerate metric. Therefore the requirement that each row of matrix
$b_{\mu\mu}{}^{ii}$ contains at least two not proportional elements is
unnecessary.
After separating the first group of coordinates, the Hamilton–Jacobi equation
becomes
$g^{\alpha\beta}c_{\alpha}c_{\beta}+g^{\mu\nu}(b_{\mu\nu}{}^{ij}d_{\ij}+k^{\alpha\beta}_{\mu\nu}c_{\alpha}c_{\beta})=2E,$
(70)
where matrix $g^{\mu\nu}$ is diagonal. We get previous expression (55) for
elements in the second block because $d_{nn}:=2E$. Therefore
$g^{\alpha\beta}c_{\alpha}c_{\beta}+g^{\mu\nu}k^{\alpha\beta}_{\mu\nu}c_{\alpha}c_{\beta}=0.$
(71)
This equality must be fulfilled for all $c$, and defines the upper block of
the inverse separable metric (68):
$g^{\alpha\beta}=-k^{\alpha\beta}_{\mu\nu}g^{\mu\nu},$ (72)
where $k^{\alpha\beta}_{\mu\mu}(y^{\mu})$ are arbitrary functions on single
coordinates providing nondegeneracy of $g^{\alpha\beta}$ and positivity of
right hand sides of Eqs. (69). Thus we have proved
###### Theorem 5.2.
If separable metric admits exactly $1\leq{\textsc{n}}<n$ and not more
commuting Killing vector fields, and coisotropic coordinates are absent,
$n-{\textsc{n}}-{\textsc{m}}=0$, then there exists such set of independent
parameters (43) and separating functions (69), that separable inverse metric
has block form (68). The lower block is diagonal with elements (55), where
$b_{ii}{}^{\mu\mu}(y)$ is the matrix inverse to matrix
$b_{\mu\mu}{}^{ii}(y^{\mu})$ whose elements in each row depend only on single
coordinates, and all elements of the last row differ from zero,
$b_{nn}{}^{\mu\mu}\neq 0$. The upper block has form (72) with arbitrary
functions
$k^{\alpha\beta}_{\mu\mu}(y^{\mu})=k^{\beta\alpha}_{\mu\mu}(y^{\mu})$
depending on single coordinates. In addition, the upper block (71) must be
nondegenerate and provide positiveness of right hand sides of Eqs. (69). In
the Hamiltonian formulation, there are $n-{\textsc{n}}$ indecomposable
quadratic conservation laws:
$\begin{split}p_{\alpha}=&c_{\alpha},\\\
b_{ii}{}^{\mu\nu}\big{(}p_{\mu}p_{\nu}-k^{\alpha\beta}_{\mu\nu}c_{\alpha}c_{\beta}\big{)}=&d_{ii}.\end{split}$
(73)
Quadratic conservation laws (73) appear after contraction of Eqs. (69) with
inverse matrix $b_{ii}{}^{\mu\mu}$.
Thus the canonical separable inverse metric of type
$[{\textsc{n}},n-{\textsc{n}},0]_{2}$ has block diagonal form
$g^{**}=\begin{pmatrix}-k^{\alpha\beta}_{\mu\nu}g^{\mu\nu}&0\\\
0&g^{\mu\nu}\end{pmatrix},$ (74)
where the lower block $g^{\mu\nu}$ is diagonal (55), and stars denote all
indices $*=(\alpha,\mu)$.
We can also simplify matrix $b_{\mu\mu}{}^{ii}$ using canonical transformation
(66) from previous section. Therefore one of nonzero elements in each row of
matrix $b$ can se set to unity without loss of generality.
###### Example 5.2.
Consider the simplest example, when separable metric admits only one
indecomposable quadratic conservation law, i.e. ${\textsc{n}}=n-1$ and
$d_{nn}=2E$. In this case, there is only one coordinate $y$, which may not be
enumerated. Matrix $b_{\mu\mu}{}^{ii}$ consists of only one element $b(y)\neq
0$, and its inverse is $1/b$. Due to theorem 5.2, the most general canonical
separable metric (74) is block diagonal:
$g^{**}=\begin{pmatrix}g^{\alpha\beta}(y)&0\\\
0&\frac{1}{b(y)}\end{pmatrix}=\begin{pmatrix}-k^{\alpha\beta}(y)\frac{1}{b(y)}&0\\\
0&\frac{1}{b(y)}\end{pmatrix},\qquad\alpha,\beta=1,\dotsc,n-1,$ (75)
where the $(n-1)\times(n-1)$ block $g^{\alpha\beta}(y)$ can be arbitrary
nondegenerate matrix if we include arbitrary function $b(y)\neq 0$ in the
definition of $k^{\alpha\beta}(y)$. Variables are separated as
$W^{\prime}_{\alpha}=c_{\alpha},\qquad W^{\prime
2}_{n}=2Eb+k^{\alpha\beta}c_{\alpha}c_{\beta}>0,$
where the inequality restricts functions $b$ and $k$ for fixed $E$ and $c$.
The respective conservation laws are
$p_{\alpha}=c_{\alpha},\qquad\frac{1}{b}\big{(}p_{n}^{2}-k^{\alpha\beta}c_{\alpha}c_{\beta}\big{)}=2E.$
(76)
To simplify metric (75) we perform the canonical transformation
$(y,p_{n})\mapsto(Y,P_{n})$ with the generating function
$S_{2}:=\int\\!\\!dy\sqrt{|b(y)|}P_{n},$
leaving remaining variables $x^{\alpha},p_{\alpha}$ untouched. Then
$p_{n}=\frac{\partial S_{2}}{\partial y}=\sqrt{|b|}P_{n},\qquad
Y=\frac{\partial S_{2}}{\partial P_{n}}=\int\\!\\!dy\sqrt{|b|},$
and the quadratic conservation law takes the form
$\pm P^{2}_{n}+g^{\alpha\beta}c_{\alpha}c_{\beta}=2E,$
After this transformation of variables, the canonical separable metric becomes
$g^{**}=\begin{pmatrix}g^{\alpha\beta}(y)&0\\\ 0&\pm 1\end{pmatrix},$ (77)
where the sign choice depends on the signature of the metric. ∎
### 5.3 Coisotropic coordinates
We showed in section 4 that linear conservation laws for the “symmetric”
choice of independent parameters correspond to Killing vector fields. However,
if energy $E$ in the right hand side of the Hamilton–Jacobi equation is
considered as independent parameter, linear conservation laws not related to
Killing vectors may appear. This possibility arises only for indefinite
metrics having coisotropic coordinates. For Riemannian positive definite
metrics linear conservation laws are always related to Killing vector fields.
Existence of these linear conservation laws can take place only when
sufficient number of commuting Killing vectors are simultaneously present.
Suppose that ${\textsc{m}}=0$, i.e. we have only Killing vector fields and
coisotropic coordinates. then we have two groups of coordinates
$(x^{\alpha},z^{\varphi})\in{\mathbb{M}}$, where indices take the following
values
$\alpha,\beta,\dotsc=1,\dotsc,{\textsc{n}};\qquad\varphi,\phi,\dotsc={\textsc{n}}+1,\dotsc,n;\qquad\frac{n}{2}\leq{\textsc{n}}<n.$
(78)
By assumption, there are independent parameters
$\\{c_{1},\dotsc,c_{\textsc{n}},a_{{\textsc{n}}+1},\dotsc,a_{n-1},a_{n}:=2E\\}.$
(79)
Separable metric of type $[{\textsc{n}},0,n-{\textsc{n}}]_{1}$ has block form
(47)
$g^{**}=\begin{pmatrix}g^{\alpha\beta}(z)&g^{\alpha\phi}(z)\\\
g^{\varphi\beta}(z)&0\end{pmatrix}.$ (80)
This metric must be nondegenerate, therefore the number of Killing vectors
cannot be less then the number of coisotropic coordinates
${\textsc{n}}\geq
n-{\textsc{n}}\qquad\Leftrightarrow\qquad{\textsc{n}}\geq\frac{n}{2},$ (81)
which was assumed at the very beginning (78). In addition, the rank of
rectangular matrix $g^{\alpha\phi}$ is equal to $n-{\textsc{n}}$, otherwise
separable metric is degenerate.
After separation of cyclic coordinates condition (6) becomes
$\det\big{(}\partial^{r}W^{\prime}_{\varphi}\big{)}\neq 0,$ (82)
and Hamilton–Jacobi equation (7) for metric (80) takes the form
$g^{\alpha\beta}c_{\alpha}c_{\beta}+2g^{\alpha\varphi}c_{\alpha}W^{\prime}_{\varphi}=2E.$
(83)
Separating functions $W^{\prime}_{\varphi}$ (45) are linear in parameters $a$:
$W^{\prime}_{\varphi}=b_{\varphi}{}^{r}(z^{\varphi},c)a_{r}+l_{\varphi}(z^{\varphi},c),$
(84)
where $b_{\varphi}{}^{r}$ is some nondegenerate matrix, whose elements in each
row depend only on single coordinate $z^{\varphi}$ and, possibly, on the first
group of parameters $(c_{\alpha})$, and $l_{\varphi}$ are some functions also
on single coordinate and first group of parameters. Equating terms with $E$ in
Eq. (83), we obtain
$2g^{\alpha\varphi}c_{\alpha}=b_{n}{}^{\varphi},$ (85)
where $b_{n}{}^{\varphi}$ is the last row of matrix $b_{r}{}^{\varphi}$, which
is inverse to $b_{\varphi}{}^{r}$:
$b_{r}{}^{\varphi}b_{\phi}{}^{r}=\delta_{\phi}^{\varphi}$. This equality
implies that the last row of matrix $b_{n}{}^{\varphi}(z,c)$, whose elements
depend in general on all coordinates $z$, are linear in $c_{\alpha}$. Let
$b_{n}{}^{\varphi}:=b_{n}{}^{\alpha\varphi}c_{\alpha},$ (86)
where $b_{n}{}^{\alpha\varphi}(z)$ is a set of arbitrary functions on $z$,
such that matrix $(b_{r}{}^{\alpha\varphi}c_{\alpha})$ is not degenerate.
Equality (85) must hold for all values of $c_{\alpha}$, therefore
$2g^{\alpha\varphi}=b_{n}{}^{\alpha\varphi}.$ (87)
Now we substitute the expression for separating functions
$W^{\prime}_{\varphi}$ (84) and use Eq. (85) in the Hamilton–Jacobi equation
(83):
$g^{\alpha\beta}c_{\alpha}c_{\beta}+b_{n}{}^{\alpha\varphi}c_{\alpha}l_{\varphi}=0.$
(88)
It implies that functions $l_{\varphi}$ are linear and homogeneous in
$c_{\alpha}$:
$l_{\varphi}=l^{\alpha}_{\varphi}(z^{\varphi})c_{\alpha},$ (89)
where $l^{\alpha}_{\varphi}(z^{\varphi})$ are some functions depending on
single coordinate. Indices here can be written one over the other because they
will be never raised or lowered. Then Eq. (88) defines the square block for
Killing vectors
$2g^{\alpha\beta}=-b_{n}{}^{\alpha\varphi}l^{\beta}_{\varphi}-b_{n}{}^{\beta\varphi}l^{\alpha}_{\varphi}.$
(90)
Thus Hamilton–Jacobi equation (83) is solved, and the separable metric is
$g^{**}=\frac{1}{2}\begin{pmatrix}-b_{n}{}^{\alpha\chi}l_{\chi}^{\beta}-b_{n}{}^{\beta\chi}l_{\chi}^{\alpha}&b_{n}{}^{\alpha\phi}\\\\[4.0pt]
b_{n}{}^{\beta\varphi}&0\end{pmatrix},$ (91)
where the star denotes all indices, $*=(\alpha,\varphi)$.
To simplify the separable metric, we perform the canonical transformation with
generating function
$S_{2}:=x^{\alpha}P_{\alpha}+z^{\varphi}P_{\varphi}+\sum_{\varphi={\textsc{n}}+1}^{n}\int\\!\\!dz^{\varphi}l^{\alpha}_{\varphi}(z^{\varphi})P_{\alpha}.$
(92)
Then
$\displaystyle p_{\alpha}=$ $\displaystyle P_{\alpha},$ $\displaystyle\quad
p_{\varphi}=$ $\displaystyle P_{\varphi}+l^{\alpha}_{\varphi}P_{\alpha},$
$\displaystyle X^{\alpha}=$ $\displaystyle
x^{\alpha}+\sum_{\varphi={\textsc{n}}+1}^{n}\int\\!\\!dz^{\varphi}\,l^{\alpha}_{\varphi},$
$\displaystyle Z^{\phi}=$ $\displaystyle
z^{\varphi},\qquad{\sf\,ns\,}(\varphi).$
The last two equalities define the coordinate transformation with the Jacobi
matrix
$\frac{\partial(X^{\beta},Z^{\phi})}{\partial(x^{\alpha},z^{\varphi})}=\begin{pmatrix}\delta_{\alpha}^{\beta}&0\\\
l^{\beta}_{\varphi}&\delta_{\varphi}{}^{\phi}\end{pmatrix}.$
The metric transforms as follows $g^{**}\mapsto\tilde{g}^{**}$ with
$\tilde{g}^{\alpha\beta}=0,\qquad\tilde{g}^{\alpha\phi}=g^{\alpha\phi},\qquad\tilde{g}^{\varphi\phi}=0,$
where equality $g^{\varphi\phi}\equiv 0$ was used. After this transformation
the separable metric is simplified
$g^{**}=\frac{1}{2}\begin{pmatrix}0&b_{n}{}^{\alpha\phi}\\\\[4.0pt]
b_{n}{}^{\beta\varphi}&0\end{pmatrix}.$ (93)
This metric is always degenerate except for $n=2{\textsc{n}}$. Then
$\det
g^{**}=\frac{(-1)^{\textsc{n}}}{2^{n}}\det{}^{2}(b_{n}{}^{\alpha\phi})\neq 0.$
Now we are left with the problem to find such matrix $b_{\varphi}{}^{r}(z,c)$
in Eq. (84), that equality (87) be satisfied for all coordinates $z$ and
parameters $c$. In other words we have to extract explicitly the dependence of
matrix elements $b_{\varphi}{}^{r}$ on parameters, because Eq. (86) contains
elements of the inverse metric.
###### Proposition 5.1.
Matrix elements $(b_{\varphi}{}^{r})$ in Eq. (84) satisfying equality (87)
must have the form
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}b_{\varphi}{}^{r}(z^{\varphi},c)=\frac{\phi_{\varphi}{}^{r}}{h_{\varphi}^{\alpha}c_{\alpha}},\qquad\qquad{\sf\,ns\,}(\varphi),$
(94)
where $\phi_{\varphi}{}^{r}(z^{\varphi})$ is the nondegenerate matrix, whose
elements of each row depend on single coordinate and contain unity, and
$h_{\varphi}^{\alpha}(z^{\varphi})$ is a set of arbitrary nonzero functions
depending on single coordinates.
###### Proof.
Parameterise matrix $(b_{\varphi}{}^{r})$ as
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}b_{\varphi}{}^{r}(z^{\varphi},c):=\frac{\phi_{\varphi}{}^{r}(z^{\varphi},c)}{h_{\varphi}(x^{\varphi},c)},\qquad\qquad{\sf\,ns\,}(\varphi),$
where $h_{\varphi}$ are some nonzero functions, and
$(\phi_{\varphi}{}^{r}):=\begin{pmatrix}1&\phi_{{\textsc{n}}+1}{}^{{\textsc{n}}+2}&\phi_{{\textsc{n}}+1}{}^{{\textsc{n}}+3}&\cdots&\phi_{{\textsc{n}}+1}{}^{n}\\\\[8.0pt]
\phi_{{\textsc{n}}+2}{}^{{\textsc{n}}+1}&1&\phi_{{\textsc{n}}+2}{}^{{\textsc{n}}+3}&\cdots&\phi_{{\textsc{n}}+2}{}^{n}\\\\[8.0pt]
\phi_{{\textsc{n}}+3}{}^{{\textsc{n}}+1}&\phi_{{\textsc{n}}+3}{}^{{\textsc{n}}+2}&1&\cdots&\phi_{{\textsc{n}}+3}{}^{n}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
\phi_{n}{}^{{\textsc{n}}+1}&\phi_{n}{}^{{\textsc{n}}+2}&\phi_{n}{}^{{\textsc{n}}+3}&\cdots&1\end{pmatrix}.$
(95)
This matrix has unities on the diagonal, and each row of matrix
$(b_{\varphi}{}^{r})$ is the quotient of the row $(\phi_{\varphi}{}^{r})$ by
$h_{\varphi}$.
Multiply Eq. (87) by matrix $b_{\varphi}{}^{r}$ and sum over $\varphi$:
$g^{\alpha\varphi}c_{\alpha}b_{\varphi}{}^{r}=\sum_{\varphi={\textsc{n}}+1}^{n}g^{\alpha\varphi}c_{\alpha}\frac{\phi_{\varphi}{}^{r}}{h_{\varphi}}=\delta_{n}^{r}.$
This equality must be fulfilled for all coordinates and parameters. Therefore
$\phi_{\varphi}{}^{r}(z^{\varphi},c)=\phi_{\varphi}{}^{r}(z^{\varphi}),\qquad
h_{\varphi}(x^{\varphi},c)=h_{\varphi}^{\alpha}(z^{\varphi})c_{\alpha},$
where $h_{\varphi}^{\alpha}(z^{\varphi})$ are some functions on single
coordinate. It means that matrix (95) must not depend on $c$, and functions
$h_{\varphi}$ be linear in $c$. ∎
It implies that matrix $(b_{\varphi}{}^{r})$ and consequently canonical
separable metric is parameterized by $2{\textsc{n}}^{2}-{\textsc{n}}$
arbitrary functions, and
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}2g^{\alpha\varphi}=b_{n}{}^{\alpha\varphi}=\phi_{n}{}^{\varphi}h_{\varphi}^{\alpha},\qquad\qquad{\sf\,ns\,}(\varphi),$
(96)
where $(\phi_{r}{}^{\varphi})$ is the matrix inverse to
$(\phi_{\varphi}{}^{r})$. Thus we have found canonical separable metric in the
case of only Killing vector fields and coisotropic coordinates.
###### Theorem 5.3.
If separable metric admit exactly n and not more commuting Killing vector
fields, and all other coordinates are coisotropic, then there exists such set
of parameters (79) and separating functions (84), that the Hamilton–Jacobi
equation admit complete separation of variables if and only if the dimension
of the manifold is even, $n=2{\textsc{n}}$, and canonical separable metric has
block form (93). The off diagonal blocks are given by Eqs. (96), where
$(\phi_{n}{}^{\varphi})$ is the last row of the matrix inverse to arbitrary
nondegenerate matrix (95), whose rows depend on single coordinates, and the
diagonal consists of unities. All conservation laws are linear:
$\begin{split}p_{\alpha}=&c_{\alpha},\\\
\sum_{\varphi}\phi_{r}{}^{\varphi}h_{\varphi}^{\alpha}c_{\alpha}p_{\varphi}=&a_{r}.\end{split}$
(97)
The second conservation law (97) is linear in momenta only after separation of
cyclic coordinates. If this is not done, then it is quadratic
$\sum_{\varphi}\phi_{r}{}^{\varphi}h_{\varphi}^{\alpha}p_{\alpha}p_{\varphi}=a_{r}.$
The respective Killing tensor is indecomposable in general.
The Hamilton–Jacobi equation for canonical separable metric (93) is
$\sum_{\varphi}\phi_{n}{}^{\varphi}h_{\varphi}^{\alpha}W^{\prime}_{\alpha}W^{\prime}_{\varphi}=2E,$
(98)
and variables are completely separated
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}W^{\prime}_{\alpha}=c_{\alpha},\qquad
W^{\prime}_{\varphi}=\frac{\phi_{\varphi}{}^{r}}{h_{\varphi}^{\alpha}c_{\alpha}}a_{r},\qquad\qquad{\sf\,ns\,}(\varphi).$
The unusual feature of this separation is that parameters $c$ appear in the
denominator.
Finally, using canonical transformations we simplify the form of the canonical
separable metric by setting one of the nonzero elements in each row of matrix
$h_{\varphi}^{\alpha}$ to unity.
### 5.4 General separation of variables
If separable metric admits simultaneously commuting Killing vector fields,
indecomposable quadratic conservation laws, and coisotropic coordinates, then
coordinates are divided into three groups $(x,y,z)\in{\mathbb{M}}$ described
in section 5. There are two possibilities for full sets of independent
parameters: (42) and (43), and separable metric must have block form (47).
Separable functions are given by Eqs. (44) and (45).
First, we consider case 2, when energy enters parameters for quadratic
conservation laws,
$d_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}:=2E$. Without loss
of generality, we set
$b_{\mu\nu}{}^{ij}\big{|}_{i\neq j}\equiv 0,\qquad
b_{\mu\nu}{}^{ij}\big{|}_{\mu\neq\nu}\equiv 0,\qquad
b_{\mu\nu}{}^{r}\big{|}_{\mu\neq\nu}\equiv 0,\qquad
k_{\mu\nu}\big{|}_{\mu\neq\nu}\equiv 0,$
because metric $g^{\mu\nu}$ and matrix of parameters $d_{ij}$ are diagonal.
The respective Hamilton–Jacobi equation is
$g^{\alpha\beta}c_{\alpha}c_{\beta}+g^{\mu\nu}\big{(}b_{\mu\nu}{}^{ij}d_{ij}+b_{\mu\nu}{}^{r}a_{r}+k_{\mu\nu}\big{)}+2g^{\alpha\phi}c_{\alpha}\big{(}b_{\phi}{}^{ij}d_{ij}+b_{\phi}{}^{r}a_{r}+l_{\phi}\big{)}=2E.$
(99)
Differentiate this equation consequently with respect to parameters $d$ and
$a$:
$\displaystyle
g^{\mu\nu}\frac{\partial(W^{\prime}_{\mu}W^{\prime}_{\nu})}{\partial
d_{ii}}+2g^{\alpha\varphi}c_{\alpha}\frac{\partial
W^{\prime}_{\varphi}}{\partial d_{ii}}=$ $\displaystyle 0,\qquad
i\neq{\textsc{n}}+{\textsc{m}},$ (100) $\displaystyle
g^{\mu\nu}\frac{\partial(W^{\prime}_{\mu}W^{\prime}_{\nu})}{\partial
d_{ii}}+2g^{\alpha\varphi}c_{\alpha}\frac{\partial
W^{\prime}_{\varphi}}{\partial d_{ii}}=$ $\displaystyle 1,\qquad
i={\textsc{n}}+{\textsc{m}},$ $\displaystyle
g^{\mu\nu}\frac{\partial(W^{\prime}_{\mu}\tilde{W}^{\prime}_{\nu})}{\partial
a_{r}}+2g^{\alpha\varphi}c_{\alpha}\frac{\partial
W^{\prime}_{\varphi}}{\partial a_{r}}=$ $\displaystyle 0,$
This system of equation is considered as the system of linear algebraic
equations for $g^{\mu\nu}$ and linear combinations
$g^{\alpha\varphi}c_{\alpha}$. Its determinant differs from zero
$\det\begin{pmatrix}\displaystyle\frac{\partial(W^{\prime 2}_{\mu})}{\partial
d_{ii}}&\displaystyle\frac{\partial W^{\prime}_{\varphi}}{\partial
d_{ii}}\\\\[8.0pt] \displaystyle\frac{\partial(W^{\prime 2}_{\mu})}{\partial
a_{r}}&\displaystyle\frac{\partial W^{\prime}_{\varphi}}{\partial
a_{r}}\end{pmatrix}=2^{{\textsc{m}}}\tilde{W}^{\prime}_{{\textsc{n}}+1}\dotsc\tilde{W}^{\prime}_{{\textsc{n}}+{\textsc{m}}}\det\begin{pmatrix}\displaystyle\frac{\partial(W^{\prime}_{\mu})}{\partial
d_{ii}}&\displaystyle\frac{\partial W^{\prime}_{\varphi}}{\partial
d_{ii}}\\\\[8.0pt] \displaystyle\frac{\partial(W^{\prime}_{\mu})}{\partial
a_{r}}&\displaystyle\frac{\partial W^{\prime}_{\varphi}}{\partial
a_{r}}\end{pmatrix}\neq 0$
due to condition (6).
Consider $(n-{\textsc{n}})\\!\times\\!(n-{\textsc{n}})$ matrix
$B:=\begin{pmatrix}b_{\mu\mu}{}^{ii}(y^{\mu},c)&b_{\mu\mu}{}^{r}(y^{\mu},c)\\\
b_{\varphi}{}^{ii}(z^{\varphi},c)&b_{\varphi}{}^{r}(z^{\varphi},c)\end{pmatrix},$
(101)
whose elements of each row depend on single coordinate and, possibly, the
first group of parameters $c$. It must be nondegenerate, as we shall see. The
inverse metric is
$B^{-1}=\begin{pmatrix}b_{ii}{}^{\mu\mu}&b_{ii}{}^{\varphi}\\\
b_{r}{}^{\nu\nu}&b_{r}{}^{\varphi}\end{pmatrix},$
where
$\displaystyle
b_{\mu\mu}{}^{ij}b_{ij}{}^{\nu\nu}+b_{\mu\mu}{}^{r}b_{r}{}^{\nu\nu}=$
$\displaystyle\delta_{\mu}^{\nu},$ $\displaystyle\qquad
b_{ii}{}^{\mu\nu}b_{\mu\nu}{}^{jj}+b_{ii}{}^{\varphi}b_{\varphi}{}^{jj}=$
$\displaystyle\delta_{i}^{j},$ $\displaystyle
b_{\mu\mu}{}^{ij}b_{ij}{}^{\phi}+b_{\mu\mu}{}^{r}b_{r}{}^{\phi}=$
$\displaystyle 0,$ $\displaystyle\qquad\qquad
b_{ii}{}^{\mu\nu}b_{\mu\nu}{}^{r}+b_{ii}{}^{\varphi}b_{\varphi}{}^{r}=$
$\displaystyle 0,$ $\displaystyle
b_{\varphi}{}^{ij}b_{ij}{}^{\nu\nu}+b_{\varphi}{}^{r}b_{r}{}^{\nu\nu}=$
$\displaystyle 0,$ $\displaystyle
b_{r}{}^{\mu\nu}b_{\mu\nu}{}^{jj}+b_{r}{}^{\varphi}b_{\varphi}{}^{jj}=$
$\displaystyle 0,$ $\displaystyle
b_{\varphi}{}^{ij}b_{ij}{}^{\phi}+b_{\varphi}{}^{r}b_{r}{}^{\phi}=$
$\displaystyle\delta_{\varphi}^{\phi},$ $\displaystyle
b_{r}{}^{\mu\nu}b_{\mu\nu}{}^{s}+b_{r}{}^{\varphi}b_{\varphi}{}^{s}=$
$\displaystyle\delta_{r}^{s}.$
Note that elements of the inverse matrix depend in general on all coordinates
$y$, $z$, and parameters $c$. Matrix $B$ must be nondegenerate in order for
Eq. (99) to have unique solution, and
$g^{\mu\mu}=b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\mu\mu}(y,z),\qquad
2g^{\alpha\phi}c_{\alpha}=b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\phi}(y,z,c),$
(102)
because constant $E$ enters the left hand side of Eq. (99) only through
parameter $d_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}$. This
implies that elements
$b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\mu\mu}$ do not
depend on $c$, and
$b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\phi}$ are linear
in $c$:
$b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\phi}=b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\alpha\phi}c_{\alpha}.$
Substitution of obtained expressions $g^{\mu\nu}$ and
$g^{\alpha\phi}c_{\alpha}$ into the Hamilton–Jacobi equation (99) yields
$g^{\alpha\beta}c_{\alpha}c_{\beta}+b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\mu\nu}k_{\mu\nu}+b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\alpha\phi}c_{\alpha}l_{\phi}=0,$
(103)
which must be fulfilled for all $c$. Therefore functions $k_{\mu\mu}$ and
$l_{\phi}$ must be quadratic and linear in $c$, respectively. It is sufficient
to choose them homogeneous
$k_{\mu\mu}=k^{\alpha\beta}_{\mu\nu}c_{\alpha}c_{\beta},\qquad
l_{\varphi}=l^{\alpha}_{\varphi}c_{\alpha},$ (104)
where functions $k^{\alpha\beta}_{\mu\nu}(y^{\mu})$ and
$l^{\alpha}_{\varphi}(z^{\varphi})$ do not depend on parameters $c$. Functions
$k^{\alpha\beta}_{\mu\nu}=k^{\beta\alpha}_{\mu\nu}$ can differ from zero for
$\alpha\neq\beta$. Indices of $k$ and $l$ are written one over the other
because they will never be raised or lowers.
To simplify separable metric, we perform canonical transformation
$(x^{\alpha},z^{\varphi},p_{\alpha},p_{\varphi})\mapsto(X^{\alpha},Z^{\varphi},P_{\alpha},P_{\varphi})$
with generating function
$\hphantom{\qquad\qquad{\sf\,ns\,}(\alpha)}S_{2}:=x^{\alpha}P_{\alpha}+z^{\varphi}P_{\varphi}+\int\\!\\!dz^{\varphi}l^{\alpha}_{\varphi}P_{\alpha},\quad\quad{\sf\,ns\,}(\varphi).$
(105)
Then
$\displaystyle p_{\alpha}=$ $\displaystyle\frac{\partial S_{2}}{\partial
x^{\alpha}}=P_{\alpha},$ $\displaystyle\qquad p_{\varphi}=$
$\displaystyle\frac{\partial S_{2}}{\partial
z^{\varphi}}=P_{\varphi}+l^{\alpha}_{\varphi}P_{\alpha},$ (106) $\displaystyle
X^{\alpha}=$ $\displaystyle\frac{\partial S_{2}}{\partial
p_{\alpha}}=x^{\alpha}+\int\\!\\!dz^{\varphi}l^{\alpha}_{\varphi},$
$\displaystyle Z^{\varphi}=$ $\displaystyle\frac{\partial S_{2}}{\partial
p_{\varphi}}=z^{\varphi},\quad\quad{\sf\,ns\,}(\varphi).$
Afterwards relation (45) in the Hamiltonian formulation becomes
$P_{\varphi}=b_{\varphi}{}^{ij}d_{ij}+b_{\varphi}{}^{r}a_{r},$
where equalities $p_{\alpha}=P_{\alpha}=c_{\alpha}$ are used. Therefore we can
set $l^{\alpha}_{\varphi}\equiv 0$ without loss of generality. Now nontrivial
blocks of inverse metric (102) take the form
$\begin{split}g^{\alpha\beta}=&-b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\mu\nu}k^{\alpha\beta}_{\mu\nu},\\\
g^{\mu\mu}=&~{}~{}b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\mu\mu},\\\
2g^{\alpha\phi}=&~{}~{}b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\alpha\varphi},\end{split}$
(107)
where functions $k^{\alpha\beta}_{\mu\nu}(y^{\mu})$ are arbitrary, and
functions
$b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\mu\mu}$ and
$b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\alpha\varphi}$
are defined by matrix (101). Note that block $g^{\alpha\beta}$ may be
degenerate due to the presence of rectangular block $g^{\alpha\varphi}$. In
section 5.2 this is not allowed because metric (74) becomes degenerate.
Moreover, using the canonical transformation of type (66), one of nonzero
elements in each row of matrix $b_{\mu\mu}{}^{ii}$ can be set to unity.
Thus we have solved the functional Hamilton–Jacobi equation.
###### Theorem 5.4.
Let separable metric be of type
$[{\textsc{n}},{\textsc{m}},n-{\textsc{n}}-{\textsc{m}}]_{2}$. Then there is
such set of parameters (43) and separating functions (44), (45), that
canonical separable metric has block form (47) with blocks (107), where
functions
$k^{\alpha\beta}_{\mu\nu}(y^{\mu})=k^{\beta\alpha}_{\mu\nu}(y^{\mu})$ are
arbitrary, and functions
$b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\mu\mu}$ and
$b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\alpha\varphi}$
are elements of the $({\textsc{n}}+{\textsc{m}})$-th row of the matrix inverse
to matrix $B$ (101). Matrix $B$ must be nondegenerate with arbitrary elements,
such that elements of each row depend on single coordinates and parameters
$(c_{\alpha})$ in such a way that all elements of the
$({\textsc{n}}+{\textsc{m}})$-th row of matrix $B^{-1}$ are nonzero, elements
$b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\mu\mu}$ do not
depend on $(c_{\alpha})$, and elements
$b_{{\textsc{n}}+{\textsc{m}}\,{\textsc{n}}+{\textsc{m}}}{}^{\phi}$ are linear
in parameters $(c_{\alpha})$. Arbitrary functions must also produce
nondegenerate separable metric and provide real solutions of Eqs. (44) for
$W^{\prime}_{\mu}$ and $W^{\prime}_{\varphi}$. In addition, conservation laws
in the Hamiltonian formulation have the form
$\begin{split}p_{\alpha}=&c_{\alpha},\\\
b_{ii}{}^{\mu\nu}\big{(}p_{\mu}p_{\nu}-k^{\alpha\beta}_{\mu\nu}c_{\alpha}c_{\beta}\big{)}+b_{ii}{}^{\varphi}p_{\varphi}=&d_{ii},\\\
b_{r}{}^{\mu\nu}\big{(}p_{\mu}p_{\nu}-k^{\alpha\beta}_{\mu\nu}c_{\alpha}c_{\beta}\big{)}+b_{r}{}^{\varphi}p_{\varphi}=&a_{r}.\end{split}$
(108)
In general, there are n linear and $n-{\textsc{n}}$ quadratic conservation
laws for separable metrics of type
$[{\textsc{n}},{\textsc{m}},n-{\textsc{n}}-{\textsc{m}}]_{2}$ for
${\textsc{m}}\geq 1$.
Let us consider separable metrics of type
$[{\textsc{n}},{\textsc{m}},n-{\textsc{n}}-{\textsc{m}}]_{1}$, i.g. energy $E$
enters the group of parameters for coisotropic coordinates (42). Then solution
of the Hamilton–Jacobi equation repeats all previous steps. The only
difference is the change of relations (102). Blocks of separable metric after
the canonical transformation (105) become
$g^{\mu\mu}=b_{n}{}^{\mu\mu}(y,z),\qquad
2g^{\alpha\phi}=b_{n}{}^{\alpha\phi}(y,z).$ (109)
Therefore we only formulate the result.
###### Theorem 5.5.
Let separable metric be of type
$[{\textsc{n}},{\textsc{m}},n-{\textsc{n}}-{\textsc{m}}]_{1}$. Then there is
such set of parameters (42) and separating functions (44), (45), that
canonical separable metric has block form (47) with blocks
$\begin{split}g^{\alpha\beta}=&-b_{n}{}^{\mu\nu}k^{\alpha\beta}_{\mu\nu},\\\
g^{\mu\mu}=&~{}~{}b_{n}{}^{\mu\mu},\\\
2g^{\alpha\varphi}=&~{}~{}b_{n}{}^{\alpha\varphi},\end{split}$ (110)
where functions
$k^{\alpha\beta}_{\mu\nu}(y^{\mu})=k^{\beta\alpha}_{\mu\nu}(y^{\mu})$ are
arbitrary, and functions $b_{nn}{}^{\mu\mu}$ and $b_{nn}{}^{\alpha\varphi}$
are elements of the $n$-th row of the matrix inverse to matrix $B$ (101).
Matrix $B$ must be nondegenerate with arbitrary elements such that elements of
each row depend on single coordinates and parameters $(c_{\alpha})$ in such a
way that all elements of the last row of matrix $B^{-1}$ are nonzero, elements
$b_{nn}{}^{\mu\mu}$ do not depend on $(c_{\alpha})$, and $b_{nn}{}^{\phi}$ are
linear in parameters $(c_{\alpha})$. Arbitrary functions must also produce
nondegenerate separable metric and provide real solutions of Eqs. (44) for
$W^{\prime}_{\mu}$ and $W^{\prime}_{\varphi}$. In addition, conservation laws
in the Hamiltonian formulation have form (108).
Thus we looked over all possible classes of separable metrics, introduced sets
of independent parameters and separating functions and explicitly separated
all variables. The used method is constructive and allows to write down all
$n$ conservation laws (108) for geodesics. We showed that there exists such
choice of independent parameters and separating functions in complete
integrals that conservation laws are at most quadratic in momenta: (i), there
are linear conservation laws corresponding to commuting Killing vector fields;
(ii), there are indecomposable quadratic conservation laws; (iii), there may
exist linear conservation laws for coisotropic coordinates which are not
related to Killing vectors. The last possibility arises only for indefinite
metrics which may have zeroes on the diagonal.
We see that constructed conservation laws are at most quadratic.
###### Theorem 5.6.
Let the Hamilton–Jacobi equation for geodesics (2) admit complete separation
of variables. Then there exists such choice of independent parameters and
separating functions in complete integrals that corresponding conservation
laws are at most quadratic in momenta.
This statement is important because for another choice of parameters and
separating functions we cannot guarantee the absence of higher order
conservation laws.
At the end we prove that any additional conservation law is a function of
conservation laws described in theorems (5.4) and (5.5). To this end we square
equality $p_{\alpha}=c_{\alpha}$. Then all conservation laws become quadratic
(remember that functions $b_{ii}{}^{\varphi}$ and $b_{r}{}^{\varphi}$ are
linear in $c_{\alpha}$ and, consequently, in momenta $p_{\alpha}$). Part of
these conservation laws may be decomposable but this is not essential. We
simplify notation:
$\begin{split}(x^{\alpha},y^{\mu},z^{\varphi})\quad\mapsto&\quad(q^{\alpha}),\\\
(p_{\alpha},p_{\mu},p_{\varphi})\quad\mapsto&\quad(p_{\alpha}),\\\
(c_{\alpha}^{2},d_{ij},a_{r})\quad\mapsto&\quad(c_{\textsc{a}}),\end{split}$
where indices in the right hand sides take all values from $1$ to $n$:
$\alpha=1,\dotsc,n$ and ${\textsc{a}}=1,\dotsc,n$. The conservation laws take
the form
$c_{\textsc{a}}=F_{\textsc{a}}(q,p)=\frac{1}{2}f_{\textsc{a}}^{\alpha\beta}p_{\alpha}p_{\beta},$
where $f_{\textsc{a}}^{\alpha\beta}(q)$ are some functions only on coordinates
$q$. They contain the Hamiltonian among themselves: for separable metrics of
type $1$ and $2$, it is $F_{n}$ and $F_{{\textsc{n}}+{\textsc{m}}}$,
respectively. Perform the canonical transformation
$(q^{\alpha},p_{\alpha})\mapsto(Q^{\textsc{a}},P_{\textsc{a}})$ with
generating functions $S_{2}:=W(q,P)$ to action-angle variables. Then
$F_{\textsc{a}}=P_{\textsc{a}},\qquad\forall{\textsc{a}}=1,\dotsc n,$
in new variables. Assume that there is additional involutive conservation law
$G:=\sum_{{\textsc{a}}=1}^{n}G^{\textsc{a}}(Q)P_{\textsc{a}}^{m}={\sf\,const},\qquad
m\in{\mathbb{R}},$
with some differentiable functions $G^{\textsc{a}}$ only from new coordinates,
and where $m$ is some real number. In particular, $G$ may be homogeneous
polynomial of any order $m$ on new momenta. Involutivity means that
$[G,F_{\textsc{a}}]=[G,P_{\textsc{a}}]=\sum_{{\textsc{b}}=1}^{n}\frac{\partial
G^{\textsc{b}}}{\partial Q^{\textsc{a}}}P_{\textsc{b}}^{m}=0.$
This equality must hold for all momenta and values of index a. Therefore
functions $G^{\textsc{a}}$ do not depend on coordinates and, consequently,
there is functional dependence between integrals $(F_{\textsc{a}},G)$ because
the number of functions $F_{\textsc{a}}$ in maximal and equals to the number
of momenta, i.e. any additional conservation law is some function on integrals
$(F_{\textsc{a}})$ at least locally, $G=G(F)$. Thus we proved
###### Theorem 5.7.
Let variables in the Hamilton–Jacobi equation for geodesics be completely
separable, and all conservation laws $F_{\textsc{a}}$ are found according to
theorems (5.4) and (5.5). Then any additional involutive conservation law $G$
for geodesics is some function $G=G(F)$ at least locally.
This solves completely the Stäckel problem for metrics of arbitrary signature
on manifolds of any dimension. All separable metrics are divided into
equivalence classes. Metrics in each class are related by canonical
transformations and nondegenerate transformations of parameters which do not
involve coordinates. There is the canonical (the simplest) separable metric in
each equivalence class. Its form is given by theorems 5.4 and 5.5.
Corresponding conservation laws are at most quadratic in momenta. For other
choices of parameters conservation laws may have higher order but they are
functionally dependent on conservation laws listed in theorems 5.4 and 5.5 due
to theorem 5.7. Matrix $B$ in general theorems is constructed as matrix (94).
The proved theorems are constructive. In next sections we list all separable
metrics in two, three, and four dimensions as examples.
## 6 Separation of variables in two dimensions
Two dimensional manifolds provide the simplest examples of separable metrics
for the Hamilton– Jacobi equation for geodesics which have important features
present in higher dimensions. Separation of variables in two dimensions is
considered in example 5.1 in detail. Therefore we only formulate results.
There are only three classes of different separable metrics.
1) Class $[2,0,0]$. Two commuting Killing vectors. Coordinates and parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})\mapsto(x^{1},x^{2}),\qquad(c_{\alpha},d_{ij},a_{r})\mapsto(c_{1},c_{2}).$
This case is described by theorem 4.2. In canonical form, the inverse
separable metric is (pseudo)Euclidean:
$g^{\alpha\beta}=\eta^{\alpha\beta},$ (111)
where $\eta^{\alpha\beta}$ is either Euclidean or Lorentzian metric. The
Hamilton–Jacobi equation has the form
$\eta^{\alpha\beta}W^{\prime}_{\alpha}W^{\prime}_{\beta}=\eta^{\alpha\beta}c_{\alpha}c_{\beta}.$
Variables are separated as
$W^{\prime}_{\alpha}=c_{\alpha}.$
The system has two linear conservation laws:
$p_{\alpha}=c_{\alpha},$ (112)
and respective coordinates $x^{1},x^{2}$ are cyclic. Separable metric (111)
has two independent commuting Killing vector fields $\partial_{1}$ and
$\partial_{2}$.
2) Class $[1,1,0]_{2}$. One Killing vector and one indecomposable quadratic
conservation law. Coordinates and parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})\mapsto(x^{1}:=x,y^{2}:=y),\qquad(c_{\alpha},d_{ij},a_{r})\mapsto(c_{1}:=c,d_{22}:=2E).$
This case in more generality is considered in example 5.2. Canonical separable
(pseudo)Riemannian metric has form (77)
$g^{**}=\begin{pmatrix}-k(y)&0\\\ 0&1\end{pmatrix},$ (113)
where $k(y)\neq 0$ is an arbitrary function. For $k<0$ and $k>0$, we have
Riemannian and Lorentzian metrics, respectively. The Hamilton–Jacobi equation
is
$-kW^{\prime 2}_{1}+W^{\prime 2}_{2}=2E.$
Variables are separated in the following way
$W_{1}=c,\qquad W^{\prime 2}_{2}=2E+kc^{2}>0.$
In the Hamiltonian formulation, we have one linear and one quadratic
conservation laws:
$p_{1}=c,\qquad p_{2}^{2}-k(y)c^{2}=2E.$
Canonical separable metric (113) is parameterized by one arbitrary function
$k(y)\neq 0$, satisfying inequality $2E+kc^{2}>0$ for fixed $E$ and $c$.
3) Class $[0,2,0]_{2}$. Tow quadratic conservation laws. Coordinates and
parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})\mapsto(y^{1},y^{2}),\qquad(c_{\alpha},d_{ij},a_{r})\mapsto(d_{11}:=d,d_{22}:=2E).$
This case is described by theorem 5.1. In canonical form, the separable metric
is given by Eq. (59). Variables are completely separated (50), where matrix
$b$ is given by Eq. (57). Conservation laws are quadratic (56). The canonical
separable metric is parameterized by four arbitrary functions
$\phi_{11}(y^{1})$, $\phi_{12}(y^{1})$, $\phi_{21}(y^{2})$, and
$\phi_{22}(y^{2})$, whose restrictions are described in example 5.1 in detail.
This class of metrics contains two dimensional Liouville systems (8).
4) Class $[1,0,1]_{1}$. One Killing vector and one coisotropic coordinate.
Coordinates and parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})\mapsto(x^{1}:=x,z^{2}:=z),\qquad(c_{\alpha},d_{ij},a_{r})\mapsto(c_{1}:=c,a_{2}:=2E).$
Appearance of zeroes on the diagonal is possible only for Lorentzian signature
metrics and one Killing vector. This case is described by theorem 5.3.
Canonical separable metric has form (93) where off diagonal block
$b_{n}{}^{\alpha\phi}$ is reduced to one nonzero function $b(z)$, which can be
transformed to unity by the canonical transformation, i.e.
$g^{**}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$ (114)
This metric produces two linear conservation laws
$p_{1}=c,\qquad cp_{2}=E$ (115)
We see that separation of variables in two dimensions corresponds to two
commuting Killing vectors $\partial_{x}$ and $\partial_{z}$, i.e. coordinates
$x$, $z$ are null. Therefore separable metric (168) is equivalent to the
Lorentz metric, and respective classes are also equivalent
$[1,0,1]_{1}\sim[2,0,0]$.
All three classes of separable metrics were known already to Stäckel [1].
However the case 4) with coisotropic coordinate was not described. Moreover,
we proved that there no other separable metrics.
## 7 Separation of variables in three dimensions
There are six different classes of separable metrics.
1) Class $[3,0,0]$. Three commuting Killing vectirs. Coordinates and
parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})\mapsto(x^{1},x^{2},x^{3}),\qquad(c_{\alpha},d_{ij},a_{r})\mapsto(c_{1},c_{2},c_{3}).$
In canonical form, the separable metric is (pseudo)Euclidean. Separation of
variables and conservation laws have the same form as in two dimensions in
case $[2,0,0]$, but now indices $\alpha=1,2,3$ take three values, and all
Cartesian coordinates $x^{\alpha}$ are cyclic.
2) Class $[2,1,0]_{2}$. Two commuting Killing vectors and one quadratic
conservation law. Coordinates and parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})\mapsto(x^{1},x^{2},y^{3}:=y),\qquad(c_{\alpha},d_{ij},a_{r})\mapsto(c_{1},c_{2},d_{33}:=2E).$
Canonical separable metric has the form (77):
$g^{**}=\begin{pmatrix}g^{\alpha\beta}(y)&0\\\
0&1\end{pmatrix},\qquad\alpha,\beta=1,2,$ (116)
where $g^{\alpha\beta}$ is an arbitrary symmetric nondegenerate matrix. The
Hamilton–Jacobi equation and separation of variables are as follows
$\begin{split}&g^{\alpha\beta}W^{\prime}_{\alpha}W^{\prime}_{\beta}+W^{\prime
2}_{3}=2E,\\\ &W^{\prime}_{\alpha}=c_{\alpha},\qquad\qquad W^{\prime
2}_{3}=2E-g^{\alpha\beta}c_{\alpha}c_{\beta}.\end{split}$
Conservation laws are
$p_{\alpha}=c_{\alpha},\qquad
g^{\alpha\beta}(y)p_{\alpha}p_{\beta}+p_{3}^{2}=2E.$ (117)
In this case, the canonical separable metric is parameterized by three
arbitrary functions of single argument in matrix $g^{\alpha\beta}$ with
restriction $\det g^{\alpha\beta}\neq 0$. Moreover, condition $W^{\prime
2}_{3}\geq 0$ also restricts arbitrary functions for fixed $E$ and $c$.
Depending on matrix $g^{\alpha\beta}$ separable metric (116) may have
arbitrary signature. In particular, when matrix $g^{\alpha\beta}$ is constant,
we return to class $[3,0,0]$.
3) Class $[1,2,0]_{2}$. One Killing vector and two quadratic conservation
laws. Coordinates and parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})\mapsto(x^{1},y^{2},y^{3}),\qquad(c_{\alpha},d_{ij},a_{r})\mapsto(c_{1}:=c,d_{22}:=d,d_{33}:=2E).$
This case is described by theorem 5.2. It is new, and we consider it in more
detail. First, we define matrix $b$:
$b_{\mu\mu}{}^{ii}=\begin{pmatrix}\phi_{22}(y^{2})&\phi_{23}(y^{2})\\\
\phi_{32}(y^{3})&\phi_{33}(y^{3})\end{pmatrix},\qquad
b_{ii}{}^{\mu\mu}=\frac{1}{\det b}\begin{pmatrix}\phi_{33}&-\phi_{23}\\\
-\phi_{32}&\phi_{22}\end{pmatrix},$
where $\det b=\phi_{22}\phi_{33}-\phi_{23}\phi_{32}\neq 0$. After canonical
transformation of type (66), One element in each row of matrix $b$ can be
transformed to $\pm 1$. Let
$\phi_{22}=1,\qquad\phi_{32}=-1.$
Then
$b_{\mu\mu}{}^{ii}=\begin{pmatrix}~{}~{}1&\phi_{23}\\\
-1&\phi_{33}\end{pmatrix},\qquad b_{ii}{}^{\mu\mu}=\frac{1}{\det
b}\begin{pmatrix}\phi_{33}&-\phi_{23}\\\ 1&1\end{pmatrix},$ (118)
where $\det b=\phi_{23}+\phi_{33}$. Diagonal metric elements corresponding to
quadratic conservation laws are
$g^{22}=b_{33}{}^{22}=\frac{1}{\phi_{23}+\phi_{33}},\qquad
g^{33}=b_{33}{}^{33}=\frac{1}{\phi_{23}+\phi_{33}}.$ (119)
Element $g^{11}$ of inverse metric has form (74)
$g^{11}=-b_{33}^{\mu\nu}k^{11}_{\mu\nu}=-\frac{1}{\phi_{23}+\phi_{33}}(k^{11}_{22}+k^{11}_{33}).$
Simplify notation
$\phi_{23}(y^{2}):=\phi_{2}(y^{2}),\quad\phi_{33}(y^{3}):=\phi_{3}(y^{3}),\qquad
k^{11}_{22}(y^{2}):=-k_{2}(y^{2})\quad k^{11}_{33}:=-k_{3}(y^{3}).$
Now the canonical separable metric takes the form
$g^{**}=\frac{1}{\phi_{2}+\phi_{3}}\begin{pmatrix}k_{2}+k_{3}&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}.$ (120)
Variables in the Hamilton–Jacobi equation
$\frac{1}{\phi_{2}+\phi_{3}}\big{[}(k_{2}+k_{3})W^{\prime 2}_{1}+W^{\prime
2}_{2}+W^{\prime 2}_{3}\big{]}=2E$
are completely separated:
$\begin{split}W^{\prime}_{1}=&~{}~{}c,\\\ W^{\prime
2}_{2}=&~{}~{}d+2\phi_{2}E-k_{2}c^{2},\\\ W^{\prime
2}_{3}=&-d+2\phi_{3}E-k_{3}c^{2}.\end{split}$ (121)
These relations yield conservation laws
$\begin{split}p_{1}=&c,\\\
\frac{1}{\phi_{2}+\phi_{3}}\left[\phi_{3}p_{2}^{2}-\phi_{2}p_{3}^{2}+\big{(}\phi_{3}k_{2}-\phi_{2}k_{3}\big{)}p_{1}^{2}\right]=&d,\\\
\frac{1}{\phi_{2}+\phi_{3}}\left[p_{2}^{2}+p_{3}^{2}+\big{(}k_{2}+k_{3}\big{)}p_{1}^{2}\right]=&2E.\end{split}$
(122)
Thus canonical separable metric (120) is parameterized by four functions of
single argument $\phi_{2,3}$ and $k_{2,3}$. They have to produce nondegenerate
metric, and Eq. (121) must admit real solutions for separating functions
$W^{\prime}_{\mu}$. In general, there are two indecomposable quadratic and one
linear conservation laws.
4) Class $[1,1,1]_{2}$. One Killing vector, one indecomposable quadratic
conservation law, and one coisotropic coordinate. Coordinates and parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})\mapsto(x^{1}:=x,y^{2}:=y,z^{3}:=z),\qquad(c_{\alpha},d_{ij},a_{r})\mapsto(c_{1}:=c,d_{22}:=2E,a_{3}:=a).$
This case is described by theorem 5.4. As always, we start with arbitrary
$(2\times 2)$ matrix (101). To simplify notation, let
$b_{22}{}^{22}\equiv 1,\qquad b_{22}{}^{3}\equiv\phi_{2}(y),\qquad
b_{3}{}^{22}\equiv\phi_{3}/c,\qquad b_{3}{}^{3}\equiv 1/c.$
The first and the last equality can be always achieved by suitable canonical
transformation. The dependence on $c$ follows from the independence of metric
components on parameters. Then
$B=\begin{pmatrix}1&\phi_{2}\\\
\phi_{3}/c&1/c\end{pmatrix}\qquad\Rightarrow\qquad
B^{-1}=\frac{1}{1-\phi_{2}\phi_{3}}\begin{pmatrix}1&-c\phi_{2}\\\
-\phi_{3}&c\end{pmatrix},$ (123)
where $\phi_{2}(y)$ and $\phi_{3}(z)$ are arbitrary functions on single
coordinates. Equations (107) imply expression for the canonical separable
metric
$g^{**}=\frac{1}{1-\phi_{2}\phi_{3}}\begin{pmatrix}-k_{2}&0&-\phi_{2}/2\\\
0&1&0\\\ -\phi_{2}/2&0&0\end{pmatrix},$ (124)
where $k_{2}(y)$ is an arbitrary function. The Hamilton–Jacobi equation after
substitution $W^{\prime}_{1}=c$ takes the form
$\frac{1}{1-\phi_{2}\phi_{3}}\left(-k_{2}c^{2}+W^{\prime
2}_{2}-\phi_{2}cW^{\prime}_{3}\right)=2E.$
Variables are separated in the following form
$\begin{split}W^{\prime 2}_{2}=&2E+\phi_{2}a+k_{2}c^{2},\\\
W^{\prime}_{3}=&\frac{1}{c}(2\phi_{3}E+a).\end{split}$ (125)
Conservation laws (108) after substitution $p_{1}=c$ are
$\begin{split}\frac{1}{1-\phi_{2}\phi_{3}}\big{(}p_{2}^{2}-k_{2}c^{2}-\phi_{2}cp_{3}\big{)}=&2E,\\\
\frac{1}{1-\phi_{2}\phi_{3}}\big{[}-\phi_{3}(p_{2}^{2}-k_{2}c^{2})+cp_{3}\big{]}=&a.\end{split}$
(126)
Canonical separable metric (124) is parameterized by three arbitrary functions
$\phi_{2}(y)$, $\phi_{3}(z)$, and $k_{2}(y)$. Determinant of the inverse
metric (124) is
$\det g^{**}=-\frac{\phi_{2}^{2}}{4(1-\phi_{2}\phi_{3})^{3}}\neq 0,\infty.$
Arbitrary functions must be restricted $\phi_{2}\neq 0$ and
$\phi_{2}\phi_{3}\neq 1$, because otherwise the determinant is degenerate.
Moreover functions $\phi_{2}$ and $k_{2}$ are to be chosen in such a way as to
provide real solution for the first equation (125) with respect to
$W^{\prime}_{2}$. Depending on arbitrary functions the separable metric may
any signature.
5) Class $[1,1,1]_{1}$. One Killing vector, one indecomposable quadratic
conservation law, and one coisotropic coordinate. Coordinates and parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})\mapsto(x^{1}:=x,y^{2}:=y,z^{3}:=z),\qquad(c_{\alpha},d_{ij},a_{r})\mapsto(c_{1}:=c,d_{22}:=d,a_{3}:=2E).$
This case is given by theorem 5.5. Matrix $B$ has the same form as in case
$[1,1,1]_{2}$ (123), however the canonical separable metric is defined by the
last row of matrix $B^{-1}$:
$g^{**}=\frac{1}{1-\phi_{2}\phi_{3}}\begin{pmatrix}\phi_{3}k_{2}&0&1/2\\\
0&-\phi_{3}&0\\\ 1/2&0&0\end{pmatrix}.$ (127)
After substitution $W^{\prime}_{1}\equiv c$, the Hamilton–Jacobi equation take
the form
$\frac{1}{1-\phi_{2}\phi_{3}}\big{(}\phi_{3}k_{2}c^{2}-\phi_{3}W^{\prime
2}_{2}+cW^{\prime}_{3}\big{)}=2E.$
Variables are separated as
$\begin{split}W^{\prime 2}_{2}=&d+2\phi_{2}E+k_{2}c^{2},\\\
W^{\prime}_{3}=&\frac{1}{c}({\phi_{3}}d+2E).\end{split}$ (128)
Conservation laws (108) take the form
$\begin{split}\frac{1}{1-\phi_{2}\phi_{3}}\big{[}p_{2}^{2}-k_{2}c^{2}-\phi_{2}cp_{3}\big{]}=&d,\\\
\frac{1}{1-\phi_{2}\phi_{3}}\big{[}-\phi_{3}(p_{2}^{2}-k_{2}c^{2})+cp_{3}\big{]}=&2E.\end{split}$
(129)
So, canonical separable metric (127) is parameterized by three arbitrary
functions $\phi_{2}(y)$, $\phi_{3}(z)$, and $k_{2}(y)$. They are restricted by
nondegeneracy of the determinant
$\det g^{**}=\frac{\phi_{3}}{4(1-\phi_{2}\phi_{3})^{3}}\neq 0,\infty$
which imply $\phi_{3}\neq 0$ and $\phi_{2}\phi_{3}\neq 1$. Moreover, functions
$\phi_{2}$ and $k_{2}$ have to provide existence of real solutions for
$W^{\prime}_{2}$ in the first equation (128). Depending on arbitrary
functions, the separable metric may have any signature.
6) Class $[0,3,0]_{2}$. Absence of Killing vectors and coisotropic
coordinates. Coordinates and parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})\mapsto(y^{1},y^{2},y^{3}),\qquad(c_{\alpha},d_{ij},a_{r})\mapsto(d_{11}:=d_{1},d_{22}:=d_{2},d_{33}:=2E).$
This case is described by theorem 5.1. After the canonical transformation with
generating function (66) matrix $b$ and its inverse are
$b_{\mu\mu}{}^{ii}=\begin{pmatrix}1&b_{12}(y^{1})&b_{13}(y^{1})\\\
b_{21}(y^{2})&1&b_{23}(y^{2})\\\
b_{31}(y^{3})&b_{32}(y^{3})&1\end{pmatrix},\qquad
b_{ii}{}^{\mu\mu}=\frac{1}{\vartriangle}\begin{pmatrix}\vartriangle_{11}&\vartriangle_{21}&\vartriangle_{31}\\\
\vartriangle_{12}&\vartriangle_{22}&\vartriangle_{32}\\\
\vartriangle_{13}&\vartriangle_{23}&\vartriangle_{33}\end{pmatrix},$ (130)
where $\vartriangle:=\det b_{\mu\mu}{}^{ii}$ and symbols $\vartriangle_{\mu
i}$ denote cofactors of elements $b_{\mu\mu}{}^{ii}$. This matrix produces the
diagonal metric (55)
$g^{**}=\frac{1}{\vartriangle}\begin{pmatrix}\vartriangle_{13}&0&0\\\
0&\vartriangle_{23}&0\\\ 0&0&\vartriangle_{33}\end{pmatrix}.$ (131)
The respective Hamilton–Jacobe equation becomes
$\frac{1}{\vartriangle}\big{(}\vartriangle_{13}W^{\prime
2}_{1}+\vartriangle_{23}W^{\prime 2}_{2}+\vartriangle_{33}W^{\prime
2}_{3}\big{)}=2E.$
Variables are completely separated in the way
$\begin{split}W^{\prime 2}_{1}=&d_{1}+b_{12}d_{2}+2b_{13}E,\\\ W^{\prime
2}_{2}=&b_{21}d_{1}+d_{2}+2b_{23}E,\\\ W^{\prime
2}_{3}=&b_{31}d_{1}+b_{32}d_{2}+2E.\\\ \end{split}$ (132)
All three conservation laws are quadratic in general
$\begin{split}\frac{1}{\vartriangle}\big{(}\vartriangle_{11}p^{2}_{1}+\vartriangle_{21}p^{2}_{2}+\vartriangle_{31}p^{2}_{3}\big{)}=&d_{1},\\\
\frac{1}{\vartriangle}\big{(}\vartriangle_{12}p^{2}_{1}+\vartriangle_{22}p^{2}_{2}+\vartriangle_{32}p^{2}_{3}\big{)}=&d_{2},\\\
\frac{1}{\vartriangle}\big{(}\vartriangle_{13}p^{2}_{1}+\vartriangle_{23}p^{2}_{2}+\vartriangle_{33}p^{2}_{3}\big{)}=&2E.\end{split}$
(133)
This case includes the Liouville system (see example 2.1). Indeed, take matrix
$b$ in the form
$b_{\mu\mu}{}^{ii}:=\begin{pmatrix}1&0&\phi_{1}(y^{1})\\\
0&1&\phi_{2}(y^{2})\\\ -1&-1&\phi_{3}(y^{3})\end{pmatrix}\quad\Rightarrow\quad
b_{ii}{}^{\mu\mu}=\frac{1}{\phi_{1}+\phi_{2}+\phi_{3}}\begin{pmatrix}\phi_{2}+\phi_{3}&-\phi_{1}&-\phi_{1}\\\
-\phi_{2}&\phi_{1}+\phi_{3}&-\phi_{2}\\\ 1&1&1\end{pmatrix}.$
Then conservation laws are
$\begin{split}p^{2}_{1}-2\phi_{1}E=&d_{1},\\\ p^{2}_{2}-2\phi_{2}E=&d_{2},\\\
\frac{1}{\phi_{1}+\phi_{2}+\phi_{3}}\big{(}p^{2}_{1}+p^{2}_{2}+p^{2}_{3}\big{)}=&2E,\end{split}$
(134)
which coincides with Eqs. (13) and (15) for $\Theta\equiv 0$ up to notation.
Five of six separable metrics in three dimensions were listed in [29], where
different technique is used. Types of metrics coincide with respect to the
number of Killing vectors, quadratic conservation laws and coisotropic
coordinates. However computations of the present section allowed us to find
explicitly separable functions $W^{\prime}_{\alpha}$, $\alpha=1,\dotsc,3$. In
addition, we give more detailed classification indicated by indices $1,2$, and
separable metric of class $[1,1,1]_{1}$ is not mentioned in paper [29].
## 8 Separation of variables in four dimensions
Separable metrics in four dimensions are of great importance in gravity
models, in particular, in general relativity. These metrics has Lorentzian
signature, and coisotropic coordinates may appear. There are ten classes of
different separable metrics in four dimensions.
1) Class $[4,0,0]$. Four Killing vectors. Variables are separated in the same
way as in lower dimensions. Canonical separable metric and conservation laws
have form (111) and (112), respectively, but now indices run over all four
values, $\alpha,\beta=1,2,3,4$.
2) Class $[3,1,0]_{2}$. Three commuting Killing vectors and one indecomposable
quadratic conservation law. As in three dimensions, the canonical separable
metric and conservation laws have form (116) and (117), but indices
$\alpha,\beta=1,2,3$ take more values. The canonical separable metric is
parameterized by six arbitrary functions.
The Kasner solution [30] lies in this class.
3) Class $[2,2,0]_{2}$. Two commuting Killing vector fields and two
indecomposable quadratic conservation laws without coisotropic coordinates.
Coordinates and parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})\mapsto(x^{1},x^{2},y^{3},y^{4}),\qquad(c_{\alpha},d_{ij},a_{r})\mapsto(c_{1},c_{2},d_{33}:=d,d_{44}:=2E).$
This case is described by theorem 5.2. Matrix $b$ and diagonal metric elements
$g^{33}$ and $g^{44}$ have form (118) and (119) with replacement
$(2,3)\mapsto(3,4)$. The block of inverse metric $g^{\alpha\beta}$,
$\alpha,\beta=1,2$, is
$g^{\alpha\beta}=-k^{\alpha\beta}_{\mu\nu}g^{\mu\nu}=\frac{1}{\phi_{3}+\phi_{4}}\big{(}k^{\alpha\beta}_{3}+k^{\alpha\beta}_{4}\big{)},\qquad\phi_{3}+\phi_{4}\neq
0,$
where $\phi_{3}(y^{3})$, $\phi_{4}(y^{4})$, and $k^{\alpha\beta}_{3}(y^{3})$,
$k^{\alpha\beta}_{4}(y^{4})$ – are arbitrary functions of single coordinates.
Thus canonical separable metric has the form
$g^{**}=\frac{1}{\phi_{3}+\phi_{4}}\begin{pmatrix}k^{\alpha\beta}_{3}+k^{\alpha\beta}_{4}&0&0\\\
0&1&0\\\ 0&0&1\end{pmatrix}.$ (135)
In addition, we have to require fulfillment of two conditions
$\det
g^{**}=\frac{(k^{11}_{3}+k^{11}_{4})(k^{22}_{3}+k^{22}_{4})-(k^{12}_{3}+k^{12}_{4})(k^{21}_{3}+k^{21}_{4})}{(\phi_{3}+\phi_{4})^{4}}\neq
0,\infty.$ (136)
After separation of two coordinates $W^{\prime}_{\alpha}\equiv c_{\alpha}$,
the Hamilton–Jacobi equation becomes
$\frac{1}{\phi_{3}+\phi_{4}}\big{(}k^{\alpha\beta}_{3}c_{\alpha}c_{\beta}+k^{\alpha\beta}_{4}c_{\alpha}c_{\beta}+W^{\prime
2}_{3}+W^{\prime 2}_{4}\big{)}=2E.$ (137)
Variables are completely separated as
$\begin{split}W^{\prime
2}_{3}=&~{}~{}d+2E\phi_{3}-k^{\alpha\beta}_{3}c_{\alpha}c_{\beta},\\\
W^{\prime
2}_{4}=&-d+2E\phi_{4}-k^{\alpha\beta}_{4}c_{\alpha}c_{\beta}.\end{split}$
(138)
Two conservation laws are quadratic in general
$\begin{split}\frac{1}{\phi_{3}+\phi_{4}}\big{[}\phi_{4}(p_{3}^{2}+k_{3}^{\alpha\beta}c_{\alpha}c_{\beta})-\phi_{3}(p_{4}^{2}+k_{4}^{\alpha\beta}c_{\alpha}c_{\beta}\big{]}=&d,\\\
\frac{1}{\phi_{3}+\phi_{4}}\big{[}p_{3}^{2}+k_{3}^{\alpha\beta}c_{\alpha}c_{\beta}+p_{4}^{2}+k^{\alpha\beta}_{4}c_{\alpha}c_{\beta}\big{]}=&2E.\end{split}$
(139)
In this case, the canonical separable metric (135) is parameterized by eight
arbitrary functions, $\phi_{3}(y^{3})$, $\phi_{4}(y^{4})$,
$k^{\alpha\beta}_{3}(y^{3})$, and $k^{\alpha\beta}_{4}(y^{4})$ satisfying
inequalities (136). Besides, they must be chosen in such a way that equations
for $W^{\prime}$ (138) have real solutions.
This class of separable metrics includes the Schwarzschild,
Reissner–Nordström, Kerr, and other famous solutions in general relativity
[24].
4) Class $[2,0,2]_{1}$. Two commuting Killing vectors and two coisotropic
coordinates. Coordinates and parameters:
$(x^{\alpha},z^{\varphi})=(x^{1},x^{2},z^{3},z^{4}),\qquad(c_{\alpha},d_{ii},a_{r})\mapsto(c_{1},c_{2},a_{3}:=a,a_{4}:=2E).$
This case is described by theorem 5.3. It is new and considered in more
detail. Matrix $\phi$ (95) has the block form
$(\phi_{\varphi}{}^{r})=\begin{pmatrix}1&\phi_{3}\\\
\phi_{4}&1\end{pmatrix},\qquad\Leftrightarrow\qquad(\phi_{\varphi}{}^{r})^{-1}=\frac{1}{1-\phi_{3}\phi_{4}}\begin{pmatrix}1&-\phi_{3}\\\
-\phi_{4}&1\end{pmatrix},$ (140)
where $\phi_{3}(z^{3})$ and $\phi_{4}(z^{4})$ are arbitrary functions of
single variables. Functions $h_{\varphi}$ are
$h_{3}(z^{3},c)=h_{3}^{\alpha}(z^{3})c_{\alpha},\qquad
h_{4}(z^{4},c)=h_{4}^{\alpha}(z^{4})c_{\alpha},\qquad\alpha,\beta=1,2,$
where $h_{3}^{\alpha}(z^{3})$ and $h_{4}^{\alpha}(z^{4})$ are linear. Then
matrix $b$ has the form
$(b_{\varphi}{}^{r})=\begin{pmatrix}\displaystyle\frac{1}{h_{3}^{\alpha}c_{\alpha}}&\displaystyle\frac{\phi_{3}}{h_{3}^{\alpha}c_{\alpha}}\\\\[10.0pt]
\displaystyle\frac{\phi_{4}}{h_{4}^{\alpha}c_{\alpha}}&\displaystyle\frac{1}{h_{4}^{\alpha}c_{\alpha}}\end{pmatrix}\qquad\Leftrightarrow\qquad(b_{r}{}^{\varphi})=\frac{1}{1-\phi_{3}\phi_{4}}\begin{pmatrix}h_{3}^{\alpha}c_{\alpha}&-\phi_{3}h_{4}^{\alpha}c_{\alpha}\\\
-\phi_{4}h_{3}^{\alpha}c_{\alpha}&h_{4}^{\alpha}c_{\alpha}\end{pmatrix}.$
(141)
It implies separable metric
$g^{**}=\frac{1}{1-\phi_{3}\phi_{4}}\begin{pmatrix}0&0&-\phi_{4}h_{3}^{1}&h_{4}^{1}\\\
0&0&-\phi_{4}h_{3}^{2}&h_{4}^{2}\\\
-\phi_{4}h_{3}^{1}&-\phi_{4}h_{3}^{2}&0&0\\\
h_{4}^{1}&h_{4}^{2}&0&0\end{pmatrix}.$ (142)
The Hamilton–Jacobi equation after separation of the first two coordinates,
$\frac{1}{1-\phi_{3}\phi_{4}}\big{(}-\phi_{3}h_{3}^{\alpha}c_{\alpha}W^{\prime}_{3}+h_{4}^{\alpha}c_{\alpha}W^{\prime}_{4}\big{)}=2E,$
(143)
is completely separated as
$\begin{split}W^{\prime}_{3}=&\frac{1}{h_{3}^{\alpha}c_{\alpha}}(a+2\phi_{3}E),\\\
W^{\prime}_{4}=&\frac{1}{h_{4}^{\alpha}c_{\alpha}}(\phi_{4}a+2E).\end{split}$
(144)
Conservation laws (97) are linear
$\begin{split}\frac{1}{1-\phi_{3}\phi_{4}}\big{(}h_{3}^{\alpha}c_{\alpha}p_{3}-\phi_{3}h_{4}^{\alpha}c_{\alpha}p_{4}\big{)}=&a,\\\
\frac{1}{1-\phi_{3}\phi_{4}}\big{(}-\phi_{4}h_{3}^{\alpha}c_{\alpha}p_{3}+h^{\alpha}_{4}c_{\alpha}p_{4}\big{)}=&2E.\end{split}$
(145)
Arbitrary functions are restricted by inequality
$\det
g^{**}=\frac{(\phi_{4})^{2}\big{(}h_{3}^{2}h_{4}^{1}-h_{3}^{1}h_{4}^{2}\big{)}^{2}}{(1-\phi_{3}\phi_{4})^{4}}\neq
0,\infty.$ (146)
Thus canonical separable metric of type $[2,0,2]_{1}$ is parameterized by six
arbitrary functions of single arguments satisfying requirement (146). Two
nonzero functions, e.g., $h_{3}^{2}$ and $h_{4}^{1}$ can be set unity. Note
that $\det g^{**}$ is always positive. Therefore signature of the separable
metric of type $[2,0,2]_{1}$ can be only $(++--)$.
5) Class $[2,1,1]_{1}$. Two commuting Killing vectors, one indecomposable
quadratic conservation law and one coisotropic coordinate. Coordinates and
parameters:
$(x^{\alpha},y^{\mu},z^{\varphi})=(x^{1},x^{2},y^{3}:=y,z^{4}:=z),\qquad(c_{\alpha},d_{ii},a_{r})\mapsto(c_{1},c_{2},d_{33}:=d,a_{4}:=2E).$
This case is described by theorem 5.5. Matrix $B$ is parameterized similar to
Eq. (123):
$B=\begin{pmatrix}1&\phi_{3}\\\
\displaystyle\frac{\phi_{4}}{h_{4}^{\alpha}c_{\alpha}}&\displaystyle\frac{1}{h_{4}^{\alpha}c_{\alpha}}\end{pmatrix}\qquad\Rightarrow\qquad
B^{-1}=\frac{1}{1-\phi_{3}\phi_{4}}\begin{pmatrix}1&-\phi_{3}h_{4}^{\alpha}c_{\alpha}\\\
-\phi_{4}&h_{4}^{\alpha}c_{\alpha}\end{pmatrix},$
where $\phi_{3}(y)$, $\phi_{4}(z)$, and $h_{4}^{\alpha}(z)$, $\alpha=1,2$, are
arbitrary functions of single coordinates. It yields separable metric
$g^{**}=\frac{1}{1-\phi_{3}\phi_{4}}\begin{pmatrix}\phi_{4}k_{3}^{\alpha\beta}&0&h_{4}^{\alpha}/2\\\
0&-\phi_{4}&0\\\ h_{4}^{\beta}/2&0&0\end{pmatrix},$ (147)
where functions $k_{3}^{\alpha\beta}(y)=k_{3}^{\beta\alpha}(y)$ are arbitrary.
After separation of the first two coordinates,
$W^{\prime}_{\alpha}=c_{\alpha}$, the Hamilton–Jacobi equation takes the form
|
# The Sign of non-Gaussianity and the Primordial Black Holes Abundance
Hassan Firouzjahi<EMAIL_ADDRESS>School of Astronomy, Institute for Research
in Fundamental Sciences (IPM)
P. O. Box 19395-5531, Tehran, Iran Antonio Riotto<EMAIL_ADDRESS>Département de Physique Théorique, Université de Genève, 24 quai E. Ansermet,
CH-1211 Geneva, Switzerland Gravitational Wave Science Center (GWSC),
Université de Genève, 24 quai E. Ansermet, CH-1211 Geneva, Switzerland
###### Abstract
The abundance of primordial black holes changes in the presence of local non-
Gaussianity. A positive non-linear parameter $f_{NL}$ increases the abundance
while a negative one reduces it. We show that in non-attractor single-field
models of inflation which enhance the curvature power spectrum and may give
rise to primordial black holes, $f_{NL}$ is always positive, when computed in
correspondence of the peak of the curvature power spectrum where the
primordial black hole abundance has its maximum. This implies that the
interpretation of the recent pulsar timing arrays data from scalar-induced
gravitational waves generated at primordial black hole formation may not be
supported by invoking non-Gaussianity within non-attractor single-field
models.
Introduction. Very recently the NANOGrav NANOGrav:2023gor ; NANOGrav:2023hde ,
EPTA EPTA:2023fyk ; EPTA:2023sfo ; EPTA:2023xxk , PPTA Reardon:2023gzh ;
Zic:2023gta ; Reardon:2023zen and CPTA Xu:2023wog collaborations have
provided evidence for a stochastic background of Gravitational Waves (GWs)
detected through the pulsar timing arrays. One immediate question is under
which circumstances such GWs can be associated to the formation of Primordial
Black Holes (PBHs) during which GWs are inevitably generated at second-order
Sasaki:2018dmp . Their amount is proportional to the the square of the
amplitude of the dimensionless curvature perturbation power spectrum ${\cal
P}_{\cal R}$, $\Omega_{\text{\tiny GW}}\sim{\cal P}_{\cal R}^{2}$. The
abundance of PBHs is exponentially sensitive to the same amplitude,
$f_{\text{\tiny PBH}}\sim{\rm exp}(-1/{\cal P}_{\cal R})$, where
$f_{\text{\tiny PBH}}$ is the PBH abundance with respect to the total dark
matter. The problem is that the observed stochastic GW background is explained
by a relatively large values of ${\cal P}_{\cal R}$, which has been claimed to
lead to a too large PBH abundance Franciolini:2023pbf ; Liu:2023ymk ;
Cai:2023dls ; Inomata:2023zup ; Zhu:2023faa . While this negative conclusion
may be invalidated by the recent observation that corrections from the non-
linear radiation transfer function and the determination of the true physical
horizon crossing decrease the PBH abundance DeLuca:2023tun , one can also rely
on the introduction of some local Non-Gaussianity (NG) in the curvature
perturbation
${\cal R}={\cal R}_{\rm g}+\frac{3}{5}f_{NL}\left({\cal R}^{2}_{\rm
g}-\langle{\cal R}^{2}_{\rm g}\rangle\right),$ (1)
where ${\cal R}_{\rm g}$ is the Gaussian component111We are adopting this
quadratic expansion to be model independent, even though in general the exact
relation between ${\cal R}$ and ${\cal R}_{\rm g}$ can be worked out model by
model. However, since typically $f_{NL}{\cal R}_{\rm g}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 0.5pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}1$,
the quadratic expansion is justified.. The short-scale power spectrum ${\cal
P}_{S}$ responsible for the PBH formation is modulated by the presence of a
long mode ${\cal R}_{L}$. The threshold ${\cal R}_{c}$ for the formation of
the PBHs is shifted approximately by Young:2015kda
${\cal R}_{c}\simeq{\cal R}_{c}^{\rm g}\left(1-\frac{3}{5}f_{NL}{\cal
R}_{c}^{\rm g}\right),$ (2)
compared to the threshold ${\cal R}_{c}^{\rm g}$ in the Gaussian theory.
Therefore, around peaks of the power spectrum of the curvature perturbation, a
positive $f_{NL}$ increases the abundance of the PBHs, while a negative
$f_{NL}$ has the opposite effect, thus helping the agreement with the recent
pulsar timing array observations. This remains true even when calculating the
abundance through a more correct variable, the averaged density contrast
Kehagias:2019eil ; DeLuca:2022rfz .
Under general assumptions, in this paper we will show that the sign of
$f_{NL}$ at the peak scale of the power spectrum, where PBHs are mostly
formed, is always positive in non-attractor single-field models. This no-go
result is intimately related to the fact that $f_{NL}$ measures the response
of the short-scale power spectrum ${\cal P}_{S}$ to the presence of a long
mode and the sign of the NG is determined by the rate of growth of ${\cal
P}_{S}$. The latter is positive if PBHs needs to be produced and this sets the
sign of $f_{NL}$. Our findings automatically imply that NG may not help non-
attractor single-field models to relax the tension between the observed
stochastic GW background in pulsar timing arrays and the overproduction of
PBHs.
Non-attractor single-field models and the sign of NG. In attractor single-
field models the curvature perturbation is constant on superhorizon scales and
is equivalent in the spatially flat gauge to a field fluctuation ${\cal
R}=-\delta\phi/\phi^{\prime}$, where primes denote derivatives with respect to
the number of e-folds. The phase-space trajectory of the long mode
perturbation follows that of the background itself. Short-scale modes evolving
in a long mode perturbation then follow the phase-space trajectory of the
background, with the only difference being the local e-folds which determines
the relation between the comoving and the physical wavenumbers. The NG is
therefore proportional to the variation of the short-scale power spectrum due
to the long-wavelength mode
$\displaystyle{\cal P}_{S}(x)$ $\displaystyle=$ $\displaystyle{\cal
P}_{S}\left[1-\frac{d\ln{\cal P}_{S}}{d\ln k_{S}}{\cal R}_{L}(x)\right]$ (3)
$\displaystyle=$ $\displaystyle\left[1+\frac{12}{5}f_{NL}{\cal
R}_{L}(x)\right].$
This modulation is zero at the peak of the short-scale power spectrum and
corresponds to a dilation of scales rather than an amplitude enhancement.
In non-attractor single-field models, the attractor condition
$\delta\phi^{\prime}=(\phi^{\prime\prime}/\phi^{\prime})\delta\phi$ is
violated. In fact, during an Ultra-Slow-Roll (USR) phase, the curvature
perturbation grows like ${\cal R}\sim a^{3}$, being $a$ the scale factor, and
therefore in the spatially flat gauge $\delta\phi=-\phi^{\prime}{\cal R}=$
constant, implying that $\delta\phi^{\prime}=0$. Because of the the dependence
of the background evolution on the initial kinetic energy, the perturbation
may not be mapped into a change in the background clock along the same phase-
space trajectory. The long mode perturbations carry no corresponding
$\delta\phi^{\prime}$ and so they shift the USR trajectory to one with a
different relationship between $\phi$ and $\phi^{\prime}$. In other words, a
local measurement is sensitive to $\phi^{\prime}$ as different observers
provide different measurements of the short-scale power spectrum depending on
their relative position in the long-wavelength mode. This implies that in USR
models the corresponding value of $f_{NL}$ can be large, even at the peak of
the short-scale power spectrum.
We consider single field models of inflation with the potential $V(\phi)$ for
a canonically normalized scalar field with the sound speed of perturbations
being equal to the speed of the gravitational waves perturbations. To be
general, we do not specify the form of the potential. We assume that inflation
has multiple stages, containing at least three distinct phases. The first
stage is a conventional slow-roll (SR) phase in which the observed large
scales, such as the CMB scales, leave the horizon. The power spectrum of these
perturbations are fixed by the CMB observations Planck to be
${\cal{P}}_{\cal{R}}\simeq 2\times 10^{-9}$ with ${\cal{R}}$ being the
curvature perturbations. The second phase is when the power spectrum
experiences a rapid growth with a prime peak in power spectrum to generate
PBHs Byrnes:2018txb ; Cole:2022xqc ; Carrilho:2019oqg ; t ; Ozsoy:2021pws . A
common mechanism for the enhancement of the power spectrum may be the USR
setup where the potential is flat Kinney:2005vj ; Namjoo:2012aa . However, we
consider a general case and for this purpose, we may call this intermediate
non-attractor phase as a “USR-type” phase. All we require from the form of the
potential to be such that the power spectrum to increase monotonically during
the second phase. The final phase is an attractor SR regime which is extended
towards the end of inflation. The transitions between the stages can be either
sharp or mild. We present our results for a three-phase setup SR $\rightarrow$
non-attractor $\rightarrow$ SR, and the extension of the results to higher
multiple phases is straightforward. We do not consider the stochastic random
motion of the background field so the behaviour of $\phi$ is monotonic. The
non-attractor phase is extended in the region $\phi_{e}<\phi<\phi_{s}$ during
the time interval $t_{s}<t<t_{e}$ and we are interested in the growth of power
spectrum for the modes which leave the Hubble radius during the non-attractor
phase.
For PBH formation, we are interested in the short-scale power spectrum and in
particular the PBH mass function will be dominated by the PBHs forming when
the scale $k_{\rm pk}$ corresponding to the peak of the power spectrum will
re-enter the Hubble radius. Let us consider therefore the effect of the long
mode $k_{L}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
0.5pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}k_{\rm pk}\sim k_{S}$. Notice that
long mode is itself suffering a period of USR phase, but it has exited the
Hubble radius earlier than the scale $k_{S}$. The measurements of the power
spectrum and the bispectrum are made at the end of inflation $t=t_{f}$ when
the modes are frozen. The effects of the long mode on the short modes can be
viewed as the modulation of the background quantities at the end of non-
attractor phase $t=t_{e}$. As in separate universe approach, one can view the
effects of the long mode as affecting nearby patches slightly differently.
Consequently, different patches approach the final attractor phase with
slightly different initial conditions modulated by the long mode at the end of
non-attractor phase. With this picture in mind the bispectrum for two short
modes under the modulation of a long mode can be written as
$\displaystyle\Big{<}{\cal R}^{f}_{L}{\cal R}^{f}_{S}{\cal
R}^{f}_{S}\Big{>}\simeq\Big{<}{\cal
R}^{f}_{L}\,\Big{<}{\cal{R}}^{f}_{S}{\cal{R}}^{f}_{S}\Big{>}_{{\cal{R}}^{e}_{L}}\Big{>}$
(4)
in which ${\cal{R}}_{S}$ and ${\cal{R}}_{L}$ represent the short and long
modes while the superscript $f$ and $e$ indicate the corresponding values at
$t=t_{f}$ and $t=t_{e}$, respectively. The assumption of having a single-field
setup is essential in writing the above relation. If there are extra light
fields, then one has to include the modulations by them in the right-hand side
of Eq. (4) as well.
In non-attractor single-field models ${\cal R}_{L}$ and $\dot{\cal R}_{L}$ are
to be treated as independent variables Namjoo:2013fka . Expanding
$\big{<}{\cal{R}}^{f}_{S}{\cal{R}}^{f}_{S}\big{>}_{{\cal{R}}^{e}_{L}}$ to
leading order yields
$\displaystyle\Big{<}{\cal R}^{f}_{L}{\cal R}^{f}_{S}{\cal R}^{f}_{S}\Big{>}$
$\displaystyle\simeq$ $\displaystyle\Big{<}{\cal R}^{f}_{L}\left({\cal
R}^{e}_{L}\frac{\partial}{\partial{\cal R}^{e}_{L}}\langle{\cal
R}^{f}_{S}{\cal R}^{f}_{S}\rangle\right.$ (5) $\displaystyle+$
$\displaystyle\left.\dot{\cal R}^{e}_{L}\frac{\partial}{\partial\dot{\cal
R}^{e}_{L}}\langle{\cal R}^{f}_{S}{\cal R}^{f}_{S}\rangle\right)\Big{>}.$
An implicit assumption in performing the above expansion is that ${\cal{R}}$
and $\dot{\cal{R}}$ to be continuous across the transition. This is the usual
assumption that one needs to impose for the continuity of the metric and the
extrinsic curvature across the transition. Having said this, we do not impose
any assumption on the potential $V(\phi)$ and its derivatives, as long as
${\cal{R}}$ and $\dot{\cal{R}}$ are continuous across the transition.
Expressing the left hand side of Eq. (5) in terms of the usual non-Gaussianity
parameter $f_{NL}$ and defining the power spectrum in Fourier space as
$\big{\langle}{\cal{R}}_{\bf k_{1}}{\cal{R}}_{\bf
k_{2}}\big{\rangle}=(2\pi)^{3}\delta^{3}({\bf{k_{1}}}+{\bf{k_{2}}})P(k_{1})$
and discarding the trivial factors of $(2\pi)^{3}\delta^{3}(\bf{k})$ which
matches automatically from the momentum conservation, we obtain
$\displaystyle\frac{12}{5}f_{NL}{P}^{f}_{L}{P}^{f}_{S}$ $\displaystyle\simeq$
$\displaystyle\langle{\cal{R}}^{f}_{L}{\cal{R}}^{e}_{L}\big{\rangle}\frac{\partial{P}^{f}_{S}}{\partial{\cal
R}^{e}_{L}}+\big{\langle}{\cal{R}}^{f}_{L}\dot{\cal{R}}^{e}_{L}\big{\rangle}\frac{\partial{P}^{f}_{S}}{\partial\dot{\cal
R}^{e}_{L}},$ (6)
in which $P^{f}_{S}$ and $P^{f}_{L}$ represents the power spectrum at the end
of inflation for the short and long modes respectively.
From the above expression we have to calculate correlations like
$\big{\langle}{\cal{R}}^{f}_{L}{\cal{R}}^{e}_{L}\big{\rangle}$ for the long
mode perturbations at two different times $t_{e}$ and $t_{f}$. As explained
before, this is because the long mode at the end of non-attractor phase
modulates the power spectrum of the short modes which are measured at the end
of inflation.
Since the long mode is far outside the horizon at the end of non-attractor
phase, we can treat it as classical and relate
$\big{\langle}{\cal{R}}^{f}_{L}{\cal{R}}^{e}_{L}\big{\rangle}$ to $P^{f}_{L}$
via the ratio of the mode functions at these two times:
$\displaystyle\big{\langle}{\cal{R}}^{f}_{L}{\cal{R}}^{e}_{L}\big{\rangle}=\left(\frac{{\cal{R}}^{e}_{L}}{{\cal{R}}^{f}_{L}}\right)P^{f}_{L},$
(7)
and similarly
$\displaystyle\big{\langle}{\cal{R}}^{f}_{L}\dot{\cal{R}}^{e}_{L}\big{\rangle}=\frac{1}{2}\left(\frac{{\cal{R}}^{f}_{L}}{{\cal{R}}^{e}_{L}}\right)\frac{dP^{e}_{L}}{dt}.$
(8)
Plugging the above relations into Eq. (6) yields
$\displaystyle\frac{12}{5}f_{NL}$ $\displaystyle=$
$\displaystyle\left(\frac{{\cal R}^{e}_{L}}{{\cal
R}^{f}_{L}}\right)\frac{\partial\ln{\cal{P}}^{f}_{S}}{\partial{\cal{R}}^{e}_{L}}+\frac{1}{2}\left(\frac{{\cal
R}^{f}_{L}}{{\cal
R}^{e}_{L}}\right)\frac{\dot{\cal{P}}^{e}_{L}}{{\cal{P}}^{f}_{L}}\frac{\partial\ln{\cal{P}}^{f}_{S}}{\partial\dot{\cal{R}}^{e}_{L}},$
in which the dimensionless power spectrum ${\cal{P}}_{\cal{R}}$ is related to
the power spectrum via
${\cal{P}}_{\cal{R}}\equiv\frac{k^{3}}{2\pi^{2}}P_{\cal{R}}.$ (10)
We should now trade the two independent variables
$({\cal{R}}_{L},\dot{\cal{R}}_{L})$ with two other variables in which then
partial derivative has a more transparent meaning. From the point of view of a
local observer within a region of size $\sim 1/k_{S}$, the long mode
perturbation evolves with time, but with negligible spatial gradients so the
metric takes the following form
$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)e^{2{\cal{R}}_{L}(t)}d{\bf x}^{2},$ (11)
We can absorb the long mode into the scale factor via $\widetilde{a}\equiv
ae^{{\cal{R}}_{L}}$ and the corresponding Hubble rate will change as
$\widetilde{H}=H+\dot{\cal{R}}_{L}$. Consequently
$\displaystyle d\ln\widetilde{a}=d{\cal{R}}_{L}$ (12)
and
$\displaystyle d\widetilde{H}=d\dot{\cal{R}}_{L}.$ (13)
Eqs. (12) and (13) are two differential relations that can be used to relate
$(d{\cal{R}},d\dot{\cal{R}})$ to $(d\ln\widetilde{a},d\ln\widetilde{H})$. More
specifically, we have
$\displaystyle d\ln{\cal{P}}_{S}$ $\displaystyle=$
$\displaystyle\frac{\partial\ln{{\cal{P}}}_{S}}{\partial{\cal
R}_{L}}d{\cal{R}}_{L}+\frac{\partial\ln{{\cal{P}}}_{S}}{\partial\dot{\cal
R}_{L}}d\dot{\cal{R}}_{L}$ (14) $\displaystyle=$
$\displaystyle\frac{\partial\ln{{\cal{P}}}_{S}}{\partial\ln\widetilde{a}}d\ln\widetilde{a}+\frac{\partial\ln{{\cal{P}}}_{S}}{\partial\widetilde{H}}d\widetilde{H}.$
Using the relations between $(d\ln\widetilde{a},d\widetilde{H})$ and
$(d{\cal{R}},d\dot{\cal{R}})$, from the second line of the above equation we
obtain
$\displaystyle
d\ln{\cal{P}}_{S}=\frac{\partial\ln{{\cal{P}}}_{S}}{\partial\ln\widetilde{a}}d{\cal{R}}_{L}+\frac{\partial\ln{{\cal{P}}}_{S}}{\partial\widetilde{H}}{d\dot{\cal{R}}_{L}}.$
(15)
Comparing this differential equation with the first line of Eq. (14) we obtain
$\displaystyle\frac{\partial\ln{{\cal{P}}}_{S}}{\partial{\cal
R}_{L}}=\frac{\partial\ln{{\cal{P}}}_{S}}{\partial\ln\widetilde{a}}$ (16)
and
$\displaystyle\frac{\partial\ln{{\cal{P}}}_{S}}{\partial\dot{\cal
R}_{L}}=\frac{\partial\ln{{\cal{P}}}_{S}}{\partial\widetilde{H}}.$ (17)
Now, plugging the above relations into formula (The Sign of non-Gaussianity
and the Primordial Black Holes Abundance) and replacing $\widetilde{a}$ and
$\tilde{H}$ simply by $a$ and $H$ yields
$\displaystyle\frac{12}{5}f_{NL}$ $\displaystyle=$
$\displaystyle\left(\frac{{\cal{R}}^{e}_{L}}{{\cal{R}}^{f}_{L}}\right)\frac{\partial\ln{{\cal{P}}}^{f}_{S}}{\partial\ln
a_{e}}+\left(\frac{{\cal{R}}^{f}_{L}}{{\cal{R}}^{e}_{L}}\right)\frac{\dot{\cal{P}}^{e}_{L}}{2H^{2}_{e}{\cal{P}}^{f}_{L}}\frac{\partial\ln{\cal
P}^{f}_{S}}{\partial\ln H_{e}}.$
One can think of Eq. (The Sign of non-Gaussianity and the Primordial Black
Holes Abundance) as an extension of Maldacena’s consistency condition
Maldacena:2002vr to the non-attractor setups (see also cr1 ; cr2 ). The
importance of this consistency condition is that we can read off the value of
$f_{NL}$ from the properties of the power spectrum and without the need to
calculate the bispectrum using either $\delta N$ or in-in formalisms for
higher orders perturbation theory.
So far our analysis was general relying only on the assumption of a single-
field inflation model undergoing non-attractor phase(s) during inflation. The
working assumption is that the power spectrum experiences rapid growth until
it reaching a peak associated to the narrow scale where PBHs are formed. For
the modes which leave the Hubble radius during the non-attractor phase and
near the peak, the power spectrum locally has the following form in momentum
space
$\displaystyle{\cal{P}}_{S}=f(a_{e})\left(\frac{k_{S}}{a_{e}H_{e}}\right)^{n_{\cal{R}}-1},$
(19)
in which $n_{\cal{R}}$ is the spectral index and $f(a)$ is a function of the
background which controls the rapid growth of the power spectrum. Technically
speaking, the factor $f(a)$ comes from the fact that the first slow-roll
parameter $\epsilon\equiv-\dot{H}/H^{2}$ falls off rapidly during the non-
attractor phase so the the power spectrum ${\cal{P}}\propto\epsilon^{-1}$
experiences a rapid growth during the non-attractor phase. For example, in the
conventional USR phase $\epsilon\propto a^{-6}$ and correspondingly
$f(a)=a^{6}$. In our analysis, we do not rely on the particular type of the
transition and the form of $f(a)$ and all we assume is that $f(a)$ is a
growing function of $a$ to ensure the rapid growth of ${\cal{P}}_{\cal{R}}$
during the non-attractor phase. We emphasize again that the form of power
spectrum given in Eq. (19) is valid only locally near the peak which is
followed by a rapid increase in power spectrum. The general form of the power
spectrum in $k$-space is more complicated and may not be even described by a
power law behaviour. For example, it can have oscillatory features after the
prime peak as in conventional USR setup Byrnes:2018txb ; Cole:2022xqc ;
Carrilho:2019oqg ; t ; Ozsoy:2021pws . However, since we are interested in
power spectrum slightly prior and around the peak associated to the narrow
scales where the PBHs are formed, then the ansatz (19) is physically
justified.
From Eq. (19) we infer
$\displaystyle\frac{\partial\ln{\cal P}^{f}_{S}}{\partial\ln
H_{e}}=-\frac{d\ln{\cal P}^{f}_{S}}{d\ln k_{S}}=1-n_{\cal R}.$ (20)
Near the peak of the power spectrum by definition $(n_{\cal R}-1)\simeq 0$ and
correspondingly we obtain
$\displaystyle f_{NL}^{\rm
pk}=\frac{5}{12}\left(\frac{{\cal{R}}^{e}_{L}}{{\cal{R}}^{f}_{L}}\right)\frac{\partial\ln{\cal
P}^{f}_{S}}{\partial\ln a_{e}}.$ (21)
We note that the prefactor $({\cal{R}}^{e}_{L}/{\cal{R}}^{f}_{L})$ appears
because the mode function in general evolves after the non-attractor phase.
This is because the transition from the non-attractor phase to the final
attractor phase may be mild so the mode keeps evolving in time until it
reaches its final attractor value c . The long mode is far outside the horizon
after the peak, evolving from its initial value ${\cal{R}}^{e}_{L}$ at
$t=t_{e}$ to its final value ${\cal{R}}^{f}_{L}$ at $t=t_{f}$. Therefore,
${\cal{R}}^{f}_{L}$ is in phase with ${\cal{R}}^{e}_{L}$ in $k$-space.
However, as the background quantities such as the slow-roll parameters are
evolving during a mild transition, the mode function may change sign so the
ratio $({\cal{R}}^{e}_{L}/{\cal{R}}^{f}_{L})$ may become negative. On the
other hand, if the transition is mild, then the peak in power spectrum will
not be significant as the power spectrum evolves in subsequent evolution so it
is not a viable model for PBHs formation in the first place. Therefore, in
what follows, we make an implicit assumption that the transition from the
intermediate non-attractor phase to the final attractor phase is sharp enough
such that $({\cal{R}}^{e}_{L}/{\cal{R}}^{f}_{L})$ remains positive. Since the
power spectrum is an increasing function of time during the intermediate non-
attractor phase, we conclude that
$f_{NL}^{\rm pk}>0.$ (22)
While our conclusion about the sign of $f_{NL}^{\rm pk}$ is general (with the
implicit assumption of a sharp enough transition), let us examine it for some
non-trivial examples. Let us consider a setup in which a USR phase is followed
by an attractor SR phase in which the transition to the final attractor phase
can be either sharp or mild. Defining the slow-roll parameter associated to
the derivative of the potential at the final attractor phase by
$\sqrt{2\epsilon_{V}}\equiv V_{\phi}/V$, the sharpness of the transition from
the intermediate USR phase to the final attractor phase is determined by the
parameter $h$ given by c
$\displaystyle h\equiv-6\sqrt{\frac{\epsilon_{V}}{\epsilon_{e}}}\,,$ (23)
in which $\epsilon_{e}$ is the value of the slow-roll parameter at the end of
USR phase. Note that in this convention $h<0$. For a very sharp transition
$|h|\gg 1$ while for a mild transition $h$ may be comparable to slow-roll
parameters. In order to have sharp enough transition such that the ratio
$({\cal{R}}^{e}_{L}/{\cal{R}}^{f}_{L})$ remains positive, we assume
$\eta_{V}\rightarrow 0$ in which $\eta_{V}$ is the second slow-roll parameter
given by $\eta_{V}=V_{\phi\phi}/V$.
The mode function for the modes which leave the horizon during the USR phase
is given by c
$\displaystyle{\cal{R}}^{f}_{k}=\left(1+\sqrt{\frac{\epsilon_{V}}{\epsilon_{e}}}\right)\frac{H}{\sqrt{4\epsilon_{V}k^{3}}}.$
(24)
Since during the USR phase the slow-roll parameter falls off like $a^{-6}$,
then $\epsilon_{e}\propto a_{e}^{-6}$. Taking the derivative with respect to
$a_{e}$ we find
$\displaystyle\frac{d\ln{\cal{P}}^{f}}{d\ln
a_{e}}=6\sqrt{\frac{\epsilon_{V}}{\epsilon_{e}}}\left(1+\sqrt{\frac{\epsilon_{V}}{\epsilon_{e}}}\right)^{-1}=\frac{6h}{h-6}.$
(25)
On the other hand, the ratio ${\cal{R}}^{e}_{L}/{\cal{R}}^{f}_{L}$ yields an
additional factor
$\displaystyle\frac{{\cal{R}}^{e}_{L}}{{\cal{R}}^{f}_{L}}=\sqrt{\frac{\epsilon_{V}}{\epsilon_{e}}}\left(1+\sqrt{\frac{\epsilon_{V}}{\epsilon_{e}}}\right)^{-1}=\frac{h}{h-6}.$
(26)
We see that the ratio $({\cal{R}}^{e}_{L}/{\cal{R}}^{f}_{L})$ is positive as
expected. Using Eqs. (25) and (26) in our formula (21) yields
$\displaystyle f_{NL}^{\rm pk}=\frac{5h^{2}}{2(6-h)^{2}}>0.$ (27)
For an infinitely sharp transition with $h\rightarrow-\infty$ in which the
mode function is frozen immediately after the transition with
${\cal{R}}^{e}_{L}={\cal{R}}^{f}_{L}$, from Eq. (27) we obtain the expected
result $f_{NL}^{\rm pk}=5/2$. The expression Eq. (27) agrees with the result
for $f_{NL}$ obtained in c where the power spectrum is scale-invariant as
well.
As a second example, now suppose we extend the above setup such that there is
an upward shift $\Delta V$ in the potential at the end of non-attractor phase,
followed by the final SR phase. As in Ref. Cai:2022erk , suppose the upward
step in the potential is instantaneous, yielding to a sudden change in
inflaton’s velocity. Imposing the conservation of energy, the inflaton
velocity at the end of upward transition $\pi_{d}$ is related to the velocity
at the end of no-attractor phase $\pi_{e}$ via
$\displaystyle\pi_{d}=-\sqrt{\pi_{e}^{2}-6\frac{\Delta V}{V}}\,,$ (28)
in which $\pi\equiv\phi^{\prime}$ with a prime denoting the derivative with
respect to the number of e-folds. The linear mode function is given by
Cai:2022erk
$\displaystyle{\cal{R}}^{f}_{k}=\left(\frac{1}{g}+\sqrt{\frac{\epsilon_{V}}{\epsilon_{e}}}\right)\frac{H}{\sqrt{4\epsilon_{V}k^{3}}},$
(29)
in which $g\equiv\pi_{d}/\pi_{e}$ with $0<g<1$. Correspondingly, this yields
$\displaystyle\frac{d\ln{\cal{P}}^{f}}{d\ln
a_{e}}=\frac{6hg^{4}+36g^{2}-36}{g^{2}(g^{2}h-6)},$ (30)
in which the sharpness parameter $h$ is now defined as
$h\equiv-(6/g)\sqrt{\epsilon_{V}/\epsilon_{e}}$. In addition, the ratio of the
mode functions is given by
$\displaystyle\frac{{\cal{R}}^{e}_{L}}{{\cal{R}}^{f}_{L}}=\frac{hg^{2}}{hg^{2}-6}>0.$
(31)
Note that if we set $g=1$ so $\Delta V=0$, Eqs. (31) and (30) reduce to Eqs.
(26) and (25) respectively. Now plugging Eqs. (31) and (30) into our master
formula Eq. (21) yields
$\displaystyle f_{NL}^{\rm pk}=\frac{5h(hg^{4}+6g^{2}-6)}{2(g^{2}h-6)^{2}},$
(32)
in exact agreement with Cai:2022erk for a scale-invariant power spectrum. If
we set $g=1$, corresponding to no bump in potential, then Eq. (32) reduces to
Eq. (27). Noting that $h<0$ and $0<g<1$, one can check that $f_{NL}^{\rm
pk}>0$ for all allowed values of $(h,g)$ as our theorem predicts. Note that
the above value of $f_{NL}$ was calculated in Cai:2022erk using the $\delta
N$ formalism to second order in perturbation theory. However, in our approach
based on consistency condition, we only need to calculate the linear mode
function without the need to go to higher orders in perturbation theory.
As a corollary, our theorem implies that in the setups where the power
spectrum experiences a suppression going through a minimum, then $f_{NL}<0$ at
the minimum as was observed in a specific setup in Domenech:2023dxx .
Conclusions. In this note we have shown that the non-linear parameter $f_{NL}$
in single-field non-attractor models is always positive if calculated for the
peak of the enhanced power spectrum. This result implies the NG always
increases the PBH abundance. The sign of the NG is fixed by the response of
the short-scale power spectrum to the presence of a long mode. If PBHs need to
be form, the short-scale power spectrum needs to grow and this set the sign of
$f_{NL}^{\rm pk}$ uniquely. This logic implies that our no-go result does not
hold in the case in which the NG is generated after the inflationary phase,
e.g. in the presence of a spectator field. Indeed, one can generate PBHs
within a spiky model where the comoving curvature power spectrum is enhanced
at small scales through a spectator isocurvature field Kawasaki:2012wr . This
isocurvature perturbation will then subsequently decay into radiation
perturbation and become a curvature mode after inflation. In such a case the
long mode cannot be reabsorbed by a redefinition of the scale factor and
therefore the sign of the NG is not defined. As a consequence, $f_{NL}$ can be
negative in models with extra fields. We comment that our conclusion about the
sign of $f_{NL}^{\rm pk}$ requires an implicit assumption that the transition
from the non-attractor phase to the final attractor phase be sharp enough so
the mode function keeps its original sign. Physically, this is the relevant
case for PBHs formation since if the transition is not sharp enough, then the
peak is not prominent and PBHs may not form in the first place.
Acknowledgments. H.F. thanks the Department of Theoretical Physics at the
University of Geneva for the kind hospitality when part of this work has been
done. We thank M. Sasaki and M. H. Namjoo for insightful discussions and
comments. A.R. thanks the Boninchi Fundation for support.
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|
††thanks: These two authors contributed equally††thanks: These two authors
contributed equally
# Coherent and dissipative coupling in a magneto-mechanical system
P. Carrara Dipartimento di Fisica, Università degli Studi di Milano, Via
Celoria 16, 20133 Milano, Italy Istituto Officina dei Materiali, Consiglio
Nazionale delle Ricerche, Strada Statale 14, km 163.5, 34149 Basovizza (TS),
Italy M. Brioschi Dipartimento di Fisica, Università degli Studi di Milano,
Via Celoria 16, 20133 Milano, Italy Istituto Officina dei Materiali,
Consiglio Nazionale delle Ricerche, Strada Statale 14, km 163.5, 34149
Basovizza (TS), Italy R. Silvani Istituto Officina dei Materiali, Consiglio
Nazionale delle Ricerche, c/o Dipartimento di Fisica e Geologia, Via A.
Pascoli, 06123 Perugia, Italy A.O. Adeyeye Department of Physics, Durham
University, South Rd, DH1 3LE Durham, United Kingdom Department of Electrical
and Computer Engineering, National University of Singapore, 4 Engineering
Drive 3, 117576 Singapore G. Panaccione Istituto Officina dei Materiali,
Consiglio Nazionale delle Ricerche, Strada Statale 14, km 163.5, 34149
Basovizza (TS), Italy G. Gubbiotti Corresponding author, email:
<EMAIL_ADDRESS>Istituto Officina dei Materiali, Consiglio Nazionale
delle Ricerche, c/o Dipartimento di Fisica e Geologia, Via A. Pascoli, 06123
Perugia, Italy G. Rossi Corresponding author, email<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16,
20133 Milano, Italy Istituto Officina dei Materiali, Consiglio Nazionale
delle Ricerche, Strada Statale 14, km 163.5, 34149 Basovizza (TS), Italy R.
Cucini Istituto Officina dei Materiali, Consiglio Nazionale delle Ricerche,
Strada Statale 14, km 163.5, 34149 Basovizza (TS), Italy
###### Abstract
Hybrid elastic and spin waves hold promises for energy-efficient and versatile
generation and detection of magnetic signals, with potentially long coherence
times. Here we report on the combined elastic and magnetic dynamics in a one-
dimensional magneto-mechanical crystal composed of an array of magnetic
nanowires. Phononic and magnonic modes are impulsively excited by an optical
ultrafast trigger and their decay is monitored by time resolved Magneto
Optical Kerr Effect, with complementary Brillouin Light Scattering
measurements and micromagnetic simulations. The strength and degree of mixing
of coherent and dissipative coupling of the quasi-particles is determined
quantitatively.
In hybrid magnonics, the coupling of magnons (the quanta of spin waves) to
other degrees of freedom is explored to achieve enhanced functionalities in
solid-state systems and devices [1]. Novel results have been achieved in
coupling magnons to microwave [2] and optical photons [3], and/or to phonons
[4, 5, 6, 7], or even to superconducting qubits [8]. This approach has found
fertile ground in the field of quantum engineering [9], where hybridized
quasi-particles can boost transduction and sensing capabilities [10, 11], down
to the single-quantum detection and manipulation [12], or allow novel
computation, simulation and storage platforms [13].
Here we report the experimental observation of magnon-phonon mixed coupling in
a 1D magnonic-phononic crystal via time-resolved Magneto-Optical Kerr Effect
(tr-MOKE). The hybridized modes entangle the coherent and dissipative
couplings of the quasi-particle subsystems, and indicate a novel energy
exchange mechanism in magnonic-phononic crystals. Experiments in the time
domain are crucial to explore the weak-coupling regime: the strength of
coherent and dissipative coupling is quantitatively assessed employing a
hamiltonian model.
The coherent and dissipative coupling [14] describes the energy exchange
between two coherently coupled systems directly [15], or towards a common
reservoir [16], respectively. Dissipative coupling is now boosting research
for its close link to non-Hermitian physics [17, 18] and non-reciprocal
transport [19].
Both coupling mechanisms can coexist in a single system, creating the
possibility of tuning one into the other. Nonetheless, only a few papers
report the coexistence of both coherent and dissipative coupling in hybrid
systems [20, 16, 21, 19], a condition we dub mixed coupling.
Figure 1: (a) SEM micrograph of the investigated sample. Inset: a cross-
sectional sketch of the bilayered NWs; the inter-NW distance $d$, NW width
$w$, and full periodicity $D$ are also indicated (not to scale). (b) Magnetic
hysteresis loop obtained via static MOKE at $\varphi=60^{\circ}$; the coercive
field $B_{c}=17$ mT is indicated. (c) Sketch of the experimental setup for tr-
MOKE. (d-e) Time- and frequency-domain maps of the tr-MOKE signal at
$\varphi=60^{\circ}$. The magnetic field is swept from positive to negative
values. The vertical dashed lines at $B_{\text{ext}}=-22$ mT highlight the
magnetization reversal, which results in phase anomalies in the time domain
and in discontinuity for the PM mode in the frequency domain. The enhancement
of MOKE amplitude at late delays in panel (d) corresponds to the condition of
PM-MEC1 coupling in panel (e).
The sample employed in this study is an array of rectangular cross-sectioned
bilayered nanowires (NWs) of Fe (10 nm thick) and Py (Ni80Fe20, 10 nm thick)
on Si (001) substrate. Details on the fabrication process can be found in
Ref.[22]. Each NW is $w=340$ nm wide, and the inter-NW spacing is $d=70$ nm;
this gives an overall periodicity $D=w+d=410$ nm (Fig.1(a)). The coercive
field is $B_{c}=18$ mT, as extracted from static MOKE hysteresis loops
(Fig.1(b)). Such system has been characterized as a magnonic crystal via
Brillouin Light Scattering (BLS), ferromagnetic resonance and micromagnetic
simulations [23, 24, 25, 26]: magnon bands form as for inter-NW magnetic
dipolar interaction. Similarly, the spatial modulation of the elastic
properties of the sample surface gives rise to standing surface phononic
modes, together with acoustic modes localized in each NW [27, 28, 29].
The optical end-station employed for pump-probe spectroscopy is sketched in
Fig.1(c): the sample is excited with an ultrafast near-infrared pulsed laser
(pump), and the magnetic and magneto-elastic dynamics are probed via tr-MOKE.
The incidence angles from the sample normal are $12^{\circ}$ and $6^{\circ}$
for the pump and probe beams, respectively; consequently, we are primarily
sensitive to the out-of-plane component of the magnetization. Further details
on the setup are reported in [30] and in the Supplemental Material [31]. The
pump excitation triggers acoustic modes and, via inverse magnetostriction, the
phononic strain fields interact with the magnetization of the NWs, leading to
coupled magneto-mechanical dynamics [32]. Moreover, it is possible to excite
pure magnetization dynamics via ultrafast heat-induced magnetic anisotropy
quenching upon ultrafast laser illumination. This thermal mechanism, firstly
described in Ref.[33] for an easy-plane magnetic anisotropy system and then
extended to different systems [34, 35], requires an external magnetic field
with strength comparable to the sample magnetic anisotropy field. The magnon
modes excited with this mechanism rely only on the magnetostatic properties of
the NWs (magnetization and shape anisotropy) and can be observed via
complementary dynamic techniques like BLS, as well as simulated with
micromagnetic models.
In Fig.1(d) we report a wide-scan map of tr-MOKE results, for $B_{\text{ext}}$
swept, from +90 to -90 mT, in approximately 1.5 mT steps. For every magnetic
field, we record the tr-MOKE signal up to a delay of 3.3 ns. An exponential
plus linear background is subtracted from each trace. The map in Fig.1(e) is
obtained performing FFT of each trace. Here, three features can be identified:
two flat modes with different intensities, featuring no frequency dispersion
with $B_{\text{ext}}$, and a third dispersive mode. We label the formers as
magneto-elastic-coupling modes (MEC1 and MEC2) and the latter as a pure
magnonic mode (PM). We now briefly discuss their excitation mechanism.
The pump photons are absorbed by the metallic NWs, resulting in thermo-elastic
expansion (the direct of Si is well above the pump photon energy). The
periodic strain field stabilizes a standing surface acoustic wave with the
wavelength matching the NW array periodicity ($D$). Pump-induced heating also
generates localized breathing modes within each NW. These acoustic modes drive
NW magnetization, producing the time-modulated magnetic contrast we observe in
tr-MOKE (flat modes in Fig.1(e)).
Figure 2: (a) Time-domain trace (black circles) and fit (red line) of time-
resolved reflectivity at $B_{\text{ext}}=0$ mT. The residuals (grey line) are
rigidly shifted for clarity. Inset: FFT of the original trace (black line) and
of the residuals (grey line). (b) The frequency of the PM mode was extracted
from a time-domain fit of tr-MOKE traces (blue circles), together with the
frequency of the lowest-order magnon mode as obtained from BLS measurements
(orange circles) and micromagnetic simulations (black line). The azimuth is
$\varphi=60^{\circ}$. The three vertical shaded bars indicate regions in which
the extraction of frequency from tr-MOKE data is not possible.
The attribution of such flat modes to magneto-elastic modes in similar systems
is consistent with the literature [34, 32]. To further confirm the assignment,
we perform time-resolved reflectivity measurements (tr-R) using identical
experimental conditions as in the tr-MOKE measurements, while detecting the
non-rotated component of the probe polarization. The time-domain trace after
background removal is reported in Fig.2(a) (black circles), for
$B_{\text{ext}}=0$ mT. The fit of the function
$y=\sum_{i}A_{i}\sin{(\omega_{i}t+\phi_{i})}e^{-\gamma_{i}t}$ (1)
is shown in red ($i=1,2$ denotes the lower- and upper-frequency phononic mode,
respectively). The parameters $A_{i},~{}\omega_{i},~{}\phi_{i},~{}\gamma_{i}$
are the amplitude, angular frequency, phase and damping parameter for $i$-th
mode. The feature-less residuals of the fitting are shown in grey and rigidly
shifted for clarity. The frequencies ($f_{i}=\omega_{i}/2\pi$) and dampings
obtained as best-fit parameters are $f_{\text{MEC1}}=(8.89\pm 0.05)$ GHz and
$f_{\text{MEC2}}=(10.75\pm 0.05)$ GHz, and $\gamma_{\text{MEC1}}=(0.73\pm
0.09)$ rad/ns and $\gamma_{\text{MEC2}}=(1.4\pm 0.1)$ rad/ns, in agreement
with the parameters for the flat modes in tr-MOKE (see below). The lower
frequency mode is assigned to the Rayleigh wave of the substrate covered with
the NW array [34], and the higher frequency mode to a localized (width)
breathing mode of each NW.
Let us now focus on the PM mode. The leading magnetic anisotropy in the NWs is
shape anisotropy, which for thin NWs, favors in-plane magnetization along the
NW longitudinal axis, which is thus the magnetic easy axis (EA). If
$B_{\text{ext}}$ is not aligned to EA and comparable in strength to the
magnetic anisotropy field, the pump absorption in NWs results in ultrafast
quenching of the magnetization and concurrently to impulsive softening of the
magnetic anisotropy, triggering magnetization precession; dynamics at the GHz
range reflects the equilibrium eigenstates of the system [31, 33].
Furthermore, the same triggered dynamics in each NW result in synchronized
precession, generating a measurable MOKE signal—a zero-wavevector magnonic
mode. Note that the PM mode’s spectral weight diminishes to zero whenever the
magnetization’s equilibrium axis aligns with EA: i) as $B_{\text{ext}}$
weakens (see Fig.1(e)), or ii) at $\varphi=0^{\circ}$ for any $B_{\text{ext}}$
value [31]. To confirm the PM mode assignment as the lowest-order
magnetization precession, we compare tr-MOKE with BLS and micromagnetic
simulations (see [31] for details). In Fig.2(b), we present the PM mode
frequency from tr-MOKE fits (blue circles) alongside the lowest-order magnonic
frequency’s field dependence from BLS (orange circles) and micromagnetic
simulations (black line). The datasets and simulations show good agreement,
especially in the concavity change at positive field and switching field
value. The grey shaded bars in Fig.2(b) highlight regions in which extraction
of tr-MOKE values is not possible, either because of the absence of the PM
mode or because of the mixing to the MEC modes. The systematic redshift of the
tr-MOKE results is compatible with a few degrees of experimental mismatch in
the azimuth angle $\varphi$.
We now focus on the region (50 - 80) mT, where the crossing of the PM and MEC1
modes (see Fig.1(e)) suggests the presence of coupling. To gain a deeper
insight, we employ a Hamiltonian $\mathcal{H}$, modeling both coherent and
dissipative coupling [16, 36]:
$\mathcal{H}/\hbar=\tilde{\omega}_{A}a^{\dagger}a+\tilde{\omega}_{B}b^{\dagger}b+g\left(a^{\dagger}b+e^{i\Phi}ab^{\dagger}\right)~{}.$
(2)
Here $a^{\dagger}$ and $b^{\dagger}$ ($a$ and $b$) are the creation
(annihilation) operators for mode $A$ and $B$, respectively (in our experiment
the modes are the phonon and the magnon);
$\tilde{\omega}_{i}=\omega_{i}-i\gamma_{i}$ is the generalized angular
frequency of uncoupled mode $i=(A,B)$, encompassing both the angular frequency
$\omega_{i}$ and the intrinsic damping $\gamma_{i}$; $g$ is the strength of
the coupling, whose nature depends on the value of the phase $\Phi$. The
coupled eigenvalues of $\mathcal{H}$ are
$\tilde{\omega}_{\pm}=\left(\frac{\tilde{\omega}_{A}+\tilde{\omega}_{B}}{2}\right)\pm\sqrt{\left(\frac{\tilde{\omega}_{A}-\tilde{\omega}_{B}}{2}\right)^{2}+g^{2}e^{i\Phi}}~{},$
(3)
where again the real part of $\tilde{\omega}_{\pm}$ gives the angular
frequency and the opposite imaginary part gives the damping parameter. In
Fig.3 the coupled (red and blue lines) and uncoupled (black dashed lines)
eigenvalues are plotted as a function of the uncoupled frequency detuning. At
$\Phi=0$ (pure coherent coupling, panels (a) and (e)), frequency gapping at
zero detuning results; this is often understood as the smoking gun for
hybridization. The dampings, on the other hand, attract each other and are
equal at zero detuning. At $\Phi=\pi$ (pure dissipative coupling, panels (c)
and (g)) the deviation from the uncoupled values are opposite: frequency
attraction extends the degeneracy to a finite region across zero detuning,
while the dampings repel each other. The eigenvalue symmetry for $\Phi=0$ and
$\pi$ is lost if both couplings are at play, i.e. for mixed coupling, as shown
for $\Phi=\pi/2$ (panels (b) and (f)) and $\Phi=3\pi/2$ (panels (d) and (h)).
The striking feature in mixed coupling is that the condition for degenerate
dampings is shifted from zero detuning. We propose this shift as a
phenomenological identifier of mixed coupling systems.
Figure 3: Modification of the eigenvalues of $\mathcal{H}$ upon changing the
coupling phase $\Phi$. (a-d) Frequency dispersions (real part). (e-h) Damping
parameters (imaginary part). All plots are shown as a function of the
uncoupled frequency detuning. The upper ($\tilde{\omega}_{+}$) and the lower
branch ($\tilde{\omega}_{-}$) results are shown as red and blue lines,
respectively. The uncoupled values for frequency and damping are shown in each
plot as black dashed lines. For comparison to the experimental results shown
later, in the calculations we assumed $\gamma_{A}=1.2$ rad/ns and
$\gamma_{B}=0.8$ rad/ns; the coupling strength was kept at $g=0.5$ rad/ns.
Figure 4: Analysis of frequency (a) and damping (b) as extracted from the
time-domain fit of tr-MOKE traces acquired in a close-up of the PM-MEC1
crossing region. The experimental data for the hybridizing modes show
deviations from the uncoupled modes, mixing phonon character (field-
independent frequency, green circles) and magnon character (field-dependent
frequency, yellow circles). The fit results for the upper hybridized branch
(red line) and for the lower hybridized branch (blue line) are also shown,
together with the values for the uncoupled modes (black dashed lines). The
arrows highlight the mismatch between the field values for zero-detuning
(upper panel) and for damping crossing (lower panel). The fit results for the
non-hybridizing MEC2 mode are also reported (orange circles).
The proposed hamiltonian model requires the time domain fit of the traces,
acquired in finely sampled intervals (approximately 0.3 mT) of
$B_{\text{ext}}$ in a close-up of the PM-MEC1 crossing region. In Fig.4 we
report, as color-coded circles with error bars, the frequency (panel (a)) and
damping (panel (b)) obtained as best-fit parameters. Note that the time-domain
analysis reveals frequency gapping of the modes, a feature not visible in the
FFT map (see also [31]).
As highlighted by the black arrows, there is a mismatch between the damping
crossing and the condition of zero detuning. We fit to the data in Fig.4 the
eigenvalues of $\mathcal{H}$ (Eq.3): the best-fit curves are reported for the
upper branch (red line) and for the lower branch (blue line). The PM mode is
assumed as linear in $B_{\text{ext}}$, a reasonable approximation of the
actual non-linear dispersion, given the reduced detuning window. The result of
the fitting gives $g=\left(0.55\pm 0.03\right)$ rad/ns and $\Phi=1.3\pm 0.1$.
These values set the investigated system as weakly coupled since the strength
$g$ is lower than the intrinsic damping of both hybridizing modes; as $\Phi$
is close to $\pi/2$, both coupling mechanisms are at play with comparable
strength.
To summarize, we derived from tr-MOKE experimental data the coupling of
phononic and magnonic modes in a 1D magnonic-phononic crystal. The time domain
data give evidence of a mixed phonon-magnon coupling mechanism at play. By
means of comparison of the experimental data with the eigenvalues of a
comprehensive hamiltonian model we can derive the coupling strength and the
competition between coherent and dissipative coupling. Such quantitative
information is crucial for identifying the experimental parameters that
continuously tune the coupling in a magneto-mechanical system, transitioning
from purely coherent to purely dissipative coupling. Our results hint to the
possibility of novel magnonic-phononic devices, in analogy to what was shown
by cavity magnonics (magnon-photon hybridization)[16, 36]. Finally, we
demonstrated time-domain spectroscopy as a key tool to perform a detailed
investigation of weakly coupled hybrids systems, allowing for reliable
quantitative analysis of the coupling even in those cases when the intrinsic
damping dominates over coupling strength, challenging for frequency-domain
techniques.
P.C. and M.B. thank Vincent Polewczyk and Giacomo Jark for valuable
discussions. P.C. is also grateful to Riccardo Panza and Alberto Scazzola for
valuable discussions. This project has received funding from the European
Union’s Horizon 2020 research and innovation program under grant agreement No
101007417. Research at IOM-CNR has been funded by the European Union - Next
Generation EU under the Italian Ministry of University and Research (MUR)
National Innovation Ecosystem grant ECS00000041 - VITALITY-CUP
B43C22000470005. G.G. and G.P. acknowledge Università degli Studi di Perugia,
CNR and MUR for support within the project Vitality. A.O.A. and G.G.
acknowledge the funding from the Royal Society through the Wolfson Fellowship
and International Exchanges IEC\R2\222074.
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|
# MS-TP-21-12 Yukawa coupling unification in an $\mathsf{SO(10)}$ model
consistent with Fermilab $(g-2)_{\mu}$ result
Amin<EMAIL_ADDRESS>, Pran
<EMAIL_ADDRESS>and Raza M<EMAIL_ADDRESS>
aInstitut für Theoretische Physik, Westfälische Wilhelms-Universität Münster,
Wilhelm-Klemm-Straße 9, 48149 Münster, Germany
bDepartment of Physics, Northeastern University, Boston, MA 02115-5000, USA
cDepartment of Physics, American University of Sharjah, P.O. Box 26666,
Sharjah, UAE444Permanent address
###### Abstract
We investigate the Yukawa coupling unification for the third generation in a
class of $\mathsf{SO(10)}$ unified models which are consistent with the 4.2
$\sigma$ deviation from the standard model of the muon $g-2$ seen by the
Fermilab experiment E989. A recent analysis in supergravity grand unified
models shows that such an effect can arise from supersymmetric loops
correction. Using a neural network, we further analyze regions of the
parameter space where Yukawa coupling unification consistent with the Fermilab
result can appear. In the analysis we take into account the contributions to
Yukawas from the cubic and the quartic interactions. We test the model at the
high luminosity and high energy LHC and estimate the integrated luminosities
needed to discover sparticles predicted by the model.
###### Contents
1. 1 Introduction
2. 2 The model
3. 3 $\mathsf{SO(10)}$ SUGRA model with Yukawa unification consistent with Fermilab $(g-2)_{\mu}$
4. 4 Sparticle hierarchies and signal region analysis
1. 4.1 Slepton pair production and event simulation at the LHC
2. 4.2 Event selection
3. 4.3 Results
5. 5 Conclusion
6. 6 Appendix: Contributions to Yukawas from higher dimensional operators
## 1 Introduction
Recently the Fermilab E989 experiment [1] has measured $a_{\mu}=(g-2)_{\mu}/2$
with significantly greater accuracy than the previous Brookhaven experiment
[2, 3]. Thus the combined Fermilab experimental data and Brookhaven
experimental data gives
$a^{\rm exp}_{\mu}=116592061(41)\times 10^{-11}\,,$ (1.1)
which is to be compared with the Standard Model (SM) prediction [4]
$a^{\rm SM}_{\mu}=116591810(43)\times 10^{-11}.$ (1.2)
The combined Fermilab and Brookhaven result shows an excess over the SM result
by an amount $\Delta a^{\rm FB}_{\mu}$ which is
$\Delta a^{\rm FB}_{\mu}=a_{\mu}^{\rm exp}-a_{\mu}^{\rm SM}=251(59)\times
10^{-11}.$ (1.3)
Eq. (1.3) records a $4.2\sigma$ deviation from the SM compared to $3.7\sigma$
for the Brookhaven result. Thus the Fermilab experiment further strengthens
the Brookhaven result on the possible existence of new physics beyond the
Standard Model. Subsequent to the Fermilab result, artificial neural network
analysis was used to explore the parameter space of supergravity (SUGRA)
unified models. It was seen that regions of the parameter space where
supersymmetric loops can give the desired correction consistent with the
Fermilab results are those where gluino-driven radiative breaking of the
electroweak symmetry occurs [5], a region referred to as $\tilde{g}$SUGRA [6,
7, 8]. Using a neutral network we investigate this region further to explore
the region where Yukawa unification in an $\mathsf{SO(10)}$ model [8] can
occur consistent with the Fermilab result.
The outline of rest of the paper is as follows: In section 2 details of the
$\mathsf{SO(10)}$ model are discussed. In section 3 an analysis of the
parameter space of SUGRA $\mathsf{SO(10)}$ model which gives Yukawa coupling
unification consistent with the Fermilab $g-2$ result is given. Here the light
and the heavy sparticle spectrum is also computed. In section 4, simulations
for the observation of the sparticles predicted by the model at HL-LHC and HE-
LHC are given. Conclusions are given in section 5. Some further details of the
model are given in the Appendix.
## 2 The model
The general class of $\mathsf{SO(10)}$ models we consider are those of [9, 10]
with [8] being one of them which are similar in spirit to the missing partner
$\mathsf{SU(5)}$ models [11, 12]. These models involve large Higgs
representations such as $\mathsf{126+\overline{126}}$, $\mathsf{210}$,
$\mathsf{120}$ for Yukawa couplings. Large Higgs representations have been
used in several early works [13, 14, 15] and also more recently, e.g., [16,
17, 18, 19, 20, 21, 22] and the references therein (for a review of
$\mathsf{SO(10)}$ models, see Ref. [23]). In the model we consider [10, 8],
the missing partner mechanism comes about as follows: The Higgs sector
consists of the fields $\mathsf{126+\overline{126}}$, $\mathsf{210}$,
$2\times\mathsf{10+120}$ set of representations. The fields
$\mathsf{126+\overline{126}}$, $\mathsf{210}$ are heavy which break the GUT
symmetry down to the SM gauge group symmetry, while the
$2\times\mathsf{10+120}$ Higgs fields are light. The heavy fields contain 3
pairs of heavy Higgs doublets while the light fields have four pairs of light
Higgs doublets. When the light and heavy fields mix, three pairs of the light
Higgs doublets become heavy while one combination of the light doublets
remains light and is identified as the Higgs field of the MSSM. We give below
further details of the model used in this analysis.
The superpotential of the $\mathsf{SO(10)}$ model is given by [8]
$\displaystyle W=W_{\rm GUT}+W_{\rm DT}+W_{\rm Yuk},$ (2.1)
where
$\displaystyle W_{\textsc{gut}}=$
$\displaystyle~{}M^{126}\Delta_{\mu\nu\rho\sigma\lambda}\overline{\Delta}_{\mu\nu\rho\sigma\lambda}+M^{210}\Phi_{\mu\nu\rho\sigma}\Phi_{\mu\nu\rho\sigma}+\eta\Phi_{\mu\nu\rho\sigma}\Delta_{\mu\nu\lambda\tau\xi}\overline{\Delta}_{\rho\sigma\lambda\tau\xi}$
$\displaystyle+\lambda\Phi_{\mu\nu\rho\sigma}\Phi_{\rho\sigma\lambda\tau}\Phi_{\lambda\tau\mu\nu}\,,$
(2.2) $\displaystyle W_{\textsc{dt}}=~{}$ $\displaystyle
a~{}{{}^{1}}\Omega_{\mu}\overline{\Delta}_{\mu\nu\rho\sigma\lambda}\Phi_{\nu\rho\sigma\lambda}+\sum_{r=1}^{2}b_{r}~{}{{}^{r}}\Omega_{\mu}{\Delta}_{\mu\nu\rho\sigma\lambda}\Phi_{\nu\rho\sigma\lambda}+c~{}\Sigma_{\mu\nu\rho}\Delta_{\nu\rho\sigma\lambda\tau}\Phi_{\mu\sigma\lambda\tau}$
$\displaystyle+\overline{c}~{}\Sigma_{\mu\nu\rho}\overline{\Delta}_{\nu\rho\sigma\lambda\tau}\Phi_{\mu\sigma\lambda\tau}\,.$
(2.3)
The notation used above is as follows: $\Delta_{\mu\nu\rho\sigma\lambda}$ and
$\overline{\Delta}_{\mu\nu\rho\sigma\lambda}$ are fields for the
$\mathsf{126}$ and $\mathsf{\overline{126}}$ representations,
$\Phi_{\mu\nu\rho\sigma}$ is the field for the $\mathsf{210}$ representation
and ${{}^{r}}\Omega_{\mu}(r=1,2)$ are the fields for the two $\mathsf{10}$ of
Higgs representations and $\Sigma_{\mu\nu\rho}$ is the field for the
$\mathsf{120}$-plet representation. In the above $W_{\rm GUT}$ breaks the
$\mathsf{SO(10)}$ GUT symmetry down to the standard model gauge group
$\mathsf{SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}}$ by VEV formations of
$\mathcal{V}_{1_{126}}$ and $\mathcal{V}_{1_{\overline{126}}}$ and the VEVs of
$\mathcal{V}_{1_{210}}$, $\mathcal{V}_{24_{210}}$, $\mathcal{V}_{75_{210}}$.
The equations that determine these VEVs are derived in [8]. Thus the
$\mathsf{126+\overline{126}}$-plet VEVs $\mathcal{V}_{1_{126}}$ and
$\mathcal{V}_{1_{\overline{126}}}$ break the $\mathsf{SO(10)}$ symmetry down
to $\mathsf{SU(5)\times U(1)}$ and the $\mathsf{210}$-plet VEVs
$\mathcal{V}_{1_{210}}$, $\mathcal{V}_{24_{210}}$, $\mathcal{V}_{75_{210}}$
further break the gauge symmetry down to $\mathsf{SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}}$. The notation for the VEVs is explicit. Thus, for
example, $\mathcal{V}_{1_{126}}$ stands for the VEV of the $\mathsf{SU(5)}$
singlet in the $\mathsf{SU(5)\times U(1)}$ decomposition of $\mathsf{126}$ and
$\mathcal{V}_{24_{210}}$ stands for the VEV of the the $\mathsf{24}$-plet of
$\mathsf{SU(5)}$ field in the $\mathsf{SU(5)\times U(1)}$ decomposition of
$\mathsf{210}$. The doublet-triplet splitting is generated by $W_{\rm DT}$
which contains $2\times\mathsf{10+120}$-plets of light fields. Thus the heavy
fields $\mathsf{126+\overline{126}}$-plet and $\mathsf{210}$-plet contain
three heavy $\mathsf{SU(2)}$ Higgs doublet pairs while the light fields
$2\times\mathsf{10+120}$-plets contain four light Higgs doublet pairs. After
mixing of the light and heavy fields, three light Higgs doublets become heavy
leaving one pair massless which we identify as the standard model Higgs
doublet.
The Yukawa couplings arise from cubic and quartic interactions. They are given
by
$\displaystyle W_{\rm Yuk}=W_{3}+W_{4},$ (2.4)
where
$W_{3}=\sum_{r=1}^{2}f^{10_{r}}~{}\langle\Psi_{(+)}^{*}|B\Gamma_{\mu}|\Psi_{(+)}\rangle~{}{{}^{r}}{\Omega_{\mu}}\,.$
(2.5)
Here $B$ and $\Gamma$’s are the $\mathsf{SO(10)}$ charge conjugation and gamma
matrices [16] and $W_{4}$ are the higher dimensional interactions discussed
below. Yukawa couplings arising from Eq. (2.5) are given by
$\displaystyle\mathcal{L}_{\textnormal{Yuk}}=+h^{0}_{\tau}~{}\epsilon^{ab}{\mathbf{H_{d}}}_{a}\mathbf{{L}}_{b}{\mathbf{E}}^{\mathtt{c}}-h^{0}_{b}~{}{\mathbf{H_{d}}}_{a}\mathbf{{Q}}^{a\alpha}{\mathbf{D}}_{\alpha}^{\mathtt{c}}-h^{0}_{t}~{}\epsilon_{ab}{\mathbf{H_{u}}}^{a}\mathbf{{Q}}^{b\alpha}{\mathbf{U}}_{\alpha}^{\mathtt{c}}+\textnormal{h.c.},$
(2.6)
where
$\displaystyle h^{0}_{\tau}$
$\displaystyle=i2\sqrt{2}\sum_{r=1}^{2}f^{10_{r}}V_{d_{r1}},~{}~{}h^{0}_{b}=-i2\sqrt{2}\sum_{r=1}^{2}f^{10_{r}}V_{d_{r1}},~{}~{}h^{0}_{t}=-i2\sqrt{2}\sum_{r=1}^{2}f^{10_{r}}U_{d_{r1}},$
(2.7)
where $U_{d_{r1}}$ and $V_{d_{r1}}$ are defined by Eq. (2.14) and evaluated
numerically in Tables 2 and 3. In addition to Yukawa couplings arising from
$W_{3}$, contributions arise from higher dimensional operators in $W_{4}$
where
$\displaystyle W_{4}=W^{(1)}_{4}+W^{(2)}_{4}+W^{(3)}_{4},$ (2.8)
and where
$\displaystyle W^{(1)}_{4}$ $\displaystyle=$
$\displaystyle-\frac{f^{(1)}}{5!M_{c}}b_{r}\langle\Psi_{(+)}^{*}|B\Gamma_{[\lambda}\Gamma_{\mu}\Gamma_{\nu}\Gamma_{\rho}\Gamma_{\sigma]}|\Psi_{(+)}\rangle~{}\left[{{}^{r}}\Omega_{\lambda}\Phi_{\mu\nu\rho\sigma}-{{}^{r}}\Omega_{\mu}\Phi_{\lambda\nu\rho\sigma}+{{}^{r}}\Omega_{\nu}\Phi_{\lambda\mu\rho\sigma}\right.$
(2.9)
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-{{}^{r}}\Omega_{\rho}\Phi_{\lambda\mu\nu\sigma}+{{}^{r}}\Omega_{\sigma}\Phi_{\lambda\mu\nu\rho}\right]\,,$
$\displaystyle W_{4}^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{f^{(2)}}{5!M_{c}}\langle\Psi_{(+)}^{*}|B\Gamma_{[\lambda}\Gamma_{\mu}\Gamma_{\nu}\Gamma_{\rho}\Gamma_{\sigma]}|\Psi_{(+)}\rangle~{}\left[\Sigma_{\lambda\alpha\beta}\Phi_{\gamma\rho\sigma\lambda}-\Sigma_{\lambda\alpha\gamma}\Phi_{\beta\rho\sigma\lambda}+\Sigma_{\lambda\alpha\rho}\Phi_{\beta\gamma\sigma\lambda}\right.$
(2.10)
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\Sigma_{\lambda\alpha\sigma}\Phi_{\beta\gamma\rho\lambda}-\Sigma_{\lambda\gamma\beta}\Phi_{\alpha\rho\sigma\lambda}+\Sigma_{\lambda\rho\beta}\Phi_{\alpha\gamma\sigma\lambda}\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\Sigma_{\lambda\sigma\beta}\Phi_{\alpha\gamma\rho\lambda}-\Sigma_{\lambda\gamma\rho}\Phi_{\beta\alpha\sigma\lambda}+\Sigma_{\lambda\gamma\sigma}\Phi_{\beta\alpha\rho\lambda}\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\Sigma_{\lambda\rho\sigma}\Phi_{\beta\alpha\gamma\lambda}\right]\,,$
$\displaystyle W_{4}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{f^{(3)}}{M_{c}}\langle\Psi_{(+)}^{*}|B\Gamma_{\mu}|\Psi_{(+)}\rangle\Sigma_{\rho\sigma\lambda}\Phi_{\rho\sigma\lambda\mu}\,.$
(2.11)
Thus $W_{4}$ gives additional contributions to the Yukawa couplings for the
third generation which we denote by $\delta h_{t},~{}\delta h_{b},~{}\delta
h_{\tau}$ which are evaluated in the Appendix. The total Yukawa couplings
arising from Eq. (2.4) is then given by
$\displaystyle h_{t}=h^{0}_{t}+\delta h_{t},~{}~{}h_{b}=h^{0}_{b}+\delta
h_{b},~{}~{}h_{\tau}=h^{0}_{\tau}+\delta h_{\tau}\,,$ (2.12)
where $h_{b},~{}h_{t},~{}h_{\tau}$ act as boundary conditions on Yukawas of
$b,t,\tau$ which are evolved down to the electroweak scale $Q$ where they are
related to $b,t,\tau$ masses so that
$\displaystyle
m_{t}(Q)=\frac{h_{t}(Q)v\sin\beta}{\sqrt{2}},~{}~{}m_{b}(Q)=\frac{h_{b}(Q)v\cos\beta}{\sqrt{2}},~{}~{}m_{\tau}(Q)=\frac{h_{\tau}(Q)v\cos\beta}{\sqrt{2}}.$
(2.13)
Here we used the relations $\langle H_{d}\rangle=\frac{v}{\sqrt{2}}\cos\beta$
and $\langle H_{u}\rangle=\frac{v}{\sqrt{2}}\sin\beta$, and where $v=246$ GeV.
As noted above there are seven Higgs doublet pairs three of which are heavy
and four are light, and after the mixing of the light and heavy fields three
pairs of light Higgs doublets become heavy and one pair remains light. To
extract the light Higgs doublets we need to diagonalize the $7\times 7$ Higgs
doublet mass matrix given in [8]. The Higgs doublet mass matrix is not
symmetric and is diagonalized by two unitary matrices $U_{d}$ and $V_{d}$.
Thus the down Higgs and the up Higgs doublet mass matrices are diagonalized by
the transformation
$\displaystyle{\cal H}_{d}=V_{d}{\cal H}_{d}^{\prime},~{}~{}{\cal
H}_{u}=U_{d}{\cal H}_{u}^{\prime}\,,$ (2.14)
where
$\displaystyle{\cal H}_{d}^{T}$ $\displaystyle=$
$\displaystyle({}^{(\overline{5}_{10_{1}})}\\!{\mathsf{D}}_{a},{}^{(\overline{5}_{10_{2}})}\\!{\mathsf{D}}_{a},{}^{(\overline{5}_{120})}\\!{\mathsf{D}}_{a},{}^{(\overline{5}_{{126}})}\\!{\mathsf{D}}_{a},{}^{(\overline{5}_{{210}})}\\!{\mathsf{D}}_{a},{}^{(\overline{45}_{120})}\\!{\mathsf{D}}_{a},{}^{(\overline{45}_{\overline{126}})}\\!{\mathsf{D}}_{a}),$
(2.15) $\displaystyle\ {\cal H}_{d}^{{}^{\prime}T}$ $\displaystyle=$
$\displaystyle({\mathbf{H_{d}}}_{a},{}^{2}\\!{\mathsf{D}}_{a}^{\prime},{}^{3}\\!{\mathsf{D}}_{a}^{\prime},{}^{4}\\!{\mathsf{D}}_{a}^{\prime},{}^{5}\\!{\mathsf{D}}_{a}^{\prime},{}^{6}\\!{\mathsf{D}}_{a}^{\prime},{}^{7}\\!{\mathsf{D}}_{a}^{\prime}),$
(2.16) $\displaystyle{\cal H}_{u}^{T}$ $\displaystyle=$
$\displaystyle({}^{(\overline{5}_{10_{1}})}\\!{\mathsf{D}}^{a},{}^{(\overline{5}_{10_{2}})}\\!{\mathsf{D}}^{a},{}^{(\overline{5}_{120})}\\!{\mathsf{D}}^{a},{}^{(\overline{5}_{{126}})}\\!{\mathsf{D}}^{a},{}^{(\overline{5}_{{210}})}\\!{\mathsf{D}}^{a},{}^{(\overline{45}_{120})}\\!{\mathsf{D}}^{a},{}^{(\overline{45}_{\overline{126}})}\\!{\mathsf{D}}^{a}),$
(2.17) $\displaystyle{\cal H}_{u}^{{}^{\prime}T}$ $\displaystyle=$
$\displaystyle({\mathbf{H_{u}}}^{a},{}^{2}\\!{\mathsf{D}}^{a\prime},{}^{3}\\!{\mathsf{D}}^{a\prime},{}^{4}\\!{\mathsf{D}}^{a\prime},{}^{5}\\!{\mathsf{D}}^{a\prime},{}^{6}\\!{\mathsf{D}}^{a\prime},{}^{7}\\!{\mathsf{D}}^{a\prime}).$
(2.18)
In the above the notation is as follows: ${}^{(\overline{5}_{10_{1}})}$ stands
for the down Higgs doublet in the $\mathsf{SU(5)}-\overline{\mathsf{5}}$-plet
in the $\mathsf{10}_{1}$ which is one of the two $\mathsf{10}$-plets of light
Higgs of $\mathsf{SO(10)}$. Further, $\mathsf{D}$’s and
${\mathsf{D}}^{\prime}$’s represent the normalized kinetic energy basis and
normalized kinetic and mass eigenbasis, respectively of the Higgs doublet mass
matrix. The pair of doublets $({\mathbf{H_{d}}}_{a},{\mathbf{H_{u}}}^{a})$ are
identified to be light and are the normalized electroweak Higgs doublets of
the minimal supersymmetric standard model (MSSM). The matrix elements of
$U_{d}$ and $V_{d}$ relevant in our analysis below are those elements that
connect the light doublets, i.e., $U_{d_{11}},U_{d_{21}},\cdots.U_{d_{71}}$,
and the elements $V_{d_{11}},V_{d_{21}},\cdots,V_{d_{71}}$. Other matrix
elements of $U_{d}$ and $V_{d}$ do not contribute in the low energy theory. As
noted above the explicit form of the $7\times 7$ Higgs doublet mass matrix is
given in [8]. The $U$ and the $V$ matrices are obtained by diagonalization of
this matrix. Numerical values of the non-zero matrix elements of $U_{d}$ and
$V_{d}$ relevant in the analysis are displayed in Tables 2 and 3 for
benchmarks of Table 1.
## 3 $\mathsf{SO(10)}$ SUGRA model with Yukawa unification consistent with
Fermilab $(g-2)_{\mu}$
Since the muon $g-2$ is one of the most accurately determined quantities in
physics even a small deviation from the standard model prediction would be a
significant indicator of new physics. For example, it is known that
supersymmetric loop corrections could be of the same size as the electroweak
corrections in the SM [24, 25, 26, 27, 28, 29]. Indeed the Brookhaven result
in 2001 [2] resulted in several works pointing out the impact on physics
expected at colliders and elsewhere [30, 31, 32, 33, 34, 35, 36, 37]. Thus the
experiment became one of the important constraints on the parameter space of
SUSY models. The discovery of the Higgs boson at 125 GeV further constrained
the parameter space implying that the size of weak SUSY scale could be large
lying in the TeV region [38, 39]. Since the Fermilab result has indicated more
strongly than the Brookhaven experiment for the existence of new physics, it
is interesting to ask how the $b-t-\tau$ unification is affected [1]. The
early work of [40] pointed out that such a unification could occur in
$\mathsf{SO(10)}$ with appropriate choice of soft parameters. Such a
unification has important effects on other phenomena such as dark matter (DM)
[41]. Thus it is of interest to ask if $b-t-\tau$ unification can come about
consistent with Fermilab data. We investigate this question using a neural
network which is found to be useful in the analysis of large parameter spaces
[42, 43]).
The analysis is done within the framework of supergravity grand unified models
[44] using non-univeralities of gaugino masses [45, 46, 47, 48, 49]. The scan
of the SUGRA parameter space is performed using an artificial neutral network
(ANN) implemented in xBIT [50]. The ANN has three layers with 25 neurons per
layer. It constructs the likelihood of a point using the three constraints on
the Higgs mass, DM relic density and muon $g-2$, i.e.,
$\displaystyle m_{h^{0}}=125\pm 2~{}\text{GeV},$ $\displaystyle\Omega
h^{2}<0.126,$ $\displaystyle\Delta a_{\mu}=(2.87\pm 0.97)\times 10^{-9}.$
The ANN first generates a set of points using the SUGRA input parameters which
are used to train the neutral network based on the constructed likelihood
function. The input parameters are $m_{0}$, $A_{0}$, $m_{1}$, $m_{2}$, $m_{3}$
and $\tan\beta$ where $m_{0}$ is the universal scalar mass, $A_{0}$ is the
universal trilinear coupling, $m_{1},m_{2},m_{3}$ are the
$\mathsf{U(1),SU(2),SU(3)}$ gaugino masses all at the GUT scale and
$\tan\beta=\langle H_{u}\rangle/\langle H_{d}\rangle$ where $H_{u}$ gives mass
to the up quarks and $H_{d}$ gives mass to the down quarks and the charged
leptons. We notice that the ANN predicts a particle spectrum consistent with
$\tilde{g}$SUGRA where the colored sparticles are heavy and the sleptons,
staus and electroweakinos are lighter. Generating the sparticle spectrum
requires evolving the renormalization group equations (RGEs) and for this we
use SPheno-4.0.4 [51, 52] which implements two-loop MSSM RGEs and three-loop
SM RGEs while taking into account SUSY threshold effects at the one-loop
level. The larger SUSY scale makes it necessary to employ a two-scale matching
condition at the electroweak and SUSY scales [53] thereby improving the
calculations of the Higgs boson mass and of the sparticle spectrum. The bottom
quark mass and $\alpha_{S}$ (the fine structure constant for the
$\mathsf{SU(3)_{C}}$) are run up to the scale of the $Z$ boson mass, $M_{Z}$,
using four-loop RGEs in the $\overline{\rm MS}$ scheme while for the top
quark, the evolution starts at the pole mass and the $\overline{\rm MS}$ mass
is computed by running down to the $M_{Z}$ scale including two-loop QCD
corrections. The tau mass is calculated at $M_{Z}$ including one-loop
electroweak corrections. The calculation of the $\overline{\rm MS}$ Yukawas at
the electroweak scale involves the first matching conditions to include SM
thresholds. Those couplings are then run using 3-loop SM RGEs to $M_{\rm
SUSY}$ where the second matching takes place to include SUSY thresholds at the
one-loop level and a shift is made to the $\overline{\rm DR}$ scheme. The
2-loop MSSM RGEs of the $\overline{\rm DR}$ Yukawas and gauge couplings are
then run to the GUT scale where the soft SUSY breaking boundary conditions are
applied. The obtained set of points are then passed to Lilith [54, 55],
HiggsSignals [56] and HiggsBounds [57] to check the Higgs sector constraints
as well as SModelS [58, 59, 60] to check the LHC constraints. Furthermore,
micrOMEGAs-5.2.7 [61] has a module which we use to check the constraints from
DM direct detection experiments.
We discuss now the results of our analysis. In Table 1 we give an analysis of
the VEVs of the heavy fields that enter in the GUT symmetry breaking for a
range of GUT parameters $\eta,\lambda$, $M^{126}$ and $M^{210}$ where the VEVs
are in general complex. The VEVs are obtained by solving the spontaneous
symmetry breaking equations using $W_{\rm GUT}$. Using the VEVs of Table 1,
one solves for the Higgs doublet mass matrix using a range of
$a,b_{1},b_{2},c,\bar{c}$ that appear in $W_{\rm DT}$. The diagonalization of
the Higgs mass matrix allows us to identify the linear combination of the
Higgs doublet fields which are massless and correspond to the pair of MSSM
Higgs.
Model | $\eta$ | $\lambda$ | $M^{126}$ | $M^{210}$ | $\mathcal{V}_{1_{{}_{{210}}}}$ | $\mathcal{V}_{24_{{}_{{210}}}}$ | $\mathcal{V}_{75_{{}_{{210}}}}$ | $\mathcal{V}_{1_{{}_{{126}}}}$
---|---|---|---|---|---|---|---|---
(a) | 2.22 | 1.96 | $5.73\times 10^{17}$ | $1.14\times 10^{15}$ | $2.00\times 10^{18}$ | $(-4.00+\imath 0.42)\times 10^{18}$ | $(-8.97+\imath 0.38)\times 10^{18}$ | $(2.82-\imath 0.09i)\times 10^{17}$
(b) | 2.85 | 2.31 | $8.10\times 10^{17}$ | $2.00\times 10^{16}$ | $2.20\times 10^{18}$ | $-4.01\times 10^{18}$ | $-2.08\times 10^{18}$ | $\imath 2.93\times 10^{18}$
(c) | 3.00 | 2.88 | $4.27\times 10^{17}$ | $2.12\times 10^{15}$ | $1.10\times 10^{18}$ | $(-2.19+\imath 0.36)\times 10^{18}$ | $(-4.94+\imath 0.32)\times 10^{18}$ | $(2.45-\imath 0.12)\times 10^{17}$
(d) | 2.62 | 0.63 | $4.31\times 10^{17}$ | $4.01\times 10^{15}$ | $1.28\times 10^{18}$ | $-2.27\times 10^{18}$ | $-1.15\times 10^{18}$ | $\imath 9.20\times 10^{17}$
(e) | 1.37 | 2.61 | $5.67\times 10^{17}$ | $1.41\times 10^{16}$ | $3.20\times 10^{18}$ | $(-6.32+\imath 1.65)\times 10^{18}$ | $(-1.45+\imath 0.15)\times 10^{19}$ | $(1.60-\imath 0.12)\times 10^{18}$
(f) | 1.11 | 2.51 | $3.03\times 10^{17}$ | $1.44\times 10^{16}$ | $2.12\times 10^{18}$ | $(-4.16+\imath 1.39)\times 10^{18}$ | $(-9.65+\imath 1.27)\times 10^{18}$ | $(1.46-\imath 0.14)\times 10^{18}$
(g) | 2.24 | 0.90 | $4.04\times 10^{17}$ | $2.89\times 10^{15}$ | $1.40\times 10^{18}$ | $-2.65\times 10^{18}$ | $-1.42\times 10^{18}$ | $\imath 1.33\times 10^{18}$
(h) | 2.98 | 2.71 | $5.09\times 10^{17}$ | $1.79\times 10^{16}$ | $1.32\times 10^{18}$ | $(-2.54+\imath 1.19)\times 10^{18}$ | $(-6.09+\imath 1.11)\times 10^{18}$ | $(7.83-\imath 1.04)\times 10^{17}$
(i) | 2.07 | 1.13 | $3.11\times 10^{17}$ | $1.21\times 10^{16}$ | $1.16\times 10^{18}$ | $-1.86\times 10^{18}$ | $-8.71\times 10^{17}$ | $\imath 1.21\times 10^{18}$
(j) | 2.99 | 0.39 | $6.61\times 10^{17}$ | $1.88\times 10^{16}$ | $1.71\times 10^{18}$ | $-1.58\times 10^{18}$ | $-4.80\times 10^{17}$ | $\imath 7.54\times 10^{17}$
Table 1: A numerical estimate of the VEVs of the Standard Model singlets in
$\mathsf{210}$, $\mathsf{126}$ and $\mathsf{\overline{126}}$-plets arising in
the spontaneous breaking of the $\mathsf{SO(10)}$ GUT gauge symmetry under the
assumption
$\mathcal{V}_{1_{{}_{{126}}}}=\mathcal{V}_{1_{{}_{\overline{126}}}}$. All VEVs
and masses are in GeV.
The diagonalization also allows for computation of non-vanishing elements of
the $U$ and $V$ matrices that connect to the light Higgs. These are the matrix
elements $U_{d_{11}}$, $U_{d_{21}},~{}U_{d_{31}},~{}U_{d_{61}}$ and the matrix
elements $V_{d_{11}}$, $V_{d_{21}},~{}V_{d_{31}},~{}V_{d_{61}}$. They are
listed in Tables 2 and 3. In Table 4 we give a list of parameters that enter
in the cubic couplings $W_{3}$ and in the quartic couplings $W_{4}$. In Table
5 we give the computations of the contributions of the cubic couplings, the
quartic couplings and their sum for $b,t,\tau$ for the model points of Table
1. Computation of $b,t,\tau$ masses using the analysis of Table 5 as boundary
conditions at the GUT scale and using RG evolution down to the electroweak
scale is given in Table 6. An analysis of the Higgs boson mass, the light
sparticle masses, the dark matter relic density and of the supersymmetric
correction to the muon anomaly is given in Table 7. A comparison between Table
6 and Table 7 shows that one has a unification of Yukawas and a $g-2$ anomaly
consistent with the Fermilab result of Eq. (1.3). One may note that the dark
matter relic density is not fully saturated by the model points of Table 7.
This implies that the dark matter may likely be multicomponent which includes
other forms of dark matter, such as dark fermions of the hidden sector [62,
63, 64] or possibly a dark photon [65] or an axion [66, 67].
Model | $a$ | $b_{1}$ | $b_{2}$ | $c$ | $\bar{c}$ | $U_{d{{}_{11}}}$ | ${U_{d{{}_{21}}}}$ | ${U_{d{{}_{31}}}}$ | ${U_{d{{}_{61}}}}$
---|---|---|---|---|---|---|---|---|---
(a) | 0.22 | 1.86 | 1.14 | 1.46 | 0.18 | $-0.034+\imath 0.051$ | $0.298+\imath 0.285$ | $0.231+\imath 0.351$ | $-0.495-\imath 0.636$
(b) | 2.03 | 2.50 | 2.70 | 0.81 | 1.15 | $-0.040-\imath 0.015$ | $0.082+\imath 0.030$ | $0.185+\imath 0.067$ | $-0.917-\imath 0.333$
(c) | 1.70 | 2.95 | 1.21 | 0.21 | 2.74 | $-0.163-\imath 0.009$ | $0.381+\imath 0.080$ | $-0.098+\imath 0.409$ | $0.091-\imath 0.798$
(d) | 0.27 | 2.55 | 2.16 | 0.51 | 2.65 | $0.487+\imath 0.003$ | $-0.600-\imath 0.003$ | $-0.121-\imath 0.001$ | $0.623+\imath 0.003$
(e) | 2.33 | 1.65 | 1.04 | 0.08 | 2.95 | $-0.101+\imath 0.165$ | $0.185-\imath 0.249$ | $0.377+\imath 0.213$ | $-0.782-\imath 0.259$
(f) | 0.16 | 1.41 | 1.53 | 0.40 | 2.46 | $-0.715-\imath 0.001$ | $0.661+\imath 0.023$ | $0.015+\imath 0.106$ | $-0.075-\imath 0.187$
(g) | 0.51 | 2.90 | 1.08 | 0.19 | 1.37 | $0.096+\imath 0.138$ | $-0.272-\imath 0.390$ | $-0.103-\imath 0.148$ | $0.482+\imath 0.693$
(h) | 2.52 | 2.91 | 0.21 | 0.25 | 2.99 | $-0.067+\imath 0.047$ | $0.850-\imath 0.440$ | $0.101+\imath 0.085$ | $-0.230-\imath 0.087$
(i) | 1.57 | 1.38 | 2.45 | 0.75 | 1.41 | $0.067-\imath 0.043$ | $-0.072+\imath 0.047$ | $-0.137+\imath 0.089$ | $0.823-\imath 0.531$
(j) | 0.68 | 2.70 | 1.01 | 0.21 | 0.49 | $0.130-\imath 0.003$ | $-0.361+\imath 0.008$ | $-0.072+\imath 0.002$ | $0.920-\imath 0.021$
Table 2: A numerical estimate of the elements of the down Higgs zero mode
eigenvector using the analysis of Table 1 and the couplings of Eq. (2.3).
Model | $a$ | $b_{1}$ | $b_{2}$ | $c$ | $\bar{c}$ | $V_{d{{}_{11}}}$ | ${V_{d{{}_{21}}}}$ | ${V_{d{{}_{31}}}}$ | ${V_{d{{}_{61}}}}$
---|---|---|---|---|---|---|---|---|---
(a) | 0.22 | 1.86 | 1.14 | 1.46 | 0.18 | $-0.273$ | $0.411+\imath 0.083$ | $-0.401$ | $0.765+\imath 0.062$
(b) | 2.03 | 2.50 | 2.70 | 0.81 | 1.15 | $-0.091$ | $0.107$ | $-0.196$ | $0.971$
(c) | 1.70 | 2.95 | 1.21 | 0.21 | 2.74 | $0.323$ | $-0.783-\imath 0.010$ | $0.246$ | $-0.467-\imath 0.057$
(d) | 0.27 | 2.55 | 2.16 | 0.51 | 2.65 | $-0.592$ | $0.708$ | $-0.074$ | $0.378$
(e) | 2.33 | 1.65 | 1.04 | 0.08 | 2.95 | $-0.357$ | $0.568+\imath 0.010$ | $-0.345$ | $0.644+\imath 0.127$
(f) | 0.16 | 1.41 | 1.53 | 0.40 | 2.46 | $0.730$ | $-0.673-\imath 0.006$ | $0.057$ | $-0.104-\imath 0.026$
(g) | 0.51 | 2.90 | 1.08 | 0.19 | 1.37 | $-0.276$ | $0.747$ | $-0.127$ | $0.591$
(h) | 2.52 | 2.91 | 0.21 | 0.25 | 2.99 | $0.086$ | $-0.977-\imath 0.055$ | $0.089$ | $-0.157-\imath 0.055$
(i) | 1.57 | 1.38 | 2.45 | 0.75 | 1.41 | $-0.120$ | 0.095 | $-0.163$ | 0.975
(j) | 0.68 | 2.70 | 1.01 | 0.21 | 0.49 | $-0.046$ | 0.163 | $-0.077$ | 0.983
Table 3: A numerical estimate of the elements of the up Higgs zero mode eigenvector using the analysis of Table 1 and the couplings of Eq. (2.3). Model | $f^{(1)}$ | $f^{(2)}$ | $f^{(3)}$ | $f^{10_{r}}$
---|---|---|---|---
(a) | 0.16 | 0.24 | 0.03 | (0.17, 0.23)
(b) | 0.12 | 0.10 | 0.12 | (0.30, 1.04)
(c) | 0.40 | 0.08 | 0.08 | (2.36, 1.06)
(d) | 0.68 | 0.35 | 0.22 | (0.43, 0.44)
(e) | 1.24 | 0.10 | 0.04 | (0.12, 0.25)
(f) | 1.58 | 0.63 | 0.10 | (2.03, 2.26)
(g) | 0.79 | 0.15 | 0.14 | (0.38, 0.24)
(h) | 1.55 | 0.38 | 0.22 | (1.55, 0.21)
(i) | 0.15 | 0.11 | 0.08 | (1.29, 2.30)
(j) | 0.44 | 0.09 | 0.22 | (0.52, 0.49)
Table 4: The GUT scale parameters in the cubic and quartic superpotentials $W_{3}$, $W_{4}^{(1)}$, $W_{4}^{(2)}$ and $W_{4}^{(3)}$ for the model points (a)$-$(j). The masses are in GeV. Model | $h_{t}^{0}$ | $h_{b}^{0}$ | $h_{\tau}^{0}$ | $\delta h_{t}^{\rm GUT}$ | $\delta h_{b}^{\rm GUT}$ | $\delta h_{\tau}^{\rm GUT}$ | $h_{t}^{\rm GUT}$ | $h_{b}^{\rm GUT}$ | $h_{\tau}^{\rm GUT}$
---|---|---|---|---|---|---|---|---|---
(a) | 0.274 | 0.148 | 0.148 | 0.204 | 0.201 | 0.063 | 0.478 | 0.073 | 0.088
(b) | 0.223 | 0.238 | 0.238 | 0.259 | 0.282 | 0.183 | 0.482 | 0.044 | 0.055
(c) | 0.190 | 0.193 | 0.193 | 0.319 | 0.190 | 0.163 | 0.501 | 0.029 | 0.036
(d) | 0.161 | 0.169 | 0.169 | 0.331 | 0.236 | 0.089 | 0.492 | 0.066 | 0.081
(e) | 0.153 | 0.278 | 0.278 | 0.400 | 0.341 | 0.231 | 0.486 | 0.062 | 0.074
(f) | 0.189 | 0.118 | 0.118 | 0.298 | 0.200 | 0.108 | 0.484 | 0.091 | 0.104
(g) | 0.149 | 0.222 | 0.222 | 0.348 | 0.272 | 0.159 | 0.497 | 0.051 | 0.062
(h) | 0.210 | 0.198 | 0.198 | 0.289 | 0.220 | 0.156 | 0.487 | 0.042 | 0.053
(i) | 0.268 | 0.179 | 0.179 | 0.216 | 0.248 | 0.094 | 0.483 | 0.068 | 0.085
(j) | 0.313 | 0.160 | 0.160 | 0.177 | 0.211 | 0.099 | 0.489 | 0.051 | 0.060
Table 5: The magnitude of the contributions to the top, bottom, and tau Yukawa couplings from cubic interactions (columns 2-4), from quartic interactions (columns 5-7) and the magnitude of their complex sum (columns 8-10) at the GUT scale for the parameter set of Table 4. The Yukawa couplings are in general complex and we add the contributions of the cubic and quartic interactions as complex numbers and exhibit only their magnitudes in the table. Model | $m_{0}$ | $A_{0}$ | $m_{1}$ | $m_{2}$ | $m_{3}$ | $\tan\beta$ | $m_{t}$ (pole) | $\overline{m}_{b}(\overline{m}_{b})$ | $m_{\tau}$ (pole)
---|---|---|---|---|---|---|---|---|---
(a) | 657 | -2228 | 661 | 526 | 7774 | 14.0 | 172.2 | 4.15 | 1.77682
(b) | 673 | 1127 | 939 | 570 | 8833 | 8.2 | 172.2 | 4.22 | 1.77682
(c) | 387 | 880 | 949 | 980 | 8118 | 5.3 | 172.8 | 4.19 | 1.77682
(d) | 164 | 197 | 632 | 1539 | 6171 | 12.2 | 172.9 | 4.20 | 1.77682
(e) | 416 | 339 | 740 | 416 | 4559 | 11.6 | 172.8 | 4.22 | 1.77682
(f) | 688 | 1450 | 852 | 634 | 8438 | 16.8 | 172.9 | 4.22 | 1.77682
(g) | 106 | 22.6 | 523 | 1309 | 5240 | 9.3 | 172.8 | 4.19 | 1.77682
(h) | 206 | 603 | 842 | 1298 | 7510 | 8.0 | 172.1 | 4.15 | 1.77682
(i) | 452 | 648 | 624 | 346 | 4843 | 13.1 | 172.8 | 4.20 | 1.77682
(j) | 196 | -803 | 828 | 1599 | 8929 | 9.4 | 172.6 | 4.22 | 1.77682
Table 6: The SUGRA parameters sets used for RG analysis where the boundary conditions for the Yukawas for the top, bottom, and the tau are taken from Table 5. In the analysis the GUT scale ranges from $8.6\times 10^{15}$ GeV to $2.0\times 10^{16}$ GeV. Model | $h^{0}$ | $\tilde{\mu}$ | $\tilde{\nu}_{\mu}$ | $\tilde{\tau}$ | $\tilde{\chi}^{0}_{1}$ | $\tilde{\chi}^{\pm}_{1}$ | $\Omega h^{2}$ | $\Delta a_{\mu}(\times 10^{-9})$
---|---|---|---|---|---|---|---|---
(a) | 123.3 | 459.0 | 452.6 | 270.8 | 243.1 | 323.0 | 0.103 | 2.30
(b) | 125.3 | 422.8 | 415.7 | 370.4 | 337.3 | 337.6 | 0.003 | 2.14
(c) | 123.3 | 427.2 | 420.5 | 379.6 | 369.8 | 707.7 | 0.125 | 1.91
(d) | 123.9 | 856.4 | 852.4 | 243.5 | 240.1 | 1227 | 0.016 | 1.94
(e) | 123.8 | 361.0 | 352.6 | 282.0 | 272.7 | 272.9 | 0.002 | 1.98
(f) | 123.0 | 508.1 | 502.3 | 331.9 | 324.2 | 404.3 | 0.004 | 2.11
(g) | 123.4 | 722.8 | 718.2 | 206.5 | 195.5 | 1038.4 | 0.103 | 2.57
(h) | 124.5 | 628.7 | 623.6 | 338.3 | 326.8 | 998.4 | 0.082 | 1.94
(i) | 123.7 | 346.8 | 338.0 | 240.3 | 205.6 | 205.8 | 0.001 | 2.67
(j) | 123.5 | 774.1 | 769.8 | 319.1 | 314.7 | 1247 | 0.016 | 2.59
Table 7: Low scale SUSY mass spectrum showing the Higgs boson, the smuon, the
muon sneutrino, the stau and the light electroweakino masses and the LSP relic
density for the benchmarks of Table 6. Also shown is $\Delta a_{\mu}$.
A scan on the parameter space using the GUT scale input of $\mathsf{SO(10)}$
results in a larger set of points than those presented in Tables 1$-$5. The
range of values the input parameters take are: $0.5<\eta,\lambda<6.0$,
$0.1<a,b_{1},b_{2},c,\bar{c}<3.0$, $1\times 10^{16}<M^{126}<9.5\times
10^{17}$, $1\times 10^{15}<M^{210}<3.5\times 10^{16}$,
$0.01<f^{(1)},f^{(2)},f^{(3)}<4.0$ and $0.1<f^{10_{r}}<5.5$. The result of the
scan is shown in Fig. 1. The left panel is a scatter plot in the variables
$\eta$ and $\lambda$ with the muon $g-2$ shown on the color axis consistent
with $\Delta{a^{\rm FB}_{\mu}}$. The right panel shows a scatter plot in the
top, bottom and tau Yukawa couplings at the GUT scale. The set of points in
the scatter plot is consistent with experimental constraints and the evolution
of the GUT scale Yukawas to the electroweak scale produces the correct top,
bottom and tau masses within experimental uncertainties.
Figure 1: Scatter plots resulting from the scan of input parameters from
$\mathsf{SO(10)}$. The left panel shows the parameters $\eta$ and $\lambda$
with the color axis being the muon $g-2$. The right panel shows the top,
bottom and tau Yukawa couplings at the GUT scale.
## 4 Sparticle hierarchies and signal region analysis
The set of data points retained after satisfying the constraints from the
Higgs sector, the DM relic density, dark matter direct detection and the LHC
is further processed and points consistent with Yukawa coupling unification
are kept. We observe that the spectrum consisting of light electroweakinos,
sleptons (selectron and smuons) and staus belong to three cases of mass
hierarchy.
#### Case 1:
The electroweakinos, $\tilde{\chi}^{0}_{2},\tilde{\chi}^{\pm}_{1}$ are almost
degenerate, with the stau being the next-to-lightest supersymmetric particle
(NLSP). The mass hierarchy here is
$m_{\tilde{\tau}_{1}}<m_{\widetilde{\rm EW}}<m_{\tilde{\ell}},$
where $\widetilde{\rm EW}=(\tilde{\chi}^{0}_{2},\tilde{\chi}^{\pm}_{1})$ and
$\tilde{\ell}$ represents the sleptons.
#### Case 2:
In this category, one of the electroweakinos ($\tilde{\chi}^{0}_{2}$ or
$\tilde{\chi}^{\pm}_{1}$) is the NLSP and the hierarchy reads
$m_{\widetilde{\rm EW}}<m_{\tilde{\tau}_{1}}<m_{\tilde{\ell}}\,.$
Here we distinguish two subcategories (I) and (II) where
$\displaystyle
m_{\tilde{\chi}^{\pm}_{1}}<m_{\tilde{\chi}^{0}_{2}}<m_{\tilde{\tau}_{1}}~{}~{}~{}\text{(I)},$
$\displaystyle
m_{\tilde{\chi}^{\pm}_{1}}<m_{\tilde{\tau}_{1}}<m_{\tilde{\chi}^{0}_{2}}~{}~{}~{}\text{(II)}.$
#### Case 3:
The last category also includes stau as the NLSP but the electroweakino and
slepton hierarchy is inverted, i.e.,
$m_{\tilde{\tau}_{1}}<m_{\tilde{\ell}}<m_{\widetilde{\rm EW}}.$
Benchmarks (a), (f) belong to Case 1, while (b), (e) and (i) belong to Case 2
and (c), (d), (g), (h) and (j) belong to Case 3. Fig. 2 shows the obtained
data set categorized according to the above three cases.
Figure 2: A scatter plot in the $M_{0}$-$A_{0}$ plane showing the three cases
(and subcases) with the chargino-second neutralino mass gap shown on the color
axis.
An illustration of such a complex spectrum is given in Fig. 3. The upper
panels correspond to benchmark (a) while the lower ones are for (d). Cascade
decays are common in high scale models which, unlike simplified models
considered by ATLAS and CMS, produce more complicated event topology. Thus,
for slepton pair production, analyses by ATLAS [68, 69] and CMS [70, 71]
consider a 100% branching ratio of $\tilde{\ell}\to\ell\tilde{\chi}^{0}_{1}$
which can happen in spectra belonging to Case 3. However, Cases 1 and 2 do not
necessarily abide by this and one can get several decay channels making the
final states more complicated.
In the next section, we select a set of benchmarks belonging to the three
cases discussed above. We study slepton pair production and decay at HL-LHC
and HE-LHC. We design a set of signal regions to target the rich final states
corresponding to the three cases of mass hierarchies. For earlier works on
SUSY discovery at HL-LHC and HE-LHC, see Refs. [72, 73] and the CERN yellow
reports [74, 75].
Figure 3: A display of the particle spectrum using PySLHA [76] for benchmarks
(a) (upper panels) and (d) (lower panels). The left panels represent the
spectrum up to 13 TeV while the right panels give the low-lying masses of the
spectrum.
### 4.1 Slepton pair production and event simulation at the LHC
The pair production cross section of sleptons (selectrons and smuons) is
proportional to the electron and muon Yukawa coupling which means that those
cross sections are small compared to staus and electroweak gauginos. For our
LHC analysis, we select six of the ten benchmarks shown in Table 6
corresponding to sleptons in the mass range of $\sim 350$ GeV to $\sim 850$
GeV. The production cross sections of the slepton pairs at 14 TeV and 27 TeV
are calculated at the aNNLO+NNLL accuracy using Resummino-3.0 [77, 78] and the
five-flavor NNPDF23NLO PDF set. The results, arranged in decreasing order of
cross section, are shown in Table 8. Also shown are the different branching
ratios of sleptons but for brevity we do not exhibit the branching ratios of
$\tilde{\chi}^{0}_{2}$ and $\tilde{\chi}^{\pm}_{1}$ for benchmarks (b), (f)
and (i). To have an idea of the decay channels involved, one can examine the
right panel of Fig. 3 which shows the low-lying spectrum of benchmark (a).
Since (a) and (f) both belong to Case 1, one can have an idea of the different
decay channels of $\tilde{\chi}^{0}_{2}$ and $\tilde{\chi}^{\pm}_{1}$ which
involve the stau. This leads to a tau-enriched final state.
Model | $\sigma(pp\rightarrow\tilde{e}_{L}\,\tilde{e}_{L})$ | $\sigma(pp\rightarrow\tilde{\mu}_{L}\,\tilde{\mu}_{L})$ | Branching ratios
---|---|---|---
| 14 TeV | 27 TeV | 14 TeV | 27 TeV | $\tilde{\ell}_{L}\to\ell\tilde{\chi}^{0}_{1}$ | $\tilde{\ell}_{L}\to\ell\tilde{\chi}^{0}_{2}$ | $\tilde{\ell}_{L}\to\nu_{\ell}\tilde{\chi}^{\pm}_{1}$
(i) | 2.896 | 9.633 | 2.909 | 9.673 | 31% | 6% | 63%
(b) | 1.242 | 4.590 | 1.244 | 4.598 | 31% | 6% | 63%
(f) | 0.541 | 2.252 | 0.543 | 2.262 | 22% | 26% | 52%
(h) | 0.194 | 0.958 | 0.194 | 0.957 | 100% | - | -
(g) | 0.094 | 0.533 | 0.094 | 0.533 | 100% | - | -
(d) | 0.037 | 0.253 | 0.037 | 0.253 | 100% | - | -
Table 8: The aNNLO+NNLL pair production cross-sections, in fb, of sleptons at
$\sqrt{s}=14$ TeV and at $\sqrt{s}=27$ TeV for benchmarks (b), (d) and
(f)$-$(i) of Table 1 arranged in decreasing order of production cross
sections. Also shown are the slepton branching ratios to electroweakinos and
leptons.
The final states which make up our signal region (SR) involve two same flavor
and opposite sign (SFOS) leptons with missing transverse energy (MET). We also
require at least two jets (N $\geq 2$) which can be used to form kinematic
variables that are effective for jetty final states. We call the signal region
SR-2$\ell$Nj. For such final states, the dominant SM backgrounds are from
diboson production, $Z/\gamma+$jets, dilepton production from off-shell vector
bosons ($V^{*}\rightarrow\ell\ell$), $t\bar{t}$ and $t+W/Z$. The subdominant
backgrounds are Higgs production via gluon fusion ($ggF$ H) and vector boson
fusion (VBF). The simulation of the signal and background events is performed
at LO with<EMAIL_ADDRESS>interfaced to LHAPDF [79] using the
NNPDF30LO PDF set. Up to two hard jets are added at generator level. The
parton level events are passed to PYTHIA8 [80] for showering and hadronization
using a five-flavor matching scheme in order to avoid double counting of jets.
For the signal events, the matching/merging scale is set at one-fourth the
mass of the pair produced sleptons. Additional jets from ISR and FSR are added
to the signal and background events. Jets are clustered with FastJet [81]
using the anti-$k_{t}$ algorithm [82] with jet radius $R=0.4$. DELPHES-3.4.2
[83] is then employed for detector simulation and event reconstruction using
the HL-LHC and HE-LHC card. The SM backgrounds are scaled to their relevant
NLO cross sections while aNNLO+NNLL cross sections are used for the signal
events.
### 4.2 Event selection
The selected SFOS leptons must have a leading and subleading transverse
momenta $p_{T}>15$ GeV for electrons and $p_{T}>10$ GeV for muons with
$|\eta|<2.5$. Each event should contain at least two non-b-tagged jets with
the leading $p_{T}>20$ GeV in the $|\eta|<2.4$ region and a missing transverse
energy $E^{\rm miss}_{T}>70$ GeV. Despite the specific preselection criteria,
the analysis cuts used for the six benchmarks cannot be the same. This is due
to the rich final states involved. To help us discriminate the signal from the
background events, we use a set of kinematic variables along with a deep
neural network (DNN) which is trained and tested on two independent sets of
signal and background samples. We list the kinematic variables that enter in
the training of the DNN:
1. 1.
$E^{\rm miss}_{T}$: the missing transverse energy in the event. It is usually
high for the signal due to the presence of neutralinos.
2. 2.
The transverse momentum of the leading non-b tagged jets, $p_{T}(j_{1})$.
Rejecting b-tagged jets reduces the $t\bar{t}$ background.
3. 3.
The transverse momentum of the leading lepton (electron or muon),
$p_{T}(\ell_{1})$.
4. 4.
$M_{\rm T2}$, the stransverse mass [84, 85, 86] of the leading and subleading
leptons
$M_{\rm T2}=\min\left[\max\left(m_{\rm T}(\mathbf{p}_{\rm
T}^{\ell_{1}},\mathbf{q}_{\rm T}),m_{\rm T}(\mathbf{p}_{\rm
T}^{\ell_{2}},\,\mathbf{p}_{\rm T}^{\text{miss}}-\mathbf{q}_{\rm
T})\right)\right],$ (4.1)
where $\mathbf{q}_{\rm T}$ is an arbitrary vector chosen to find the
appropriate minimum and the transverse mass $m_{T}$ is given by
$m_{\rm T}(\mathbf{p}_{\rm T1},\mathbf{p}_{\rm T2})=\sqrt{2(p_{\rm T1}\,p_{\rm
T2}-\mathbf{p}_{\rm T1}\cdot\mathbf{p}_{\rm T2})}.$ (4.2)
5. 5.
The quantity $M^{\rm min}_{\rm T}$ defined as $M^{\rm min}_{\rm
T}=\text{min}[m_{\rm T}(\textbf{p}_{\rm T}^{\ell_{1}},\textbf{p}^{\rm
miss}_{\rm T}),m_{\rm T}(\textbf{p}_{\rm T}^{\ell_{2}},\textbf{p}^{\rm
miss}_{\rm T})]$. The variables $M_{\rm T2}$ and $M^{\rm min}_{\rm T}$ are
effective when dealing with large MET in the final state.
6. 6.
The dilepton invariant mass, $m_{\ell\ell}$, helps in rejecting the diboson
background with a peak near the $Z$ boson mass which can be done by setting
$m_{\ell\ell}>100$ GeV.
7. 7.
The opening angle between the MET system and the dilepton system,
$\Delta\phi(\textbf{p}_{\rm T}^{\ell},\textbf{p}^{\rm miss}_{\rm T})$, where
$\textbf{p}_{\rm T}^{\ell}=\textbf{p}_{\rm T}^{\ell_{1}}+\textbf{p}_{\rm
T}^{\ell_{2}}$.
8. 8.
The smallest opening angle between the first three leading jets in an event
and the MET system, $\Delta\phi_{\rm min}(\textbf{p}_{\rm
T}(j_{i}),\textbf{p}^{\rm miss}_{\rm T})$, where $i=1,2,3$.
We use the DNN implementation in the ‘Toolkit for Multivariate Analysis’
(TMVA) [87] framework within ROOT6 [88]. The DNN employed has three dense
hidden layers with 128 neurons per layer and $\tanh$ as an activation function
to define the output neurons given the input values. The DNN trains on the
signal and background events using the above set of kinematic variables in
three phases with a decreasing learning rate. After the ‘learning’ process is
over, the DNN tests the predictions on another set of signal and background
samples. Despite having one background set, the training and testing must be
done every time a signal sample is used, i.e., six times in our case. During
the testing stage, the DNN creates a new discriminator which is called the DNN
response or the DNN score. Cuts on this new variable maximizes the signal
($S$) to background ($B$) ratio, $S/\sqrt{S+B}$.
We give in Table 9 the set of analysis cuts on a select number of kinematic
variables along with the new ‘DNN response’ variable. Variations in cuts are
used for our six benchmarks depending on the hierarchy of the spectrum which
allows us to put them in three categories with (b),(i) as the first, (f) as
the second and (d),(g),(h) as the third. The values shown in parentheses are
the modified cuts at 27 TeV which are essential to improving the
$S/\sqrt{S+B}$ ratio.
Variable | (b), (i) | (f) | (d), (g), (h)
---|---|---|---
$m_{\ell\ell}~{}\text{[GeV]}>$ | 136 (110) | 150 | 150 (110)
$E^{\rm miss}_{T}/\textbf{p}^{\ell}_{\rm T}>$ | 1.9 (2.8) | - | -
$\Delta\phi_{\rm min}(\textbf{p}_{\rm T}(j_{i}),\textbf{p}^{\rm miss}_{\rm T})~{}\text{[rad]}>$ | - | 0.85 (1.5) | -
$p_{T}^{\ell_{2}}~{}\text{[GeV]}>$ | - | - | 190 (370)
$M_{T2}~{}\text{[GeV]}>$ | \- (140) | \- (120) | 200 (300)
DNN response $>$ | 0.9 | 0.9 | 0.9
$\mathcal{L}$ at 14 TeV [fb-1] | NV, 1887 | 1262 | NV, 2074, 1738
$\mathcal{L}$ at 27 TeV [fb-1] | 2804, 1320 | 694 | 1031, 689, 1194
Table 9: The analysis cuts on a set of kinematic variables at 14 TeV (27 TeV)
grouped by the benchmarks of Table 6. Notice that with the exception of
$m_{\ell\ell}$ harder cuts are applied at 27 TeV. Entries with a dash (-) mean
that no requirement on the variable is considered. Also shown at the bottom
are the required integrated luminosities for discovery at 14 TeV and 27 TeV.
Entries with ‘NV’ mean that the point is not visible at the corresponding
center-of-mass energy.
### 4.3 Results
We begin by discussing the benchmarks (d), (g) and (h) which belong to Case 3.
Here the mass splitting between the slepton and the neutralino is large,
ranging from 300 GeV to 600 GeV, which produces very energetic leptons. For
those benchmarks, the sleptons decay to a light lepton and a neutralino with a
100% branching ratio (see Table 8) which makes for a clean final state. The
most effective kinematic variables for this case are $M_{T2}$ and
$p_{T}^{\ell_{2}}$ where the latter is the transverse momentum of the
subleading lepton. We present two-dimensional plots in these variables in the
middle panels of Fig. 4. The left panel depicts point (d) and the right one is
the dominant diboson background. One can clearly see that the largest number
of background events (color axis) are concentrated at small $M_{T2}$ and
$p_{T}^{\ell_{2}}$ while for the signal larger values are highly populated as
well due to the energetic final states. A hard cut on $M_{T2}$ and
$p_{T}^{\ell_{2}}$ as well as the ‘DNN response’ can reject most of the
background events.
Next, we discuss benchmarks (b) and (i) which belong to Case 2. Here the
branching ratios to a lepton and a neutralino are smaller, at 31% and the
slepton-neutralino mass gaps are at 85 GeV and 140 GeV, respectively. Such a
mass gap is not enough to allow harder cuts on $p_{T}^{\ell_{2}}$ and that’s
why it has been omitted in Table 9. For this reason, we make use of the
leading and subleading transverse momenta of the leptons to reconstruct the
total momentum of the system, $\textbf{p}^{\ell}_{\rm T}$, to form the new
variable $E^{\rm miss}_{T}/\textbf{p}^{\ell}_{\rm T}$. Two-dimensional plots
in the $E^{\rm miss}_{T}/\textbf{p}^{\ell}_{\rm T}$ and the dilepton invariant
mass, $m_{\ell\ell}$, variables are shown in the top panels of Fig. 4. The
left panel shows the distributions for point (b) while the right one is for
dilepton production from off-shell vector bosons. For the background, most of
the events lie in the region $E^{\rm miss}_{T}/\textbf{p}^{\ell}_{\rm T}<2$
and $m_{\ell\ell}<100$ GeV which is the reason for the choice of cuts in Table
9.
Figure 4: Two dimensional plots in select kinematic variables with the number
of events on the color axis. Top panels: $E^{\rm
miss}_{T}/\textbf{p}^{\ell}_{\rm T}$ vs the dilepton invariant mass for
benchmark (b) (left) and off-shell vector boson background (right). Middle
panels: the subleading lepton transverse momentum vs $M_{T2}$ for benchmark
(d) (left) and the diboson background (right). Bottom panels: $\Delta\phi_{\rm
min}(\textbf{p}_{\rm T}(j_{i}),\textbf{p}^{\rm miss}_{\rm T})$ vs the dilepton
invariant mass for benchmark (f) (left) and $Z/\gamma+\text{jets}$ background
(right).
Finally, for point (f) which belongs to Case 1, the branching fraction to a
lepton and a neutralino is the smallest compared to its decay to a second
neutralino and a chargino. The second neutralino and chargino decay
predominantly to a stau which in turn decays to a neutralino and a tau. Hence
we are faced with a case of tau-enriched final state which can hadronize
forming jets. In our selection, we have rejected b-tagged jets but made no
special requirements on tau-tagged jets. For this particular case, jets (tau-
tagged or not) can be used to reject the SM background through the variable
$\Delta\phi_{\rm min}(\textbf{p}_{\rm T}(j_{i}),\textbf{p}^{\rm miss}_{\rm
T})$ defined above. In the bottom panels of Fig. 4 we show this variable
plotted against $m_{\ell\ell}$ for point (f) (left panel) and the
$Z/\gamma$+jets background (right panel). Excluding the region formed by
$\Delta\phi_{\rm min}(\textbf{p}_{\rm T}(j_{i}),\textbf{p}^{\rm miss}_{\rm
T})<1$ rad and $m_{\ell\ell}<100$ GeV is effective in reducing the SM
background.
Figure 5: Distributions in the DNN response variable at 14 TeV (left) and 27
TeV (right) for benchmarks (b) (top panels) and (g) (bottom panels).
Along with cuts on the variables discussed thus far, the ‘DNN response’ plays
an important role. We show in Fig. 5 distributions in this variable after the
above cuts have been implemented. The top panel depicts benchmark (b) which
shows clearly that at 14 TeV this point cannot be discovered with 3000 fb-1
while the signal is in excess over the background near 1 for 2800 fb-1 at 27
TeV. The bottom panels show point (g) also at 14 TeV (left) and 27 TeV
(right). The benchmark is discoverable at both HL-LHC and HE-LHC but requires
smaller integrated luminosity for discovery at HE-LHC (700 fb-1) than at HL-
LHC (2100 fb-1). The evaluated integrated luminosities for discovery at both
machines are summarized in the lower part of Table 9. Entries with ‘NV’
indicate that the benchmark is not discoverable at the corresponding machine.
Note that there is a modest improvement in the integrated luminosity at HE-LHC
in comparison to HL-LHC but the former is expected to gather data at the rate
of $\sim 820$ fb-1 per month, so most of those points will be discoverable
within the first two to three months of run. Note that points (f), (g), (h)
and (i) are discoverable at both machines while (b) and (d) can only be
discoverable at HE-LHC.
We note that recently several works have come out regarding a SUSY explanation
of the Fermilab muon $g-2$ [89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100,
101, 102, 103].
## 5 Conclusion
In this work we have investigated if high scale models can produce Yukawa
coupling unification consistent with the Fermilab muon $g-2$ result. We used a
neural network to investigate the parameter space of a class of
$\mathsf{SO(10)}$ models where Yukawa couplings arise from the cubic as well
as the quartic interactions. As in a recent work it is found that the
preferred parameter space lies in a region where gluino-driven radiative
breaking of the electroweak symmetry occurs. The model produces a split
spectrum consisting of a light sector and a heavy sector. The light sector
contains light sleptons and light weakinos, and the heavy sector contains the
gluino, the squarks and the heavy Higgs. The masses of the light sparticles
lie in the few hundred GeV range and are accessible at the LHC. With the help
of a deep neural network, we carried out a dedicated search of sleptons in the
two-lepton final state at HL-LHC and HE-LHC. It is found that most of the
considered benchmarks are discoverable within the optimal integrated
luminosity of HL-LHC while all of them are discoverable at HE-LHC with less
integrated luminosities.
Acknowledgments: The research of AA was supported by the BMBF under contract
05H18PMCC1, while the research of PN was supported in part by the NSF Grant
PHY-1913328.
## 6 Appendix: Contributions to Yukawas from higher dimensional operators
In this appendix we give the contributions $\delta h_{b},\delta h_{t},\delta
h_{\tau}$ to the Yukawas that arise from higher dimensional operators where
$\displaystyle\delta h_{t}=\delta h_{t}^{(1)}+\delta h_{t}^{(2)}+\delta
h_{t}^{(3)},~{}~{}\delta h_{b}=\delta h_{b}^{(1)}+\delta h_{b}^{(2)}+\delta
h_{b}^{(3)},~{}~{}\delta h_{\tau}=\delta h_{\tau}^{(1)}+\delta
h_{\tau}^{(2)}+\delta h_{\tau}^{(3)}\,.$ (6.1)
Here $\delta h^{(1)}$ is the contribution arising from $W_{4}^{(1)}$, $\delta
h^{(2)}$ is the contribution arising from $W_{4}^{(2)}$, and $\delta h^{(3)}$
is the contribution arising from $W_{4}^{(3)}$. The explicit forms of these
are given below [8].
Thus $W_{4}^{(1)}$ gives the following contribution to the third generation
Yukawas
$\displaystyle\delta h^{(1)}_{t}$ $\displaystyle=$
$\displaystyle\frac{if^{(1)}}{60\sqrt{2}M_{c}}\left(\sum_{r=1}^{2}b_{r}U_{d_{r1}}\right)\left[\frac{5\sqrt{3}}{2}\mathcal{V}_{75_{{}_{{210}}}}-4\sqrt{15}\mathcal{V}_{24_{{}_{{210}}}}-8\sqrt{15}\mathcal{V}_{1_{{}_{{210}}}}\right],$
(6.2) $\displaystyle\delta h^{(1)}_{b}$ $\displaystyle=$
$\displaystyle\frac{if^{(1)}}{60\sqrt{2}M_{c}}\left(\sum_{r=1}^{2}b_{r}V_{d_{r1}}\right)\left[\frac{\sqrt{20}}{3}\mathcal{V}_{75_{{}_{{210}}}}-20\sqrt{\frac{5}{3}}\mathcal{V}_{24_{{}_{{210}}}}\right],$
(6.3) $\displaystyle\delta h^{(1)}_{\tau}$ $\displaystyle=$
$\displaystyle\frac{if^{(1)}}{60\sqrt{2}M_{c}}\left(\sum_{r=1}^{2}b_{r}V_{d_{r1}}\right)\left[20\sqrt{3}\mathcal{V}_{75_{{}_{{210}}}}-20\sqrt{15}\mathcal{V}_{24_{{}_{{210}}}}\right],$
(6.4)
The contribution of $W_{4}^{(2)}$ to the third generation Yukawas is given by
$\displaystyle\delta h^{(2)}_{t}=$
$\displaystyle-\frac{if^{(2)}}{120M_{c}}\left[\frac{10}{3}\sqrt{\frac{2}{3}}\mathcal{V}_{75_{{}_{{210}}}}U_{d_{61}}+\frac{5}{3}\sqrt{\frac{10}{3}}\mathcal{V}_{24_{{}_{{210}}}}U_{d_{61}}+6\sqrt{5}\mathcal{V}_{24_{{}_{{210}}}}U_{d_{31}}-8\sqrt{5}\mathcal{V}_{1_{{}_{{210}}}}U_{d_{31}}\right],$
(6.5) $\displaystyle\delta h^{(2)}_{b}=$
$\displaystyle-\frac{if^{(2)}}{120M_{c}}\Bigg{[}-\frac{20}{3}\sqrt{\frac{2}{3}}\mathcal{V}_{75_{{}_{{210}}}}V_{d_{61}}-\frac{20}{3}\mathcal{V}_{75_{{}_{{210}}}}V_{d_{31}}-\frac{1}{3}\sqrt{\frac{10}{3}}\mathcal{V}_{24_{{}_{{210}}}}V_{d_{61}}-\frac{10\sqrt{5}}{3}\mathcal{V}_{24_{{}_{{210}}}}V_{d_{31}}$
$\displaystyle\hskip
56.9055pt-4\sqrt{\frac{10}{3}}\mathcal{V}_{1_{{}_{{210}}}}V_{d_{61}}\Bigg{]},$
(6.6) $\displaystyle\delta h^{(2)}_{\tau}=$
$\displaystyle-\frac{if^{(2)}}{120M_{c}}\Bigg{[}-20\sqrt{\frac{2}{3}}\mathcal{V}_{75_{{}_{{210}}}}V_{d_{61}}-20\mathcal{V}_{75_{{}_{{210}}}}V_{d_{31}}-\sqrt{\frac{10}{3}}\mathcal{V}_{24_{{}_{{210}}}}V_{d_{61}}-10\sqrt{5}\mathcal{V}_{24_{{}_{{210}}}}V_{d_{31}}$
$\displaystyle\hskip
56.9055pt-4\sqrt{30}\mathcal{V}_{1_{{}_{{210}}}}V_{d_{61}}\Bigg{]}.$ (6.7)
Finally, the contribution of $W_{4}^{(3)}$ to the third generation Yukawas is
given by
$\displaystyle\delta h_{t}^{(3)}$
$\displaystyle=-\frac{3i}{8}\frac{f^{(3)}}{M_{c}}\left[\frac{2}{3}\sqrt{\frac{2}{3}}\mathcal{V}_{75_{210}}U_{d_{61}}+\frac{1}{3}\sqrt{\frac{10}{3}}\mathcal{V}_{24_{210}}U_{d_{61}}-\frac{2}{\sqrt{5}}\mathcal{V}_{24_{210}}U_{d_{31}}+\frac{8}{3\sqrt{5}}\mathcal{V}_{1_{210}}U_{d_{31}}\right],~{}~{}~{}$
(6.8) $\displaystyle\delta h_{b}^{(3)}$
$\displaystyle=-\frac{3i}{8}\frac{f^{(3)}}{M_{c}}\left[\frac{2}{3}\sqrt{\frac{2}{3}}\mathcal{V}_{75_{210}}V_{d_{61}}+\frac{1}{3}\sqrt{\frac{10}{3}}\mathcal{V}_{24_{210}}V_{d_{61}}-\frac{2}{\sqrt{5}}\mathcal{V}_{24_{210}}V_{d_{31}}+\frac{8}{3\sqrt{5}}\mathcal{V}_{1_{210}}V_{d_{31}}\right],$
(6.9) $\displaystyle\delta h_{\tau}^{(3)}$
$\displaystyle=\frac{3i}{8}\frac{f^{(3)}}{M_{c}}\left[\frac{2}{3}\sqrt{\frac{2}{3}}\mathcal{V}_{75_{210}}V_{d_{61}}+\frac{1}{3}\sqrt{\frac{10}{3}}\mathcal{V}_{24_{210}}V_{d_{61}}-\frac{2}{\sqrt{5}}\mathcal{V}_{24_{210}}V_{d_{31}}+\frac{8}{3\sqrt{5}}\mathcal{V}_{1_{210}}V_{d_{31}}\right].$
(6.10)
The total Yukawas are the sum of the contributions from the cubic and from the
quartic terms at the GUT scale as given in Eq. (2.12).
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Open-loop potential difference games with inequality constraints[This work is supported by Science and Engineering Research Board (SERB), Government of India, Grant no. SERB–EMR/2017/001267.]
1]Aathira Prasad
1]Puduru Viswanadha Reddy
[1]Department of Electrical Engineering, Indian Institute of Technology - Madras, India
[ ],
Static potential games are non-cooperative games which admit a fictitious function, also referred to as a potential function, such that the minimizers of this function constitute a subset (or a refinement) of the Nash equilibrium strategies of the associated non-cooperative game. In this paper we study a class $N$-player non-zero sum difference games with inequality constraints which admit a potential game structure. In particular, we provide conditions for the existence of an optimal control problem (with inequality constraints) such that the solution of this problem yields an open-loop Nash equilibrium strategy of the corresponding dynamic non-cooperative game (with inequality constraints). Further, we provide a way to construct potential functions associated with this optimal control problem. We specialize our general results to a linear-quadratic setting and provide a linear complementarity problem based approach for computing the refinements of the open-loop Nash equilibria obtained in [Reddy and Zaccour, 2015]. We illustrate our results with an example inspired by energy storage incentives in a smart grid.
Potential games; dynamic games with inequality constraints; open-loop Nash equilibrium;
linear complementarity problem.
§ INTRODUCTION
Multi-agent control systems and related distributed architectures are becoming increasingly popular with emerging applications such as smart grids, Internet of Things (IoT) systems, intelligent traffic networks and cyber-security. These systems are characterized by the presence of interdependent multiple decision making entities, which are large-scale, distributed, networked and heterogeneous in nature. Game theory has emerged as a powerful tool for analyzing multi-agent systems; see [Manshaei et al., 2013], [Saad et al., 2012] and [Zhu and Başar, 2015] for applications of game theory in the areas mentioned above. In particular, dynamic game theory provides a mathematical framework for modeling multi-agent interactions which evolve over time, and has been successfully used in analyzing a variety of decision problems arising in engineering, economics and management science; see [Başar and Olsder, 1999], [Dockner et al., 2000], [Başar et al., 2018]. A significant share of dynamic game models are formulated in an unconstrained setting, that is, where the state and control variables are unconstrained; except for the state equation which captures the evolution of the interaction environment. Constraints appear naturally in real world applications in the form of production-capacity, environmental, market and budget constraints. For example, electric vehicle charging station with limited energy resources impose joint constraints on players. Further, decision problems related to network congestion, which are used in modeling traffic systems, inherently involve capacity constraints. Recently, [Reddy and Zaccour, 2015] and [Reddy and Zaccour, 2016] study a class of non-zero sum difference games with inequality constraints and provide conditions for the existence of Nash equilibria with open-loop and feedback information structures.
The novelty of this paper lies in studying a class of dynamic games with inequality constraints
which admit a potential game structure. Static potential games were first introduced in [Rosenthal, 1973] and further studied in [Monderer and Shapley, 1996]. Loosely speaking, a potential game is a non-cooperative game that satisfies the property that a Nash equilibrium of the game is obtained when players jointly optimize a fictitious function, also referred to as a potential function. In other words, a Nash equilibrium of the game is obtained by solving an optimization problem as opposed to a fixed point problem. An important property associated with a potential game is that the strategy profile that provides the optimum for the potential function is a pure strategy Nash equilibrium, and as a result, the existence of pure strategy Nash equilibrium is guaranteed in a potential game. It is well known ([Quint and Shubik, 1997]) that a non-cooperative game can admit more than one Nash equilibrium. The multiplicity of equilibria naturally poses a selection problem, and to address this, certain refinements of Nash equilibira have been proposed; see [Myerson, 1997]. These refinements provide a way of selecting a subset of equilibria based on additional properties that a Nash equilibrium is required to satisfy. In a potential game, the Nash equilibrium is refined with the property that players are jointly optimizing the potential function in a cooperative manner, even though they are acting strategically optimizing their individual
objective functions. Due to this property, potential games find applications in the study of network
congestion games <cit.>, decentralized learning algorithms ([Marden et al., 2009]) and
in utility function design ([Marden and Shamma, 2018]) for multi-agent systems. Further, in [Slade, 1994]
this property of potential games has been explored in the context of oligopolistic markets.
In summary, the static potential games have been studied extensively in literature.
The objective of this paper is to extend the notion of a potential game in a dynamic setting, and in particular, for dynamic games where inequality constraints appear jointly in control and state variables. Besides providing the conditions on the existence of this class of games, another important objective of our paper is towards computing the refinements of the open-loop Nash equlibria. We consider a class of finite horizon $N$-player non-zero sum nonlinear difference games with constraint structure similar to [Reddy and Zaccour, 2015] and [Reddy and Zaccour, 2016]. Our contributions are summarized as follows.
* We define the notion of open-loop potential difference game and associate an inequality constrained optimal control problem with the dynamic non-cooperative game with inequality constraints. Further, in Lemma <ref> and Theorem <ref> we provide conditions under which the non-cooperative game admits a potential game structure.
* When the potential functions associated with the optimal control problem are not specified, we provide, in Theorem <ref>, a method for constructing the potential functions from the objective functions of the players using the theory of conservative vector fields.
* We specialize the obtained results to a linear quadratic setting and characterize in Theorem <ref> a class of linear quadratic potential difference games with inequality constraints. Further, in Theorem <ref> we provide a linear complementarity problem based method for computing a refinement of the open-loop Nash equilibria.
The paper is organized as follows. In section <ref>, we introduce the dynamic game model with the separable structure of the objective function and inequality constraints jointly in state and decision variables. In section <ref>, we define the open-loop dynamic potential games by introducing an optimal control problem associated with the dynamic potential game. Further, we provide the structure of the dynamic game for the existence of potential functions. We demonstrate the equivalence of the solution of the optimal control problem as an open-loop Nash equilibrium of the dynamic game. When the potential functions are specified before hand, we also illustrate a procedure for the construction of potential functions. In section <ref>, we specialize these results to the linear quadratic setting. In section <ref> we illustrate our results with an application motivated by energy storage incentives in a smart grid. Finally, in section <ref> we provide conclusions and future work.
§.§ Literature review
The literature on dynamic potential games is limited compared to their static counter part. In a static non-cooperative game the objective functions of the players are required to satisfy certain conditions to admit a potential function. Quite naturally, an extension of these conditions to a dynamic setting imposes structural constraints on the instantaneous and terminal objective functions. In [Slade, 1994], the author studies oligopolistic markets, and provides conditions under which a single optimization problem provides a Nash equilibrium, both in the static and dynamic settings. The objective function of this optimization problem in this work is referred to as a fictitious function. In particular, in the dynamic case, existence of these functions under the open-loop and feedback information structures has been explored. In [Dragone et al., 2015], the authors consider a non-cooperative differential game with open-loop information structure, and provide conditions for the existence of Hamiltonian potential functions. Potential differential games were studied in [Fonseca-Morales and Hernández-Lerma, 2018] in an open-loop setting. The authors primarily focused on separable structure of the payoff function from which the potential function could be derived. The unconstrained discrete time game is considered in [González-Sánchez and Hernández-Lerma, 2014] in a stochastic setting. In [González-Sánchez and Hernández-Lerma, 2016], the authors provide a survey on static and dynamic potential games. All these works study dynamic potential games in discrete and continuous time settings without additional constraints on the state and control variables. In [Zazo et al., 2016], the authors consider a discrete time infinite horizon non-cooperative difference games with inequality constraints and provide conditions for the existence of potential functions. They also mention a way for constructing potential functions using the theory of conservative vector fields; an approach followed in static games [Monderer and Shapley, 1996]. Our work is closer to [Zazo et al., 2016] in spirit, but differs considerably in the model and approach. The constraint structure in our paper is inspired by our previous works [Reddy and Zaccour, 2015] and [Reddy and Zaccour, 2016]. In particular, we assume that the players have two types of control variables, namely, 1) variables that (directly) affect the dynamics but do not enter the constraints, and 2) variables that enter the constraints jointly with the state variables but do not appear in the dynamics. Further, we assume that the objective functions are separable in the two types of control variables, that is, that there are no cross-terms between them. As was shown in [Reddy and Zaccour, 2015] and [Reddy and Zaccour, 2016], we also demonstrate in this paper that these assumptions are crucial in obtaining a semi-analytical characterizations for Nash equilibria, and in providing a linear complementarity problem based method for computing them.
More importantly, the linear quadratic dynamic potential games analyzed in this paper provides refinements of the open-loop Nash equilibria obtained in [Reddy and Zaccour, 2015], and a procedure for computing them.
§.§ Notation
We shall use the following notation. The $n$-dimensional Euclidean space is denoted by $\mathbb R^n$, $n\geq 1$. $A^\prime$ denotes the transpose of a matrix or a vector $A$. $A_1\oplus A_2\oplus \cdots \oplus A_n$ represents the block diagonal matrix obtained by taking the matrices $A_1, A_2,\cdots,A_n$ as diagonal elements in this sequence. The matrix with all entries as zeros is denoted by $\mathbf{0}$, all entries as one is denoted by $\mathbf{1}$, and the identity matrix is represented by $\mathbf{I}$. We call two vectors $x,y\in \mathbb R^n$ complementary if $x\geq 0$, $y\geq0$ and $x^\prime y=0$, and $0\leq x \perp y \geq 0$ denotes this condition. Let $A$ be a $n\times n$ matrix and $a$ be a $n\times 1$ vector. Let $n$ be partitioned as $n=n_1+n_2+\cdots+n_K$. We represent $[A]_{ij}$ as the $n_i\times n_j$ sub-matrix associated with indices $n_i$ (row) and $n_j$ (column), and $[a]_i$ as the $n_i\times 1$ sub-vector associated with indices $n_i$. $[A]_{i\bullet}$ and $[A]_{\bullet i }$ represent the $i^{th}$ row block matrix of dimensions $n_i \times n $ and $i^{th}$ column matrix of dimension $n \times n_i$ of the matrix $A$ respectively. The notation $\mathbf{e}_i$ represents a column vector with $i^{th}$ element as 1 and the rest of the elements as zero. For differentiation with respect to a vector or matrix variable we follow the convention from [Lütkepohl, 1996].
§ DYNAMIC GAME MODEL
In this section, we introduce a class of finite horizon discrete time non-zero sum games with inequality constraints. Let $\mathcal N=\{1,2,\cdots,N\}$ be the set of players and $\mathcal K=\{0,1,2,\cdots,K\}$ be the set of time periods. At each time instant, player $i \in \mathcal N$ chooses the following two types of actions (control variables); variables that enter the dynamics of the system, but not the constraints, denoted by $u^i_k\in U^i_k\subset \mathbb R^{m_i}$; and variables that do not affect the dynamics, but do constrain the decision making process, denoted by $v_k^i\in V_k^i \subset \mathbb R^{s_i}$. Here, $U_k^i$ and $V_k^i$ are the sets of admissible values for the two types of variables. We denote the vector of actions of all players at time period $k$ by $\mathbf u_k:=\begin{bmatrix} {u^1_k}^\prime&
\end{bmatrix} ^\prime$ and $\mathbf{v}_k:=\begin{bmatrix} {v^1_k}^\prime&
{v^2_k}^\prime&\cdots& {v^N_k}^\prime
\end{bmatrix}^\prime$.
The state $ x_k \in {X}_k \subset \mathbb R^n$ evolves according to the following discrete time dynamics
\begin{align}
{x}_{k+1}=f_k( x_k ,\mathbf{u}_k),~k\in \mathcal K\backslash \{K\},~x_0 \text{ given}.
\label{eq:statedynamics}
\end{align}
Here, $X_k$ denotes the set of admissible state vectors at time period $k$. We consider the following joint inequality constraints (also called coupled constraints) associated with players strategies
\begin{align}
h_k( x_k ,\mathbf{v}_k)\geq 0,~\mathbf{v}_k\geq 0,~ k\in \mathcal K.
\label{eq:constraints}
\end{align}
Notice, the decision variables $\{u^i_k,~k\in \mathcal K \backslash \{K\},~i\in \mathcal N\}$ do not enter the constraints directly but affect them indirectly through the state variables.
Let the strategies of player $i$, that is, the plan of actions for the entire planning period, be denoted by $\tilde{u}^i:=\{u^i_k,~k\in \mathcal K\backslash \{K\}\}$ and
$\tilde{v}^i:=\{v^i_k,~k\in \mathcal K \}$. The set of players excluding player $i$ is denoted by $-i:=\mathcal N\backslash \{i\}$. The joint strategy profiles of the players be denoted by $\tilde{\mathbf{u}}:=(\tilde{u}^i,\tilde{u}^{-i})$ and $\tilde{\mathbf{v}}:=(\tilde{v}^i,\tilde{v}^{-i})$, and the corresponding strategy sets are denoted by $\mathcal U$ and $\mathcal V$ respectively. Each player $i\in \mathcal N$ uses her strategies $\tilde{u}^i$ and $\tilde{v}^i$ to minimize the following objective function given by
\begin{align}
J^{i}(x_0,(\tilde{u}^i,\tilde{u}^{-i}),(\tilde{v}^i,\tilde{v}^{-i}))& =
g^i_K( x_k ,\mathbf{v}_K) + \sum_{k=0}^{K-1} g^i_k( x_k ,\mathbf{u}_k,\mathbf{v}_k).
\label{eq:objectives}
\end{align}
We have the following assumptions related to the dynamic game (<ref>)-(<ref>).
* The admissible action sets $\{U_k^i,~k\in \mathcal K\backslash \{K\},~i\in \mathcal N\}$, are such that the sets of state vectors $\{X_k,~k\in \mathcal K\}$ are convex, and feasible action sets $\{V^i_k( x_k ,v_k^{-i})=\{v_k^i\in \mathbb R^{s_i}~|~ h^i( x_k ,\mathbf{v}_k)\geq 0,~ \mathbf{v}_k\geq 0\},~\forall x_k\in X_k\}$ are non-empty, convex and bounded for all $k\in \mathcal K$, $i\in \mathcal N$. A strategy pair $(\tilde{\bu},\tilde{\bv})$ associated with these actions sets is an admissible strategy pair.
* The matrices $\left\{\frac{\partial h_k}{\partial v_k^i},~k\in \mathcal K,~i\in \mathcal N\right \}$ have full rank, so as to satisfy constraint qualification conditions.
* The instantaneous cost functions in (<ref>) admit a separable structure, that is, they do not contain cross terms involving the strategies $\mathbf{\tilde{u}}$ and $\mathbf{\tilde{v}}$ and are represented by
$g_k^i( x_k ,\mathbf{u}_k,\mathbf{v}_k)=
gu_k^i( x_k ,\mathbf{u}_k)+gv_k^i( x_k ,\mathbf{v}_k)$.
* The partial derivatives of the functions $f_k$, $h_k$, and the players' cost functions $\{g_k^i,~k\in \mathcal K,~i\in \mathcal N\}$ exist in their arguments, and are twice continuously differentiable in their arguments.
Note that in (<ref>) the cost incurred by player $i$ not only depends on his actions but also by the actions of other players $-i$. So, (<ref>)-(<ref>) constitutes a dynamic game with inequality constraints, and we refer to it as NZDG from here on. Then, the Nash equilibrium strategies of players, denoted by $(\mathbf{\tilde{u}}^*,\mathbf{\tilde{v}}^*)$, for this class of games is defined as follows.
The strategy profile $(\tilde{\mathbf{u}}^*,\tilde{\mathbf{v}}^*)$ is Nash equilibrium if for every player $i\in \mathcal N$ the strategies $(\tilde{\mathbf{u}}^*,\tilde{\mathbf{v}}^*)$ solves
\begin{align}
\min_{(\tilde{u}^i,\tilde{v}^i)}J^{i}(x_0,(\tilde{u}^i,{\tilde{u}^{-i*}}),(\tilde{v}^i,{\tilde{v}^{-i*}})) \text{ subject to \eqref{eq:statedynamics} and \eqref{eq:constraints}}.
\end{align}
From the above definition the Nash equilibrium strategy is stable against a player's unilateral deviation. In multistage games the interaction environment is dynamic, which is embedded in state variables and their evolution. It is well known that the Nash equilibrium solution varies with the information used by the players during the (dynamic) decision making process. So, in a dynamic game an information structure must be specified when players design their strategies. In an open-loop information structure, the players design their strategies using only the knowledge of the time $k$ (and initial state $ {x}_0$). Whereas in a feedback information structure, players design their equilibrium strategies using the knowledge of the state variable. In this paper, we assume open-loop information structure. This implies, that the decision variables entering the dynamics are functions of time. Next, as it is clear from the inequality constraints (<ref>) that the admissible action set of a player depends on the state variable. Therefore player $i$ takes action $v_k^i\in V^i_k( x_k ,v_k^{-i})$ as a function of time $k$, and the feasible set $V^i_k( x_k ,v_k^{-i})$ is parametrized by the state variable and the decision variables of players excluding player $i$; see also [Reddy and Zaccour, 2015] which considers open-loop information structure for this class of games.
§ OPEN-LOOP DYNAMIC POTENTIAL GAMES
In this section we introduce the notion of a dynamic potential game and seek to find
conditions under which NZDG is a dynamic potential game. It is well known that the classical potential game is a static concept, ([Monderer and Shapley, 1996]). A (static) game is said to be a potential game if there exists a potential function, and the minimum of this function provides a pure strategy Nash equilibrium of the game, there by providing refinement (or selection) of Nash equilibria. In other words, when a potential function exists for a game, a Nash equilibrium can be obtained by solving a minimization problem. So, when we extend this notion of potential game in a dynamic context, it natural to associate an optimal control problem with NZDG. We consider the following
optimal control problem with inequality constraints.
\begin{align}
\mathrm{OCP}:\quad \quad & \min_{\tilde{\mathbf{u}},\tilde{\mathbf{v}}} J( {x}_0,\tilde{\mathbf{u}},\tilde{\mathbf{v}})\label{eq:OCP1}\\
\text{subject to}~ & {x}_{k+1}=f_{k}( x_k ,\mathbf{u}_k),~ {x}_0 \text{ is given},\label{eq:OCP2}\\
& h_k( x_k ,\mathbf{v}_k)\geq 0,~\mathbf{v}_k\geq 0, ~ k\in \mathcal K,\label{eq:OCP3}\\
\text{where}~& J(x_0,\tilde{\mathbf{u}},\tilde{\mathbf{v}})=P_K( x_k ,\mathbf{v}_K)+\sum_{k=0}^{K-1}P_k( x_k ,\mathbf{u}_k,\mathbf{v}_k),\notag
\end{align}
where $P_k:\mathbb{R}^n \times \mathbb {R}^{m} \times
\mathbb R^s \rightarrow \mathbb R$ and $P_K:\mathbb{R}^n \times \mathbb {R}^s\rightarrow \mathbb R$ are instantaneous and terminal cost functions, which are continuous and twice continuously differentiable
in their arguments.
The dynamic game NZDG is referred to as open-loop potential difference game (OLPDG) if there exist cost functions $\{P_k,~k\in \mathcal K\}$ such that the optimal solution of OCP provides an open-loop Nash equilibrium of NZDG. Whenever such cost functions exist, $\{P_k,~k\in \mathcal K\}$ are referred to as potential cost functions.
In the next two theorems we discuss the necessary and sufficient conditions for
an admissible pair $(\tilde{\bu}^*,\tilde{\bv}^*) \in \mathcal U \times \mathcal V$ to be an optimal solution of OCP. Towards this end,
we define the instantaneous and terminal Lagrangian functions as follows
\begin{align}
&\cL_k(x_k,\bu_k,\bv_k,\lambda_{k+1},\mu_k) =P_k(x_k,\bu_k,\bv_k)+\lambda_{k+1}^\prime f_k(x_k,\bu_k)-\mu_k^\prime h_k(x_k,\bv_k),\\
&\cL_K(x_K,\bv_K,\mu_K)=P_K(x_K,v_K)-\mu_K^\prime h_K(x_K,\bv_K).
\end{align}
Let Assumption <ref> holds true.
Let $\mathbf{(\tilde{\mathbf{u}}^*, \tilde{\mathbf{v}}^*)}$ be an optimal admissible pair for OCP, and $\{x_k^*,~k\in \mathcal K\}$ be state trajectory generated by $\tilde{\bu}^*$, the then there exist co-states $\{\lambda_k^*,~k\in \mathcal K\}$ and a multipliers $\{\mu_k^*,~k\in \mathcal K\}$ such that the following conditions hold true:
\begin{align}
&\text{for $k\in \mathcal K\backslash \{K\}$}\notag \\
\bu_k^*&=\argmin_{\bu_k \in \bU_k}\mathcal L_k(x^*_k,\bu_k,\bv^*_k,\lambda_{k+1}^*,\mu_k^*)\label{eq:nceq1}\\
x^*_{k+1}&=\frac{\partial \mathcal L_k}{\partial \lambda_{k+1}}(x^*_k,\bu^*_k,\bv^*_k,\lambda_{k+1}^*,\mu_k^*),~ x^*_0=x_0 \label{eq:nceq2}\\
\lambda^*_k&=\frac{\partial \mathcal L_k}{\partial x_k}(x^*_k,\bu^*_k,\bv^*_k,\lambda_{k+1}^*,\mu_k^*),\label{eq:nceq3a}\\ \lambda^*_K&=\frac{\partial \cL_K}{\partial x_K}(x_K^*,\bv^*_K,\mu_K^*),\label{eq:nceq3b}\\
&\text{and for $k\in \mathcal K$}\notag \\
& 0\leq h(x_k^*,\bv_k^*) \perp \mu_k^* \geq 0 \label{eq:nceq4}\\
&0\leq \frac{\partial \mathcal L_k}{\partial \bv_k} (x^*_k,\bu^*_k,\bv^*_k,\lambda_{k+1}^*,\mu_k^*) \perp \bv^*_k \geq 0.\label{eq:nceq5}
\end{align}
The necessary conditions follow directly by applying the discrete-time maximum principle.
The following theorem provides conditions under which (<ref>)-(<ref>) are also sufficient for optimality of $\mathbf{(\tilde{\mathbf{u}}^*, \tilde{\mathbf{v}}^*)}$.
Let Assumption <ref> holds true.
Let the pair of control strategies $(
\tilde{\bu}^*,\tilde{\bv}^*)\in \mathcal U \times \mathcal V$ and the collection of trajectories $\{x_k^*,\lambda^*_k,\mu^*_k,~k
\in \mathcal K\}$ satisfy (<ref>). Assume that the Lagrangian $\cL_k(x_k,\bu_k,\bv_k,\lambda_{k+1},\mu_k)$ has a minimum with respect to $(\bu_k,\bv_k)$ for all $k\in \mathcal K\backslash \{K\}$. Let the minimized Lagrangian be given by
$\cL_k^*(x_k,\lambda_{k+1},\mu_k)=\min_{(\bu_k,\bv_k)}\cL_k(x_k,\bu_k,\bv_k,\lambda_{k+1},\mu_k)$ for $k\in \mathcal K\backslash \{K\}$. Assume that terminal Lagrangian $\cL_K(x_K,\bv_K,\mu_K)$ has a minimum with respect to $\bv_K$, and denote the minimum terminal cost function by $
\cL^*_K(x_K,\mu_K)=\min_{\bv_K} \cL_K(x_K,\bv_K,\mu_K)$. Then, if $\cL_k^*(x_k,\lambda_{k+1},\mu_k)$ is convex with respect to $x_k$ for all $k\in \mathcal K\backslash \{K\}$ and $\cL^*_K(x_K,\mu_K)$ is convex with respect to $x_K$, then the pair $(\tilde{\bu}^*,\tilde{\bv}^*)$ is optimal for OCP.
For any admissible $(\tilde{\bu},\tilde{\bv})\in \mathcal U\times \mathcal V$ we consider the difference
\begin{align}
J(x_0,\tilde{\bu},\tilde{\bv})-J(x_0,\tilde{\bu}^*,\tilde{\bv}^*)=&P_K( {x}_K,\mathbf{v}_K)+\sum_{k=0}^{K-1}P_k( {x}_k,\mathbf{u}_k,\mathbf{v}_k) -P_K( {x}^*_K,\mathbf{v}^*_K)-\sum_{k=0}^{K-1}P_k({x}_k^*,\mathbf{u}^*_k,\mathbf{v}^*_k)\notag\\
=& \cL_K(x_K,\bv_K,\mu_K^*)-\cL_K(x_K^*,\bv^*_K,\mu_K^*)+{\mu_K^*}^\prime\left(h(x_K,\bv_K)-h(x_K^*,\bv^*_K)\right)\notag \\ +& \sum_{k=0}^{K-1} \cL_k(x_k,\bu_k,\bv_k,\lambda^*_{k+1},\mu^*_k) -\cL_k(x^*_k,\bu^*_k,\bv^*_k,\lambda^*_{k+1},\mu^*_k)\notag \\+&\sum_{k=0}^{K-1}
-{\lambda_{k+1}^*}^\prime(x_{k+1}-x^*_{k+1})+{\mu^*_k}^\prime\left(h(x_k,\bv_k)-h(x_k^*,\bv_k^*)\right)\notag \\
\geq &\cL_K^*(x_K,\mu_K^*)-\cL_K^*(x_K^*,\mu_K^*)+ \sum_{k=0}^{K-1}-{\lambda_{k+1}^*}^\prime (x_{k+1}-x^*_{k+1}) \notag \\ +&\sum_{k=0}^{K-1}\cL_k^*(x_k,\lambda_{k+1}^*,\mu_k^*)-\cL^*(x_k^*,\lambda_{k+1}^*,\mu_k^*)\notag\\ +&\sum_{k=0}^K {\mu_k^*}^\prime \left(h(x_k,\bv_k)-h(x_k^*,\bv_k^*)\right).
\label{eq:objdiff}
\end{align}
From the envelope theorem we have $\frac{\partial \cL^*_k}{\partial x_k}(x_k^*,\lambda_{k+1}^*,\mu_k^*)=\frac{\partial \cL_k}{\partial x_k}(x_k^*,\bu_k^*,\bv_k^*,\lambda_{k+1}^*,\mu_k^*)=
\lambda_k^*$
and $\frac{\partial \cL^*_K}{\partial x_K}(x_K^*,\mu_K^*)=
\frac{\partial \cL_K}{\partial x_K}(x_K^*,\bv_K^*,\mu_K^*)=\lambda_K^*$. Then using this and from the convexity of minimized Lagrangian and terminal Lagrangian with respect to state variables we get
\begin{align}
&\cL_k^*(x_k,\lambda_{k+1}^*,\mu_k^*)-\cL_k^*(x_k^*,\lambda_{k+1}^*,\mu_k^*)\geq \left(\frac{\partial \cL^*_k}{\partial x_k}(x_k^*,{\lambda_{k+1}}^*,\mu_k^*)\right)^\prime (x_k-x_k^*)={\lambda_k^*}^\prime(x_k-x_k^*), \\
&\cL_K^*(x_K,\mu_K^*)-\cL_K^*(x_K^*,\mu_K^*)\geq \left(\frac{\partial \cL^*_K}{\partial x_K}(x_K^*,\mu_K^*)\right)^\prime (x_K-x_K^*)=
\end{align}
Using (<ref>) in (<ref>) we get
\begin{align*}
\sum_{k=0}^K {\lambda_{k}^*}^\prime(x_{k}-x^*_{k})-\sum_{k=0}^{K-1}{\lambda_{k+1}^*}^\prime (x_{k+1}-x^*_{k+1}) +\sum_{k=0}^K {\mu_k^*}^\prime \left(h(x_k,\bv_k)-h(x_k^*,\bv_k^*)\right)\\
&={\lambda_0^*}^\prime (x_0-x_0^*)+\sum_{k=0}^K {\mu_k^*}^\prime \left(h(x_k,\bv_k)-h(x_k^*,\bv_k^*)\right)\\ &=\sum_{k=0}^K {\mu_k^*}^\prime h(x_k,\bv_k) \geq 0 \hfill.
\end{align*}
The last but one equality follows from the
complementary condition ${\mu_k^*}^\prime h_k(x_k^*,\bv_k^*)=0$ for $k\in \mathcal K$, and
the initial condition $x_0=x_0^*$. The last inequality follows as multipliers and constraints satisfy
the conditions $\mu_k^*\geq0$ and $h_k(x_k,\bv_k)\geq 0$ for all $k\in \mathcal K$.
The sufficient condition provided in Theorem <ref> is an adaptation of
Arrow type sufficient condition <cit.> to a discrete-time setting with inequality constraints, and differs from the nonlinear programming based methods ( [Pearson and Sridhar, 1966]).
§.§ Structure of OLPDG
In this subsection we provide conditions under which the optimal solution of OCP provides
an open-loop Nash equilibrium of NZDG. Toward this end, we have the following assumption.
The cost functions $\{P_k,~k\in \mathcal K\}$ associated with OCP satisfy the following conditions for every $i\in \mathcal N$
\begin{align}
&\frac{\partial P_k}{\partial u_{k}^{i}} = \frac{\partial gu_{k}^{i}}{\partial u_{k}^{i}},~ \frac{\partial P_k}{\partial v_{k}^{i}} = \frac{\partial gv_{k}^{i}}{\partial v_{k}^{i}},~\frac{\partial P_k}{\partial {x}_{k}} = \frac{\partial g_{k}^{i}}{\partial {x}_{k}},~k\in \mathcal K\backslash \{K\}, \label{eq:gradcond1}\\
&\frac{\partial P_K}{\partial {x}_{K}} = \frac{\partial g_{K}^{i}}{\partial {x}_{K}},~
\frac{\partial P_K}{\partial \mathbf{v}_{K}^i} = \frac{\partial g_{K}^{i}}{\partial \mathbf{v}_{K}^i}.
\label{eq:gradcond2}
\end{align}
Let Assumption <ref> holds true.
Then, the cost functions of the OCP and NZDG satisfy the following relation for each player $i\in \mathcal N$.
\begin{align}
J(x_0,(\tilde{u}^i,\tilde{u}^{-i}),(\tilde{v}^i,\tilde{v}^{-i})) - J(x_0,(\tilde{w}^i,\tilde{u}^{-i}),(\tilde{z}^i,\tilde{v}^{-i})) &=J^{i}(x_0,(\tilde{u}^i,\tilde{u}^{-i}),(\tilde{v}^i,\tilde{v}^{-i}))\notag \\ &- J^{i}(x_0,(\tilde{w}^i,\tilde{u}^{-i}),(\tilde{z}^i,\tilde{v}^{-i}))
\label{eq:Difference Condition}
\end{align}
$\forall \tilde{u}^i:=\{u^i_k\in U_k^i,~k\in \mathcal{K}\backslash \{K\}\},~\forall\tilde{w}^i:=\{w^i_k\in U_k^i,~k\in \mathcal{K}\backslash \{K\}\},~ \forall \tilde{v}^i:=\{v^i_k \in V_k^i,~k\in \mathcal K \}$ and $\forall \tilde{z}^i:=\{z^i_k \in V_k^i,~k\in \mathcal K \}$.
From (<ref>) and the separable structure of cost functions as assumed in Assumption <ref>, we have
for every $k\in \mathcal K\backslash \{K\}$
\begin{align}
&\frac{\partial }{\partial u_k^i}\left(P_k(x_k,\mathbf{u}_k,\mathbf{v}_k)-g_k^i(x_k,\mathbf{u}_k,\mathbf{v}_k)\right) = \frac{\partial }{\partial u_k^i}\left(P_k(x_k,\mathbf{u}_k,\mathbf{v}_k)-gu_k^i(x_k,\mathbf{u}_k)\right) = 0, \\
&\frac{\partial }{\partial v_k^i}\left(P_k(x_k,\mathbf{u}_k,\mathbf{v}_k)-g_k^i(x_k,\mathbf{u}_k,\mathbf{v}_k)\right)= \frac{\partial }{\partial v_k^i}\left(P_k(x_k,\mathbf{u}_k,\mathbf{v}_k)-gv_k^i(x_k,\mathbf{v}_k)\right) = 0, \\
&\frac{\partial }{\partial x_{k}}\left(P_k(x_k,\mathbf{u}_k,\mathbf{v}_k)-g_k^i(x_k,\mathbf{u}_k,\mathbf{v}_k) \right) = 0.
\end{align}
From (<ref>) we have
\begin{align}
&\frac{\partial }{\partial x_{K}}\left(P_K(x_K,\mathbf{v}_K)-g_K^i(x_K,\mathbf{v}_K) \right) = 0 \\ &
\frac{\partial }{\partial v_{K}^i}\left(P_K(x_K,\mathbf{v}_K)-g_K^i(x_K,\mathbf{v}_K) \right) = 0.
\end{align}
Clearly, from (<ref>) it follows that the difference
function $P_k(x_k,\bu_k,\bv_k)-g_k^i(x_k,\bu_k,\bv_k)$ is independent of (or does not contain) the variables $x_k$, $u_k^i$ and $v_k^i$. Similarly, from (<ref>) it follows that the
difference function $P_K(x_K,\bv_k)-g_K^i(x_K,\bv_K)$ does not contain the variables $x_K$ and $v_K^i$. This implies that these difference functions can be expressed as
\begin{align}
& P_k( {x}_k,\mathbf{u}_k,\mathbf{v}_k)-g_k^i( {x}_k,\mathbf{u}_k,\mathbf{v}_k) = \Theta^i_k(u_k^{-i},v_{k}^{-i}),~ \forall k \in \mathcal K \backslash \{K\},\\
& P_K( {x}_K,\mathbf{v}_K)-g_K^i( {x}_K,\mathbf{v}_K) = \Theta^i_K(v_K^{-i}).
\end{align}
$\forall u_k^i \in U_k^i$ and $\forall v_k^i \in V_k^i$. Since (<ref>) is satisfied by every $u_k^i \in U_k^i$ and $v_k^i \in V_k^i$, for any $u_k^i,w_k^i \in U_k^i$ and $v_k^i,z_k^i \in V_k^i$ we obtain
\begin{align*}
P_k(x_k,(u_k^i,u_k^{-i}),(v_k^i,v_k^{-i}))-g_k^i(x_k,(u_k^i,u_k^{-i}),(v_k^i,v_k^{-i})) &=P_k(x_k,(w_k^i,u_k^{-i}),(z_k^i,v_k^{-i}))\\&-g_k^i(x_k,(w_k^i,u_k^{-i}),(z_k^i,v_k^{-i})),\\
P_K(x_K,(v_K^i,v_K^{-i}))-g_K^i(x_K,(v_K^i,v_K^{-i})) &= P_K(x_K,(z_K^i,v_K^{-i}))-g_K^i(x_K,(z_K^i,v_K^{-i})). %\label{eq:Lemma1 eq4}
\end{align*}
Upon rearranging the above equations and adding we get
\begin{align}
P_k(x_k,(u_k^i,u_k^{-i}),(v_k^i,v_k^{-i}))- P_k(x_k,(w_k^i,u_k^{-i}),(z_k^i,v_k^{-i}) &=g_k^i(x_k,(u_k^i,u_k^{-i}),(v_k^i,v_k^{-i}))\notag\\&-g_k^i(x_k,(w_k^i,u_k^{-i}),(z_k^i,v_k^{-i})),
\label{eq:instdiff}\\
\end{align}
Taking summation of (<ref>) for all time steps $k \in \mathcal{K}\backslash \{K\}$ and using (<ref>), we obtain (<ref>).
Lemma <ref> provides the dynamic counterpart of the principle of exact potential games introduced by Monderer and Shapely <cit.>.
Let Assumptions <ref> and <ref> hold true.
Let the admissible pair $(\tilde{\bu}^*,\tilde{\bv}^*)$ be the optimal solution of OCP. Then
$(\tilde{\bu}^*,\tilde{\bv}^*)$ is an open-loop Nash equilibrium of NZDG, that is, NZDG is an OLPDG with potential functions $\{P_k,~k\in \mathcal K\}$.
Let $\{x^*_k,~k\in \mathcal K\}$ be the state trajectory generated by $\tilde{\bu}^*$. As $(\tilde{\bu}^*,\tilde{\bv}^*)$ is optimal for OCP there exist co-states $\{\lambda^*_k,~k\in \mathcal K\}$ and multipliers
$\{\mu^*_k,~k\in \mathcal K\}$ such that the necessary conditions (<ref>) hold true. Expanding these conditions in terms of cost functions we get
\begin{align}
\intertext{for $~k\in \mathcal K\backslash \{K\}$}
\label{eq:OCP Cond1} & \frac{\partial P_k}{\partial u_{k}^i}(x^*_k,\bu_k^*,\bv_k^*)+\left(\frac{\partial f_k}{\partial u_k^i}(x^*_k,\bu_k^*)\right)^\prime \lambda_{k+1}^{*} = 0,~i\in \mathcal N, \\
\label{eq:OCP Cond2} & x_{k+1}^{*} = f_k(x^*_k,\bu_k^*),\quad x_0^*=x_0, \\
\label{eq:OCP Cond3} & \lambda_k^*=\frac{\partial P_k}{\partial x_{k}}(x^*_k,\bu_k^*,\bv_k^*)+\left(\frac{\partial f_k}{\partial x_k}(x^*_k,\bu_k^*)\right)^{\prime} \lambda_{k+1}^*\notag \\ &\hspace{1.25in}-\left(\frac{\partial h_k}{\partial x_k}(x_k^*,\bv_k^*)\right)^\prime \mu_k^*, \\
\label{eq:OCP Cond4} & \lambda_{K}^{*}= \frac{\partial P_K}{\partial x_{K}}(x_K^*,\bv_K^*)-\left(\frac{\partial h_K}{\partial x_K}(x^*_K,\bv_K^*)\right)^\prime \mu_{K}^{*}, \\
\label{eq:OCP Cond5} & 0 \leq \left(\frac{\partial P_k}{\partial v_{k}^i}(x^*_k,\bu_k^*,\bv_k^*)-\left(\frac{\partial h_k}{\partial v_k^i}(x^*_k,\bv_k^*) \right)^\prime \mu_{k}^{*}\right) \perp v_k^{i^{*}} \notag \\&\hspace{1.85in}\geq 0,~~i\in \mathcal N,\\
\label{eq:OCP Cond6} & 0 \leq \left(\frac{\partial P_K}{\partial v_{K}^i}(x^*_K,\bv_K^*)-\left(\frac{\partial h_K}{\partial v_K^i}(x^*_K,\bv_K^*)\right)^\prime \mu_{K}^{*}\right) \perp v_K^{i^{*}}\notag \\ &\hspace{1.85in} \geq 0,~i\in \mathcal N, \\
\text{for}& ~k\in \mathcal K, ~
0\leq h_k(x^*_k,\bv_k^*)\perp \mu_k^* \geq 0. \label{eq:OCP Cond7}
\end{align}
Next, we write $\bu_k^*=(u_k^{i*},u_k^{-i*})$ and $\bv_k^*=(v_k^{i*},v_k^{-i*})$ and for each player $i\in \mathcal N$ we define the multipliers
\begin{align}
{\Lambda_k^{i*}}:=\lambda_k^*,k\in \mathcal K\backslash \{K\},~\delta_k^{i*}:=\mu_k^*,~k\in \mathcal K.
\label{eq:samemultipliers}
\end{align}
Now from Assumption <ref> and using the above notation we write (<ref>)
as follows.
\begin{align}
\text{for} &~k\in \mathcal K\backslash \{K\} \notag\\
&\frac{\partial g_k^i(x^*_{k},(u_{k}^{i},u_{k}^{-i*}),(v_{k}^{i},v_{k}^{-i*}))}{\partial u_k^i}\Big|_{u_k^i = u_k^{i*}}+\left(\frac{\partial f_k(x^*_{k},(u_{k}^{i},u_{k}^{-i*}))}{\partial u_k^i}\Big|_{u_k^i = u_k^{i*}}\right)^{\prime}\Lambda_{k+1}^{i*} = 0, \label{eq:OLNEcond1} \\
& x_{k+1}^* = f(x_{k}^{*},\mathbf{u}_{k}^{*}),~x_0^*=x_0, \label{eq:OLNEcond2} \\
\Lambda_k^{i*} &= \frac{\partial g_k^i(x_{k},(u_{k}^{i*},u_{k}^{-i*}),(v_{k}^{i*},v_{k}^{-i*}))}{\partial x_k}\Big|_{x_k = x_k^{*}}+\left(\frac{\partial f_k(x_{k},(u_{k}^{i*},u_{k}^{-i*}))}{\partial x_k}\Big|_{x_k = x_k^{*}}\right)^{\prime}\Lambda_{k+1}^{i*}\notag \\
&\qquad-\left(\frac{\partial h_k(x_{k},(v_{k}^{i*},v_{k}^{-i*}))}{\partial x_k}\Big|_{x_k = x_k^{*}}\right)^{\prime}\delta_{k}^{*},\\
\Lambda_K^{i*}& = \frac{\partial g_K^i(x_{K},(v_{K}^{i*},v_{K}^{-i*}))}{\partial x_K}\Big|_{x_K = x_K^{*}} -\left(\frac{\partial h_K(x_{K},(v_{K}^{i*},v_{K}^{-i*}))}{\partial x_K}\Big|_{x_K = x_K^{*}}\right)^{\prime}\delta_K^{i*},\\
&0 \leq \frac{g_k^i(x^*_{k},(v_{k}^{i},v_{k}^{-i*}))}{\partial v_k^i}\Big|_{v_k^i =v_k^{i*}}\notag -\left(\frac{\partial h_k(x^*_{k},(v_{k}^{i},v_{k}^{-i*}))}{\partial v_k^i}\Big|_{v_k^i = v_k^{*}}\right)^{\prime}\delta_k^{i*} \perp v_k^{i*} \geq 0,\\
& 0 \leq \frac{g_K^i(x^*_{K},(v_{K}^{i},v_{K}^{-i*}))}{\partial v_K^i}\Big|_{v_K^i =v_K^{i*}}-\left(\frac{\partial h_K(x^*_{K},(v_{K}^{i},v_{K}^{-i*}))}{\partial v_K^i}\Big|_{v_K^i = v_K^{*}}\right)^{\prime}\delta_k^{i*} \perp v_K^{i*} \geq 0,\\
\text{for} &~k\in \mathcal K, ~ 0 \leq h_{k}(x_{k}^*,\mathbf{v}_{k}^*) \perp \delta_k^{i*} \geq 0.
\end{align}
Next, we consider Player $i$'s optimal control problem
(<ref>) when players in $-i=\mathcal N\backslash \{i\}$ use strategies $(\tilde{u}^{-i*},\tilde{v}^{-i*})$. Taking the co-state vectors as
$\{\Lambda_k^i,~k\in \mathcal K\}$ and Lagrange multipliers as $\{\delta_k^i,~k\in \mathcal K\}$ we write the instantaneous Lagrangian and terminal Lagrangian functions associated with Player $i$'s problem are given as
\begin{align}
&\mathcal L _k^i(x_k,(u_k^i,u_k^{-i*}),(v_k^{i},v_k^{-i*}),\Lambda_{k+1}^{i},\delta_k^{i})=g^i_k(x_k,(u_k^i,u_k^{-i*}),(v_k^{i},v_k^{-i*})) +{\Lambda^i_{k+1}}^\prime f_k(x_k,(u_k^i,u_k^{-i*}))\notag \\&\hspace{4.4in}-{\delta^i_k}^\prime h_k(x_k,(v_k^{i},v_k^{-i*})),\\
& \cL^i_K(x_K,(v_K^{i},v_K^{-i*}),\mu_K)=g^i_K(x_K,(v_k^{i},v_K^{-i*}))-{\delta^i_K}^\prime h_K(x_K,(v_K^{i},v_K^{-i*})).
\end{align}
Representing the equations (<ref>) in terms of the instantaneous Lagrangian and terminal Lagrangian functions (<ref>)
associated with Player $i$'s optimal control problem (<ref>) we get
\begin{align}
\text{for } &k\in \mathcal K\backslash \{K\}\notag \\
&u_k^{i*}=\argmin_{u^i_k\in U^i_k}\mathcal L^i_k(x^*_k,(u_k^i,u_k^{-i*}),(v_k^{i},v_k^{-i*}),\Lambda_{k+1}^{i*},\delta_k^{i*}), \\
&x^*_{k+1}=\frac{\partial \cL^i_k}{\partial \Lambda^{i*}_{k+1}}(x^*_k,(u_k^{i*},u_k^{-i*}),(v_k^{i*},v_k^{-i*}),\Lambda_{k+1}^{i*},\delta_k^{i*}),\quad x^*_0=x_0,\\
&\lambda^*_k=\frac{\partial \mathcal L^i_k}{\partial x_k}(x^*_k,(u_k^{i*},u_k^{-i*}),(v_k^{i*},v_k^{-i*}),\Lambda_{k+1}^{i*},\delta_k^{i*}),\\& \lambda^*_K=\frac{\partial \cL^i_K}{\partial x_K}(x_K^*,(v_k^{i*},v_k^{-i*}),\delta_K^{i*}),\\
&0\leq \frac{\partial \mathcal L^i_k}{\partial v^i_k} (x^*_k,(u_k^{i},u_k^{-i*}),(v_k^{i},v_k^{-i*}),\lambda_{k+1}^*,\mu_k^*) \perp v^{i*}_k \geq 0,\\
&0\leq \frac{\partial \cL^i_K}{\partial v^i_K} (x^*_K,(v_K^{i},v_K^{-i*}),\Lambda_{k+1}^{i*},\delta_K^{i*}) \perp v^{i*}_K \geq 0,\\
\text{for } &k\in \mathcal K,~ 0\leq h(x_k^*,(v_k^{i},v_k^{-i*})) \perp \delta_k^{i*} \geq 0.
\end{align}
Clearly, (<ref>) constitute the necessary conditions for optimality of associated with
Player i's optimal control problem (<ref>) where strategies of players in $-i$ are fixed at
$(\tilde{u}^{-i*},\tilde{v}^{-i*})$. In other words, $(\tilde{u}^{i*},\tilde{v}^{i*})$ is a candidate best response to $(\tilde{u}^{-i*},\tilde{v}^{-i*})$.
Next, we show that the strategy $(\tilde{u}^{i*},\tilde{v}^{i*})$ indeed minimizes the objective $J^i(x_0,(\tilde{u}^{i^*},\tilde{u}^{-i*}),(\tilde{v}^{i^*},\tilde{v}^{-i*}))$ subject to
the dynamics $x_{k+1}=f_k(x_k,(u_k^i,u_k^{-i*}))$, $k\in \mathcal K \backslash \{K\}$ and constraints $h_k(x_k,(v_k^i,v_k^{-i*}))\geq 0$, $v_k^i\geq 0$, $k\in \mathcal K$. Since $(\mathbf{\tilde{u}^*,\tilde{v}^*})$ is an optimal solution of the OCP, the following inequality holds true
\begin{align}
J(x_0,(\tilde{u}^{i^*},\tilde{u}^{-i*}),(\tilde{v}^{i^*},\tilde{v}^{-i*})) \leq J(x_0,(\tilde{u}^{i},\tilde{u}^{-i*}),(\tilde{v}^{i},\tilde{v}^{-i*})),~\forall (\tilde{u}^i,\tilde{v}^i)
\label{eq:OCP max}
\end{align}
with $\tilde{v}_k^i$ satisfying the constraint $h_k( x_k ,({v}_k^i,{v}_k^{\mbox{-}i^*}))\geq 0 ~\forall k \in \mathcal{K}$, where $ x_k $ is the state trajectory evolved according to the action set $(\tilde{u}^i,\tilde{u}^{\mbox{-}i^*})$. From lemma <ref>, we have
\begin{multline*}
J(x_0,(\tilde{u}^{i^*},\tilde{u}^{-i*}),(\tilde{v}^{i^*},\tilde{v}^{-i*})) - J(x_0,(\tilde{u}^{i},\tilde{u}^{-i*}),(\tilde{v}^{i},\tilde{v}^{-i*}))\\ = J^i(x_0,(\tilde{u}^{i^*},\tilde{u}^{-i*}),(\tilde{v}^{i^*},\tilde{v}^{-i*}))- J^i(x_0,(\tilde{u}^{i},\tilde{u}^{-i*}),(\tilde{v}^{i},\tilde{v}^{-i*})).
\end{multline*}
Using the above observation in (<ref>) we get
\begin{align}
J^i(x_0,(\tilde{u}^{i^*},\tilde{u}^{-i*}),(\tilde{v}^{i^*},\tilde{v}^{-i*})) \leq J^i(x_0,(\tilde{u}^{i},\tilde{u}^{-i*}),(\tilde{v}^{i},\tilde{v}^{-i*})) \quad \forall ((\tilde{u}^{i},\tilde{u}^{-i*}),(\tilde{v}^{i},\tilde{v}^{-i*}))
\end{align}
with $\tilde{v}_k^i$ satisfying the constraint $h_k( x_k ,({v}_k^i,{v}_k^{\mbox{-}i^*}))\geq 0 ~\forall k \in \mathcal{K}$. This implies $(\tilde{u}^{i^*},\tilde{v}^{i^*})$ is indeed a best response to $(\tilde{u}^{-i*},\tilde{v}^{-i*})$. So, $(\mathbf{\tilde{u}^*,\tilde{v}^*})$ is an open-loop Nash equilibrium of NZDG. Following Definition <ref> we have NZDG is an OLPDG with potential functions
$\{P_k,~k\in \mathcal K\}$.
The Nash equilibira correspond to fixed points of best response mapping defined over
joint strategy sets of players, and as a result there can exist more than one equilibria in a non-comparative game. When OCP associated with OLPDG has an optimal solution, then from Theorem <ref> this solution is an open-loop Nash equilibrium of NZDG, thereby providing a refinement of the open-loop Nash equilibria. This implies that, solution of the OCP provides a way for selecting one among possibly many open-loop Nash equiliria.
Notice, the constraints in (<ref>) are coupled. Rosen in [Rosen, 1965] studied non-cooperative games with couple constraints. The equilibria in these games are referred to as normalized Nash equlibria, and are characterized by the property that
multiplier vector associated with constraints in each player's individual optimization problem is co-linear with a common multiplier vector. In our work, we observe a similar feature by construction, that is, in (<ref>) the obtained open-loop Nash equilibrium has the property that the associated multipliers and co-state variables are same for all players.
§.§ Construction of potential cost functions
In section <ref>, a sufficient condition is provided to verify if NZDG is an OLPDG given the potential cost functions $\{P_k,~k\in \mathcal K\}$ of the OCP. However, in practice these functions are not available before hand. In these settings, it is desirable to construct potential functions using players' cost functions. This construction procedure involves two steps. First, to verify if NZDG is an OLPDG, and then to construct the potential functions. Toward this end, we recall the following necessary and sufficient condition existence of conservative vector fields from multi-variable calculus ([Apostol, 1969]).
Let $\Omega$ be a convex open subset in $\mathbb R^n$. Let $F:\Omega \rightarrow \mathbb R^n$ be a vector field with continuous derivatives defined over $\Omega$. The following conditions on $F$ are equivalent
a.There exists a scalar potential function $\Pi:\Omega\rightarrow \mathbb R$ such that
$F(\omega)=\nabla \Pi(\omega)$ for all $\omega \in \Omega$, where $\nabla$ denotes the gradient operator.
b. The partial derivatives satisfy
\begin{align}
\frac{\partial [F(\omega)]_i}{\partial [\omega]_j} = \frac{\partial [F(\omega)]_j}{\partial [\omega]_i},~\forall \omega \in \Omega,~i,j=1,2,\cdots,n.
\label{eq:symcond}
\end{align}
c. Let $a$ be a fixed point in $\Omega$, and $\mathcal{C}\subset \Omega$ be a piecewise smooth curve
joining $a$ with an arbitrary point $\omega \in \Omega$. Then the potential function $\Pi$ satisfies
\begin{align}
\Pi(\omega) -\Pi(a)=\int_\mathcal{C} F(\omega)\bigcdot d\omega = \int_0^1 F(\alpha(z)) \bigcdot \frac{\partial \alpha}{\partial z} ~dz,
\end{align}
where $\bigcdot$ is the dot product, and $\alpha: [0, 1] \rightarrow \mathcal{C}$ is a bijective parametrization of the curve $\mathcal{C}$ such that $\alpha(0)=a$ and $\alpha(1)=\omega$.
A vector field $F:\Omega\rightarrow \mathbb R^n$ satisfying these conditions is called a conservative vector field.
See <cit.>.
A consequence of the condition (<ref>) is that the Jacobian matrix of the vector field $F:\Omega \rightarrow \mathbb R^n$ evaluated at $\omega \in \Omega $ is symmetric for all $\omega\in \Omega$.
Let Assumption <ref>.(1) holds true. Let the players' utility functions satisfy the following conditions, for all $i,j\in \mathcal N$,
\begin{align}
& \frac{\partial^2 g_k^i}{\partial (u_k^j)^\prime \partial u_k^i} =\left( \frac{\partial^2 g_k^j}{\partial (u_k^i)^\prime \partial u_k^j}\right)^\prime,~k\in \mathcal K\backslash \{K\}, \label{eq:ux}\\
&\frac{\partial g_k^i}{\partial x_k} = \frac{\partial g_k^j}{\partial x_k} \label{eq: partial x},~k\in \mathcal K,\\
& \frac{\partial^2 g_k^i}{\partial (v_k^j)^\prime \partial v_k^i} = \left(\frac{\partial^2 g_k^j}{\partial (v_k^i)^\prime \partial v_k^j}\right)^\prime,~k\in \mathcal K, \label{eq:vxx}
\end{align}
then NZDG is a OLPDG.
Let the vector fields $F_k({x}_k,\mathbf{u}_k,\mathbf{v}_k)$ at $(x_k,\bu_k,\bv_k)\in {\Omega}_k$, where ${\Omega}_k := {X}_k \times \prod_{i \in \mathcal{N}}U_k^i \times \prod_{i \in \mathcal{N}}V_k^i$ of dimension $(n+m+s) \times 1$ and $F_K({x}_K,\mathbf{v}_K)$ at $(x_K,\bv_K)\in {\Omega}_K$ where $\Omega_K:= {X}_K \times \prod_{i \in \mathcal{N}}V_K^i$ of dimension $(n+s) \times 1$ be defined by
\begin{align*}
&F_k({x}_k,\bu_k,\bv_k) = \begin{bmatrix}\frac{\partial g_k^1}{\partial {x}_k}^\prime & \frac{\partial g_k^1}{\partial u_k^1}^\prime & \cdots & \frac{\partial g_k^N}{\partial u_k^N}^\prime & \frac{\partial g_k^1}{\partial v_k^1}^\prime& \cdots & \frac{\partial g_k^N}{\partial v_k^N}^\prime \end{bmatrix}^\prime,~ k \in \mathcal{K}\backslash \{K\}, \\
&F_K(x_K,\bv_K) = \begin{bmatrix}\frac{\partial g_K^1}{\partial {x}_k}^\prime & \frac{\partial g_K^1}{\partial v_K^1}^\prime& \cdots & \frac{\partial g_K^N}{\partial v_K^N}^\prime \end{bmatrix}^\prime,
\end{align*}
then the instantaneous and terminal potential functions are given by
\begin{align}
&P_k(x_k,\bu_k,\bv_k) =c_k+\int_0^1 F_k(\alpha_k(z)) \bigcdot \frac{\partial \alpha_k(z)}{\partial z}dz,~k\in \mathcal K, \label{eq:Pfk}\\
&P_K(x_K,\bv_K) =c_K+\int_0^{1} F_K(\alpha_K(z)) \bigcdot \frac{\partial \alpha_K(z)}{\partial z}dz,
\label{eq:PfK}
\end{align}
where $\alpha_k:[0,1]\rightarrow \mathcal{C}_k$ ($k\in \mathcal K$) is a bijective parametrization
of a piece-wise smooth curve $\mathcal{C}_k\subset {\Omega}_k$ such that
$\alpha_k(0)= (x_{0k},\bu_{0k},\bv_{0k})$, $\alpha_k(1)=(x_{k},\bu_{k},\bv_{k})$ for $k\in\mathcal K \backslash \{K\}$, $\alpha_K(0)= (x_{0K},\bv_{0K})$, $\alpha_K(1)=(x_{K},\bv_{K})$, and $\{c_k,k\in \mathcal K\}$ are constants.
We write the vector field $F_k,~k \in \mathcal{K} \backslash K $ as $F_k = \begin{bmatrix}{F_k^x}^\prime & {F_k^u}^\prime & {F_k^v}^\prime \end{bmatrix}^\prime_{(n+m+s) \times 1} $ such that
\begin{align*}
& F_k^x = \left[\frac{\partial g_k^1}{\partial x_k }\right]_{n \times 1},
~ F_k^u =\begin{bmatrix} \frac{\partial g_k^1}{\partial u_k^1}^\prime & \frac{\partial g_k^2}{\partial u_k^2}^\prime & \hdots & \frac{\partial g_k^N}{\partial u_k^N}^\prime
\end{bmatrix}^\prime_{m \times 1},\\& F_k^v = \begin{bmatrix} \frac{\partial g_k^1}{\partial v_k^1}^\prime & \frac{\partial g_k^2}{\partial v_k^2}^\prime & \hdots & \frac{\partial g_k^N}{\partial v_k^N}^\prime
\end{bmatrix}^\prime_{s \times 1}.
\end{align*}
Let $\mathbf{w}_k,~k \in \mathcal{K} \backslash \{K\} $ be the $(n+m+s) \times 1$ vector given by $\mathbf{w}_k = \begin{bmatrix}
x_k^\prime & \bu_k^\prime & \bv_k^\prime
\end{bmatrix}^\prime_{(n+m+s) \times 1}$. Therefore, we can write the Jacobian matrix, $ \mathcal{J}_k,~k \in \mathcal{K} \backslash\{k\}$ as
\begin{align}
\label{eq:Jacobiank}
\mathcal{J}_k = \frac{\partial F_k}{\partial (\mathbf{w}_k)^\prime } & = \begin{bmatrix}\frac{\partial F_k^x}{\partial (x_k)^\prime} & \frac{\partial F_k^x}{\partial (\bu_k)^\prime} & \frac{\partial F_k^x}{\partial (\bv_k)^\prime} \\[0.2cm]
\frac{\partial F_k^u}{\partial (x_k)^\prime} & \frac{\partial F_k^u}{\partial (\bu_k)^\prime} & \frac{\partial F_k^u}{\partial (\bv_k)^\prime} \\[0.2cm]
\frac{\partial F_k^v}{\partial (x_k)^\prime} & \frac{\partial F_k^v}{\partial (\bu_k)^\prime} & \frac{\partial F_k^v}{\partial (\bv_k)^\prime}
\end{bmatrix}_{(n+m+s) \times (n+m+s)},
\end{align}
\begin{align*}
&\frac{\partial F_k^x}{\partial (x_k)^\prime } = \left[ \frac{\partial^2 g_k^1}{\partial (x_k)^\prime \partial x_k}\right]_{n \times n },~\frac{\partial F_k^u }{\partial (x_k)^\prime } = \begin{bmatrix} \frac{\partial^2 g_k^1}{\partial (x_k)^\prime \partial u_k^1} \\[0.2cm] \frac{\partial^2 g_k^2}{\partial (x_k)^\prime \partial u_k^2} \\ \vdots \\ \frac{\partial^2 g_k^N}{\partial (x_k)^\prime \partial u_k^N} \end{bmatrix}_{m \times n }, ~\frac{\partial F_k^v }{\partial (x_k)^\prime } = \begin{bmatrix} \frac{\partial^2 g_k^1}{\partial (x_k)^\prime \partial v_k^1} \\[0.2cm] \frac{\partial^2 g_k^2}{\partial (x_k)^\prime \partial v_k^2} \\ \vdots \\ \frac{\partial^2 g_k^N}{\partial (x_k)^\prime \partial v_k^N} \end{bmatrix}_{s \times n },
\end{align*}
\begin{align*}
& \frac{\partial F_k^x}{\partial (\bu_k)^\prime } = \begin{bmatrix}
\frac{\partial^2 g_k^1}{\partial (u_k^1)^\prime \partial x_k }& \frac{\partial^2 g_k^1}{\partial (u_k^2)^\prime \partial x_k } & \hdots & \frac{\partial^2 g_k^1}{\partial (u_k^N)^\prime \partial x_k }
\end{bmatrix}_{n \times m},~\frac{\partial F_k^x}{\partial (\bv_k)^\prime } = \begin{bmatrix}
\frac{\partial^2 g_k^1}{\partial (v_k^1)^\prime \partial x_k }& \frac{\partial^2 g_k^1}{\partial (v_k^2)^\prime \partial x_k } & \hdots & \frac{\partial^2 g_k^1}{\partial (v_k^N)^\prime \partial x_k }\end{bmatrix}_{n \times s},
\end{align*}
\begin{align*}
& \frac{\partial F_k^u}{\partial (\bu_k)^\prime } = \begin{bmatrix} \frac{\partial^2 g_k^1}{\partial (u_k^1)^\prime \partial u_k^{1}} & \frac{\partial^2 g_k^1}{\partial (u_k^2)^\prime \partial u_k^{1}}& \hdots & \frac{\partial^2 g_k^1}{\partial (u_k^N)^\prime \partial u_k^{1}} \\[0.2cm]
\frac{\partial^2 g_k^2}{\partial (u_k^{1})^\prime \partial u_k^2} & \frac{\partial^2 g_k^2}{\partial (u_k^2)^\prime \partial u_k^{2}}& \hdots & \frac{\partial^2 g_k^2}{\partial (u_k^N)^\prime \partial u_k^{2}} \\
\vdots & \vdots & \ddots & \vdots \\
\frac{\partial^2 g_k^N}{\partial (u_k^{1})^\prime \partial u_k^N} & \frac{\partial^2 g_k^N}{\partial (u_k^{2})^\prime \partial u_k^N}& \hdots & \frac{\partial^2 g_k^N}{\partial (u_k^{N})^\prime \partial u_K^N}
\end{bmatrix}_{m \times m},~\frac{\partial F_k^v}{\partial (\bu_k)^\prime} = \begin{bmatrix} \frac{\partial^2 g_k^1}{\partial (u_k^{1})^\prime \partial v_k^1 }& \frac{\partial^2 g_k^1}{\partial (u_k^2)^\prime \partial v_k^{1}}& \hdots & \frac{\partial^2 g_k^1}{\partial (u_k^N)^\prime \partial v_k^{1}} \\[0.2cm]
\frac{\partial^2 g_k^2}{\partial (u_k^{1})^\prime \partial v_k^2} & \frac{\partial^2 g_k^2}{\partial (u_k^{2})^\prime \partial v_k^2 }& \hdots & \frac{\partial^2 g_k^2}{\partial (u_k^N)^\prime \partial v_k^{2}} \\
\vdots & \vdots & \ddots & \vdots \\
\frac{\partial^2 g_k^N}{\partial (u_k^{1})^\prime \partial v_k^N} & \frac{\partial^2 g_k^N}{\partial (u_k^{2})^\prime \partial v_k^N}& \hdots & \frac{\partial^2 g_k^N}{\partial (u_k^{N})^\prime \partial v_k^N}
\end{bmatrix}_{s \times m},
\end{align*}
\begin{align*}
&\frac{\partial F_k^u}{\partial (\bv_k)^\prime } = \begin{bmatrix} \frac{\partial^2 g_k^1}{\partial (v_k^{1})^\prime \partial u_k^1 }& \frac{\partial^2 g_k^1}{\partial (v_k^2)^\prime \partial u_k^{1}}& \hdots & \frac{\partial^2 g_k^1}{\partial (v_k^N)^\prime \partial u_k^{1}} \\[0.2cm]
\frac{\partial^2 g_k^2}{\partial (v_k^{1})^\prime \partial u_k^2} & \frac{\partial^2 g_k^2}{\partial (v_k^{2})^\prime \partial u_k^2 }& \hdots & \frac{\partial^2 g_k^2}{\partial (v_k^N)^\prime \partial u_k^{2}} \\
\vdots & \vdots & \ddots & \vdots \\
\frac{\partial^2 g_k^N}{\partial (v_k^{1})^\prime\partial u_k^N} & \frac{\partial^2 g_k^N}{\partial (v_k^{2})^\prime \partial u_k^N}& \hdots & \frac{\partial^2 g_k^N}{\partial (v_k^{N})^\prime \partial u_k^N}
\end{bmatrix}_{m \times s},~\frac{\partial F_k^v}{\partial (\bv_k)^\prime } = \begin{bmatrix} \frac{\partial^2 g_k^1}{\partial (v_k^1)^\prime \partial v_k^{1}} & \frac{\partial^2 g_k^1}{\partial (v_k^2)^\prime \partial v_k^{1}}& \hdots & \frac{\partial^2 g_k^1}{\partial (v_k^N)^\prime \partial v_k^{1}} \\
\frac{\partial^2 g_k^2}{\partial (v_k^{1})^\prime \partial v_k^2} & \frac{\partial^2 g_k^2}{\partial (v_k^2)^\prime \partial v_k^{2}}& \hdots & \frac{\partial^2 g_k^2}{\partial (v_k^N)^\prime \partial v_k^{2}} \\[0.2cm]
\vdots & \vdots & \ddots & \vdots \\
\frac{\partial^2 g_k^N}{\partial (v_k^{1})^\prime \partial v_k^N} & \frac{\partial^2 g_k^N}{\partial (v_k^{2})^\prime \partial v_k^N}& \hdots & \frac{\partial^2 g_k^N}{\partial (v_k^N)^\prime \partial v_k^{N}}
\end{bmatrix}_{s \times s}.
\end{align*}
Firstly, as $\frac{\partial F_k^x}{\partial (x_k)^\prime } = \frac{\partial^2 g_k^1}{\partial (x_k)^\prime \partial x_k}$ we have that $\frac{\partial F_k^x}{\partial (x_k)^\prime }$ is a symmetric matrix.
The off-diagonal block matrices in $\frac{\partial F_u}{\partial (\bu_k)^\prime}$ satisfy (<ref>)
and this implies $\frac{\partial F_u}{\partial (\bu_k)^\prime}$ is symmetric matrix. Similarly, from
(<ref>), we have that $\frac{\partial F_v}{\partial (\bv_k)^\prime}$ is also a symmetric matrix.
Next, from (<ref>), $ \frac{\partial g_k^i}{\partial x_k} = \frac{\partial g_k^j}{\partial x_k}$, and as cost functions are twice continuously differentiable, we have $\frac{\partial^2 g_k^i}{\partial (u_k^j)^\prime \partial x_k} = \frac{\partial^2 g_k^j}{\partial (u_k^j)^\prime \partial x_k}$.
Next, from the symmetry property of mixed partials we get $\frac{\partial^2 g_k^j}{\partial (u_k^j)^\prime \partial x_k} = \left(\frac{\partial^2 g_k^j}{\partial (x_k)^ \prime \partial u_k^j}\right)^\prime $, and then using (<ref>) we have
$ \frac{\partial^2 g_k^i}{\partial (u_k^j)^\prime \partial x_k} = \left( \frac{\partial^2 g_k^j}{\partial (x_k)^\prime \partial u_k^j}\right)^\prime$, which implies $\frac{\partial F_k^x}{\partial (\bu_k)^\prime} = \left(\frac{\partial F_k^u}{\partial (x_k)^\prime} \right) ^\prime $. Again, using similar arguments we can show that
$\frac{\partial F_k^x}{\partial (\bv_k)^\prime } = \left(\frac{\partial F_k^v}{\partial (x_k)^\prime } \right) ^\prime$. From the separable structure of the cost functions we have
$\frac{\partial^2 g_k^i}{\partial (v_k^j)^\prime \partial u_k^i} =\left( \frac{\partial^2 g_k^j}{\partial (u_k^i)^\prime \partial v_k^j}\right)^\prime = \mathbf{0}$, which implies $\frac{\partial F_k^u}{\partial (\bv_k)^\prime } = \left(\frac{\partial F_k^v}{\partial (\bu_k)^\prime } \right) ^\prime = \mathbf{0}$. Clearly, from these observations we see that Jacobian matrix (<ref>) is a symmetric matrix, and as a result $F_k$ is a conservative vector field for all $k\in \mathcal K \backslash \{K\}$.
At the terminal instant we write $F_K = \begin{bmatrix}F_K^x & F_K^v \end{bmatrix}_{(n+s) \times 1}$ and define $\mathbf{w}_K = \begin{bmatrix} x_K^\prime & \bv_K^\prime \end{bmatrix}^\prime_{(n+s) \times 1}$. From (<ref>) and using the same reasoning as above we can show that the Jacobian $ \mathcal{J}_K = \frac{\partial F_K}{\partial (\mathbf{w}_K)^\prime } = \begin{bmatrix}\frac{\partial F_K^x}{\partial (x_K)^\prime} & \frac{\partial F_K^x}{\partial (\bv_K)^\prime } \\[0.2cm]
\frac{\partial F_K^v}{\partial (x_K)^\prime} & \frac{\partial F_K^v}{\partial (\bv_K)^\prime }
\end{bmatrix}_{(n+s) \times (n+s)}$ is a symmetric matrix. This implies, that $F_K$ is a conservative vector field.
As $\alpha_k:[0,1]\rightarrow \mathcal C_k \subset \Omega_k$ is bijective parametrization of a piece-wise path connecting a fixed point $(x_{0k},\bu_{0k},\bv_{0k})\in \Omega_k$ to an arbitrary point $(x_k,\bu_k,\bv_k)\in \Omega_k$.
Following Lemma <ref>, the instantaneous potential function satisfies
\begin{align*}
P_k(x_k,\bu_k,\bv_k)= c_k+\int_{0}^{1}F_k(\alpha_k(z)) \bigcdot \frac{\partial{\alpha_k(z)}}{\partial z}dz,~k \in \mathcal{K}\backslash \{K\},
% \label{eq:PF_k}
\end{align*}
where $c_{k}=P_k(x_{0k},\bu_{0k},\bv_{0k})$ is value of the potential function evaluated at $(x_{0k},\bu_{0k},\bv_{0k})$. Similarly, the terminal potential function is given by
\begin{align}
P_K(x_K,\bv_K) = c_{K}+\int_{0}^{1}F_K(\alpha_K(z)) \bigcdot \frac{\partial{\alpha_K(z)}}{\partial
z}dz. \notag %\label{eq:PF_K}
\end{align}
where $c_{K} = P_K(x_{0K},\bv_{0K})$.
From (<ref>) and (<ref>), we note that the instantaneous potential function and terminal potential function at a given point are not unique, but are unique up to a constant, and depend upon the choice of the initial fixed points $\{\alpha_k(0),~k\in \mathcal K\}$. This implies that we obtain a family of potential functions, and as a result, several optimal control problems associated with OLPDG. However, as the objective functions of these problems differ by a constant, they have the same optimal solution.
§ OPEN-LOOP LINEAR QUADRATIC POTENTIAL DIFFERENCE GAME
In this section, we specialize the results obtained from the previous section to a linear quadratic setting and provide a numerical method for computing the open-loop Nash equilibrium associated with OLPDG. Toward this end, we introduce the following $N$-player non-zero sum finite horizon linear quadratic difference game as follows. Each Player $i\in \mathcal N$ solves
\begin{align}
\text{NZDG1}:~& \min_{\tilde{u}^i,\tilde{v}^i} J^i(x_0,(\tilde{u}^i,\tilde{u}^{-i}),(\tilde{v}^{i},\tilde{v}^{-i})),\\
&\text{subject to}\notag \\
& x_{k+1}=A_kx_k +\sum_{i\in \mathcal N} B_k^i u_k^i,k\in \mathcal K\backslash \{K\},~ x_0 \text{ (given)}, \label{eq:LQStatedynamics}\\
& M_kx_k+N_k \bv_k +r_k \geq 0,~\bv_k \geq 0,~ k\in \mathcal K, \label{eq:LQconstraints}
\end{align}
\begin{align}
J^i(x_0,(\tilde{u}^i,\tilde{u}^{-i}),(\tilde{v}^{i},\tilde{v}^{-i}))& = \frac{1}{2}x_K^{\prime}Q_K^ix_K+p_K^{i^\prime}x_K + \sum_{k=0}^{K-1}\left(\frac{1}{2}x_k^{\prime}Q^i_kx_k+{p_k^i}^\prime x_k+\frac{1}{2}\mathbf{u}^\prime_kR^i_k\mathbf{u}_k\right)\nonumber\\&+\sum_{k=0}^{K}\left(\frac{1}{2}\mathbf{v}_k^{\prime}D^i_k\mathbf{v}_k+{d^{i}_k}^\prime\mathbf{v}_k+x_k^{\prime}L^i_k\mathbf{v}_k \right),
\label{eq:LQobjective}
\end{align}
where the matrices $Q^i_k\in \mathbb{R}^{n \times n}$, $i\in \mathcal N$, $k\in \mathcal K$ are symmetric, $R_k^i\in \mathbb R^{m_i\times m_i},~i\in \mathcal N,~k\in \mathcal K\backslash \{K\}$ are symmetric and positive definite, $D_k^i\in \mathbb R^{s\times s},~i\in \mathcal N, k\in \mathcal K$ are symmetric, $p_k^i\in \mathbb R^n$, $i\in \mathcal N$, $k\in \mathcal K \backslash \{K\}$, and $d_k^i\in \mathbb R^s$, $L_k^i\in \mathbb R^{n\times s}$, $i\in \mathcal N,~k\in \mathcal K$.
Associated with NZDG1 we introduce the following optimal control problem
\begin{align}
\mathrm{OCP1}:\quad \quad& \min_{\tilde{\mathbf{u}},\tilde{\mathbf{v}}} J(x_0,\tilde{\mathbf{u}},\tilde{\mathbf{v}}),\label{eq:LQOCP1}\\
&\text{subject to \eqref{eq:LQStatedynamics} and \eqref{eq:LQconstraints}}\notag
\end{align}
\begin{align} J(x_0,\tilde{\mathbf{u}},\tilde{\mathbf{v}})&=\frac{1}{2} x_k ^{\prime}Q_K x_k +p_K^{\prime} x_k + \sum_{k=0}^{K-1}\left(\frac{1}{2} x_k ^{\prime}Q_k x_k +p^{\prime}_k x_k +\frac{1}{2}\mathbf{u}_k^{\prime}R_k \mathbf{u}_k\right)\nonumber\\& +\sum_{k=0}^{K}\left(\frac{1}{2}\mathbf{v}_k^{\prime}D_k\mathbf{v}_k+d^{\prime}_k\mathbf{v}_k+ x_k ^{\prime}L_k\mathbf{v}_k \right)
\label{eq:OCP1obj}
\end{align}
with $Q_k\in \mathbb{R}^{n \times n}$, $d_k\in \mathbb R^s$, $p_k\in \mathbb R^n$, $D_k\in \mathbb R^{s\times s}$, $L_k\in \mathbb R^{n\times s}$,
$i\in \mathcal N$, $k\in \mathcal K$, and $R_k\in \mathbb R^{m \times m}$, $i\in \mathcal N$, $k\in \mathcal K \backslash \{K\}$.
The admissible action sets $\{U_k^i,~k\in \mathcal K\backslash \{K\},~i\in \mathcal N\}$, are such that the sets of state vectors $\{X_k,~k\in \mathcal K\}$, obtained from (<ref>), are convex, and feasible action sets $\{V^i_k( x_k ,v_k^{-i})=\{v_k^i\in \mathbb R^{s_i}~|~ M_kx_k+N_k \bv_k+r_k\geq 0,~ \mathbf{v}_k\geq 0\},~\forall x_k\in X_k\}$ are non-empty, convex and bounded for all $k\in \mathcal K$, $i\in \mathcal N$.
In the next theorem we provide conditions under which NZDG1 is an open-loop dynamic potential game, and using Theorem <ref> we construct the potential functions associated optimal control problem (OCP1).
Let Assumption <ref> holds true. Let the parameters associated with NZDG1 satisfy the following conditions
\begin{align}
&[R_{k}^i]_{ij}=[R_k^j]_{ij}, ~i,j\in \mathcal N,~i\neq j, ~k\in \mathcal K\backslash \{K\}, \label{eq:Ri Rj}\\
&Q_k^i=Q_k^j,~p_k^i=p_k^j,~L_k^i=L_k^j,[D_k^i]_{ij}=[D_k^j]_{ij}, ~i,j\in \mathcal N,~i\neq j, ~k\in \mathcal K, \label{eq:Di Dj}
\end{align}
then NZDG is a OLDPG. Further, the OCP associated with the related OLDPG is described by
\begin{align} \label{eq:PF payoff1}
~i\in \mathcal N,~k\in \mathcal K,\\
&[R_k]_{i\bullet}=[R_k^i]_{i\bullet},~i\in \mathcal N,~k\in \mathcal K\backslash \{K\}. \label{eq:PF payoff2}
\end{align}
We first consider the conditions in (<ref>) and verify the condition (<ref>),
\begin{align*}
\frac{\partial g_k^i}{\partial x_k} = Q^i_kx_k+p^i_k+L^i_k\mathbf{v}_k = Q^j_kx_k+p^j_k+L^j_k\mathbf{v}_k = \frac{\partial g_k^j}{\partial x_k},~k\in \mathcal K.
\end{align*}
$Q_k^i=Q_k^j$, $p_k^i=p_k^j$ and $L_k^i=L_k^j, \forall i,j \in \mathcal{N}$, the condition (<ref>) holds true. Next, by using (<ref>) and (<ref>) and the symmetric structure of $R_k^i$ and $D_k^i$, we verify the conditions (<ref>) and (<ref>) as follows
\begin{align*}
&\frac{\partial^2 g_k^i}{\partial (u_k^j)^\prime \partial u_k^i} = [R^i_k]_{ij} = [R^j_k]_{ij} =\left([R^j_k]_{ji}\right)^\prime = \left(\frac{\partial^2 g_k^j}{\partial (u_k^i)^\prime \partial u_k^j}\right)^\prime,~k \in \mathcal{K}\backslash\{K\}, \\
&\frac{\partial^2 g_k^i}{\partial (v_k^j)^\prime \partial v_k^i} = [D^i_k]_{ij} = [D^j_k]_{ij} =\left([D^j_k]_{ji}\right)^\prime =\left(\frac{\partial^2 g_k^j}{\partial (v_k^i)^\prime \partial v_k^j}\right)^\prime,~k \in \mathcal{K}.
\end{align*}
So, NZDG1 is a OLPDG. Next, we proceed to construct the potential functions associated with OLPDG.
The gradient vector field $F_k=\begin{bmatrix} {F_k^x}^\prime& {F_k^u}^\prime& {F_k^v}^\prime \end{bmatrix}^\prime$ is calculated as
\begin{align*}
&F_k^x= Q_k^1x_k+p_k^1+L_k^1\mathbf{v}_k,~ F_k^u= \begin{bmatrix}
\vdots\\
\end{bmatrix} \bu_k,~F_k^v= \begin{bmatrix}
\vdots\\
\end{bmatrix} \bv_k + \begin{bmatrix}[d_k^1]_1\\ [d_k^2]_2\\ \vdots \\ [d_k^N]_N \end{bmatrix}
[L^1_k]^\prime_{\bullet 1}\\
[L^2_k]_{\bullet 2}\\
\vdots\\
[L^N_k]^\prime_{\bullet N}
\end{bmatrix}x_k.
\end{align*}
Since, $F_k$ is conservative, the instantaneous potential function (<ref>) evaluated as a line integral, along an arbitrary piecewise path in $\Omega_k$, depends only on the initial and final points. We consider a straight line connecting the origin in $\Omega_k$ and an arbitrary point $(x_k,\bu_k,\bv_k)\in \Omega_k$, and the associated bijective
parametrization of this line is given by $\alpha_k(z)=z\begin{bmatrix}x_k^\prime & \bu_k^\prime & \bv_k^\prime \end{bmatrix}^\prime$ with $z\in [0, 1]$. The instantaneous potential function is computed as
\begin{align}
P_k(x_k,\bu_k,\bv_k)&= c_k+\int_0^1 F_k (\alpha_k(z)) \bigcdot \frac{d \alpha_k(z)}{dz}~dz \notag \\
&=c_k+\int_0^1 \left(x_k^\prime ~F_k^x(\alpha_k(z)) + \bu_k^\prime~F_k^u(\alpha_k(z))+\bv_k^\prime~ F_k^v(\alpha_k(z))\right) dz.
\label{eq:Pkeq}
\end{align}
We define matrices $R_k$, $D_k$, $Q_k$, $L_k$, $p_k$ and $d_k$ such that $[R_k]_{i\bullet}=[R_k^i]_{i\bullet}$,
$[D_k]_{i\bullet}=[D_k^i]_{i\bullet}$, $Q_k=Q_k^i$, $[L_k]_{\bullet i}=[L_k^i]_{\bullet i}$, $p_k=p_k^i$, and $[d_k]_i=[d_k^i]_i$ for all $i\in \mathcal N$, $k\in \mathcal K\backslash \{K\}$. Then, from (<ref>) it follows that $R_k$ and $D_k$ are symmetric matrices and $Q_k=Q_k^i=Q_k^j$, $L_k=L_k^i=L_k^j$, $p_k=p_k^i=p_k^j$ for all $i,j\in \mathcal N$, $k\in \mathcal K \backslash \{K\}$. Using this (<ref>) can be written as
\begin{multline}
P_k(x_k,\bu_k,\bv_k)=c_k+ \int_0^1 \left(x_k^\prime \left[Q_k (zx_k)+p_k+L_k (z\bv_k)\right]+\bu_k^\prime R_k (z\bu_k)+
\bv_k^\prime \left[ D_k (z\bv_k) + d_k + L_k (zx_k)\right] \right) dz \\
=c_k+ \frac{1}{2}x_k^\prime Q_k x_k+p_k^\prime x_k +\frac{1}{2} \bu_k^\prime R_k \bu_k +\frac{1}{2} \bv_k^\prime D_k \bv_k + d_k^\prime \bv_k + x_k^\prime L_k \bv_k.
\label{eq:potfn1}
\end{multline}
Similarly, the terminal vector field $F_K=\begin{bmatrix}{F_K^x}^\prime & {F_K^v}^\prime \end{bmatrix}^\prime$ is calculated as
\begin{align*}
F_K^x= Q_K^1x_K+p_K^1+L_K^1\mathbf{v}_K,~
F_K^v= \begin{bmatrix}
\vdots\\
\end{bmatrix} \bv_K + \begin{bmatrix}[d_K^1]_1\\ [d_K^2]_2\\ \vdots \\ [d_K^N]_N \end{bmatrix}
[L^1_K]^\prime_{\bullet 1}\\
[L^2_K]_{\bullet 2}\\
\vdots\\
[L^N_K]^\prime_{\bullet N}
\end{bmatrix}x_K.
\end{align*}
We define matrices $D_K$, $Q_K$, $L_K$, $p_K$ and $d_K$ such that
$[D_K]_{i\bullet}=[D_K^i]_{i\bullet}$, $Q_K=Q_K^i$, $[L_K]_{\bullet i}=[L_K^i]_{\bullet i}$, $p_K=p_K^i$, and $[d_K]_i=[d_K^i]_i$ for all $i\in \mathcal N$. Then, from (<ref>) it follows that $D_K$ is a symmetric matrix and $Q_K=Q_K^i=Q_K^j$, $L_K=L_K^i=L_K^j$, $p_K=p_K^i=p_K^j$ for all $i,j\in \mathcal N$. Using this, and using the same procedure as before, the terminal potential function (<ref>) is calculated as
\begin{align}
P_K(x_K,\bv_K)&=c_K+\int_0^1 \left(x_K^\prime F_K^x(\alpha_K(z))\bv_K^\prime F_K^v(\alpha_K(z)) \right) dz \notag\\
&=c_K+\frac{1}{2}x_K ^\prime Q_K x_K+ p_K^\prime x_K +\frac{1}{2}\bv_K^\prime D_K \bv_K + x_K^\prime L_K \bv_K+ d_K^\prime \bv_K.
\label{eq:potfn2}
\end{align}
The instantaneous and terminal potential function given by (<ref>) and (<ref>), respectively, constitute the objective function (<ref>) associated with the OCP1. The parameters associcated with this objective function satisfy the conditions (<ref>).
§.§ Computation of open-loop Nash equilibrium associated with OLPDG
In this section, under a few assumptions on the parameters we transform the necessary conditions associated with OCP1 to a large-scale linear complementarity problem, there by providing a way to compute the open-loop Nash equilibrium.
Let $(\mathbf{\tilde{u}^{*},\tilde{v}^{*}})$ be the optimal solution of the OCP1 and $\{x_k ^*,~k\in \mathcal K\}$ be the state trajectory generated by $\mathbf{\tilde{u}^{*}}$. The necessary conditions of optimality of
are then given by
\begin{align}
\text{for }& k\in \mathcal K\backslash \{K\} \notag \\
\label{eq:LQOCP Cond1} & R_k\mathbf{u}_k^*+ \mathbf{B}_k^\prime\lambda^*_{k+1} = 0,~ \mathbf{B}_k=\begin{bmatrix}B_k^1 & B_k^1 & \cdots & B_k^N \end{bmatrix}, \\
\label{eq:LQOCP Cond2} & \mathbf{x}_{k+1}^{*} = A_k x_k ^*+\sum_{l \in \mathcal N}B_k^lu_k^{l^*}, \quad x_{0}~ \text{is given},\\
\label{eq:LQOCP Cond3} & \lambda_k^* = Q_k x_k ^{*}+p_k+L_k\mathbf{v}_k^{*}+ A_k^\prime \lambda_{k+1}^{*}-M_k^{\prime}\mu_k^{*},\\
\label{eq:LQOCP Cond4} & \lambda_K^{*} = Q_K x_k ^*+p_K+L_K\mathbf{v}_K^*-M_K^{\prime}\mu_K^*,\\
\text{for }& k\in \mathcal K \notag \\
\label{eq:LQOCP Cond5} & 0 \leq \left( D_k\mathbf{v}_k^{*}+d_k+L_k^\prime x_k ^* - N_k^\prime\mu_k^*\right) \perp v_k^* \geq 0, \\
\label{eq:LQOCP Cond7} & 0 \leq \left( M_k x_k ^*+ N_k\mathbf{v}_k^*+r\right) \perp \mu_k^* \geq 0.
\end{align}
The above set of necessary conditions lead to a weakly coupled system of a parametric two-point boundary value problem (<ref>)-(<ref>) and a parametric linear complimentarity problem
(<ref>)-(<ref>). We have the following assumption.
The co-state variable $\lambda_k$ is assumed to be affine in the state variable $ x_k $ for $k \in \mathcal K$ i.e., $\lambda_k^* = H_k x_k ^*+\beta_k$ where $H_k \in \mathbb{R}^{n \times n}$ and $\beta_k \in \mathbb{R}^{n \times 1}$.
Using Assumption <ref> it can be shown that the two-point boundary value problem (<ref>)-(<ref>) can be solved if the following backward equations for $k \in \mathcal K\backslash \{K\}$ and $i \in \mathcal N$ has a solution
\begin{align}
&\label{eq: Backward1} \Gamma_{k+1}^{k} = \mathbf{I}+S_k H_{k+1}, \\
& \label{eq: Backward2} H_{k}=Q_k+A_k^\prime H_{k+1}(\Gamma^{k}_{k+1})^{\mbox{-}1}A_k,\\
&\label{eq: Backward3} \beta_k =p_k-M^{\prime}\mu_k^*+L_k\mathbf{v}_k^*+A_k^{\prime}\beta_{k+1}-A_k^{\prime}H_{k+1}(\Gamma_{k
\end{align}
where $S_k = \mathbf{B}_kR_k^{\mbox{-}1}\mathbf{B}_k^{\prime}$, $H_K = Q_K$ and $\beta_K = p_K+L_K\mathbf{v}_K^*-M_K^{\prime}\mu_K^*$. Assuming $\Gamma_{k+1}^{k}$ to be invertible for $k = K-1,\dots,1$, we obtain $H_k$ and $\beta_k$ for $k = K-1,\dots,0$, and the state vector $ x_k ^*$ and the joint control vector $\mathbf{u}_k^*$ are given by
\begin{align}
\label{eq:State DE} \mathbf{x}_{k+1}^* & = \left(\Gamma^k_{k+1} \right)^{\mbox{-}1}\left(A_k x_k ^*-S_k\beta_{k+1} \right), ~k \in \mathcal K\backslash \{K\}, \\
\label{eq:decision Eq} \mathbf{u}_k^* & = -R_k^{\mbox{-}1}\mathbf{B}_k^{\prime}\left(H_{k+1}\mathbf{x}_{k+1}^{*}+\beta_{k+1} \right).
\end{align}
Suppose that the set of backward equations (<ref>)-(<ref>) admits a solution, i.e., the matrix $\Gamma^k_{k+1}$ is invertible for all $k \in \mathcal K \backslash \{K\}$, then the two point boundary value problem in (<ref>)-(<ref>) has a unique solution. To show this, let $\bar{\lambda_k} = \lambda_k^* - (H_k x_k ^*+\beta_k)$ be another solution for the two point boundary value problem (<ref>)-(<ref>). Substituting $\lambda_k^* = \bar{\lambda}_k+ H_k x_k ^*+\beta_k$ in (<ref>) and (<ref>), we get a decoupled system of equations as
\begin{align}
\mathbf{x}_{k+1}& = (\Gamma^k_{k+1})^{\mbox{-}1}\left(A_k\mathbf{x}_{k}^*-S_k\bar{\lambda}_{k+1}-S_k\beta_{k+1} \right),\\
\bar{\lambda}_{k} & = A_k^{\prime}\bar{\lambda}_{k+1}-A_k^{\prime}H_{k+1}(\Gamma^k_{k+1})^{\mbox{-}1}S_k\bar{\lambda}_{k+1}.
\end{align}
From the terminal condition, $\bar{\lambda}_{K}=0$ which results in $\bar{\lambda}_{k}=0 ~\forall k \in \mathcal K$. This proves that the solution for the two point boundary value problem described by (<ref>)-(<ref>) is unique.
In view of Remark <ref> we have the following standing assumption in the remaining part of the paper.
The set of matrices $\{\Gamma^k_{k+1},~ k \in \mathcal K\backslash \{K\}\}$ are invertible.
Next, we note that the backward equations (<ref>) and (<ref>) are coupled but evolve independently of (<ref>). Taking $G_{k+1} = A_k^{\prime}-A_k^{\prime}H_{k+1}(\Gamma_{k
+1})^{\mbox{-}1}S_k$, (<ref>) can be represented in the vector form as follows
\begin{align} \label{eq:Beta DE}
\beta_k = \sum_{\tau = k}^{K}\psi(k,\tau)\left( p_\tau+\begin{bmatrix}L_\tau &-M_\tau^{\prime}\end{bmatrix} \begin{bmatrix}\mathbf{v}_\tau^{*^\prime} & \mu_\tau^{*^\prime}\end{bmatrix}^{\prime}\right),~\forall k \in \mathcal K,
\end{align}
where the transition matrix associated with (<ref>) is $\psi(k,\tau)=G_{k+1}\cdots G_{\tau-1}G_{\tau}$, if $\tau > k$ and $\psi(k,k) = \mathbf{I}$. Thus, (<ref>) represents a parametric linear backward difference equation parametrized by $\{\mathbf{v}_k^*,\mu_k^*,~k \in \mathcal K, ~i \in \mathcal N\}$. Now, we analyse the forward difference equation for the state trajectory (<ref>). Suppose
\begin{align*}
\mathbf{x}_{k+1} = \bar{A}_{k} x_k ^*+\bar{B}_{k}\beta_{k+1}, \quad x_0~ \text{is given},
\end{align*}
where $\bar{A}_k = (\Gamma^k_{k+1})^{\mbox{-}1}A_k$ and $\bar{B}_k = -(\Gamma^k_{k+1})^{\mbox{-}1}S_k$ $\forall k \in \mathcal K\backslash\{K\}$. Denoting the transition matrix as $\phi(\rho,k) = \bar{A}_{k-1}\bar{A}_{k-2}\cdots \bar{A}_{\rho}$ for $\rho < k$ and $\phi(k,k) = \mathbf{I}$, and from (<ref>) we have
for $k \in \mathcal K\backslash\{0\}$,
\begin{multline}
x_k ^* = \phi(0,k)x_0+\sum_{\tau= 1}^{K}\left(\left(\sum_{\rho=1}^{\operatorname{min}(k,\tau)}\phi(\rho,k)\bar{B}_{\rho-1}\psi(\rho,\tau) \right)\left( p_\tau+\begin{bmatrix}L_\tau &-M_\tau^{\prime}\end{bmatrix} \begin{bmatrix}\mathbf{v}_\tau^{*^\prime} & \mu_\tau^{*^\prime}\end{bmatrix}^{\prime}\right)\right).
\label{eq:State ForwardEq2}
\end{multline}
Further, we combine the variables in (<ref>) as $p_{\mathbf{K}} = \begin{bmatrix}p_{1}^{\prime}& \hdots & p_{K}^{\prime} \end{bmatrix}^{\prime} $, $~\mathbf{x}_{\mathbf{K}}^* = \begin{bmatrix}\mathbf{x}_{1}^{*^\prime}& \hdots ~ \mathbf{x}_{K}^{*^\prime} \end{bmatrix}^{\prime} $ and $y_{\mathbf{K}}^{*} =\left[\mathbf{v}_{1}^{*^\prime}~\mu_1^{*^\prime}\right.$
$\left.\hdots ~ \mathbf{v}_{K}^{*^\prime}~\mu_K^{*^\prime} \right]^{\prime}$. As a result, (<ref>) can be written as:
\begin{align}\label{eq:State ForwardEq3}
{x}_{\mathbf{K}}^* =\mathbf{\Phi}_{0}x_0+\mathbf{\Phi}_1p_{\mathbf{K}}+\mathbf{\Phi}_2y_{\mathbf{K}}^*,
\end{align}
where $[\mathbf{\Phi}_0]_k = \phi(0,k),~ [\mathbf{\Phi}_1]_{k\tau} =\sum_{\rho=1}^{\operatorname{min}(k,\tau)}\phi(\rho,k)\bar{B}_{\rho-1}\psi(\rho,\tau) $ and $[\mathbf{\Phi}_2]_{k\tau} = \sum_{\rho=1}^{\operatorname{min}(k,\tau)}\phi(\rho,k)\bar{B}_{\rho-1}\psi(\rho,\tau)\\ \begin{bmatrix}L_\tau &-M_\tau^{\prime}\end{bmatrix}$ for $k,\tau \in \mathcal K\backslash \{0\}.$ Thus, we expressed the state trajectory paramterized with $\{\mathbf{v}_k^*,\mu_k^*,~k \in \mathcal K\backslash \{0\}\}$. Next, we analyse the parametric linear complimentarity problem in (<ref>)-(<ref>) in detail. First, the vector representation of these problems is given by
\begin{align}\label{eq:pLCP1}
\mathrm{pLCP}( x_k ^*):~\begin{bmatrix}D_k & -N_k^{\prime} \\ N_k & 0 \end{bmatrix}\begin{bmatrix}\mathbf{v}_k^* \\ \mu_k^* \end{bmatrix}+\begin{bmatrix}L_k^{\prime} \\ M_k\end{bmatrix} x_k ^* + \begin{bmatrix}d_k \\ r_k \end{bmatrix} \perp \begin{bmatrix}\mathbf{v}_k^* \\ \mu_k^* \end{bmatrix}.
\end{align}
Let $\tilde{\mathbf{M}} = \begin{bmatrix}D_1 & -N_1^{\prime} \\ N_1 & 0 \end{bmatrix}\oplus \cdots \oplus \begin{bmatrix}D_K & -N_K^{\prime} \\ N_K & 0 \end{bmatrix} , \tilde{\mathbf{q}} =\begin{bmatrix}L_1^{\prime} \\ M_1\end{bmatrix} \oplus \cdots \oplus \begin{bmatrix}L_K^{\prime} \\ M_K\end{bmatrix}$ and $\tilde{\mathbf{s}} = \begin{bmatrix}d_1^\prime &r_1^\prime &\cdots &d_K^\prime& r_K^\prime \end{bmatrix}^\prime$.
Aggregating these parametric problems for all time steps $k \in \mathcal K \backslash \{0\}$, we obtain a single parametric linear complementarity problem as
\begin{align}\label{eq:pLCP2}
\mathrm{pLCP(\mathbf{x}_\mathbf{K}^*}):\quad \tilde{\mathbf{M}}y_{\mathbf{K}}^*+\tilde{\mathbf{q}}\mathbf{x}_{\mathbf{K}}^* + \tilde{\mathbf{s}} \perp y_{\mathbf{K}}^*.
\end{align}
Substituting (<ref>) in (<ref>) results in the following large scale linear complimentarity problem:
\begin{align}\label{eq:LCP1}
\mathrm{LCP}(x_0):\quad \mathbf{M}y_{\mathbf{K}}^*+\mathbf{q} \perp y_{\mathbf{K}}^*,
\end{align}
where $\mathbf{M} = \tilde{\mathbf{M}} + \tilde{\mathbf{q}}\mathbf{\Phi}_2$ and $\mathbf{q} =\tilde{\mathbf{q}}(\mathbf{\Phi}_{0}x_0+\mathbf{\Phi}_1p_{\mathbf{K}})+\tilde{\mathbf{s}}$.
Thus, we formulated the necessary conditions (<ref>)-(<ref>) as a single large scale linear complimentarity problem with the aid of Assumptions <ref> and <ref>. Solving (<ref>) we obtain candidate optimal solution of OCP1 there by a candidate open-loop Nash equilibrium of NZDG1.
Next, we study sufficient conditions under which the solution of $\mathrm{LCP}(x_0)$ and $\mathrm{pLCP}(x_0)$ indeed minimize OCP1, and as a result provide an open-loop Nash equilibrium of NZDG1. The sufficient conditions provided in Theorem <ref> requires both the minimized instantaneous and terminal Lagrangian functions to be convex in the state variables. Since the solutions $\bv_k^*$ are obtained from solving the parametric linear complementarity problem (<ref>), upon substitution, the minimized Lagrangian functions may not be convex in the state variables. So, to derive the required sufficient conditions we transform the OCP1 as static optimization problem in the decision variables $(\tilde{\bu},\tilde{\bv})$.
Toward this end, the objective function associated with the OCP1 can be written as
\begin{align}
J(x_0,\tilde{\mathbf{u}},\tilde{\mathbf{v}})&=\sum_{k=0}^{K-1}\frac{1}{2}\bu_k^\prime R_k \bu_k +\sum_{k=0}^K\frac{1}{2}\left( x_k ^{\prime}Q_k x_k +\left(p_k+L_k\bv_k^{*}
-M_k^\prime \mu_k^*\right)^\prime x_k \right) \notag\\
&+\sum_{k=0}^{K}\left(\frac{1}{2}\mathbf{v}_k^{\prime}D_k\mathbf{v}_k+d^{\prime}_k\mathbf{v}_k+ x_k ^{\prime}L_k (\bv_k-\bv_k^*)+{\mu_k^*}^\prime M_k x_k \right),
\label{eq:objOCP}
\end{align}
where $\{(\mathbf{v}_k^*,\mu_k^*),~k \in \mathcal K\}$ is the solution of $\mathrm{LCP}(x_0)$ and $\mathrm{pLCP}(x_0)$.
We define value function $W_k,~k\in \mathcal K$ as
\begin{align*}
& W_k=\frac{1}{2}x_k^\prime E_k x_k + e_k^\prime x_k + w_k,
\end{align*}
where $E_k \in \mathbb{R}^{n \times n},~e_k \in \mathbb{R}^n,~w_k \in \mathbb{R}$ and $i \in \mathcal{N}$. Let the matrices $T_k:=R_k+ \mathbf{B}_k^\prime E_{k+1}\mathbf{B}_k$ be invertible for all $k\in \mathcal K\backslash \{K\}$ with the matrices $E_k,~ k\in \mathcal K$ computed as the solution of the following backward Ricatti difference equations
\begin{align}
&E_k= A_k^\prime E_{k+1} A_k +Q_k -A_k^\prime E_{k+1}\mathbf{B}_k T_k^{-1} \mathbf{B}_k^\prime E_{k+1}A_k,~ E_K=Q_K,\label{eq:LQDGSC}\\
&e_k=A_k^\prime e_{k+1}-A_k^\prime E_{k+1}\mathbf{B}_k T_k ^{-1}\mathbf{B}_k^\prime e_{k+1}+p_k+L_k\bv_k^* -M_k^\prime \mu_k^*,~e_K=p_K+L_K\bv_K^*-M_K^\prime \mu_K^*, \label{eq:lqdge}\\
&w_k=w_{k+1}-e^\prime_{k+1} \mathbf{B}_kT_k^{-1}\mathbf{B}_k^\prime e_{k+1},~w_K=0.
\end{align}
Then, by denoting $\Delta_k = W_{k+1}-W_{k}$, and using the sum $\sum_{K=0}^{K-1}\Delta_k$ and (<ref>) we can write the objective function (<ref>) as
\begin{align}
J(x_0,\tilde{\mathbf{u}},\tilde{\mathbf{v}})& = W_0 +\sum_{k=0}^{K-1} \frac{1}{2}||\mathbf{u}_k +T_k^{-1} \mathbf{B}_k^\prime\left( E_{k+1}A_kx_k+ e_{k+1}\right) ||^2_{T_k}\notag \\
&+\sum_{k=0}^{K}\left(\frac{1}{2}\mathbf{v}_k^{\prime}D_k\mathbf{v}_k+d^{\prime}_k\mathbf{v}_k+ x_k ^{\prime}L_k (\bv_k-\bv_k^*)+{\mu_k^*}^\prime M_k x_k \right).\label{eq:static}
\end{align}
The next lemma relates how the candidate optimal control (<ref>) obtained by solving the two-point boundary value problem (<ref>)-(<ref>) is related to the minimizer of the objective function (<ref>).
Let the set of matrices $\{T_k,~k\in \mathcal K\backslash \{K\}\}$ be invertible and
the solutions $E_k$ of the symmetric matrix Riccati difference equation
(<ref>) exist for all $k\in \mathcal K$. If the two-point boundary value problem
\begin{align}
\bar{\lambda}_k &= A^\prime_k \bar{\lambda}_{k+1}+Q_k \bar{x}_k+p_k+L_k\mathbf{v}_k^{*} -M^{\prime}_k\mu_k^{*},\label{eq:lbareq1}\\ \bar{\lambda}_K &= Q_K x_k+p_K+L\mathbf{v}_K^*-M_K^{\prime}\mu_K^*,\label{eq:lbareq2}\\
\bar{x}_{k+1}&=A_k \bar{x}_k - \mathbf{B}_kR_k^{-1} \mathbf{B}_k^\prime \bar{\lambda}_{k+1},~\bar{x}_0=x_0
\end{align}
has a unique solution, then we set $\bar{\bu}_k=-{R}^{-1}_k \mathbf{B}_k^\prime \bar{\lambda}_{k+1}$ and $\bar{e}_k:=\bar{\lambda}_k-E_k \bar{x}_k$. Then the sequences
$\{\bar{x}_k, \bar{e}_k\}$ solve equations $\bar{\mathbf{u}}_k+T_k^{-1}\mathbf{B}_k^\prime \left(E_{k+1}A_k \bar{x}_k+\bar{e}_{k+1}\right)=0$ and (<ref>).
Firstly, we have
\begin{align*}
\bar{\bu}_k+T_k^{-1} \mathbf{B}_k^\prime\left( E_{k+1} A_k \bar{x}_k+\bar{e}_{k+1}\right) &=-R_k^{-1} \mathbf{B}_k^\prime\bar{\lambda}_{k+1}+
T_k^{-1}\mathbf{B}_k^\prime E_{k+1}
\left(\bar{x}_{k+1}+\mathbf{B}_k R_k^{-1}\mathbf{B}_k^\prime \bar{\lambda}_{k+1}\right)+T_k^{-1}\mathbf{B}_k^\prime \bar{e}_{k+1}\\
&= \left(-R_k^{-1}+T_k^{-1}\left(
R_k+\mathbf{B}_k ^\prime E_{k+1}\mathbf{B}_k\right)R_k^{-1} \right) \mathbf{B}_k^\prime \bar{\lambda}_{k+1}=0.
\end{align*}
To prove (<ref>) we have
\begin{align*}
& A^\prime_k \bar{e}_{k+1}-A^\prime_k E_{k+1}\mathbf{B}_k T_k^{-1}\mathbf{B}_k^\prime\bar{e}_{k+1}+p_k+L_k\bv_k^*-M_k^\prime \mu_k^*-\bar{e}_k \\ & \hspace{1.5in} =p_k+L_k\bv_k^*-M_k^\prime \mu_k^*+A_k^\prime \bar{\lambda}_{k+1}-\bar{e}_{k}-A_k^\prime E_{k+1} \mathbf{B}_k T_{k}^{-1}\mathbf{B}_k^\prime \bar{\lambda}_{k+1}\\& \hspace{1.5in}+
A_k^\prime E_{k+1}\left(\mathbf{B}_k T_k^{-1}\mathbf{B}_k^\prime E_{k+1}-\mathbf{I}\right) \left(A_k\bar{x}_k-\mathbf{B}_k R_k^{-1}\mathbf{B}_k^\prime \bar{\lambda}_{k+1}\right)\\ &\hspace{1.5in}
=\left(p_k+L_k\bv_k^*-M_k^\prime \mu_k^*\right)+A_k^\prime \bar{\lambda}_{k+1}+Q_kx_k-\bar{\lambda}_k \\
&\hspace{1.5in}-\left(A_k^\prime E_{k+1}A_k-A_k^\prime E_{k+1}
\mathbf{B}_k T_k^{-1} \mathbf{B}_k^\prime E_{k+1}A_k+Q_k-E_k \right)x_k\\
&\hspace{1.5in}+A_k^\prime E_{k+1}B_k \left(R_k^{-1}-T_k^{-1}
-T_k^{-1}\mathbf{B}_k^\prime E_{k+1} \mathbf{B}_k R_k^{-1}\right) B_k^\prime \bar{\lambda}_{k+1}=0.
\end{align*}
The last step in the above expression is obtained by using (<ref>) and (<ref>),(<ref>).
In the following lemma we provide conditions under which the objective function (<ref>) is a strictly convex function of $(\tilde{\bu},\tilde{\bv})$.
Let the solution $E_k$ of the symmetric Ricatti equation (<ref>) exist. Let
\begin{align}
\Upsilon_{k}^{\tau} =
\begin{cases}
\mathbf{B}_\tau E_{\tau+1} A_{\tau}A_{\tau-1}\dots A_{k+1} &0 \leq k < K-1 \\
\mathbf{0}& k = K-1
\end{cases}
\end{align}
Now, we define the matrix
\begin{align}
\mathbf{H}&=\begin{bmatrix} \mathbf{Y} & \mathbf{C}^\prime \\ \mathbf{C} & \mathbf{D} \end{bmatrix},
\label{eq:Hessian}
\end{align}
\begin{multline*}
~\mathbf{D}=\oplus_{k=0}^{K} D_k,\quad
{\mathbf{C} = \begin{bmatrix}
\mathbf{0} & \mathbf{B}_0^\prime L_1 & \mathbf{B}_0^\prime A_1^\prime L_2 & \hdots & \mathbf{B}_0^\prime A_{1}^\prime \cdots A_{K-2}^\prime A_{K-1}^\prime L_K\\ \mathbf{0} & \mathbf{0} & \mathbf{ B}_1^\prime L_2 & \hdots & \mathbf{B}_1^\prime A_{2}^\prime \cdots A_{K-2}^\prime A_{K-1}^\prime L_K \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \hdots & \mathbf{B}_2^\prime A_{3}^\prime \cdots A_{K-2}^\prime A_{K-1}^\prime L_K \\
\vdots & \vdots & \vdots & \mathbf{0} & \ddots & \vdots \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \hdots & \mathbf{B}_{K-1} ^\prime L_K
\end{bmatrix}}
\end{multline*}
and $\mathbf{Y}$ is a $Km \times Km$ matrix where each block submatrix $[Y]_{lk}$ is a $m \times m$ matrix such that for $k \in \mathcal{K}\backslash \{K\}$
\begin{align*}
[Y]_{lk}& = \begin{cases} \mathbf{B}_k^\prime E_{k+1}A_k\dots A_{l+1}\mathbf{B}_l+ \mathbf{B}_k^\prime \left( \sum_{\tau = k+1}^{K-1}( \Upsilon_{k}^{\tau})^\prime T_{
\tau}^{-1}( \Upsilon_{k}^{\tau}) A_k\dots A_{l+1}\right)\mathbf{B}_{l},~0 \leq l < k, \\
T_k + \mathbf{B}_k^\prime \left[\sum_{\tau = k+1}^{K-1}( \Upsilon_{k}^{\tau})^\prime T_{
\tau}^{-1}( \Upsilon_{k}^{\tau}) \right]\mathbf{B}_k,~ l = k,
\end{cases} \\ [Y]_{kl}& = [Y]_{lk}^\prime.
\end{align*}
If the matrix $\mathbf{H}$ is positive definite then the objective function (<ref>) is
a strictly convex function of $(\tilde{\bu},\tilde{\bv})$.
We compute the Hessian matrix of the objective function (<ref>) with respect the decision variables $(\tilde{\bu},\tilde{\bv})$ by eliminating the state variable $x_k$. Now, we have
$\frac{\partial^2 J}{\partial (\bv_k)^\prime \partial \bv_k}=D_k$, $\frac{\partial^2 J}{\partial (\bv_k)^\prime \partial v_l}=0$, for $k\neq l$, $k,l\in \mathcal K$. Calculating the remaining second order partial derivatives we get
\begin{align*}
\quad [Y]_{lk} &= \frac{\partial^2 J}{\partial (\bu_l)^\prime \partial \bu_k} = \mathbf{B}_k^\prime E_{k+1}A_k\dots A_{l+1}\mathbf{B}_l + \mathbf{B}_k^\prime \left[ \sum_{\tau = k+1}^{K-1}( \Upsilon_{k}^{\tau})^\prime T_{
\tau}^{-1}( \Upsilon_{k}^{\tau}) \right] A_k\dots A_{l+1}\mathbf{B}_{l},\; 0 \leq l < k, \\
[Y]_{kk}& = \frac{\partial^2 J}{\partial (\bu_k)^\prime \partial \bu_k} = T_k + \mathbf{B}_k^\prime \left[\sum_{\tau = k+1}^{K-1}( \Upsilon_{k}^{\tau})^\prime T_{
\tau}^{-1}( \Upsilon_{k}^{\tau}) \right]\mathbf{B}_k. \\
[\mathbf{C}]_{lk}& = \frac{\partial^2 J}{\partial (\mathbf{u}_l)^\prime \partial \mathbf{v}_k} = \begin{cases} \mathbf{B}_l^\prime A_{l+1}^\prime \cdots A_{k-1}^\prime L_{k},~ l < k-1 \\
\mathbf{B}_{l}^\prime L_{k},~ l = k-1~\text{and}\\
\mathbf{0},~ l \geq k
\end{cases}
\end{align*}
Then, if the Hessian matrix $\mathbf{H}$ is positive definite, the objective function (<ref>) is strictly convex in $(\tilde{\bu},\tilde{\bv})$.
Next, using the above result in the next theorem we show that the solutions of $\mathrm{LCP}(x_0)$ and
$\mathrm{pLCP}(x_0)$ indeed provide the optimal solution of the OCP1.
Let Assumptions <ref>, <ref> and
<ref> hold true. Let the Hessian matrix $\mathbf{H}$ given by
(<ref>) is positive definite. Then the solutions of $\mathrm{LCP}(x_0)$ and $\mathrm{pLCP}{(x_0)}$ constitute an open-loop Nash equilibrium of the NZDG1.
Let $\{(\mathbf{v}_k^*,\mu_k^*),~k \in \mathcal K\}$ be the solution of $\mathrm{LCP}(x_0)$ and $\mathrm{pLCP}(x_0)$.
Then transforming the objective function of the OCP1 as (<ref>) and consider the minimization
problem subject to the state dynamics $x_{k+1}=A_kx_k +\mathbf{B}_k \bu_k$ and the constraints $M_k x_k+N_k \bv_k+r_k \geq 0$, $\bv_k\geq 0$. Since $\mathbf{H}$ is positive definite, we have that the objective function $J(x_0,\tilde{\bu},\tilde{\bv})$ is strictly convex in $(\tilde{\bu},\tilde{\bv})$ from Lemma <ref>. Also, from Assumption <ref>, the sets $\{U_k^i,~k\in \mathcal K \backslash \{K\}\}$ and $\{V_k^i,~k\in \mathcal K\}$ for all $i\in \mathcal N$ are non-empty, convex and bounded. Therefore, by solving the KKT conditions, we obtain the solution of the static optimization. The Lagrangian associated with this optimization problem is given by
\begin{align}
\mathcal L= J(x_0,\tilde{\bu},\tilde{\bv})-\sum_{k=0}^{K}\mu_k^\prime \left(M_k x_k +N_k \bv_k+r_k\right).
\end{align}
The KKT conditions are then given by
\begin{align}
&T_k \left(\bu_k+T_k^{-1}\mathbf{B}_k^\prime (E_{k+1}A_kx_k+e_{k+1})\right)\notag+\mathbf{B}_k^\prime \sum_{\tau = k+1}^{K-1} \left(( \Upsilon_{k}^{\tau})^{\prime }\left(\bu_\tau + T_\tau^{-1} \mathbf{B}_\tau^\prime ( E_{\tau+1}A_\tau x_\tau +e_{\tau+1}\right)\right)\\& +\mathbf{B}_k^\prime \left(L_{k+1} (\bv_{k+1}-\bv_{k+1}^*)-M_{k+1}^\prime (\mu_{k+1}-\mu_{k+1}^*)
\right)+\sum_{\tau=k+1}^{K-1}\left(A_{\tau}A_{\tau-1}\cdots A_{k+1} \mathbf{B}_k\right)^\prime \left(
L_{\tau+1}(\bv_{\tau+1}-\bv_{\tau+1}^*)\right)\notag\\&-\sum_{\tau=k+1}^{K-1}\left(M^\prime_{\tau+1}(\mu_{\tau+1}-\mu_{\tau+1}^*)\right) =0, \label{eq:kkt1} \\
&0\leq D_k \bv_k -N_k^\prime \mu_k +L_k^\prime x_k +d_k \perp \bv_k\geq 0, \label{eq:kkt2}\\
&0\leq M_k x_k+N_k \bv_k +r_k \perp \mu_k \geq 0. \label{eq:kkt3}
\end{align}
Let $x_k^*,~k\in \mathcal K$ be the state trajectory generated by the solution $\{(\mathbf{v}_k^*,\mu_k^*),~k \in \mathcal K\}$ using (<ref>). Next, the
co-state defined by $\lambda_k^*=H_kx_k^*+\beta_k$ along with the state vector $x_k^*$ solve
the two point boundary value problem (<ref>). Then, from Lemma <ref>,
it follows that $ \bu_k^*+T_k^{-1}\mathbf{B}_k^\prime \left(E_{k+1} A_k x_k^*+e_{k+1}\right) =0$
for all $k\in \mathcal K \backslash \{K\}$. Using this in (<ref>)
we obtain
\begin{align*}
\mathbf{B}_k^\prime \left(L_{k+1} (\bv_{k+1}-\bv_{k+1}^*)-M_{k+1}^\prime (\mu_{k+1}-\mu_{k+1}^*)
\right)&+ \sum_{\tau=k+1}^{K-1}\left(A_{\tau}A_{\tau-1}\cdots A_{k+1} \mathbf{B}_k\right)^\prime (
L_{\tau+1}(\bv_{\tau+1}-\bv_{\tau+1}^*)\\&-\sum_{\tau=k+1}^{K-1} M^\prime_{\tau+1}(\mu_{\tau+1}-\mu_{\tau+1}^*)=0.
\end{align*}
The above equation and the remaining equations (<ref>) and (<ref>) are satisfied by $(\tilde{\bv}_k^*,\mu^*_k)$ as they are solutions of $\mathrm{pLCP}(x_k^*)$.
This implies, we have shown that the solutions are of the $\mathrm{LCP}(x_0)$ along with $\mathrm{pLCP}(x_0)$ indeed provide the optimal solution of the OCP1. Since the OCP1
is associated with OLDPG we have that $(\tilde{\bu}^*,\tilde{\bv}^*)$, obtained from solutions of $\mathrm{LCP}(x_0)$ and $\mathrm{pLCP}(x_0)$, provides
an open-loop Nash equilibrium of NZDG1.
§ ILLUSTRATION : SMART GRID SYSTEM WITH ENERGY STORAGE
To illustrate our results, we consider a smart grid system with energy storage. Smart grids provide opportunities for exploring distributed renewable energy sources. However, integrating solar and wind based sources has been a challenge in meeting the demand due to their intermittent nature. Energy storage systems become critical in providing continuous power in the case of interruption and are often used as an emergency power supply during unforeseen outages ( [Oh, 2011] ). Power utilities can cut their generation costs by storing energy during the off‐peak hours and releasing during the peak hours.
So, installing energy storage systems is crucial for an efficient, reliable and resilient smart grid, see [Kolokotsa et al., 2019]. However, setting up a centralized energy storage unit for all the smart-grid users is not practical due to high set up costs and maintenance. One alternative would be to incentivize prosumers to install energy storage units at their homes; see Figure <ref> for an illustration. In this way, the smart grid storage system becomes decentralized. Additionally, the storage for each user is limited by the total resources available in the grid as well as the battery capacity. In [Zazo et al., 2016], the authors study an energy demand problem in smart-grids without energy storage, and model the decision problem as a dynamic non-cooperative game without constraints.
We build upon the model studied in [Zazo et al., 2016] by incorporating energy storage incentives for the prosumers, and model the decision problem as a dynamic game with inequality constraints.
A smart grid system with energy storage
Consider a smart grid with $N$ users who utilize the smart grid resources for different activities like heating, lighting, and other home appliances. Let $\mathcal{N} = \{1,2,\cdots,N\}$ be the set of users and the total time period be $\mathcal{K} = \{1,2,\cdots,K\}$. The final time $K$ is determined by the uniform interval with which the electrical data is processed in the grid in a day. For instance, if the data is processed every two hours in a day, $K =12$. The smart grid consists of $S$ type of energy resources such as solar, hydroelectric, coal etc., and each user $i \in \mathcal{N}$ consumes energy for $m_i$ activities. All resources are shared by all users. The state of the game at each time step, $X_k \in \mathbb{R}^S$ is the total amount of consumable resources in the smart grid. In the smart grid, users act as prosumers, implying that they not only consume energy, but also contribute the excess energy produced by renewable resources back to the grid. Furthermore, the resources can be autonomously recharged. Therefore, state of the game is governed by the discrete time dynamics
\begin{align}
X_{k+1} = \tilde{A}_k X_k +\sum_{i=1}^{N}\tilde{B}_k^i I_k^i,~k \in \mathcal{K} \backslash \{K\},
\end{align}
where, at time step $k$, $\tilde{A}_k \in \mathbb{R}^{S \times S} $ governs the energy which is autonomously depleted or replenished, $I_k^i \in \mathbb{R}^{m_i} $ denotes the amount of resources consumed or contributed by user $i$ and $\tilde{B}_k^i \in \mathbb{R}^{S\times m_i}$ is the weight associated with the resource expenditure or contribution. The smart grid authority has provided a battery storage unit for each user as a secondary storage unit. This is provided for the purpose of island-ed mode, operation under unforeseen disconnection of the smart grid from the main power grid; see [Ray and Biswal, 2020]. The amount of resource stored in each player's battery unit at time instant $k$ is $K_k^i \geq 0 $. The total energy storage in batteries is limited to be $\epsilon_k >0$ units lower than the total resources of the grid, since excess of battery storage only results in higher storage costs. The limiting factor $\epsilon_k$ is chosen in such a way that the maximum storage limit represents the maximum energy required for the users to perform the essential activities during island-ed mode operation. Further, the amount of energy stored in the battery is limited by it's maximum capacity denoted by $K_{max}^i$. Therefore the constraints on the battery storage are given by
\begin{align}
& \sum_{i \in \mathcal{N}} K_k^i \leq \sum_{i \in S}X_k^i - \epsilon_k, \label{eq:totalstorage}\\
& 0 \leq K_k^i \leq K_{max}^i, ~ i \in \mathcal{N}.
\end{align}
The costs incurred by users are attributed to unsatisfied demand, unbalanced resources and battery storage. Each user has a target demand to meet which is given by $P_k^i X_k $, where $P_k^i \in \mathbb{R}^{m_i \times S}$ denotes the demand matrix. The cost associated with unsatisfied demand is characterized by the following quadratic function
\begin{align}
(\Pi_k^i)_{ud} = \frac{1}{2}\left(P_k^i X_k -I_k^i\right)^\prime \tilde{ R}_k^i\left(P_k^i X_k -I_k^i\right),~k \in \mathcal K \backslash \{K\},
\end{align}
where $\tilde{R}_k^i ={ r}_k^i \mathbf{I} \in \mathbb{R}^{m_i \times m_i}$ with $r_k^i>0$.
For higher values of the parameter $r_k^i$ the user prioritizes in minimizing the unsatisfied demand cost compared to other costs. Next, if the resources at a time step are higher than at the previous step, then there is a cost associated with storage. Similarly, there are also costs associated with productivity loss between consecutive time periods. These costs are modeled as
\begin{align}
(\Pi_k^i)_{ur} = \frac{1}{2} \left( X_k -{X}_{k-1}\right)^\prime \tilde{Q}_k\left(X_k -{X}_{k-1}\right),
\end{align}
where $\tilde{Q}_k = q_k \mathbf{I} \in \mathbb{R}^{S \times S}$ with $q_k^i>0$.
For higher values of the parameter $q_k^i$ the user prioritizes in keeping the resources at steady state levels without spikes. Each player incurs a battery storage cost which is given by
\begin{align}
(\Pi_k^i)_{bs} = \frac{1}{2} K_k^{i^2} b_k^i,
\end{align}
where $b_k^i$ represents the battery storage cost per energy unit for the user $i$. Along with these costs, there is an incentive provided to users for energy storage, which has two components: a player specific incentive which depends only on the player's battery resource, and a common incentive which depends on the total grid resource. The total incentive for Player $i$ is given by
\begin{align}
(\Pi_k^i)_{ic} = a_k^{i} K_k^i + X_k^\prime \tilde{L}_k\begin{bmatrix} {K}_k^1 & \hdots & K_k^N \end{bmatrix}^{\prime} ,
\end{align}
where the parameter ${a}_k^i$ reflects the player specific incentive and the parameter $\tilde{L}_k \in \mathbb{R}^{S \times N}$ the common incentive. The salvage cost incurred by Player $i$ for the amount of resources in the last stage $K$ which is given by
\begin{align}
\Pi^i_K = \frac{1}{2} X_K ^\prime \tilde{Q}_K X_K,~\text{with}~\tilde{Q}_K = q_K \mathbf{I} \in \mathbb{R}^{S \times S}.
\end{align}
Player $i$ using the consumption (or contribution) and storage schedules $\{I_k^i,~k\in \mathcal K \backslash \{K\},~K_k^i,~k\in \mathcal K\}$ seeks to minimize the total cost given by
\begin{align*}
J^i &= \Pi_K^i+\sum_{k=0}^{K-1} \left[(\Pi_k^i)_{ud} +(\Pi_k^i)_{ur}\right] +\sum_{k=0}^K \left[(\Pi_k^i)_{bs}-(\Pi_k^i)_{ic}\right]\\
\frac{1}{2}X_k ^\prime \tilde{Q}_K X_k + \sum_{k=0}^{K-1}\left(\frac{1}{2}\left(P^i X_k -I_k^i\right)^\prime \tilde{R}_k^i\left(P^i X_k -I_k^i\right)+\frac{1}{2}\left( X_k -{X}_{k-1}\right)^\prime \tilde{Q}_k\left( X_k -{X}_{k-1}\right)\right)\\ &+\sum_{k=0}^{K}\left(\frac{1}{2}K_k^{i^2} b_k^i - \left( a_k^iK_k^i +X_k ^\prime \tilde{L}_k^i \begin{bmatrix} {K}_k^1 & \hdots & K_k^N \end{bmatrix}^{\prime} \right)\right).
\end{align*}
We transform the above dynamic game problem with inequality constraints to the standard form NZDG1 as follows.
\begin{align*}
& x_k = \begin{bmatrix} X_k ^\prime & {X}_{k-1}^\prime\end{bmatrix}^\prime, ~\mathbf{v}_k = \begin{bmatrix}
K_k^{1} & \dots & K_k^{N}
\end{bmatrix}^\prime,\\ &{u}_k^i = P^i X_k - I_k^i,~ \mathbf{u}_k = \begin{bmatrix}
u_k^{1^\prime} & \dots & {u}_k^{N^\prime}
\end{bmatrix}^\prime.
\end{align*}
The smart grid resource allocation problem with energy storage is modeled as NZDG1 with parameters
defined as follows
\begin{align*}
&A_k \triangleq \begin{bmatrix} \tilde{A}_k+ \sum_{i = 1}^{N} \tilde{B}_k^i P^i & \mathbf{0}_{S \times S} \\ \mathbf{I}_S & \mathbf{0}_{S \times S}
\end{bmatrix}, \quad {B}_k^i \triangleq \begin{bmatrix}
-\tilde{B}_k^i \\ \mathbf{0}_{S \times m_i}\end{bmatrix}, \\& {Q}_K = \begin{bmatrix} 1 & 0 \\ 0 & 0
\end{bmatrix} \otimes \tilde{Q}_K ,\quad {Q}_k = \begin{bmatrix} 1 & -1 \\ -1 & 1
\end{bmatrix} \otimes \tilde{Q}_k , ~d_k^i = -a_k^i \mathbf{e}_i, \\ &D_k^i = b_k^i (\mathbf{e}_i \mathbf{e}_i^\prime),~R_k^i = \mathbf{0}_{m_1 \times m_1}\oplus \cdots \oplus \tilde{R}_k^i \oplus \mathbf{0}_{m_N \times m_N} ,\\
& M_k = \begin{bmatrix}
\mathbf{1}_{1 \times S} & \mathbf{0}_{1 \times S} \\ \mathbf{0}_{N \times S} & \mathbf{0}_{N \times S}
\end{bmatrix},~ N_k = -\begin{bmatrix} \mathbf{1}_{1 \times N}\\ \mathbf{I}_{N} \end{bmatrix}, \\&r_k=\begin{bmatrix} -\epsilon_k & K_{max}^1 & K_{max}^2 & \hdots & K_{max}^N \end{bmatrix}^\prime.
\end{align*}
Here, we observe that cost matrices $Q_k,~D_k^i,~d_k^i,~L_k,~k \in \mathcal{K}$ and $R_k^i,~k \in \mathcal{K} \backslash \{K\}$ satisfy the conditions in (<ref>). Therefore we can obtain the OCP1 associated with the related OLPDG by (<ref>) from Theorem <ref>.
For illustration purpose, we assume $K = 12$, that is, that the data is processed every two hours in a day. We assume $S=4$ resources, $N=2$ users and $m_{i} = 2$ for both the users. The remaining parameters are taken as follows
\begin{align*}
&q_K = 2.5 ,~q_k = 1 ,~ r_k^1 = r_k^2 = 0.7,~b_k^1 = b_k^2 = 1.6,\\ &a_k^1 = 3.4,~a_k^2 = 4,~\epsilon_k = 3.5,~ K_{max}^1 = 11.2,~ K_{max}^2 = 12.2,\\
&\tilde{ L}_k =0.5 *\mathbf{1}_{3 \times 2},~ \tilde{A}_k = \mathbf{I}_S,~
P_k^1 = P_k ^2 =0.375* \mathbf{1}_{2 \times 3},\\
&\tilde{B}_k^1 = \tilde{B}_k^2 = \begin{bmatrix} 0.75 & 0 \\ 0 & 0.375 \\ 0 & 0.75\end{bmatrix},~
X_0=\begin{bmatrix}4 \\ 4 \\ 4 \end{bmatrix} ,~X_{-1} = \begin{bmatrix}0 \\ 0\\ 0\end{bmatrix}.
\end{align*}
We assume that both the users are identical in terms of energy demand. We can consider two households with similar energy requirements. However, we assume that user 2 with a
higher user specific incentive for storage, and reflected in the parameter values $a_k^1 = 3.4$ and $a_k^2 = 4$. Subsequently, the storage capacity of user 1's batteries are set to $K_{max}^{1} = 11.2$ units, and for user 2 are set as $K_{max}^2 = 12.2.$ units. It can be verified that the sufficient conditions in Lemma <ref> and Theorem <ref> are satisfied with the chosen parameters. We have used the freely available software, the PATH solver (see <http://pages.cs.wisc.edu/ ferris/path.html>), for solving the linear complementarity problem (<ref>).
Figure <ref> illustrates the evolution of resources in the grid. We have assumed three sources of energy in the grid. The consumption or contribution to the grid is represented in Figure <ref>. When a user consumes resources from the grid for an activity, the actual utility of the user corresponding to this activity is negative. Likewise, contribution to the grid results in positive utility. Here, $I_k^{i1},I_k^{i2}$ denote the consumption or contribution by the user $i$ for $m_i =2 $ activities. We note that the consumption or contribution of both the users are identical due to the identical demand costs. Further, we observe that both the users switch from contribution to consumption for the first activity at time period $k = 8$ and for the second activity at time period $k = 6$. As the users start to consume from the grid, the grid resources decrease which is evident from Figure <ref>. The evolution of battery storage decision for both the users is shown in Figure <ref>. User 2 has a higher user specific incentive as well as higher maximum storage capacity in comparison with user 1. Consequently, user 2 stores higher amount of energy in its battery. User 1 utilizes the full battery storage capacity from $k =3$ to $k =11$, and user 2 utilizes full battery storage capacity from $k =4$ to $k =10$. At the final time step, $K =12$, the total grid resources fall to a lower value, due to the salvage cost. Since the total battery storage is also limited by the total grid resources and $\epsilon_k$, the battery storage of both the users also decreases at this stage.
Evolution of resources
Consumption or contribution
Battery storage
State (grid resources) and decision (consumption/contribution and battery storage) variables for the smart grid system with energy storage.
Battery storage: user 1
Battery storage: user 2
20% decrease in player specific incentive
20% decrease in player specific incentive
Panels (a) and (b) illustrate the variation in battery storage with change in storage incentive. Panels (c) and (d) indicate the storage limit and total battery storage with 20%
variation in the incentive towards battery storage.
Next, we analyze the effect of incentive parameter $a_k^i$ on the energy storage behavior of the users. We vary the player specific incentive parameter, $a_k^i$ for both the users. As the incentive parameter $a_k^i$ is varied with a 20% variation around the baseline values, without varying the battery storage cost parameters, we deduce that both the users store higher amount of energy in their batteries with higher values of the incentive parameter. Figures <ref> and <ref> illustrate that the users utilize higher capacities for a longer time when the incentive parameter is higher. However, since the battery capacity of each player is limited, the players are unable to increase the storage continually with increase in the incentive. Besides, full capacity is utilized for a shorter period when we lower the incentive parameter. Finally, the constraint on total storage (<ref>) also makes sure that the total energy stored in the battery is lower than the total resources by at least $\epsilon_k$. This result is illustrated in Figures <ref> and <ref>.
§ CONCLUSIONS
In this paper, we studied the conditions under which a class of $N$-player non-zero sum discrete time dynamic games with inequality constraints admits a potential game structure. Drawing motivations from the theory of static potential games, we associated an optimal control problem with inequality constraints, and derived conditions under which the solution of this optimal control problem provides a (constrained) open-loop Nash equilibrium. When the potential functions are not specified before hand, we derived conditions under which the potential functions can be constructed using the problem data. We specialized these results to a linear quadratic setting and provided a linear complementarity problem based approach for computing the open-loop Nash equilibrium. In particular, the computed equilibrium is a refinement of the open-loop Nash equilibria obtained in [Reddy and Zaccour, 2015]. We illustrated our results with an example inspired by resource allocation in a smart grid network with energy storage. For future work, we plan to investigate the existence of potential functions for this class of games under feedback information structure.
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# A New Error in Variables Model for Solving Positive Definite Linear System
Using Orthogonal Matrix Decompositions
Negin Bagherpour Faculty of Mathematical Sciences, Sharif University of
Technology, Tehran, Iran, ([email protected]). Nezam Mahdavi-Amiri
Faculty of Mathematical Sciences, Sharif University of Technology, Tehran,
Iran, ([email protected]).
###### Abstract
The need to estimate a positive definite solution to an overdetermined linear
system of equations with multiple right hand side vectors arises in several
process control contexts. The coefficient and the right hand side matrices are
respectively named data and target matrices. A number of optimization methods
were proposed for solving such problems, in which the data matrix is
unrealistically assumed to be error free. Here, considering error in measured
data and target matrices, we present an approach to solve a positive definite
constrained linear system of equations based on the use of a newly defined
error function. To minimize the defined error function, we derive necessary
and sufficient optimality conditions and outline a direct algorithm to compute
the solution. We provide a comparison of our proposed approach and two
existing methods, the interior point method and a method based on quadratic
programming. Two important characteristics of our proposed method as compared
to the existing methods are computing the solution directly and considering
error both in data and target matrices. Moreover, numerical test results show
that the new approach leads to smaller standard deviations of error entries
and smaller effective rank as desired by control problems. Furthermore, in a
comparative study, using the Dolan-Moré performance profiles, we show the
approach to be more efficient.
###### keywords:
Error in variables models, positive definiteness constraints, overdetermined
linear system of equations, multiple right hand side vectors
###### AMS:
65F05, 65F20, 49M05
## 1 Introduction
Computing a symmetric positive definite solution of an overdetermined linear
system of equations arises in a number of physical problems such as estimating
the mass inertia matrix in the design of controllers for solid structures and
robots; see, e.g., [9], [17], [14]. Modeling a deformable structure also leads
to such a mathematical problem; e.g., see [25]. The problem turns into finding
an optimal solution of the system
(1) $DX\simeq T,$
where $D,T\in{\mathbb{R}}^{m\times n}$, with $m\geq n$, are given and a
symmetric positive definite matrix $X\in{\mathbb{R}}^{n\times n}$ is to be
computed as a solution of (1). In some special applications, the data matrix
$D$ has a simple structure, which may be taken into consideration for
efficiently organized computations. Estimation of the covariance matrix and
computation of the correlation matrix in finance are two such examples where
the data matrices are respectively block diagonal and the identity matrix;
e.g., see [31].
A number of least squares formulations have been proposed for physical
problems, which may be classified as ordinary and error in variables (EIV)
models.
Also, single or multiple right hand side least squares may arise. With a
single right hand side, we have an overdetermined linear system of equations
$Dx\simeq t$, where $D\in{\mathbb{R}}^{m\times n}$, $t\in{\mathbb{R}}^{m\times
1}$, with $m\geq n$, are known and the vector $x\in{\mathbb{R}}^{n\times 1}$
is to be computed. In an ordinary least squares formulation, the error is only
attributed to $t$. So, to minimize the corresponding error, the following
mathematical problem is devised:
(2) $\displaystyle\min$ $\displaystyle\|\Delta t\|$ $\displaystyle s.t.$
$\displaystyle Dx=t+\Delta t.$
There are a number of methods for solving (2), identified as direct and
iterative methods. A well known direct method is based on using the QR
factorization of the matrix $D$ [27]. An iterative method has also been
introduced in [7] for solving (2) using the GMRES algorithm.
In an EIV model, however, errors in both $D$ and $t$ are considered; e.g., see
[3]. Total least squares formulation is a well-known EIV model, where the goal
is to solve the following mathematical problem (e.g., see [6] and [16]):
(3) $\displaystyle\min$ $\displaystyle\|[\Delta D,\Delta t]\|$ $\displaystyle
s.t.$ $\displaystyle(D+\Delta D)x=t+\Delta t.$
We note that $\|\cdot\|$ in (2) and (3) respectively denote the vector 2-norm
and the matrix Frobenius norm. Both direct [24] and iterative [12] methods
have been presented for solving (3). Moreover, the scaled total least squares
formulation has been considered to unify both ordinary and total leats squares
formulation; e.g., see [24]. In a scaled toal least squares formulation, the
mathematical problem
$\displaystyle\min\|[\Delta D,\Delta t]\|$ (4) $\displaystyle s.t.(D+\Delta
D)x=\lambda t+\Delta t$
is to be solved for an arbitrary scalar $\lambda$. Zhou [19] has studied the
effect of perturbation and gave an error analysis of such a formulation.
A least squares problem with multiple right hand side vectors can also be
formulated as an overdetermined system of equations $DX\simeq T$, where
$D\in{\mathbb{R}}^{m\times n}$, $T\in{\mathbb{R}}^{m\times k}$, with $m\geq
n$, are given and the matrix $X\in{\mathbb{R}}^{n\times k}$ is to be computed.
With ordinary and total least squares formulations, the respective
mathematical problems are:
(5) $\displaystyle\min$ $\displaystyle\|\Delta T\|$ $\displaystyle s.t.$
$\displaystyle DX=T+\Delta T$ $\displaystyle X\in{\mathbb{R}}^{n\times k}$
and
(6) $\displaystyle\min$ $\displaystyle\|[\Delta D,\Delta T]\|$ $\displaystyle
s.t.$ $\displaystyle(D+\Delta D)X=T+\Delta T$ $\displaystyle
X\in{\mathbb{R}}^{n\times k}.$
Common methods for solving (5) are similar to the ones for (2); see, e.g.,
[7], [27]. Solving (6) is possible by using the method described in [8], based
on the SVD factorization of the matrix $[D,\hskip 2.84544ptT]$. Connections
between ordinary least squares and total least squares formulations have been
discussed in [11].
Here, we consider a newly defined EIV model for solving a positive definite
linear problem. Our goal is to compute a symmetric positive definite solution
$X\in{\mathbb{R}}^{n\times n}$ to the overdetermined system of equations
$DX\simeq T$, where both matrices $D$ and $T$ may contain errors. We refer to
this problem as positive definite linear system of equations later. No EIV
model, even the well-known total least squares formulation, is considered for
solving the positive definite linear system of equations in the literature.
Several approaches have been proposed for this problem, commonly considering
the ordinary least squares formulation and minimizing the error ${\|\Delta
T\|}_{F}$ over all $n\times n$ symmetric positive definite matrices, where
${\|.\|}_{F}$ is the Frobenious norm; see e.g. [10, 23]. Larson [13] discussed
a method for solving a positive definite least squares problem considering the
corresponding normal system of equations. He considered both symmetric and
positive definite least squares problems. Krislock [25] proposed an interior
point method for solving a variety of least squares problems with positive
semi-definite constraints. Woodgate [18] described a new algorithm for solving
a similar problem in which a symmetric positive semi-definite matrix $P$ is
computed to minimize $\|F-PG\|$, with known $F$ and $G$. Hu [10] presented a
quadratic programming approach to handle the positive definite constraint. In
her method, the upper and lower bounds for the entries of the target matrix
can be given as extra constraints. In real measurements, however, both the
data and target matrices may contain errors; hence, the total least squares
formulation appears to be appropriate.
The rest of our work is organized as follows. In Section 2, we define a new
error function and discuss some of its characteristics. A method for solving
the resulting optimization problem with the assumption that $D$ has full
column rank is presented in Section 3. In Section 4, we generalize the method
to the case of data matrix having an arbitrary rank. In Section 5, a detailed
discussion is made on computational complexity of both methods. Computational
results and comparisons with available methods are given in Section 6. Section
7 gives our concluding remarks.
## 2 Problem Formulation
Consider a single equation $ax\simeq b$, where $a,b\in{\mathbb{R}}^{n}$ and
$x\in{\mathbb{R}}^{+}$. Errors in the $i$th entry of $b$ and $a$ are
respectively equal to $\mid a_{i}x-b_{i}\mid$ and $\mid
a_{i}-\frac{b_{i}}{x}\mid$; e.g., see [24].
In [24], ${\sum}_{i=1}^{n}L_{i}$ was considered as a value to represent errors
in both $a$ and $b$. As shown in Figure LABEL:f1, $L_{i}$ is the height of the
triangle ABC which turns to be equal to
$L_{i}=\frac{|b_{i}-a_{i}x|}{\sqrt{1+x^{2}}}$. Here, to represent the errors
in both $a$ and $b$, we define the area error to be
(7) ${\sum}_{i=1}^{n}|b_{i}-a_{i}x||a_{i}-\frac{b_{i}}{x}|,$
which is equal to
${\sum}_{i=1}^{n}(b_{i}-a_{i}x)(a_{i}-\frac{b_{i}}{x}),$
for $x\in{\mathbb{R}}^{+}$.
Considering the problem of finding a symmetric and positive definite solution
to the overdetermined system of linear equations $DX\simeq T$, in which both
$D$ and $T$ include error, the values $DX$ and $TX^{-1}$ are predicted values
for $T$ and $D$ from the model $DX\simeq T$; hence, vectors ${\Delta
T}_{j}={(DX-T)}_{j}$ and ${\Delta D}_{j}={(D-TX^{-1})}_{j}$ are the entries of
errors in the $j$th column of $T$ and $D$, respectively. Extending the error
formulation (7), the value
$E={\sum}_{j=1}^{n}(DX_{j}-T_{j})^{T}(D_{j}-(TX^{-1})_{j}$
seems to be an appropriate measure of error. We also have
(8)
$E={\sum}_{j=1}^{n}{\sum}_{i=1}^{m}{(DX-T)}_{ij}{(D-TX^{-1})}_{ij}=\mathop{\mathrm{tr}}((DX-T)^{T}(D-TX^{-1})),$
with $\mathop{\mathrm{tr}}(.)$ standing for trace of a matrix. Therefore, the
problem can be formulated as
(9) $\min\limits_{X\succ 0}\mathop{\mathrm{tr}}((DX-T)^{T}(D-TX^{-1})),$
where $X$ is symmetric and by $X\succ 0$, we mean $X$ is positive definite.
Problem (9) poses a newly defined EIV model for solving the positive definite
linear system of equations.
In Lemma 2, we represent an equivalent formulation for the error, $E$. First,
consider to a well-known property of positive definite matrices. Note A matrix
$X\in{\mathbb{R}}^{n\times n}$ is positive definite if and only if there
exists a nonsingular matrix $Y\in{\mathbb{R}}^{n\times n}$ such that
$X=YY^{T}$.
The following results about the trace operator are also well-known; e.g., see
[21].
###### Lemma 1.
For an nonsingulartible matrix $P\in{\mathbb{R}}^{n\times n}$ and arbitrary
matrices $Y\in{\mathbb{R}}^{n\times n}$, $A\in{\mathbb{R}}^{m\times n}$ and
$B\in{\mathbb{R}}^{n\times m}$ we have
$\indent(1)\indent\mathop{\mathrm{tr}}(Y)=\mathop{\mathrm{tr}}(P^{-1}YP).$
$\indent(2)\indent\mathop{\mathrm{tr}}(AB)=\mathop{\mathrm{tr}}(BA).$
###### Lemma 2.
The error $E$, defined by (8), is equal to
(10) $E=\|DY-TY^{-T}\|_{F}^{2}$
where $X=YY^{T}$ and $\|.\|_{F}$ denotes the Frobenius norm of a matrix.
###### Proof.
Substituting $X=YY^{T}$ in (8) and using Lemma 1, we get
$\displaystyle E$ $\displaystyle=$
$\displaystyle\mathop{\mathrm{tr}}((DX-T)^{T}(D-TX^{-1}))=\mathop{\mathrm{tr}}((DX-T)^{T}(DX-T)X^{-1})$
$\displaystyle=$
$\displaystyle\mathop{\mathrm{tr}}((DX-T)^{T}(DX-T)Y^{-T}Y^{-1})=\mathop{\mathrm{tr}}(Y^{-1}(DX-T)^{T}(DX-T)Y^{-T})$
$\displaystyle=$
$\displaystyle\mathop{\mathrm{tr}}({(DXY^{-T}-TY^{-T})}^{T}(DXY^{-T}-TY^{-T}))$
$\displaystyle=$ $\displaystyle\mathop{\mathrm{tr}}({(DY-TY^{-T})}^{T}(DY-
TY^{-T}))$ $\displaystyle=$ $\displaystyle\|DY-TY^{-T}\|_{F}^{2}.$
∎
Considering this new formulation for $E$, it can be concluded that by use of
our newly defined EIV model, computing a symmetric and positive definite
solution to the over-determined system of equations $DX\simeq T$ is equivalent
to computing a nonsingular matrix $Y\in{\mathbb{R}}^{n\times n}$ to be the
solution of
$\min\|DY-TY^{-T}\|_{F}^{2},$
and letting $X=YY^{T}$. A similar result is obtained by considering the over-
determined system $DX\simeq T$ with $X=YY^{T}$ and multiplying both sides by
$Y^{-T}$. We have,
$DYY^{T}\simeq T,$
or equivalently,
(11) $DY\simeq TY^{-T}.$
Now, to assign a solution to (11), it makes sense to minimize the norm of
residual. Thus, to compute $X=YY^{T}$, it is sufficient to let $Y$ to be the
solution of
$\min\|DY-TY^{-T}\|_{F}^{2}.$
Note An appropriate characteristic of the error formulation proposed by (8) is
that for a symmetric and positive definite matrix $X$, the value of $E$ is
nonnegative and it is equal to zero if and only if $DX=T$.
## 3 Mathematical Solution: Full Rank Data Matrix
Here, we are to develop an algorithm for solving (9) with the assumption that
$D$ has full column rank.
Using Lemma 1, with $X$ being symmetric, we have
$\mathop{\mathrm{tr}}({(DX-T)}^{T}(D-TX^{-1}))=\mathop{\mathrm{tr}}(D^{T}DX+X^{-1}T^{T}T)-2\mathop{\mathrm{tr}}(T^{T}D).$
So, (9) can be written as
(12) $\min\mathop{\mathrm{tr}}(AX+X^{-1}B),$
where $A=D^{T}D$ and $B=T^{T}T$ and the symmetric and positive definite matrix
$X$ is to be computed.
###### Corollary 3.
For each $X^{\ast}$ satisfying the first order necessary conditions of (12),
the sufficient optimality conditions described in Theorem LABEL:14 are
satisfied and since $\Phi(X)=\mathop{\mathrm{tr}}(AX+X^{-1}B)$ is convex on
the cone of symmetric positive definite matrices, we can confirm that the
symmetric positive definite matrix satisfying the KKT necessary conditions
mentioned in Theorem LABEL:13 is the unique global solution of (12).
### Computing the positive definite matrix satisfying KKT conditions
As mentioned in Theorem LABEL:13, the KKT conditions lead to the nonlinear
matrix equation
(13) $XAX=B.$
Note that (13) is an special case of the continuous time Riccati equation
(CARE), [22]
(14) $A^{T}XE+E^{T}XA-(E^{T}XB+S)R^{-1}(B^{T}XE+S^{T})+Q=0,$
with $R=0$, $E=\frac{A}{2}$ and $Q=-B$. There is a MATLAB routine to solve
CARE for arbitrary values of $A$, $E$, $B$, $S$, $R$ and $Q$. To use the
routine, it is sufficient to type the command
X=care(A,B,Q,R,S,E),
for the input arguments as in (14). Higham [22] developed an effective method
for computing the positive definite solution to this special CARE when $A$ and
$B$ are symmetric and positive definite using well-known decompositions.
Lancaster and Rodman ([28]) also discussed solving different types of
algebraic Riccati equations. Moreover, they derived a perturbation analysis
for these matrix equations.
Note (QR decomposition) The QR decomposition [27] of a matrix
$A\in{\mathbb{R}}^{m\times n}$ with $m\geq n$, is a decomposition of the form
$A=QR$, where $R$ is an $m\times n$ upper triangular matrix and $Q$ satisfies
$QQ^{T}=Q^{T}Q=I$. Moreover, if $A$ has full column rank, then $R$ also has
full column rank.
Note (Cholesky decomposition) A Cholesky decomposition [27] of a symmetric
positive definite matrix $A\in{\mathbb{R}}^{n\times n}$ is a decomposition of
the form $A=R^{T}R$, where $R$, known as the Cholesky factor of $A$, is an
$n\times n$ nonsingular upper triangular matrix.
Note (Spectral decomposition) [27] All eigenvalues of a symmetric matrix,
$A\in{\mathbb{R}}^{n\times n}$, are real and there exists an orthonormal
matrix with columns representing the corresponding eigenvectors. Thus, there
exist an orthonormal matrix $U$ with columns equal to the eigenvectors of $A$
and a diagonal matrix $D$ containing the eigenvalues such that $A=UDU^{T}$.
Also, if $A$ is positive definite, then all of its eigenvalues are positive,
and so we can set $D=S^{2}$. Thus, spectral decomposition for a symmetric
positive definite matrix $A$ is a decomposition of the form $A=US^{2}U^{T}$,
where $U^{T}U=UU^{T}=I$ and $S$ is a diagonal matrix.
###### Theorem 4.
[22] Assume $D,T\in{\mathbb{R}}^{m\times n}$ with $m\geq n$ are known and
$rank(D)=rank(T)=n$. Let $D=QR$ be the QR factorization of $D$. Let $A=D^{T}D$
and $B=T^{T}T$. Define the matrix $\tilde{Q}=RBR^{T}$ and compute its spectral
decomposition, that is, $\tilde{Q}=RBR^{T}=U{\tilde{S}}^{2}U^{T}$. Then, (12)
has a unique solution, given by
$X^{\ast}=R^{-1}U\tilde{S}U^{T}R^{-T}.$
###### Proof.
Based on Theorem LABEL:14 and the afterwards discussion, it is sufficient to
show that $X^{\ast}$ satisfies the necessary optimality conditions,
$X^{\ast}AX^{\ast}=B$. Note that from $D=QR$, we have
$A=D^{T}D=R^{T}Q^{T}QR=R^{T}R.$
Substituting $X^{\ast}$, we have
$\displaystyle X^{\ast}AX^{\ast}$ $\displaystyle=$ $\displaystyle
R^{-1}U\tilde{S}U^{T}R^{-T}R^{T}RR^{-1}U\tilde{S}U^{T}R^{-T}$ $\displaystyle=$
$\displaystyle R^{-1}U{\tilde{S}}^{2}U^{T}R^{-T}=R^{-1}RBR^{T}R^{-T}=B.$
∎
Note To compute $R$, it is also possible to first compute $A=D^{T}D$ and then
calculate the Cholesky decomposition for $A$. However, because of more
stability, in Theorem 4 the QR decomposition of $D$ is used.
We are now ready to outline the steps of our proposed algorithm.
Solving the EIV model for positive definite linear system using QR
decomposition.
Compute the QR decomposition for D and let $D=QR$.
Let $\tilde{Q}=RBR^{T}$, where $B=T^{T}T$ and compute the spectral
decomposition of $\tilde{Q}$, that is, $\tilde{Q}=U{\tilde{S}}^{2}U^{T}.$
Set $X^{\ast}=R^{-1}U\tilde{S}U^{T}R^{-T}.$
Set $E=\mathop{\mathrm{tr}}((DX^{\ast}-T)^{T}(D-T{X^{\ast}}^{-1}))$.
Note that Algorithm 1 computes the solution of (9) directly.
The following theorem shows that by use of spectral decomposition of $A$ a
method similar to the one introduced in [22] is in hand for solving the
continuous time Riccati equation.
###### Theorem 5.
Let $A=D^{T}D$ and $B=T^{T}T$ with $D,T\in{\mathbb{R}}^{m\times n}$, $m\geq n$
and $rank(D)=n.$ Let the spectral decomposition of $A$ be $A=US^{2}U^{T}$.
Define the matrix $\tilde{Q}=SU^{T}BUS$ and compute its spectral
decomposition, $\tilde{Q}=SU^{T}BUS=\bar{U}{\bar{S}}^{2}{\bar{U}}^{T}$. Then,
the unique minimizer of (12) is
$X^{\ast}=US^{-1}\bar{U}\bar{S}{\bar{U}}^{T}S^{-1}U^{T}.$
###### Proof.
Similar to the proof of Theorem 4, it is sufficient to show that the mentioned
$X^{\ast}$ satisfies $X^{\ast}AX^{\ast}=B$. Substituting $X^{\ast}$, we have
$\displaystyle X^{\ast}AX^{\ast}$
$\displaystyle=US^{-1}\bar{U}\bar{S}{\bar{U}}^{T}S^{-1}U^{T}US^{2}U^{T}US^{-1}\bar{U}\bar{S}{\bar{U}}^{T}S^{-1}U^{T}$
$\displaystyle=US^{-1}\bar{U}{\bar{S}}^{2}{\bar{U}}^{T}S^{-1}U^{T}=US^{-1}SU^{T}BUSS^{-1}U^{T}=B.$
∎
Next, based on Theorem 5, we outline an algorithm for solving (9).
Solving the EIV model for positive definite linear system using spectral
decomposition.
Let $A=D^{T}D$ and compute its spectral decomposition: $A=US^{2}U^{T}.$
Let $\tilde{Q}=SU^{T}BUS$, where $B=T^{T}T$ and compute the spectral
decomposition of $\tilde{Q}$, that is,
$\tilde{Q}=\tilde{U}{\tilde{S}}^{2}{\tilde{U}}^{T}.$
Set $X^{\ast}=US^{-1}\tilde{U}\tilde{S}{\tilde{U}}^{T}S^{-1}U^{T}.$
Set $E=\mathop{\mathrm{tr}}((DX^{\ast}-T)^{T}(D-T{X^{\ast}}^{-1}))$.
In Section 4 we generalize our proposed method for solving positive definite
linear system of equations when the data matrix is rank deficient.
## 4 Mathematical Solution: Rank Deficient Data Matrix
Since the data matrix $D$ is usually produced from experimental measurements,
we may have $rank(D)<n$. Here, we are to generalize Algorithm 1 for solving
(9), assuming that $rank(D)=r<n$. In Section 4.1 we outline two algorithms to
compute the general solution of (9). It will be shown that, in general, (9)
may not have a unique solution. Hence, in section 4.2 we discuss how to find a
particular solution of (9) having desirable characteristics for control
problems.
### 4.1 General solution
Based on theorems LABEL:13 and LABEL:14, a symmetric positive definite matrix
$X^{\ast}$ is a solution of (9) if and only if
(15) $X^{\ast}AX^{\ast}=B.$
Therefore, in the following, we discuss how to find a symmetric positive
definite matrix $X^{\ast}$ satisfying (15).
First we note that in case $D$ and $T$ are rank deficient, there might be no
solution for (15), and if there is any, it is not necessarily a unique
solution; see, e.g., [22]. Higham [22] considered to
$X=B^{\frac{1}{2}}{(B^{\frac{1}{2}}AB^{\frac{1}{2}})}^{-\frac{1}{2}}B^{\frac{1}{2}}$
as a solution of (15), which is symmetric and positive semidefinite. However,
we are interested in finding a symmetric positive definite solution to (15).
Hence, in the following, first the necessary and sufficient conditions on $A$
and $B$ to guarantee the existence of positive definite solution to (15) are
discussed. We then outline two algrithms to compute such a solution.
Let the spectral decomposition of $A$ be
$A=U\left(\begin{array}[]{cc}S^{2}&0\\\ 0&0\\\ \end{array}\right)U^{T}$, where
$S^{2}\in{\mathbb{R}}^{r\times r}$ is a diagonal matrix having the positive
eigenvalues of $A$ as its diagonal entries. Substituting the decomposition in
(15), we get
(16) $X^{\ast}U\left(\begin{array}[]{cc}S^{2}&0\\\ 0&0\\\
\end{array}\right)U^{T}X^{\ast}=B.$
Since $U$ is orthonormal, (16) can be written as
$U^{T}X^{\ast}U\left(\begin{array}[]{cc}S^{2}&0\\\ 0&0\\\
\end{array}\right)U^{T}X^{\ast}U=U^{T}BU.$
Then, letting $\tilde{X}=U^{T}XU$ and $\tilde{B}=U^{T}BU$, we have
(17) $\tilde{X}\left(\begin{array}[]{cc}S^{2}&0\\\ 0&0\\\
\end{array}\right)\tilde{X}=\tilde{B}.$
Thus, the matrix $X=U\tilde{X}U^{T}$ is a solution of (9) if and only if
$\tilde{X}$ is symmetric positive definite and satisfies (17). Substituting
the block form
$\tilde{X}=\left(\begin{array}[]{cc}{\tilde{X}}_{rr}&{\tilde{X}}_{r,n-r}\\\
{\tilde{X}}_{n-r,r}&{\tilde{X}}_{n-r,n-r}\\\ \end{array}\right)$, where
${\tilde{X}}_{rr}\in{\mathbb{R}}^{r\times r}$,
${\tilde{X}}_{r,n-r}={\tilde{X}}_{n-r,r}^{T}\in{\mathbb{R}}^{r\times(n-r)}$
and ${\tilde{X}}_{n-r,n-r}\in{\mathbb{R}}^{(n-r)\times(n-r)}$, in (17) leads
to
$\left(\begin{array}[]{cc}{\tilde{X}}_{rr}S^{2}{\tilde{X}}_{rr}&{\tilde{X}}_{rr}S^{2}{\tilde{X}}_{r,n-r}\\\
{\tilde{X}}_{n-r,r}S^{2}{\tilde{X}}_{rr}&{\tilde{X}}_{n-r,r}S^{2}{\tilde{X}}_{r,n-r}\\\
\end{array}\right)=\tilde{B}=\left(\begin{array}[]{cc}{\tilde{B}}_{rr}&{\tilde{B}}_{r,n-r}\\\
{\tilde{B}}_{n-r,r}&{\tilde{B}}_{n-r,n-r}\\\ \end{array}\right),$
which is satisfied if and only if
(18a) ${\tilde{X}}_{rr}S^{2}{\tilde{X}}_{rr}={\tilde{B}}_{rr},$ (18b)
${\tilde{X}}_{rr}S^{2}{\tilde{X}}_{r,n-r}={\tilde{B}}_{r,n-r},$ (18c)
${\tilde{X}}_{n-r,r}S^{2}{\tilde{X}}_{r,n-r}={\tilde{B}}_{n-r,n-r}.$
Before discussing how to compute $\tilde{X}$, we show that if (15) has a
symmetric and positive definite solution, then ${\tilde{B}}_{rr}$ must be
nonsingular. The matrix ${\tilde{X}}_{rr}$ as a main minor of the positive
definite matrix $\tilde{X}$ is nonsingular. $S$ is also supposed to be
nonsingular. Hence, it can be concluded from (18a) that ${\tilde{B}}_{rr}$ is
nonsingular.
Let $\bar{D}=S$ and suppose $\bar{T}$ satisfies
${\bar{T}}^{T}{\bar{T}}={\tilde{B}}_{rr}$. Consider problem (9) corresponding
to the data and target matrices $\bar{D}$ and $\bar{T}$ as follows:
(19) $\min\limits_{\bar{X}\succ
0}\mathop{\mathrm{tr}}((\bar{D}\bar{X}-\bar{T})^{T}(\bar{D}-\bar{T}{\bar{X}}^{-1})).$
We know from theorems LABEL:13 and LABEL:14 that the necessary and sufficient
optimality conditions for the unique solution of problem (19) implies (18a).
Thus, ${\tilde{X}}_{rr}$ can be computed using Algorithm 1 for the input
arguments $\bar{D}$ and $\bar{T}$. Substituting the computed
${\tilde{X}}_{rr}$ in (18b), the linear system of equations
(20) ${\tilde{X}}_{rr}S^{2}{\tilde{X}}_{r,n-r}={\tilde{B}}_{r,n-r}$
arises, where ${\tilde{X}}_{rr},S^{2}\in{\mathbb{R}}^{r\times r}$ are known
and ${\tilde{X}}_{r,n-r}\in{\mathbb{R}}^{r\times(n-r)}$ is to be computed.
Since ${\tilde{X}}_{rr}$ is positive definite and $S^{2}$ is nonsingular, the
coefficient matrix of the linear system (20) is nonsingular and
${\tilde{X}}_{r,n-r}$ can be uniquely computed.
It is clear that since $\tilde{X}$ is symmetric, ${\tilde{X}}_{n-r,r}$ is the
same as ${{\tilde{X}}_{r,n-r}}^{T}$. Now, we check whether the computed
${\tilde{X}}_{n-r,r}$ and ${\tilde{X}}_{r,n-r}$ satisfy (18c). Inconsistency
of (19) means that there is no symmetric positive definite matrix satisfying
(18a)-(18c), and if so, (9) has no solution. Thus, in solving an specific
positive definite system with rank deficient data and target matrices using
the presented EIV model, a straightforward method to investigate the existence
of solution is to check whether (18c) holds for the given data and target
matrices. On the other hand, for numerical results, it is necessary to
generate meaningful test problems. Hence, in the following two lemmas, we
investigate the necessary and sufficient conditions for satisfaction of (18c).
###### Lemma 6.
Let the spectral decomposition of $A$ be determined as
$A=U\left(\begin{array}[]{cc}S^{2}&0\\\ 0&0\\\ \end{array}\right)U^{T}$
where $S^{2}\in{\mathbb{R}}^{r\times r}$ and $rank(A)=rank(B)=r$. The
necessary and sufficient condition for satisfaction of (18c) is
$BU_{r}{({U_{r}}^{T}BU_{r})}^{-1}{U_{r}}^{T}B-B\in\mathop{\mathrm{Null}}({U_{n-r}}^{T}).$
###### Proof.
From (18a), we have
(21)
${{\tilde{X}}_{rr}}^{-1}S^{-2}{{\tilde{X}}_{rr}}^{-1}={{\tilde{B}}_{rr}}^{-1},$
and from (18c), we get
(22) ${\tilde{X}}_{r,n-r}=S^{-2}{{\tilde{X}}_{rr}}^{-1}{{\tilde{B}}_{r,n-r}},$
${\tilde{X}}_{n-r,r}={\tilde{B}}_{n-r,r}{{\tilde{X}}_{rr}}^{-1}S^{-2}.$
Manipulating (18c) with (21) and (22), we get
(23)
${\tilde{B}}_{n-r,r}{{\tilde{B}}_{rr}}^{-1}{\tilde{B}}_{r,n-r}={\tilde{B}}_{n-r,n-r}.$
Considering the block form $U=\left(\begin{array}[]{cc}U_{r}&U_{n-r}\\\
\end{array}\right)$, where $U_{r}\in{\mathbb{R}}^{n\times r}$ and
$U_{n-r}\in{\mathbb{R}}^{n\times(n-r)}$, we have
(27) $\displaystyle\tilde{B}=U^{T}BU$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}{U_{r}}^{T}\\\ {U_{n-r}}^{T}\\\
\end{array}\right)B\left(\begin{array}[]{cc}U_{r}&U_{n-r}\\\
\end{array}\right)$ (30) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}{U_{r}}^{T}BU_{r}&{U_{r}}^{T}BU_{n-r}\\\
{U_{n-r}}^{T}BU_{r}&{U_{n-r}}^{T}BU_{n-r}\\\ \end{array}\right).$
Rewriting (23) results in
(31)
${U_{n-r}}^{T}BU_{r}{({U_{r}}^{T}BU_{r})}^{-1}{U_{r}}^{T}BU_{n-r}={U_{n-r}}^{T}BU_{n-r},$
which is equivalent to (e.g., see [20])
(32) $BU_{r}{({U_{r}}^{T}BU_{r})}^{-1}{U_{r}}^{T}B=B+Z,$
where $Z\in{\mathbb{R}}^{n\times n}$ is in the null space of ${U_{n-r}}^{T}$.
Thus, (15) has a positive definite solution if and only if
(33)
$BU_{r}{({U_{r}}^{T}BU_{r})}^{-1}{U_{r}}^{T}B-B\in\mathop{\mathrm{Null}}({U_{n-r}}^{T}).$
∎
Note For real problems with arbitrary values of $D$ and $T$, the necessary and
sufficient condition given in Lemma 6 may not be satisfied, in general. Hence,
we are to propose a threshold to determine if
(34)
$F={U_{n-r}}^{T}\left(BU_{r}{({U_{r}}^{T}BU_{r})}^{-1}{U_{r}}^{T}B-B\right)$
is close enough to zero. In the following, we show that if $\|F\|<\delta$, for
a sufficiently small scalar $\delta$, then $X_{r(n-r)}$ computed from (18b) is
a proper approximation for the solution of (18c). Substituting $F$ in (31), we
have
(35)
${\tilde{B}}_{n-r,r}{{\tilde{B}}_{rr}}^{-1}{\tilde{B}}_{r,n-r}-{\tilde{B}}_{n-r,n-r}=FU_{n-r},$
and
(36)
${\tilde{X}}_{n-r,r}S^{2}{\tilde{X}}_{r,n-r}-{\tilde{B}}_{n-r,n-r}=FU_{n-r}.$
Let $X^{\ast}$ satisfy (18c), that is,
(37) ${X^{\ast}}_{n-r,r}S^{2}{X^{\ast}}_{r,n-r}-{\tilde{B}}_{n-r,n-r}=0.$
Then, we have
(38)
${\tilde{X}}_{n-r,r}S^{2}{\tilde{X}}_{r,n-r}-{X^{\ast}}_{n-r,r}S^{2}{X^{\ast}}_{r,n-r}=FU_{n-r}.$
Letting $\tilde{Y}=S{\tilde{X}}_{r,n-r}$ and $Y^{\ast}=S{X^{\ast}}_{r,n-r}$,
(38), we get
(39) ${\tilde{Y}}^{T}\tilde{Y}-{Y^{\ast}}^{T}Y^{\ast}=FU_{n-r}$
and
(40)
${\tilde{y}}_{i}^{T}{\tilde{y}}_{j}-{y_{i}^{\ast}}^{T}y_{j}^{\ast}=(FU_{n-r})_{ij},$
where ${\tilde{y_{i}}}$ and ${y_{i}^{\ast}}$ are the $i$th column of
$\tilde{Y}$ and $Y^{\ast}$ respectively. Now, since the 2 norm of each column
of $U_{n-r}$ is equal to one, every entry of $U_{n-r}$ is less than or equal
to one. Moreover, under the assumption $\|F\|<\delta$, none of the entries of
$F$ are greater than $\delta$. Hence, we have
(41)
$|{(FU_{n-r})}_{ij}|=|f_{i}^{T}u_{j}|\leq|f_{i1}+\cdots+f_{i(n-r)}|<(n-r)\delta,$
where $f_{i}^{T}$ and $u_{j}$ are the $i$th row of $F$ and the $j$th column of
$U_{n-r}$ respectively. Now, (40) together with (41) gives
(42)
$|{\tilde{y_{i}}}^{T}\tilde{y_{j}}-{y_{i}^{\ast}}^{T}y_{j}^{\ast}|<{(n-r)}\delta.$
Hence, there is a constant $c_{ij}$ such that
(43) $|{\tilde{y}}_{ij}-y_{ij}^{\ast}|<c_{ij},$
where $\tilde{y}_{ij}$ and $y_{ij}^{\ast}$ are the $(i,j)$th entry of
$\tilde{Y}$ and $Y^{\ast}$ respectively. Letting $S=diag(s_{1},\cdots,s_{r})$,
from (43) we get
(44) $|s_{i}||({\tilde{X}}_{n-r,r})_{ij}-(X^{\ast}_{n-r,r})_{ij}|\leq c_{ij},$
for $i=1,\cdots,r$ and $j=1,\cdots,n-r$ and
${\|{\tilde{X}}_{r,n-r}-{X^{\ast}}_{r,n-r}\|}\leq C.$
Hence, assuming
${\tilde{X}}_{r,r}={X^{\ast}}_{r,r},$
we have $\|\tilde{X}-X^{\ast}\|<\alpha$ which means that if $FU_{n-r}$ is
close enough to zero, then the computed solution from the approximate
satisfaction of (18c) would be close enough to the exact solution.
In the following lemma, we give a sufficient condition which guarantees the
existence of a solution for (15). We later use this result to generate
consistent test problems in Section 6.
###### Lemma 7.
Let the spectral decomposition of $B$ be
$B=V\left(\begin{array}[]{cc}{\sum}^{2}&0\\\ 0&0\\\ \end{array}\right)V^{T},$
where ${\sum}^{2}\in{\mathbb{R}}^{r\times r}$ and $rank(A)=rank(B)=r$. A
sufficient condition for satisfaction of (18c) is that
(45) $V=U\left(\begin{array}[]{cc}Q&0\\\ 0&P\\\ \end{array}\right),$
where $Q\in{\mathbb{R}}^{r\times r}$ and $P\in{\mathbb{R}}^{(n-r)\times(n-r)}$
satisfy $QQ^{T}=Q^{T}Q=I$ and $PP^{T}=P^{T}P=I$.
###### Proof.
A possible choice for $Z$ in Lemma 7 is zero, for which (32) is equivalent to
(46) $U_{r}{({U_{r}}^{T}BU_{r})}^{-1}{U_{r}}^{T}=B^{+}+W,$
with $W\in{\mathbb{R}}^{n\times n}$ in the null space of $B$. To obtain a
simplified sufficient condition for existence of a positive definite solution
to (15), we let $W=0$. Multiplying (46) by ${U_{r}}^{T}$ and $U_{r}$
respectively on the left and right, and substituting the spectral
decomposition of $B$, we get
(47)
${({U_{r}}^{T}V_{r}{\sum}^{2}{V_{r}}^{T}U_{r})}^{-1}={U_{r}}^{T}B^{+}U_{r}={U_{r}}^{T}V_{r}{\sum}^{-2}{V_{r}}^{T}U_{r}.$
Letting $M={U_{r}}^{T}V_{r}$, we get
${(M{\sum}^{2}M^{T})}^{-1}=M{\sum}^{-2}M^{T}.$
Since $M$ has full rank, we get
(48) $M^{-T}{\sum}^{-2}M^{-1}=M{\sum}^{-2}M^{T}.\\\ $
Now, since ${\sum}^{-2}$ is nonsingular, (48) holds if and only if
(49) $M^{T}M=I.$
This leads to
(50)
${({U_{r}}^{T}V_{r})}^{T}{U_{r}}^{T}V_{r}={V_{r}}^{T}U_{r}{U_{r}}^{T}V_{r}=I.$
Since $U$ is orthonormal, we have
$UU^{T}=U_{r}{U_{r}}^{T}+U_{n-r}{U_{n-r}}^{T}=I$. Hence, we get
(51) $U_{r}{U_{r}}^{T}=I-U_{n-r}{U_{n-r}}^{T}.$
Substituting (51) in (50), we get
${V_{r}}^{T}(I-U_{n-r}{U_{n-r}}^{T})V_{r}=I-{V_{r}}^{T}U_{n-r}{U_{n-r}}^{T}V_{r}=I,$
which is satisfied if and only if ${U_{n-r}}^{T}V_{r}=0$. Since the columns of
$U_{r}$ form an orthogonal basis for the null space of ${U_{n-r}}^{T}$ [27],
it can be concluded that each column of $V_{r}$ is a linear combination of the
columns of $U_{r}$. Thus,
(52) $V_{r}=U_{r}Q$
is a necessary condition for (49) to be satisfied, and since both $U_{r}$ and
$V_{r}$ have orthogonal columns, $Q\in{\mathbb{R}}^{r\times r}$ satisfies
$QQ^{T}=Q^{T}Q=I$. On the other hand, we know from the definition of the
spectral decomposition that $VV^{T}=UU^{T}=I$. Thus,
$\displaystyle V_{r}{V_{r}}^{T}+V_{n-r}{V_{n-r}}^{T}=I,$ (53) $\displaystyle
U_{r}{U_{r}}^{T}+U_{n-r}{U_{n-r}}^{T}=I.$
Manipulating (52) with (53), we get
(54) $V_{n-r}{V_{n-r}}^{T}=U_{n-r}{U_{n-r}}^{T},$
which holds if and only if there exists a matrix
$P\in{\mathbb{R}}^{(n-r)\times(n-r)}$ such that $PP^{T}=P^{T}P=I$ and
(55) $V_{n-r}=PU_{n-r}.$
It can be concluded from (52) and (55) that
$V=U\left(\begin{array}[]{cc}Q&0\\\ 0&P\\\ \end{array}\right)$, where
$QQ^{T}=Q^{T}Q=I$ and $PP^{T}=P^{T}P=I$. ∎
###### Corollary 8.
The matrices $P$ and $Q$ defined in Lemma 6 can set to be rotation matrices
[27] to satisfy
$\displaystyle PP^{T}=P^{T}P=I,$ $\displaystyle QQ^{T}=Q^{T}Q=I.$
Thus, to compute a target matrix, $T$, satisfying Lemma 6, it is sufficient to
first compute $V$ from (45) with $Q\in{\mathbb{R}}^{r\times r}$ and
$P\in{\mathbb{R}}^{(n-r)\times(n-r)}$ arbitrary rotation matrices and $U$ as
defined in Lemma 6 and then set $T=\bar{U}\left(\begin{array}[]{cc}\sum&0\\\
0&0\\\ \end{array}\right)V^{T}$, where $\bar{U}\in{\mathbb{R}}^{m\times m}$
and $\sum\in{\mathbb{R}}^{r\times r}$ are arbitrary orthonormal and diagonal
matrices.
Thus, problem (9) has a solution if and only if the data and target matrices
satisfy (33). In this case, ${\tilde{X}}_{rr}$, ${\tilde{X}}_{r,n-r}$ and its
transpose, ${\tilde{X}}_{n-r,r}$, are respectively computed from (18a) and
(18b). Hence, the only remaining step is to compute ${\tilde{X}}_{n-r,n-r}$ so
that $\tilde{X}$ is symmetric and positive definite.
We know that $\tilde{X}$ is symmetric positive definite if and only if there
exists a nonsingular lower triangular matrix $L\in{\mathbb{R}}^{n\times n}$ so
that
(56) $\tilde{X}=LL^{T},$
where $L$ is lower triangular and nonsingular. Considering the block forms
$\tilde{X}=\left(\begin{array}[]{cc}{\tilde{X}}_{rr}&{\tilde{X}}_{r,n-r}\\\
{\tilde{X}}_{n-r,r}&{\tilde{X}}_{n-r,n-r}\\\ \end{array}\right)$ and
$L=\left(\begin{array}[]{cc}L_{rr}&0\\\ L_{n-r,r}&L_{n-r,n-r}\\\
\end{array}\right)$, where $L_{n-r,r}$ is an $(n-r)\times r$ matrix and
$L_{rr}\in{\mathbb{R}}^{r\times r}$ and
$L_{n-r,n-r}\in{\mathbb{R}}^{(n-r)\times(n-r)}$ are nonsingular lower
triangular matrices, we get
(64)
$\displaystyle\left(\begin{array}[]{cc}{\tilde{X}}_{rr}&{\tilde{X}}_{r,n-r}\\\
{\tilde{X}}_{n-r,r}&{\tilde{X}}_{n-r,n-r}\\\ \end{array}\right)$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}L_{rr}&0\\\
L_{n-r,r}&L_{n-r,n-r}\\\
\end{array}\right)\left(\begin{array}[]{cc}{L_{rr}}^{T}&{L_{n-r,r}}^{T}\\\
0&{L_{n-r,n-r}}^{T}\\\ \end{array}\right).$
Thus,
(65a) ${\tilde{X}}_{rr}=L_{rr}{L_{rr}}^{T},$ (65b)
${\tilde{X}}_{r,n-r}=L_{rr}{L_{n-r,r}}^{T},$ (65c)
${\tilde{X}}_{n-r,r}=L_{n-r,r}{L_{rr}}^{T},$ (65d)
${\tilde{X}}_{n-r,n-r}=L_{n-r,r}{L_{n-r,r}}^{T}+L_{n-r,n-r}{L_{n-r,n-r}}^{T}.$
Therefore, to compute a symmetric positive definite $\tilde{X}$, (65a)–(65d)
must be satisfied. Let ${\tilde{X}}_{rr}=\tilde{L}{\tilde{L}}^{T}$ be the
Cholesky decomposition of ${\tilde{X}}_{rr}$. $L_{rr}=\tilde{L}$ satisfies
(65a). Substituting $L_{rr}$ in (65b), ${L_{n-r,r}}^{T}$ is computed uniquely
by solving the resulting linear system. Since (65c) is transpose of (65b), it
does not give any additional information. Finally, to compute a matrix
${\tilde{X}}_{n-r,n-r}$ to satisfy (65d), it is sufficient to choose an
arbitrary lower triangular nonsingular matrix $L_{n-r,n-r}$ and substitute it
in (65d). The resulting ${\tilde{X}}_{n-r,n-r}$ gives a symmetric positive
definite $\tilde{X}$ as follows:
$\tilde{X}=\left(\begin{array}[]{cc}{\tilde{X}}_{rr}&{\tilde{X}}_{r,n-r}\\\
{\tilde{X}}_{n-r,r}&{\tilde{X}}_{n-r,n-r}\\\ \end{array}\right).$
Now, based on the above discussion, we outline the steps of our algorithm for
solving (9) in the case $rank(D)=r<n.$
Solving the EIV model for positive definite linear system with rank deficient
data and target matrices using spectral decomposition.
$\delta$ as the upper bounds for absolute error is taken to be close to the
machine (or user’s) zero.
Let $A=D^{T}D$ and compute its spectral decomposition:
$A=U\left(\begin{array}[]{cc}S^{2}&0\\\ 0&0\end{array}\right)U^{T}.$
Let $B=T^{T}T$ and $\tilde{B}=U^{T}BU$.
Compute $rank(D)=r$ and let
$\displaystyle{\tilde{B}}_{rr}=\tilde{B}(1:r,1:r),$
$\displaystyle{\tilde{B}}_{r,n-r}=\tilde{B}(1:r,r+1:n),$
$\displaystyle{\tilde{B}}_{n-r,n-r}=\tilde{B}(r+1:n,r+1:n)$
Let $\bar{D}=S$, assume $\bar{T}$ satisfies
${\tilde{B}}_{rr}={\bar{T}}^{T}{\bar{T}}$.
Perform Algorithm 1 with input parameters $D=\bar{D}$ and $T=\bar{T}$, and let
${\tilde{X}}_{rr}=X^{\ast}$.
Solve the linear system (18b) to compute ${\tilde{X}}_{r,n-r}$ and let
${\tilde{X}}_{n-r,r}={{\tilde{X}}_{r,n-r}}^{T}$.
Compute the spectral decomposition for $B$, that is,
$B=V\left(\begin{array}[]{cc}D^{2}&0\\\ 0&0\\\ \end{array}\right)V^{T}.$
Compute $M={U_{r}}^{T}V_{r}$.
If
$\|{U_{n-r}}^{T}(BU_{r}{({U_{r}}^{T}BU_{r})}^{-1}{U_{r}}^{T}B-B)\|\geq\delta$
stop ((9) has no solution)
Else
Let the Cholesky decomposition of ${\tilde{X}}_{rr}$ be
${\tilde{X}}_{rr}=\tilde{L}{\tilde{L}}^{T}$ and set $L_{rr}=\tilde{L}$.
Solve the lower triangular system (65b) to compute $L_{n-r,r}$.
Let $L_{n-r,n-r}\in{\mathbb{R}}^{(n-r)\times(n-r)}$ be an arbitrary
nonsingular lower triangular matrix and compute ${\tilde{X}}_{n-r,n-r}$ using
(65d).
Let
$\tilde{X}=\left(\begin{array}[]{cc}{\tilde{X}}_{rr}&{\tilde{X}}_{r,n-r}\\\
{\tilde{X}}_{n-r,r}&{\tilde{X}}_{n-r,n-r}\\\ \end{array}\right)$ and
$X^{\ast}=U\tilde{X}U^{T}$.
Compute $E=\mathop{\mathrm{tr}}((DX^{\ast}-T)(D-T{X^{\ast}}^{-1})).$
EndIf.
Next, we show how to use the complete orthogonal decomposition of the data
matrix $D$ instead of the spectral decomposition of $A$.
Note (Complete Orthogonal Decomposition) [27] Let $A\in{\mathbb{R}}^{m\times
n}$ be an arbitrary matrix with $rank(A)=r$. There exist
$R\in{\mathbb{R}}^{r\times r}$, $U\in{\mathbb{R}}^{m\times m}$ and
$V\in{\mathbb{R}}^{n\times n}$ so that $R\in{\mathbb{R}}^{r\times r}$ is upper
triangular, $UU^{T}=U^{T}U=I$, $VV^{T}=V^{T}V=I$ and
$A=U\left(\begin{array}[]{cc}R&0\\\ 0&0\\\ \end{array}\right)V^{T}.$
Next, Algorithm 4 is presented using the complete orthogonal decomposition of
$D$.
Solving the EIV model for positive definite linear system with rank deficient
data and target matrices using complete orthogonal decomposition.
$\delta$ as the upper bounds for absolute error is taken to be close to the
machine (or user’s) zero.
Compute the complete orthogonal decomposition of $D$, that is,
$D=U\left(\begin{array}[]{cc}R&0\\\ 0&0\\\ \end{array}\right)V^{T}.$
Let $A=D^{T}D=V_{r}R^{T}R{V_{r}}^{T}$, $B=T^{T}T$ and $\tilde{B}=V^{T}BV$,
where $V_{r}$ consists of the first $r$ columns of $V$.
Compute $rank(D)=r$ and let
$\displaystyle{\tilde{B}}_{rr}=\tilde{B}(1:r,1:r),$
$\displaystyle{\tilde{B}}_{r,n-r}=\tilde{B}(1:r,r+1:n),$
$\displaystyle{\tilde{B}}_{n-r,n-r}=\tilde{B}(r+1:n,r+1:n).$
Let $\bar{D}=R$, assume $\bar{T}$ satisfies
${\tilde{B}}_{rr}={\bar{T}}^{T}{\bar{T}}$.
Perform Algorithm 1 with input parameters $D=\bar{D}$ and $T=\bar{T}$, and let
${\tilde{X}}_{rr}=X^{\ast}$.
Solve the linear system (18b) to compute ${\tilde{X}}_{r,n-r}$ and let
${\tilde{X}}_{n-r,r}={{\tilde{X}}_{r,n-r}}^{T}$.
Compute the spectral decomposition for $B$, that is,
$B=V\left(\begin{array}[]{cc}D^{2}&0\\\ 0&0\\\ \end{array}\right)V^{T}.$
Compute $M={U_{r}}^{T}V_{r}$.
If
$\|{U_{n-r}}^{T}(BU_{r}{({U_{r}}^{T}BU_{r})}^{-1}{U_{r}}^{T}B-B)\|\geq\delta$
stop ((9) has no solution)
Else
Let the Cholesky decomposition of ${\tilde{X}}_{rr}$ be
${\tilde{X}}_{rr}=\tilde{L}{\tilde{L}}^{T}$ and set $L_{rr}=\tilde{L}$.
Solve the lower triangular system (65b) to compute $L_{n-r,r}$.
Let $L_{n-r,n-r}\in{\mathbb{R}}^{(n-r)\times(n-r)}$ be an arbitrary
nonsingular lower triangular matrix and compute ${\tilde{X}}_{n-r,n-r}$ using
(65d).
Let
$\tilde{X}=\left(\begin{array}[]{cc}{\tilde{X}}_{rr}&{\tilde{X}}_{r,n-r}\\\
{\tilde{X}}_{n-r,r}&{\tilde{X}}_{n-r,n-r}\\\ \end{array}\right)$ and
$X^{\ast}=U\tilde{X}U^{T}$.
Compute $E=\mathop{\mathrm{tr}}((DX^{\ast}-T)(D-T{X^{\ast}}^{-1})).$
EndIf.
Thus, based on the above study, the computational complexity of PDEIV-QR is
lower than that of PDEIV-Spec, for all matrix sizes. But, for the case of rank
deficient data matrix, depending on the matrix size and rank, one of the
algorithms PDEIV-RD-Spec and PDEIV-RD-COD may have a lower computational
complexity.
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|
# Dummy Prototypical Networks for Few-Shot Open-Set Keyword Spotting
###### Abstract
Keyword spotting is the task of detecting a keyword in streaming audio.
Conventional keyword spotting targets predefined keywords classification, but
there is growing attention in few-shot (query-by-example) keyword spotting,
e.g., $N$-way classification given $M$-shot support samples. Moreover, in
real-world scenarios, there can be utterances from unexpected categories
(open-set) which need to be rejected rather than classified as one of the $N$
classes. Combining the two needs, we tackle few-shot open-set keyword spotting
with a new benchmark setting, named splitGSC. We propose episode-known dummy
prototypes based on metric learning to detect an open-set better and introduce
a simple and powerful approach, Dummy Prototypical Networks (D-ProtoNets). Our
D-ProtoNets shows clear margins compared to recent few-shot open-set
recognition (FSOSR) approaches in the suggested splitGSC. We also verify our
method on a standard benchmark, miniImageNet, and D-ProtoNets shows the state-
of-the-art open-set detection rate in FSOSR.
Index Terms: Few-shot learning, Open-set Recognition, Keyword Spotting, Dummy
Prototype, Prototypical Networks
## 1 Introduction
Keyword spotting (KWS) detects keywords like “Hey, Google” and “Hey, Siri” in
streaming audio. KWS systems usually target edge devices such as mobile phones
and smart speakers, and previous studies have concentrated on better network
designs in terms of detection rate [1, 2] and computational cost [3, 4] while
targeting multi-keyword classifications with Google speech commands dataset
(GSC) version 1 and 2 [5].
Recently, there has been growing attention in query-by-example (few-shot)
keyword spotting systems [6, 7, 8, 9]. Few-shot learning (FSL) absorbs
knowledge from a training dataset and leverages the knowledge to adapt to
evaluation tasks of unseen categories using only a few labeled (support)
samples. However, on top of FSL, real-world scenarios naturally meet
utterances of unexpected categories without support examples, and neural
networks tend to be over-confident [10] and can misjudge those unexpected
samples by one of the FSL classes. Thus, there is a need to detect those
unseen open-set classes (Open-Set Recognition (OSR) [11]). This work
introduces OSR to few-shot keyword spotting that is a more challenging
setting, few-shot open-set recognition (FSOSR) [12] for KWS.
Figure 1 shows example episodes of FSL and FSOSR while using an FSL method,
Prototypical Networks (ProtoNets) [13]. In an $N$-way $M$-shot episode, the
goal of FSL is correctly classifying $N$ classes that are unseen during
training but known using $M$ support samples for each. FSL does not consider
open-set classes out of the $N$ classes. On the other hand, FSOSR needs to
distinguish an unknown open-set from the known classes while still performing
FSL. FSOSR is more challenging than conventional OSR because an open-set
changes over episodes based on the choice of $N$ classes (Figure 1 bottom).
Thus, a desirable FSOSR method needs to adapt to the varying open-set. We
predict episode-specific (episode-known) dummies based on support examples in
each episode and classify an open-set as the dummies. Using the episode-known
dummies, we propose Dummy Prototypical Networks (D-ProtoNets).
For few-shot open-set keyword spotting (FSOS-KWS), we introduce a benchmark
setting named splitGSC, a subset of GSC ver2. Our D-ProtoNets achieves state-
of-the-art (SOTA) performance in splitGSC. We also verify D-ProtoNets on
miniImageNet [14], a widely used FSL benchmark, and D-ProtoNets is better in
detecting open-set than other baselines.
Figure 1: Example episodes: In few-shot learning, decision boundaries classify
classes given few support samples. Few-shot open-set recognition is required
to distinguish samples of unknown categories (open-set) from those of known
classes. Open-set can vary over episodes in FSOSR. Figure 2: Dummy
Prototypical Network. $f_{\phi}(\cdot)$ and $g_{\varphi}(\cdot)$ are the
encoder and the dummy generator, respectively. ‘K’ and ‘U’ stand for a set of
known classes having support samples and unknown open-set, respectively, and
$d(\cdot)$ is a distance metric.
## 2 Related Works
The literature on few-shot learning and open-set recognition are vast, thus we
focus on the most relevant works.
Few-Shot Learning. Few-shot learning has three popular branches, adaptation,
hallucination, and metric learning methods. The adaptation methods [15] make a
model easy to fine-tune in the low-shot regime, and the hallucination methods
[16] augment training examples for data starved classes. Our approach aligns
with the last one, metric-based learning [13, 14], which learns a metric space
in which distance metrics can classify samples. Especially our method is
designed on top of Prototypical Networks (ProtoNets) [13]. Recently, FEAT [17]
shows that it is helpful to make support samples task-specific using a set-to-
set function, Transformer [18], in FSL. Also, some approaches have addressed
few-shot KWS [7, 8, 9], but to the best of our knowledge, this is the first
work introducing few-shot open-set keyword spotting (FSOS-KWS).
Open-Set Recognition. [11] introduced OSR to deep learning, and discriminative
or generative approaches have been proposed by [19, 20, 21]. Recently, [22]
suggests placeholders for both data and classifiers using manifold mixup [23]
and learnable dummy classifiers, respectively. Their dummy classifiers are
learnable, fixed, and thus suitable for OSR with fixed closed-set. On the
other hand, our D-ProtoNets suggests episode-known, varying dummies for FSOSR.
Few-shot open-set recognition. There is growing attention to the FSOSR due to
its importance, and the previous studies [12, 24] concentrate on better OSR
while preserving FSL performance. PEELER [12] suggests entropy maximization
loss for an open-set and Gaussian Embedding for flexible decision boundaries.
Based on a set-to-set transformation method [17], [24] introduces SnaTCHer,
the SOTA FSOSR approach that compares the distance between transformed and
modified prototypes and detects open-set using transformation consistency.
Those works try to detect abnormality of open-set while our method directly
learns a dummy class to detect open-set.
## 3 Method
### 3.1 Preliminaries
Notations. An FSOSR setting consists of seen training data
$\mathcal{D}_{\text{train}}$ and unseen evaluation data
$\mathcal{D}_{\text{eval}}$ that do not overlap classes.
$\mathcal{D}_{\text{train}}$ is composed of labeled samples,
$\\{(\bm{x}_{i},y_{i})\\}_{i=1}^{|\mathcal{D}_{\text{train}}|}$, where
$\bm{x_{i}}$ is an input feature, and $y_{i}$ is its corresponding label.
During training, a model learns from $N$-way $M$-shot pseudo-FSOSR episodes,
each of which has $N$ known classes offering $M$ support examples for each
class and pseudo-unknown (pseudo-open-set) classes without any support
examples. We denote a support set and a query set as $S$ and $Q$,
respectively, in an episode. $S$ contains samples from $N$ classes, and $Q$
contains queries from both $N$ known and $N_{U}$ unknown classes where
$|S|=NM$, $|Q|=(N+N_{U})M_{Q}$, and $M_{Q}$ is the number of queries for each
class. At inference time, all classes of $\mathcal{D}_{\text{eval}}$ are
unseen, and again an episode consists of $N$ known classes with support
samples and $N_{U}$ unknown open-set classes.
Prototypical Networks. Our work is in line with metric-learning-based
approaches and is based on the representative work, Prototypical Networks
(ProtoNets) [13]. In an N-way M-shot episode, ProtoNets gets $N$ prototypes,
$\\{\bm{c}_{n}\\}_{n=1}^{N}$, using the average of the support samples of each
class, $n$ , where
$\bm{c}_{n}=\frac{1}{M}\sum_{(\bm{x}_{i},y_{i})\in
S_{n}}{f_{\phi}(\bm{x}_{i})},$ (1)
$S_{n}$ is a subset of $S$ whose labels are $n$, $|S_{n}|=M$, and $f_{\phi}$
is an encoder with $\phi$ parameters. Based on the prototypes, ProtoNets gets
distribution over $N$ classes,
$p_{\phi}(y=n|\bm{x})=\frac{\exp(-d(f_{\phi}(\bm{x}),\bm{c}_{n}))}{\sum_{n^{\prime}=1}^{N}{\exp(-d(f_{\phi}(\bm{x}),\bm{c}_{n^{\prime}})})},$
(2)
where $d(\cdot)$ is a distance metric, e.g., Euclidean distance,
$d(z,z^{\prime})=||z-z^{\prime}||^{2}$. Using Eq. 2, ProtoNets minimizes
negative log-probability, $-\log p_{\phi}(y=n|\bm{x})$, of the true class $n$.
### 3.2 Few-shot Open-set Keyword Spotting
Here, we introduce a benchmark-setting of FSOS-KWS using Google speech
commands dataset (GSC) ver2 [5]. There are 35 keywords in total, and
conventional KWS does 12 class classifications following the settings of [5].
The 12 classes consist of 10 keywords ( “Yes,” “No,” “Up,” “Down,” “Left,”
“Right,” “On,” “Off,” “Stop,” and “go”) and two additional classes “Unknown
words” which refers to the remaining 25 keywords, and “Silence” (background
noise only). We split the dataset by class label like miniImageNet [14]. Our
split has 15, 10, and 10 keywords for train, validation, and test set,
respectively, as follows:
* •
Train keywords: “Happy,” “House,” “Bird,” “Bed,” “Backward,” “Sheila,”
“Marvin,” “Wow,” “Tree,” “Follow,” “Dog,” “Visual,” “Forward,” “Learn,” and
“Cat”.
* •
Validation keywords (numbers): “Zero,” “One,” “Two,” “Three,” “Four,” “Five,”
“Six,” “Seven,” “Eight,” and “Nine”.
* •
Test keywords (10 keyword classes used in conventional 12 class KWS): “Yes,”
“No,” “Up,” “Down,” “Left,” “Right,” “On,” “Off,” “Stop,” and “go”.
These fixed keyword splits can prevent possible performance variance from
split changes over trials. On top of the split, we add the particular class
“Silence” which can only be included in an open-set as a background noise
class. For example, in a 5-way 5-shot episode, we randomly choose five known
classes without “Silence” and then choose the same number of open-set classes
from the remaining classes, including “Silence”. We name this specific setting
as split Google speech commands dataset (splitGSC). More details are available
in Section 4.1.
### 3.3 Dummy Prototypical Network
We suggest episode-known dummy prototypes and introduce a simple but powerful
system named Dummy Prototypical Networks (D-ProtoNets) to handle varying open-
sets over episodes.
Episode-known Dummy Prototype. Let us consider $N$ prototypes,
$\\{\bm{c}_{n}\\}_{n=1}^{N}$, in an episode, where $\bm{c}\in R^{1\times D}$
with the output dimension of $f_{\phi}$, $D$. The prototypes are permutation
invariant each other for ProtoNets. Thus, we suggest a dummy generator,
$g_{\varphi}$, with parameters $\varphi$ based on DeepSets [25], which is
inherently permutation invariant. In detail, $g_{\varphi}$ generates a dummy
$\bm{c}_{d}$ using given $N$ prototypes
$C=[\bm{c}_{1};\bm{c}_{2};\cdots;\bm{c}_{N}]\in R^{N\times D}$ as an input,
hence episode-known:
$\bm{c}_{d}=g_{\varphi}(C)=\operatorname*{Maxpool}(g_{1}(C))W_{g},$ (3)
where $g_{1}$ consists of fully-connected (FC) layers with nonlinearity, and
$g_{1}(C)\in R^{N\times H}$ with a hidden dimension $H$.
$\operatorname*{Maxpool}$ operates over $N$ and outputs a feature in
$R^{1\times H}$. $W_{g}$ is a learnable $H\times D$ matrix, and $\bm{c}_{d}\in
R^{1\times D}$. Finally, we get an augmented prototype set
$\\{\bm{c}_{1},\cdots,\bm{c}_{N},\bm{c}_{d}\\}$. We set the labels of open-set
queries to the N+1-th label $y_{d}$ which corresponds to the dummy,
$\bm{c}_{d}$. Then, the Eq. 2 becomes
$p_{\theta}(y=n|\bm{x})=\frac{\exp(-d(f_{\phi}(\bm{x}),\bm{c}_{n})/\tau_{n})}{\sum_{n^{\prime}=1}^{N+1}{\exp(-d(f_{\phi}(\bm{x}),\bm{c}_{n^{\prime}})/\tau_{n})}},$
(4)
where $\theta$ consists of the encoder $\phi$ and the dummy generator
$\varphi$, and $\tau_{n}$ is a softmax temperature which is usually same over
classes. Here we use a larger $\tau_{N+1}$ compared to other $\tau_{n\neq
N+1}$ to let the dummy easily reduce its loss, i.e.,
$\tau_{N+1}=\gamma\cdot\tau_{n\neq N+1}$, where $\gamma>1$. The N+1
classification is learned by cross entropy loss as below:
$\displaystyle\mathcal{L}^{K}_{CE}=\sum_{(\bm{x}_{i},y_{i})\in Q_{K}}{-\log
p_{\theta}(y=y_{i}|\bm{x}_{i})},$
$\displaystyle\mathcal{L}^{U}_{CE}=\sum_{(\bm{x}_{i},y_{d})\in Q_{U}}{-\log
p_{\theta}(y=y_{d}|\bm{x}_{i})},$ (5)
where $Q_{K}$ and $Q_{U}$ are known and unknown queries, respectively. We
balance the two losses by $\lambda$, and the total loss is
$\mathcal{L}_{CE}=\mathcal{L}^{K}_{CE}+\lambda\cdot\mathcal{L}^{U}_{CE}$.
During a test, we classify $N$ known classes by
$\hat{y_{i}}=\operatorname*{arg\,max}_{n\in\\{1,\cdots,N\\}}p_{\theta}(y_{i}=n|\bm{x}_{i})$.
We verify whether a $x_{i}$ is an open-set by comparing
$p_{\theta}(y_{i}=y_{d}|\bm{x}_{i})$ to a threshold $\delta$. Overall system
is described in Figure 2.
Multiple Dummies. We expand D-ProtoNets with multi $L$ dummies by changing
$W_{g}$ to a $H\times(L\cdot D)$ matrix. The model can naively choose most
probable dummy for an input $\bm{x}_{i}$ by
$\operatorname*{arg\,max}_{l}(-d(\bm{x}_{i},\bm{c}_{l}))$. Instead, we use
Gumbel softmax [26] to replace the non-differentiable sample,
$\operatorname*{arg\,max}$, with a differentiable sample. The probability of
choosing dummy $l$ is
$p(y^{L}=l|\bm{x})=\frac{\exp((-d(f_{\phi}(\bm{x}),\bm{c}_{l})+\epsilon_{l})/\tau)}{\sum_{l^{\prime}=1}^{L}{\exp((-d(f_{\phi}(\bm{x}),\bm{c}_{l^{\prime}})+\epsilon_{l^{\prime}})/\tau})},$
(6)
where $\epsilon_{1},\cdots,\epsilon_{L}$ are i.i.d samples drawn from the
standard Gumbel distribution of $\mu=0$ and $\beta=1$ following [26], and
$y^{L}$ is a temporal dummy label among L dummies. During training, we get
$\bm{c}_{d}=\sum_{l}{p(y^{L}=l|\bm{x})\cdot\bm{c}_{l}}$, and we choose $y^{L}$
by $\operatorname*{arg\,max}_{l}p(y^{L}=l|\bm{x})$ at inference time.
## 4 Experiments
### 4.1 Experimental Settings
Dataset. We use Google speech commands (GSC) dataset ver2 [5], containing
105,829 utterances from 2,618 speakers. The dataset is first split to train,
validate, and test sets with 84,843, 9,981, and 11,005 utterances,
respectively, using the official split (using a hash function on the name of
each utterance file) [5]. Then, we chose the samples based on our splitGSC
split and got 22,916, 3,643, and 4,074 samples for train, validation, and
test, respectively. Then following the settings of [5], we add “Silence”
samples to each split by the average number of utterances per class of each
split, and finally, splitGSC has 24,444, 4,007, and 4,482 utterances for
train, validation and test, respectively. Especially, we use the official test
set that [5] offers and apply our split to it. During the training, we use
minimal data augmentation, commonly used in GSC tasks [1, 4, 5]: Adding
official background noise offered by GSC with the probability of 0.8.
Backbones. We experiment with two widely used backbones in previous FSL
benchmarks [13, 17], Conv4-64 [14] and ResNet-12 [27], and one backbone
designed for KWS, BCResNet-8 [4]. Each corresponds to the encoder, $f_{\phi}$,
and their output dimensions are 768, 512, and 256 for Conv4-64, ResNet12, and
BCResNet-8, respectively. Conv4-64 does not have global average pooling at the
end, and thus we get 768 dimensions, larger than its number of channels, 64.
Implementation Details. Each utterance in GSC is 1 sec long, and the sampling
rate is 16 kHz. We use input features of 40-dimensional log Mel-spectrograms
with frameshift and window length of 10 and 30 ms, respectively, following
[4]. We train a model for 100 epochs with Adam optimizer [28] of initial
learning rate of 0.001. The learning rate is step decayed by multiplying 0.5
at every 20 epochs. Each epoch consists of 100 episodes, and an episode has 5
known (5-way) and 5 open-set classes. We use 5 support examples (5-shot), and
all the classes have 5 and 15 queries for each during training and test,
respectively. We use early-stop by few-shot validation accuracy and evaluate a
trained model with 1,000 episodes.
We design the dummy generator, $g$, as simple as possible. We use $g_{1}$ of
FC-ReLU-FC with hidden $D=32$. We use Euclidean distance for $d$ and set
hyperparameter $\lambda=0.1$ as a default setting. During training, the
softmax temperatures $\tau_{n\neq N+1}$ are fixed to 1, and $\gamma=3$, i.e.,
$\tau_{N+1}=3$, for $\mathcal{L}_{CE}$. The $\tau$ in Gumbel softmax is cosine
annealed from 2 to 0.5 in Eq. 6 following [26]. We use $L=3$ dummies as
default.
Table 1: splitGSC: 5-way {1, 5}-shot FSOSR results. The numbers are mean (std)
over 5 trials (%). (bold: best)
| | 1-shot | 5-shot
---|---|---|---
Model | Backbone | Accuracy | AUROC | Accuracy | AUROC
ProtoNet | Conv4-64 | 43.2 (0.7) | 55.3 (0.5) | 67.6 (1.0) | 63.8 (0.6)
FEAT | Conv4-64 | 46.9 (1.2) | 61.1 (0.3) | 65.6 (1.3) | 65.5 (0.7)
PEELER | Conv4-64 | 42.8 (1.5) | 57.5 (1.0) | 66.3 (1.8) | 66.7 (1.0)
SnaTCHer-F | Conv4-64 | 47.3 (1.2) | 51.1 (0.7) | 67.0 (1.6) | 53.1 (1.4)
D-ProtoNet, L=3 | Conv4-64 | 45.3 (0.9) | 65.9 (0.3) | 69.6 (0.8) | 73.9 (0.7)
ProtoNet | ResNet-12 | 68.3 (1.0) | 60.7 (0.7) | 85.9 (0.7) | 68.4 (1.2)
FEAT | ResNet-12 | 68.8 (1.1) | 65.6 (0.7) | 84.0 (0.9) | 71.0 (0.7)
PEELER | ResNet-12 | 65.4 (1.5) | 66.2 (0.9) | 82.4 (1.1) | 72.1 (1.7)
SnaTCHer-F | ResNet-12 | 70.4 (1.0) | 73.2 (1.0) | 84.7 (0.6) | 83.3 (0.8)
D-ProtoNet, L=3 | ResNet-12 | 69.7 (1.0) | 78.8 (0.3) | 86.9 (0.3) | 86.7 (0.3)
ProtoNet | BCResNet-8 | 66.7 (0.7) | 60.9 (0.7) | 83.1 (0.5) | 68.8 (0.6)
D-ProtoNet, L=3 | BCResNet-8 | 65.2 (1.4) | 75.7 (0.9) | 81.9 (1.1) | 82.3 (1.5)
Baselines. We compare our method to other notable approaches: ProtoNet [13],
FEAT [17], PEELER [12], and SOTA FSOSR method, SnaTCHer [24] based on their
available official implementations. There are the following changes to fit the
baselines better to our splitGSC carefully. We set the hidden dimension of 16
for the Transformers in FEAT and SnaTCHer-F. Also, we set dropout rates in the
Transformers of 0.5 and 0 for Conv4-64 and 0.6 and 0.1 for ResNet-12
experiments.
### 4.2 Results
Table 1 shows overall results in splitGSC. ‘Acc’ stands for FSL accuracy, and
we use the threshold-free area under the receiver-operating characteristics
(AUROC) as an OSR measure following previous FSOSR approaches [12, 24]. By
introducing dummy prototypes and additional losses to various backbones, our
D-ProtoNet significantly improves vanilla ProtoNet in AUROC and shows better
FSL accuracies. Other baselines also improve vanilla ProtoNet, but D-ProtoNets
shows clear margins. SnaTHCer is the strong baseline, and they suggest
directly using distance metric instead of softmax output to detect open-set,
i.e., $\bm{x}_{i}$ is an open-set sample if
$\max_{n\in{1,\cdots,N}}\\{-d(f_{\phi}(\bm{x}_{i}),\bm{c}_{n})\\}<\delta$
while other approaches [12, 29, 30] usually use
$\max_{n\in{1,\cdots,N}}p(y_{i}=n|\bm{x}_{i})$. However, we observed that the
unnormalized distance metric does not always hold for detecting open-set and
shows poor AUROC with Conv4-64 in splitGSC and our training details.
Table 2: miniImageNet: 5-way {1, 5}-shot FSOSR results. The numbers are mean
(std) over 5 trials (%). *SnaTCHer results are quoted from the paper.
| | 1-shot | 5-shot
---|---|---|---
Model | Backbone | Accuracy | AUROC | Accuracy | AUROC
ProtoNet | ResNet-12 | 63.1 (0.4) | 55.3 (0.1) | 82.2 (0.2) | 61.0 (0.4)
FEAT | ResNet-12 | 67.1 (0.4) | 59.6 (0.5) | 82.3 (0.4) | 62.5 (0.2)
PEELER | ResNet-12 | 62.1 (0.2) | 59.2 (0.3) | 81.8 (0.2) | 67.4 (0.3)
SnaTCHer-F | ResNet-12 | 67.4 (0.4) | 69.4 (0.8) | 82.4 (0.1) | 76.4 (0.7)
*SnaTCHer-F [24] | ResNet-12 | 67.0 | 68.3 | 82.0 | 77.4
*SnaTCHer-T [24] | ResNet-12 | 66.6 | 70.2 | 81.8 | 76.7
*SnaTCHer-L [24] | ResNet-12 | 67.6 | 69.4 | 82.4 | 76.2
D-ProtoNet, L=1 | ResNet-12 | 63.2 (0.3) | 69.7 (0.3) | 82.1 (0.1) | 76.6 (0.5)
D-ProtoNet, L=3 | ResNet-12 | 63.4 (0.4) | 70.2 (0.4) | 81.8 (0.2) | 77.8 (0.3)
D-ProtoNet, L=3 + FEAT | ResNet-12 | 65.1 (0.2) | 70.6 (0.6) | 81.9 (0.1) | 78.5 (0.9)
### 4.3 miniImageNet
We further verify D-ProtoNets on widely used mini-ImageNet [14]. The dataset
is the subset of ImageNet [31] and contains 100 classes with 600 84$\times$84
images for each. Following [32], we split the data into train, validation, and
test sets with 64, 16, and 20 classes, respectively. We follow the
implementations and training details of FEAT and SnaTCHer’s official
implementations for miniImageNet. The backbone is the ResNet-12 introduced by
[33], whose output dimension is 640. Backbones are pretrained by adding a
softmax layer to classify all 64 seen classes [17, 24]. Then, the model is
trained for 200 epochs with 100 episodes for each. In each 5-way episode, we
randomly choose 5 open-set classes, and 15 queries for all classes. We use
softmax temperature $\tau=64$ for FEAT and SnaTCHer, following their best
settings. We optimized $\tau$ of vanilla ProtoNet in
$\\{0.1,1,16,32,64,128\\}$ following [17] and found that $\tau=16$ works best.
D-ProtoNets use $\tau_{n\neq N+1}=16$ and $\gamma=3$.
Table 2 shows the comparison between D-ProtoNets and various baselines. Our
dummy prototypes improve vanilla ProtoNets in detecting open-set samples while
preserving FSL accuracies, and achieve SOTA level open-set detection. FEAT and
SnaTCHer show better FSL accuracies than ours, especially for 1-shot settings.
We interpret that their additional set-to-set transformation for support
examples makes the extreme low shot regime more robust by using the relation
between a few support examples. However, both methods do additional
transformations, and SnaTCHer requires heavy computations due to the
transformations for all the queries.
We also tried the transformation method FEAT to our D-ProtoNets to improve
further. At inference time, we observed that using an unnormalized distance
metric, $-d(f_{\phi}(\bm{x}_{i}),\bm{c}_{N+1})$, instead of softmax output
[24] increases the 1-shot and 5-shot AUROC from 68.5 to 70.6 and from 78.2 to
78.5, respectively, while using the D-ProtoNets+FEAT. The result implies that
D-ProtoNets can be used with a complementary concept like FEAT.
### 4.4 Ablation Study
Table 3: Ablations. splitGSC: 5-way {1, 5}-shot FSOSR results using ResNet-12.
The numbers are mean (std) over 5 trials (%). ‘Gum.’ stands for Gumbel
softmax.
Methods | 1-shot | 5-shot
---|---|---
L | $\mathcal{L}^{U}_{CE}$ | $\gamma$ | Gum. | Accuracy | AUROC | Accuracy | AUROC
- | | - | - | 68.3 (1.0) | 60.7 (0.7) | 85.9 (0.7) | 68.4 (1.2)
3 | ✓ | 1 | | 69.9 (1.1) | 76.4 (1.1) | 86.4 (1.1) | 82.7 (1.2)
3 | ✓ | 3 | | 69.2 (1.4) | 78.7 (0.9) | 86.5 (0.7) | 86.2 (0.9)
3 | ✓ | 3 | ✓ | 69.7 (1.0) | 78.8 (0.3) | 86.9 (0.3) | 86.7 (0.3)
1 | ✓ | 3 | - | 70.0 (0.5) | 78.7 (0.1) | 86.7 (0.3) | 85.9 (0.4)
2 | ✓ | 3 | ✓ | 69.1 (0.5) | 78.5 (0.4) | 86.4 (0.2) | 86.1 (0.6)
3 | ✓ | 3 | ✓ | 69.7 (1.0) | 78.8 (0.3) | 86.9 (0.3) | 86.7 (0.3)
4 | ✓ | 3 | ✓ | 70.3 (0.7) | 78.8 (0.6) | 86.9 (0.7) | 86.5 (1.2)
5 | ✓ | 3 | ✓ | 69.2 (0.5) | 78.4 (0.7) | 86.4 (0.4) | 86.7 (0.5)
3 | ✓ | 1 | ✓ | 70.1 (2.0) | 76.8 (0.8) | 86.4 (1.1) | 82.2 (0.6)
3 | ✓ | 2 | ✓ | 69.8 (1.0) | 78.3 (0.4) | 87.0 (0.7) | 85.9 (0.6)
3 | ✓ | 3 | ✓ | 69.7 (1.0) | 78.8 (0.3) | 86.9 (0.3) | 86.7 (0.3)
3 | ✓ | 5 | ✓ | 69.8 (1.1) | 78.7 (0.6) | 86.5 (0.6) | 86.4 (0.8)
3 | ✓ | 10 | ✓ | 70.3 (1.0) | 79.1 (0.5) | 86.3 (0.6) | 86.7 (0.3)
Ablations. We do some ablation studies for D-ProtoNets, and Table 3 shows the
effect of our loss terms, number of dummies, $\tau_{N+1}$, and Gumbel softmax.
The top row implies the baseline, ProtoNet. By adding dummies, there are
considerable improvements of more than 10 % in AUROC. We could get further
improvements through introducing Gumbel softmax and the $\tau_{N+1}=3$ while
$\tau_{n\neq N+1}$ are fixed to 1. We also experiment with how the number of
dummies, L, affects D-ProtoNets. D-ProtoNets shows better results as L
increases from $1$ to $3$ and does not show further improvements with $L=4$
and $5$ (seem converged at $L=3$). We further experiment the effect of various
$\gamma\in\\{1,2,3,5,10\\}$ for $\tau_{N+1}$. The AUROC increases as $\gamma$
increases from $1$ to $3$ and seems to converge near $\gamma=3$.
Erase undesirable instance discrepancy. Recently, [34] shows the importance of
making few-shot models concentrate on foreground objects rather than
backgrounds in images. Motivated by [34], we use an explicit normalization
along the frequency-axis, named Relaxed instance Frequency-wise Normalization
(RFN) [35, 36], to reduce undesirable instance discrepancy in audio features.
The RFN module operates its input $\bm{x}$ and outputs
$\lambda\cdot\text{LN}(\bm{x})+(1-\lambda)\cdot\text{IFN}(\bm{x})$, where IFN
is instance normalization [37] along frequency-axis, and LN is layer
normalization [38] for relaxing the effect of IFN. Here we use the relaxation
$\lambda=0.5$ and apply RFN at the input of the encoder $f_{\phi}$. We expect
RFN to make the model concentrate on keywords more than other discrepancies,
e.g., speaker ID. Table 4 shows that RFN consistently improves ProtoNet and
D-ProtoNet with various backbones. The results indicate that erasing
undesirable instance discrepancy is highly important in FSOS-KWS.
Table 4: Using RFN. splitGSC: 5-way {1, 5}-shot FSOSR results. The numbers are
mean (std) over 5 trials (%).
| | 1-shot | 5-shot
---|---|---|---
Model | Backbone | Accuracy | AUROC | Accuracy | AUROC
ProtoNet | Conv4-64 | 43.2 (0.7) | 55.3 (0.5) | 67.6 (1.0) | 63.8 (0.6)
\+ RFN | Conv4-64 | 46.3 (0.9) | 57.3 (0.7) | 69.9 (1.0) | 65.3 (0.6)
D-ProtoNet, L=3 | Conv4-64 | 45.3 (0.9) | 65.9 (0.3) | 69.6 (0.8) | 73.9 (0.7)
\+ RFN | Conv4-64 | 48.8 (0.6) | 69.3 (0.5) | 71.9 (0.7) | 76.9 (0.4)
ProtoNet | ResNet-12 | 68.3 (1.0) | 60.7 (0.7) | 85.9 (0.7) | 68.4 (1.2)
\+ RFN | ResNet-12 | 70.5 (1.0) | 62.9 (0.7) | 87.3 (0.8) | 70.9 (1.4)
D-ProtoNet, L=3 | ResNet-12 | 69.7 (1.0) | 78.8 (0.3) | 86.9 (0.3) | 86.7 (0.3)
\+ RFN | ResNet-12 | 72.6 (0.5) | 80.3 (1.0) | 88.3 (0.6) | 87.8 (0.9)
ProtoNet | BCResNet-8 | 66.7 (0.7) | 60.9 (0.7) | 83.1 (0.5) | 68.8 (0.6)
\+ RFN | BCResNet-8 | 71.1 (1.6) | 65.2 (1.0) | 86.3 (0.7) | 74.2 (1.0)
D-ProtoNet, L=3 | BCResNet-8 | 65.2 (1.4) | 75.7 (0.9) | 81.9 (1.1) | 82.3 (1.5)
\+ RFN | BCResNet-8 | 69.7 (0.5) | 78.3 (0.3) | 85.5 (0.3) | 85.4 (0.6)
## 5 Conclusion
This work tackles few-shot open-set recognition in keyword spotting (FSOS-KWS)
and suggests a new benchmark, splitGSC. To adapt to the varying open-set, we
introduce episode-known dummies to Prototypical Networks (ProtoNets), named
Dummy Prototypical Networks (D-ProtoNets). D-ProtoNets shows clear margins
compared to recent baselines in splitGSC and achieves SOTA open-set detection
in miniImageNet. We also suggest future research, erasing undesirable instance
discrepancy, for FSOS-KWS.
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|
# The effect of magnetic field on the inner Galactic rotation curve
Man Ho Chan1 , Antonino Del Popolo2,3
1Department of Science and Environmental Studies, The Education University of
Hong Kong, Tai Po, Hong Kong
2Dipartimento di Fisica e Astronomia, University of Catania, Viale Andrea
Doria 6, 95125, Catania, Italy
3Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya str. 48,
119017 Moscow, Russia<EMAIL_ADDRESS>
(Accepted XXXX, Received XXXX)
###### Abstract
In the past few decades, some studies pointed out that magnetic field might
affect the rotation curves in galaxies. However, the impact is relatively
small compared with the effects of dark matter and the baryonic components. In
this letter, we revisit the impact of magnetic field on the rotation curve of
our Galaxy. We show that the inner Galactic rotation curve could be affected
significantly by the magnetic field. The addition of the inner bulge
component, which has been proposed previously to account for the inner
rotation curve data, is not necessary. The magnetic field contribution can
fully account for the excess of the inner rotation velocity between 5 pc to 50
pc from the Galactic Centre. Our analysis can also constrain the azimuthal
component of the central regular magnetic field strength to $B_{0}\sim 50-60$
$\mu$G, which is consistent with the observed range.
###### keywords:
Galaxy: centre; Galaxy: kinematics and dynamics
††pagerange: The effect of magnetic field on the inner Galactic rotation
curve–References††pubyear: XXXX
## 1 Introduction
The rotation curves of galaxies are important indicators of mass distribution
in galaxies. Many rotation curves were revealed by the neutral atomic gas
(e.g. HI, CO) which is believed to be a very good tracer of the gravitational
field (e.g. Spitzer Photometry and Accurate Rotation Curves (Lelli, McGaugh &
Schombert, 2016)). In particular, many past studies have shown that large-
scale magnetic field can affect the gas dynamics in spiral galaxies
(Piddington, 1964; Nelson, 1988; Battaner et al., 1992; Battaner & Florido,
1995). Some studies even show that this magnetic field effect can explain the
flatness of rotation curves in galaxies without the need of dark matter
(Nelson, 1988; Battaner et al., 1992; Battaner & Florido, 1995). This is known
as the ‘magnetic alternative to dark matter’ (Sánchez-Salcedo & Reyes-Ruiz,
2004). Nevertheless, later studies have shown that the boost of rotation
curves due to magnetic field contribution is less than 20 km/s at the
outermost point of rotation curves (Sánchez-Salcedo & Reyes-Ruiz, 2004;
Sánchez-Salcedo & Santillán A., 2013). Since then, the ‘magnetic alternative
to dark matter’ was no longer a popular model.
In the past decade, the idea of the magnetic field effect on galactic rotation
curves was revived. Some studies have shown that the magnetic field effect can
explain why rotation curves in some galaxies start to rise again at the outer
edges of the HI discs, such as our Galaxy (Ruiz-Granados et al., 2012) and the
M31 galaxy (Ruiz-Granados et al., 2010). Considering the effects of magnetic
field can somewhat improve the fits of the outer part of galactic rotation
curves. However, some other studies have argued that the effect in the outer
rotation curve region is not very significant (Sánchez-Salcedo & Santillán A.,
2013; Elstner, Beck & Gressel, 2014).
Although the effect of magnetic field in the outer rotation curve region has
been greatly debated, such effect in the inner region of a galaxy has not been
discussed thoroughly. In this letter, we particularly investigate the magnetic
field effect on the inner rotation curve of our Galaxy. There is a small
rotation velocity excess range (between 5 pc to 50 pc from the Galactic
Centre) which could not be accounted by the contributions of the supermassive
black hole and the central bulge component (Sofue, 2013). An extra inner bulge
has to be added to account for this abnormal excess range. We show that this
small excess range could be explained by the magnetic field effect so that
adding the extra inner bulge component is not necessary.
## 2 Magnetic field effect on rotation curve
In a gaseous disc in equilibrium, magnetic field effects on the gas can be
modelled as a pressure term in the asymmetric drift (Sánchez-Salcedo &
Santillán A., 2013). Such asymmetric drift is a consequence of the support by
thermal, turbulent, cosmic ray and magnetic pressures. The dynamical effects
of the regular magnetic field can significantly boost the gravitational
orbital velocity due to the magnetic tension (Nelson, 1988). The total
magnetic field in a galaxy can be simply expressed as a sum of the regular
field term (the azimuthal component) $B_{\phi}$ and a random field term (the
turbulent magnetic field component) $B_{\rm ran}$ (Sánchez-Salcedo & Santillán
A., 2013; Elstner, Beck & Gressel, 2014). In particular, the random field can
be isotropic or anisotropic. The contribution of the regular magnetic field
component to the circular velocity is given by (Ruiz-Granados et al., 2010)
$v_{\rm
B1}^{2}=\frac{r}{4\pi\rho_{g}}\left(\frac{B_{\phi}^{2}}{r}+\frac{1}{2}\frac{dB_{\phi}^{2}}{dr}\right),$
(1)
where $\rho_{g}$ is the gas density and $r$ is the radial distance from the
Galactic Centre. The random magnetic field component would contribute to the
circular velocity via the magnetic pressure term $P_{B}$ as (Sánchez-Salcedo &
Santillán A., 2013)
$v_{\rm
B2}^{2}=\frac{r}{\rho_{g}}\frac{dP_{B}}{dr}=\frac{r}{\rho_{g}}\frac{d}{dr}\frac{\langle
B_{\rm ran}^{2}\rangle}{8\pi},$ (2)
where $\langle B_{\rm ran}^{2}\rangle$ is the mean-square value of the random
magnetic field strength. Therefore, the total contribution of the magnetic
field to the circular velocity is:
$v_{\rm mag}^{2}=v_{\rm B1}^{2}+v_{\rm
B2}^{2}=\frac{r}{8\pi\rho_{g}}\left[\frac{2B_{\phi}^{2}}{r}+\frac{d}{dr}(B_{\phi}^{2}+\langle
B_{\rm ran}^{2}\rangle)\right].$ (3)
In the inner Galactic Centre region ($r\leq 500$ pc), the rotation curve is
also contributed by the supermassive black hole $v_{\rm BH}$ and the baryonic
bulge $v_{\rm bulge}$ components. Therefore, including the magnetic field
contribution, the observed total rotation curve is
$v^{2}=v_{\rm BH}^{2}+v_{\rm bulge}^{2}+v_{\rm mag}^{2}.$ (4)
The supermassive black hole rotation curve contribution is
$v_{\rm BH}^{2}=\frac{GM_{\rm BH}}{r},$ (5)
where $M_{\rm BH}=(4.154\pm 0.014)\times 10^{6}M_{\odot}$ is the mass of the
supermassive black hole (Abuter et al., 2020). The bulge mass density can be
modeled by the exponential spheroid model with scale radius $a$ as (Sofue,
2013):
$\rho_{\rm bulge}(r)=\rho_{c}e^{-r/a},$ (6)
where $\rho_{c}$ is the central bulge mass density. Therefore, the bulge
rotation curve contribution is given by
$v_{\rm bulge}^{2}=\frac{GM_{0}}{r}F\left(\frac{r}{a}\right),$ (7)
where $M_{0}=8\pi a^{3}\rho_{c}$ and $F(x)=1-e^{-x}(1+x+x^{2}/2)$ (Sofue,
2013). We take the values $M_{0}=8.4\times 10^{9}M_{\odot}$ and $a=0.12$ kpc
obtained in Sofue (2013) to perform our analysis.
To match the inner Galactic rotation curve data without considering the
magnetic field contribution, two bulge components (inner bulge and outer
bulge) were assumed in previous studies (Sofue, 2013). However, in the
followings, we will investigate whether the magnetic field contribution can
mimic the effect of the inner bulge component. Therefore, we assume that there
is only one bulge component to minimise the number of parameters in our
analysis.
Moreover, the dark matter contribution is not considered here because it is
not significant in the inner Galactic Centre region ($r\leq 500$ pc). Assuming
the Navarro-Frenk-White (NFW) profile (Navarro, Frenk & White, 1997), the
total mass of dark matter is less than 5% of the bulge mass inside 500 pc,
with the fitted parameters in Sofue (2015). Therefore, we neglect the
contribution of dark matter for simplicity. We also neglect the gravitational
contribution of gas in the inner Galactic Centre region as the gas mass is
less than 3% of the bulge mass inside 500 pc (Ferrière, Gillard & Jean, 2007).
Over $\sim 300$ pc along the Galactic plane and $\sim 150$ pc in the vertical
direction at the Galactic Centre, the magnetic field is approximately
horizontal and the strength is very strong, which can range from $B\sim
0.01-1$ mG for general intercloud medium (Ferrière, 2009). The equipartition
between magnetic field energy and turbulent energy suggests that
$B\propto\rho_{g}^{1/2}$ (Schleicher et al., 2010). This is also supported by
the observed relation between magnetic field and star-formation rate (Heesen
et al., 2014; Tabatabaei et al., 2017). Therefore, we assume that the regular
magnetic field profile follows the exponential spheroid model as
$B_{\phi}=B_{0}e^{-r/2a},$ (8)
where $B_{0}$ is the central regular magnetic field strength. Note that Eq.
(8) follows from Eq. (6) only if gas density is assumed to be proportional to
the baryonic bulge mass density $\rho_{\rm bulge}(r)$. For the random magnetic
field, it is an important component in the interstellar medium of galaxies
(Beck et al., 2019). We define $\eta$ as the ratio of the regular magnetic
field to the total magnetic field so that the random magnetic field strength
can be expressed as $\langle B_{\rm
ran}^{2}\rangle=(\eta^{-2}-1)B_{\phi}^{2}$. Although some earlier studies
found that the ratio is around $\eta\sim 0.6-0.7$ in some galaxies (Fletcher
et al., 2004; Beck, 2007; Sánchez-Salcedo & Santillán A., 2013), recent
studies have shown that these ordered fields are dominated by anisotropic
random fields and the ratio should be $\eta\sim 0.01-0.3$ in galactic disk
region (Beck et al., 2019). Nevertheless, the actual value of $\eta$ at the
Galactic Centre is uncertain and the value of $\eta$ may be larger. In the
followings, we will first assume $\eta=0.65$. Then, we will also demonstrate
the cases for $\eta=0.3$ and $\eta=0.9$ for comparison.
For the gas density, observational data show that the gas number density is
close to $n_{H}\sim 10-100$ cm-3 at the inner Galactic Centre and $n_{H}\sim
1$ cm-3 out to $\sim 220$ pc along the Galactic plane (Ferrière, Gillard &
Jean, 2007; Ferrière, 2009). We follow Ferrière, Gillard & Jean (2007) to
assume that the gas density decreases exponentially with $r$ for small $r$ and
then approaches a constant value $\rho_{0}^{\prime}$ when $r$ becomes large.
Therefore, we write the gas density profile as
$\rho_{g}=\rho_{0}\exp\left(-\frac{r}{1~{}\rm pc}\right)+\rho_{0}^{\prime}.$
(9)
Putting Eq. (8) and Eq. (9) into Eq. (3), we get
$v_{\rm mag}^{2}=\frac{v_{0}^{2}e^{-r/a}[1-\eta^{-2}(r/2a)]}{\exp(-r/1~{}{\rm
pc})+y},$ (10)
where $v_{0}^{2}=B_{0}^{2}/(4\pi\rho_{0})$, $y=\rho_{0}^{\prime}/\rho_{0}$.
Here, $v_{0}$ and $y$ are free parameters in our model.
## 3 Data analysis
The inner rotation curve data have been obtained in Sofue (2013). We will
focus on the region $r\leq 360$ pc because the rotation curve attains its
maximum at around $r=360$ pc. At this position, the percentage contribution of
the bulge component is maximised. Larger than $r=360$ pc, the dark matter and
disc components start to contribute more to the Galactic rotation curve. We
fit our predicted total rotation curve $v(r)$ with the observed data $v_{\rm
obs}(r)$. To quantify the goodness of fits, we calculate the reduced
$\chi^{2}$ value of the fits, which is defined as
$\chi_{\rm red}^{2}=\frac{1}{N-M}\sum_{i=1}^{N}\frac{(v_{i}-v_{{\rm
obs},i})^{2}}{\sigma_{i}^{2}},$ (11)
where $\sigma_{i}$ is the uncertainty of the observed rotation curve data, $N$
is the total number of data points and $M$ is the number of free parameters.
Here, we have $M=2$.
In Fig. 1, we present the best fit of our model (with $\eta=0.65$) and
separate the corresponding rotation curve contributions. The best-fit values
are $v_{0}=16.4$ km/s and $y=0.023$, with $\chi_{\rm red}^{2}=0.14$. We can
see that including the magnetic field contribution could provide an excellent
fit to the observed inner rotation curve without introducing any inner bulge
component suggested in Sofue (2013). The reduced $\chi^{2}$ value for adding
an inner bulge component without magnetic field contribution is $\chi_{\rm
red}^{2}=0.12$, almost the same goodness of fit. Note that the contribution of
the magnetic field effect to the rotation velocity could be slightly negative
when $r>2\eta^{2}a\approx 0.1$ kpc. Fig. 2 shows the comparison of the
residual plots between the Galactic total rotation curve data and the two
scenarios. No systematic trend of the residuals is shown for both scenarios.
At the Galactic Centre, the gas density is $\rho_{g}\approx 1.4m_{p}n_{H}$,
where $m_{p}$ is the proton mass. If we take the asymptotic number density
$n_{H}=1$ cm-3 (Ferrière, Gillard & Jean, 2007), we have
$\rho_{0}^{\prime}\approx 2.3\times 10^{-24}$ g/cm3. Using the best-fit values
$y=0.023$ and $v_{0}=16.4$ km/s, we get $\rho_{0}=1.0\times 10^{-22}$ g/cm-3
(i.e. $n_{H}\approx 43$ cm-3) and $B_{0}=58$ $\mu$G. These values are
consistent with the number density observed and the central total magnetic
field constrained ($B\sim 10-100$ $\mu$G) (Ferrière, Gillard & Jean, 2007;
Ferrière, 2009; Guenduez et al., 2020).
As the actual value of $\eta$ at the Galactic Centre is uncertain, we also
investigate the cases for $\eta=0.3$ and $\eta=0.9$. For $\eta=0.9$, we also
get a very good fit with $v_{0}=15.2$ km/s ($B_{0}=54$ $\mu$G) and $y=0.023$
($\chi_{\rm red}^{2}=0.11$). However, for $\eta=0.3$, a relatively poor fit is
obtained ($\chi_{\rm red}^{2}=1.46$). We plot the corresponding components and
the total rotation curves fitted in Fig. 3. Generally speaking, for
$\eta>0.6$, a very good fit would be obtained ($\chi_{\rm red}^{2}<0.17$).
Figure 1: The black dots with error bars represent the inner Galactic rotation
curve data for $r=1-360$ pc (Sofue, 2013). The red, green, and blue solid
lines indicate the best-fit rotation curve components of the supermassive
black hole, bulge, and magnetic field contribution respectively. The orange
line is the best-fit total rotation curve in our model. The violet dotted line
represents the inner bulge contribution assumed in Sofue (2013). Here, we have
assumed $\eta=0.65$.
Figure 2: The red circles and blue triangles represent the residuals of the
best-fit total rotation curve due to the contributions of the magnetic field
effect and the inner bulge component respectively compared with the inner
Galactic rotation curve data.
Figure 3: The black dots with error bars represent the inner Galactic rotation
curve data for $r=1-360$ pc (Sofue, 2013). The red, green, and blue lines
indicate the best-fit rotation curve components of the supermassive black
hole, bulge, and magnetic field contribution respectively (blue solid:
$\eta=0.3$; blue dashed: $\eta=0.9$). The orange lines are the best-fit total
rotation curve in our model (orange solid: $\eta=0.3$; orange dashed:
$\eta=0.9$).
## 4 Discussion
In this letter, we show that adding the magnetic field contribution can
satisfactorily explain the inner Galactic rotation curve data without invoking
an inner bulge component. The magnetic field effect on rotation curve is a
predicted effect in magneto-hydrodynamics (MHD). Our results provide an
indirect evidence that magnetic field can affect the galactic rotation curve
significantly. Previous studies have shown that the magnetic field
contribution cannot boost the gas rotating speed by more than 20 km/s in the
outermost region (Sánchez-Salcedo & Santillán A., 2013). Now we show that such
effect can be large in the central region of a galaxy. The maximum
contribution of the magnetic field on the rotation curve is 97 km/s at $r\sim
12$ pc.
In our analysis, we have assumed that magnetic field strength traces the
baryonic distribution (i.e. the bulge density), which is predicted by
theoretical models. For example, numerical simulations and the equipartition
theory show that the magnetic field strength follows the baryonic density in
galaxy clusters and galaxies (Dolag et al., 2001; Govoni et al., 2017;
Schleicher et al., 2010) and it is supported by observational data (Heesen et
al., 2014; Tabatabaei et al., 2017; van Weeren et al., 2019). Our constrained
magnetic field strength at the Galactic Centre ($B_{0}\sim 50-60$ $\mu$G) is
also consistent with the order of magnitude of the observed total magnetic
field strength $B\sim 10-100$ $\mu$G (Ferrière, 2009; Guenduez et al., 2020).
These could be verified by future observational data of magnetic field at the
Galactic Centre. Moreover, we have first taken the value of $\eta$ to be a
constant $\eta=0.65$. Some other studies have found that the magnetic field
strength might be dominated by anisotropic random field rather than the
regular field (Houde et al., 2013; Beck et al., 2019). The value of $\eta$ can
be as small as $\sim 0.1$ for the disk region in spiral galaxies (Beck et al.,
2019). However, some studies have revealed a very large regular field strength
$\sim 1$ mG at the Galactic Centre (Eatough et al., 2013). Therefore, the
actual value of $\eta$ in the Galactic Centre region is uncertain. We have
particularly investigated the cases of $\eta=0.3$ and $\eta=0.9$ for
comparison. We have found that $\eta=0.9$ can also give a good fit for the
data. Generally, $\eta>0.6$ could provide good fits with the rotation curve
data without invoking the inner bulge component. Further radio observations
are definitely required to examine the value of $\eta$ as well as our model
presented.
On the other hand, we have also assumed that the gas density follows the
exponential density profile in the deep central region and approaches a
constant value at a relatively large $r\sim 100$ pc. The constant gas density
at $r\sim 100$ pc is supported by observational data (Ferrière, Gillard &
Jean, 2007; Ferrière, 2009). We have also tried the isothermal density
profile, which is commonly used as a model for interstellar medium
(Kalashnikov & Chechetkin, 2022), to model the gas density distribution. A
good fit can still be obtained (with $B_{0}=62$ $\mu$G and $\chi_{\rm
red}^{2}=0.12$). Therefore, our results are not very sensitive to the gas
density profile assumed.
To conclude, we show that magnetic field can affect inner rotation curve
significantly. We anticipate that such effect can also be seen in the inner
region of other galaxies. Future high-quality observations of the inner
galactic rotation curves could verify our suggestion.
## 5 acknowledgements
We thank the anonymous referee for useful constructive feedback and comments.
This work was partially supported by the Seed Funding Grant (RG 68/2020-2021R)
and the Dean’s Research Fund (activity code: 04628) from The Education
University of Hong Kong.
## 6 Data availability statement
The data underlying this article will be shared on reasonable request to the
corresponding author.
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|
# Has Telescope Array Discovered Electroweak Monopole?
Y. M. Cho<EMAIL_ADDRESS>School of Physics and Astronomy, Seoul National
University, Seoul 08826, Korea Center for Quantum Spacetime, Sogang
University, Seoul 04107, Korea Franklin H. Cho<EMAIL_ADDRESS>Center for Quantum Nano Science, Ewha Woman’s University, Seoul 03766, Korea
###### Abstract
We propose the ultra high energy cosmic ray recently detected by Telescope
Array to be the electroweak monopole, and present theoretical arguments which
support this. This strongly motivates the necessity for the “cosmic” MoEDAL
experiment which could back up our proposal. To confirm this we propose
Telescope Array to measure the magnetic charge of the ultra high energy cosmic
ray particles with SQUID.
ultra high energy cosmic ray, Greisen-Zatsepin-Kuzmin (GZK) limit, electroweak
monopole as ultra high energy cosmic ray, enhanced Chrenkov radiation of
electroweak monopole, cosmological production of electroweak monopole, remnant
electroweak monopole density
Recently the Telescope Array Group (TAG) has announced the detection of an
ultra high energy cosmic ray (UHECR) of energy 244 EeV ($244\times 10^{18}$
eV) [1]. This is the most recent confirmation of the existence of the UHECR
particles, following the 320 EeV particle in 1991, 213 EeV particle in 1993,
and 280 EeV particle in 2001 [2]. This tells that the UHECR particles exist in
nature.
The TAG (and similar previous) report is very interesting and remarkable in
two aspects. First, the energy of the cosmic ray exeeds the Greisin-Zatsepin-
Kuzmin (GZK) energy limit [3]. Second, the arrival direction of the UHECR
implies that it came from the Local Void. This is puzzling, because there are
few known particles in nature which could produce such UHECR.
A natural candidate for the UHECR is the proton, but it is very difficult for
proton to generate such high energy. A relativistic proton moving through the
cosmic microwave background, after collision with the 3 K microwave photons
becomes $\Delta^{*}$ and decays to nucleons and pions,
$\displaystyle p+\gamma\rightarrow\Delta^{*}\rightarrow N+\pi.$ (1)
And the mean free path for this process is known to be about 6 Mpc. This
resonant scattering degrades the energy of the relativistic protons and
prevent them to acquire the energy above $5\times 10^{19}$ eV. This is the GZK
limit [3]. This implies that, if the UHECR of TAG is proton, it should have
originated nearby, or should have energy far above the GZK limit. But these
possibilities seems unlikely.
The other point is that the UHECR observed by TAG appears to come from the
void, which suggests that the origin of this UHECR is not galactic center or
other astronomical objects like the neutron stars. This is another puzzling
feature of this UHECR [1].
From this we may conclude that the UHECR of TAG is not likely to be an ultra
relativistic proton, or any known particle produced by the astronomical
objects. This lack of a possible explanation for the UHECR is disappointing,
and it has been suggested that this could be due to “an incomplete knowledge
of particle physics” [1]. The purpose of this Letter is to argue that the
UHECR observed by TA could be the remnant electroweak monopole produced in the
early universe during the electroweak phase transition. We present theoretical
reasons to support this, and discuss how to confirm this proposal
experimentally.
The proposition that the monopoles could be the source of the UHECR is not new
[4]. Obviously the monopole, if exists, becomes an ideal candidate for the
UHECR particle. It has the strong magnetic interaction, stronger than the
electric interaction of proton by the factor $1/\alpha$. It has the absolute
stability guaranteed by the $\pi_{2}(S^{2})$ monopole topology, which is
required for the UHECR particles. Moreover, it may have the mass much heavier
than the proton. For this reason it has been asserted that the grand
unification monopole could generate the UHECR [5]. But in this paper we argue
that the electroweak (“Cho-Maison”) monopole could be the UHECR particles. To
do that we must understand why the electroweak monopole, not others, should be
viewed as the UHECR of TAG.
With the advent of the Dirac monopole, the magnetic monopole has become an
obsession in physics [6, 7]. After the Dirac monopole we have had the Wu-Yang
monopole [8], the ’tHooft-Polyakov monopole [9], the grand unification
monopole [10], and the electroweak monopole [11, 12]. But the electroweak
monopole stands out as the most realistic monopole that exists in nature and
could actually br detected by experiment [13, 15, 16, 14, 17, 18].
This is because the Dirac monopole in electrodynamics transforms to the
electroweak monopole after the unification of the electromagnetic and weak
interactions, and the Wu-Yang monopole in QCD becomes unobservable after the
monopole condensation which confines the color. Moreover, the ’tHooft-Polyakov
monopole exists only in a hypothetical theory, and the grand unification
monopole which could have been amply produced at the grand unification scale
in the early universe probably has become completely irrelevant at present
universe after inflation.
Unlike other monopoles the electroweak monopole has the following unique
features [11]. First, the magnetic charge is not $2\pi/e$ but $4\pi/e$, twice
that of the Dirac monopole. This is because the period of the electromagnetic
U(1) subgroup of the standard model is $4\pi$, since the electromagnetic U(1)
comes from the U(1) subgroup of SU(2). Second, the mass is of the order of
several TeV, probably between 4 to 10 TeV. This is because the mass basically
comes from the same Higgs mechanism which makes the W boson massive, except
that here the coupling is magnetic (i.e., $4\pi/e$). This makes the monopole
mass $1/\alpha$ times heavier than the W boson mass. In spite of this, the
size of the monopole is set by the weak boson masses. This is because the
monopole has the Higgs and W boson dressing which has the exponential damping
fixed by the weak boson masses. Third, this is the monopole which exists
within (not beyond) the standard model as the electroweak generalization of
the Dirac monopole, as a hybrid between Dirac and ’tHooft-Polyakov monopoles.
Finally, this monopole is absolutely stable. The topological stability must be
obvious, but it also has the dynamical stability [18].
The importance of the electroweak monopole comes from the fact that it must
exist if the standard model is correct. This means that the discovery of the
monopole, not the Higgs particle, should be viewed as the final (and
topological) test of the standard model. Moreover, if discovered, it will
become the first topologically stable magnetically charged elementary particle
in the history of physics [13]. Furthermore, the monopoles produced in the
early universe could play important roles in physics, in particular in
cosmology [14]. Indeed, when coupled to gravity, they automatically turn to
the primordial magnetic black holes which could account for the dark matter,
become the seed of stellar objects and galaxies, creating the large scale
structures of the universe. As importantly, they could generate the
intergalactic magnetic field, and could be the source of the ultra high energy
cosmic rays [14].
This makes the experimental detection of the electroweak monopole a most
urgent issue after the discovery of the Higgs particle. For this reason MoEDAL
and ATLAS at CERN are actively searching for the monopole [19, 20, 21]. Since
the electroweak monopole has unique characteristics, they could detect the
monopole without much difficulty, if LHC could produce them. If the monopole
mass exceeds 7 TeV, however, the present 14 Tev LHC may not be able to produce
the monopoles. In this case we may have to wait for the next upgrading of LHC,
or else look for the remnant monopoles produced in the early universe during
the electroweak phase transition.
How can we detect the remnant electroweak monopoles? Obviously we could design
a “cosmic” MoEDAL experiment located at high mountains to detect them, and
this is in planning now. At this point one might wonder if the underground
experiments like IceCube or Antares could be helpful. Unfortunately the
underground experiments may have difficulty to detect them because the
penetration length of the monopole in matters is expected to be very short (of
the order of 10 meters in aluminum) because of the strong magnetic interaction
[22, 23].
Another way to detect the electroweak monopole is to detect UHECR particles
which could be generated by the remnant monopoles, using the cosmic ray
experiments. The primary purpose of this type of experiments, of course, is to
detect the high energy cosmic rays (not the monopoles). Nevertheless this type
of experiments could be used to detect the remnant monopoles, and in this
connection the TA experiment could play an important role. The question here
is why and how the remnant electroweak monopoles could be identified as the
UHECR particles. Now, we are ready to discuss how they can become the UHECR
particles.
To see this we first notice that the remnant electroweak monopoles could
easily acquire the energy above the GZK limit from the intergalactic magnetic
field. Since the average intergalactic magnetic field $B$ is about $3\times
10^{-6}$ G with the coherent length $L$ of the order of 300 pc, we could
estimate the monopole energy gain traveling through the intergalactic magnetic
field to be [14]
$\displaystyle\Delta E\simeq\frac{4\pi}{e}~{}BL\simeq 1.2\times
10^{20}~{}\text{eV}.$ (2)
Moreover, we can easily show that the monopole energy loss due to the linear
acceleration is completely negligible. This confirms that they could acquire
the energy above the GZK limit without any problem.
Moreover, unlike the proton, the monopole retains its energy traveling through
the cosmic microwave background. This is because the photon-monopole
scattering cross section is givden by the classical Thompson scattering cross
section, which is many orders of magnitude down the photon-proton cross
section described by (1).
Notice that (2) is independent of the monopole mass. So, depending on the mass
the monopole (even with the above energy) could be relativistic or non-
relativistic. For example, if it is the grand unification monopole of mass of
$10^{17}$ GeV, it becomes non-relativistic and thus can not generate
relativistic secondaries in the cosmic ray shower. In contrast, (2) makes the
electroweak monopole with mass of $M_{W}/\alpha$ extremely relativistic, so
that it could easily generates the relativistic showers. And in reality we do
have the relativistic showers in the UHECR. This strongly indicates that the
UHECR could be the electroweak monopole. In the following we will assume the
monopole mass to be $M=M_{W}/\alpha$ for simplicity.
How is the electromagnetic shower of the electroweak monopole? The magnetic
field of the monopole $B=(4\pi/e)\hat{r}/r^{2}$, when boosted to the energy
(2), generates the electric field $\vec{E}=\gamma\beta\vec{v}\times\vec{B}$.
So, the electromagnetic energy loss of the relativistic electroweak monopole
should be similar to that of a heavy charged particle of mass $M_{W}/\alpha$
with similar $\gamma$ factor (for our monopole with energy (2), we have
$\gamma\geq 10^{7}$) and charge $1/\alpha$. Moreover, the Cherenkov radiation
of the monopole is enhanced by the factor $(1/\alpha)^{2}$, compared to that
of the proton. This enhanced Cherenkov radiation is an important feature of
the UHECR generated by the electroweak monopole, which could be useful in
identifying the UHECR particle as the electroweak monopole.
As for the hadronic shower of the electroweak monopole, notice that the
maximum fraction of the energy transferred from our monopole of mass $M$ to
the target particles with mass $m$ is given by [5]
$\displaystyle\frac{E^{\prime}_{m}}{E_{M}}=\frac{m}{E_{M}}\Big{(}\frac{4E_{M}^{2}-2M^{2}+m^{2}}{4mE_{M}+2M^{2}}\Big{)}.$
(3)
Now, for our case we have $E_{M}^{2}>>M^{2}>>m^{2}$, so that
$\displaystyle\frac{E^{\prime}_{m}}{E_{M}}\simeq\frac{2mE_{M}}{2mE_{M}+M^{2}}\simeq
1.$ (4)
This should be compared to the maximum energy transfer of the proton (with
$M\simeq m$)
$\displaystyle\frac{E^{\prime}_{m}}{E_{M}}=1-\frac{m}{2E_{M}}\simeq 1,$ (5)
which is not so different from (4). From this we may conclude that our
monopoles transfer most of the energy in the first forward hadronic
scattering, and thus produce an air shower resembling a typical hadron
initiated shower. This implies that trying to identify the UHECR with hadronic
shower pattern may not be a wise strategy.
Now, we have to discuss the remnant electroweak monopole density at present
universe. This is a complicated issue, but fortunately this has already been
studied before [14]. To summarize the results we start from the temperature
dependent effective action of the standard model
$\displaystyle
V_{eff}(\rho)=V_{0}(\rho)-\frac{C_{1}}{12\pi}\rho^{3}T+\frac{C_{2}}{2}\rho^{2}T^{2}-\frac{\pi^{2}}{90}g_{*}T^{4}+\delta
V_{T},$ $\displaystyle
V_{0}(\rho)=\frac{\lambda}{8}(\rho^{2}-\rho_{0}^{2})^{2},$ $\displaystyle
C_{1}=\frac{6M_{W}^{3}+3M_{Z}^{3}}{\rho_{0}^{3}}\simeq 0.36,$ $\displaystyle
C_{2}=\frac{4M_{W}^{2}+2M_{Z}^{2}+M_{H}^{2}+4m_{t}^{2}}{8\rho_{0}^{2}}\simeq
0.37,$ (6)
where $V_{0}$ is the zero-temperature potential, $g_{*}$ is the total number
of distinct helicity states of the particles with mass smaller than $T$,
$C_{1}$ and $C_{2}$ are the contributions from the weak bosons and fermions,
$m_{t}$ is the top quark mass, and $\delta V_{T}$ is the slow-varying
logarithmic corrections and the lighter quark contributions which we will
neglect from now on.
Figure 1: The temperature dependent effective potential (6) at various
temperatures, where $T_{G}$ is the Ginzburg temperature. Notice that the
potential at $T_{1},~{}T_{c},~{}T_{2}$ are almost indistinguishable.
The potential has three local extrema at
$\displaystyle\rho_{s}=0,$
$\displaystyle\rho_{\pm}(T)=\Big{\\{}\frac{C_{1}}{4\pi\lambda}\pm\sqrt{\Big{(}\frac{C_{1}}{4\pi\lambda}\Big{)}^{2}+\frac{\rho_{0}^{2}}{T^{2}}-\frac{2C_{2}}{\lambda}}\Big{\\}}~{}T^{.}$
(7)
The first extremum $\rho_{s}=0$ represents the Higgs vacuum of the symmetric
(unbroken) phase, the second one $\rho_{-}(T)$ represents the local maximum,
and the third one $\rho_{+}(T)$ represent the local minimum Higgs vacuum of
the broken phase. It is charactrized by three temperatures,
$\displaystyle T_{1}=\frac{4\pi\lambda}{\sqrt{32\pi^{2}\lambda
C_{2}-C_{1}^{2}}}~{}\rho_{0}\simeq 146.13~{}\text{GeV},$ $\displaystyle
T_{c}=\sqrt{\frac{18}{36\pi^{2}\lambda
C_{2}-C_{1}^{2}}}~{}\pi\lambda\rho_{0}\simeq 146.09~{}{\rm GeV},$
$\displaystyle T_{2}=\sqrt{\frac{\lambda}{2C_{2}}}~{}\rho_{0}\simeq
145.82~{}\text{GeV}.$ (8)
Above $T_{1}$ only $\rho_{s}=0$ becomes the true vacuum of the effective
potential, and the electroweak symmetry remains unbroken. At the critical
temperature $T_{c}$, the two vacua $\rho_{s}$ and $\rho_{+}$ are degenerate
and the electroweak phase transition starts. At $T_{2}$ we have only one
vacuum $\rho_{+}$ with $\rho_{0}=\rho_{-}$, and the phase transition ends. So,
in principle the electroweak phase transition is the first order. Since
$T_{1}$, $T_{c}$, and $T_{2}$ are very close, however, the phase transition
practically becomes the second order [14]. The effective potential (6) is
graphically shown in Fig. 1.
The monopole production in the second order phase transition is supposed to be
described by the Kibble-Zurek mechanism, so that the monopole production start
from $T_{c}$. However, the thermal fluctuations of the Higgs vacuum which
create the seed of the monopoles continue till the universe cools down to the
Ginzburg temperature $T_{G}\simeq 57.6~{}\text{GeV}$, where the monopole
production stops[14, 24]. The Ginzburg temperature is shown in Fig. 1. So, the
electroweak monopole formation takes place between $T_{c}$ and $T_{G}$, or in
average around $T_{i}$,
$\displaystyle T_{i}=\frac{T_{c}+T_{G}}{2}\simeq 101.7~{}\text{GeV}.$ (9)
In time scale, we can say that the electroweak monopole production start from
$1.8\times 10^{-11}sec$ to $1.2\times 10^{-10}sec$ after the big bang for the
period of $10.3\times 10^{-11}~{}sec$, or around $3.5\times 10^{-11}sec$ after
the big bang in average.
Two important parameters of the electroweak phase transition are the
temperature dependent Higgs boson mass $\bar{M}_{H}$ which determine the
correlation length $\xi=1/\bar{M}_{H}$ and the W-boson mass $\bar{M}_{W}$
which determines the monopole mass $\bar{M}\simeq\bar{M}_{W}/\alpha$. The
Higgs boson acquires its minimum mass 5.54 GeV at $T=T_{c}$ which approaches
to the zero temperature value 125.2 GeV as the universe cools down. The
W-boson which is massless before the symmetry breaking becomes massive toward
the value 6.76 GeV at $T_{c}$ and 73.2 GeV at $T_{G}$. This tells that the
infant monopole masses at $T_{c}$ and $T_{G}$ are around 1.4 TeV and 10 TeV
(with the adolescent mass 10.7 TeV).
According to the Kibble-Zurek mechanism the initial monopole density is
determined by the mean value of two correlation lengths at $T_{c}$ and $T_{G}$
[25, 26],
$\displaystyle\xi_{i}=\frac{\xi(T_{c})+\xi(T_{G})}{2}\simeq 9.4\times
10^{-16}~{}\text{cm}.$ (10)
From this we can estimate the initial density of the monopoles to be
$n_{i}\simeq T_{i}^{3}/\xi_{i}^{3}\simeq 0.2~{}T_{i}^{3}$. This estimate,
however, has a defect that the energy within one correlation volume is not
enough to provide the monopole mass. This is because the monopole mass is
$M_{W}/\alpha$, but the size is fixed by $M_{W}$. A natural way to cure this
defect is to adopt a new correlation length $\bar{\xi}_{i}$ which satisfies
the energy constraint,
$\displaystyle\bar{\xi}_{i}=\Big{(}\frac{1}{\alpha}\Big{)}^{1/3}\xi_{i}\simeq
5.16\times\xi_{i}.$ (11)
With this the initial monopole density is given by
$\displaystyle
n_{i}\simeq\frac{T_{i}^{3}}{\bar{\xi}_{i}^{3}}=\alpha\times\frac{T_{i}^{3}}{\xi_{i}^{3}}\simeq
1.5\times 10^{-3}~{}T_{i}^{3}.$ (12)
This is smaller than the Kibble-Zurek estimate by the factor $\alpha$.
Figure 2: The cosmic evolution of the electroweak monopole density
$n_{m}/T^{3}$ against $\tau=M_{m}/T$.
To determine the remnant monopole density at present universe, however, we
have to know how it evolves in cosmology. The evolution of the monopoles is
determined by the Boltzmann’s equation [27]
$\displaystyle\frac{dn}{dt}+3Hn=-\sigma n^{2},$ (13)
where $H$ is the Hubble parameter and $\sigma\simeq 1/3\alpha T^{2}$ is the
monopole annihilation cross section. The solution of the evolution equation is
shown in Fig. 2. Notice that the monopoles, as soon as produced, quickly
annihilate each other. This is because at the initial stage of the monopole
production, the capture radius of the monopole and anti-monopole is much
bigger than the correlation length. This quickly reduces the initial monopole
density by the factor $10^{-6}$. Moreover, the final value of the monopole
density becomes independent of the initial value and approaches to
$n\rightarrow 18.25~{}(T/M_{P})\times T^{3}$ regardless of the initial
condition, where $M_{P}$ is the Planck mass [27]. And the monopole-
antimonopole annihilation ceases at the temperature $T_{f}\simeq
60~{}{\text{MeV}}$. This is below the muon decoupling temperature, which tells
that the annihilation of the monopoles continues very long time.
Below $T_{f}$ the monopoles are free streaming, and we can estimate the
remnant monopole density at the present universe. The monopole density at
$T_{f}$ becomes
$\displaystyle n_{f}\simeq 0.9\times 10^{-19}~{}T_{f}^{3},$ (14)
which is much lower than the initial density given by (12). The number of
monopole within the co-moving volume is conserved thereafter. But they still
interact with the electron pairs in the hot plasma before decouple around
$T_{d}\simeq 0.5~{}\text{MeV}$, when the electron pairs disappear and the
interaction rate becomes less than the Hubble expansion rate. Since the
decoupling temperature of the electroweak monopole is much less than the
monopole mass, the free streaming monopoles just after the decoupling start as
completely non-relativistic. But eventually they become extremely relativistic
by the acceleration of the intergalactic magnetic field.
Assuming that the expansion is adiabatic, the current number density and the
energy density of the monopole is given by [14]
$\displaystyle
n_{0}=\frac{g_{0}}{g_{f}}~{}\Big{(}\frac{T_{0}}{T_{f}}\Big{)}^{3}~{}n_{f},$
$\displaystyle\rho_{m}=n_{0}~{}M=\frac{g_{0}}{g_{f}}~{}\Big{(}\frac{T_{0}}{T_{f}}\Big{)}^{3}~{}n_{f}~{}M,$
(15)
where $T_{0}=2.73~{}\text{K}=2.35\times 10^{-13}~{}\text{GeV}$ is the
temperature of the universe today and $g$ is the effective number of degrees
of freedom in entropy. So we have the current density of monopole
$\displaystyle\Omega_{m}~{}h^{2}=\frac{\rho_{m}}{\rho_{c}}~{}h^{2}\simeq
1.2\times 10^{-10},$ (16)
where $\rho_{c}$ is the critical density of present universe and $h\simeq
0.678$ is the scaled Hubble constant in the unit
$H_{0}/(100~{}\text{km}~{}\text{s}^{-1}~{}\text{Mpc}^{-1})$. This is about
$1.3\times 10^{-9}$ of the baryon density. This assures that the electroweak
monopole does not alter the standard big bang cosmology in any significaly
way.
In terms of the number density, this translates to about $6.1\times
10^{-5}/~{}\text{Km}^{3}$, or about $2.3\times 10^{-13}~{}n_{b}$, where
$n_{b}$ is the number density of the baryons. This is roughly $10^{5}$ times
bigger than the monopole density set by the Parker bound [28], which implies
that (15) is an over estimation.
Actually there are reasons that the real remnant monopole density could be
much less [14]. First, as the only heavy stable particle with mass about
$10^{4}$ times heavier than the proton, they can easily generate the density
perturbation and might have been buried in galactic centers. Second, they have
a very short penetration length in matters, so that most of them could have
been trapped and filtered out by the stellar objects after collision with
them. Third (and most importantly), when coupled to gravity, they
automatically turn to Reissner-Nordstrom type black holes and become the
premordial magnetic black holes. This strongly implies that indeed (15) could
be made consistent with the Parker bound.
With this remark we can confidently say that the UHRCR particle observed by TA
could be the remnant electroweak monopole produced in the early universe, as
one of us has pointed out in an earlier work [14]. In particular, our estimate
of the monopole density appears to be consistent with the UHECR event rate at
TA, and it comes from the void as TA indicated.
Unfortunately, this proposition could only be confirmed indirectly at TA at
the moment, with the enhanced Cherenkov radiation of the monopole. To confirm
this directly, TA should be able to measure the magnetic charge of the UHECR.
In principle this could be done by installing SQUID to each of the surface
detectors at TA. We hope that TAG could measure the magnetic charge of the
UHECR with SQUID in the near future. The details of our discussions will be
published in a separate paper [29].
ACKNOWLEDGEMENT
The work is supported in part by the National Research Foundation funded by
the Ministry of Education (Grant 2018-R1D1A1B0-7045163), the Ministry of
Science and Technology (Grant 2022-R1A2C1006999), and by Center for Quantum
Spacetime, Sogang University, Korea.
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|
# The Kohn-Luttinger effect in dense matter and its implications for neutron
stars.
Mia Kumamoto<EMAIL_ADDRESS>Institute for Nuclear Theory, University of
Washington, Seattle, WA USA Department of Physics, University of Washington,
Seattle, WA USA Sanjay Reddy<EMAIL_ADDRESS>Institute for Nuclear Theory,
University of Washington, Seattle, WA USA
###### Abstract
Repulsive short-range interactions can induce p-wave attraction between
fermions in dense matter and lead to Cooper pairing at the Fermi surface. We
investigate this phenomenon, well-known as the Kohn-Luttinger effect in
condensed matter physics, in dense matter with strong short-range repulsive
interactions. We find that repulsive interactions required to stabilize
massive neutron stars can induce p-wave pairing in neutron and quark matter.
When massive vector bosons mediate the interaction between fermions, the
induced interaction favors Cooper pairing in the ${}^{3}P_{2}$ channel. For
the typical strength of the interaction favored by massive neutron stars, the
associated pairing gaps in neutrons can be in the range of 10 keV to 10 MeV.
Strong and attractive spin-orbit and tensor forces between neutrons can result
in repulsive induced interactions that greatly suppress the ${}^{3}P_{2}$
pairing gap in neutron matter. In quark matter, the induced interaction is too
small to result in pairing gaps of phenomenological relevance.
## I Introduction
The discovery of massive neutron stars by radio observations of pulsars
Cromartie _et al._ (2019); Antoniadis _et al._ (2013); Demorest _et al._
(2010) confirmed that the maximum mass of neutron stars
$M_{\mathrm{max}}>2~{}M_{\odot}$, and gravitational wave and x-ray
observations constrain the radius of a neutron star with mass $~{}\simeq
1.4~{}M_{\odot}$ to the range $11-13$ km Abbott _et al._ (2018); De _et al._
(2018); Capano _et al._ (2020); Miller _et al._ (2019); Riley _et al._
(2019). These constraints and theoretical calculations of the EOS of neutron-
rich matter at $n_{B}\lesssim 2n_{\mathrm{sat}}$ Hebeler and Schwenk (2010);
Gandolfi _et al._ (2014); Tews _et al._ (2013); Lynn _et al._ (2016);
Drischler _et al._ (2020), taken together strongly suggest a rapid increase
in the pressure and the speed of sound in the NS core Tews _et al._ (2018).
This, in turn, implies strong repulsive interactions are necessary for any
putative phase of high-density matter in the core. This article addresses
whether such repulsion can have other observable consequences. In particular,
we investigate if such repulsion can lead to Cooper pairing between fermions
with non-zero angular momentum due to the Kohn-Luttinger(KL) effect Kohn and
Luttinger (1965) in the cores of neutron stars.
The KL effect, which arises because the interaction at the Fermi surface is
modified due to screening in the medium, implies that the Cooper pairing
instability in high angular momentum states is inevitable and occurs even when
the bare interaction is repulsive Kohn and Luttinger (1965). The effect has
been discussed extensively in condensed matter physics (For a recent pedagogic
review, see Ref. Kagan (2013)). In the context of dense nuclear matter, early
work in Fay and Layzer (1968); Pines (1971); Clark _et al._ (1976) recognized
that the interaction between nucleons induced by polarization effects in the
medium would significantly alter the pairing gaps (for recent reviews, see
Dean and Hjorth-Jensen (2003); Gezerlis _et al._ (2014)). The induced
interaction, typically calculated in second-order perturbation theory or Fermi
liquid theory, naturally incorporates the KL effect. In dilute Fermi systems
with attractive s-wave short-range interactions, it has been known since the
work of Gor’kov and Melik-Barkhudarov that the induced interaction suppresses
the s-wave pairing gap relative to the BCS prediction Gorkov and Melik-
Barkhudarov (1961). In neutron matter, when the s-wave interaction is
repulsive, the induced interaction was initially expected to increase the
p-wave attraction between neutronsFay and Layzer (1968). However, more recent
work in Ref. Schwenk and Friman (2004) finds that the induced spin-orbit
interaction can dominate and result in a net suppression instead at modest
density. Here, we revisit calculating the induced interaction in high-density
matter characterized by a large sound speed to study its implications for
${}^{3}P_{2}$ pairing. We consider short-range interactions that contain
central and non-central components and study the competition between the
attractive and repulsive components of the induced p-wave interaction and its
density dependence.
In quark matter, when the Fermi surfaces of up, down, and strange quarks are
split due to charge neutrality and a larger strange quark mass, the KL effect
provides a mechanism to pair quarks of the same flavor and color. However, in
this case, we find that p-wave interaction induced by short-range repulsion
introduced to increase the pressure of quark matter is too small to be of
phenomenological relevance.
Our study, which relies on extrapolating results derived from perturbation
theory to strong coupling, provides order-of-magnitude estimates for the
pairing gaps. Although the method we employ is inadequate to make quantitative
predictions, it identifies a mechanism for ${}^{3}P_{2}$ pairing in dense
Fermi systems with large repulsive interactions mediated by short-range
interactions mediated by heavy vector bosons.
In section II, we review the KL mechanism for non-relativistic fermions. In
section III, we derive the induced interaction between neutrons at high
density by assuming that the bare interaction is due to the exchange of heavy
vector mesons. In section IV, we consider the possible effects of the KL
mechanism in quark matter. We discuss the implications for neutron star
cooling in section V, summarize our main findings, and discuss open questions
in section VI.
## II Kohn-Luttinger Mechanism
Kohn and Luttinger showed that a short-range repulsive potential can induce
attraction in large odd partial waves due to medium effects that can
overscreen the effective interaction between fermions at finite density Kohn
and Luttinger (1965). There has been renewed interest in studying the KL
effect in condensed matter systems because calculations suggest that the
induced pairing gaps in $p$-waves and low-order partial waves could be large
enough to be realized in experiments (see, for example, Baranov _et al._
(1992); González (2008); Nandkishore _et al._ (2014); González and Stauber
(2019)). KL’s original calculation included terms at second order in the
potential; more recent analysis Efremov _et al._ (2000) calculates the
potential up to fourth order in a constant potential characterized by a large
scattering length as well as including retardation effects where pairing
occurs away from the Fermi surface, also contributing at fourth order.
In weak coupling, the KL effect arises naturally at second order in the
potential by evaluating the diagrams in Fig. 1. We refer to these diagrams
from left to right as the screening, vertex, and exchange diagrams,
respectively. The vertex diagram also has a mirror image, which must be
included. We consider interactions that occur at the Fermi surface, so
$|k|=|k^{\prime}|=k_{F}$. The momentum transfer is labeled $q=k^{\prime}-k$.
Figure 1: Irreducible second order diagrams for Kohn-Luttinger mechanism
diagb (80,80) t1,t2 b1,b2 l1 r1 fermionb1,v1 fermionv1,t1 fermionb2,v4
fermionv4,t2 photon,tension=0.2v1,v2
fermion,left,label=$\ell+q$,tension=0.1v2,v3
fermion,left,label=$\ell$,tension=0.1v3,v2 photon,tension=0.2v3,v4
label=$-k;1$b1 label=$k;2$b2 label=$-k^{\prime};3$t1 label=$k^{\prime};4$t2
diagc (80,80) l1,l2,l3,l4,l5 r1,r2,r3,r4,r5 fermionl1,l3 fermionl3,l5
fermionr1,r2 fermionr4,r5 fermion,label=$\ell$,label.side=leftr2,v1
fermion,label=$\ell+q$,label.side=leftv1,r4 photonr2,r4 photonl3,v1
label=$-k;1$l1 label=$-k^{\prime};3$l5 label=$k;2$r1 label=$k^{\prime};4$r5
diagd (80,80) l1,l2,l5,l3,l4 r1,r2,r5,r3,r4 fermionl1,l2
fermion,label=$\ell+q$,label.side=leftl2,l3 fermionl3,l4 fermionr1,r2
fermion,label=$\ell$r2,r3 fermionr3,r4 photonr2,l3 photonl2,r3 label=$-k;1$l1
label=$k^{\prime};4$l4 label=$k;2$r1 label=$-k^{\prime};3$r4
For a short-range potential with zero range and strength denoted by $U_{0}$,
the non-relativistic calculation of these diagrams is straightforward. In this
case, the screening diagram cancels the contribution from the two vertex
diagrams, and only the exchange diagram contributes. The exchange diagram also
gets an overall sign since it is crossed and is given by
$V_{KL}(q)=U_{0}^{2}\frac{1}{\beta}\sum_{\ell_{0}}\int\frac{d^{3}\ell}{(2\pi)^{3}}\frac{1}{\ell_{0}-\ell^{2}/2m}\frac{1}{\ell_{0}-(\ell+q)^{2}/2m}$
(1)
Notice that since $|k|=|k^{\prime}|$, the frequency transfer $q_{0}$ is just
zero. Taking the Matsubara sum and simplifying it gives a singular
contribution to the potential. Since we are considering the effects of
interactions with the medium, the loop integral has a factor
$n_{F}(\ell^{2}/2m)=1/(e^{\beta(\ell^{2}/2m-\mu)}+1)$ and does not need to be
regulated. At the low temperatures we consider ($T\ll E_{F}=k_{F}^{2}/2m$)
this simplifies $n_{F}(\ell^{2}/2m)\approx\Theta(k_{F}-\ell)$. The momentum
integral in Eq. 1 yields the Lindhard function defined by
$\begin{split}U(q)=-\frac{m}{4\pi^{3}q}\int\ell\,d\ell\,d\Omega_{\ell}\frac{\Theta(k_{F}-\ell)}{\cos\theta_{q\ell}-q/2\ell}=\frac{mk_{F}}{4\pi^{2}}\left[1-\frac{1}{\overline{q}}\left(1-\frac{\overline{q}^{2}}{4}\right)\mbox{log}\left|\frac{1-\overline{q}/2}{1+\overline{q}/2}\right|\right]\end{split}$
(2)
where $\bar{q}=q/k_{\text{F}}$. Thus, Eq. 1 can be written as
$V_{KL}=-U_{0}^{2}~{}U(q)$ (3)
For any potential, the contribution to the induced potential from the
singularity at $q=2k_{F}$ of the Lindhard function scales as $(-1)^{L}L^{-4}$
for large L Kohn and Luttinger (1965); Maiti and Chubukov (2013) where $L$ is
the angular momentum quantum number. Since the regular contributions to the
total potential falls off exponentially with $L$, attraction is guaranteed for
large odd partial waves.
Although these results are only generically true for large $L$, they persist
for relatively low partial waves for some potentials. It was shown by Baranov
_et al._ (1992); Efremov _et al._ (2000) that the constant potential
calculated above would result in p-wave attraction. The p-wave contribution
from the potential in Eq. 3 can be found easily by making a change of
integration variable from $\int_{-1}^{1}d\cos\theta$ to
$\int_{0}^{2}\overline{q}d\overline{q}$. The matrix element in the Born
approximation should be doubled due to a diagram with outgoing momenta
switched, but we will absorb this normalization into the gap equation to match
the literature. The p-wave potential from the exchange diagram and its crossed
counterpart is then given by
$V_{\ell=1}=-U_{0}^{2}\frac{mk_{F}}{4\pi}\frac{4}{5\pi}(2\log 2-1)$ (4)
The superfluid gap due to the induced attraction in p-waves was calculated
several decades earlier in Ref. Fay and Layzer (1968); Kagan and Chubukov
(1988). In the BCS approximation, the p-wave gap
$\Delta_{p}\simeq\epsilon_{F}~{}\exp{\left(\frac{2}{N(0)V_{\ell=1}}\right)}=\epsilon_{F}~{}\exp{\left(\frac{-5\pi^{2}}{4(2\ln{2}-1)(ak_{\text{F}})^{2}}\right)}\,,$
(5)
where $\epsilon_{F}=k_{\text{F}}^{2}/2m$ is the Fermi energy,
$N(0)=mk_{\text{F}}/2\pi^{2}$ is the density of states at the Fermi surface
for each spin, and the scattering length $a=mU_{0}/4\pi$ in weak coupling.
In strong coupling, we cannot calculate the effective interaction at the Fermi
surface reliably, and in the following, we shall assume that Eq. 5 provides a
useful estimate. Further, we shall also assume that the s-wave scattering
amplitude between quasi-particles at the Fermi surface, denoted by $f_{0}$, is
directly related to the strength of the bare interaction $U_{0}$. In Fermi
liquid theory (FLT), the sound speed
$c_{s}=\frac{k_{F}}{\sqrt{3mm^{*}}}\sqrt{(1+F_{0})}\,,$ (6)
where $F_{0}=N(0)f_{0}$ is dimensionless measure of the quasi-particle
interaction and $m^{*}$ is the fermion effective mass at the Fermi surface.
Using this relation, we can estimate the interaction strength $U_{0}\approx
f_{0}$ at a given density if $c_{s}$ and $m^{*}$ are known. If $m^{*}\approx
m$ and $U_{0}=f_{0}$, then the induced p-wave gap in Eq. 5 can be rewritten as
$\Delta_{p}\approx\epsilon_{F}~{}\exp{\left(-\frac{5}{(2\ln{(2)}-1)F_{0}^{2}}\right)}\,,$
(7)
to illustrate its extreme sensitivity to $F_{0}$ and the sound speed through
Eq. 6. For example, models of high-density neutron matter typically predict
$F_{0}\gtrsim 2$ for $n_{B}\gtrsim 3~{}n_{\mathrm{sat}}$ Friman and Weise
(2019). Under these conditions, Eq.7 predicts robust p-wave pairing with gaps
$\Delta_{p}\gtrsim 1$ MeV due to the induced interaction.
In the next section, we will calculate the induced interaction between
neutrons in more realistic scenarios where the bare potential is momentum-
dependent and contains central and non-central components.
## III Induced p-wave pairing in dense neutron matter
The s-wave potential at the Fermi surface becomes repulsive in the neutron
star core when $n_{B}\gtrsim n_{\mathrm{sat}}/2$. At these higher densities,
${}^{3}P_{2}$ pairing is favored because the bare potential in this channel
remains attractive, and non-central components of the interaction, especially
the spin-orbit interaction, favor the alignment of spin and orbital angular
momentum. Calculations of the ${}^{3}P_{2}$ pairing gap in the BCS
approximation reported in Refs.Baldo _et al._ (1998); Gezerlis _et al._
(2014) show that the pairing gaps are model dependent, especially for
$n_{B}>2n_{\mathrm{sat}}$ because the nucleon-nucleon potentials at the
relevant momenta are not well constrained by scattering data. In these
calculations, the maximum value of the gap $\Delta_{{}^{3}P_{2}}\simeq 1-2$
MeV occurs between $2-3~{}n_{\mathrm{sat}}$ and decreases rapidly with
increasing density. At lower density, when the nucleon momenta
$p\ll\Lambda_{\chi}$ where $\Lambda_{\chi}\simeq 500$ MeV is the breakdown
scale of chiral EFT, a recent study used chiral EFT potentials and found that
the maximum value $\Delta_{{}^{3}P_{2}}\simeq 0.4$ MeV was reached at
$n_{B}\simeq 1.3~{}n_{\mathrm{sat}}$ and its decrease at higher density was
found to be sensitive to the details of the short-distance physics Drischler
_et al._ (2017). Together, these findings suggest that if neutron matter
persists at the highest densities encountered in neutron stars, the bare
${}^{3}P_{2}$ potential could be small, and $\Delta_{{}^{3}P_{2}}$ depends on
model assumptions about the nuclear interaction at short distances.
When the bare ${}^{3}P_{2}$ potential weakens, the gap is especially sensitive
to corrections due to induced interactions in the medium (see, for example,
the discussion in section 3.4 of Ref. Gezerlis _et al._ (2014)). In early
work, the interaction induced by the central components of the nuclear force
was found to increase the ${}^{3}P_{2}$ gap Fay and Layzer (1968); Pethick and
Ravenhall (1991), as would be expected from the discussion of the KL mechanism
in section II. However, as mentioned earlier, calculations that employed
realistic low-energy nuclear potentials with significant non-central
components found that the interference between the central and spin-orbit
component of the nuclear force led to significant suppression of the
${}^{3}P_{2}$ gap for $n_{B}<2n_{\mathrm{sat}}$ Schwenk and Friman (2004).
At $n_{B}\gtrsim 2n_{\mathrm{sat}}$, the description of nuclear interactions
relies on model assumptions since the typical nucleon momenta
$p\gtrsim\Lambda_{\chi}$. To investigate the competition between a strong and
repulsive central force and the spin-orbit component of the nuclear force at
high density, we revisit the calculation of the induced interaction in simple
models. In what follows, we shall assume that the dominant contribution to the
nucleon-nucleon interaction at short distances is due to the exchange of heavy
vector mesons such as the $\omega$ and $\rho$ mesons with masses
$m_{\omega}\approx m_{\rho}\simeq 800$ MeV. For $n_{B}\lesssim
4n_{\mathrm{sat}}$, $k_{\text{F}}/\Lambda$ where $\Lambda\simeq m_{\omega}$
remains a useful expansion parameter. In this case, including terms up to
${\cal O}[(k_{\text{F}}/\Lambda)^{2}]$, the interaction can be described by
the potential
$\begin{split}V(q,q^{\prime})&=C_{0}+\tilde{C}_{0}\sigma_{1}\cdot\sigma_{2}+C_{2}(q^{2}+q^{\prime
2})+C^{\prime}_{2}(q^{\prime 2}-q^{2})\\\
&+\left[\tilde{C}_{2}(q^{2}+q^{\prime 2})+\tilde{C}^{\prime}_{2}(q^{\prime
2}-q^{2})\right]\sigma_{1}\cdot\sigma_{2}+iV_{SO}~{}q\times
q^{\prime}\cdot(\sigma_{1}+\sigma_{2})\\\
&+V_{T}q\cdot\sigma_{1}q\cdot\sigma_{2}\,.\end{split}$ (8)
In neutron matter, due to the Pauli principle, only the combinations
$\bar{C}_{0}=C_{0}-3\tilde{C}_{0}$, $\bar{C}_{2}=C_{2}-3\tilde{C}_{2}$, and
$\bar{C}^{\prime}_{2}=C^{\prime}_{2}+\tilde{C}^{\prime}_{2}$ are relevant. In
the full expansion of the vector meson potential, there is also a term
proportional to $(q\times q^{\prime}\cdot\sigma_{1})(q\times
q^{\prime}\cdot\sigma_{2})$. An exchange tensor term proportional to
$q^{\prime}\cdot\sigma_{1}q^{\prime}\cdot\sigma_{2}$ is also allowed by the
symmetries of the interaction but is not present in the vector exchange. In
this exploratory study, we shall neglect the spin-orbit squared and tensor
exchange components, and in this case, 5 LECs denoted by $\bar{C}_{0}$,
$\bar{C}_{2}$, $\bar{C}^{\prime}_{2}$, $V_{SO}$, and $V_{T}$ are adequate. The
large and attractive spin-orbit interaction, whose strength is set by $V_{SO}$
plays an important role, as discussed below. Here, $q=k_{1}-k_{3}$ and
$q^{\prime}=k_{1}-k_{4}$, and the momenta of neutrons in the initial state are
given by $k_{1}$ and $k_{2}$, and the final state momenta are $k_{3}$ and
$k_{4}$.
It is straightforward to repeat the calculation of the diagrams described in
the preceding section with the potential in Eq. 8. However, it is simpler to
define the anti-symmetrized potential
$\begin{split}V(q,q^{\prime})&=C_{0}(\delta_{13}\delta_{24}-\delta_{14}\delta_{23})+C_{2}(q^{2}+q^{\prime
2})(\delta_{13}\delta_{24}-\delta_{14}\delta_{23})\\\
&+C^{\prime}_{2}(q^{\prime
2}-q^{2})(\delta_{13}\delta_{24}+\delta_{14}\delta_{23})+\tilde{C}_{0}(\sigma_{13}\cdot\sigma_{24}-\sigma_{14}\cdot\sigma_{23})\\\
&+\tilde{C}_{2}(q^{2}+q^{\prime
2})(\sigma_{13}\cdot\sigma_{24}-\sigma_{14}\cdot\sigma_{23})+\tilde{C}^{\prime}_{2}(q^{\prime
2}-q^{2})(\sigma_{13}\cdot\sigma_{24}+\sigma_{14}\cdot\sigma_{23})\\\
&+2iV_{SO}q\times
q^{\prime}\cdot(\sigma_{13}\delta_{24}+\sigma_{24}\delta_{13})+V_{T}(q\cdot\sigma_{13}q\cdot\sigma_{24}-q^{\prime}\cdot\sigma_{14}q^{\prime}\cdot\sigma_{23})\end{split}$
(9)
that includes the effect of the exchange processes. We use the notation
$\delta_{ij}=\chi_{j}^{\dagger}\chi_{i}$ and
$\sigma_{ij}=\chi_{j}^{\dagger}\sigma\chi_{i}$ for incoming and outgoing two-
component spinors $\chi_{i}$ and $\chi_{j}^{\dagger}$. In this case, the
induced interaction at second-order is calculated by evaluating the diagrams
depicted in Fig. 2.
first-diagram (80,80) t1,t2 b1,b2 l1 r1 fermionb1,v1 fermionv1,t1 fermionb2,v2
fermionv2,t2 fermion,left,label=$\ell+q;b$,tension=0.4v1,v2
fermion,left,label=$\ell;a$,tension=0.4v2,v1
decoration.shape=circle,decoration.filled=shaded,label=$V_{L}$,label.dist=40v1
decoration.shape=circle,decoration.filled=shaded,label=$V_{R}$,label.dist=40v2
label=$-k;1$b1 label=$k;2$b2 label=$-k^{\prime};3$t1 label=$k^{\prime};4$t2
second-diagram (80,80) t1,t2 b1,b2 l1 r1 fermionb1,v1 fermionv1,t1
fermionb2,v2 fermionv2,t2
fermion,left,label=$\ell+q^{\prime};b$,tension=0.4v1,v2
fermion,left,label=$\ell;a$,tension=0.4v2,v1
decoration.shape=circle,decoration.filled=shaded,label=$\bar{V}_{L}$,label.dist=40v1
decoration.shape=circle,decoration.filled=shaded,label=$\bar{V}_{R}$,label.dist=40v2
label=$-k;1$b1 label=$k;2$b2 label=$k^{\prime};4$t1 label=$-k^{\prime};3$t2
Figure 2: ZS (left) and ZS’ (right) diagrams. The hatched blobs represent the
anti-symmetrized interaction defined in Eq. 9.
The diagram on the left is called the zero-sound diagram and denoted as ZS,
and the diagram on the right is called the exchange zero-sound diagram and is
denoted by the symbol ZS’. A detailed derivation of the total induced
interaction $V_{ind}=V_{ind}^{ZS}-V_{ind}^{ZS^{\prime}}$ is presented in
Appendix A.
First, we present the result obtained by neglecting the momentum-dependent
components of the bare central interaction. In this case, the induced
potential
$\begin{split}V^{\rm
ind}&=-(C_{0}^{2}+3\tilde{C}_{0}^{2})(U(q)\delta_{14}\delta_{23}-U(q^{\prime})\delta_{13}\delta_{24})+6C_{0}\tilde{C}_{0}(U(q)\delta_{13}\delta_{24}-U(q^{\prime})\delta_{14}\delta_{23})\\\
&+(-\tilde{C}_{0}^{2}+2C_{0}\tilde{C}_{0})(\sigma_{13}\cdot\sigma_{24}-\sigma_{14}\cdot\sigma_{23})(U(q)+U(q^{\prime}))\\\
&-3\tilde{C}_{0}^{2}(\sigma_{13}\cdot\sigma_{24}+\sigma_{14}\cdot\sigma_{23})(U(q)-U(q^{\prime}))\end{split}$
(10)
The s- and p-wave potentials given by this interaction are
$\begin{split}V^{\rm
ind}_{{}^{1}S_{0}}(0)=\bar{C}^{2}_{0}\frac{mk_{F}}{3\pi^{2}}(2\log{2}+1)\\\
V^{\rm
ind}_{{}^{3}P_{J}}(0)=-\bar{C}^{2}_{0}\frac{mk_{F}}{5\pi^{2}}(2\log{2}-1)\,.\end{split}$
(11)
where again $\bar{C}_{0}=C_{0}-3\tilde{C}_{0}$ is the momentum-independent
bare ${}^{1}S_{0}$ potential.
The calculation of the induced potential, including the momentum-dependent
central interactions, is tedious, and the analytic results contain a large
number of terms. Details of the intermediate expressions can be found in
Appendix A.
We find analytic results for the second-order induced potentials in the $s$
and $p$ induced potentials. The induced ${}^{1}S_{0}$ and ${}^{3}P_{J}$
potentials calculated to $\mathcal{O}(mk_{F}^{3})$ are given by
$\begin{split}V^{\rm
ind}_{{}^{1}S_{0}}&=mk_{F}\bar{C}_{0}^{2}\frac{1}{3\pi^{2}}(1+2\log{2})+mk_{F}^{3}[\bar{C}_{0}\bar{C}_{2}\frac{2}{3\pi^{2}}(5+4\log{2})\\\
&+\bar{C}_{0}\bar{C}^{\prime}_{2}\frac{2}{5\pi^{2}}(7-4\log{2})-\bar{C}_{0}V_{T}\frac{16}{15\pi^{2}}(2+\log
2)]\,,\end{split}$ (12)
and
$\begin{split}V^{\rm
ind}_{{}^{3}P_{J}}&=mk_{F}\bar{C}_{0}^{2}\frac{1}{5\pi^{2}}(1-2\log{2})+mk_{F}^{3}[\bar{C}_{0}\bar{C}_{2}\frac{2}{105\pi^{2}}(59-68\log{2})\\\
&-\bar{C}_{0}\bar{C}^{\prime}_{2}\frac{2}{105\pi^{2}}(29+52\log{2})\\\
&+\bar{C}_{0}V_{T}\frac{1}{105\pi^{2}}(91J^{2}-221J+50+(224J^{2}-544J+220)\log
2)]\,,\end{split}$ (13)
respectively.
When momentum dependence of the central interaction is neglected, the bare
spin-orbit force does not contribute to the induced interaction, and the
leading order dependence of the induced potential on $J$ is determined by the
tensor interaction. See Appendix A for a detailed discussion.
These contributions have the same behavior as the leading order KL result. The
contribution to the induced interaction in a particular partial wave arising
from terms in the bare interaction that do not contribute to that partial wave
is strongly influenced by the KL singularity at $q=2k_{\text{F}}$. For this
reason, their contribution is suppressed relative to other terms in the
interaction at the same order in the expansion. Notice, for example, that in
the p-wave induced potential, the term proportional to
$\bar{C}_{0}\bar{C}^{\prime}_{2}$ has a numerical factor five times larger
than $\bar{C}_{0}\bar{C}_{2}$ and fifteen times larger than $\bar{C}_{0}V_{T}$
for ${}^{3}P_{2}$. This implies that the singularity at $q=2k_{\text{F}}$ does
not play an essential role when $\bar{C}^{\prime}_{2}$ is of modest size. By
comparing the relevant terms in Eq. 13, we find that the KL singularity plays
an essential role only when
$\bar{C}^{\prime}_{2}\ll\bar{C}_{0}/(16k_{\text{F}}^{2})$. In what follows, we
shall continue to use the term Kohn-Luttinger effect to refer to the induced
interaction, but it should be borne in mind that the singularity at
$q=2k_{\text{F}}$ does not play a dominant role for typical values of
$\bar{C}^{\prime}_{2}$ we explore in this study.
Since the spin-orbit force is strong and important in the ${}^{3}P_{2}$
channel, we expect that it may contribute to the induced interaction even
though it enters at a higher order in the momentum expansion. To investigate
the impact of the spin-orbit coupling, we calculate a subset of the terms that
contain the central and spin-orbit interactions at $\mathcal{O}(mk_{F}^{5})$.
These corrections to the induced potentials are given by:
$\begin{split}V_{{}^{1}S_{0}}^{(5)}&=mk_{F}^{5}[\bar{C}_{2}^{2}\frac{8}{315\pi^{2}}(277+96\log{2})-\bar{C}_{2}^{\prime
2}\frac{8}{105\pi^{2}}(43+24\log{2})\\\
&+\bar{C}_{2}\bar{C}^{\prime}_{2}\frac{32}{105\pi^{2}}(37+6\log{2})+V_{SO}^{2}\frac{8}{35\pi^{2}}(17+16\log
2)]\,,\end{split}$ (14)
and
$\begin{split}V_{{}^{3}P_{J}}^{(5)}&=mk_{F}^{5}[\bar{C}_{2}^{2}\frac{16}{567\pi^{2}}(83-24\log{2})+\bar{C}_{2}^{\prime
2}\frac{64}{567\pi^{2}}(34-3\log{2})\\\
&-\bar{C}_{2}\bar{C}^{\prime}_{2}\frac{16}{2835\pi^{2}}(523+204\log{2})+\bar{C}^{\prime}_{2}V_{SO}[J(J+1)-4]\frac{32}{945\pi^{2}}(43+24\log{2})\\\
&+V_{T}V_{SO}[J(J+1)-4]\frac{16}{945\pi^{2}}(43+24\log 2)\\\
&+V_{SO}^{2}(7J^{2}-17J+10)\frac{32}{4725\pi^{2}}(43+24\log 2)]\,.\end{split}$
(15)
These are not all of the terms that contribute at $\mathcal{O}(mk_{F}^{5})$.
Not included are terms proportional to $\bar{C}_{2}V_{T}$,
$\bar{C}^{\prime}_{2}V_{T}$, and $V_{T}^{2}$.
From Eq. 15, we deduce that when the bare spin-orbit interaction is
attractive, it could enhance ${}^{3}P_{2}$ pairing if
$2\bar{C}^{\prime}_{2}+V_{T}+\frac{4}{5}V_{SO}>0\,.$ (16)
The relation of this result to the suppression of ${}^{3}P_{2}$ pairing at low
density due to spin-orbit interactions found in Ref. Schwenk and Friman
(2004), which employed a realistic low-momentum nucleon-nucleon potential fit
to scattering data warrants further study.
We consider three scenarios to study the implications of these results for
dense neutron matter. Each scenario is defined by a specification of the LECs
that appear in the bare potential defined in Eq. 8. In scenario A, we shall
assume that the exchange of heavy vector mesons mediates the interactions
between neutrons. When the mass of the vector meson is large compared to the
neutron Fermi momentum, and interaction is described by a current-current
four-fermion Lagrangian
$\mathcal{L}_{\rm
int}=-\frac{G_{V}}{2}(\overline{n}\gamma_{\mu}n)(\overline{n}\gamma^{\mu}n)\,.$
(17)
Retaining only the leading terms in the $p/m_{n}$ expansion, the LECs
appearing in Eqs. 12 and 13 are given by
$\bar{C}_{0}=G_{V},\quad\bar{C}_{2}=\frac{5G_{V}}{8m_{n}^{2}},\quad\bar{C}^{\prime}_{2}=\frac{3G_{V}}{8m_{n}^{2}},\quad
V_{SO}=-\frac{3G_{V}}{8m_{n}^{2}},\quad V_{T}=\frac{G_{V}}{4m_{n}^{2}}$ (18)
Although the simple vector interaction cannot capture the complex nature of
interactions between neutrons, which could involve a richer operator structure
due to pion exchange and many nucleon forces, it is able to describe the
qualitative aspects of the nucleon-nucleon interaction at high momentum; it
predicts negative phase shifts in the ${}^{1}S_{0}$, ${}^{3}P_{0}$, and
${}^{3}P_{1}$ channels. The phase shift in the ${}^{3}P_{2}$ channel vanishes
because $V_{SO}=-\bar{C}^{\prime}_{2}$, and the spin-orbit interaction exactly
cancels the contribution from the central force. This aspect of short-range
vector interactions that leads to a vanishing bare potential in the
${}^{3}P_{2}$ channel is a generic feature of any four-fermion interaction
without derivative couplings since initial and final states constructed only
from spin and helicity operators cannot be combined to form a tensor of rank
greater than 1. As a result, such an interaction cannot generate potentials in
channels with $J\geq 2$ at the tree level. Including momentum dependence in
the meson propagator, explicit derivative couplings (i.e., momentum dependence
beyond that found in the Dirac spinors), or momentum dependence from loops
lifts this restriction.
Figure 3: Total potential for heavy vector boson exchange at
$\mathcal{O}(mk_{F}^{3})$ without (left) and with (right) $V_{SO}$ corrections
at $\mathcal{O}(mk_{F}^{5})$. The coupling constant is tuned to $F_{0}=3$ at
$3n_{\mathrm{sat}}$.
The couplings are related to the Fermi liquid parameters $F_{0}$ and $G_{0}$.
In the mean-field theory, $F_{0}$ and $G_{0}$ depend only on the central
components of the interaction and are given by
$\displaystyle
F_{0}=N(0)\left(\bar{C}_{0}+2k_{\text{F}}^{2}(\bar{C}_{2}+3\bar{C_{2}}^{\prime})\right)\,,$
(19) $\displaystyle
G_{0}=-N(0)\left(\bar{C}_{0}+2k_{\text{F}}^{2}(\bar{C}_{2}-\bar{C_{2}}^{\prime})\right)\,,$
(20)
where $N(0)=\sqrt{k_{\text{F}}^{2}+m_{n}^{2}}k_{\text{F}}/2\pi^{2}$ is the
density of states for each spin at the Fermi surface. Thus, if $F_{0}$ and
$G_{0}$ are specified, the strength of the s-wave components of the
interaction are constrained by the equation
$(2\bar{C}_{0}+4k_{F}^{2}\bar{C}_{2})=(F_{0}-3G_{0})/2N(0)$ and the p-wave
component is obtained using the relation
$\bar{C}^{\prime}_{2}=(F_{0}+G_{0})/(8N(0)k_{F}^{2})$. In scenario A, the
interaction contains just one parameter, $G_{V}$. In this case, $F_{0}$ and
$G_{0}$ are not independent and $G_{V}$ is determined by specifying $F_{0}$,
which we take to be in the range $2-4$ at $n_{B}=3~{}n_{\mathrm{sat}}$.
Figure 4: The ${}^{3}P_{2}$ gap corresponding to the short-range vector
interaction in Eq. 17. Results at $\mathcal{O}(mk_{F}^{3})$ are shown in the
left panel and at $\mathcal{O}(mk_{F}^{5})$ in the right panel. The coupling
constant is tuned to the given value of $F_{0}$ at $3n_{\mathrm{sat}}$. (color
online)
The induced and total p-wave interactions at the Fermi surface are shown in
Fig. 3 for $F_{0}=3$ at $n_{B}=3n_{\mathrm{sat}}$. The actual value shown
$VN(0)/2$ is the quantity in the exponent of the BCS equation. The model
naturally prefers ${}^{3}P_{2}$ pairing because although the induced
interaction at the Fermi surface is attractive for all values of the total
angular momentum $J=0,1,2$, the sum $V^{\rm bare}+V^{\rm ind}$ is only
attractive for $J=1,2$ and the net attraction is larger for $J=2$. In Fig. 4,
we show the ${}^{3}P_{2}$ pairing gaps calculated using the BCS formula in Eq.
5. Results are shown for three choices of the coupling $G_{V}$ obtained by
setting $F_{0}=2,3$ and $4$ at $n_{B}=3~{}n_{\mathrm{sat}}$.
To study the interplay between the central p-wave and the spin-orbit
interactions, we consider scenario B, in which we introduce parameters
$\xi_{\rm p}$ and $\xi_{\rm SO}$ to control the strength of the central p-wave
interaction and the spin-orbit interaction, respectively. In this case, we
neglect the tensor coupling, and the LEC constants are given by:
$\bar{C}_{0}=G_{V},\quad\bar{C}_{2}=\frac{G_{V}}{\Lambda^{2}},\quad\bar{C}^{\prime}_{2}=\xi_{\rm
p}\frac{G_{V}}{\Lambda^{2}},\quad
V_{SO}=-\xi_{SO}\frac{G_{V}}{\Lambda^{2}},\quad V_{T}=0$ (21)
As a generic choice at the same order as the nucleon and meson masses, we take
$\Lambda=1\,\mathrm{GeV}$. Fig.5 shows curves of constant $VN(0)/2$ in the
space of $\xi_{p}$ and $\xi_{SO}$ for $G_{V}=20\,\mathrm{GeV}^{-2}$ and
$G_{V}=40\,\mathrm{GeV}^{-2}$. Corresponding values of $F_{0}$ for each value
of $\xi_{p}$ are shown on the right axis.
Figure 5: Contours of constant ${}^{3}P_{2}$ potential in model B in the
$\xi_{p}-\xi_{SO}$ plane for two choices of $G_{V}$ at $3n_{\mathrm{sat}}$.
$F_{0}$, calculated using Eq. 19 is also shown. The black dashed line shows
where the bare interaction vanishes.
The black line shows where the bare interaction is zero. Above this line, the
bare interaction is repulsive, and below it is attractive. The general
behavior of the induced interaction is set by terms proportional to
$\bar{C}^{\prime 2}_{2}$, $\bar{C}^{\prime}_{2}V_{SO}$, and $V_{SO}^{2}$ with
some secondary effects from terms proportional to
$\bar{C}_{0}\bar{C}^{\prime}_{2}$ and $\bar{C}_{2}\bar{C}^{\prime}_{2}$. Even
though some of these terms enter at higher order in the gradient expansion,
they are generally more important than lower-order terms in the induced
interaction because they do not rely on the KL singularity to contribute to
the ${}^{3}P_{2}$ potential. Of the five terms listed, the only ones that can
be attractive are $\bar{C}_{0}\bar{C}^{\prime}_{2}$ and
$\bar{C}_{2}\bar{C}^{\prime}_{2}$ when $\xi_{p}$ is positive, and
$\bar{C}^{\prime}_{2}V_{SO}$ when $\xi_{SO}$ and $\xi_{p}$ are of the same
sign. As a result, for stronger couplings, the overall interaction is
repulsive when $\xi_{p}$ is negative and of reasonable size, even though that
corresponds to a more attractive bare interaction. There is significant net
attraction only when $\xi_{p}$ and $\xi_{SO}$ are both positive and $\xi_{p}$
is not much larger than $\xi_{SO}$ as this leads to a more repulsive bare
interaction without producing enough induced attraction to match.
Fig.6 shows the ${}^{3}P_{2}$ gap as a function of $\xi_{SO}$ for a few
choices of $\xi_{p}$ and the same choices of $G_{V}$. This interplay between
the relative size of $\xi_{p}$ and $\xi_{SO}$ sensitively determines the size
of the gap. This model is not detailed enough to make quantitative
predictions, but the general trend remains that high-order terms in the
expansion play an important role in determining the size of the gap. An
attractive spin-orbit appears to be necessary to have gaps of a reasonable
size. An attractive bare p-wave potential precludes pairing even though the
bare interaction is stronger because of the repulsion due to the induced
interaction.
Figure 6: ${}^{3}P_{2}$ gap as a function of $\xi_{SO}$ in model B for a few
choices of $\xi_{p}$ and $G_{V}$ at $3n_{\mathrm{sat}}$.
To incorporate trends observed in the nucleon-nucleon phase shifts that have
been measured up to $E_{\rm lab}\simeq 300$ MeV which correspond $p_{\rm
CM}=\sqrt{m_{n}E_{\rm lab}/2}\simeq 375$ MeV, we consider scenario C in which
we incorporate a non-zero $V_{T}$ by introducing a parameter $\xi_{T}$ that
sets the strength of the tensor interaction, and
$V_{T}=-G_{V}\xi_{T}/\Lambda^{2}$. This allows us to obtain any desired
ordering of the p-wave phase shifts for $J=0,1,2$ and match scattering data
that require a weakly attractive bare ${}^{3}P_{2}$ interaction and repulsive
interactions in the ${}^{3}P_{0}$ and ${}^{3}P_{1}$ channels. Since the
induced interaction at ${\cal O}(mk_{\text{F}}^{5})$ in Eq. 15 neglected the
part of the potential proportional to $\bar{C}^{\prime}_{2}V_{T}$, the results
we present here are strictly only valid for $\xi_{T}\ll\xi_{SO}$. Thus,
scenario C must be viewed as an initial exploration into the effects of
$V_{T}$ to be continued in future work.
To obtain the correct ordering of the p-wave interactions, the parameters must
satisfy the following conditions. To have an attractive bare ${}^{3}P_{2}$
potential, $\xi_{SO}>\xi_{p}$ and to have a repulsive bare ${}^{3}P_{0}$,
$\xi_{p}+2\xi_{SO}-3\xi_{T}/2>0$. The condition
$2\xi_{SO}/5<\xi_{T}<2\xi_{SO}$ ensures that the ${}^{3}P_{1}$ potential is
most repulsive and ${}^{3}P_{2}$ is most attractive. We define $\alpha$ and
$\beta$ to be the ratio between the bare potentials given by
$\begin{split}\alpha\equiv\frac{V_{{}^{3}P_{2}}}{V_{{}^{3}P_{0}}}=\frac{\xi_{p}-\xi_{SO}}{\xi_{p}+2\xi_{SO}-3\xi_{T}/2}\\\
\beta\equiv\frac{V_{{}^{3}P_{1}}}{V_{{}^{3}P_{0}}}=\frac{\xi_{p}+\xi_{SO}+\xi_{T}}{\xi_{p}+2\xi_{SO}-3\xi_{T}/2}\end{split}$
(22)
Phase shifts for lab energies between $250\,\mathrm{MeV}$ and
$350\,\mathrm{MeV}$ favor $\alpha$ between $-1$ and $-3$ and $\beta$ between
$2$ and $4$. The blue and orange bands in Fig. 7 and Fig. 8 identify regions
$-3<\alpha<-1$ and $2<\beta<4$, respectively with the correct sign for each
bare potential. We only show regions with $\xi_{SO}>0$ and $\xi_{T}>0$ since
this is required to obtain the correct ordering of p-waves phase shifts. These
are also the signs favored by a tensor interaction arising from pion exchange
and a spin-orbit force from heavy meson exchange.
Figure 7: The bare ${}^{3}P_{2}$ potential, the induced potential at
$\mathcal{O}(mk_{F}^{3})$, and the induced $V_{SO}$ at
$n_{B}=3n_{\mathrm{sat}}$. $G_{V}=40\,\mathrm{GeV}^{-2}$ in the upper panel
and $G_{V}=20\,\mathrm{GeV}^{-2}$ in the lower panel.
To illustrate the relevance of the induced interaction, Fig. 7 shows the
interaction for scenario C broken down into bare potential,
$\mathcal{O}(mk_{F}^{3})$ induced potential, and $\mathcal{O}(mk_{F}^{5})$
corrections to the induced potential for a few choices of $\xi_{SO}$ and
$\xi_{T}$ and the same two values of $G_{V}$ used for scenario B. In the right
panel, $\xi_{SO}$ is fixed to match the value of $\xi_{p}$ so that the bare
interaction is always zero, as is found in the meson exchange model. The range
of $\xi_{p}$ chosen is motivated by the observation that meson exchange models
predict $0.5\lesssim\xi_{p}\lesssim 1.5$ and matching to phase shifts between
250 and 350 MeV predicts $-0.5\lesssim\xi_{p}\lesssim 0$. As seen before, the
$\mathcal{O}(mk_{F}^{3})$ induced interaction is dominated by the term
proportional to $\bar{C}_{0}\bar{C}^{\prime}_{2}$ and gives more attraction
for positive $\xi_{p}$ and repulsion for negative $\xi_{p}$. The induced
interaction does not exactly go through the origin for $\xi_{p}=0$, but the KL
suppression of the terms that do not include $\bar{C}^{\prime}_{2}$ or
$V_{SO}$ is sufficient that it is not visible on this scale. As expected from
Eq. 23, negative values of $\xi_{p}$ lead to more repulsive
$\mathcal{O}(mk_{F}^{5})$ corrections, with this effect being stronger for
larger values of $\xi_{SO}$ and $\xi_{T}$.
Figure 8: Curves of constant total potential for different choices of
$\xi_{p}$ at $3n_{\mathrm{sat}}$. The black dashed line shows where the spin-
orbit correction vanishes. The coupling constant is
$G_{V}=40\,\mathrm{GeV}^{-2}$. (color online)
Fig. 8 shows the contours of constant potential for scenario C. As in scenario
B, negative or zero $\xi_{p}$ corresponds to repulsion for most of the
parameter space. However, unlike scenario B, when $\xi_{T}$ is of reasonable
size, increasing $\xi_{SO}$ results in repulsion much more quickly. This is a
result of the term proportional to $V_{T}V_{SO}$. The contribution of the
spin-orbit terms can be easily summarized in terms of the constants of this
model:
$V^{(5SO)}_{{}^{3}P_{2}}=\xi_{SO}\left(\xi_{T}+\frac{4\xi_{SO}}{5}-2\xi_{p}\right)\frac{G_{V}^{2}}{\Lambda^{4}}\frac{32mk_{F}^{5}}{945\pi^{2}}(43+24\log
2)$ (23)
In scenario A, $\xi_{T}$ is negative and $\xi_{SO}=\xi_{p}$, so the term in
parentheses is always negative, and the spin-orbit corrections give additional
attraction. However, when we allow the constants to vary and take $\xi_{T}$
positive as is favored by pion exchange and phase shift data, much of the
favored region of the phase diagram shows suppression due to these
corrections. The black dashed line in Fig. 8 shows where
$\xi_{T}+4\xi_{SO}/5-2\xi_{p}=0$. The spin-orbit corrections suppress the
potential above and to the right of this line, while the potential is enhanced
below and to the left.
Figure 9: ${}^{3}P_{2}$ gap as a function of $\xi_{p}$ for a few choices of
$G_{V}$, $\xi_{SO}$, and $\xi_{T}$ at $n_{B}=3n_{\mathrm{sat}}$.
Fig. 9 shows the ${}^{3}P_{2}$ gap at $3n_{\mathrm{sat}}$ as a function of
$\xi_{p}$ for a few choices of $\xi_{SO}$ and $\xi_{T}$. For negative
$\xi_{p}$, even though the bare interaction is more attractive, the induced
interaction strongly suppresses the gap. The contribution of positive
$\xi_{T}$ suppresses the gap. For positive $\xi_{p}$, MeV-scale gaps are
possible. These results are sensitive to the value of $F_{0}$, as can be
inferred by comparing the upper and lower panels of Fig.7. For smaller
$F_{0}$, the relative importance of the induced interaction is diminished, and
for most of the parameter space, the induced interaction suppresses an
attractive bare interaction. This still has a significant effect as the gap is
exponentially sensitive to the potential. For almost all the explored
parameter space, the induced interaction suppresses the gap by an order of
magnitude or more.
## IV Induced p-wave pairing in quark matter
The inner cores of neutron stars may be made up of deconfined quark matter.
Quarks are expected to form a color superconductor at asymptotic densities,
with the dominant pairing being in the color antitriplet channel
(antisymmetric). See Alford (2001) for a review. Color symmetric pairing is
generally not considered since gluon exchange in the color sextet (symmetric)
is repulsive, while the antitriplet channel is attractive. At asymptotic
densities, pairing all three colors and flavors (up, down, and strange) in
color and flavor antisymmetric pairs is expected (the CFL phase). At lower
densities where the strange quark mass cannot be neglected, up and down quarks
can pair in the color antisymmetric channel. This pairing involves two colors,
typically denoted as red and green, and is called the 2SC phase. In this
phase, the strange quarks and blue-up and blue-down quarks are unpaired. The
possibility that these quarks could pair due to the KL mechanism in QCD was
first studied in Ref. Schäfer (2006). This study investigated the KL effect in
gauge theories where fermions interact via long-range forces that are
dynamically screened due to Landau damping of the magnetic gauge bosons. Here,
the energy dependence of the interaction plays a critical role, and the
results of Ref. Schäfer (2006) indicate that a gap arises via a mechanism
analogous to the Kohn-Luttinger effect but conclude that it is too small to be
phenomenologically relevant.
To assess if the KL mechanism could be relevant in quark matter with short-
range interactions that are independent of energy, we shall calculate the
induced interaction in the ${}^{3}P_{2}$ channel between quarks of the same
flavor and color due to the short-range, flavor and color-independent,
repulsive vector interaction. Such pairing would include the strange and the
up-and-down blue quarks. In what follows, we shall focus on the induced
interaction between strange quarks of the same color at moderate density when
$m_{s}\gg k_{Fs}$ The specific question we address here is if repulsive short-
range interactions introduced to stabilize quark matter inside neutron stars
Baym _et al._ (2018) can lead to pairing gaps of phenomenological relevance.
For concreteness, we consider a description of quark matter within the purview
of the Nambu-Jona-Lasino(NJL) models (see Buballa (2005) for a comprehensive
review). In these models, defined by the interaction Lagrangian Buballa
(2005); Baym _et al._ (2018); Song _et al._ (2019)
${\cal
L}_{V}=G(\bar{q}q)^{2}+H(\bar{q}\bar{q})(qq)-g_{V}~{}(\bar{q}\gamma_{\mu}q)^{2}\,,$
(24)
where $G$ and $H$ are the four-fermion scalar quark-antiquark and diquark
coupling strengths, and vector coupling $g_{V}$ is introduced to generate
higher pressures as noted earlier. The scalar interaction between quarks and
antiquarks leads to a non-trivial vacuum with $\langle\bar{q}q\rangle\neq 0$
that spontaneously breaks chiral symmetry and the coefficient, $G$, is
determined by hadron masses and the pion decay constant in the vacuum. For
typical momentum cut-off $\Lambda_{\rm NJL}\approx 600$ MeV, $G\Lambda_{\rm
NJL}^{2}\simeq 2$ Buballa (2005). At densities of interest to neutron stars,
chiral symmetry remains broken. The constituent strange quark mass is expected
to be $300-500$ MeV, while the up and down quark masses can be significantly
smaller. The diquark coupling $H$ and the vector coupling $g_{V}$ are expected
to be of similar size because they can be thought of as arising from the same
underlying high-energy color current-current interactions in QCD Song _et
al._ (2019). Their values at the densities of interest to neutron stars are
determined phenomenologically. The diquark coupling $H$, which encodes the
attraction in the color antisymmetric channel, leads to s-wave pairing between
quarks. For $H\simeq G$, the s-wave pairing gap between up and down quarks is
about $50$ MeV and is typically inadequate to induce pairing between strange
quarks and light quarks Alford _et al._ (2008), as mentioned above. The
analysis of the quark matter EOS in Baym _et al._ (2018, 2019) concluded that
vector coupling needed to be of moderate size with $g_{V}\simeq G$ to support
the large sound speed needed to support a two solar mass neutron star.
First, we note that the contribution to the induced interaction from the
closed fermion loop (the first diagram in Fig. 1) is enhanced by a factor
$N_{f}N_{c}$. This is because the bare vector interaction introduced to
stiffen the quark matter EOS is independent of color and flavor. Thus, in
contrast to the one-component fermi system, where the contribution from the
closed fermion loop was canceled by the diagram that encoded the vertex
corrections, in quark matter with $N_{f}=N_{c}=3$, the first diagram in Fig. 1
makes the dominant contribution to the induced potential. In computing this
diagram, the up and down quarks must be treated as relativistic particles,
leading to a somewhat more complicated expression. After doing the Matsubara
sum and noting that $\bar{u}_{3}\not{q}u_{1}=\bar{u}_{4}\not{q}u_{2}$ for
$u_{1}$ and $u_{2}$ incoming and $\bar{u}_{3}$ and $\bar{u}_{4}$ outgoing
spinors, the induced potential from the first diagram is given by:
$V^{\rm
ind}=g_{V}^{2}(\bar{u}_{3}\gamma_{\mu}u_{1})(\bar{u}_{4}\gamma_{\nu}u_{2})\left(\frac{E_{k}+m_{s}}{2m_{s}}\right)^{2}\sum_{f,c}\int\frac{\ell
d\ell
d\Omega_{\ell}}{4\pi^{3}qE_{\ell}}\frac{\Theta(k_{fc}-\ell)}{c_{q\ell}-q/2\ell}(2\ell^{\mu}\ell^{\nu}-g^{\mu\nu}\vec{\ell}\cdot\vec{q})$
(25)
Figure 10: Induced p-wave potential between strange quarks for
$g_{V}=2/\Lambda^{2}_{NJL}$, $\Lambda_{NJL}=600\,\mathrm{MeV}$, and
$m_{s}=350\,\mathrm{MeV}$.
The calculation of the induced interaction, including both the electric
($\bar{u}\gamma^{i}u$ for $i=0$) and magnetic ($\bar{u}\gamma^{i}u$ for
$i=(1,2,3)$) components is unwieldy. In what follows, we shall focus on the
electric component as the magnetic component is suppressed by the strange
quark mass. In this case, setting $\mu=\nu=0$ in Eq. 25 we find that
$V^{\rm
ind}=\frac{g_{V}^{2}\delta_{13}\delta_{24}}{2\pi^{2}q}\sum_{c}\int\frac{d\ell
dc_{q\ell}}{c_{q\ell}-q/2\ell}\left[\sum_{f=u,d}(2\ell^{2}-\ell
qc_{q\ell})\Theta(k_{fc}-\ell)+2\ell m_{s}\Theta(k_{sc}-\ell)\right]\,.$ (26)
After performing the momentum integrals, we find that
$V^{\rm
ind}=g_{V}^{2}\delta_{13}\delta_{24}\sum_{c}\left[\sum_{f=u,d}\left(-\frac{k_{fc}^{2}}{2\pi^{2}}+\frac{q^{2}}{2}U_{0}^{\rm
rel}\left(\frac{q}{k_{fc}}\right)-2k_{fc}^{2}U_{2}^{\rm
rel}\left(\frac{q}{k_{fc}}\right)\right)-2U(q)\right]$ (27)
where the relativistic Lindhard functions $U_{0}^{\rm rel}$ and $U_{2}^{\rm
rel}$ defined in Appendix B.
Note that by not including the magnetic part of the vector integration
($\bar{u}\gamma^{i}u$ for $i=(1,2,3)$), explicit dependence on $J$ has been
removed, and all the p-waves have the same potential. Nonetheless, the
${}^{3}P_{2}$ gap remains of primary interest because the bare interaction
vanishes for $J=2$, while it is repulsive for $J=0,1$. Fig. 10 shows the
induced p-wave potential and ${}^{3}P_{2}$ gap for strange quarks of mass
$350$ MeV, for $g_{V}=2/\Lambda_{NJL}^{2}$ and
$\Lambda_{NJL}=600\,\mathrm{MeV}$. Densities of each color and flavor are
determined assuming 2SC pairing of up and down quarks, charge neutrality, and
beta equilibrium. This value of $g_{V}$ suggested by comparison with NJL
models is smaller than the smallest coupling we considered for nucleons.
However, it should be noted that even the smallest values of $G_{V}$
considered for nucleons would result in a sound speed greater than one for the
lightest quarks so this is a physically reasonable choice.
## V Implications for Neutron Stars
The thermal evolution of neutron stars, especially those that are reheated by
accretion from a companion at late times, is sensitive to heat capacity and
neutrino emissivity in their cores Yakovlev and Pethick (2004); Page _et al._
(2009); Potekhin _et al._ (2015). The neutrino emissivity and the specific
heat of dense matter are both strongly modified by Cooper pairing. When the
pairing gap is large compared to the temperature, the neutrino emissivity and
the specific heat are exponentially suppressed by the factor
$\exp{(-\Delta/kT)}$. Additionally, in the vicinity of the critical
temperature, Cooper pair breaking and formation (PBF) processes enhance the
neutrino emissivity, and this enhancement is especially important for neutron
Cooper pairing in the ${}^{3}P_{2}$ channel in the core of the neutron star
Page _et al._ (2009); Potekhin _et al._ (2015). Studies of isolated neutron
star cooling reported in Ref. Page _et al._ (2009) that include the modified
URCA ($nn\rightarrow npe^{-}\bar{\nu}_{e}$ and $e^{-}pn\rightarrow nn\nu_{e}$)
reactions and the PBF process but discount the possibility of other more rapid
neutrino emission processes such as direct URCA Lattimer _et al._ (1991) find
that a critical temperature for ${}^{3}P_{2}$ pairing,
$T_{c}\approx\Delta_{{}^{3}P_{2}}/1.7$, that is larger than $5\times 10^{8}$ K
($\approx 50$ keV) throughout the inner core would be disfavored by
observations. This favors a scenario in which the $\Delta_{{}^{3}P_{2}}$ is
suppressed at the modest density encountered in the outer core due to the
competition between the interactions induced by the central and spin-orbit
components of the nuclear forces Schwenk and Friman (2004) but is insensitive
to the behavior of the gap at higher density.
Accreting neutron stars exhibit a diversity of cooling behaviors, and a few
neutron stars show behavior that requires rapid neutrino cooling Yakovlev and
Pethick (2004); Brown _et al._ (2018). Such rapid neutrino cooling can be
realized in the dense nuclear matter when the proton fraction in the core
exceeds about $11\%$ to lift kinematic restrictions on the direct URCA
reactions $e^{-}+p\rightarrow n+\nu_{e}$ and $n\rightarrow
e^{-}+p+\bar{\nu}_{e}$ Lattimer _et al._ (1991). In addition, rapid cooling
would also require ${}^{3}P_{2}$ pairing to be absent at high density. Our
finding that the induced interaction disfavors ${}^{3}P_{2}$ when the spin-
orbit and tensor forces are strong and attractive provides some insight into
the conditions necessary to realize unpaired neutron matter at high density
characterized by a high sound speed. On the other hand, if the central
component of the p-wave interaction is strongly repulsive and the non-central
components are weak, the induced interaction favors ${}^{3}P_{2}$ pairing
between neutrons, and rapid neutrino cooling cannot be realized in nuclear
matter at high density. In this scenario, rapid cooling in neutron stars would
require new ungapped fermionic excitations, such as hyperons or quarks, to
enable the direct URCA reaction.
Figure 11: Contours for $\Delta_{{}^{3}P_{2}}=10\,\mathrm{keV}$ for
$G_{V}=40\,\mathrm{GeV}^{-2}$ at $3n_{\mathrm{sat}}$ for a few choices of
$\xi_{p}$. Below and to the right of the contour, the gap is larger than 10
keV.
In transiently accreting neutron stars, inference of the heat deposition due
to deep crustal heating from observations of accretion outbursts and the
inference of the core temperature from subsequent observation in quiescence
have been used to derive a lower limit to the neutron star core heat capacity
Cumming _et al._ (2017); Brown _et al._ (2018). Further, if the neutron
stars cooling can be observed during quiescence, an upper limit on the core
heat capacity can also be deduced from observations Cumming _et al._ (2017);
Brown _et al._ (2018). For neutron stars in the low mass x-ray binaries KS
1731-260, MXB 1659-29, and XTE J1701-462, with core temperatures in the range
$10^{7}-10^{8}$ K, the lower limit was found to be a factor of a few below the
core heat capacity expected if neutrons and protons in the core are paired.
However, upper limits from future cooling observations in these systems could
constrain the extent of neutron pairing in the neutron star core. For example,
the analysis in Ref. Brown _et al._ (2018) suggests that if the neutron star
in MXB 1659-29 cools by about 4% during a 10-year period, a very large
fraction of the neutrons in the neutron star core must be superfluid with a
gap that is much larger than a few keV. If observed, it would disfavor a large
attractive tensor interaction and would require an attractive spin-orbit
interaction as shown in Fig.11. Repulsive bare p-wave interactions permit
larger regions of parameter space, while attractive bare p-wave interactions
are more restrictive.
## VI Conclusion
We have calculated the induced potential between fermions at the Fermi surface
to study the role of polarization effects in the dense medium. We find that
short-range repulsive interactions due to the exchange of heavy vector mesons
between neutrons, whose strength is related to the Fermi liquid parameter
$F_{0}$ and the sound speed at high density, induce an attractive p-wave
potential. Using a model that allows us to independently vary the strength of
central and non-central p-wave interactions, we have investigated the
competition between the bare and induced interactions to determine the
conditions necessary to realize ${}^{3}P_{2}$ pairing in neutron matter at
high density. When neutron matter is characterized by a large speed of sound
$c_{s}^{2}>1/3$ and $F_{0}\gtrsim 2$, the induced interaction plays an
important role. We find that
* •
The contribution to the induced interaction in a particular partial wave
arising from terms in the bare interaction that do not contribute to that
partial wave are suppressed because their contribution is strongly influenced
by the KL singularity at $q=2k_{\text{F}}$. For this reason, the bare p-wave
and the spin-orbit interactions are generally more important than lower-order
terms for the induced ${}^{3}P_{2}$ potential.
* •
The induced interaction favors ${}^{3}P_{2}$ pairing if the central component
of the s-wave and p-wave interaction are strongly repulsive and the non-
central components are small. The resulting gap, $\Delta_{{}^{3}P_{2}}$, can
be in the range $0.1-10$ MeV in the neutron star core and is exponentially
sensitive to the induced potential.
* •
When the central p-wave and the spin-orbit interaction are both strong and
attractive, the induced interaction is repulsive. Although the bare
interaction is strongly attractive, the induced repulsion can preclude pairing
or suppress $\Delta_{{}^{3}P_{2}}$ by orders of magnitude.
* •
In the presence of a strongly attractive spin-orbit interaction, the induced
interaction favors ${}^{3}P_{2}$ pairing when the central p-wave is repulsive.
Pairing persists even when the strength of the central p-wave repulsion is
greater than the attractive spin-orbit interaction.
An important caveat to these findings is our assumption that the bare
interaction at the Fermi surface is well represented by Eq. 8. Further, as
noted earlier, at the high momenta of relevance when
$n_{B}>2n_{\mathrm{sat}}$, the nucleon-nucleon potential, and thereby the
parameters of our model, are not well constrained by scattering data.
Nonetheless, results obtained within the purview of the model allowed us to
explore the connection between pairing and the strong repulsive central
interactions needed to generate a high sound speed and large $F_{0}$ at
densities expected in the cores of massive neutron stars. As noted earlier,
our calculation, which includes the effect due to strong spin-orbit forces,
provides simple formulae to gauge the interplay between repulsive central
interaction and attractive spin-orbit interactions. However, a study of the
role of strong tensor interactions warrants further study.
Another aspect that warrants mention is the role of many-body forces. Although
we have not explicitly accounted for them in our study here, earlier work has
demonstrated that three-body forces can be incorporated through a density-
dependent two-body potential that can then be constructed by normal ordering
the three-body force with respect to a convenient reference state, such as the
ground state of the non-interacting many-body system Hebeler and Schwenk
(2010); Holt _et al._ (2020). Including the three-body force would thereby
introduce a density dependence to the parameters of our model that set the
strength of the two-body s-wave and p-wave interactions in dense matter. We
believe the large range of parameter values we explored should be sufficient
to account for corrections due to many-body forces partially. The density-
dependence of the two-nucleon partial-wave matrix elements at the Fermi
surface and the correlation between the parameters induced by the 3-body
forces will be explored in future work.
## VII Acknowledgements
The U.S. DOE supported the work of M. K. and S. R. under Grant No. DE-FG02-
00ER41132. We thank Silas Beane, Roland Farrell, Bengt Friman, Yuki Fujimoto,
and Achim Schwenk for their suggestions and helpful discussions. S. R. also
thanks the members of the N3AS Physics Frontier Center, funded by the NSF
Grant No. PHY-2020275 for useful conversations.
## Appendix A Induced interactions
In this appendix, we derive analytic results for the induced interaction in
two steps. First, for the sake of simplicity and clarity, we assume that the
bare potential only contains a momentum-independent s-wave interaction
characterized by the $C_{0}$ and $\tilde{C}_{0}$, and spin-orbit force with
strength $V_{SO}$. In this case, the ZS diagram involves the product
$V_{L}\times V_{R}$, where
$\begin{split}V_{L}&=C_{0}(\delta_{13}\delta_{ab}-\delta_{1b}\delta_{a3})+\tilde{C}_{0}(\sigma_{13}\cdot\sigma_{ab}-\sigma_{1b}\cdot\sigma_{a3})-V_{SO}2iq\times(\ell+k^{\prime})\cdot(\sigma_{13}\delta_{ab}+\sigma_{ab}\delta_{13})\\\
V_{R}&=C_{0}(\delta_{24}\delta_{ba}-\delta_{2a}\delta_{b4})+\tilde{C}_{0}(\sigma_{24}\cdot\sigma_{ba}-\sigma_{2a}\cdot\sigma_{b4})+V_{SO}2iq\times(\ell-k)\cdot(\sigma_{24}\delta_{ba}+\sigma_{ba}\delta_{24})\,.\end{split}$
(28)
Evaluating term-by-term, we find that the $C_{0}^{2}$ contribution is given by
$\begin{split}C_{0}^{2}\sum_{ab=\\{\uparrow,\downarrow\\}}(\delta_{13}\delta_{ab}-\delta_{1b}\delta_{a3})(\delta_{24}\delta_{ba}-\delta_{2a}\delta_{b4})=C_{0}^{2}\delta_{14}\delta_{23}\end{split}$
(29)
To calculate the $\tilde{C}^{2}_{0}$ contribution, we use the following
identities:
$\begin{split}\sum_{ab=\\{\uparrow,\downarrow\\}}\sigma_{ab}^{i}\sigma_{ba}^{j}&=2\delta^{ij}\\\
\sum_{b=\\{\uparrow,\downarrow\\}}\sigma_{bc}^{j}\sigma_{ab}^{i}&=\chi^{\dagger}_{b}\sigma^{j}\sigma^{i}\chi_{a}=\delta_{ab}\delta^{ij}-i\varepsilon^{ijk}\sigma_{ab}^{k}\\\
\sum_{bc=\\{\uparrow,\downarrow\\}}\sum_{i=1}^{3}\sigma_{cd}^{i}\sigma_{bc}^{j}\sigma_{ab}^{i}&=\sum_{i=1}^{3}\chi_{d}^{\dagger}\sigma^{i}(2\delta^{ij}-\sigma^{i}\sigma^{j})\chi_{a}=-\sigma_{ad}^{j}\end{split}$
(30)
to find that
$\begin{split}\tilde{C}^{2}_{0}\sum_{ab=\\{\uparrow,\downarrow\\}}(\sigma_{13}\cdot\sigma_{ab}-\sigma_{1b}\cdot\sigma_{a3})(\sigma_{24}\cdot\sigma_{ba}-\sigma_{2a}\cdot\sigma_{b4})=\tilde{C}_{0}^{2}(4\sigma_{13}\cdot\sigma_{24}+3\delta_{14}\delta_{23}+2\sigma_{14}\cdot\sigma_{23})\end{split}$
(31)
The $C_{0}\tilde{C}_{0}$ contribution is calculated by noting that
$\sum_{ab}\delta_{ab}\sigma_{ba}=\mbox{Tr}[\sigma]=0$ and
$\sum_{b}\sigma_{bc}\cdot\sigma_{ab}=3\delta_{ac}$. Explicitly,
$\begin{split}C_{0}\tilde{C}_{0}&\sum_{ab=\\{\uparrow,\downarrow\\}}[(\delta_{13}\delta_{ab}-\delta_{1b}\delta_{a3})(\sigma_{24}\cdot\sigma_{ba}-\sigma_{2a}\cdot\sigma_{b4})+(\sigma_{13}\cdot\sigma_{ab}-\sigma_{1b}\cdot\sigma_{a3})(\delta_{24}\delta_{ba}-\delta_{2a}\delta_{b4})]\\\
&=C_{0}\tilde{C}_{0}[-2(3\delta_{13}\delta_{24}+\sigma_{13}\cdot\sigma_{24})+2\sigma_{14}\cdot\sigma_{23}]\end{split}$
(32)
We have calculated the leading order contributions from the spin-orbit
interaction, proportional to $C_{0}~{}V_{SO}$ and $\tilde{C}_{0}~{}V_{SO}$ and
find that their contributions vanish. First, consider the $C_{0}~{}V_{SO}$
term
$\begin{split}C_{0}V_{SO}&\sum_{ab=\\{\uparrow,\downarrow\\}}[(\delta_{13}\delta_{ab}-\delta_{1b}\delta_{a3})2iq\times(\ell-k)\cdot(\sigma_{24}\delta_{ba}+\sigma_{ba}\delta_{24})\\\
&+(\delta_{24}\delta_{ba}-\delta_{2a}\delta_{b4})2iq\times(-\ell-k^{\prime})\cdot(\sigma_{13}\delta_{ab}+\sigma_{ab}\delta_{13})]\\\
&=2iC_{0}V_{SO}[q\times(\ell-k)\cdot(2\delta_{13}\sigma_{24}-\sigma_{24}\delta_{13}-\sigma_{13}\delta_{24})\\\
&+q\times(-\ell-k^{\prime})\cdot(2\delta_{24}\sigma_{13}-\sigma_{13}\delta_{24}-\sigma_{24}\delta_{13})]\,.\end{split}$
(33)
Eq. 33 can be simplified further by noting that terms proportional to
$q\times\ell$ vanish upon integrating over the angle $\theta_{q\ell}$ and
using the fact that $q\times k=q\times k^{\prime}=-q\times q^{\prime}/2$. We
find the induced interaction proportional to $C_{0}~{}V_{SO}$
$\begin{split}iC_{0}V_{SO}[q\times
q^{\prime}\cdot(2\delta_{13}\sigma_{24}-\sigma_{24}\delta_{13}-\sigma_{13}\delta_{24}+2\delta_{24}\sigma_{13}-\sigma_{13}\delta_{24}-\sigma_{24}\delta_{13})]=0\end{split}$
(34)
To see that $q\times\ell$ terms vanish, notice that the only angular
dependence from the loop integral is on the angle between $q$ and $\ell$.
Consider the integral $\int
d\Omega_{\ell}\hat{\ell}\cdot\hat{u}f(\hat{\ell}\cdot\hat{q})$ where
$f(\hat{\ell}\cdot\hat{q})$ contains the angular dependence of the loop
integral and $\hat{\ell}\cdot\hat{u}$ corresponds to terms like
$q\times\ell\cdot\sigma$. Rotate $\Omega_{\ell}$ so that $\hat{q}=\hat{z}$ and
$\phi_{\ell}=0$ corresponds to the azimuthal angle of $\hat{u}$ calling these
angles $\theta_{q\ell}$ and $\phi_{u\ell}$. Also define the polar angle of
$\hat{u}$ as $\theta_{uq}$ Now
$\hat{\ell}\cdot\hat{u}=\sin{\theta}_{q\ell}\cos{\phi}_{u\ell}\sin{\theta}_{uq}+\cos{\theta}_{q\ell}\cos{\theta}_{uq}$.
Doing the integral $\int d\phi_{u\ell}\cos{\phi}_{u\ell}=0$ so the only term
that survives is proportional to $\cos{\theta}_{uq}$. In the spin-orbit terms,
$\ell$ always enters as $q\times\ell\cdot\sigma=\ell\cdot(\sigma\times q)$
with $\sigma\times q$ orthogonal to $q$, so this contribution always vanishes.
Similarly, the contribution proportional to $\tilde{C}_{0}V_{SO}$ can also be
simplified by making the substitutions $2iq\times(\ell-k)\rightarrow iq\times
q^{\prime}$ and $2iq\times(-\ell-k^{\prime})\rightarrow iq\times q^{\prime}$
and using the identities in Eq. 30. We find that
$\begin{split}\tilde{C}_{0}V_{SO}&\sum_{ab=\\{\uparrow,\downarrow\\}}[(\sigma_{13}\cdot\sigma_{ab}-\sigma_{1b}\cdot\sigma_{a3})iq\times
q^{\prime}\cdot(\sigma_{24}\delta_{ba}+\sigma_{ba}\delta_{24})\\\
&+(\sigma_{24}\cdot\sigma_{ba}-\sigma_{2a}\cdot\sigma_{b4})iq\times
q^{\prime}\cdot(\sigma_{13}\delta_{ab}+\sigma_{ab}\delta_{13})]\\\
&=\tilde{C}_{0}V_{SO}~{}iq\times
q^{\prime}\cdot(2\sigma_{13}\delta_{24}-3\sigma_{24}\delta_{13}+\sigma_{13}\delta_{24}\\\
&+2\sigma_{24}\delta_{13}-3\sigma_{13}\delta_{24}+\sigma_{24}\delta_{13})\\\
&=0\,.\end{split}$ (35)
Thus, spin-orbit terms do not contribute to the induced interaction at leading
order in $V_{SO}$. Up to this order, including all of the non-zero terms
associated with the product $V_{L}\times V_{R}$ and performing the particle-
hole loop integration, we find that the induced interaction due to the ZS
diagram is given by
$\begin{split}V^{\rm
ind}_{ZS}&=-U(q)\left[(C_{0}^{2}+3\tilde{C}_{0}^{2})\delta_{14}\delta_{23}-6C_{0}\tilde{C}_{0}\delta_{13}\delta_{24})\right]\\\
&-U(q)\left[(4\tilde{C}_{0}^{2}-2C_{0}\tilde{C}_{0})\sigma_{13}\cdot\sigma_{24}+(2\tilde{C}_{0}^{2}+2C_{0}\tilde{C}_{0})\sigma_{14}\cdot\sigma_{23})\right]\end{split}$
(36)
where
$\begin{split}U(q)&=-\frac{1}{\beta}\sum_{\ell_{0}}\int\frac{d^{3}\ell}{(2\pi)^{3}}\frac{1}{\ell_{0}-\ell^{2}/2m}\frac{1}{\ell_{0}-(\ell+q)^{2}/2m}\\\
&=-\frac{m}{2\pi^{2}q}\int_{0}^{k_{F}}\ell
d\ell\int_{-1}^{1}\frac{d\cos{\theta}_{q\ell}}{\cos{\theta}_{q\ell}-q/2\ell}\\\
&=-\frac{mk_{F}^{2}}{2\pi^{2}q}\left[-\frac{q}{2k_{F}}+\frac{1}{2}\left(1-\frac{q^{2}}{4k_{F}^{2}}\right)\mbox{log}\left|\frac{1-q/2k_{F}}{1+q/2k_{F}}\right|\right]\end{split}$
(37)
is the positive Lindhard function.
The contribution from the ZS’ diagram is obtained by switching indices 3 and 4
and by replacing $q$ by $q^{\prime}$ in the loop integral. Explicitly,
$\begin{split}V^{\rm
ind}_{ZS^{\prime}}&=-U(q^{\prime})\left[(C_{0}^{2}+3\tilde{C}_{0}^{2})\delta_{13}\delta_{24}-6C_{0}\tilde{C}_{0}\delta_{14}\delta_{23})\right]\\\
&-U(q^{\prime})\left[(4\tilde{C}_{0}^{2}-2C_{0}\tilde{C}_{0})\sigma_{14}\cdot\sigma_{23}+(2\tilde{C}_{0}^{2}+2C_{0}\tilde{C}_{0})\sigma_{13}\cdot\sigma_{24})\right]\,.\end{split}$
(38)
The calculation of the momentum-dependent part of the induced potential is
similar but a bit more tedious and the analytic results involves a large
number of terms. To obtain useful formula with fewer terms we present results
for the spin singlet and spin-triplet contributions. These will require the
second and fourth moments of the Lindhard function denoted $U_{2}$ and
$U_{4}$. $U_{2}$ is defined as follows:
$\begin{split}U_{2}(q)&=-\frac{m}{2\pi^{2}q}\int_{0}^{k_{F}}\ell^{3}d\ell\int_{-1}^{1}\frac{d\cos{\theta}_{q\ell}}{\cos{\theta}_{q\ell}-q/2\ell}\\\
&=-\frac{mk_{F}^{4}}{2\pi^{2}q}\left[-\frac{q}{12k_{F}}-\frac{q^{3}}{16k_{F}^{3}}+\frac{1}{4}\left(1-\frac{q^{4}}{16k_{F}^{4}}\right)\log\left|\frac{1-q/2k_{F}}{1+q/2k_{F}}\right|\right]\end{split}$
(39)
$U_{4}$ is defined analagously and is given by:
$U_{4}(q)=-\frac{mk_{F}^{6}}{2\pi^{2}q}\left[-\frac{q}{30k_{F}}-\frac{q^{3}}{72k_{F}^{3}}-\frac{q^{5}}{96k_{F}^{5}}+\frac{1}{6}\left(1-\frac{q^{6}}{64k_{F}^{6}}\right)\log\left|\frac{1-q/2k_{F}}{1+q/2k_{F}}\right|\right]$
(40)
Five momentum structures appear corresponding to the five pairings of the
combinations of constants given above. The momentum-dependent parts of $V_{L}$
and $V_{R}$ take the following form for the spin-independent terms. The spin-
dependent terms are analagous.
$\begin{split}V_{L}&\supset
C_{2}((-\ell-k^{\prime})^{2}+q^{2})(\delta_{13}\delta_{ab}-\delta_{1b}\delta_{a3})+C^{\prime}_{2}((-\ell-k^{\prime})^{2}-q^{2})(\delta_{13}\delta_{ab}+\delta_{1b}\delta_{a3})\\\
V_{R}&\supset
C_{2}((\ell-k)^{2}+q^{2})(\delta_{24}\delta_{ba}-\delta_{2a}\delta_{b4})+C^{\prime}_{2}((\ell-k)^{2}-q^{2})(\delta_{24}\delta_{ba}+\delta_{2a}\delta_{b4})\end{split}$
(41)
The contributions of the momentum dependence to the induced potential are:
$\begin{split}\xi_{a}(q)&=\frac{1}{\beta}\sum_{\ell_{0}}\int\frac{d^{3}\ell}{(2\pi)^{3}}\Delta(\ell)\Delta(\ell+q)[2q^{2}+(-\ell-k^{\prime})^{2}+(\ell-k)^{2}]\\\
\xi_{b}(q)&=\frac{1}{\beta}\sum_{\ell_{0}}\int\frac{d^{3}\ell}{(2\pi)^{3}}\Delta(\ell)\Delta(\ell+q)[-2q^{2}+(-\ell-k^{\prime})^{2}+(\ell-k)^{2}]\\\
\xi_{c}(q)&=\frac{1}{\beta}\sum_{\ell_{0}}\int\frac{d^{3}\ell}{(2\pi)^{3}}\Delta(\ell)\Delta(\ell+q)[(-\ell-k^{\prime})^{2}+q^{2}][(\ell-k)^{2}+q^{2}]\\\
\xi_{d}(q)&=\frac{1}{\beta}\sum_{\ell_{0}}\int\frac{d^{3}\ell}{(2\pi)^{3}}\Delta(\ell)\Delta(\ell+q)[(-\ell-k^{\prime})^{2}-q^{2}][(\ell-k)^{2}-q^{2}]\\\
\xi_{e}(q)&=\frac{1}{\beta}\sum_{\ell_{0}}\int\frac{d^{3}\ell}{(2\pi)^{3}}\Delta(\ell)\Delta(\ell+q)[((-\ell-k^{\prime})^{2}-q^{2})((\ell-k)^{2}+q^{2})\\\
&+((\ell-k^{\prime})^{2}+q^{2})((\ell-k)^{2}-q^{2})]\end{split}$ (42)
$\Delta(\ell)=(\ell_{0}-\ell^{2}/2m)^{-1}$ is the fermion propagator. After
doing the loop integral, these give:
$\begin{split}\xi_{a}(q)&=-\frac{2mk_{F}^{3}}{3\pi^{2}}-(q^{2}+2k_{F}^{2})U(q)-2U_{2}(q)\\\
\xi_{b}(q)&=-\frac{2mk_{F}^{3}}{3\pi^{2}}+(3q^{2}-2k_{F}^{2})U(q)-2U_{2}(q)\\\
\xi_{c}(q)&=-\frac{mk_{F}^{3}}{\pi^{2}}\left(\frac{11}{15}k_{F}^{2}+\frac{7}{12}q^{2}\right)-\left(k_{F}^{4}+\frac{3}{2}q^{2}k_{F}^{2}+\frac{q^{4}}{8}\right)U(q)\\\
&-\frac{3}{2}q^{2}U_{2}(q)-U_{4}(q)\\\
\xi_{d}(q)&=-\frac{mk_{F}^{3}}{\pi^{2}}\left(\frac{11}{15}k_{F}^{2}-\frac{3}{4}q^{2}\right)-\left(k_{F}^{4}-\frac{5}{2}q^{2}k_{F}^{2}+\frac{17}{8}q^{4}\right)U(q)\\\
&+\frac{5}{2}q^{2}U_{2}(q)-U_{4}(q)\\\
\xi_{e}(q)&=-\frac{mk_{F}^{3}}{\pi^{2}}\left(\frac{22}{15}k_{F}^{2}-\frac{q^{2}}{6}\right)-\left(2k_{F}^{4}-q^{2}k_{F}^{2}-\frac{7}{4}q^{4}\right)+q^{2}U_{2}(q)-2U_{4}(q)\end{split}$
(43)
The total central induced potential in the spin triplet channel:
$\begin{split}V^{\rm
ind}_{S=1}&=-\bar{C}_{0}^{2}(U(q)-U(q^{\prime}))+\bar{C}_{0}\bar{C}_{2}(\xi_{a}(q)-\xi_{a}(q^{\prime}))+\bar{C}_{0}\bar{C}^{\prime}_{2}(\xi_{b}(q)-\xi_{b}(q^{\prime}))\\\
&+\bar{C}_{2}^{2}(\xi_{c}(q)-\xi_{c}(q^{\prime}))+5\bar{C}_{2}^{\prime
2}(\xi_{d}(q)-\xi_{d}(q^{\prime}))+\bar{C}_{2}\bar{C}^{\prime}_{2}(\xi_{e}(q)-\xi_{e}(q^{\prime}))\end{split}$
(44)
The total central induced potential in the spin singlet channel:
$\begin{split}V^{\rm
ind}_{S=0}&=\bar{C}_{0}^{2}\left(U(q)+U(q^{\prime})\right)-\bar{C}_{0}\bar{C}_{2}(\xi_{a}(q)+\xi_{a}(q^{\prime}))+3\bar{C}_{0}\bar{C}^{\prime}_{2}(\xi_{b}(q)+\xi_{b}(q^{\prime}))\\\
&-\bar{C}_{2}^{2}\left(\xi_{c}(q)+\xi_{c}(q^{\prime})\right)+3\left(\bar{C}_{2}^{\prime
2}(\xi_{d}(q)+\xi_{d}(q^{\prime}))+\bar{C}_{2}\bar{C}^{\prime}_{2}(\xi_{e}(q)+\xi_{e}(q^{\prime})\right)\,.\end{split}$
(45)
This gives s- and p-wave central potentials:
$\begin{split}{}^{1}S_{0}&:\bar{C}_{0}^{2}\frac{mk_{F}}{3\pi^{2}}(1+2\log
2)+mk_{F}^{3}[\bar{C}_{0}\bar{C}_{2}\frac{2}{3\pi^{2}}(5+4\log{2})\\\
&+\bar{C}_{0}\bar{C}^{\prime}_{2}\frac{2}{5\pi^{2}}(7-4\log{2})]+mk_{F}^{5}[\bar{C}_{2}^{2}\frac{8}{315\pi^{2}}(277+96\log{2})\\\
&-\bar{C}_{2}^{\prime
2}\frac{8}{105\pi^{2}}(43+24\log{2})+\bar{C}_{2}\bar{C}^{\prime}_{2}\frac{32}{105\pi^{2}}(37+6\log{2})]\end{split}$
(46) $\begin{split}{}^{3}P_{J}&:\bar{C}_{0}^{2}\frac{mk_{F}}{5\pi^{2}}(1-2\log
2)+mk_{F}^{3}[\bar{C}_{0}\bar{C}_{2}\frac{2}{105\pi^{2}}(59-68\log{2})\\\
&-\bar{C}_{0}\bar{C}^{\prime}_{2}\frac{2}{105\pi^{2}}(29+52\log{2})]+mk_{F}^{5}[\bar{C}_{2}^{2}\frac{16}{567\pi^{2}}(83-24\log{2})\\\
&+\bar{C}_{2}^{\prime
2}\frac{64}{567\pi^{2}}(34-3\log{2})-\bar{C}_{2}\bar{C}^{\prime}_{2}\frac{16}{2835\pi^{2}}(523+204\log{2})]\end{split}$
(47)
The spin-orbit potential gives an additional contribution to the p-waves:
$\begin{split}2\bar{C}^{\prime}_{2}V_{SO}\frac{1}{\beta}&\sum_{\ell_{0}}\int\frac{d^{3}\ell}{(2\pi)^{3}}\Delta(\ell)\Delta(\ell+q)[((-\ell-k^{\prime})^{2}-q^{2})iq\times(\ell-k)\cdot(3\delta_{13}\sigma_{24}+\delta_{24}\sigma_{13})\\\
&+((\ell-k)^{2}-q^{2})iq\times(-\ell-k^{\prime})\cdot(3\delta_{24}\sigma_{13}+\delta_{13}\sigma_{24})]\\\
&=\bar{C}^{\prime}_{2}V_{SO}iq\times
q^{\prime}\cdot(\sigma_{13}\delta_{24}+\sigma_{24}\delta_{13})\xi_{f}(q)\end{split}$
(48)
The function $\xi_{f}(q)$ is given by:
$\xi_{f}(q)=-\frac{2mk_{F}^{3}}{3\pi^{2}}+(5q^{2}-4k_{F}^{2})U(q)$ (49)
This gives a contribution to the p-waves after including the ZS’ diagram:
${}^{3}P_{J}:[J(J+1)-4]\bar{C}^{\prime}_{2}V_{SO}\frac{32mk_{F}^{5}}{945\pi^{2}}(43+24\log{2})$
(50)
We calculate the contribution of the tensor interaction only to
$\mathcal{O}(mk_{F}^{3})$. The term proportional to $C_{0}V_{T}$ gives:
$\begin{split}C_{0}V_{T}\frac{1}{\beta}&\sum_{\ell_{0}}\int\frac{d^{3}\ell}{(2\pi)^{3}}\Delta(\ell)\Delta(\ell+q)[-\delta_{13}\delta_{24}((-\ell-k^{\prime})^{2}+(\ell-k)^{2})-2q\cdot\sigma_{13}q\cdot\sigma_{24}\\\
&+(-\ell-k^{\prime})\cdot\sigma_{23}(-\ell-k^{\prime})\cdot\sigma_{14}+(\ell-k)\cdot\sigma_{23}(\ell-k)\cdot\sigma_{14}]\end{split}$
(51)
Doing the calculation for the $\tilde{C}_{0}V_{T}$ term gives $-3$ times the
result for the term proportional to $C_{0}V_{T}$ after reducing to spin
singlet or triplet. The potentials in these channels after including the ZS’
diagram are given by:
$V^{\rm ind}_{S=0}=-\bar{C}_{0}V_{T}(2q^{2}U(q)+2q^{\prime 2}U(q^{\prime}))$
(52) $\begin{split}V^{\rm
ind}_{S=1}&=\bar{C}_{0}V_{T}\Bigl{[}\frac{mk_{F}^{3}}{6\pi^{2}}(-\hat{q}\cdot\sigma_{13}\hat{q}\cdot\sigma_{24}+\hat{q}^{\prime}\cdot\sigma_{13}\hat{q}^{\prime}\cdot\sigma_{24})+U(q)\left(\frac{3}{4}q\cdot\sigma_{13}q\cdot\sigma_{24}\right.\\\
&-\left.\frac{1}{2}q^{\prime}\cdot\sigma_{13}q^{\prime}\cdot\sigma_{24}+\frac{q^{2}}{4}\right)-U(q^{\prime})\left(\frac{3}{4}q^{\prime}\cdot\sigma_{13}q^{\prime}\cdot\sigma_{24}-\frac{1}{2}q\cdot\sigma_{13}q\cdot\sigma_{24}+\frac{q^{\prime
2}}{4}\right)\\\
&+\bigl{.}U_{2}(q)(1+\hat{q}\cdot\sigma_{13}\hat{q}\cdot\sigma_{24})-U_{2}(q^{\prime})(1+\hat{q}^{\prime}\cdot\sigma_{13}\hat{q}^{\prime}\cdot\sigma_{24})\Bigr{]}\end{split}$
(53)
where we define the unit vector $\hat{q}=\vec{q}/|q|$. For the spin triplet,
outgoing spin indices are exchanged on some terms to simplify the equations.
Doing the integrals gives:
$\begin{split}{}^{1}S_{0}&:-\bar{C}_{0}V_{T}\frac{16mk_{F}^{3}}{15\pi^{2}}(2+\log
2)\\\ {}^{3}P_{2}&:-\bar{C}_{0}V_{T}\frac{4mk_{F}^{3}}{15\pi^{2}}(1-\log 2)\\\
{}^{3}P_{1}&:-\bar{C}_{0}V_{T}\frac{4mk_{F}^{3}}{21\pi^{2}}(4+5\log 2)\\\
{}^{3}P_{0}&:\bar{C}_{0}V_{T}\frac{2mk_{F}^{3}}{21\pi^{2}}(5+22\log
2)\end{split}$ (54)
The part of the interaction proportional to $V_{SO}^{2}$ takes the form:
$\begin{split}-8V_{SO}^{2}\frac{1}{\beta}&\sum_{\ell_{0}}\int\frac{d^{3}\ell}{(2\pi)^{3}}\Delta(\ell)\Delta(\ell+q)[(q\times(-\ell-k^{\prime})\cdot\sigma_{13})(q\times(\ell-k)\cdot\sigma_{24})\\\
&+\delta_{13}\delta_{24}(q\times(-\ell-k^{\prime}))\cdot(q\times(\ell-k))]\\\
&=-\frac{2mk_{F}^{3}}{3\pi^{2}}(q^{2}(\sigma_{13}\cdot\sigma_{24}+2\delta_{13}\delta_{24})-(q\cdot\sigma_{13})(q\cdot\sigma_{24}))+U(q)[q^{4}\sigma_{13}\cdot\sigma_{24}\\\
&-q^{2}(q\cdot\sigma_{13})(q\cdot\sigma_{24})+2(q\times
q^{\prime}\cdot\sigma_{13})(q\times
q^{\prime}\cdot\sigma_{24})+8\delta_{24}\delta_{24}q^{2}k_{F}^{2}]\\\
&-U_{2}(q)[4(q^{2}\sigma_{13}\cdot\sigma_{24}-(q\cdot\sigma_{13})(q\cdot\sigma_{24}))+8\delta_{13}\delta_{24}q^{2}]\end{split}$
(55)
A tedious calculation gives the following contribution:
$\begin{split}{}^{1}S_{0}&:V_{SO}^{2}\frac{8mk_{F}^{5}}{35\pi^{2}}(17+16\log
2)\\\ {}^{3}P_{2}&:V_{SO}^{2}\frac{128mk_{F}^{5}}{4725\pi^{2}}(43+24\log 2)\\\
{}^{3}P_{1}&:0\\\
{}^{3}P_{0}&:V_{SO}^{2}\frac{64mk_{F}^{5}}{945\pi^{2}}(43+24\log
2)\end{split}$ (56)
## Appendix B Induced interaction between quarks
Treating quarks relativistically, the screening diagram is given by
$\begin{split}V^{\rm
ind}&=g_{V}^{2}(\bar{u}_{3}\gamma_{\mu}u_{1})(\bar{u}_{4}\gamma_{\nu}u_{2})\left(\frac{E_{k}+m_{s}}{2m_{s}}\right)^{2}\sum_{f,c}\frac{1}{\beta}\sum_{\ell_{0}}\int\frac{d^{3}\ell}{(2\pi)^{3}}\\\
&\times\frac{\mathrm{Tr}[\gamma^{\mu}(\not{\ell}+\not{q}+m_{f})\gamma^{\nu}(\not{\ell}+m_{f})]}{(\ell_{0}^{2}-\ell^{2}-m_{f}^{2})(\ell_{0}^{2}-(\ell+q)^{2}-m_{f}^{2})}\end{split}$
(57)
where $f=u,d,s$ and $c=r,g,b$ denotes the flavor and color of quarks that
appear in the particle-hole loop. Since the screening diagram is enhanced by
the number of flavors and colors and the other diagrams are not, we calculate
only this part of the potential. We neglect anti-particle contributions by
discarding the Matsubara sums that produce terms proportional to
$(\exp[\beta(E+\mu)]+1)^{-1}$, which are negligible at small temperatures.
Doing the trace and noticing that
$\bar{u}_{3}\not{q}u_{1}=\bar{u}_{4}\not{q}u_{2}=0$, Eq. 57 can be written as
$V^{\rm
ind}=g_{V}^{2}(\bar{u}_{3}\gamma_{\mu}u_{1})(\bar{u}_{4}\gamma_{\nu}u_{2})\left(\frac{E_{k}+m_{s}}{2m_{s}}\right)^{2}\sum_{f,c}\int\frac{\ell
d\ell
d\Omega_{\ell}}{4\pi^{3}qE_{\ell}}\frac{\Theta(k_{fc}-\ell)}{\cos\theta_{q\ell}-q/2\ell}(2\ell^{\mu}\ell^{\nu}-g^{\mu\nu}\vec{\ell}\cdot\vec{q})$
(58)
$k_{fc}$ is the Fermi momentum of flavor $f$ and color $c$ with the normal
subscript $F$ suppressed for readability. Since the Fermi momentum of strange
quarks is approximately the same for all colors, also suppress the color label
on $k_{s}$. Expanding to zeroth order in $k_{s}/m_{s}$, only the $\mu=\nu=0$
components contribute and we get:
$V^{\rm
ind}=g_{V}^{2}\delta_{13}\delta_{24}\frac{1}{2\pi^{2}q}\sum_{f,c}\int\frac{\ell
d\ell}{\sqrt{\ell^{2}+m_{f}^{2}}}\frac{dc_{q\ell}}{\cos\theta_{q\ell}-q/2\ell}\Theta(k_{fc}-\ell)(2\ell^{2}+2m_{f}^{2}-2\ell
q\cos\theta_{q\ell})$ (59)
Setting $m_{u}=m_{d}=0$ and discarding components from the strange quark that
are not proportional to $m_{s}$ gives:
$V^{\rm
ind}=g_{V}^{2}\delta_{13}\delta_{24}\sum_{c}\left[\sum_{f=u,d}\left(-\frac{k_{fc}^{2}}{2\pi^{2}}+\frac{q^{2}}{2}U_{0}^{\rm
rel}\left(\frac{q}{k_{fc}}\right)-2k_{fc}^{2}U_{2}^{\rm
rel}\left(\frac{q}{k_{fc}}\right)\right)-2U(q)\right]$ (60)
where the mass in the Lindhard function $U(q)$ is the strange quark mass and
we define relativistic dimensionless Lindhard functions in analogy with the
relativistic ones (defining $\tilde{q}=q/k_{fc}$ to be distinguished from
$\bar{q}=q/k_{s}$):
$\begin{split}U_{0}^{\rm
rel}(\tilde{q}=q/k_{fc})&=-\frac{1}{2\pi^{2}\tilde{q}}\int
d\bar{\ell}\log\left|\frac{1-\tilde{q}/2\bar{\ell}}{1+\tilde{q}/2\bar{\ell}}\right|\\\
&=\frac{1}{2\pi^{2}\tilde{q}}\left[\log\left|1+\frac{2}{\tilde{q}}\right|-\left(1-\frac{\tilde{q}}{2}\right)\log\left|\frac{1-\tilde{q}/2}{1+\tilde{q}/2}\right|\right]\\\
U_{2}^{\rm
rel}(\tilde{q})&=-\frac{1}{2\pi^{2}\tilde{q}}\int\bar{\ell}^{2}d\bar{\ell}\log\left|\frac{1-\tilde{q}/2\bar{\ell}}{1+\tilde{q}/2\bar{\ell}}\right|\\\
&=\frac{1}{6\pi^{2}\tilde{q}}\left[\frac{\tilde{q}}{2}+\frac{\tilde{q}^{3}}{4}\log\left|1+\frac{2}{\tilde{q}}\right|-\left(1-\frac{\tilde{q}^{3}}{8}\right)\log\left|\frac{1-\tilde{q}/2}{1+\tilde{q}/2}\right|\right]\end{split}$
(61)
Analytical expressions for s- and p-wave potentials can easily be found with
Mathematica or equivalent, but are long and unenlightening so we do not
reproduce them here.
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|
# Simulation of Attacker Defender Interaction in a Noisy Security Game
Erick Galinkin, Emmanouil Pountourakis, John Carter, Spiros Mancoridis
###### Abstract
In the cybersecurity setting, defenders are often at the mercy of their
detection technologies and subject to the information and experiences that
individual analysts have. In order to give defenders an advantage, it is
important to understand an attacker’s motivation and their likely next best
action. As a first step in modeling this behavior, we introduce a security
game framework that simulates interplay between attackers and defenders in a
noisy environment, focusing on the factors that drive decision making for
attackers and defenders in the variants of the game with full knowledge and
observability, knowledge of the parameters but no observability of the state
(“partial knowledge”), and zero knowledge or observability (“zero knowledge”).
We demonstrate the importance of making the right assumptions about attackers,
given significant differences in outcomes. Furthermore, there is a measurable
trade-off between false-positives and true-positives in terms of attacker
outcomes, suggesting that a more false-positive prone environment may be
acceptable under conditions where true-positives are also higher.
## Introduction
The use of artificial intelligence in cybersecurity holds tremendous promise,
as time is of the essence in dealing with an intrusion. However, while tasks
like malware detection (Ucci, Aniello, and Baldoni 2019; Raff et al. 2017) and
network anomaly detection (Fernandes et al. 2019; Carter and Mancoridis 2021)
have seen substantial progress, the automation of mitigation and response
remains a challenge. Part of this challenge lies in the fact that there are
many possible ways to represent highly heterogeneous security data, while
other challenges involve finding appropriate responses to particular events.
In general, defenders seek to find a response that has a minimal effect on the
legitimate use of the information systems they are tasked with defending,
while being maximally impactful to potential attackers. This sort of min-max
problem can be viewed through a game theoretic lens.
In the development of game theoretic models, there is a fundamental tension
between constructing a model that is both simple enough to be tractable and
descriptive enough to be useful. In cybersecurity, this tension is exacerbated
due to the high-stakes associated with decisions in conjunction with the fact
that the generation, transmission, and response to information happens rapidly
and with little human intervention such that on-system activity is effectively
instantaneous. In these games, a common simplifying assumption is that
defenders are operating against some adversary that operates according to a
fixed strategy (Liang and Xiao 2013; Manshaei et al. 2013), often represented
using the SIRE model from epidemiology (Khouzani, Sarkar, and Altman 2012;
Liu, Peng, and Zhong 2020; Tambe 2011). In practice, these sorts of
adversaries are rare, and there are human decision-making aspects to how an
attacker behaves within a victim network.
Understanding attacker decision-making is crucial for the mitigation of
attacks. Ideally, defenders desire near real-time responses to attacker
activity as a way to minimize an attacker’s ability to achieve their goals. In
order to generate these responses, it is beneficial to predict the next best
action for the attacker and prevent that action from being taken. This
necessitates modeling both the attacker’s uncertainty about the defender – not
knowing what technologies are available to the defender or what actions they
are likely to take – and the defender’s uncertainty about the attacker. To the
best of our knowledge, ours is the first work that prioritizes the attacker’s
decision-making process and places it within a game theoretic framework.
We leverage the framework of Stochastic Bayesian Games (Albrecht and
Ramamoorthy 2013) (SBG) as our foundation and consider three different models
of attacker knowledge: full knowledge about the state, actions, history, and
parameter space; partial knowledge, where the state is hidden but the
parameters of the environment are known; and zero-knowledge, where the
attacker knows nothing but the actions they have taken and what actions they
can take. In prior work, the full state and action space is always known to
all players, as well as the history of play – only the strategies and utility
function, which are determined by the type of the attacker are unknown. The
modification to include hidden information necessitates some adaptation of the
most common solution method for Stochastic Bayesian Games: Harsanyi-Bellman ad
hoc coordination, which assumes that information about the states and actions
is public knowledge. A key contribution of this work is the development of
direct solutions to a partially observable SBG with limited state and action
spaces given some conditions on the environment.
In order to study the impact of attacker knowledge, we begin by considering
our model with a highly restricted state and action space and simulate
attacker interactions with the model. We find that these scenarios show that a
full-knowledge assumption yields much higher expected outcomes for attackers –
an assumption that demands a more aggressive response from defenders. However,
in limited-knowledge scenarios, attackers must conduct some level of
reconnaissance or training against a target to estimate their ability to
operate, otherwise they perform very poorly. Moreover, we find that attackers
outcomes are largely indifferent to overall alert rates when the rate of
detecting malicious activity is high, suggesting that there is a quantifiable
trade off against alerts unrelated to malicious activity.
## Related Work
Recent challenges like CAGE (TTCP 2021) have encouraged development of models
like CybORG (Foley et al. 2022) that use reinforcement learning to train
autonomous agents that defend against cyber attacks. The Ph.D thesis of
Campbell (Campbell 2022) also considers a very similar problem space and
solution. These models approach the same problem we wish to consider – the
development of a defensive agent that disrupts an adversary while minimizing
impact to network users. Our work approaches a similar concept from first
principles and contributes observations and insights that emerge even in
simple models, particularly from the attacker’s perspective.
This paper builds primarily on prior work by Albrecht and Ramamoorthy
(Albrecht and Ramamoorthy 2013) on Stochastic Bayesian Games (SBG), and the
Ph.D dissertation of Maccarone (Maccarone 2021) on applications of SBG to
nuclear power plant cybersecurity. While many security games use Nash
equilibria or some comparable equilibrium concept, the complexity of
cybersecurity means that often, there are many equilibria and due to hidden
information, it is unlikely that all players will identify the same
equilibrium. HBA relies on players selecting actions according to the observed
history of the game and some set of decision criteria as the game is played
and is not “predictive” in the sense of e.g., Nash equilibrium. We expand the
scope of Maccarone’s work using the HBA concept in a more general
cybersecurity landscape.
The partial observability of the proposed SBG draws on the work of Tomášek,
Bošanský, and Nguyen (Tomášek, Bošanskỳ, and Nguyen 2020) on one-sided
partially observable stochastic games. Their work considered sequential
attacks on some target and develops scalable algorithms for solving zero-sum
security games in this setting. These algorithms consider upper and lower
value bounds on a subgame, and we couple this partial observability with the
aforementioned SBG to more closely approach a real-world setting.
## Game Model
###### Definition 1.
A Partially Observable Stochastic Bayesian security game is defined as:
* •
$S$, the state space, with initial state $s^{0}$ and terminal states
$\overline{S}\subseteq S$
* •
$N=\\{\alpha,\delta\\}$, the players of the game: in our case, the attacker
$\alpha$ and the defender $\delta$. For each player $i$, we define:
* –
action space $A_{i}$
* –
type space $\Theta_{i}$
* –
utility function $u_{i}:S\times A\times S\rightarrow\mathbb{R}$
* –
strategy $\pi_{i}:\mathbb{H}\times A_{i}\times\Theta_{i}\rightarrow[0,1]$
* •
$T:S\times A\times S\rightarrow[0,1]$, the state transition function
* •
$\Delta:\mathbb{N}_{0}\times\Theta\rightarrow[0,1]$, the distribution of types
* •
$p\coloneqq[0,1]\times A_{\alpha}$ The alert probability vector
The element $\mathbb{H}$ contains the history of the game, a sequence of prior
states and actions. Given the speed at which actions occur, we assume that the
attacker and defender move simultaneously at each time step, and let the
action $a^{t}=(a_{\alpha}^{t},a_{\delta}^{t})$ denote the joint action taken
at time $t$. The history $H^{t}=\langle
s^{0},a^{0},s^{1},a^{1},...,s^{t}\rangle$ is a concatenation of all states and
actions taken from time $0$ until time $t$. Our type distributions, $\Delta$
are static in the sense that each player has the same type throughout all time
steps of the game. In our game, the transition function $T$ is always not
known by the players, though in some of the discussed cases, parameters of $T$
may be known.
The type space $\Theta$ is the type space of Harsanyi (Harsanyi 1967), meaning
that a player’s utility and strategies are determined by their type. We assume
that strategies for particular types are known, despite a player’s type being
known only to themselves. In alignment with Albrecht and Ramamoorthy (Albrecht
and Ramamoorthy 2013), a type in this context can be thought of as a “program”
that governs a player’s behavior. In the cybersecurity context, we can also
contextualize the player’s type as the sort of adversary we are playing
against. This can be defined at a high level e.g.,
$\Theta_{\alpha}=\\{$nation-state, cybercriminal, insider threat$\\}$ or at a
low level e.g., $\Theta_{\alpha}=\\{$APT28, LockBit, Greg, …$\\}$. A highly
granular $\Theta$ will offer more specificity, but increases the complexity of
playing the game.
###### Remark 2.
Although the model presented in Definition 1 contains a single attacker and a
single defender, the model can support multiple attackers at the cost of
additional state space complexity, since each attacker has their own unique
values for the hidden information. Throughout this paper, all examples assume
there is only a single attacker.
###### Remark 3.
We use the term “know” to describe information that a player believes with
probability 1. Thus, when considering $H$, each player may have beliefs – even
very strong ones – about the other player’s actions, but knows only the
actions they have taken and the elements of $s^{t}$ that they are able to
directly observe. It is this feature that makes the game “partially
observable” in contrast to conventional SBG where it is assumed that states
and actions are common knowledge.
### Harsanyi-Bellman Ad Hoc Coordination
Ad hoc coordination is derived from the notion of private information in
Bayesian games (Harsanyi 1967) coupled with the inclusion of state,
probabilistic state transition, and time from Stochastic games (Shapley 1953).
If every player in a HBA knows the type distribution exactly, then the game
admits a Bayesian Nash equilibrium (Harsanyi 1968). Ad hoc coordination then,
is based on the assertion that each player does not know the type space
$\Theta_{j}$ of the other players and is thus ignorant of $\Delta$. Since the
players are ignorant of $\Delta$, they may identify different posterior
distributions and cannot guarantee convergence to a Nash equilibrium (Dekel,
Fudenberg, and Levine 2004), even a sub-optimal one. Thus, Albrecht, and
Ramamoorthy (Albrecht and Ramamoorthy 2013) use a best-response rule that
maximizes expected utility with respect to each player’s belief about the type
of the other players.
###### Definition 4.
Harsanyi-Bellman Ad Hoc Coordination (HBA) is defined as
$a_{i}^{t}\sim\operatorname*{arg\,max}_{a_{i}}E_{s^{t}}^{a_{i}}\left[H^{t}\right]$
where:
$\begin{split}E_{s}^{a_{i}}\left[\hat{H}\right]&=\sum_{\theta_{-i}^{*}\in\Theta_{-i}^{*}}Pr(\theta_{-i}^{*}|H^{t})\\\
&\quad\sum_{a_{-i}\in A_{-i}}Q_{s}^{a_{i,-i}}(\hat{H})\\\ &\quad\prod_{j\neq
i}\pi_{j}(\hat{H},a_{j},\theta_{j}^{*})\end{split}$ (1)
is the expected long-term payoff for player $i$ taking action $a_{i}$ in state
$s$, given history up to time $t$ $H^{t}$ and
$\begin{split}Q_{s}^{a}(\hat{H})&=\sum_{s^{\prime}\in
S}T(s,a,s^{\prime})(u_{i}(s,a,s^{\prime})\\\
&+\gamma\max_{a_{i}}E_{s^{\prime}}^{a_{i}}\left[\langle\hat{H},a,s^{\prime}\rangle\right])\end{split}$
(2)
is the expected long-term payoff for player $i$ when the joint action $a$ is
taken in state $s$ and discount factor $\gamma$. We denote the “projected
history” of the game, $\hat{H}$, to be the entire history of the game from
$0\leq\tau\leq t$ including the current time step, $t$, assuming that a
particular action is taken and concatenated to the history. This solution
concept is very effective in conventional SBG settings, but under our
restrictions of partial observability, there are components of $s$ and $a$
that are not seen by both players, creating uncertainty about $H$ for players
at any time step. We propose non-HBA solutions for our simplified settings in
their respective sections.
## Simplified Setting
In our simplified game, we assume an attacking agent and a defending agent who
choose their moves simultaneously. The game is played on an infinite graph,
and the attacker’s objective is to control as many nodes as possible. In this
game our state space $S$ is a tuple of two integers: one that serves as a
count of “alerts” in the environment and a second that consists of the number
of nodes the attacker has infected. The action space for defenders consists of
two actions: a pass action, where they do nothing, and a shutdown action, that
ends the game. The attacker’s action space consists of two actions: infecting
an adjacent node and ending the game.
The game is characterized by:
* •
$p$, the probability that an alert occurs given that the attacking player has
taken an action.
* •
$q$, the probability that an alert occurs whether or not the attacking player
has taken an action.
* •
$Th$, the defender-selected threshold of alerts that ends the game, selected
before play begins.
Let $p$ and $q$ parameterize two Bernoulli random variables,
$X_{m}\sim\text{Bernoulli}(p)$, $X_{n}\sim\text{Bernoulli}(q)$. We define the
indicator function
$\mathbf{1}_{X}=\begin{cases}1~{}&{\text{ if }}X_{m}=1\text{ or }X_{n}=1\\\
0~{}&{\text{ if }}X_{n}=0\text{ and }X_{n}=0\end{cases}$
Let $s_{A}$ be the total number of alerts that have occurred. Formally,
$s_{A}=\sum_{i=1}^{t}\mathbf{1}_{X}$. This random variable parameterized by
$p$ and $q$ also represents part of our transition function $T$, in that the
resultant state $s^{\prime}=(s_{A},s_{I})$ follows from the change in $s_{A}$.
If the attacking player ends the game before the defending player, they get a
reward, equal to $s_{I}$, the number of nodes they have infected and the
defender loses exactly this amount. If the defending player ends the game and
the attacking player has gained control of at least one node, then both
players get zero reward. Clearly, in this setting, the optimal first-move for
the defender is to end the game immediately and set $Th=0$. Thus, in our
simplified environment, we also assume that if the defender ends the game when
the attacker is not present, they incur a very large cost that demands $Th>0$.
We analyze three cases of this game:
1. 1.
All parameters are common knowledge and $s_{A}$ is observable to both players
2. 2.
All parameters are common knowledge but $s_{A}$ is not observable to the
attacker
3. 3.
Parameters are hidden from the attacker and $s_{A}$ is not observable to the
attacker
In all cases of the game, the actions taken by the attacking player and
$s_{I}$ are unknown to the defender. This means that in the full knowledge
setting, the parameters $p$ and $q$ of $T$ are known to all players, $s_{A}$
but not $s_{I}$ is observable by the defender, and the full state at each time
step is observed by the attacker. In the partial knowledge setting, $p$ and
$q$ are known by all players, but only the defender can observe $s_{A}$ and
only the attacker can observe $s_{I}$. In the zero knowledge setting, only the
defender knows $p$ and $q$ and can observe $s_{A}$, and the attacker can
observe only $s_{I}$.
### Full knowledge
In the full knowledge setting, the attacker and defender can both observe
$s_{A}$, and know $p$, $q$, and $Th$. Since the cost to the defender of ending
the game when they are not under attack is so large, they must choose $Th$
such that the probability an attacker is present is close to 1. Since $Th$ is
known and $s_{A}$ is observable, the attacker-optimal strategy is to end the
game when $SA=Th$.
###### Proposition 5.
The defender-optimal threshold, $Th$ is given by
$\left\lfloor\left\lceil\frac{1}{p}\right\rceil(p+q-pq)\right\rfloor$.
###### Proof.
At each time step from 1 to $t$, there is probability $q$ that an alert fires
independent of any other action. If the attacking player takes an action,
there is probability $p$ that an alert fires. This lets us treat $p$ as the
rate of arrival for alerts and view $X_{m}$ as a Bernoulli process. We wish to
find the number of trials required until the first success, which follows the
Geometric distribution and consequently, the expected number of trials before
the first success is $\frac{1}{p}$. However, $t$, $X_{n}$, and $X_{m}$ cannot
be directly observed.
Since only $s_{A}$ can be observed, we must find the expected number of alerts
that occur at time $t=\frac{1}{p}$. As our events are discrete, we’ll need to
take the ceiling of $t$. The four possible outcomes of $\mathbf{1}_{X}$ are:
$\displaystyle(X_{m}=0,X_{n}=0),$ $\displaystyle(X_{m}=1,X_{n}=0),$
$\displaystyle(X_{m}=0,X_{n}=1),$ $\displaystyle(X_{m}=1,X_{n}=1)$
and all but the first outcome yields an alert. We can therefore find the
probability that an alert occurs at some time $\tau$ by taking the complement
of the probability of the first event. Since $X_{m}$ and $X_{n}$ are
independent, that probability is $(1-p)(1-q)$. So our expected number of
alerts at time $t=\left\lceil\frac{1}{p}\right\rceil$ is:
$\displaystyle((1-p)(1-q))t$
$\displaystyle=(1-p)(1-q)\left\lceil\frac{1}{p}\right\rceil$
$\displaystyle=(pq-p-q+1)\left\lceil\frac{1}{p}\right\rceil$
$\displaystyle=\left\lceil\frac{1}{p}\right\rceil(1-(pq-p-q+1))$
$\displaystyle=\left\lceil\frac{1}{p}\right\rceil(1-pq+p+q-1)$
$\displaystyle=\left\lceil\frac{1}{p}\right\rceil(p+q-pq)$
We cannot guarantee that this number is an integer, and since the defender
wants the minimal threshold, we seek the floor of the number of alerts and
set:
$Th=\left\lfloor\left\lceil\frac{1}{p}\right\rceil(p+q-pq)\right\rfloor$ (3)
∎
The attacker-optimal strategy then, is dependent upon how ties are broken. If
the attacker always wins ties, they should act until $s_{A}=Th$. If the
defender always wins ties or if ties are broken randomly, the attacker should
act until $Th-1<s_{A}\leq Th$. For simplicity, we assume that the attacker
always wins ties throughout this work.
### Zero Observability
In the second case, the parameters of the game are known to both players –
that is, $p,q$ are common knowledge. Since $Th$ is chosen a priori and known
by the defender, the attacking player can conclude the value of $Th$ for a
rational opponent by Proposition 5. However, $s_{A}$, cannot be observed by
the attacker. The defending player sets
$Th=\left\lfloor\left\lceil\frac{1}{p}\right\rceil(p+q-pq)\right\rfloor$ as in
Equation 3. The attacking player must therefore determine the maximum number
of actions they can expect to take before $Th$ is reached.
###### Proposition 6.
The optimal number of actions for the attacker to take when $s_{A}$ cannot be
observed is $\frac{1-p-q+pq}{p}$.
###### Proof.
$p$ and $q$ being known, the attacker seeks to find the expected arrival time
of the $Th$th alert generated by the Bernoulli process described by the joint
distribution of $X_{m}$ and $X_{n}$. Since the arrival rate for the joint
process is $p+q-pq$ as demonstrated in Proposition 5, the attacker seeks to
find $t$ such that they are able to maximize their actions. We want to know
the number of trials, or actions, required before $Th$ alerts occur, which can
be represented by the negative binomial distribution (Casella and Berger 2021)
with probability mass function:
$\binom{k+r-1}{r-1}(1-p)^{k}p^{r}$
Where $r$ is the number of successes, $k$ is the number of failures, $p$ is
the probability of success, and $\binom{k+r-1}{r-1}$ is the binomial
coefficient. We call this random variable $X\sim NB(k;r,p)$. The expected
value of $X$ is given by $\frac{r(1-p)}{p}$, so the expected number of actions
that can be taken by an attacker before the threshold is reached is given by:
$\displaystyle t$ $\displaystyle=E[X]$ $\displaystyle=\frac{r(1-p)}{p}$
$\displaystyle=\frac{T(1-(p+q-pq)}{p+q-pq}$
$\displaystyle=Th\frac{1-p-q+pq}{p+q-pq}$
$\displaystyle\approx(\frac{1}{p}p+q-pq)\frac{1-p-q+pq}{p+q-pq}$
$\displaystyle=\frac{1-p-q+pq}{p}$ (4)
∎
The attacker, not knowing the value of $s_{A}$, rationally draws the same
conclusion as the defender – since the threshold is defined as a function of
the Bernoulli processes by Equation 3, they can operate until the expected
number of alerts generated is equal to the expected threshold, given by
Equation 4. In our simulation, this value is chosen by the attacker a priori.
Once the attacker has taken all their allocated actions, they end the game –
if they choose to take another action when the defender decides to end the
game, they receive no reward.
### Unknown parameters
In the setting where $p$ and $q$ are unknown, the attacker cannot make a
rational assessment about $Th$. Since the attacker is still utility
maximizing, we must establish some criteria by which the attacker decides
whether or not to stop. In the zero-shot setting, where both the count of
alerts and the environment parameters are unknown, the attacking agent has no
way to reasonably choose a number of actions to take. Attacker-defender
interaction here would demand a more complex model, an example of which we
discuss in the conclusion. If the attacker is able to conduct multiple trials
against a defender, they can use the number of actions $t$ they’ve taken along
with a Bayesian updating rule to estimate $Th$ as a function of $t$.
Over time, this attacker should optimize to approach both the partial
knowledge and full knowledge attacker. However, our goal here is to understand
the impact of limited information on the agents, and so we choose an arbitrary
small number of attempts for the experimental setting. Our simple Bayesian
learning agent is given 10 attempts to interact with the defender, and for
each experiment, we return their maximum score and average win rate.
## Simulation and Discussion
We evaluate the three models described over 20 evenly spaced values of $p$ and
$q$ between 0.01 and 0.99, for a total of 400 combinations of parameters. In
each model, play proceeds as follows:
1. 1.
The defender calculates $Th$ according to Equation 3
2. 2.
Players take an action, and $s_{I}$ is incremented unless either player ends
the game.
3. 3.
Bernoulli trials are run for $X_{m}$ and $X_{n}$, returning 1 with probability
$p$ and $q$, respectively.
4. 4.
If either trial returns 1, increment $s_{A}$
5. 5.
The defender and attacker choose their next action and repeat steps 2-5 until
the game ends
The defender’s choice of action is always to pass unless $s_{A}=Th$, in which
case, they end the game. The attacker’s choice of action varies between
models. In the full knowledge model, the attacker ends the game if $s_{A}=Th$,
otherwise they attack another node. In the partial knowledge model, the
attacker chooses a lifespan based on Equation 4. If $s_{I}$ is less than that
lifespan, they attack another node; otherwise, they end the game. In the zero
knowledge model, the attacker is following the same rules as the partial
knowledge model, but is trying to find the optimal lifespan – at the end of
each one of their attempts, they update their expected value of the joint
distribution of $p$ and $q$, determine a new optimal lifespan, and try until
they have exhausted their 10 total attempts. The average win rate and average
score of those 10 attempts are then taken as one trial. Each combination of
$p$ and $q$ was run over 1 million trials, and the results of those trials –
the attacker score and wins – averaged.
Figures 1, 2, and 3 show the average score of each model over 1 million trials
as a function of $p$ and $q$. Within these figures, we can see two emergent
phenomena. First, we can see that the general shape of the surface is the same
across all three models, peaking, as we would expect, when both $p$ and $q$
are very low. Second, looking at specifically the index of the Z-axis, we see
that the highest attained value decreases monotonically as more information is
removed from the attacker.
Figure 1: Average score over 1 million trials for full knowledge model Figure
2: Average score over 1 million trials for partial knowledge model
In the partial knowledge trials shown in Figure 2, there is a curious dip in
the score achieved when both $p$ and $q$ are very low. Looking closely at both
the scores and win rates shows that this is because while the win rate is very
high in the partial knowledge setting in general, there is a win rate decrease
at very low detection rates. Despite boasting an average win rate of .9823
across all parameter settings and trials, the partial knowledge attacker’s win
rate at the lowest rate of alerting, $p=q=0.01$ is only .41012. This occurs
due to the high number of actions that the attacker expects to achieve in the
average case combined with the large variance that is observed when $p$ and
$q$ are low.
Figure 3: Average score over 1 million trials for zero knowledge model
The first phenomenon, that the achieved score is a relatively direct function
of the alert rates, follows directly from our assumptions about the threshold
and how attackers gain rewards: the longer an attacker is present, the higher
their potential reward; if the rate of alerts is higher, then the attacker has
less time to accomplish whatever task they aim to. Although the values of the
attacker’s score where $p$ and $q$ are larger are still usually greater than
zero, they are dwarfed by the values achieved when the probability of alerts
is very low. We can also observe, by looking at the X-axis of the figures,
that the value of $p$ is much more meaningful than the value of $q$. This
makes sense, since $p$ is the parameter with the largest effect in both
Equations 3 and 4.
More interesting is the fact that for most values of $p$, the impact of $q$ is
meaningful only when the attacker’s knowledge is limited. Specifically, in the
full knowledge scenario, the difference between some fixed $p$ across all
values of $q$ is small. However, as is most obvious in Figure 2, the impact of
a high $q$ when $p$ is low significantly changes the expectations of an
attacker to persist in an environment – information that a savvy defender
could potentially exploit.
The second phenomenon is most pronounced by looking at the Z-axis across the
three figures, which differs meaningfully across them. Examining the values,
we find that across all parameters and trials, the full-knowledge attacker
attains a mean score of 9.4701, the partial knowledge attacker attains a mean
score of 3.8224, and the zero knowledge attacker attains a mean score of
2.3751. At the high end, the full knowledge attacker attains a maximum score
of 113.0872, the partial knowledge attacker attains a maximum score of
72.0665, and the zero knowledge attacker attains a maximum score of 16.2920.
The value of additional information to the attacker is therefore quite
substantial.
## Conclusion and Future Work
This work considers a partially observable stochastic Bayesian game that
offers useful insights even under strong assumptions about the state and
action space. In particular, we can quantify the value of total alert rate
versus the true positive rate, finding that the value of an alert, from an
defender’s perspective, is almost entirely about how effective it is at
detecting attacker activity, while overall alert rate is essentially
meaningless to the attacker, except at the extremes where the overall alert
rate far exceeds the true positive rate. This is most pronounced in the
scenarios where an attacker’s knowledge about the parameters and the state of
the world are limited – scenarios that map more closely to real-life. An
extension of this work where the state and action spaces were expanded would
offer more useful parallels and insights about these phenomena.
In this work, we have found optimal solutions for our particular limited state
and action spaces. Since Harsanyi-Bellman ad hoc coordination depends heavily
on observing the history and there is substantial uncertainty about how to
choose a strategy or predict what actions an opponent may have taken –
particularly since any attacker action having been taken would demand a
defender response and any defender response would end the game – a
generalization of this concept to the partially observable case of this game
is difficult to assess in our limited game. Future work in this area will
generalize the HBA solution concept to arbitrary partially observable SBGs.
This work has also demonstrated that an attacker’s potential success in an
environment is monotonically linked to their knowledge of the environment.
Crucially, this means that when we are emulating attackers or building game
theoretic models of attack and defense, the assumptions about what an attacker
knows are nontrivial, and making incorrect assumptions about attacker
knowledge and behavior can have negative impacts on defender optimization – to
wit, if we assume that attackers are omnipotent, then our optimal defensive
strategy is very aggressive, while an optimal defensive strategy under better
assumptions may yield more uptime for users. Here, our defender admits only a
fixed strategy against an attacker for whom they have no options. However, a
more dynamic defender would be able to anticipate an attacker’s actions given
their knowledge, and optimize against that.
Finally, our zero-knowledge attacker is given a small, arbitrary number of
opportunities to learn the environment and optimize their score. In order to
conduct a more reasonable few-shot approach to the zero-knowledge environment,
we should consider a model with a larger state and action space. In order for
the attacker to be a learning agent, some local feedback – feedback aside from
winning or losing the game – would allow for a more robust interplay and allow
for the incorporation of attacker learning dynamics. Given the extant
literature on the use of reinforcement learning for analogous problems, we
would seek to apply that same technique in few-shot settings of a stochastic
Bayesian game.
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|
# Cool Gaseous Exoplanets: surveying the new frontier with Twinkle
Luke Booth,1 Subhajit Sarkar,1 Matt Griffin,1 and Billy Edwards,2
1Cardiff Hub for Astrophysics Research and Technology (CHART), School of
Physics and Astronomy, Cardiff University, 5 The Parade, CF24 3AA, United
Kingdom
2SRON, Netherlands Institute for Space Research, Niels Bohrweg 4, NL-2333 CA,
Leiden, The Netherlands<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
Cool gaseous exoplanets ($1.75\ R_{\oplus}<R_{\text{p}}<3\ R_{\text{J}}$,
$200$ K $<T_{\text{eq}}<1000$ K) are an as-yet understudied population, with
great potential to expand our understanding of planetary atmospheres and
formation mechanisms. In this paper, we outline the basis for a homogeneous
survey of cool gaseous planets with Twinkle, a 0.45-m diameter space telescope
with simultaneous spectral coverage from 0.5-4.5 $\mu$m, set to launch in
2025. We find that Twinkle has the potential to characterise the atmospheres
of 36 known cool gaseous exoplanets (11 sub-Neptunian, 11 Neptunian, 14
Jovian) at an SNR $\geq$ 5 during its 3-year primary mission, with the
capability of detecting most major molecules predicted by equilibrium
chemistry to > $5\sigma$ significance. We find that an injected mass-
metallicity trend is well-recovered, demonstrating Twinkle’s ability to
elucidate this fundamental relationship into cool regime. We also find Twinkle
will be able to detect cloud layers at 3$\sigma$ or greater in all cool
gaseous planets for clouds at $\leq$ 10 Pa pressure level, but will be
insensitive to clouds deeper than $10^{4}$ Pa in all cases. With these results
we demonstrate the capability of the Twinkle mission to greatly expand the
current knowledge of cool gaseous planets, enabling key insights and
constraints to be obtained for this poorly-charted region of exoplanet
parameter space.
###### keywords:
exoplanets – planets and satellites: gaseous planets – planets and satellites:
atmospheres – techniques: spectroscopic – instrumentation: spectrographs
††pubyear: 2024††pagerange: Cool Gaseous Exoplanets: surveying the new
frontier with Twinkle–A
## 1 Introduction
Over the past three decades, the field of exoplanet science has progressed
rapidly, from the first detections in the 1990s (Wolszczan, 2012; Mayor &
Queloz, 1995) and the first atmospheric spectrum in 2002 (Charbonneau et al.,
2002), to the revolution in exoplanet demographics resulting from Kepler
(Lissauer et al., 2014; Fulton et al., 2017) and over a decade of transmission
spectra from the Hubble Space Telescope (HST) (Edwards et al., 2022). Most
recently, the James Webb Space Telescope (JWST) is now returning high
precision transmission spectra, resulting in new discoveries such as the first
detection of sulphur-bearing species in an atmosphere (Rustamkulov et al.,
2023; Alderson et al., 2023; Tsai et al., 2023) and the first detection of
carbon-bearing molecules in a habitable zone planet (Madhusudhan et al.,
2023). Today over 5500111NASA Archive Planetary Systems Composite Table
https://exoplanetarchive.ipac.caltech.edu/cgi-bin/TblView/nph-
tblView?app=ExoTbls&config=PSCompPars (27/09/2023) confirmed exoplanets are
known, the majority of which have been detected by transit photometry using
large dedicated space-based surveys such as Kepler, K2, CHEOPS and TESS or
ground-based surveys such as WASP, HATNet and NGTS. Such transiting planets
provide potential targets for atmospheric characterization through
transmission and/or eclipse spectroscopy. Using these techniques, in the next
decade, the upcoming Ariel and Twinkle missions will perform the first
dedicated population-level surveys of exoplanet atmospheres (Tinetti et al.,
2018; Edwards et al., 2019a).
The currently known transiting exoplanet population is highly diverse in both
radius and temperature, containing planets with radii that vary from less than
that of Mercury to several times the size of Jupiter and equilibrium
temperatures, $T_{\text{eq}}$, that span the range from less than $200$ to
over $4000$ K. Selection effects of the two most prolific detection methods
(transit and radial velocity) bias the currently discovered planetary
population towards shorter period planets. Super-Earths and sub-Neptunes are
the most frequent type of planet, often found in closely packed multi-planet
systems, e.g. the TRAPPIST-1 (Gillon et al., 2017), Kepler-296 (Barclay et
al., 2015), Kepler-32 (Swift et al., 2013) and K2-384 (Christiansen et al.,
2022) systems. A large population of "hot Jupiters" is also known, their large
size, short periods and high transit probabilities positively biasing their
transit detectability. Notably, some planet populations appear more sparse.
These include the "hot Neptune desert" (Szabó & Kiss, 2011; Mazeh et al.,
2016; Edwards et al., 2023) as well as colder gas giants. Wittenmyer et al.
(2020), relying on long-duration radial velocity data, determined that the
occurrence rates of giant planets around solar-type stars was fairly constant
below 300 days and was increased at longer periods. The occurrence rates of
hot Jupiters and temperate gas giants would thus be similar ($\sim$ 1%) but
very cold giants, analogous to Jupiter or Saturn, are more frequent ($\sim$
7%). Despite this, in practice there is a dearth of known transiting gas
giants at cooler temperatures, with about half as many transiting giants
having $T_{\text{eq}}$ less than 1000 K as those with $T_{\text{eq}}$ greater
than 1000 K (around all types of star) (Table 1).
To date only a small fraction of all known transiting exoplanets, $\sim$ 180
planets222ExoAtmospheres community database (13/11/2023)
http://research.iac.es/proyecto/exoatmospheres/index.php, have had their
atmospheres characterised through a combination of transmission, emission,
cross-correlation and direct imaging spectroscopy, with specific molecular
detections reported in about half these cases333Defined as where one or more
molecules have definitive detections reported in the ExoAtmospheres community
database, but excluding cases where only upper limits are given.. The most
successful method applied to date has been transmission spectroscopy. This
technique is most sensitive to high scale height atmospheres and large
planetary radii which tend to augment spectral feature amplitudes. The
amplitude of spectral features in transmission $A_{\text{p}}$ can be
approximated by:
$A_{\text{p}}=\frac{2R_{\text{p}}\cdot nH}{R_{\text{s}}^{2}}$ (1)
where $R_{\text{p}}$ is the radius of the planet, $R_{\text{s}}$ is the radius
of the host star and $H$ is the pressure scale height. $n$ is commonly taken
to have a value of 5, and gives the number of scales heights of a typical
spectral features. The scale height is given by:
$H=\frac{k_{\text{B}}T_{\text{eq}}}{\mu g}$ (2)
where $\mu$ is the mean molecular weight of the atmosphere, commonly taken to
be $\sim 2.3$ for hydrogen/helium-dominated (H2-He) atmospheres, $g$ is the
surface gravity of the planet, $k_{\text{B}}$ is Boltzmann’s constant and
$T_{\text{eq}}$ is the equilibrium temperature of the planet. As well as
making the best targets for transmission spectroscopy, large, very hot planets
also make better targets for day-side emission spectroscopy due to their
higher thermal flux. As a result, currently almost two thirds of all planets
that have had their atmospheres analysed have $T_{\text{eq}}$ > $1000$ K, with
this population accounting for $\sim$70% of planets that have molecular
detections33footnotemark: 3.
Early searches for trends in exoplanet atmospheres (Sing et al., 2016; Tsiaras
et al., 2018; Edwards et al., 2022) have also typically been dominated by hot
Jupiters and warm Neptunes. Trends reported include temperature vs cloud/hazes
(Crossfield & Kreidberg, 2017; Libby-Roberts et al., 2020; Guilluy et al.,
2021; Estrela et al., 2022) and planet mass vs metallicity ([e.g. Wakeford et
al., 2018; Welbanks et al., 2019). Color-magnitude diagrams and trends
regarding phase-curve properties and day-night temperature variations with
equilibrium temperature have also been reported (e.g. Zhang, 2020). Such
population level studies are a start to understanding how atmospheric
properties and planet composition relate to fundamental initial conditions,
and are key to a full understanding of planet formation and evolution. However
the lower temperature planets are poorly represented due to a paucity of
atmospheric spectra in this regime. This in turn is due to a combination of
few strong candidates for spectroscopic follow-up and the intrinsic challenge
in obtaining transmission spectra for cooler atmospheres (which will have
smaller scale heights) at sufficient signal-to-noise ratio (SNR).
In this paper, we examine the capability of the upcoming Twinkle space mission
to advance the understanding of gaseous planets in the "cool" regime (which we
define as being between 200 and 1000 K), through a dedicated spectroscopic
survey consisting of a statistically meaningful sample of such planets. The
sample will cover a wide range of temperatures, and include planet sizes
ranging from sub-Neptunes to Jovian planets. Twinkle will obtain transmission
spectra in the $0.5$-$4.5\mu$m wavelength range. Such a survey would have the
potential to verify and extend trends into the cool regime, providing key
observational constraints for for atmospheric models and planet formation
theories.
The paper is structured as follows: Section 2 outlines the scientific case for
studying cool gaseous exoplanets, which is followed by a brief description of
the Twinkle Space Telescope in Section 3. Section 4 describes the construction
of a preliminary candidate list based on known exoplanets spanning the
parameter space of the cool gaseous planet population. Sections 5 and 6
describe two simulated studies performed on planets from the preliminary
candidate list. Section 5 explores the ability of Twinkle to constrain
atmospheric metallicity and recover an injected metallicity trend, whilst
Section 6 examines Twinkle’s sensitivity to detecting clouds in gaseous
planets over a range of sizes and temperatures.
## 2 Cool gaseous planets
Spectroscopic observations of “cool” gaseous planets provide the opportunity
to shed light on the physical and chemical processes that govern H2-He
dominated atmospheres at low temperatures and the formation histories of this
population.
### 2.1 Categorisation
In this paper we sub-categorise the cool gaseous planets into nine sub-groups
based on size and temperature. A survey of cool gaseous planets across the
full parameter space of size and temperature will allow the possibility of
trends to be elucidated with respect to both parameters. We limit the lower
bound of temperature we consider to 200 K. Below this level, the temperatures
are generally beyond the outer limits of the "habitable zone", corresponding
closer to those of the cold gas and ice giants of our solar system and in
practice, there are hardly any transiting planets below this lower limit. Our
upper temperature bound is 1000 K, which previous studies have used to define
as boundary between hot Jupiters and cool giants (Thorngren et al., 2019;
Wallack et al., 2019). Avoiding terms like "temperate" or "warm" which have no
consensus definitions, we call all planets in this temperature range "cool".
We choose to further sub-divide the large cool temperature regime into three
distinct temperature brackets: C1 (200-500 K), C2 (500-750 K) and C3 (750-1000
K) as distinctive patterns of chemical and physical processes are likely to
occur with temperature. The C1 category would encompass planets in the
"habitable zone". In terms of size, we include established planetary classes
with radii above the Kepler radius valley mid-point ($1.75~{}R_{\oplus}$):
sub-Neptunes ($1.75~{}R_{\oplus}-3~{}R_{\oplus}$), Neptunes
($3~{}R_{\oplus}-0.5~{}R_{\text{J}}$) and Jovians
($0.5~{}R_{\text{J}}-3~{}R_{\text{J}}$). These are planets where primary H2-He
dominated atmospheres are likely to be the norm (Fulton et al., 2017; Gupta &
Schlichting, 2019). Their low mean molecular weights will mitigate some of the
challenges associated with observing cool atmospheres, and hence they are
favoured over rocky planets in this temperature regime for transmission
spectroscopy.
### 2.2 The workings of cool atmospheres
In the low temperature regime we would expect molecules such as NH3 and CH4 to
dominate over N2, CO2 and CO in thermochemical equilibrium, with final
abundances modulated by bulk elemental composition (higher metallicities tend
to favour CO and particularly CO2 over CH4 and N2 over NH3) or the C/O ratio
(Madhusudhan, 2012; Moses, 2014). However at lower temperatures reaction rates
slow, such that the timescale for reaching chemical equilibrium increases
compared to hotter atmospheres. Competing disequilibrium processes such as
transport processes (e.g convection and eddy diffusion) and photochemistry
would be expected to have stronger effects than in hotter atmospheres (Prinn &
Barshay, 1977; Zahnle & Marley, 2014). Molecules that tend to occur at low
temperatures, like CH4 and NH3, are also more sensitive to photochemistry than
their hot counterparts (CO and N2) (Moses, 2014). Such processes can change
the composition, radiative balance and temperature-pressure profiles (Moses,
2014). HCN may be a significant molecule in cooler atmospheres as a result of
coupled NH3-CH4 photochemistry, and CO may occur in the IR photosphere through
CH4-H2O photochemistry or transport-induced quenching, but always at lower
abundances than CH4 (Moses, 2014).
The quench level is the atmospheric level where the chemical equilibrium time
scale for a given reaction just falls below that for vertical (or horizontal)
mixing, and the molecular abundances at this level are then transported to
higher altitudes, resulting in a complex disequilibrium composition (also
modulated by photochemistry) in the upper layers potentially probed in
transmission. In these upper layers, photochemical timescales may be expected
to be shorter than chemical equilibrium so we might expect to see the
byproducts of photochemistry at greater levels than for hotter atmospheres.
From breakdown of the photosensitive molecules CH4 and NH3, complex
hydrocarbons and nitriles are more likely to occur in colder than in hotter
atmospheres (Moses, 2014). This could include high-altitude photochemically-
produced hazes which may modulate the energy balance of the planet.
Photochemical reactions can also result in species that cause atmospheric
warming and inversions or conversely could act as coolants (e.g. cooling by
C2H2 and C2H6, byproducts of methane photolysis in the atmosphere of Jupiter).
Cloud condensates may form, reflecting the condensation profiles of low
temperature molecules, giving rise to water, methane or ammonia clouds, which
will also impact albedo and energy balance. The presence of clouds will also
impact the measured quantities of the condensed species at higher altitudes
reflected in the spectrum. The exact mechanisms and chemical pathways for
disequilibrium chemistry remain an area of active research, as highlighted by
the detection and subsequent interpretation of SO2 in the atmosphere of the
hot Saturn WASP-39 b by the JWST Early Release Science (ERS) team (Tsai et
al., 2023; Rustamkulov et al., 2023; Alderson et al., 2023). Therefore, while
disequilibrium processes are expected and sophisticated atmospheric models
exist to simulate the transport-induced quenching (e.g Moses et al., 2011;
Drummond et al., 2020; Zamyatina et al., 2023) and photochemistry (e.g. Tsai
et al., 2017) there are few if any observational constraints on the chemical
kinetics from exoplanet observations. Further to this, there is a present lack
of data on how planet and therefore gaseous envelope size would affect these
processes. A large and diverse spectroscopic survey of cool gaseous planets is
ideally placed to find such "smoking guns".
### 2.3 Planet formation quandaries
Clues to the formation history of an exoplanet are encoded in its composition
and therefore its atmospheric spectrum. Elemental ratios, such as the C/O
ratio, may be able to locate a planet origin location relative to different
"ice lines" in the protoplanetary disk (e.g. Öberg et al., 2011) and could be
measurable in an exoplanet atmosphere. The C/O ratio may be complicated by
planet migration and/or planetesimal pollution. It may be possible to
disentangle such pollution effects from origin location effects by examining a
range of element ratios (e.g. Turrini et al., 2021; Pacetti et al., 2022).
Measurement of the atmospheric metallicity may also provide evidence for the
formation mechanism. In core-accretion scenarios, the atmospheric metallicity
is expected to be increased compared to the host star (Thorngren et al.,
2016), whereas in gravitational instability scenarios we would expect a near
stellar metallicity. Current characterisations of the mass-metallicity
relationship (e.g Wakeford et al., 2018; Welbanks et al., 2019) are supportive
of core-accretion, but have large uncertainties and are derived mostly from
hot giant exoplanets. Theoretical structural evolution models (Thorngren et
al., 2019) indicate that the mass-metallicity trend should continue in planets
$<$ 1000 K; however, there is currently limited data to test and confirm this.
Cool gaseous planets may present a challenge to planet formation theories. To
hold onto a H2-He atmospheres requires rapid formation of a massive core
$\geq$ 10 $M_{\oplus}$ (e.g. Pollack et al., 1996). Traditional core accretion
theory holds that such cores are more likely to form beyond the ice line where
water ice adds to bulk and adhesion. However many gaseous planets ranging from
sub-Neptunes, through Neptune-sizes to Jupiter-sizes are found with
equilibrium temperatures that would put them within the ice line. Indeed Hill
et al. (2018) found $>$ 70 planets of size $>$ 3 $R_{\oplus}$ in the habitable
zones of G, K and M-type stars, with occurrence rates ranging from 6 to 11.5%
depending on the stellar class. A planet formation quandary therefore exists
in explaining the formation of these planets, requiring some modifications to
basic core accretion models. Another problem is the presence of gas giants
around M-dwarf stars (many of which are in the cool regime) (e.g. Kanodia et
al., 2023), where core accretion models predict slower accretion rates (e.g.
Ida & Lin, 2005; Burn et al., 2021) that would make large core formation
challenging on the timescale of the disc life time. Gravitational stability is
also a potential pathway to forming giant exoplanets (Boss, 1997) and has seen
renewed interest due to its ability to explain the existence of the growing
number of M-dwarf gas giant exoplanets. For gaseous planets within the water
ice-line, formation scenarios include core-accretion beyond the water-ice line
followed by disc-migration (Paardekooper & Johansen, 2018), or core formation
interior to the water-ice line, followed by in-situ enrichment via gas, dust
and pebble accretion close to the host star (Knierim et al., 2022), along with
other variations of the above models proposed in recent years.
Compositional information including atmospheric metallicity and elemental
ratios such as the carbon-to-oxygen ratio (C/O) may therefore shed light on
planetary formation and evolution processes in this regime.
### 2.4 Moons and habitability
Giant planets have long been postulated to be likely exomoon hosts (Heller &
Pudritz, 2015; Spalding et al., 2016; Hill et al., 2019; Saillenfest et al.,
2023), and although multiple exomoon candidates have been identified, there
has yet to be a definitive detection (Sucerquia et al., 2020; Heller, 2020;
Rovira-Navarro et al., 2021; Kipping & Yahalomi, 2023). Though moons can form
around planets of any size, cool gaseous planets may have a higher probability
of hosting exomoons, owing to their having probably migrated inwards a shorter
distance than hotter planets of a similar size and therefore being less likely
to have undergone disruption the orbits of, or ejection of, their moons
(Spalding et al., 2016). Furthermore, if such moons are sufficiently large,
they themselves may have atmospheres, generated through mass transfer from
their parent planet or, more likely, outgassing or volcanism as a result of
tidal heating. Transmission spectroscopy may therefore provide a method by
which exomoons could be detected around cool gaseous planets, with typical
products of volcanism (including sodium- and potassium-bearing species)
(Rovira-Navarro et al., 2021) unlikely to feature in cooler gaseous
atmospheres. Thus cool giant spectroscopy may potentially yield evidence for
the presence of exomoons. However, investigating this fascinating possibility
requires a determination of the abundance of potential volcanic tracers and
subsequently their detection feasibility with current and future
instrumentation (eg: Twinkle, Ariel, JWST and ELTs). This is beyond the scope
of the current paper and is left to future work. We also note that such
exomoons could be in the habitable zone in some cases, and therefore could be
locations for potential habitability. The habitability of cold giants
themselves is an unexplored topic, and while liquid water layers or oceans
such as those postulated for Hycean sub-Neptunes (Madhusudhan et al., 2021)
can be ruled out on the basis of pressure and temperature, the possibility of
aerial biospheres could be explored. This has previously been raised in the
context of Jupiter (Sagan & Salpeter, 1976), and more recently in brown dwarfs
(Yates et al., 2017; Lingam & Loeb, 2019), sub-Neptunes (Seager et al., 2021)
and Venus (Greaves et al., 2021).
### 2.5 Previous observations
Relatively few cool gaseous planets have been studied spectroscopically to
date. Sub-Neptunes are ubiquitous but small in size (reducing the
$A_{\text{p}}$ factor in Equation 1), so despite the large number, there are
few targets suitable for spectroscopic follow-up. Nonetheless, several sub-
Neptunes in the "cool" regime have been studied spectroscopically. A non-
exhaustive list includes: K2-18 b (Tsiaras et al., 2019; Benneke et al.,
2019b; Bézard et al., 2020; Madhusudhan et al., 2023), GJ 1214 b (Kreidberg et
al., 2014; Gao et al., 2023; Kempton et al., 2023), GJ 9827 d (Roy et al.,
2023), HD 3167 c (Guilluy et al., 2021), HD 97658 b (Knutson et al., 2014b)
and TOI-270 d (Mikal-Evans et al., 2023). Spectra of these planets have
revealed greatly contrasting atmospheres. Observations conducted by HST, and
more recently JWST, have shown the spectrum of GJ 1214 b to be flat well into
the mid-infrared (Kreidberg et al., 2014; Gao et al., 2023), requiring
significant cloud / haze production to explain, whilst HST spectra of K2-18 b
showed clear absorption features at 1.4 $\mu$m relatively unimpeded by the
presence of cloud.
This feature has been inferred to be due to the presence of H2O (Tsiaras et
al., 2019; Benneke et al., 2019b) or CH4 (Bézard et al., 2020; Blain et al.,
2021), with recent JWST/NIRISS and NIRSpec observations strongly detecting CH4
(Madhusudhan et al., 2023). Absorption features at $\sim$1.4 $\mu$m,
interpreted as due to H2O have also been seen in HST spectra of GJ 9827 d, HD
3167 c and TOI-270 d, though observations spanning broader wavelengths are
required to fully resolve the known degeneracy between H2O and CH4.
Neptune to Jupiter-sized giant planets in the cool regime are also poorly
characterised spectroscopically. In terms of temperature, such planets provide
a "missing link" between the two well-studied planetary populations of hot
Jupiter exoplanets and the giants of our own Solar System. Previous spectra of
such planets include those for GJ 436 b (Knutson et al., 2014a; Hu et al.,
2015), GJ 3470 b (Benneke et al., 2019a), HD 106315 c (Guilluy et al., 2021;
Kreidberg et al., 2022), HIP 41378 f (Alam et al., 2022), Kepler-51 b, d
(Libby-Roberts et al., 2020), K2-33 b (Thao et al., 2023), WASP-29 b (Wong et
al., 2022), WASP-80 b (Bell et al., 2023a), and WASP-107 b (Kreidberg et al.,
2018; Piaulet et al., 2019; Spake et al., 2021), though many additional
planets in this size regime have been the target of ground-based searches for
metastable helium absorption (Vissapragada et al., 2022; Allart et al., 2023).
Despite the increased number of spectra in the cool giant regime, these
atmospheres remain poorly understood. Multiple planets, including HIP-41378 f,
Kepler-51 b, d, K2-33 b and WASP-29 b, exhibit spectra that are flat and
featureless across the HST/WFC3 wavelength range, while others exhibit clear
or muted absorption features at $\sim$1.4 $\mu$m.
### 2.6 Surveys of cool gaseous planets
There is a scarcity of systematic surveys of cool gaseous planets with high
precision spectra. Kammer et al. (2015) obtained Spitzer secondary eclipse
measurements at at 3.6 and 4.5 µm for five gas giants in the temperature range
980-1184 K. They used the atmospheric CH4/CO ratio as a marker of atmospheric
metallicity, with results somewhat supportive of increased metallicity with
lower masses. Wallack et al. (2019) performed a similar Spitzer study on five
further gas giants with $T_{\text{eq}}<1000$ K. They found no evidence for a
solar-system-like mass-metallicity relationship but did find a relationship
between inferred CH4/(CO+CO2) and stellar metallicity. More recently, Baxter
et al. (2021) performed transmission photometry of 33 gaseous planets at 3.6
and 4.5 µm using Spitzer, of which 13 had temperatures between 500 and 1000 K.
There was some evidence of a mass-metallicity relation: the cool planets ($<$
1000 K) were generally biased with lower mass and appeared to have higher
metallicity as well as lower eddy diffusion coefficients and a lack of methane
compared to expectations. A lack of methane had previously been noted on a
number of cool planets compared to equilibrium expectations constituting the
so-called “missing methane problem”. Methane has now recently been detected in
two “cool” gaseous planets with JWST: K2-18 b and WASP-80 b (Madhusudhan et
al., 2023; Bell et al., 2023b). More recently a Cycle 2 JWST survey of seven
giant planets in the “cool” regime orbiting M-dwarf stars has been planned
(JWST Proposal 3171, PI: S. Kanodia).
There is therefore a need for a homogeneous cool gaseous planet survey with
wide wavelength coverage to further explore and constrain the relationship
between planet mass and atmospheric metallicity, and open up this field of
study. Cooler gaseous planets may provide more robust metallicity measurements
than hotter gaseous planets as they will be significantly less affected by the
degeneracy in radius between poorly-understood radius inflationary effects and
increasing metallicity (and thus mean-molecular weight) which acts to suppress
the atmospheric extent (Thorngren et al., 2016). Improved metallicity
measurements may also help to validate and extend reported mass-metallicity
trends (Wakeford et al., 2018; Welbanks et al., 2019) or support the absence
of a trend (Edwards et al., 2022). Furthermore, the temperature range spanned
by cool gaseous planets is likely to aid in the exploration of cloud and haze
coverage trends predicted from theoretical and laboratory work, hints of which
have been seen in early HST observations (Dymont et al., 2022; Estrela et al.,
2022; McGruder et al., 2023).
Whilst the number of known cool giant planets is comparatively small, it has
been steadily growing. In the last 18 months alone, TESS photometry has lead
to the discovery of intriguing cool giants such as the low-density warm-Jovian
TOI-1420 b (Yoshida et al., 2023), a warm Saturn TOI-199 b (Hobson et al.,
2023) and several around M-dwarf stars (e.g. Powers et al., 2023; Harris et
al., 2023; Han et al., 2023). Re-observation of TESS sectors during the second
extended mission has enabled single transit and “duotransit” planetary
candidates (Hawthorn et al., 2023), many with orbital periods longer than
$\sim$15 days, to be confirmed, with follow-up efforts by ground-based surveys
such as the Next Generation Transit Survey (NGTS) and the CHEOPS space
telescope subsequently hunting down and constraining the true orbital periods
of these candidates. Examples of “cool” planets found in this manner include
the sub-Neptunes HD 22946 d (Garai et al., 2023) and HD 15906 b and c (Tuson
et al., 2023), the Neptunes HD 9618 b and c (Osborn et al., 2023), TOI-5678 b
(Ulmer-Moll et al., 2023) and the Jupiters TOI-4600 b and c (Mireles et al.,
2023) (planet c having a $T_{\text{eq}}$ of 191K).
However, compared to planets with hot atmospheres, cooler gaseous planets will
have smaller scale heights and thus spectral features will give lower SNRs.
Co-adding of multiple transit observations is frequently used to improve SNR,
but increases total observing time required. This is even more problematic for
planets orbiting Sun-like stars where orbital distances are greater and
periods longer. As such it is observationally more intensive to characterise
cool gaseous planets (especially around Sun-like stars), and this is one
reason why they are a challenging population. Dedicated spectroscopic surveys
permitting repeat observations over several years would be key to opening up
this population to detailed characterisation and obtaining sufficient
homogenized samples for population-level studies. Two such dedicated surveys
are planned in the coming decade: the ESA Ariel mission due for launch in 2029
and the Twinkle mission which will precede it with a planned launch in 2025.
## 3 Twinkle
Developed commercially by Blue Skies Space Ltd. (BSSL), Twinkle is a fast-
tracked satellite based on heritage components, and that is expected to
characterise many exoplanetary and solar system targets during its nominal
7-year lifetime (Edwards et al., 2019a; Stotesbury et al., 2022). Expected to
launch in late 2025, the spacecraft will carry a 0.45-m diameter primary
mirror, with an inner sanctum that is actively cooled to < 90 K. Twinkle has a
spectrometer with two simultaneously operating channels across the visible and
infrared: CH0 covering 0.5-2.4 $\mu$m at R $\leq$ 70 and CH1 covering 2.4-4.5
$\mu$m at R$\leq$50, each channel having its own grism element (Stotesbury et
al., 2022). Twinkle will therefore expand on the total spectral wavelength
coverage of HST WFC3 G102 (0.8-1.15 $\mu$m) and G141 (1.075-1.7 $\mu$m)
gratings by a factor of $\sim 5$, whilst retaining similar spectral
resolution. This will open up the opportunity to potentially break
degeneracies between H2O and CH4 that are known exist in WFC3 observations,
whilst also enabling the detection of strong absorption features from
molecules such as CO2 (2.0, 2.7, 4.3 $\mu$m), CO (2.34, 4.67 $\mu$m) and NH3
(1.5, 2.0, 2.3, 3.0 $\mu$m) which lie outside the wavelength range covered by
WFC3.
Twinkle has a field of regard centred on the anti-Sun vector encompassing $\pm
40^{\circ}$ about the ecliptic (Edwards et al., 2019a). The primary exoplanet
survey mission will take place in the first 3 years of operation. Twinkle will
therefore be uniquely positioned to provide homogeneous spectroscopic
characterisation of a large number of exoplanetary atmospheres, something that
will be challenging to achieve with JWST due to competition with other
astrophysical disciplines for valuable telescope time.
Launching several years prior to the European Space Agency’s M4 mission,
Ariel, which has a 1-m class primary mirror and wavelength coverage from
0.5-7.8 $\mu$m (Tinetti et al., 2022), Twinkle will additionally act as a
useful precursor, observing many targets that fall within the current
realisation of the Ariel target list over a substantially shared region of
wavelength space. Consequently, insights gained from Twinkle may be useful for
informing future iterations of the Ariel target list, allowing the combined
science output of the two missions to be optimised.
With the capability to provide the first large homogeneous survey of exoplanet
atmospheres, we seek to explore Twinkle’s ability to identify key molecules
and elucidate trends in a population level study of cool gaseous planets,
which we propose to integrate into the science plan of the Twinkle Extrasolar
Survey (Twinkle collaboration, in prep.).
## 4 Establishing a candidate list for the Twinkle Cool Gaseous Planet Survey
Candidate list construction for the proposed Twinkle cool gaseous planet
survey uses the database of confirmed planets from the NASA Exoplanet
Archive444Planetary Systems Composite Table accessed 22/07/2022 to establish a
preliminary target list of known planets. Transiting planets are selected
based on three criteria:
* 1)
the existence of "transit" listed as the discovery method;
* 2)
the presence of a non-zero transit depth;
* 3)
the presence of a transit duration value,
with any planet not meeting one or more of these criteria being filtered out.
Planets with radii < 1.75 $R_{\oplus}$ are also removed, resulting in a list
of transiting sub-Neptunian, Neptunian and Jovian class planets (see Table 1).
To obtain the sample observable with Twinkle, we perform a cut that eliminates
planets with host stars outside $\pm 40^{\circ}$ of the ecliptic. The
remaining 383 planets with radii between $1.75~{}R_{\oplus}$ and 3
$R_{\text{J}}$ and $T_{\text{eq}}$ between 200 to 1000 K form an initial
Twinkle cool gaseous planet candidate list. The candidate list will be
modified prior to launch as new discoveries are made.
The initial candidate list is subjected to an SNR study, used to identify the
number of transits required for each planet to achieve atmospheric
detectability with Twinkle. The findings of this study are then used to
further filter the candidate list, leaving only planets that can be observed
at or above the target SNR threshold during Twinkle’s mission lifetime. This
includes a cautious estimate of Twinkle’s observing efficiency, scaling up the
number of transits required to meet the SNR threshold by a uniform factor for
each planet and resulting in our final lists of suitable and preferred
candidates for the primary 3-year and extended 7-year Twinkle exoplanet
surveys.
Table 1: Population statistics for transiting exoplanets [derived from NASA Exoplanet Archive Planetary Systems Composite Table accessed 22/07/2022]. Total numbers are shown in black, and the corresponding numbers of cool gaseous planets accessible within the field-of-regard (FOR) of the Twinkle space telescope are shown bracketed in gray. The equilibrium temperatures for this table were obtained from the NASA archive where available or otherwise calculated from stellar and orbital parameters (assuming an albedo of 0.3). At the time of submission, the number of cool gaseous planets in the Twinkle FOR had increased from the 383 shown to 416. | | | Sub-Neptunian
---
($1.75~{}R_{\oplus}$ $\leq$ $R_{\text{p}}$ < $3~{}R_{\oplus}$)
| Neptunian
---
(3 $R_{\oplus}$ $\leq$ $R_{\text{p}}$ < 0.5 $R_{\text{J}}$)
| Jovian
---
(0.5 $R_{\text{J}}$ $\leq$ $R_{\text{p}}$ < 3 $R_{\text{J}}$)
Hot | ($T_{\text{eq}}\geq$ 1000 K) | 222 | 78 | 499
C3 | (750 < $T_{\text{eq}}$ <1000 K) | 388 (75) | 117 (30) | 84 (40)
C2 | (500 < $T_{\text{eq}}$ $\leq$ 750 K) | 528 (109) | 166 (32) | 54 (22)
C1 | (200 < $T_{\text{eq}}$ $\leq$ 500 K) | 309 (59) | 118 (7) | 75 (9)
Cold | ($T_{\text{eq}}$ $\leq$ 200 K) | 1 | 3 | 6
### 4.1 Establishing candidate planet SNR
Before examining the number of transits needed for each planets, we need to
decide on a threshold SNR for spectral feature detection where the SNR is the
ratio of the amplitude of a typical spectral feature $A_{\text{p}}$ to the
noise on the transit depth (1$\sigma$ error bar) $\sigma_{\text{p}}(\lambda)$
at a given spectral binning. If we assume a typical spectral feature
corresponds to 5 scale heights, then we can use Equation 1 to find
$A_{\text{p}}$ with $n=5$, and $H$ for a given planet is obtained from
Equation 2. The error bar for a given target SNR is given by
$\centering|\text{error bar}|=\frac{2\cdot 5\cdot H\cdot
R_{\text{p}}}{R_{\text{s}}^{2}}\cdot\frac{1}{\text{SNR}_{\text{target}}}\@add@centering$
(3)
We wished to verify that SNRs calculated this way corresponded to
detectability of prominent molecules at high significance when simulated using
an atmospheric radiative transfer code with parameters retrieved via Bayesian
parameter estimation ("spectral retrieval"). The latter reflects the method of
analysis that would be applied to a real observed transmission spectrum. We
decided to investigate nominal SNRs of 3 and 5. To this end a subset of the
Twinkle cool gaseous candidates (listed in Table 2) have their atmospheres
simulated using TauREx3 (hereafter TauREx) (Waldmann et al., 2015; Al-Refaie
et al., 2021) as described further below. For a given planet, in the
calculation of $H$, $T_{\text{eq}}$ and $g$ are obtained and derived
respectively from system values used by the Twinkle radiometric tool,
TwinkleRad (Stotesbury et al., 2022), based on ExoRad (Mugnai et al., 2022),
whilst $\mu$ is obtained directly from the TauREx atmospheric model. This
allows calculation of $A_{\text{p}}$ for each planet, and error bars for SNRs
of 3 and 5 obtained using Equation 2.
For each planet, the atmospheric model and resulting model transmission
spectrum are obtained as follows. TauREx was run initially using a model with
100 plane-parallel atmospheric layers spanning pressures from $10^{-6}$ to
$10^{5}$ Pa, an isothermal temperature-pressure (T-P) profile at equilibrium
temperature, $T_{\text{eq}}$, and equilibrium chemistry set by the taurex_ace
plugin based on the ACE equilibrium chemistry regime of Agúndez et al. (2012,
2020), generated using solar C/O-ratio and metallicity values obtained for
each planet using the trend found in Welbanks et al. (2019). Altitude-
dependent volume mixing ratios (VMRs) obtained this way for each molecule were
then simplified to a single VMR value by taking the average across the profile
(Figure 1) with this single value subsequently being used to set the free
chemistry in the final model. The model was thus run again with the same
initial conditions except this time with free chemistry, i.e. the fixed VMRs
for the most abundant molecules (VMR $>$ $10^{-8}$), to give a final
transmission spectrum. We use opacity cross-sections from ExoMol (Tennyson &
Yurchenko, 2016) that include H2O, CH4, CO2, CO and NH3. In addition to
molecular absorption, contributions from Rayleigh scattering and collision-
induced absorption (CIA) between H2-H2 and H2-He were included555This work
utilizes molecular cross-sections from 0.3-50 $\mu$m sampled at R=15000, which
can be found at
https://www.dropbox.com/sh/13y33d02vh56jh2/AACh03L5h1QEbDYN7_-jMjBza/xsec/xsec_sampled_R15000_0.3-50?dl=0&subfolder_nav_tracking=1.
Rayleigh scattering data for all included atmospheric molecules and CIA files
from HITRAN 2011 available at the above link are also used.. To simulate an
observed spectrum, the resulting near-local thermal equilibrium (near-LTE)
cloud-free atmospheric spectra were then binned across the wavelength range
covered by both Twinkle spectroscopic channels to a fixed spectral resolution
of $R$=50, approximating the performance of the instrument as detailed above.
To these binned points error bars were then added according to Equation 1, for
the SNR = 3 and SNR = 5 cases. An example of such a simulated observed
spectrum is shown in Figure 2 together with the different contributions to the
spectrum.
Figure 1: Modelled VMRs for HD 106315 c. Solid lines denote altitude-dependent
chemical profiles under equilibrium conditions, whilst dashed vertical lines
denote profile-averaged VMRs. Figure 2: Forward modelled spectrum for
candidate HD 106315 c, binned to R=50 from 0.5-4.5 $\mu$m. Molecular
absorption components are shown at the binned resolution of the final
spectrum. Also shown are the total contributions to the final spectrum from
CIA (between H2-H2 and H2-He) and Rayleigh scattering (all atmospheric
molecules), with the final spectrum shown in black.
We note here that although disequilibrium chemistry processes and clouds and
hazes are expected to be present in sub-$1000$ K planetary atmospheres
(Fortney et al., 2020; Dymont et al., 2022; Fleury et al., 2023), such
processes are poorly constrained at present, and hence the extent to which
they may impact an individual planet cannot be evaluated with accuracy at this
time. We have therefore not attempted to include these processes in the
atmospheric models created for this study, which should suffice for the
purpose of establishing detectability of general spectral features.
Bayesian spectral retrievals are then conducted on the binned forward-modelled
spectra produced using the nested sampling algorithm
nestle666https://github.com/kbarbary/nestle initiated in ‘single’ mode with
150 live points. For each planet (with error bars corresponding to SNR = 3 or
SNR = 5), the following retrievals are performed. Firstly a baseline retrieval
with the “full atmospheric” model, containing all atmospheric constituents
with VMRs $>$ $10^{-8}$ (which generally included H2O, CH4, CO2, CO and NH3).
The remaining three retrieval models are each initiated in the same manner as
the full atmospheric model, but without one of either H2O, CH4 or CO. We
select these molecules as they are found to consistently have the highest VMRs
in our forward models. By comparing the Bayesian evidence obtained to that of
the full atmospheric model, detection significances could be ascertained for
each molecule. The detection significance of the three molecules is obtained
via the log Bayes factor (Trotta, 2008; Benneke & Seager, 2013), which is the
difference in the log Bayesian evidence between full atmospheric model and the
model with the molecule omitted.
We find that for all the selected planets (Table 2), CH4 is detected at >
5$\sigma$ in all cases whether the error bars are derived from an SNR of 5 or
3. With SNR=5, H2O is detected to $\geq$ 5$\sigma$ in all cases, but at SNR=
3, water detection significance falls to a minimum of $\sim 3$ (Table 2). Our
molecular detectability study also reveals that for the eight planets studied,
CO is never detected above $2\sigma$, irrespective of SNR. Given the high VMR
of this molecule in the forward models (log[VMR${}_{\text{CO}}$] = -3 to -4),
we attribute the weakness of detection to two factors. As can be seen in
Figure 2, CO features at shorter wavelengths ($\sim$1.6 and 2.34 $\mu$m) are
masked by spectral features from CH4 and H2O when all three molecules have
similar atmospheric abundances, owing to CH4 and H2O having larger cross-
sections. This challenge in robustly detecting CO is further compounded by the
fact that the strongest observable band peaks at 4.7$\mu$m, just beyond the
wavelength range covered by Twinkle (Edwards et al., 2019a).
Given that when we utliize error bars derived from an assumed SNR of 5 we
obtain excellent detection significances for H2O and CH4 in all cases studied,
we proceed by adopting SNR = 5 as the threshold to attain for all planets in
the Twinkle initial candidate list.
### 4.2 Preliminary candidate list
We next take the initial 383 candidates and estimate the number of transits
needed in each case to reach an SNR of 5. To calculate SNR we again find
$A_{\text{p}}$ for each planet using Equation 1, but this time we obtain the
transit depth errors from the radiometric tool, TwinkleRad (Stotesbury et al.,
2022) [via B. Edwards, private communication]. TwinkleRad gives the 1$\sigma$
errorbar values on the transit depth for for a single transit. These values
account for photon noise and instrumental effects and assume 100% observing
efficiency. The error bars from TwinkleRad are given at the “native”
wavelength grid of its Twinkle model, which has a median resolving-power of 42
(ranging from 18-70). Model transmission spectra are thus binned to this
native grid.
Table 2: Detection significances (in $\sigma)$ of H2O and CO obtained from retrievals conducted on simulated atmospheres for eight planets spanning the cool gaseous planets parameter space. Detection significance of CH4 is >5 $\sigma$ in all cases. Planet name | Detection significance of individual molecules
---|---
| Water (H2O) | Carbon Monoxide (CO)
| SNR $\geq$ 3 | SNR $\geq$ 5 | SNR $\geq$ 3 | SNR $\geq$ 5
WASP-11 b | 4.4 | >5.0 | 1.8 | 1.7
HD 63935 b | >5.0 | >5.0 | 1.7 | 1.7
HD 106315 c | >5.0 | >5.0 | 1.8 | 1.3
TOI-1130 c | 3.5 | >5.0 | 1.8 | 1.7
GJ 436 b | 4.2 | >5.0 | 1.7 | 1.9
HD 136352 c | 4.5 | >5.0 | 1.6 | 2.0
AU Mic c | 3.1 | 5.0 | 1.8 | 1.9
TOI-178 g | 4.3 | >5.0 | 2.0 | 2.0
Since the error on the transit depth is in reality wavelength dependent, so is
the SNR. However a single representative value was needed for a given planet
for calculation of the number of transits. For this we use the lower quartile
value for SNR across the full wavelength range covered by both channels. This
ensures that $75\%$ or more of the spectrum achieves or exceeds the target SNR
of the observation and that individual planets are not negatively biased by a
single low-impact data point.
In order to account for loss of data during Earth-occultation events that will
arise due Twinkle’s low-Earth orbit, we scale the TwinkleRad error bars by a
factor of 1/$\sqrt{0.75}$. This is done to simulate a conservative observing
efficiency of 75%, assumed to be the case for all planets within our observing
sample. Consequently, we re-calculate the representative single transit SNR of
each planet, then obtain the number of transits, $N_{\text{t}}$, required to
reach a threshold SNR of 5, rounded up to the nearest integer:
$N_{t}=\left(\frac{5}{\text{SNR}_{1}}\right)^{2}$ (4)
where $\text{SNR}_{1}$ is the lower quartile SNR for a single transit. We
combine this information with the orbital period for each planet to compute if
the required number of transits could be observed within 3 or 7 years. This
way we obtain a final candidate list of 36 and 57 planets for the primary
(3-year) and extended (7-year) missions respectively. We show the distribution
in the radius-temperature plane of the 57 candidates in the Twinkle 7-year
extended mission list in Figure 3. The final candidates are further separated
into five distinct tiers using the following assigned criteria based on
$N_{\text{t}}$ and the total integration time, $T_{\text{int}}$. The latter is
calculated assuming a single transit observation lasts 3 $\times
T_{\text{14}}$, where $T_{\text{14}}$ is the transit duration. The expectation
is that a subset of higher tier candidates will be observed by Twinkle during
its lifetime.
* •
Tier 1: $N_{\text{t}}$ < 25 and $T_{\text{int}}\leq 25$ days
* •
Tier 2: $N_{\text{t}}$ < 50
* •
Tier 3: $N_{\text{t}}$ < 100
* •
Tier 4: $N_{\text{t}}$ < 150
* •
Tier 5: any other candidates
The candidates are listed with their $N_{\text{t}}$, $T_{\text{int}}$ and
tierings in Table 5 and Table 6. These planets therefore give the preliminary
candidate list for the Twinkle cool gaseous survey and for the studies in this
paper. We note that 26/36 planets of the 3-year survey and 46/57 for the
7-year survey do not have any current transmission spectra. However since this
analysis was completed, the number of cool gaseous targets in the Twinkle FOR
has increased by 8% (Table 1), and will continue to do so up to the time of
the mission. Our sample size thus represents a conservative value, with the
final target list likely to be somewhat larger. While it is possible that some
of these planets will be observed with JWST prior to the launch of Twinkle,
the Twinkle survey being a homogeneous survey, inclusion of JWST planets in
the sample would still be important. We note that large numbers of transits
are required for many planets in the sample which may ultimately not be
practical in the real mission. However, we expect Tier 1 planets (which range
in $N_{\text{t}}$ from 1 to 22 transits) would be practical. Tier 2 planets
may also be quite possible. Thus while the full sample shown here may not be
ultimately adopted, a significant sub-sample of high-tier planets exists,
covering a wide range of size and temperature, which would make a sizeable
survey.
Figure 3: Population of known exoplanets [NASA Exoplanet Archive, accessed
18/05/23], overlain by candidates targeted by this study for the primary 3-yr
and extended 7-year Twinkle exoplanet surveys. The different temperature
regimes are indicated: Hot, C3, C2, C1 and cold. C1, C2 and C3 are three sub-
categories of the "cool" regime. The bounded region indicated the parameter
space of cool gaseous planets as defined in this paper. The equilibrium
temperatures shown for Twinkle 3- and 7-year mission candidates are obtained
from methodologies outlined in Edwards et al. (2019b) and used in the
TwinkleRad database. For all other planets, the equilibrium temperatures shown
are obtained from the NASA archive where available or otherwise calculated
from stellar and orbital parameters (assuming an albedo of 0.3).
## 5 Metallicity Trend detection study
One of the key properties for planet characterisation is metallicity, which is
commonly split into two regimes: bulk, describing the heavy-element content of
the total planet and atmospheric, pertaining to the atmosphere alone. As
mentioned previously, atmospheric metallicity and elemental ratios can provide
clues to the formation mechanism, location and migration history of the
planet. Atmospheric metallicity and elemental ratios, e.g. the C/O ratio, also
set the initial elemental mix that controls the abundances of molecular
species seen in thermochemical equilibrium. The possible inverse relationship
between planet mass and metallicity is consistent with core accretion
scenarios. In this study we take the Twinkle cool gaseous planet preliminary
sample and investigate Twinkle’s ability to elucidate a mass-metallicity trend
in this sample, if one exists.
### 5.1 Elucidation of a mass-metallicity trend
We explore here whether an injected atmospheric metallicity trend can be
recovered from a simulated atmospheric survey of cool gaseous planets using
the Twinkle 7-year candidate list (Table 5 and Table 6). The candidate list is
well-suited for this investigation spanning two orders of magnitude in mass.
We construct atmospheric forward models for each planet on the list, using
TauREx, binning these to the native Twinkle wavelength grid and adding re-
scaled wavelength-dependent error bars obtained from TwinkleRad to the
resulting spectrum. The scaling accounts for the assumed observing efficiency
of Twinkle (75%) and the unique number of transits, $N_{\text{t}}$, required
for each planet in the sample to reach the desired SNR threshold of 5. Our
forward models have molecular abundances dictated solely by ACE equilibrium
chemistry initialised with C/O=0.54 (solar) and a unique metallicity value for
each planet obtained using the H2O abundance mass-metallicity trend (including
WASP-39 b) of Wakeford et al. (2018). We use this reference since the
metallicity values are given in units of solar metallicity, and the retrieved
metallicities from TauREx are given in the same units. Ten planets in the
sample do not have currently measured masses but their estimated masses given
in the NASA Exoplanet Archive are used for this purpose (these are planets for
which there are no errors given in Table 5 and Table 6).
Model atmospheres are again generated using a simple isothermal T-P profile
spanning 100 plane-parallel layers $10^{-6}$ to $10^{5}$ Pa (10-11 bar to 1
bar) and assume cloud-free conditions; however unlike the final models used in
subsection 4.1, we retain the full, altitude-dependent, VMR profiles for each
chemical species. Transmission spectra are then generated from this
atmospheric model by including molecular absorption, CIA from both H2-H2 and
H2-He, and Rayleigh scattering. We subsequently perform Bayesian spectral
self-retrievals to retrieve for atmospheric metallicity and C/O ratio. Each
retrieval is initiated with the parameters and priors listed below in Table 3,
with atmospheric metallicity and C/O retaining the same initialisation value
and prior range for the full sample.
Results are obtained for all candidates in the Twinkle 7-year candidate list,
with the exception of K2-138 f, which is subsequently. excluded from any
analysis completed on the sample thereafter. We found persistent errors
halting the retrieval of K2-138 f. While the exact cause of the error was not
established, we note that K2-138 f has a very low mass, low gravity and an
extremely high scale height of 1000 km, which could possibly be related to the
computational failure.
We plot the results obtained, along with their 1$\sigma$ error bars (mass from
literature, metallicity from retrieval results) in Figure 4, along with the
injected mass-metallicity trend. In this figure the central points are the
medians of the posterior distributions and the errors bars encompass the
16th-84th percentile ranges. Although retrieved metallicity is found to be
over-estimated in all but four cases across the sample, this is a small
effect, with 51/56 planets (91%) of the population results having the truth
value within the $1\sigma$ confidence interval.
This study indicates that Twinkle has the capability of recovering a mass-
metallicity trend in cloud-free cool gaseous planet atmospheres. However, we
acknowledge that some planets may have hazes and clouds. Also our assumption
of equilibrium chemistry is likely a simplification given the unknown nature
of such atmospheres and the likelihood of disequilibrium processes. Further
studies could therefore examine the robustness of recovering a mass-
metallicity trend under a more complex variety of atmospheric scenarios,
including a mix of cloudy, cloudless and disequilibrium chemistry effects.
Table 3: Initial values and prior ranges of free parameters in retrieval models used in the mass-metallicity study. Where factor bounds are used, values specified are multiplied by the truth value. Parameter | Input | Prior Type | Bounds
---|---|---|---
Metallicity, Z | 50 | Uniform, Linear | [0.01, 750]
C/O ratio | 0.54 | Uniform, Linear | [0.1, 5.0]
Radius | $R_{\text{p}}$ | Uniform, Factor | [0.8 $R_{\text{p}}$, 1.2 $R_{\text{p}}$]
Temperature | $T_{\text{eq}}$ | Uniform, Factor | [0.8 $T_{\text{eq}}$, 1.2 $T_{\text{eq}}$]
Figure 4: Retrieved metallicity plotted against literature mass. Inset shows
retrieved C/O ratio against literature mass. Central points are the medians of
the posterior distributions while error bars denoted the 1$\sigma$ confidence
interval (16th-84th percentile ranges). Here we plot planets with currently
measured masses as solid dots, whilst planets with masses estimated from M-R
relations are plotted as crosses. Residuals with respect to input trend are
shown in panel 2, whilst the residual normalised by the truth value is shown
in panel 3.
### 5.2 C/O ratio retrieval
In addition to retrieving for atmospheric metallicity, our study also
retrieves for the carbon-to-oxygen ratio (C/O). Retrievals use the truth value
as the input value in all cases, but implement a broad, uninformative prior as
shown in Table 3. We find that the truth value is recovered within the
1$\sigma$ confidence interval in all cases (as shown in the inset in Figure
4). We do note that for 7 planets, the retrieved median values deviate
strongly (by $\geq$ 0.135) from the injected truth, with large error bars. We
find that in the majority of these deviated cases, the posterior distributions
were asymmetric with an extended positive tail, which may in part explain the
over-estimation given by the median.
### 5.3 Exploring sources of bias in metallicity retrievals
To explore the slight over-estimation in retrieved metallicity seen for the
bulk of the Twinkle 7yr candidate population, we first explore the posterior
distributions generated from the nested sampling retrieval results. Although
there is some evidence of non-Gaussianity in the posterior distributions, as
shown for TOI-1130 c in Figure 5, we find no evidence for systematic over-
estimation of the median due to effects of sampling an asymmetric
distribution.
We therefore elect to investigate the effect, if any, that the modelling and
retrieval process has in creating this bias, by varying combinations of the
wavelength grid and spectral resolution of the modelled spectra. This approach
is taken as atmospheres are initially generated in TauREx using cross sections
that span the wavelength range 0.3-50 $\mu$m at a spectral resolution of
$R=15000$, resulting in a substantial loss of information from the model
inputs when spectra are binned down to the specifications of the observing
instrument. Three additional sets of models (Table 4, cases 2, 3 and 4) are
generated using the wavelength ranges and spectral resolutions as listed, for
each planet in the Twinkle 7-year candidate list.
Figure 5: Retrieved joint posterior distribution for Tier 1 candidate TOI-1130 c. Table 4: Metallicity bias study test cases. Case Number & Name | Wavelength Range | Wavelength Grid
---|---|---
1 Twinkle native | 0.5-4.5 $\mu$m | Twinkle native (average $R$ $\sim$ 42)
2 Twinkle \- HST WFC3/G141 range | 1.1-1.7 $\mu$m | Twinkle native (average $R$ $\sim$ 44)
3 Twinkle $R=70$ | 0.5-4.5 $\mu$m | fixed $R=70$
4 Ariel $R=70$ | 0.5-7.8 $\mu$m | fixed $R=70$
Figure 6: The upper plot shows statistical significance of cloud detection,
derived from the Bayes factor preference for a model with clouds over model
without clouds, for eight planets. The lower plot shows the temperature-radius
plane for those eight planets with red circles indicating cases where
detection significance is $>3\sigma$ while black circles denote cases at
$<3\sigma$.
Here, case 1 is considered to be our baseline case, representing the
performance of Twinkle based on current knowledge and modelling. We base our
case 2 model on the widely-used HST WFC3/G141 instrument configuration to
examine the effect of reducing the wavelength coverage compared to case 1 on
retrieved metallicity, retaining the wavelength resolution grid of Twinkle
such that any changes in retrieved metallicity can be attributed solely to the
reduction in wavelength coverage. We use cases 3 and 4 to see the effect of
increasing wavelength coverage from that of Twinkle to that of Ariel
(approximately doubling the wavelength range). For consistency cases 3 and 4
utilize a fixed $R$ of 70. This $R$ value does not reflect the true $R$ for
Twinkle or Ariel (which will vary will wavelength), but is chosen as a nominal
value for the purposes of this comparison. Increased wavelength coverage would
be expected to boost the sensitivity to of the retrieval to molecules such as
CO (see subsection 4.1). This approach is taken rather than accurately
modelling an observation of these targets with Ariel due to the differences in
native spectral resolution between Twinkle and Ariel, which would inhibit the
ability to isolate the dependence on wavelength coverage on any systematic
findings. We keep the error bars on the spectra the same for each individual
planet across all cases, and perform self-retrievals on the semi-physically-
motivated models of cases 2, 3 and 4 in the same manner described in section
5.
Our findings are as follows: for case 2, average retrieved 1$\sigma$ error
ranges are just over a factor of 3 higher for atmospheric metallicity compared
to case 1, yet despite this, values are typically found to be proportionately
more over-estimated, with only 35/56 planets (62.5% of the population) having
the truth value within the $1\sigma$ confidence interval. Comparisons between
the baseline case (case 1) and case 3 show that the average retrieval error
bars are comparable; the average 1$\sigma$ confidence interval is 5% greater
in case 3 compared to case 1. The number of planets with truth values within
the 1$\sigma$ range is the same. Similar results are seen when comparing cases
3 and 4: in both cases 51/56 planets have truth values within 1$\sigma$ of the
retrieved atmospheric metallicity values. However, the average retrieved
1$\sigma$ range is 10% smaller in case 4 compared to case 3.
Importantly, we find that in a little over two thirds of the sample (38/56
planets), case 4 spectra, with broader wavelength coverage, yield metallicity
values that lie closer to the truth than in case 3. While this suggests that
increased wavelength coverage leads to an improvement in the ability to
recover atmospheric metallicity from spectra, further investigation shows that
typically the absolute difference between retrieved metallicity and the truth
value,$|Z_{\text{ret}}-Z_{\text{truth}}|$, is only marginally different
between cases 3 and 4. We calculate this difference for each of the 56 planets
in our sample using:
$\delta=|Z_{\text{ret}}-Z_{\text{truth}}|_{3}-|Z_{\text{ret}}-Z_{\text{truth}}|_{4}$
(5)
The average value for $\delta$ is 1.3, which is well within the quadrature sum
of the 1$\sigma$ errors of the retrieved metallicities for each case (where
the 1$\sigma$ error is taken to be half of the 1$\sigma$ confidence interval
which is itself the 16th-84th percentile range). This suggests that the
differences between cases 4 and 3 are not significant. We conclude that the
slight bias seen in retrieved metallicity is likely at least in part due to
information loss from reduced wavelength coverage, although other possible
causes may exist.
## 6 Cloud Detection Study
Clouds are thought to be present to some extent in the atmospheres of all
planets (Helling, 2021) and are known to make a significant contribution to
the overall planetary albedo, a key parameter in the determination of
equilibrium temperature (Estrela et al., 2022). Capable of influencing
temperature, and hence the energy budget of an atmosphere, clouds are
intrinsically coupled to T-P structure of the atmosphere and its chemical
composition (Madhusudhan et al., 2016), with clouds of varying compositions
expected to form as atmospheric species cross condensation fronts. Although
the basic mechanisms by which cloud formation occurs are well-formulated, the
precise details and balance of pathways are currently debated (Helling, 2019,
2021). Consequently, the detection of clouds in in cool gaseous planets and
subsequent inference of the altitudes at which they are present, has the
potential to provide insight into atmospheric processes governing cloud
formation in this regime.
We therefore investigate Twinkle’s capability to detect global cloud layers
for a sub-sample of planets taken from the candidate list (Table 5 and Table
6). We selected one representative planet from each of the nine major sub-
divisions shown in Figure 3 with the exception of C1-Jovian planets for which
no examples exist in our list. The selected planets are indicated by name in
Figure 3.
Our goal here is to find constraints on the pressure levels at which cloudy
atmospheres can be distinguished from cloud-free ones, functioning also as a
top-level search for potential trends in cloud detectability across the
different planetary types. Forward models are constructed using TauREx for
each of the eight planets selected. Models are generated as previously
described with isothermal profiles. ACE equilibrium chemistry is initiated
with C/O=0.54 (solar) and unique metallicity values for each planet from the
H2O mass-metallicity trend (including WASP-39 b) of Wakeford et al. (2018).
The resulting altitude-dependent chemical profiles are then simplified to an
individual altitude-independent VMR as described in subsection 4.1 for H2O,
CH4, CO2, CO, NH3 and N2, and the model run again with fixed VMRs to generate
the final transmission spectra. We include opacity contributions from
molecules, Rayleigh scattering and CIA. In addition, an optically thick grey
cloud layer is modelled at a given pressure level, which is varied between one
of four pressures: 104, 103, 102 and 10 Pa in different runs. In each case,
the resulting forward model spectra are then subjected to two Bayesian
spectral retrievals. Both retrievals are conducted with identical initial
input parameters and bounds, retrieving for all molecule VMRs, temperature and
planet radius, with one retrieval model retrieving for cloud pressure level,
while the other does not include clouds.
Planetary radius and equilibrium temperature are fit with the same priors from
Table 3, whilst log-uniform priors with bounds [$10^{-12}$, $10^{-1}$] are
used for all molecule VMRs (H2O, CH4, CO2, CO, NH3 and N2). Where clouds are
included in the retrieval model, log-uniform priors with bounds [$10^{-6}$,
$10^{5}$] Pa are used. We use the difference in the log Bayesian evidence from
each pair of retrievals to determine a detection significance (Trotta, 2008;
Benneke & Seager, 2013) for cloud (at the given pressure level) in each case.
The results are illustrated in Figure 6.
We find that high-altitude clouds at 10 Pa are detectable above $3\sigma$
irrespective of the planetary or temperature regime they are in. Deeper clouds
at 100 Pa can be detected above $3\sigma$ across all planet sizes, but only in
C3 (500-1000 K) planetary atmospheres.
At cloud pressure of 1000 Pa, we obtain $3\sigma$ detections only in the two
Jovian planets, suggesting it may be possible to probe physical processes and
T-P structure in the deeper layers down to 0.01 bar. At $10^{4}$ Pa (0.1 bar),
we are unable to distinguish cloudy and cloud-free atmospheres at $\geq$
3$\sigma$ in any of the cases.
These results indicate that Twinkle will be unlikely to detect clouds deeper
than 0.1 bar in any cool gaseous planets, and should be able to detect clouds
at 0.0001 bar (10 Pa) or higher in all cases. We find a rough indication of a
trend of improved detectability with temperature and size of the planet. We
also observe a tentative trend of decreasing sensitivity with planetary size
across the sub-Neptune group and sub-Neptune-Neptune boundary; however this
may also be explicable by reduction in temperature in the lower detection
planets. The robustness of these conclusions is limited due to the small sub-
sample size used here. These initial results can be built on by further
investigation using a larger sub-sample or a large grid of completely
simulated planets (providing exact temperature and radius controls).
## 7 Conclusions
In summary, we present a first realisation of a tiered candidate list for the
proposed Twinkle cool gaseous planets survey, based on currently confirmed
exoplanets. We find that Twinkle has the potential to characterise the
atmospheres of up to 36 and 57 planets in the primary 3-year and extended
7-year survey respectively. Candidates identified include 27 and 46 planets
respectively that, to the best of the authors’ knowledge at the time of
submission, do not have precise transmission spectra. A survey using just Tier
1 and Tier 2 planets, would yield 20 planets ranging in mass from 0.0079 to
0.97 $M_{\text{J}}$ and 480 to 941 K in temperature. Due to growing numbers of
discoveries in the cool gaseous regime, e.g. by TESS, these sample sizes can
be considered conservative and will be greater by the time of Twinkle’s
launch. The final candidate list used for the Twinkle mission will be updated
to include new discoveries
Twinkle is well-positioned to provide the first opportunity to garner insights
into cool gaseous planets at a population level. The 3-year baseline survey
includes all 15 Tier 1 candidates (7 Jovians, 4 Neptunians, 4 sub-Neptunians,
collectively spanning $\sim$480-934 K) and all 5 Tier 2 candidates. These
planets will be the highest priority candidates for the survey. If planets up
to Tier 3 are included, a 3-year survey would have 31 planets, and a 7-year
survey would have 35 planets.
Our study predicts that major molecular species expected to be present under
cloud-free near-equilibrium conditions in sub-1000 K atmospheres can be
detected to high significance, and that if present, trends in atmospheric
metallicity can be reliably identified across the sample of surveyed planets.
The C/O ratio was also recovered to within 1$\sigma$. These studies show that
Twinkle will provide valuable data capable of contributing towards the
understanding of trends within the cool gaseous planet demographic and beyond,
as shown in Figure 4. Such data has the potential to inform atmospheric models
and further planet formation theories for giant planets in this understudied
temperature regime.
We also predict that based on a simple grey cloud model, Twinkle has the
potential to detect high-altitude cloud decks at or above 10 Pa in all
atmospheric spectra in the cool regime, but would be insensitive to clouds
below $10^{4}$ Pa in all cases. We find a rough and tentative trend in cloud
detectability with planetary temperature and size.
The cool gaseous planets represent a new frontier in exoplanet science,
promising to expand our understanding of atmospheric physics and chemistry, as
well as planet formation and evolution. The Twinkle cool gaseous planet survey
has the potential to open up this uncharted territory of exoplanet parameter
space with paradigm-shifting results. 20 additional survey candidates present
across tiers 2 and 3, and many new candidates, typically discovered around
bright stars, hence amenable for transmission spectroscopy, confirmed and
being actively refined by TESS and other facilities (including 32 new cool
gaseous planets in the Twinkle field-of-regard) this proposed survey to be
conducted by Twinkle promises to deliver a small, but statistically meaningful
sample of homogeneous spectra.
## Acknowledgements
We thank Ben Wilcock of Blue Skies Space Ltd. for his comments on the
manuscript. L.B. is supported by a STFC doctoral training grant. Additionally,
we thank the anonymous reviewer for their helpful comments.
## Data Availability
The data underlying this article will be shared on reasonable request to the
corresponding author.
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## Appendix A Candidate Lists
Table 5: Candidate planets in Tiers $1$, $2$ and $3$ for the Twinkle cool gaseous planet survey. Planetary and stellar parameters listed are used throughout this work and were obtained from the NASA Exoplanet Archive or calculated based on assumptions presented in Edwards et al. (2019b). Where unavailable in the archive, host star spectral type uses SIMBAD values (‡) or estimation based on comparable stars (†). Planet name | $\bm{N_{\text{t}}}$ | $\bm{T_{\text{int}}}$ | $\bm{R_{\text{p}}}$ | $\bm{M_{\text{p}}}$ | $\bm{T_{\text{eq}}}$ | In 3yr | In 7yr | Tier | Transmission | $\bm{R_{\text{s}}}$ | $\bm{T_{\text{s}}}$ | Spectral
---|---|---|---|---|---|---|---|---|---|---|---|---
| | [days] | [$R_{\text{J}}$] | [$M_{\text{J}}$] | [K] | List | List | | Spectra | [$\bm{R_{\odot}}$] | [K] | Type
AU Mic b | 2 | 0.904 | 0.363 | $0.0368_{-0.0157}^{+0.0157}$ | 626.428 | Y | Y | 1 | False | 0.75 | 3700 | M1
AU Mic c | 20 | 11.359 | 0.289 | $0.0699_{-0.0211}^{+0.0211}$ | 479.593 | Y | Y | 1 | False | 0.75 | 3700 | M1
GJ 1214 b | 15 | 1.581 | 0.245 | $0.0257_{-0.0014}^{+0.0014}$ | 603.949 | Y | Y | 1 | True | 0.21 | 3250 | M4
GJ 3470 b | 7 | 1.602 | 0.408 | $0.0396_{-0.0040}^{+0.0040}$ | 702.732 | Y | Y | 1 | True | 0.55 | 3600 | M1.5
GJ 436 b | 4 | 0.590 | 0.372 | $0.0799_{-0.0063}^{+0.0066}$ | 708.023 | Y | Y | 1 | True | 0.46 | 3586 | M2.5
HD 136352 c | 17 | 6.975 | 0.260 | $0.0354_{-0.0020}^{+0.0021}$ | 701.029 | Y | Y | 1 | False | 1.06 | 5564 | G4
HD 63433 b | 20 | 8.014 | 0.192 | $0.0166$ | 934.262 | Y | Y | 1 | False | 0.91 | 5640 | G5
HD 63433 c | 17 | 8.584 | 0.238 | $0.0239$ | 698.391 | Y | Y | 1 | False | 0.91 | 5640 | G5
K2-141 c | 1 | 0.333 | 0.624 | $0.0233$ | 720.185 | Y | Y | 1 | False | 0.68 | 4599 | K7
K2-24 c | 22 | 19.008 | 0.669 | $0.0485_{-0.0057}^{+0.0060}$ | 610.027 | Y | Y | 1 | False | 1.16 | 5625 | G9
TOI-1130 c | 8 | 2.056 | 1.500 | $0.9740_{-0.0440}^{+0.0430}$ | 658.620 | Y | Y | 1 | False | 0.69 | 4250 | K7
V1298 Tau b | 22 | 18.094 | 0.916 | $0.2360$ | 695.412 | Y | Y | 1 | False | 1.34 | 4970 | K0
WASP-107 b | 1 | 0.351 | 0.940 | $0.0960_{-0.0050}^{+0.0050}$ | 719.812 | Y | Y | 1 | True | 0.67 | 4425 | K6
WASP-69 b | 2 | 0.556 | 1.110 | $0.2600_{-0.0185}^{+0.0185}$ | 928.605 | Y | Y | 1 | True | 0.86 | 4700 | K5
WASP-80 b | 10 | 2.695 | 0.999 | $0.5400_{-0.0350}^{+0.0360}$ | 799.369 | Y | Y | 1 | True | 0.59 | 4143 | K7-M0
HD 183579 b | 42 | 23.804 | 0.317 | $0.0352_{-0.0170}^{+0.0170}$ | 736.129 | Y | Y | 2 | False | 0.99 | 5788 | G2
TOI-1064 c | 36 | 10.697 | 0.237 | $0.0079_{-0.0057}^{+0.0063}$ | 653.762 | Y | Y | 2 | False | 0.73 | 4734 | K3-K5†
TOI-178 d | 46 | 13.424 | 0.229 | $0.0095_{-0.0032}^{+0.0025}$ | 708.942 | Y | Y | 2 | False | 0.65 | 4316 | K7†
TOI-421 c | 27 | 10.674 | 0.454 | $0.0517_{-0.0033}^{+0.0033}$ | 717.787 | Y | Y | 2 | False | 0.87 | 5325 | G7
WASP-29 b | 29 | 9.770 | 0.770 | $0.2450_{-0.0220}^{+0.0230}$ | 941.816 | Y | Y | 2 | True | 0.79 | 4800 | K4‡
GJ 9827 d | 79 | 12.006 | 0.180 | $0.0127_{-0.0026}^{+0.0026}$ | 705.214 | Y | Y | 3 | False | 0.60 | 4340 | K6
HATS-72 b | 57 | 21.946 | 0.722 | $0.1254_{-0.0039}^{+0.0039}$ | 714.490 | Y | Y | 3 | False | 0.72 | 4656 | K5†
HD 106315 c | 96 | 67.606 | 0.388 | $0.0478_{-0.0116}^{+0.0116}$ | 858.332 | N | Y | 3 | False | 1.30 | 6327 | F5
HD 63935 b | 71 | 29.872 | 0.267 | $0.0340_{-0.0057}^{+0.0057}$ | 877.525 | Y | Y | 3 | False | 0.96 | 5534 | G5‡
HD 63935 c | 97 | 59.066 | 0.259 | $0.0349_{-0.0076}^{+0.0076}$ | 701.589 | N | Y | 3 | False | 0.96 | 5534 | G5‡
HD 73583 b | 54 | 14.314 | 0.249 | $0.0321_{-0.0098}^{+0.0107}$ | 688.149 | Y | Y | 3 | False | 0.65 | 4511 | K4
HD 97658 b | 62 | 21.795 | 0.189 | $0.0261_{-0.0035}^{+0.0035}$ | 720.334 | Y | Y | 3 | True | 0.73 | 5212 | K1
K2-406 b | 71 | 41.930 | 0.411 | $0.0603$ | 693.947 | N | Y | 3 | False | 0.96 | 5784 | G4
TOI-1130 b | 83 | 21.998 | 0.326 | $0.0407$ | 786.160 | Y | Y | 3 | False | 0.69 | 4250 | K7
TOI-178 g | 94 | 25.339 | 0.256 | $0.0124_{-0.0051}^{+0.0041}$ | 483.212 | N | Y | 3 | False | 0.65 | 4316 | K7†
TOI-620 b | 61 | 9.190 | 0.335 | $0.0428$ | 620.967 | Y | Y | 3 | False | 0.55 | 3708 | M2.5
TOI-674 b | 68 | 10.035 | 0.468 | $0.0743_{-0.0104}^{+0.0104}$ | 698.867 | Y | Y | 3 | False | 0.42 | 3514 | M2
V1298 Tau c | 83 | 48.488 | 0.499 | $0.0839$ | 932.639 | Y | Y | 3 | False | 1.34 | 4962 | K0
V1298 Tau d | 65 | 44.450 | 0.572 | $0.1060$ | 814.103 | Y | Y | 3 | False | 1.34 | 4962 | K0
WASP-11 b | 57 | 17.901 | 1.110 | $0.5320_{-0.0200}^{+0.0210}$ | 918.615 | Y | Y | 3 | False | 0.89 | 4800 | K3
Table 6: Candidate planets in Tiers $4$ and $5$ for the Twinkle cool gaseous planet survey. Planetary and stellar parameters listed are used throughout this work and were obtained from the NASA Exoplanet Archive or calculated based on assumptions presented in Edwards et al. (2019b). Where unavailable in the archive, host star spectral type uses SIMBAD values (‡) or estimation based on comparable stars (†). Planet name | $\bm{N_{\text{t}}}$ | $\bm{T_{\text{int}}}$ | $\bm{R_{\text{p}}}$ | $\bm{M_{\text{p}}}$ | $\bm{T_{\text{eq}}}$ | In 3yr | In 7yr | Tier | Transmission | $\bm{R_{\text{s}}}$ | $\bm{T_{\text{s}}}$ | Spectral
---|---|---|---|---|---|---|---|---|---|---|---|---
| | [days] | [$R_{\text{J}}$] | [$M_{\text{J}}$] | [K] | List | List | | Spectra | [$\bm{R_{\odot}}$] | [K] | Type
HD 73583 c | 136 | 60.023 | 0.213 | $0.0305_{-0.0054}^{+0.0057}$ | 510.880 | N | Y | 4 | False | 0.65 | 4511 | K4
K2-287 b | 143 | 80.315 | 0.847 | $0.3150_{-0.0270}^{+0.0270}$ | 790.363 | N | Y | 4 | False | 1.07 | 5695 | G8
K2-32 b | 126 | 54.426 | 0.473 | $0.0472_{-0.0054}^{+0.0057}$ | 808.226 | N | Y | 4 | False | 0.86 | 5271 | G9
LP 714-47 b | 143 | 27.265 | 0.419 | $0.0969_{-0.0047}^{+0.0047}$ | 686.753 | Y | Y | 4 | False | 0.58 | 3950 | M0
TOI-561 c | 142 | 66.441 | 0.257 | $0.0220_{-0.0072}^{+0.0072}$ | 789.277 | N | Y | 4 | False | 0.85 | 5455 | G9
TOI-776 b | 140 | 42.583 | 0.165 | $0.0126_{-0.0028}^{+0.0028}$ | 530.605 | N | Y | 4 | False | 0.54 | 3709 | M1
WASP-132 b | 134 | 53.011 | 0.897 | $0.4100_{-0.0300}^{+0.0300}$ | 736.759 | Y | Y | 4 | False | 0.75 | 4714 | K4
WASP-84 b | 123 | 42.392 | 0.942 | $0.6940_{-0.0470}^{+0.0490}$ | 773.076 | Y | Y | 4 | False | 0.75 | 5314 | K0
G 9-40 b | 375 | 58.498 | 0.181 | $0.0150$ | 454.032 | N | Y | 5 | False | 0.31 | 3348 | M2.5
K2-121 b | 309 | 96.569 | 0.671 | $0.1390$ | 786.419 | N | Y | 5 | False | 0.67 | 4690 | K5‡
K2-138 f | 181 | 72.914 | 0.259 | $0.0051_{-0.0037}^{+0.0067}$ | 715.911 | N | Y | 5 | False | 0.86 | 5356 | G8
LTT 3780 c | 174 | 38.081 | 0.204 | $0.0198_{-0.0019}^{+0.0020}$ | 363.418 | N | Y | 5 | False | 0.37 | 3331 | M4
TOI-1201 b | 234 | 44.232 | 0.215 | $0.0198_{-0.0028}^{+0.0026}$ | 682.774 | Y | Y | 5 | False | 0.51 | 3476 | M2
TOI-1422 b | 173 | 96.480 | 0.353 | $0.0283_{-0.0063}^{+0.0072}$ | 838.973 | N | Y | 5 | False | 1.02 | 5840 | G2
TOI-1478 b | 208 | 108.544 | 1.060 | $0.8510_{-0.0470}^{+0.0520}$ | 889.609 | N | Y | 5 | False | 1.05 | 5597 | G8
TOI-1634 b | 1888 | 244.729 | 0.160 | $0.0319_{-0.0030}^{+0.0030}$ | 894.156 | N | Y | 5 | False | 0.45 | 3550 | M2
TOI-178 e | 231 | 71.965 | 0.197 | $0.0121_{-0.0030}^{+0.0039}$ | 616.710 | N | Y | 5 | False | 0.65 | 4316 | K7†
TOI-3714 b | 693 | 150.617 | 1.010 | $0.7000_{-0.0300}^{+0.0300}$ | 749.785 | N | Y | 5 | False | 0.51 | 3660 | M2
TOI-421 b | 322 | 50.217 | 0.239 | $0.0226_{-0.0021}^{+0.0021}$ | 982.024 | N | Y | 5 | False | 0.87 | 5325 | G9
WASP-156 b | 280 | 84.476 | 0.510 | $0.1280_{-0.0090}^{+0.0100}$ | 938.278 | Y | Y | 5 | False | 0.76 | 4910 | K3
WASP-8 b | 236 | 94.124 | 1.130 | $2.1320_{-0.0810}^{+0.0800}$ | 896.199 | N | Y | 5 | False | 1.03 | 5600 | G8
Wolf 503 b | 271 | 69.390 | 0.182 | $0.0197_{-0.0022}^{+0.0022}$ | 764.360 | N | Y | 5 | False | 0.69 | 4716 | K3.5
|
Medium-induced radiative kernel with the Improved Opacity Expansion
a]João Barata,
[a]Instituto Galego de Fisica de Altas Enerxias (IGFAE), Universidade de Santiago de Compostela,E-15782 Galicia, Spain
b,c]Yacine Mehtar-Tani,
[b]Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
[c]RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA
d]Alba Soto-Ontoso,
[d]Institut de Physique Théorique, Université Paris-Saclay, CNRS, CEA, F-91191, Gif-sur-Yvette, France
e]and Konrad Tywoniuk
[e]Department of Physics and Technology, University of Bergen, 5007 Bergen, Norway
We calculate the fully differential medium-induced radiative spectrum at next-to-leading order (NLO) accuracy within the Improved Opacity Expansion (IOE) framework. This scheme allows us to gain analytical control of the radiative spectrum at low and high gluon frequencies simultaneously. The high frequency regime can be obtained in the standard opacity expansion framework in which the resulting power series diverges at the characteristic frequency $\omega_c\sim \hat q L^2$. In the IOE, all orders in opacity are resumed systematically below $\omega_c$ yielding an asymptotic series controlled by logarithmically suppressed remainders down to the thermal scale $T \ll \omega_c$, while matching the opacity expansion at high frequency.
Furthermore, we demonstrate that the IOE at NLO accuracy reproduces the characteristic Coulomb tail of the single hard scattering contribution as well as the Gaussian distribution resulting from multiple soft momentum exchanges. Finally, we compare our analytic scheme with a recent numerical solution, that includes a full resummation of multiple scatterings, for LHC-inspired medium parameters. We find a very good agreement both at low and high frequencies showcasing the performance of the IOE which provides for the first time accurate analytic formulas for radiative energy loss in the relevant perturbative kinematic regimes for dense media.
§ INTRODUCTION
High-energy collisions of heavy nuclei provide the necessary conditions for creating an extended medium of hot and dense nuclear matter, referred to as quark-gluon plasma (QGP). The appearance of a short-lived stage of this exotic state of matter leaves a strong imprint on particle production at all momentum scales and is quantified by high-precision experimental measurements. In this work, particular attention is devoted to the high-energy particles that traverse the medium and can be used as perturbatively well controlled probes of the microscopic properties of the QGP <cit.>. In terms of experimental observables, these objects emerge in the detectors as collimated sprays of particles and energy, colloquially referred to as jets. Jet modifications, quantified with respect to a baseline obtained in proton-proton collisions (or “vacuum”), have been intensively studied at RHIC <cit.> and LHC <cit.> since more than a decade.
The suppression and modification of jets produced in heavy ion collisions, commonly known as jet quenching, is driven by two main phenomena: transverse momentum broadening and energy loss. The former refers to the acquisition of transverse momentum by the highly energetic partons that make up the jet through elastic interactions with the medium following Brownian motion. This diffusion in momentum space is characterized by the transport coefficient:
q̂ ≡⟨k_⊥^2 ⟩_typ/t ,
where $\langle k_\perp^2 \rangle_{\rm typ}$ is the typical squared transverse momentum transfer. An important role is also played by induced energy loss, such as caused by drag and inelastic, or radiative, processes. The latter component is a result of bremsstrahlung radiation triggered by collisions with medium constituents. The associated mean energy loss was found to scale as $L^2$, where $L$ is the length of the plasma, and thus constitutes an important, if not the dominant, source of jet quenching for large media, even though it is “naively” suppressed by a power of the coupling constant <cit.>.
The above physical picture applies for the regime of multiple scattering during the passage through the medium. This is the case when the opacity $\chi = L/\ell_{\rm mfp}$, defined as the ratio between the medium length, $L$, and the mean free path, $\ell_{\rm mfp}=(\rho\sigma_{\rm el})^{-1}$ (where $\rho$ is the density of scattering centers and $\sigma_{\rm el}$ the total elastic cross section), is of order one or larger, i.e. $\chi \gtrsim 1$. In effect, many interactions, i.e. those that occur within the formation time, coherently participate in inducing gluon radiation, in an analogous way to the well-known Landau-Pomeranchuk-Migdal (LPM) effect in QED <cit.>. It was soon understood that neglecting these interference effects would fail in adequately describing the radiative spectrum in a substantial region of phase space in such media. Focusing on the diffusion approximation, and thereby neglecting the Coulomb tail that captures the physics of rare hard momentum transfers, the radiative spectrum could be analytically computed albeit with an ambiguity in setting the upper bound for the Coulomb logarithm in $\hat q$ <cit.>.
More precise calculations can be performed in the dilute regime, where at most a few scatterings contribute and the full parton-medium interaction potential can be used <cit.>. However, their domain of applicability is limited to low opacity $\chi \ll 1$ or large momentum transfer. Due to the finite radius of convergence of the opacity expansion such an approach diverges <cit.> and thus cannot be applied in the case of a dense medium.
As we have mentioned, in both low and high opacity regimes, under a set of physically motivated assumptions that will be revisited in what follows, the medium-induced spectrum can be computed analytically. However, the scenario explored in current experimental facilities, such as the LHC or RHIC, is expected to be the one where the jet undergoes a handful of scatterings, $\mathcal{O}(1-50)$, with the medium <cit.>. As such, neither of the above limiting forms is in its exact domain of applicability, thus hindering quantitative theory-to-data comparisons and the extraction of the medium parameters through the jet physics program in current colliders.
These limitations in the analytic front have motivated the investigation of the radiative spectrum numerically <cit.>. However, the proposed approaches are in general less transparent and potentially more computationally costly as compared to their analytic counterparts for computing jet observables where multiple gluon radiation is to be resummed to all orders <cit.>, for applications to jet suppression <cit.>, or as a building block for Monte Carlo event generators <cit.>.
A first step towards a more precise control over the accuracy of analytic calculations was first taken in Ref. <cit.>, where the next-to-leading logarithmic corrections to $\hat q $ were computed in the multiple soft scattering regime or, equivalently, in the infinite length medium limit. More recently, substantial progress was made in unifying the low and large opacity regimes while recovering the results of Ref. <cit.> in the soft regime. This new scheme, dubbed the Improved Opacity Expansion (IOE), has been shown to successfully meet this goal when applied to the description of the single particle broadening probability <cit.> and to the medium-induced gluon energy spectrum <cit.>, which constitute the major tools used in a multitude of well established phenomenological models of jet quenching <cit.>. In a nutshell, this framework is built as a series expansion of the in-medium scattering cross section where the zeroth order term encodes the multiple scattering solution and higher N$^n$LO orders[The nomenclature used here to denote the orders in the IOE should not be confused with the more familiar perturbative expansion in powers of the coupling constant.]
in the series account for $n$ hard scatterings with the medium.
This paper aims at computing the fully differential medium-induced spectrum at NLO accuracy in the IOE framework. For different values of the gluon energy and transverse momentum, ($\omega,\k$),\footnote{Bold letters denote 2D transverse vectors in this paper, while their modulus is written as $|\k|\equiv k_\perp$.} we provide an analytic formula for an arbitrary medium profile that requires numerical integrations of the same order in computational complexity as the ones encountered in the multiple soft scattering limit~\cite{ASW1,ASW2}. In the simplified scenario in which the medium is treated as a brick of constant density, we have obtained closed, analytic formulas for the asymptotic behavior of the spectrum computed in the three different setups considered in this work: the IOE at NLO, the single-hard scattering approximation ($SH$) and the multiple soft scattering regime ($MS$). In addition, we make a phenomenologically oriented comparison with the all-orders numerical spectrum presented in Ref.~\cite{CarlotaFabioLiliana}. The numerical routines used in this publication are provided as ancillary files.
The remainder of this paper is structured as follows. Section~\ref{sec:gluon_generic_IOE} revisits the Improved Opacity Expansion framework in full generality, including its application to the single particle momentum broadening and the medium-induced energy spectrum calculations. The core of this paper is Section~\ref{sec:IOE_radiation_spectrum}, where the fully differential spectrum is calculated with a high level of detail. Those readers more interested in the final result and not so much in the technicalities can find a summary of the ready-to-use formulas in Section~\ref{sec:summary}. Finally, numerical results for LHC-motivated medium parameters are presented in Section~\ref{sec:numerics}, including a comparison to BDMPS-Z, GLV and the resummed to all orders spectrum. Further details on the analytic calculations can be found in Appendices~\ref{app:cK_appendix}, \ref{app:Q}, \ref{app:vacuum-derivation} and \ref{app:In_Out_example}.
\section{The Improved Opacity Expansion: an overview }
\label{sec:gluon_generic_IOE}
The Improved Opacity Expansion draws on the seminal 1948 work by Moli\`ere~\cite{Moliere}, where the transverse momentum broadening of charged particles in QED was described in such a way that the multiple soft scattering solution, well described by a Gaussian distribution, and the Coulomb power-law tail were reproduced in the appropriate limits. The IOE program consists in extending Moli\`ere's original approach to QCD and to more complex observables. So far, this strategy has been successfully applied to compute the medium-induced gluon energy spectrum~\cite{IOE1,IOE2,IOE3} and the transverse momentum broadening distribution of an energetic parton propagating through a dense QCD plasma~\cite{broadening_paper}.
As an introduction to the IOE, it will be instructive to first revisit how it applies to the two aforementioned observables: transverse momentum broadening and medium-induced gluon radiative spectrum. This will also serve us to lay the basis of the calculation of the fully differential spectrum.
\subsection{Transverse momentum broadening }
\label{sec:momentum-broadening}
The elementary in-medium process that underlies the observables that we discuss in this work is the elastic collision rate $γ_el≡ σ_el/^2$, where $≡(q_1,q_2)$ corresponds to the transverse momentum transfer in the $t$-channel between the hard probe and the medium. At leading order in the coupling the rate reads $γ_el ∼g^4n/q_⊥^4$, where $n$ corresponds to the density of scattering centers in the medium and the $1/q_⊥^4$ dependence denotes that, at short distances, the interaction is Coulomb-like. On the other hand, when $q_⊥→0$, the power law divergence should be screened by the medium at, roughly speaking, the Debye mass $m_D^2$ in the plasma.
Equipped with $γ_el$, we can readily write a rate equation for the transverse momentum broadening distribution $(;̨t)$, which gives the probability for a parton in color representation $R$ to acquire transverse momentum $$̨ due to in-medium propagation during a time $t$,
(,̨t)/t = C_R∫_ γ_el() [(-̨,t) -(,̨t) ] ,
where the final time corresponds to the length of the medium $t=L$ and $C_R$ is the color factor associated to a representation $R$ of SU$(3)$.[In this paper we use the notation $\int_\q = \int \frac{\rmd^2 \q}{(2\pi)^2}$ to describe transverse momentum space integrals and $\int_\x= \int \rmd^2 \x$ for integration in position space.] The boundary condition at initial time $t=0$ is simply $\cP(\k,0) = (2\pi)^2 \delta^{(2)}(\k)$. The first term in Eq. (<ref>) accounts for the gain in transverse momentum of the initial parton while the second term reflects the loss of probability for finding said parton with the measured momentum $\k$. Notice that due to rotational symmetry the broadening probability is a function of the modulus of the transverse momentum vector, i.e. $k_⊥≡||̨$.
The integral of the collision rate yields the inverse mean-free-path between two collisions, i.e $ℓ^-1_mfp ≡∫_ γ_el()$. At low opacity, $χ∼L/ℓ_mfp ≪1$, the distribution is dominated by at most a single hard scattering (SH) and one finds
\beq\label{eq:oe-lo-br}
\cP^{\rm SH}(\k,L)= C_R\gamma_{\rm el}(\k)\, L \sim (4\pi)^2\frac{\alpha_s^2C_R nL}{k_\perp^4}\, .
\eeq
Conversely, at high opacity, multiple (soft) scatterings occur with order one probability and Eq.~\eqref{eq:rate-eq} can be approximated by a diffusion equation for which analytic solutions exist. This is done by expanding in gradients for $q_⊥≪k_⊥$. The first non-vanishing contribution involves the jet quenching parameter $q̂$,
\beq
\hat q = C_R\, \int^{q_{\rm max}}\frac{\rmd^2 \q }{(2\pi)^2} \, q_\perp^2 \, \frac{\rmd \sigma_{\rm el}}{\rmd^2 \q } \approx 4\pi \alpha_s^2 C_R n(t)\log\frac{q_{\rm max}^2}{\mu_\ast^2} \,,
\eeq
where the integral over $$ is divergent in the ultraviolet and thus must be regulated, giving rise to the standard Coulomb logarithm, while the infrared region is cut-off by the screening mass that we denote by $μ^2_∗$. Assuming $q̂ $ to be constant in time, the solution to the diffusion equation is a Gaussian~\cite{broadening_paper} and the associated broadening distribution reads
\beq\label{eq:gaussian}
\cP^{\rm MS}(\k,L) =\frac{4\pi }{\hat q L } \rme^{-\frac{k_\perp^2}{
\hat q L }} \, .
\eeq
Although this result describes the physics of multiple soft scattering (MS) of the probe in the medium, the diffusion approximation has two major drawbacks: (i) it misses the heavy $1/q_⊥^4$ tail associated with large momentum exchanges and (ii) the transport coefficient depends, logarithmically, on an undetermined ultraviolet cutoff scale.
The IOE overcomes these two limitations by shifting the expansion point of the opacity scheme from the vacuum to the harmonic oscillator potential, resulting in the Gaussian distribution presented in Eq.~\eqref{eq:gaussian}. This shift in the expansion is easily performed in position space and thus we should consider the Fourier pair of $(,̨t)$,
\beq\label{eq:rate-momentum}
\cP(\x,t) = \int_\k \cP(\k,t) \, \rme^{i\x\cdot \k }\, .
\eeq
In position space, \eqn{eq:rate-eq} becomes local
\beq\label{eq:rate-position}
\frac{\del \cP(\x,t)}{\del t } = - \, v(\x) \cP(\x,t) \,,
\eeq
implying $ (,t)=^-v() t$,
where the scattering potential $v()$ combines the gain and loss terms and is thus ultraviolet finite
\beq \label{eq:v-llog}
v(\x) = C_R\int_\q \, \gamma_{\rm el}(\q) \left(1- \rme^{i\q\cdot \x}\right) \propto x_\perp^2 \log \frac{1}{x_\perp^2 \mu_\ast^2 } \, .
\eeq
In this example, \eqn{eq:rate-position} can be directly integrated, but this is not generally possible as we shall see in the case of the radiative spectrum. Furthermore, one still needs to invert the Fourier transform and this is where the IOE scheme will be particularly useful as it allows us to reduce the Fourier transform to a sum of standard integrals.
Let us recall the main difference between the Improved Opacity Expansion procedure and the usual Opacity Expansion (OE) strategy~\cite{Gyulassy:2000fs,Wiedemann,Guo:2000nz}. The latter performs an expansion directly in powers of $v()$ of \eqn{eq:rate-position}, yielding a series in powers of $q_⊥^-2$ once introduced in \eqn{eq:rate-momentum}, with the leading contribution given by \eqn{eq:oe-lo-br}. In the IOE, one shifts the expansion point to be a solution to Eq.~\eqref{eq:rate-position} with the potential $v = ≡q̂ x_⊥^2/4$, whose Fourier transform can be carried out to yield \eqn{eq:gaussian}. If we denote such a solution by $^LO$, then the aforementioned shift of the expansion point leads to
\begin{equation}
\cP(\x,L) =\big[ 1- \delta v(\x) \big]\cP^{\rm LO}(\x,L) +\mathcal{O}(\delta v^2) \, .
\end{equation}
Here the scattering potential is split into two terms, i.e. $v= +δv$, such that
$|δv| ≪|| $, in which case $δv$ can be regarded as a perturbation around the potential $$. In doing so we aim to tame the divergence of the plain Opacity Expansion series at low enough $$, typically when $q_⊥^2 < q̂ L$. This separation of $v$ into $$ and $δv$ is in general arbitrary and requires the introduction of a matching scale $Q$. Clearly, truncating the IOE series at a fixed order introduces a residual dependence on the separation scale that is of the order of the remainder and thus can be safely neglected. It can nevertheless be used to gauge the uncertainty associated with the fixed order calculation very much like scale dependence encountered in \emph{standard} perturbation theory calculations.
To illustrate this point consider the leading logarithmic form given in \eqn{eq:v-llog}. One would trivially write
\begin{equation}\label{eq:ppp}
x_\perp^2 \log \frac{1}{x_\perp^2 \mu_\ast^2 } = x_\perp^2 \left[ \log \frac{Q^2}{ \mu_\ast^2 } + \log \frac{1}{ x_\perp^2 Q^2 } \right]
\end{equation}
and define $∝x_⊥^2 logQ^2/ μ_∗^2 $ and $ δv ∝x_⊥^2 log1/ x_⊥^2 Q^2 $, up to overall time-dependent factors. A natural candidate for the separation scale in the case of momentum broadening is $Q^2 ∼q̂ t$, corresponding to the average momentum squared accumulated by the probe due to multiple soft momentum exchanges with the medium. In general, when considering other observables, the LO provides guidance to what scale should be chosen for $Q^2$. We shall see below how to make this observation more precise, in particular regarding the ultraviolet behavior of $q̂$.
Not only the IOE allows to fix the diverging behavior of the Opacity Expansion, but also provides a good approximation of the exact result at low transverse momentum provided the following hierarchy of scales is met
\beq
Q^2 \gg \mu_\ast^2 \,.
\eeq
This ensures that the Coulomb logarithm is large, i.e. $log(Q^2/μ_∗^2) ≫1$, and since at low $k_⊥$, the $$ integral is dominated by the region $x_⊥^2 ∼1/Q^2$, we also have that $log1/ x_⊥^2 Q^2 ∼1$. On the other hand, at large $k_⊥$ the rapidly oscillating Fourier phase implies that $x_⊥≪k_⊥^-1 ≪Q^-1$ which flips the relative order of the LO and its correction, i.e. $| δv| ≫|| $, and thus the logarithmic function in $v()$ can no longer be neglected. Since large momentum transfers are associated with steeply falling cross sections, such a case is associated with rare hard scatterings in the medium, and perturbation theory is applicable, recovering the standard Opacity Expansion.
Following this more qualitative discussion that highlights the strengths of the IOE approach, let us make the discussion more quantitative and rigorous by recalling some of the results presented in Ref.~\cite{broadening_paper}. First, in jet quenching phenomenology two models for the in-medium scattering rate are typically considered. One option, referred to as Gyulassy-Wang (GW) model~\cite{GW}, is to describe the medium as an ensemble of static scattering centers with Yukawa like potentials, with the in-medium rate given by
\beq\label{eq:GW}
\gamma_{\rm el}^{\rm GW} (\q,t)= \frac{g^4 n(t)}{(q_\perp^2+\mu^2)^2} \, ,
\eeq
where $μ$ is the GW screening mass and $n$ the density of scattering centers in the medium. This leads, see Eq.~\eqref{eq:v-llog}, to a scattering potential of the form
\begin{align}
\label{eq:v_GW_text}
v^{\rm GW}(\x,t)&=\frac{\hat{q}_0(t)}{\mu^2} \big[ 1-\mu x_\perp K_1(\mu x_\perp)\big] \, ,
\end{align}
where we have introduced the \textit{bare} jet quenching parameter $q̂_0(t)=4πα_s^2 C_A n(t)$ and $K_1$ is the modified Bessel function of the second kind of order $1$. Another popular choice is to describe the medium as a thermal bath, so that the scattering potential can be perturbatively computed using Hard Thermal Loop (HTL) effective theory~\cite{HTL}, with
\beq\label{eq:HTL}
\gamma_{\rm el}^{\rm HTL} (\q,t)= \frac{g^2m_D^2(t)T}{q_\perp^2\,\big[q_\perp^2+m_D^2(t) \big]} \,.
\eeq
Here $T$ is temperature of the medium and $m_D$ the Debye screening mass. The corresponding scattering potential reads
\begin{align}
\label{eq:v_HTL_text}
v^{\rm HTL}(\x,t)= \frac{2\hat{q}_0(t)}{m_D^2(t)}\left[ K_0(m_D(t)x_\perp)+\log\left(\frac{m_D(t)x_\perp}{2}\right)+\gamma_E \right] \, ,
\end{align}
where now $q̂_0(t)=α_s C_A m_D^2(t) T$, $γ_E = 0.577216…$ is the Euler-Mascheroni constant and $K_0$ is the modified Bessel function of the second kind of order $0$.\footnote{Here we defined the jet quenching parameter for gluons, i.e. $C_R=C_A$.} The differences and similarities between these two models have been extensively discussed in Refs.~\cite{IOE3,broadening_paper}. To leading logarithmic order, they can be unified in an universal form, in accordance with \eqn{eq:v-llog},
\begin{equation}\label{eq:v_LL}
v(\x,t)\equiv \frac{1}{4} \hat{q}_0(t) x_\perp^2 \log \frac{1}{x_\perp^2\mu_\ast^2} +\cO(x_\perp^4\mu_\ast^2)\, ,
\end{equation}
where $μ_∗$ is a universal screening mass that can be mapped to the masses of both models considered above.\footnote{The GW mass $\mu$ is related to the universal mass $\mu_\ast$ by $4\mu_\ast^2=\mu^2 \rme^{-1+2\gamma_E}$, and the Debye mass $m_D$ in HTL corresponds to $4\mu_\ast^2= m_D^2 \rme^{-2+2\gamma_E} $ \cite{broadening_paper}.}
Applying the IOE prescription to split the potential as $v=+δv$, see \eqn{eq:ppp}, and inserting it back into \eqn{eq:rate-position}, we obtain, after expanding in powers of the perturbative potential $δv$,
\begin{equation}\label{eq:expli_cP_series}
\begin{split}
\cP(\k,L)&=\int_\x \, \rme^{-i \x \cdot \k }\rme^{-\frac{1}{4}x_\perp^2Q^2}\sum_{n=0}^{n_{\rm max}} \, \frac{(-1)^n Q_{s0}^{2n}}{4^nn!} \, x_\perp^{2n} \log^{n}\frac{1}{x_\perp^2Q^2} \\
&\equiv \cP^{\rm LO}(\k,L) + \cP^{\rm NLO}(\k,L) + \cP^{\rm NNLO}(\k,L) + \ldots \, ,
\end{split}
\end{equation}
where we identify the next-to-leading order (NLO) term with the contribution $𝒪(δv)$, the next-to-next-to-leading order (NNLO) with the $𝒪(δv^2)$ term, and so on.\footnote{The series is truncated at $n_{\rm max}\sim Q_{s0}^2/\mu_\ast^2$ since formally this is a divergent asymptotic series; the divergence is physically associated to the fact that $x_\perp$ can not be smaller than $1/\mu_\ast$ --- see Ref.~\cite{Iancu:2004bx} for a further discussion on this truncation.} In Eq.~\eqref{eq:expli_cP_series}, we have introduced the \emph{bare} saturation scale
\begin{align}\label{eq:old_Qs0}
&Q_{s0}^2(L) = \int_{0}^{L} \rmd t\, \hat q_0(t) \, ,
\end{align}
where we allow the \textit{bare} jet quenching parameter to vary in time. In addition, we define the \textit{effective} jet quenching parameter $q̂(t)=q̂_0(t) logQ^2/μ_∗^2$, where the logarithmic dependence appears naturally from the splitting of $v()$. As discussed above, the definition of the matching scale, $Q$, can not be cast in a closed form, since it enters the definition of $q̂$ as well as depends on it directly. In turn, it is obtained by solving the transcendental equation
\begin{align}\label{eq:old_Qb}
&Q_b^2\equiv Q_s^2(L) = \int_0^{ L} \rmd t\, \hat q_0(t) \,\log \frac{Q_b^2( L)}{\mu_\ast^2}\, ,
\end{align}
where, following our previous reasoning, we have identified $Q^2 ≡Q^2_b$ with the \emph{effective} saturation scale $Q_s^2$.
We truncate \eqn{eq:expli_cP_series} at NLO accuracy, since already at this order both the hard and soft regimes should be well described. The resulting broadening distribution reads~\cite{broadening_paper}
\begin{equation}\label{eq:golden}
\cP^{\rm{LO+NLO}}(\k,L)= \frac{4\pi}{Q_s^2} \rme^{-x} - \frac{4\pi}{Q_s^2} \lambdaq \left\{1-2 \rme^{-x} + \left(1-x\right) \left[{\rm Ei}\left( 4x\right)-\log 4x\right] \right\}\, ,
\end{equation}
where $x = k_⊥^2/Q_s^2$, and
\beq
\lambdaq \equiv \frac{\hat{q}_0}{\hat{q}}=\frac{1}{\log \frac{Q^2}{\mu_\ast^{ 2}}} \ll 1\,,
\eeq
is the expansion parameter of the series in the regime $k_⊥^2 ≲Q_s^2$.\footnote{The exponential integral function is defined as ${\rm {Ei}}(x)=\displaystyle\int_{-\infty}^x \rmd t \, \frac{\rme^{t}}{t}$.} At large momentum exchanges, $k_⊥^2 ≫Q_s^2$, one obtains from the NLO term, and in accordance with \eqn{eq:oe-lo-br}, that
\begin{align}\label{eq:plot_ppp}
\cP(\k,L)^{\rm{NLO}}\Big\vert_{k_\perp^2\gg Q_{s}^2}=4\pi\frac{Q_{s0}^2}{k_\perp^4}+\mathcal{O}\left(\frac{Q_{s0}^4}{k_\perp^6}\right) \, ,
\end{align}
while the LO term is exponentially suppressed. In this high momentum limit, we recover the Coulomb tail encoded in a single scattering in the medium. On the other end, when $k_⊥^2 ≪Q_s^2$, we find
\beq
\label{eq:assist_1}
\cP(\k,L)^{\rm{LO+NLO}}\Big\vert_{k_\perp^2\ll Q_{s}^2} = \frac{4\pi}{Q_s^2}\left( 1+ \lambdaq\log 4 \rme^{1-\gamma_E} \right)+ \cO\left( \lambdaq^2\right)\, .
\eeq
The first term corresponds to the LO contribution. Thus, the NLO term, up to a small constant logarithm, is of the same functional form as the LO but power suppressed by $≪1$. In fact, one can show that, in this regime, perturbative corrections in the IOE scale as the LO term, each increasing order suppressed by an extra power of $=q̂_0/q̂$. Hence, in this limit the LO term dominates and one recovers the multiple soft solution, which correctly describes the physics at play.
In Fig.~\ref{fig:broad} we numerically compare the broadening distribution $(,̨L)$, for a medium with constant $q̂_0$, computed up to LO and NLO in the IOE, with the full $$ obtained using Eqs.~\eqref{eq:rate-momentum}, \eqref{eq:rate-position} and the GW potential in \eqn{eq:v_GW_text}. The result follows the above discussion: at large momentum transfers, $k_⊥^2≫q̂L$, the NLO term dominates and converges to the full result dominated by the single hard scattering result ($k^-4_⊥$). On the other hand, at low momentum transfers the LO and LO+NLO become comparable, reproducing the full result within an uncertainty band associated to the remaining freedom in the definition of $Q_b^2$. The biggest mismatch between the LO+NLO result and the full distribution happens near the peak of the distribution and could be eventually improved by adding more orders in the series. Nonetheless, it is clear that the IOE approach provides a neat interpolation between the soft and hard regime, instead of properly describing just one of these regions.
\begin{figure}[t!]
\centering
\includegraphics[scale=.8]{plot-broadening.pdf}
\caption{Comparison between the broadening probability distribution for the IOE at LO (dashed, green), at LO+NLO (solid, red) and the exact GW model result (solid, navy). In addition, we provide the single hard scattering solution given by \eqn{eq:plot_ppp}, which we denote by $k^{-4}_\perp$(dotted, purple). The ratio to the full solution is presented in the bottom panels. The uncertainty band arises from variations in the matching scale by factors of $2$ and $1/2$. The medium parameters are $\hat q_0\!=\!0.16$~GeV$^3$, $L=6$~fm and $\mu_\ast=0.355$~GeV. They are identical to the ones used in Section~\ref{sec:numerics}.}
\label{fig:broad}
\end{figure}
\subsection{The energy spectrum}
\label{sec:energy-spectrum}
As a second illustrative example, we consider the application of the IOE to compute the medium-induced gluon energy spectrum. The in-medium emission spectrum of a soft gluon with energy $ω$ from a hard parton with energy $E≫ω$ in color representation $R$ can be compactly cast as \cite{Blaizot:2015lma}
\begin{align}
\label{eq:BDMPS_spec_energy}
\omega\frac{\rmd I}{\rmd \omega } =\frac{2\bar{\alpha} \pi}{\omega^2} \Re \int_0^\infty \rmd t_2 \int_0^{t_2} \rmd t_1 \,
\bdel_\y\cdot \bdel_\x \big[\cK(\x,t_2;\y,t_1) - \cK_0(\x,t_2;\y,t_1)\big]_{\x=\y=0} \, .
\end{align}
Here $α̅=α_sC_R/π$ and $(,t_2;,t_1)$ is an effective emission kernel describing the broadening of the emitted gluon during its formation. It corresponds to the evolution operator of a quantum particle immersed in the imaginary potential $iv()$ in 2+1 dimensions and obeys the Schr\"odinger equation
\begin{equation}
\label{eq:cK_Sch}
\left[i\frac{\partial}{\partial t}+\frac{\bdel^2_\x}{2\omega}+iv(\x,t)\right]\cK(\x,t;\y,t_1)=i\delta^{(2)}(\x-\y)\delta(t-t_1) \,,
\end{equation}
which resums multiple scatterings of the radiated gluon with the medium between the emission times $t_1$ and $t_2$ in the amplitude and its complex conjugate, respectively.
For a general potential $v(,t)$ that includes the Coulomb tail at large momentum transfers, a closed form solution to Eq.~\eqref{eq:cK_Sch} is not known. An analytical solution can nevertheless be obtained for two special choices of the potential: vacuum and harmonic oscillator. In the vacuum case, setting $v(,t)=0$ leads to the following solution of Eq.~\eqref{eq:cK_Sch}
\begin{align}\label{eq:cK_vac}
\cK_0(\Delta\x,\Delta t) = \frac{\omega }{2\pi i \Delta t} \exp\left(i\frac{\omega\Delta\x^2}{2 \Delta t}\right) \, ,
\end{align}
where $Δ= -$ and $Δt = t_2-t_1$. Note that this contribution is explicitly removed in Eq.~\eqref{eq:BDMPS_spec_energy} so that the result is only sensitive to the purely medium-induced contribution. In fact, we can also express the resummed propagator, given by the solution of Eq.~\eqref{eq:cK_Sch}, as a Dyson-like iterative equation that resums multiple interactions around the vacuum solution, namely
\begin{align}
\label{eq:cK-opacity-expansion}
\cK(\x,t_2;\y,t_1) &= \cK_0(\x-\y, t_2-t_1)
- \int_{t_1}^{t_2} \rmd s \int_{\z} \, \cK_0(\x-\z,t_2 -s) v(\z,s) \cK(\z,s; \y,t_1) \,.
\end{align}
From the structure of the equation, we immediately see that, for a time independent rate $v(,t) = v()$, the function $$ only depends on $τ≡t_2-t_1$. This equation is equivalent to an expansion in medium opacity $χ$, defined as $χ= L/ℓ_mfp$. Computing the radiative spectrum by truncating the expansion in \eqn{eq:cK-opacity-expansion} at a fixed order in $v(,t)$, or $χ$, corresponds to the Opacity Expansion introduced in the previous section. Consistently, the $n-$th term in the OE scales as $I^n/ω∼𝒪 ( χ^n )$. The single scattering solution corresponds to the $n=1$ truncation of the expansion and is often referred to as the GLV spectrum~\cite{GLV,Wiedemann}.
The other special case where Eq.~\eqref{eq:cK_Sch} is analytically solvable is when $v(,t) = (,t) = 1/4q̂(t) x_⊥^2$, that is, when the potential reduces to that of an harmonic oscillator. We recall that the IOE splits the leading logarithmic potential given in \eqn{eq:v_LL} as
\begin{equation}\label{eq:v_IOE}
v(\x,t)\equiv \vLO+\delta v =\frac{1}{4} \hat{q}_0(t) x_\perp^2 \log \frac{Q^2}{\mu_\ast^2}+\frac{1}{4} \hat{q}_0(t)x_\perp^2 \log \frac{1}{x_\perp^2Q^2} \, ,
\end{equation}
where $Q$ is for now an undetermined matching scale, different from the one used for the broadening case. Yet, the \textit{effective} jet quenching parameter is $q̂(t) = q̂_0(t) logQ^2/μ_∗^2$. Thus, similarly to transverse momentum broadening discussed in the previous section, the solution to Eq.~\eqref{eq:cK_Sch} with a quadratic potential, that we denote as $ = ^LO$, corresponds to the leading order (LO) term in the Improved Opacity Expansion. It reads
\begin{align}
\label{eq:cK_BDMPS_2}
\cK^{\rm LO}(\x,t_2;\y,t_1)&= \frac{\omega}{2\pi i S(t_2,t_1)}\exp\left( \frac{i\omega}{2S(t_2,t_1)} \left[ C(t_1,t_2)\,\x^2+C(t_2,t_1)\,\y^2-2 \x\cdot\y\right] \right) \,.
\end{align}
Here, $C(t_2,t_1)$ and $S(t_2,t_1)$ are purely time dependent functions which are solutions to the initial condition problems~\cite{Arnold_simple}
\begin{equation}
\label{eq:cs-ho-equations}
\begin{split}
&\left[\frac{\rmd^2}{\rmd^2 t}+\Omega^2(t)\right]S(t,t_0)=0 \, ,\quad S(t_0,t_0)=0 \,,\quad \partial_t S(t,t_0)_{t=t_0}=1 \, , \\
&\left[\frac{\rmd^2}{\rmd^2t}+\Omega^2(t)\right]C(t,t_0)=0 \, ,\quad C(t_0,t_0)=1 \,,\quad \partial_t C(t,t_0)_{t=t_0}=0\, ,
\end{split}
\end{equation}
with the complex harmonic oscillator frequency $Ω(t)$ given by
\begin{equation}
\Omega(t)=\frac{1-i}{2}\sqrt{\frac{\hat{q}(t)}{\omega}} \, .
\end{equation}
More details on the properties of these functions can be found in Appendix~\ref{app:cK_appendix}.
Inserting \eqn{eq:cK_BDMPS_2} back into Eq.~\eqref{eq:BDMPS_spec_energy} and performing the time integrals, one obtains the spectrum at leading order in the IOE (or equivalently in the harmonic approximation). The final expression reads,
\begin{equation}
\label{eq:dIdw_BDMPS}
\omega\frac{\rmd I^{\rm LO}}{\rmd \omega} = 2\Bar{\alpha}\log \big\vert C(0,L) \big\vert\,,
\end{equation}
and is often referred to as the BDMPS-Z spectrum \cite{BDMPS3,BDMPS2}. The LO contribution to the IOE spectrum takes a particularly simple form in the case where the medium has an extension $L$ with a constant density $n$; we refer to this simple medium model as the plasma brick model. In the brick model one can simply define the jet quenching parameter as $q̂(t) =q̂ Θ(L-t) $, which allows one to write the $C$ and $S$ functions as
\beq
\label{eq:S-and-C-functions}
S(t_2,t_1) = \frac{1}{\Omega} \sin \Omega (t_2-t_1) \,, \qquad \text{and} \qquad C(t_2,t_1)= \cos \Omega (t_2-t_1) \,.
\eeq
In this case, the well-known behavior of the spectrum at asymptotically at low and high frequencies is
\beq
\label{eq:dIdw_BDMPS_cases}
\omega \frac{\rmd I^{\rm LO}}{\rmd \omega} \simeq 2\bar \alpha
\begin{dcases}
\,\,\sqrt{\frac{\omega_c}{2\omega}} \quad\qquad\text{for}\quad \omega \ll \omega_c\\
\,\,\frac{1}{12} \left(\frac{\omega_c}{\omega} \right)^2\quad\text{for} \quad \omega \gg \omega_c \,,\\
\end{dcases}
\eeq
where the characteristic gluon energy $ω_c = q̂ L^2/2$ corresponds to gluons with maximal formation time, i.e. $
t_f = L$. The behaviour in the soft limit highlights the Landau-Pomeranchuk-Migdal (LPM) interference~\cite{LPM1,LPM2} that occurs since the gluon is formed over timescales involving multiple interactions with the medium. The strong suppression at high gluon energies follows directly from the approximation of multiple soft interactions, implicit in the harmonic form. At these frequencies, i.e. $ω> ω_c$, the contribution from a single, hard scattering can be shown to dominate, as we will discuss below.
Let us now construct the contributions to the IOE beyond the LO term. Adopting the decomposition provided by \eqn{eq:v_IOE} that allows us to separate the harmonic part from the $$ dependent Coulomb logarithm, and in analogy to the resummation around the vacuum solution given by Eq.~\eqref{eq:cK-opacity-expansion}, the full kernel can be written as
\begin{equation}
\label{eq:cK_ful_IOE}
\cK(\x,t_2;\y,t_1) = \cK^{\rm LO}(\x,t_2;\y,t_1)- \int_{t_1}^{t_2} ds \int_\z \, \cK^{\rm LO}(\x,t_2;\z,s)\, \delta v(\z,s)\cK(\z,s;\y,t_1) \,.
\end{equation}
Truncating this relation at $𝒪(δv^2)$, it is easily seen that the LO kernel is given by $^LO$ in Eq.~\eqref{eq:cK_BDMPS_2}. The NLO kernel reads
\begin{align}
\cK^{\rm NLO}(\x,t_2;\y,t_1) &= - \int_{t_1}^{t_2} \rmd s \int_\z \, \cK^{\rm LO}(\x,t_2;\z,s) \delta v(\z,s) \cK^{\rm LO}(\z,s;\y,t_1) \, ,
\end{align}
which can be used in Eq.~\eqref{eq:BDMPS_spec_energy} to compute the NLO contribution to the IOE spectrum, as was done for the LO term. Like in the broadening case, we do not consider higher order terms since truncating the series at NLO is enough to reproduce the single hard and multiple soft regimes. At this order, the spectrum reads~\cite{IOE1,IOE2,IOE3}
\beq
\label{eq:dIdw_LO_p_NLO}
\omega\frac{\rmd I^{{\rm LO+NLO}}}{\rmd \omega} = 2\bar{\alpha}\log \big\vert C(0,L) \big\vert +\frac{1}{2}\bar{\alpha}\hat{q}_0 \,\Re \int_0^L \rmd s\, \frac{-1}{k^2(s)} \log\frac{-k^2(s)}{Q^2 \rme^{-\gamma_E}} \,,
\eeq
\begin{equation}\label{eq:k_NLO}
k^2(s)= -\frac{i\omega}{2} \left[{\rm Cot}(s,\infty)+{\rm Cot}(0,s)\right] \, .
\end{equation}
and we defined the ratio $Cot(t_2,t_1) ≡C(t_1,t_2)/S(t_2,t_1)$.
We now analyze the asymptotic forms of the spectrum by considering the brick model. In this case \eqn{eq:k_NLO} reduces to
\begin{equation}
\label{eq:k_NLO_brick}
k^2(s) = \frac{i\omega \Omega}{2} \big[ {\cot}\,\Omega s - \tan\Omega(L-s) \big] \, .
\end{equation}
At high frequencies, that is $ω≫ω_c$ or $ΩL≪1$, one finds that Eq.~\eqref{eq:k_NLO_brick} leads to $k^2(s)≃iω/(2s)$. The high-frequency behavior of the NLO term is given by
\begin{equation}
\label{eq:NLO_scaling_high_energy}
\omega \frac{\rmd I^{\rm NLO}}{\rmd \omega}\simeq \Bar{\alpha}\hat{q}_0\frac{\pi}{4}\frac{L^2}{2\omega}
=\frac{ \Bar{\alpha} \pi}{4} \chi\,\frac{\bar{\omega}_c}{\omega} \, .
\end{equation}
It dominates the spectrum, given the quadratic $ω$ suppression of the LO term, see Eq.~\eqref{eq:dIdw_BDMPS}. In this last equation, we recall that the medium opacity parameter is $χ≡q̂_0 L / μ_∗^2 ∼L/ℓ_mfp$ and we introduced the high energy cut frequency $ω_c=1/2μ_∗^2 L$. Higher-order terms are all suppressed by at least one additional power of $1/ω$ as well~\cite{IOE3}. Thus, similar to the discussion for the broadening distribution $()̨$, one observes that the dominant term, given in Eq.~\eqref{eq:NLO_scaling_high_energy}, comes solely from the NLO contribution and it can be shown to exactly match the medium-induced spectrum obtained by considering a single hard scattering in the medium~\cite{IOE1,GLV}, i.e. $n=1$ in the traditional Opacity Expansion. Furthermore, Eq.~\eqref{eq:NLO_scaling_high_energy} is independent of the matching scale, analogous to what was observed for $()̨$ in \eqn{eq:plot_ppp}.
At low frequencies, i.e. for $ω≪ω_c$ or $ΩL ≫1$, the NLO term, containing the single hard scattering physics, can be simplified by noticing that $k^2(s) ≃-ωΩ$, leading to~\cite{IOE1,IOE2,IOE3}
\begin{equation}
\label{eq:NLO_smallw_maineq}
\omega \frac{\rmd I^{\rm NLO}}{\rmd\omega}
\simeq \Bar{\alpha} \lambdaq \sqrt{\frac{\omega_c}{2\omega}}\left[\gamma_E+\log\left(\frac{\sqrt{\omega \hat{q}}}{\sqrt{2}Q^2}\right)+\frac{\pi}{4}\right] \,,
\end{equation}
which is equivalent to the next-to-leading logarithmic result derived in Ref.~\cite{Arnold:2008zu}. Again, as observed for the broadening distribution, in the soft regime higher order terms in the IOE scale as the LO contribution, see \eqn{eq:dIdw_BDMPS_cases}, with increasing power suppression by
$= (logQ^2/μ_∗^2 )^-1 ≪1$.
In fact, one can show that
\begin{equation}
\label{eq:expansion_IOE_small_frequency}
\left.\frac{\rmd I/\rmd \omega}{\rmd I^{\rm LO}/\rmd \omega} \right|_{\omega \ll \omega_c} = 1+\lambdaq \left(a_0+a_1\log\frac{\sqrt{\omega \hat{q}}}{Q^2} \right)
+ \lambdaq^2 \left(b_0 + b_1 \log\frac{\sqrt{\omega \hat{q}}}{Q^2} + b_2\log^2\frac{\sqrt{\omega \hat{q}}}{Q^2}\right) +\ldots \, ,
\end{equation}
where ${a_i, b_i}$ are purely numerical coefficients \cite{IOE3}. This result implies that, in the soft limit, the full spectrum can be written in terms of the LO result with an effective jet quenching coefficient $q̂_eff$ that absorbs the additional logarithmic dependencies.
More importantly, Eq.~\eqref{eq:expansion_IOE_small_frequency} imposes further constraints on the matching scale $Q^2$. To see this, let us first assume that the matching scale associated to the radiation spectrum
$Q^2 = Q_r^2$ is independent of $ω$. Then, in \eqn{eq:expansion_IOE_small_frequency} the logarithms in the denominators would be frozen. However, the numerator logarithms would evolve quite rapidly for $μ_∗^2≪ω^2 ≪(q̂^2Q_r^4)^-1$, leading to a divergent series (notice that the LO contribution would be negligible in this case). Thus, one concludes that $Q_r = Q_r(ω)$ in order for the spectrum to be free of unphysical divergences. In addition, one sees that the natural way to regulate the numerators is to take\footnote{Here we assume that $\hat{q}$ is time independent to simplify the discussion. The generic form for \eqn{eq:Q_r} is discussed in the next section.}
\begin{equation}\label{eq:Q_r}
Q_r^2=\sqrt{\hat{q}_0\omega\log{\frac{Q_r^2}{\mu_\ast^2}}} \, .
\end{equation}
Moreover, it can be shown~\cite{IOE3} that this form follows directly from the fact that once all orders in the IOE are resummed the spectrum takes the functional form of the LO term. For the present paper and the following calculations, the main message is that Eq.~\eqref{eq:Q_r} ensures that at low energies the spectrum is well behaved and non-physical divergences are absent. Also, and again in analogy to the broadening, at leading logarithmic order $Q_r^2 ∼√(q̂_0ω)$, which using the above relations for the gluon formation time and the average accumulated momentum, can be translated into the typical momentum acquired by a gluon with frequency $ω≪ω_c$. The solutions of Eq.~\eqref{eq:Q_r} are discussed in Appendix~\ref{app:Q}.
Finally, we still need to ensure that $Q_r^2≫μ_∗^2$ in order to justify the expansion. Ignoring the logarithmic dependence in the matching scale, we observe that the IOE approach only works if
\begin{equation}
\omega_{\rm BH} \ll \omega \,,
\end{equation}
where we defined the characteristic Bethe-Heitler (BH) frequency as $ω_BH = μ_∗^4/q̂_0$.
This condition means that the current scheme is not valid in the BH regime~\cite{Bethe:1953va}, see Ref.~\cite{Andres:2020kfg} for a similar conclusion and further discussion regarding the analytic treatment of the BH region. This regime is characterized by gluons with a formation time of the order of the mean free path in the medium, acquiring a momentum $k_⊥^2∼q̂_0 ℓ_fmp ∼μ_∗^2$ and with a typical energy $ω_BH ∼T$ of the order of the medium temperature. At this scale, non-linear dissipation effects take place \cite{Baier:2000sb} such as gluon absorption. However, in the case of large or dense enough media (such that $Q^2 ≫m_D^2$) the BH regime is power suppressed and radiative energy loss is dominated by frequencies in the deep LPM regime in the calculation of inclusive jet observables \cite{Baier:2001yt}.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.8]{plot-integrated-spectrum.pdf}
\caption{Comparison between the energy spectrum computed with GLV (dotted, purple), the IOE at LO (dashed, green), at LO+NLO (solid, red) and the all-order spectrum (solid, navy) as computed in~\cite{CarlotaFabioLiliana}. The ratio to the full solution is presented in the bottom panels. The uncertainty band arises from variations in the matching scale and the gray region indicates the regime in which Eq.~\eqref{eq:Q_r} does not have a solution. The parameters used are identical to those of Fig.~\ref{fig:broad} and $\omega_{c0}\!\equiv\!\hat q_0 L^2$.}
\label{fig:spectrum}
\end{figure}
In Fig.~\ref{fig:spectrum}, we compute the medium-induced single gluon spectrum up to NLO in the IOE, comparing with a full numerical solution to Eq.~\eqref{eq:BDMPS_spec_energy}~\cite{CarlotaFabioLiliana} and the GLV spectrum, corresponding to the limit of single scattering in the medium. The gray band indicates the region in which Eq.~\eqref{eq:Q_r} does not have a valid solution, i.e. where the IOE approach is not valid. A similar numerical comparison was previously carried out in Ref.~\cite{Andres:2020kfg}. As discussed above, we numerically observe that in the soft sector, $ω_BH ≪ω≪ω_c$, the difference between the full result and the LO contribution is small, and including the NLO provides a very good approximation. In addition, the IOE has no divergences since the matching scale is chosen for each $ω$ by solving Eq.~\eqref{eq:Q_r}. At frequencies $ω≫ω_c$, we observe that the LO is power suppressed, but the NLO term matches the full result, even faster than the GLV approximation. Overall, the agreement between the LO+NLO result and the full numerical solution is outstanding.
\section{The medium-induced radiative kernel with the IOE}\label{sec:IOE_radiation_spectrum}
After having revised the building blocks of the IOE, we proceed to compute the fully differential medium-induced spectrum for a gluon with energy $ω$ and transverse momentum $$̨. We assume that the emitted gluon is soft, $\omega \ll E$, and collinear, $\theta^2\sim k_\perp^2/\omega^2\ll 1$, with $E$ being the energy of the emitter. The emitter follows an eikonal trajectory and its kinematics are frozen. Regarding the medium properties, which are encapsulated by the jet quenching parameter $\hat{q}$, we assume that it has a smooth time profile almost everywhere and that, at large distances, the system reaches the vacuum sufficiently fast, i.e. $\lim\limits_{t\to \infty}\hat{q}(t)= 0$. Then, we study a particular scenario where the medium has a simple time dependence: up to a distance $L$ the jet quenching parameter is positive and constant, while for times larger than $L$, $\hat{q}\!=\!0$. This corresponds to the previously mentioned plasma brick model, where the medium is a slab with longitudinal size $L$, after which there is vacuum; mathematically it corresponds to defining the jet quenching parameter as $\hat{q}(t)
\!=\! \hat{q}\,\Theta(L-t) $.
Under these assumptions, the purely medium-induced spectrum can be expressed as a convolution between the broadening probability distribution, $ \cP$, and the splitting kernel, $\cK$, that we have introduced in the previous section. It reads,
\begin{align}\label{eq:spectrum}
(2\pi)^2\omega\frac{\rmd I}{\rmd \omega \rmd^2 \k}&=\lim_{\epsilon\to 0}\frac{2\bar{\alpha}\pi}{\omega^2} \Re \int_0^\infty \rmd t_2 \, \rme^{- \epsilon (t_2+t_1)} \int_0^{t_2} \rmd t_1 \int_\x \, \rme^{-i \k \cdot \x}\, \cP(\x,\infty;t_2) \nn
& \bdel_\x\cdot \bdel_\y \cK(\x,t_2;\y,t_1)_{\y=0} - (2\pi)^2\omega\frac{ \rmd I^{\rm vac}}{\rmd\omega \rmd^2\k} \, ,
\end{align}
where $t_1$ and $t_2$ correspond to the gluon splitting light-cone times in amplitude and conjugate amplitude respectively, and span from the creation point inside the medium at $t_1\!=\!t_2\!=\!0$ up to any possible in-vacuum or in-medium splitting time. In eq:spectrum, we explicitly denote the starting time $t_2$ in the broadening distribution so, compared to our formulas in Section <ref>, $\cP(\x,L) \equiv \cP(\x,L;0)$.
Also, in eq:spectrum we employ the adiabatic turn-off prescription <cit.>, which prevents the emission of purely vacuum-like radiation at asymptotically large times, with the $\epsilon\to 0$ limit being implicit for the rest of the paper. The last term in the formula subtracts a contribution corresponding to purely vacuum radiation off the hard emitter given by (see Appendix <ref>)
(2π)^2ωI^vac/ ω^2 = 4α̅π/k_⊥^2.
Before proceeding further with the explicit analytic evaluation of eq:spectrum, we anticipate a subtlety when carrying out the time integrals with the adiabatic turn-off prescription. Ignoring the $\rme^{-\epsilon t_1}$ suppression factor, the $t_1$ integral can be performed using eq:id_1. Keeping the prescription yields only an additional negative vacuum-like term, $-(2\pi)^2\omega\frac{ \rmd I^{\rm vac}}{\rmd\omega \rmd^2\k}$, so that eq:spectrum can be expressed in a more convenient form as follows
\begin{align}\label{eq:spectrum-2}
(2\pi)^2\omega\frac{\rmd I}{\rmd \omega \rmd^2 \k}&=\frac{2\bar{\alpha}\pi}{\omega^2} \Re \int_0^\infty \rmd t_2 \, \rme^{- \epsilon t_2} \int_0^{t_2} \rmd t_1 \int_\x \, \rme^{-i \k \cdot \x}\, \cP(\x,\infty;t_2) \nn
& \bdel_\x\cdot \bdel_\y \cK(\x,t_2;\y,t_1)_{\y=0} - \frac{8\bar{\alpha}\pi}{k_\perp^2} \,,
\end{align}
where now the $\epsilon$ for the $t_1$ integral prescription has been removed at the cost of a factor 2 multiplying the vacuum term. The limit $\epsilon \to 0$ has to be taken after the integral over $t_2$.
The details regarding the treatment of the adiabatic prescription are discussed in Appendix <ref>.
In what follows, we will compute Eq. (<ref>) in the IOE approach, including all terms up to $\mathcal{O}(\delta v)$ (NLO). For that, we extend eq:rate-position for a generic medium and express the broadening distribution $\cP$ as
(,t;t_0) = ^-∫_t_0^t ds (,s) ^-∫_t_0^t ds δv(,s) =^LO(,t;t_0) ^-∫_t_0^t ds δv(,s) .
Similarly, the emission kernel $\cK$ can be expanded as in eq:cK_ful_IOE:
\begin{align}
\label{eq:k-expansion}
\cK(\x,t_2;\y,t_1) = \cK^{\rm LO}(\x,t_2;\y,t_1)-\int_\z \int_{t_1}^{t_2} \dd s \, \cK^{\rm LO}(\x,t_2;\z,s)\delta v(\z,s)\cK(\z,s;\y,t_1) \,.
\end{align}
Truncating these relations up to NLO accuracy and inserting them into Eq. (<ref>), we obtain the spectrum which we write in the following way,
I/ ω^2 = I^LO/ ω^2 + I^NLO/ ω^2 + 𝒪(δv^2) .
To reiterate, the LO and NLO terms resum arbitrary number of soft medium interactions, encoded in $\vLO$, and a fixed number (zero for LO, and one for NLO) number of hard interactions with the medium, through the potential $\delta v$.
While the vacuum spectrum is already given in Eq (<ref>), the medium read as follows,
\begin{align}
\label{eq:spectrum-ioe-lo}
(2\pi)^2\omega\frac{\rmd I^{\rm LO}}{ \rmd \omega \rmd^2 \k}& = \frac{2\bar{\alpha}\pi}{\omega^2} \Re \int_0^\infty \rmd t_2\, \rme^{-\epsilon t_2} \int_0^{t_2} \rmd t_1 \int_\x\, \rme^{-i \k \cdot \x} \nn
& \times \cP^{\rm LO}(\x,\infty;t_2) \bdel_\x\cdot \bdel_\y \cK^{\rm {LO}}(\x,t_2;\y,t_1)_{\y=0} - \frac{8 \bar \alpha \pi}{k_\perp^2} \,, \\
\label{eq:spectrum-ioe-nlo}
(2\pi)^2\omega\frac{\rmd I^{\rm NLO}}{ \rmd \omega \rmd^2 \k}& = \frac{2\bar{\alpha}\pi}{\omega^2} \Re \int_0^\infty \rmd t_2\, \rme^{-\epsilon t_2} \int_0^{t_2} \rmd t_1 \int_\x\, \rme^{-i \k \cdot \x} \nn
&\times \Big[ \cP^{\rm LO}(\x,\infty;t_2) \bdel_\x\cdot \bdel_\y \cK^{\rm {NLO}}(\x,t_2;\y,t_1)_{\y=0} \nn
&+ \cP^{\rm NLO}(\x,\infty;t_2) \bdel_\x\cdot \bdel_\y \cK^{\rm {LO}}(\x,t_2;\y,t_1)_{\y=0} \Big]\,,
\end{align}
\begin{align}\label{eq:mmmm_1}
\cP^{\rm NLO}(\x, \infty;t) &= - \cP^{\rm LO}(\x,\infty;t) \int_t^\infty \rmd s \, \delta v(\x,s) \, ,
\end{align}
\begin{align}\label{eq:mmmm_2}
\cK^{\rm NLO}(\x,t_2;\y,t_1) &= - \int_\z \int_{t_1}^{t_2} \rmd s\, \cK^{\rm LO}(\x,t_2;\z,s) \delta v(\z,s) \cK^{\rm LO}(\z,s;\y,t_1) \,.
\end{align}
The LO term captures the physics associated with the production of gluon radiation due to multiple soft scattering in the medium, thus recovering the BDMPS-Z solution. The first term in the NLO term eq:spectrum-ioe-nlo includes the possibility of producing the gluon due to a hard scattering in the medium and when integrated over $\k$ gives the NLO contribution to the integrated spectrum studied in the previous section, Eq.~\eqref{eq:dIdw_LO_p_NLO} \cite{IOE1}. Finally, the last term in \eqn{eq:spectrum-ioe-nlo} arises from expanding the final state broadening distribution $$. Thus, it only affects the redistribution of the radiated gluon transverse momentum and it vanishes upon integration over $$̨.
In the following sections, we proceed to explicitly compute Eqs. (<ref>) and (<ref>).
§.§ Leading order contribution
The leading order contribution to the spectrum is captured by Eq. (<ref>). The broadening distribution $\cP^{\rm LO}$, implicitly given in eq:p-expansion, reads
\begin{align} \label{eq:P_LO_x}
\cP^{\rm LO}(\x,t;t_0) = \exp\left[ -\frac{1}{4}Q_{s0}^2(t,t_0) \log \frac{Q_b^2}{\mu_\ast^2}\, x_\perp^2 \right]\, ,
\end{align}
where we define the bare saturation scale as a slight generalization of eq:old_Qs0, reading
\begin{align}
Q_{s0}^2(t,t_0) = \int_{t_0}^{t} \rmd s\, \hat q_0(s) \, ,
\end{align}
and the matching scale, $Q_b$, satisfies (following eq:old_Qb)
\begin{align}\label{eq:Qb_new}
Q_b^2\equiv \int_0^{ \infty} \rmd t\, \hat q_0(t) \,\log \frac{Q_b^2}{\mu_\ast^2}\, .
\end{align}
Furthermore, the kernel $\cK^{ \rm LO}$ can be found in Eq. (<ref>) but, for consistency, we also repeat it here in a slightly different form, namely
\begin{align}\label{eq:cK_BDMPS_3}
\cK^{\rm LO}(\x,t_2;\y,t_1) = \frac{\omega}{2\pi i S(t_2,t_1)} \exp \left[i\frac{\omega}{2} \left( {\rm Cot}(t_2,t_1) \, \x^2 - {\rm Cot}(t_1,t_2) \, \y^2 - \frac{2}{S(t_2,t_1) }\, \x\cdot \y \right) \right] \,,
\end{align}
where we recall that (see Appendix <ref> for further details)
Cot(t_2,t_1) = C(t_1,t_2)/S(t_2,t_1) .
Also, in eq:cK_BDMPS_3 we have taken advantage of the anti-symmetry of $S$, i.e. $S(t_2,t_1) = - S(t_1,t_2)$.
In all these functions, the value of $\hat q$ enters as an argument and thus they are sensitive to the definition of the matching scale for radiation, $Q_r$, that, as discussed in Section <ref>, is obtained by solving the transcendental equation (see eq:Q_r)
Q^2_r(t)=√(q̂(t) ω)=√(q̂_0(t) ωlogQ^2_r(t)/μ_∗^2).
At this point, an important remark is in order. The functional form of the matching scale is constrained by making the spectrum finite in the infrared. This leads to the $\omega$-dependence in Eq. (<ref>), which is constrained in this way up to an overall numerical coefficient. As was shown in Ref. <cit.>, the dependence in such a factor is sub-leading for a fixed order calculation in the IOE. As such, and since all the time dependence of $Q_r^2$ emerges from $\hat{q}_0$, it is more convenient to define $Q_r^2$ as a static scale. This simplifies the time integrations needed for the spectrum calculation without downgrading its accuracy.
In order to further simplify Eq. (<ref>), we make use of a series of identities satisfied by the functions $C(t_2,t_1)$ and $S(t_2,t_1)$ that enter in the definition of $\cK^{ \rm LO}$, see eq:cs-ho-equations. In particular, we use Eq. (<ref>) to obtain
\begin{align}\label{eq:id_1}
\int_0^{t_2} \rmd t_1 \, \partial_\y \cK^{\rm LO}(\x,t_2;\y,t_1)_{\y=\0} &= -\frac{\omega^2}{2\pi}\int_0^{t_2} \rmd t_1\, \frac{\x}{S^2(t_2,t_1)}\rme^{\frac{i\omega}{2} {\rm Cot}(t_2,t_1) \x^2} \nn
&= \frac{\omega}{\pi i} \frac{\x}{\x^2} \rme^{\frac{i\omega }{2} {\rm Cot}( t_2,0) \x^2} \, ,
\end{align}
where in the second step we dropped an infinite phase which has already been accounted for in the vacuum subtraction term in eq:spectrum-2. A careful treatment of this technical point is presented in Appendix <ref>, see eq:t1-int and related discussion.
The identity eq:id_1 together with Eq. (<ref>) lead to the following expression for the leading-order spectrum
\begin{align}
\label{eq:HO_any_med_1a}
(2\pi)^2\omega\frac{\rmd I^{\rm LO}}{\rmd \omega \rmd^2 \k} &=2\bar{\alpha} \Re \int_0^\infty \rmd t_2 \, \rme^{-\epsilon t_2}\, {\rm Cot}(t_2,0)\int_\x \rme^{-i \k \cdot \x} \, \cP^{\rm LO}(\x,\infty;t_2) \, \rme^{\frac{i\omega }{2} {\rm Cot}(t_2,0)\, \x^2} \nn
&-\frac{8\pi \abar}{k_\perp^2} \, ,
\end{align}
where we used that $\partial_\x \cdot \frac{\x}{\x^2}=0$.
The result obtained in Eq. (<ref>), although compact, is somewhat obscure from a physical perspective. A more intuitive description of the in-medium emission can be achieved by using a momentum space representation
\begin{align}\label{eq:HO_any_med_2a}
(2\pi)^2\omega\frac{\rmd I^{\rm LO}}{\rmd \omega \rmd^2 \k}
&=\frac{4\bar{\alpha}\pi}{\omega} \Re\, i\int_0^\infty \rmd t_2 \, \rme^{-\epsilon t_2}\,
\int_{\p} \, \cP^{\rm LO}(\k-\p,\infty;t_2) \rme^{-i\frac{\p^2 }{2 \omega {\rm Cot}(t_2,0)} }-\frac{8\pi \abar}{k_\perp^2} \, .
\end{align}
In this form, we can interpret the first term in Eq. (<ref>) as describing the emission of a gluon via some effective kernel at time $t_2$ followed by final state broadening. The second term corresponds to a vacuum-like subtraction contribution.
Furthermore, since $\cP^{\rm LO}(\p)$ is Gaussian, the remaining momentum integral can be performed
∫_ ^LO(-̨,∞;t) ^- i^2 /2ωCot(t,0) = -2iωCot(t,0) ^-^̨2/^2(t,0) /^2(t,0) ,
where we introduced the function
^2(t_2,t_1) = Q^2_s(∞,t_2) -2iωCot(t_2,t_1) ,
and the effective saturation scale is defined as $Q_s^2(t_2,t_1) = \int_{t_1}^{t_2} \rmd t \, \hat{q}_0(t) \log \frac{Q_b^2}{\mu_\ast^2}$, with the logarithmic dependence in $\hat{q}$ determined by $Q_b$. Inserting this result into Eq. (<ref>), we finally obtain
\begin{align}\label{eq:HO_any_med_3}
(2\pi)^2\omega\frac{\rmd I^{\rm LO}}{\rmd \omega \rmd^2 \k}=8\Bar{\alpha} \pi \, \Re \int_0^\infty \rmd t \, \rme^{-\epsilon t}\, \frac{{\rm Cot}(t,0)}{\khat^2(t,0)} \rme^{-\frac{k_\perp^2}{\khat^2(t,0)}} -\frac{8\pi \abar}{k_\perp^2} \,.
\end{align}
This form of the spectrum was first derived in Ref. <cit.>. Integrating over $\k$ and using $∫_()̨=1$, we recover the LO contribution to the energy spectrum discussed in the previous section \cite{IOE1,Arnold_simple}, see Eq.~\eqref{eq:dIdw_BDMPS}.
As a sanity check, one can verify that \eqn{eq:HO_any_med_3} vanishes in the vacuum, i.e. in the limit $Ω→0$, so that
\beq
{\rm Cot}(t,0) \to \frac{1}{t} \, \quad \text{and} \quad \khat^2(t_2,t_1) \to -\frac{2i\omega}{ t_2-t_1}\,.
\eeq
Thus, we obtain
\begin{align}\label{eq:HO_any_vac_3}
(2\pi)^2\omega\frac{\rmd I^{\rm LO}}{\rmd \omega \rmd^2 \k}=8\Bar{\alpha} \pi \, \Re \int_0^\infty \rmd t \, \rme^{-\epsilon t} \,\frac{1}{2i \omega } \rme^{-i \frac{k_\perp^2}{2\omega}t} -\frac{8\pi \abar}{k_\perp^2}=0 \,.
\end{align}
\paragraph*{Plasma brick model.}
We proceed to evaluate the previous expressions for a concrete medium model, namely the brick where $q̂(t)= q̂ Θ(L-t)$. Inside the medium, i.e. when both $t_1<L$ and $t_2<L$, the $C$ and $S$ functions take simple forms (see Appendix~\ref{app:cK_appendix})
\begin{align}\label{eq:SC-brick}
S(t_2,t_1)=\frac{\sin(\Omega(t_2-t_1))}{\Omega} \, , \quad C(t_2,t_1)=\cos(\Omega(t_2-t_1)) \, ,
\end{align}
and $Cot(t_2,t_1) = Ω(Ω(t_2-t_1))$, where $Ω=1-i/2√(q̂_0/ ω logQ_r^2/μ_∗^2)$.
On the other hand, for both $t_1>L$ and $t_2 >L$, the system evolves as in vacuum and the $C$ and $S$ are obtained by setting $Ω→0$ in the previous equations such that
\begin{align}\label{eq:SC-vacuum}
S(t_2,t_1) = t_2-t_1 \, , \quad C(t_2,t_1)=1 \,.
\end{align}
This sharp separation of the problem into processes happening inside the medium and outside of it, suggests that an efficient way to evaluate Eq.~\eqref{eq:HO_any_med_3} consists in splitting the time integral into two regions: one where $0<t<L$ and a vacuum-like region where $t>L$.
We refer to the first region as \textbf{in-in} since the gluon emission occurs inside the medium in both the amplitude and its conjugate. It can be easily obtained by replacing the upper limit in the integral in Eq.~\eqref{eq:HO_any_med_3} by $L$ and employing Eq.~\eqref{eq:SC-brick}. Since the integral over $t_2$ has a finite extension, we can safely neglect the adiabatic suppression factor. This contribution to the total medium-induced LO spectrum reads
\begin{align}\label{eq:HO_brick_InIn}
(2\pi)^2\omega\frac{\rmd I^{\rm LO}_{\text{in-in}}}{\rmd \omega \rmd^2 \k}&=8\Bar{\alpha} \pi \, \Re \int_0^L \rmd t \, \Omega {\rm cot}(\Omega t) \frac{ \rme^{-\frac{k_\perp^2}{\hat{q}(L-t) -2i \omega \Omega \cot(\Omega t)} }}{\hat{q}(L-t) - 2i \omega \Omega {\rm cot}(\Omega t)}\, ,
\end{align}
where we have used that the saturation scale reduces to $Q_s^2(∞,t)=q̂(L-t)$.
The remaining region of phase space is obtained by imposing $t>L$ in Eq.~\eqref{eq:HO_any_med_3}. In this case, there are two types of contributions: (i) a purely vacuum term corresponding to the scenario where the gluon is outside the medium both in the amplitude and its conjugate, in this situation the first and second terms in Eq.~\eqref{eq:HO_any_med_3} cancel by construction, and (ii) an interference term where the amplitude gluon is emitted inside the medium while its conjugate counterpart is emitted in the vacuum (or vice-versa). The latter contribution, which we shall refer to as \textbf{in-out}, requires further manipulations.
We begin by constructing the $C$ and $S$ functions which have support inside and outside the medium. This is done by using the decomposition for the $C$ and $S$, given in Eq.~\eqref{eq:linear_rel_C_S}~\cite{Arnold_simple}
\begin{equation}
\begin{split}
&S(t_2,t_1)=C(t_1,t_0)S(t_2,t_0)-S(t_1,t_0)C(t_2,t_0) \, ,\\
& C(t_2,t_1)=-\partial_{t_1}C(t_1,t_0)S(t_2,t_0)+\partial_{t_1}S(t_1,t_0)C(t_2,t_0) \, .
\end{split}
\end{equation}
Taking $t_1=0$, $t_2=t>L$, $t_0=L$ and using the appropriate form for the $C$ and $S$ in each region (see Eqs.~\eqref{eq:SC-brick} and \eqref{eq:SC-vacuum}), we obtain
\begin{equation}
\begin{split}
&S(t,0)=(t-L) \cos \Omega L + \frac{\sin \Omega L}{\Omega }\, ,\\
&C(0,t)=\cos(\Omega L)\, ,
\end{split}
\end{equation}
which yields
\beq
{\rm Cot} (t,0)= \frac{ \Omega \cot\Omega L}{\Omega (t-L) \cot \Omega L +1 }\, .
\eeq
In addition, in Eq.~\eqref{eq:SC-brick} the broadening term (encapsulated in $Q_s^2$) has support in $(t,L)$ which is the vacuum region. Then, there is no final state broadening and one can take
\beq
\cP^{\rm LO}(\k-\p,\infty;t) \big|_{t>L}= (2\pi)^2 \delta^{(2)}(\k) \, ,
\eeq
or, equivalently, $Q_s^2=0$ in Eq.~\eqref{eq:SC-brick}. Combining all these results, we find that
\begin{align}\label{eq:HO_brick_out_2}
(2\pi)^2\omega\frac{\rmd I_{\text{in-out}}^{\rm LO}}{\rmd \omega \rmd^2 \k}
&=\frac{4\bar{\alpha}\pi}{\omega} \Re \,i \int_L^\infty \rmd t
\, \rme^{-i\frac{k_\perp^2 }{2\omega }(t-L)- i \frac{k_\perp^2}{2\omega \Omega \, \cot\Omega L } } -\frac{8\pi \abar}{k_\perp^2} \, \nn
&= \frac{8\bar{\alpha}\pi }{k_\perp^2}\Re\left(\rme^{-i\frac{k_\perp^2}{2\omega \Omega \, \cot\Omega L } }- 1\right) \, ,
\end{align}
where we have included the vacuum subtraction term, and where the implicit adiabatic prescription $∼^-ϵt$ in the first line allowed to drop the contribution from $t→∞$.
This contribution, together with Eq.~\eqref{eq:HO_brick_InIn}, constitute the medium-induced leading order spectrum, analogous to the BDMPS-Z result. The medium-induced spectrum at LO in the IOE reads then,
\beq
\frac{\rmd I^{\rm LO}}{\rmd \omega \rmd^2 \k} = \frac{\rmd I^{\rm LO}_{\text{in-in}}}{\rmd \omega \rmd^2 \k} + \frac{\rmd I^{\rm LO}_{\text{in-out}}}{\rmd \omega \rmd^2 \k} \,.
\eeq
\subsection{Next-to-leading order contribution}\label{sec:nlo}
The computation of the next-to-leading order contribution to the spectrum can be done using similar manipulations to the ones performed for the LO term in the previous section. The NLO spectrum is defined in Eq.~\eqref{eq:spectrum-ioe-nlo}. The first term corresponds to a genuine correction to the emission kernel, referred below to as the \textbf{in} contribution, while the second term, which we shall refer to as \textbf{broad} contribution, introduces the possibility of a hard scattering in the final state broadening process. Also, the vacuum-like subtraction terms that appeared in the LO contribution are absent at $𝒪(δv)$.
To summarize, the two contributions to the NLO spectrum read
\begin{align}
\label{eq:spectrum-ioe-nlo-in}
(2\pi)^2\omega\frac{\rmd I^{\rm NLO}_{\rm in}}{\rmd \omega \rmd^2 \k} &= -\frac{2\bar{\alpha}\pi}{\omega^2} \Re \int_0^\infty \rmd t_2 \int_0^{t_2} \rmd t_1 \int_0^{t_1} \rmd s \int_{\x,\z}\, \rme^{-i \k \cdot \x} \, \cP^{\rm LO}(\x,\infty ;t_2) \nn
& \times \delta v(\z,t_1) \bdel_\x \cK^{\rm {LO}}(\x,t_2;\z,t_1) \cdot \bdel_\y \cK^{\rm {LO}}(\z,t_1;\y,s) \,,\\
\label{eq:spectrum-ioe-nlo-broad}
(2\pi)^2\omega\frac{\rmd I^{\rm NLO}_{\rm broad}}{\rmd \omega \rmd^2 \k} &= -\frac{2\bar{\alpha}\pi}{\omega^2} \Re \int_0^\infty \rmd s \int_0^s \rmd t_2 \int_0^{t_2} \rmd t_1 \int_\x \rme^{-i \k \cdot \x} \, \cP^{\rm LO}(\x,\infty,t_2) \nn
& \times \delta v(\x,s) \bdel_\y \cdot \bdel_\x \cK^{\rm LO}(\x,t_2;\y,t_1) \bigg)_{\y=0} \,,
\end{align}
where we have rearranged the time integrations. \footnote{Note that in \eqn{eq:spectrum-ioe-nlo-in} $t_1$ is the time at which the hard interaction occurs, while we have labelled this time as $s$ in other equations.}
Let us begin by considering Eq.~\eqref{eq:spectrum-ioe-nlo-in}. Note, that both the $t_1$ and $s$ integrals have support only inside the medium, hence the naming as the \textbf{in} contribution. By using Eq.~\eqref{eq:id_1}, we can directly perform the $s$-integral such that
\begin{align}\label{eq:IOE_med_2}
(2\pi)^2\omega\frac{ \rmd I_{\text{in}}^{\rm NLO}}{\rmd\omega \rmd^2 \k}&=\frac{2\bar{\alpha}}{\omega} \,\Re\, i \int_0^\infty \rmd t_2 \int_\x \,\rme^{-i \k \cdot \x} \, \cP^{\rm LO}(\x,\infty; t_2)
\nn
&\times \int_0^{t_2} \rmd t_1 \int_\z \,\delta v(\z,t_1) \frac{\z}{\z^2}\cdot \partial_\x \cK^{\rm LO}(\x,t_2;\z,t_1) \rme^{\frac{i\omega}{2}{\rm Cot}(t_1,0)\z^2}
\, .
\end{align}
The remaining derivative operator gives
\begin{align}
\partial_\x \cK^{\rm LO}(\x,t_2;\z,t_1) &= \frac{\omega^2}{2\pi S^2(t_2,t_1)} \big(\x C(t_1,t_2)-\z \big) \nn
&\times \exp\left[\frac{i\omega}{2S(t_2,t_1)} \big( C(t_1,t_2) \x^2 + C(t_2,t_1)\z^2 - 2 \x\cdot\z \big)\right] \, ,
\end{align}
so that the spectrum further simplifies to (adopting hereafter the more compact notation $Cot_21 ≡Cot(t_2,t_1)$, $C_12≡C(t_1,t_2)$ and $S_12≡S(t_2,t_1)$)
\begin{align}
\label{eq:IOE_med_3}
(2\pi)^2\omega\frac{\rmd I_{\text{in}}^{\rm NLO}}{\rmd \omega \rmd^2 \k} & =\frac{\bar{\alpha}\omega}{\pi}\Re\, i \int_0^\infty \rmd t_2\int_0^{t_2} \rmd t_1 \int_{\x,\z}\, \rme^{-i \k \cdot \x}\rme^{-\frac{1}{4}Q_s^2(\infty,t_2) \x^2} \nn
&\times \frac{1}{S^2_{21}} \delta v(\z,t_1) \frac{\z}{\z^2}\cdot (\x C_{12}-\z)
\nn
&\times \rme^{\frac{i\omega}{2S_{21}} \left(C_{12}\x^2+C_{21}\z^2-2\x\cdot\z\right)} \rme^{\frac{i\omega}{2}{\rm Cot}_{10} \z^2} \, .
\end{align}
The remaining integration in $$ is Gaussian, and can be executed to obtain
\begin{align}
\int_\x \rme^{-i \k \cdot \x}\rme^{-\frac{1}{4}Q_s^2(\infty,t_2) \x^2} & \rme^{\frac{i\omega}{2S_{21}} \left(C_{12}\x^2 - 2\x\cdot\z \right)}(\x C_{12}-\z)
\nn
=&-\frac{4\pi}{\khat^2_{21}} \rme^{-\frac{\left(\k-\frac{\omega }{S_{12}} \z\right)^2}{\khat^2_{21}}}\left[\z+\frac{2iC_{12}}{\khat^2_{21}}\left(\k-\frac{\omega}{S_{12}} \z\right)\right] \, .
\end{align}
Let us re-emphasize that, in the previous expression, the matching scale associated with $Q_s$ in $_21$ is $Q_b$. Replacing $δv$ by its explicit definition, that depends on $Q_r$, the $in$ contribution reads
\begin{align}
\label{eq:IOE_med_4}
&(2\pi)^2\omega\frac{\rmd I_\text{in}^{\rm NLO}}{\rmd \omega \rmd^2 \k} =-\bar{\alpha}\omega \Re\, i\int_0^\infty \rmd t_2 \int_0^{t_2} \rmd t_1 \, \hat{q}_0(t_1) \frac{\rme^{-\frac{k_\perp^2}{\khat^2_{21}}}}{S^2_{21}\khat^4_{21}} \nn
&\times \int_\z \, \rme^{\frac{i\omega}{2}(-{\rm Cot}_{12}+{\rm Cot}_{10}+\frac{2i\omega}{S^2_{21}\khat^2_{21}}) \z^2} \rme^{-\frac{2\omega}{\khat^2_{21}S_{21}} \k\cdot \z}
\log \frac{1}{Q_r^2 \z^2} \left(Q^2_s(\infty,t_2) \z^2+2iC_{12}\k\cdot \z\right) \, .
\end{align}
Notice that we tend to write the largest time as the first argument in the functions. Nonetheless, this is not always possible since in general the $C$ function has no definite parity under the exchange of the arguments, unlike the $S$ function which is always odd.
Comparing Eq.~\eqref{eq:IOE_med_4} to the LO contribution given by Eq.~\eqref{eq:HO_any_med_1a}, we observe an additional transverse integral in the $$ variable which is no longer Gaussian due to the logarithmic dependence in $δv$. Nevertheless, this integration can be performed analytically too. The angular part can be performed by recalling the definitions of the Bessel functions of the first kind,
\begin{align}
\int_0^{2\pi} \frac{\rmd \theta}{2\pi } \rme^{-i z \cos\theta} = J_0(z) \, , \quad \int_0^{2\pi} \frac{\rmd \theta}{2\pi } \cos\theta \, \rme^{-i z \cos\theta} = -i J_1(z) \,,
\end{align}
that lead to
\begin{align}
\label{eq:nlo-in}
&(2\pi)^2\omega\frac{\rmd I_\text{in}^{\rm NLO}}{\rmd \omega \rmd^2 \k}=\frac{\bar{\alpha}\pi}{2\omega k_\perp^4} \Re \, i\int_0^\infty \rmd t_2 \,\int_0^{t_2} \rmd t_1 \, \frac{\hat{q}_0(t_1)}{\rhat_{21}^2} \rme^{-\frac{k_\perp^2}{\khat^2_{21}}}
\nn
&\times \int_0^\infty \rmd z_\perp \, z_\perp \, \rme^{-\jhat_{21}\frac{z_\perp^2}{4 k_\perp^2} } \log \left(\frac{k_\perp^2\rhat^2_{21}}{Q_r^2z_\perp^2}\right) \left[Q^2_s(\infty,t_2) z_\perp^2J_0(z_\perp)+2 C_{12}\rhat_{21} k_\perp^2 z_\perp J_1( z_\perp) \right] \, ,
\end{align}
where we have introduced the auxiliary functions
\begin{align}
&\jhat(t_2,t_1)\equiv \jhat_{21}=\frac{i}{2\omega}\left(-{\rm Cot}_{12}+{\rm Cot}_{10}+\frac{2i\omega}{S^2_{21}\khat^2_{21}} \right)S_{21}^2 \khat_{21}^4\, , \nn
&\rhat(t_2,t_1)\equiv\rhat_{21} =-\frac{2i\omega}{\khat^2_{21}S_{21}} \, .
\end{align}
The more challenging radial integral can be solved by using a convenient decomposition of the logarithmic function. Namely, the relation
\begin{equation}\label{eq:Moliere_log}
\log\frac{1}{u^2}=\lim_{\epsilon\to0} \int_\epsilon^\infty \frac{\rmd t}{t} \, \left(\rme^{-u^2t}-\rme^{-t}\right) \, ,
\end{equation}
allows us to transform the original integral into a sum of Gaussian integrations that can be readily performed. In particular, the $z$-integrals in Eq.~\eqref{eq:nlo-in} can be compactly expressed as
\beq
I_a(x,y ) = \int_0^\infty \rmd z J_0(z)\, z^3 \,\log\frac{y}{z^2} \, \rme^{ - \frac{z^2}{4x}}\, ,
\eeq
\begin{align}
I_b(x,y )=\int_0^\infty \rmd z \, z^2 \log \frac{y}{z^2} J_1\left(z \right)\rme^{-\frac{z^2}{4x}} \, .
\end{align}
Then, replacing the logarithms in the previous equations by the decomposition in Eq.~\eqref{eq:Moliere_log} allows one to write $I_a$ and $I_b$ in terms of the exponential integral function $Ei$, leading to
\beq\label{eq:bbS_alpha}
I_a\left(x,y \right)
&=&8 x^2\, \rme^{-x}(-2+\rme^{x}) + 8 x^2 \, \rme^{-x}(1-x)\left[{\rm Ei}\left(x\right)-\log\frac{4x^2 }{y}\right]\, ,
\eeq
\begin{align}\label{eq:bbS_beta}
&I_b\left(x,y\right)=-4x\left(1-\rme^{-x}\right)+4x^2\rme^{-x}\left[{\rm Ei}(x)-\log \frac{4x^2}{y}\right]\, .
\end{align}
Taking advantage of these further simplifications, the $𝐢𝐧$ contribution to the NLO gluon spectrum can be compactly written as
\begin{align}\label{eq:IOE_NLO_IN}
(2\pi)^2\omega\frac{\rmd I_{\text{in}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}&=\frac{\bar{\alpha}\pi}{2\omega k_\perp^4} \Re\, i\int_0^\infty \rmd t_2 \int_0^{t_2} \rmd t_1 \, \frac{\hat{q}_0(t_1)}{\rhat_{21}^2} \rme^{-\frac{ k_\perp^2}{\khat^2_{21}}}
\nn
&\times \left[Q^2_s(\infty,t_2) I_a\left(\frac{k_\perp^2}{\jhat_{21}},\frac{k_\perp^2\rhat^2_{21}}{Q_r^2}\right)+2 C_{12}\rhat_{21} k_\perp^2 I_b\left(\frac{k_\perp^2}{\jhat_{21}},\frac{k_\perp^2\rhat^2_{21}}{Q_r^2}\right)\right] \,.
\end{align}
Hence, we have managed to reduce the number of integrals over transverse positions and times down to two time-integrations.
Turning now to the \textbf{broad} contribution, given in \eqn{eq:spectrum-ioe-nlo-broad}, we can perform the derivatives on $^LO$ and integrate over $t_1$ using \eqn{eq:id_1}. Then it reads
\begin{align}\label{eq:IOE_med_broad1}
d I_{\rm \text{broad}}^{\rm NLO}}{\rmd \omega \rmd^2\k}&=-2\bar{\alpha} \Re \int_0^\infty \rmd t_2 \int_{t_2}^\infty \rmd s \int_\x \rme^{-i \k \cdot \x} \, \cP^{\rm LO}(\x,\infty;t_2)\nn
&\times \delta v(\x,s)\, {\rm Cot}_{20} \, \rme^{\frac{i\omega}{2}{\rm Cot}_{20}\x^2} \, ,
\end{align}
with $^LO(,∞;t_2) $ introduced in Eq.~\eqref{eq:P_LO_x}. Using the definition of the function $I_b(x,y)$ in \eqn{eq:bbS_alpha}, the $𝐛𝐫𝐨𝐚𝐝$ contribution to the spectrum can finally be written as
\begin{align}\label{eq:IOE_med_broad2}
\rmd I_{\rm \text{broad}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}
&= -\frac{\pi \bar{\alpha}}{k_\perp^4} \Re \int_0^\infty \rmd t_2 \, {\rm Cot}_{20}\, Q^2_{s0}(\infty,t_2) \, I_a\left(\frac{k_\perp^2}{\khat^2_{20}},\frac{k_\perp^2}{Q_b^2}\right)\bigg]\, ,
\end{align}
where $^2_20 ≡^2(t_2,0)$ given by \eqn{eq:Lambda21}.
\paragraph*{Plasma brick model.} So far, we have made no approximation regarding the time profile of the medium. As for the LO contribution, we now assume the plasma brick model, i.e. $q̂(t)=q̂ Θ(L-t)$. As in the previous case, this simple model allows one to further simplify Eqs.~\eqref{eq:IOE_NLO_IN} and \eqref{eq:IOE_med_broad2} by splitting the time integrations appropriately.
We start with the \textbf{broad} term since it has only one time integral left. In addition, the spectrum is proportional to $Q_s0^2 $, and thus this term only has support inside the medium $t_2<L$. As a consequence, the $C$ and $S$ functions are given directly by Eq.~\eqref{eq:SC-brick}, and Eq.~\eqref{eq:IOE_med_broad2} reduces to
\begin{align}\label{eq:IOE_med_broad_brick}
\rmd I_{\rm \text{broad}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}
&= -\frac{\hat{q}_0\pi \bar{\alpha}}{k_\perp^4} \Re\int_0^L \rmd t_2 \, \Omega \, {\rm cot}(\Omega t_2)\, (L-t_2) \,I_a\left(\frac{k_\perp^2}{\khat^2(t_2,0)},\frac{k_\perp^2}{Q_b^2}\right) \, ,
\end{align}
\beq\label{eq:help_1}
\khat^2(t_2,0)= Q^2_s(L,t_2) -2i\omega\Omega{\rm cot}(\Omega t_2)\, , \quad Q^2_s(L,t_2)=\hat{q}_0(L-t_2) \log \frac{Q_b^2}{\mu_\ast^2} \, .
\eeq
Next, let us analyze the \textbf{in} contribution given by Eq.~\eqref{eq:IOE_NLO_IN}. It has support both inside and outside of the medium and thus two contributions appear. One option is that the gluon is emitted inside the medium in the amplitude and its conjugate, which we identify as the \textbf{in-in} contribution. The second term refers to the case in which one of the emissions happens outside of the medium, that we denote as \textbf{in-out}. The \textbf{in-in} term obeys the time ordering $∫_0^L t_2 ∫_0^t_2 t_1$ in Eq.~\eqref{eq:IOE_NLO_IN}, with the $C$ and $S$ having support only inside the medium and thus are given by Eq.~\eqref{eq:SC-brick}. Therefore, this contribution reads
\begin{align}\label{eq:IOE_NLO_ININ}
(2\pi)^2\omega\frac{\rmd I_{\text{in-in}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}&=\frac{\bar{\alpha}\pi\hat{q}_0}{2\omega k_\perp^4} \Re\, i\int_0^L \rmd t_2 \,\int_0^{t_2} \rmd t_1 \, \frac{\rme^{-\frac{k_\perp^2}{\khat^2_{21}}}}{\rhat_{21}^2}
\nn
&\times \left[Q^2_s(L,t_2) I_a\left(\frac{k_\perp^2}{\jhat_{21}},\frac{k_\perp^2\rhat^2_{21}}{Q_r^2}\right)+2 C_{12}\rhat_{21} k_\perp^2 I_b\left(\frac{k_\perp^2}{\jhat_{21}},\frac{k_\perp^2\rhat^2_{21}}{Q_r^2}\right)\right] \,,
\end{align}
where again $Q^2_s(L,t_2)=q̂_0(L-t_2)logQ_b^2/μ_∗^2 $, and the auxiliary functions reduce to
\begin{align}\label{eq:help_2}
& \khat^2_{21} = Q^2_s(L,t_2) -2i\omega \Omega \, {\rm cot}(\Omega (t_2-t_1))\, , \nn
& \jhat_{21}=\frac{i }{2\omega}\left(\Omega{\rm cot}(\Omega(t_2-t_1))+\Omega{\rm cot}(\Omega t_1)+\frac{2i\omega}{S^2_{21}\khat^2_{21}} \right)S_{21}^2 \khat_{21}^4\, , \nn
&\rhat_{21} =-\frac{2i\omega}{\khat^2_{21}S_{21}} \, .
\end{align}
The \textbf{in-out} contribution can be further simplified following similar steps as in the LO case. Now, the time integrals read $∫_L^∞ t_2 ∫_0^Lt_1 $ and one can set $Q_s^2=0$ everywhere in Eq.~\eqref{eq:IOE_NLO_IN} since this term only has support outside of the medium. Then, the spectrum reads
\begin{align}\label{eq:IOE_NLO_INPOUT_1}
&(2\pi)^2\omega\frac{\rmd I_{\text{in-out}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}=\frac{\bar{\alpha}\hat{q}_0\pi}{\omega k_\perp^2} \Re\, i\int_L^\infty \rmd t_2 \,\int_0^{L} \rmd t_1 \, \frac{C_{12}}{\rhat_{21}} \rme^{-\frac{ k_\perp^2}{\khat^2_{21}}}
I_b\left(\frac{k_\perp^2}{\jhat_{21}},\frac{k_\perp^2\rhat^2_{21}}{Q_r^2}\right) \, ,
\end{align}
where we recall that the $C$ and $S$ functions are distinct from the ones used in the \textbf{in-in} term, since they have support both inside and outside of the medium. Nonetheless, as for the LO case, they can be written in terms of the purely in-medium and in-vacuum $C$ and $S$ functions by using Eq.~\eqref{eq:linear_rel_C_S}. Taking $t_0=L$, we find that for the above time ordering
\begin{align}
& S_{21}=C_{1L}S_{2L}-S_{1L}C_{2L}=\cos(\Omega (L-t_1))(t_2-L)+\frac{\sin(\Omega(L-t_1))}{\Omega}\, , \nn
& C_{12}=-\partial_2 S_{12}=\partial_2 S_{21}=\cos(\Omega(L-t_1)) \, ,
\end{align}
such that
\begin{align}
{\rm Cot}_{21}&=\frac{C_{12}}{S_{21}}=\frac{\Omega}{\Omega (t_2-L) + \tan (\Omega(L-t_1))} \, .
\end{align}
For the reversed time ordering, we find
\begin{align}
S_{12}&=-S_{21}=-\cos(\Omega (L-t_1))(t_2-L)-\frac{\sin(\Omega(L-t_1))}{\Omega} \, , \nn
C_{21}&=-\partial_1 S_{21}=\cos(\Omega(L - t_1)) -\Omega (t_2-L) \sin(\Omega(L - t_1)) \, ,
\end{align}
leading to
\begin{equation}
{\rm Cot}_{12}=\frac{C_{21}}{S_{12}}=-\frac{\Omega - \Omega^2(t_2-L)\tan \Omega (L-t_1)}{\Omega(t_2-L)+\tan \Omega (L-t_1)} \, .
\end{equation}
Combining all these results, the auxiliary functions now read
\begin{align}
\khat^2_{21} &= -2i\omega {\rm Cot}_{21}=\frac{-2i\omega \Omega}{\Omega (t_2-L) + \tan (\Omega(L-t_1))} \, ,\\
\jhat_{21} &= 2i\omega \Omega \cos^2(\Omega(L-t_1)) \big[\tan(\Omega(L-t_1))-\cot(\Omega t_1) \big] \, , \\
\rhat_{21} &= \frac{2i\omega }{ \khat^2_{21}S_{12}} =\frac{1}{\cos (\Omega(L-t_1))} \,.
\end{align}
Inserting these expressions into Eq.~\eqref{eq:IOE_NLO_INPOUT_1}, one realizes that the remaining $t_2$ integral can be carried out
\begin{align}
\int_L^\infty \rmd t_2 \,\rme^{-\frac{k_\perp^2}{\khat^2_{21}}} =\int_L^\infty \rmd t_2 \, \rme^{-i\frac{k_\perp^2}{2\omega}\left[(t_2-L)+\frac{\tan(\Omega(L-t_1))}{\Omega}\right]} =\frac{2\omega}{ik_\perp^2} \rme^{-i\frac{k_\perp^2}{2\omega\Omega}\tan(\Omega(L-t_1))} \, .
\end{align}
As a consequence, the \textbf{in-out} contribution to the NLO spectrum can be finally written as
\begin{align}\label{eq:In_out_NLO_final}
(2\pi)^2\omega\frac{\rmd I_{\text{in-out}}^{\rm NLO}}{\rmd \omega \rmd^2 \k} &= \frac{2\bar{\alpha}\hat{q}_0\pi}{k_\perp^4} \Re \int_0^{L} \rmd t_1 \, \cos^2(\Omega(L-t_1)) \nn
&\times I_b\left(\frac{k_\perp^2}{\jhat_{21}},\frac{k_\perp^2}{Q_r^2 \cos^2(\Omega(L-t_1))}\right) \rme^{-i \frac{k_\perp^2}{2\omega \Omega}\tan(\Omega(L-t_1))} \, .
\end{align}
\subsection{Final formulas}\label{sec:summary}
At this point, we summarize the main results obtained in the two previous sections. Our aim is to provide a set of compact equations which can be directly used in phenomenological studies or implemented in jet quenching Monte-Carlo codes. In what follows, we first present the results for a generic medium profile and then take the brick limit.
\subsubsection{Spectrum at LO+NLO for a generic medium profile}
Up to NLO in the IOE, the purely medium-induced gluon spectrum, i.e. after subtracting vacuum radiation, can be written as
\begin{align}\label{eq:spec_res_generic_1}
\frac{\rmd I^{\rm LO+NLO}}{\rmd \omega \rmd^2 \k} = \frac{\rmd I^{\rm LO}}{\rmd\omega \rmd^2 \k}+\frac{\rmd I_{\text{in}}^{\rm NLO}}{\rmd \omega \rmd^2 \k} +\frac{\rmd I_{\text{broad}}^{\rm NLO}}{\rmd \omega \rmd^2 \k} \, .
\end{align}
The LO term reads
\begin{align}\label{eq:spec_res_generic_2}
(2\pi)^2\omega\frac{\rmd I^{\rm LO}}{\rmd \omega \rmd^2 \k}&=8\Bar{\alpha} \pi \, \Re \int_0^\infty \rmd t \, {\rm Cot}(t) \frac{\rme^{-\frac{k_\perp^2}{\khat^2(t,0)} } }{\khat^2(t,0)} -\frac{8\pi \abar}{k_\perp^2} \, .
\end{align}
In addition, the NLO contributions are given by
\begin{align}\label{eq:spec_res_generic_3}
(2\pi)^2\omega\frac{\rmd I_{\text{in}}^{\rm NLO}}{\rmd \omega \rmd^2 \k} &=\frac{\bar{\alpha}\pi}{2\omega k_\perp^4} \Re\, i\int_0^\infty \rmd t_2 \,\int_0^{t_2} \rmd t_1 \, \frac{\hat{q}_0(t_1)}{\rhat_{21}^2} \rme^{-\frac{k_\perp^2}{\khat^2_{21}}}
\nn
&\times \left[Q^2_s(\infty,t_2) I_a\left(\frac{k_\perp^2}{\jhat_{21}},\frac{k_\perp^2\rhat^2_{21}}{Q_r^2}\right)+2 C_{12}\rhat_{21} k_\perp^2 I_b\left(\frac{k_\perp^2}{\jhat_{21}},\frac{k_\perp^2\rhat^2_{21}}{Q_r^2}\right)\right] \, ,
\end{align}
\begin{align}\label{eq:spec_res_generic_4}
\rmd I_{\rm \text{broad}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}&= -\frac{\pi \bar{\alpha}}{k_\perp^4} \Re\int_0^\infty \rmd t_1 \, {\rm Cot}(t_1)\, Q^2_{s0}(\infty,t_1) \, I_a\left(\frac{k_\perp^2}{\khat^2(t_1)},\frac{k_\perp^2}{Q_b^2}\right)\, .
\end{align}
In the above equations we introduced
\begin{align}
{\rm Cot}(t_2,t_1)= {\rm Cot}_{21}\equiv \frac{C(t_1,t_2)}{S(t_2,t_1)}=\frac{C_{12}}{S_{21}} \,,
\end{align}
where the $C$ and $S$ functions are described in \eqn{eq:Abel} and the functions $I_a$ and $I_b$ are given in Eqs.~\eqref{eq:bbS_alpha} and \eqref{eq:bbS_beta}, respectively. Further, the accumulated transverse momentum scales are $Q_s0^2(t_2,t_1)=∫_t_1^t_2 t q̂_0(t) $ and $Q_s^2(t_2,t_1) = Q_s0^2(t_2,t_1) logQ_b^2/μ^2_∗$. The remaining functions are defined as follows,
\begin{align}
\khat^2_{21} &= Q^2_s(\infty,t_2) -2i\omega{\rm Cot}_{21} \, ,\nn
\jhat_{21} &=\frac{i}{2\omega}\left(-{\rm Cot}_{12}+{\rm Cot}_{10}+\frac{2i\omega}{S^2_{21}\khat^2_{21}} \right)S_{21}^2 \khat_{21}^4\, , \nn
\rhat_{21} & =-\frac{2i\omega}{\khat^2_{21}S_{21}} \, .
\end{align}
The matching scale $Q_b$, that enters everywhere into the $broad$ and only into the $Q_s$ definition for the $in$ case, is obtained by solving the transcendental equation
\beq\label{eq:Qb_final}
Q_b=\displaystyle\int_0^{\tilde L} \rmd t \, \hat q_0(t)\log\frac{Q^2_b}{\mu^2_\ast} \, ,
\eeq
with $L̃$ some effective medium length. The exact value of $L̃$ is not important, as long as it is taken such that the relevant support for the integration of $q̂$ is covered. The radiative matching scale, $Q_r$, appears in all other terms related to the kernel expansion and is the solution of
\beq\label{eq:Qr_final}
Q^2_r=\sqrt{\hat q_0(t)\omega \log\frac{Q^2_r}{\mu^2_\ast}} \, .
\eeq
\subsubsection{Spectrum at LO+NLO for the brick model}\label{subsubsec:LO_NLO_brick}
The previous results are simplified when the medium is modelled as a plasma brick of length $L$ with
\begin{equation}
\hat{q}(t)=\hat{q} \, \Theta(L-t) \, .
\end{equation}
In this case, the full medium-induced spectrum at NLO in the IOE can be written as
\begin{align}\label{eq:spec_res_brick_1}
\frac{\rmd I^{\rm LO+NLO}}{\rmd \omega \rmd^2\k} = \frac{\rmd I_{\text{in-in}}^{\rm LO}}{\rmd \omega \rmd^2 \k}+ \frac{\rmd I_{\text{in-out}}^{\rm LO}}{\rmd \omega \rmd^2 \k}+\frac{\rmd I_{\text{in-in}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}+\frac{\rmd I_{\text{in-out}}^{\rm NLO}}{\rmd \omega \rmd^2 \k} +\frac{ \rmd I_{\text{broad}}^{\rm NLO}}{\rmd \omega \rmd^2 \k} \, .
\end{align}
The leading order terms read
\begin{align}\label{eq:spec_res_brick_2}
(2\pi)^2\omega\frac{\rmd I^{\rm LO}_{\text{In-In}}}{\rmd \omega \rmd^2 \k}&=8\Bar{\alpha} \pi \, \Re \int_0^L \rmd t \, \Omega \, {\rm cot}(\Omega t) \frac{\rme^{-\frac{k_\perp^2}{Q_s^2(L,t) -2i \omega \Omega {\rm cot}(\Omega t)} }}{Q_s^2(L,t) - 2i \omega \Omega {\rm cot}(\Omega t)} \,,
\end{align}
\begin{align}\label{eq:spec_res_brick_3}
(2\pi)^2\omega\frac{\rmd I_{\text{In-Out}}^{\rm LO}}{\rmd \omega \rmd^2 \k}
&= \frac{8\bar{\alpha}\pi }{k_\perp^2}\Re \left(\rme^{-\frac{ik_\perp^2}{2\omega \Omega \, \cot\Omega L } } -1\right) \, .
\end{align}
Here we used
\begin{align}
&\Omega = \frac{1-i}{2}\sqrt{\frac{\hat{q}_0}{\omega} \log\frac{Q_r^2}{\mu^2_\ast}} \, , \\
&Q_s^2(L,t) = \hat{q}_0 \log\frac{Q_b^2}{\mu^2_\ast} (L-t_2) \, .
\end{align}
The NLO \textbf{in-in} term can be written as
\begin{align}\label{eq:spec_res_brick_4}
(2\pi)^2\omega\frac{\rmd I_{\text{in-in}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}&=\frac{\bar{\alpha}\pi\hat{q}_0}{2\omega k_\perp^4} \Re\, i\int_0^L \rmd t_2 \,\int_0^{t_2} \rmd t_1 \, \frac{\rme^{-\frac{k_\perp^2}{\khat^2_{21}}}}{\rhat_{21}^2}
\nn
&\times \left[ Q^2_s(L,t_2) I_a\left(\frac{k_\perp^2}{\jhat_{21}},\frac{k_\perp^2\rhat^2_{21}}{Q_r^2}\right)+2 C_{12}\rhat_{21} k_\perp^2 I_b\left(\frac{k_\perp^2}{\jhat_{21}},\frac{k_\perp^2\rhat^2_{21}}{Q_r^2}\right)\right] \, ,
\end{align}
where $Q^2_s(L,t_2)$ is defined as above, the $C$ and $S$ functions are given in Eq.~\eqref{eq:SC-brick}, and
\begin{align}
& \khat^2_{21} = Q^2_s(L,t_2) -2i\omega \Omega \, {\rm cot}(\Omega (t_2-t_1))\, , \nn
& \jhat_{21}=\frac{i }{2\omega}\left(\Omega{\rm cot}(\Omega(t_2-t_1))+\Omega{\rm cot}(\Omega t_1)+\frac{2i\omega}{S^2_{21}\khat^2_{21}} \right)S_{21}^2 \khat_{21}^4\, , \nn
&\rhat_{21} =-\frac{2i\omega}{\khat^2_{21}S_{21}} \, .
\end{align}
The NLO \textbf{in-out} piece is
\begin{align}
(2\pi)^2\omega\frac{\rmd I_{\text{in-out}}^{\rm NLO}}{\rmd \omega \rmd^2 \k} &= \frac{2\bar{\alpha}\hat{q}_0\pi}{k_\perp^4} \Re \int_0^{L} \rmd t_1 \, \cos^2(\Omega(L-t_1)) \nn
&\times I_b\left(\frac{k_\perp^2}{\jhat_{21}},\frac{k_\perp^2}{Q_r^2 \cos^2(\Omega(L-t_1))}\right) \rme^{-i \frac{k_\perp^2}{2\omega \Omega}\tan(\Omega(L-t_1))} \, .
\end{align}
\begin{align}
&\khat^2_{21}=\frac{-2i\omega \Omega}{\Omega (t_2-L) + \tan (\Omega(L-t_1))} \, ,\nn
&\jhat_{21}=2i\omega \Omega \cos^2(\Omega(L-t_1)) \big[\tan(\Omega(L-t_1))-\cot(\Omega t_1) \big]\, , \nn
&\rhat_{21} = \frac{1}{\cos (\Omega(L-t_1))} \, .
\end{align}
Finally, the NLO \textbf{broad} contribution is given by
\begin{align}
\rmd I_{\rm \text{broad}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}
&= -\frac{\hat{q}_0\pi \bar{\alpha}}{k_\perp^4} \Re \int_0^L \rmd t_2 \, \Omega \, {\rm cot}(\Omega t_2)\, (L-t_2) \, I_a\left(\frac{k_\perp^2}{\khat^2(t_2,0)},\frac{k_\perp^2}{Q_b^2}\right) \, ,
\end{align}
\beq
\khat^2(t_1,0)= \hat{q}_0 \log\frac{Q_s^2}{\mu^2_\ast}(L-t_1) -2i\omega\Omega{\rm cot}(\Omega t_1)\, .
\eeq
\subsection{Asymptotic behavior}\label{sec:asymp_beahvior}
The complete expressions for the in-medium branching kernel, that we have summarized in the previous section, are written in terms of a few integrations that we did not manage to solve analytically. Before presenting their numerical implementation, we would like to give further analytical insight into the discussion. To that end, we analyze the behavior of the IOE spectrum up to NLO in two physically relevant asymptotic regimes: when the emitted gluon is either (i) soft ($ω≪ω_c$) and collinear ($k_⊥^2≪q̂ L$) or (ii) hard ($ω≫ω_c$) and wide angled ($k_⊥^2≫q̂ L$). That is, the regime of validity of BDMPS-Z and GLV approaches, respectively. Our results below are obtained by taking the brick limit, although similar conclusions are obtained for other choices of medium profile. In addition, we neglect the purely vacuum radiation as it is completely irrelevant for this discussion.
\subsubsection{Multiple soft scattering regime}\label{sec:MS_IOE}
We begin by analyzing the regime in which the emitted gluon is soft, i.e. $ω≪ω_c$, and its typical formation time is much shorter than the medium length $t_f∼√(ω/q̂)≪L$. The latter condition can be translated into a constraint on the transverse momentum off the emission, $q_⊥^2∼q̂ t_f ∼√(q̂ω)≪q̂L$,\footnote{Notice that this condition refers to the momentum of the in-medium vertex, rather than the final momentum of the gluon. Even for soft gluon emissions, final state broadening can lead to a final momentum $k_\perp^2\sim \hat{q }L$. } which implies that short formation time gluons typically acquire most of their transverse momentum due to final state broadening. Under these conditions, the IOE spectrum simplifies significantly. The general formula for the medium-induced spectrum given by Eq.~\eqref{eq:spectrum} can be re-written in momentum space as\footnote{Strictly speaking, the upper bound of the integrals should be $L$. We take $L\to \infty$ to facilitate analytical manipulations.}
\begin{align}
\label{eq:didwdkt-ms}
(2\pi)^2\omega\frac{\rmd I}{\rmd \omega \rmd^2 \k}&=\frac{2\bar{\alpha}\pi}{\omega^2} \Re\bigg[\int_0^\infty \rmd t_2 \int_0^{t_2} \rmd \tau \int_{\x,\q} \rme^{-i \q\cdot \x}\cP(\k-\q;L-t_2)
\nn &\times\bdel_\y\cdot \bdel_\x \, \cK(\x,t_2;\y,t_2-\tau)_{\y=0} \bigg] \, .
\end{align}
where we have exploited that $$ and $$ are invariant under time translations, i.e. depend only on time differences, when the plasma is homogenous. Eq.~\eqref{eq:didwdkt-ms} can be simplified noting that $τ∼t_f ≪t_2$ and thus one can set $t_2→∞$ in the $τ$ integration upper limit. That is, the two time integrations decouple. In addition, we note that $q_⊥∼1/x_⊥$ corresponds to the transverse momentum acquired in the branching process, $q_⊥∼√(q̂ω)$, that, as we have anticipated, is small with respect to the characteristic broadening momentum, i.e. $q_⊥≪q̂L$. As a consequence, we neglect $$ with respect to $$̨ inside $\cP$. Then, the $\q$ integral acts solely on $\cK$ and the $\x$ and $\q$ integrals yield
\begin{align}
\label{eq:factorized}
(2\pi)^2\omega\frac{\rmd I}{\rmd \omega \rmd^2 \k}&=\frac{2\bar{\alpha}\pi}{\omega^2} \Re\bigg[\int_0^\infty \rmd t_2 \int_0^{\infty} \rmd \tau \, \cP(\k;L-t_2) \bdel_\y\cdot \bdel_\x \cK(\x,t_2;\y,t_2-\tau)_{\x=\y=0} \bigg] \, .
\end{align}
A familiar element in the previous equation is the integrated spectrum defined as
ωI/ω = 2α̅π/ω^2[ ∫_0^∞ τ
_·_ (,t_2;,t_2-τ)_==0 ] .
Then, Eq. (<ref>) can be finally written as
(2π)^2ωI/ω^2 = ∫_0^∞t_2 𝒫(;̨ L-t_2) ωI/ω .
That is, in the soft and collinear limit, the spectrum is given by the product of the time integral of the broadening distribution and the energy spectrum. This result is not tied with the IOE approach and has been previously obtained in the literature in the context of BDMPS-Z calculations <cit.> and exploited in Monte-Carlo simulations <cit.>. We proceed to compute Eq.(<ref>) in the IOE.
The soft limit of the IOE energy spectrum at all orders was computed in Refs. <cit.> and reads
\begin{align}\label{eq:ll_1}
\omega \frac{ \rmd I^{\rm IOE}}{\rmd \omega}= \omega \frac{\rmd I^{\rm LO}}{d\omega}(\hat{q}\to \hat{q}_{\rm eff}) \, ,
\end{align}
where $\omega \frac{\rmd I^{\rm LO}}{\rmd \omega}= \bar{\alpha }\sqrt{\frac{\hat{q}L^2}{\omega}}$ corresponds to the well known BDMPS-Z result. The effective jet quenching parameter is given at leading-logarithmic order by <cit.>[As shown in Ref. <cit.>, if all terms in $\hat{q}_{\rm eff}$ are resummed, Eq. (<ref>) gives the full energy spectrum for $\omega \ll \omega_c$.]
\begin{align}\label{eq:ll_3}
\hat{q}_{\rm eff}=\hat{q}_0 \log\left(\frac{Q_r^2}{\mu_\ast^2}\right) \, \left(1+\frac{1.016}{\log\left(\frac{Q_r^2}{\mu_\ast^2}\right)}+\mathcal{O}\left(\frac{1}{\log^2\left(\frac{Q_r^2}{\mu_\ast^2}\right)}\right)\right) \, .
\end{align}
Eqs. (<ref>) and (<ref>) show that the IOE energy spectrum is governed by the LO result with higher orders suppressed by a logarithmic power which can be written in terms of the ratio $\frac{\hat{q}_0}{\hat{q}}$. A similar conclusion can be reached regarding the broadening distribution in the kinematical limit $k_\perp^2\ll \hat{q}L$, see for example eq:assist_1 for the result up to NLO. Combining these two results, one concludes that the soft and collinear limit of the fully differential spectrum in Eq. (<ref>) also obeys this functional form. Note that the spectrum will consist of terms where the matching scale is given by $Q_r$ and others where it is $Q_b$, depending if the terms come from the expansion of the kernel or of the broadening distribution.
In Appendix <ref> we explicitly show that the in-out NLO contribution in the IOE spectrum scales as the LO term multiplied by a logarithm that arises from the ratio $\frac{\hat{q}_0}{\hat{q}}$.
§.§.§ Rare, hard scattering regime
Let us now consider the orthogonal regime with respect to the previous section. Here, the gluon is hard, $\omega\gg \omega_c$, and carries a large transverse momentum $k_\perp^2\gg \hat{q} L$, i.e. the intrinsic momentum of the gluon is significantly larger than what it typically acquires through broadening in the medium. In this case, the multiple soft scattering contribution is suppressed by the LPM effect and the emission spectrum is dominated by single hard scattering in the medium. This corresponds to the truncation of the opacity expansion, considered by GLV, at first order <cit.>, leading to an emission spectrum reading
\begin{equation}\label{eq:glv}
\begin{split}
(2\pi)^2\omega\frac{\rmd I^{\rm GLV}}{\rmd \omega \rmd^2 \k} &= \frac{2\Bar{\alpha}\hat{q}_0L^3\pi}{\omega^2}\int_0^\infty \rmd x \, \frac{x-\sin(x)}{x^2}
\frac{\gamma+u-x}{(u^2+2u(\gamma-x)+(\gamma+x)^2)^{3/2}} \, ,
\end{split}
\end{equation}
where $u=\frac{L}{2\omega}k_\perp^2$ and $\gamma=\frac{\mu^2L}{2\omega}$. The medium potential was taken to be the GW model and thus $\mu$ is its infrared regulator. It can be related to the universal physical mass $\mu_\ast$, as mentioned in Section <ref> and detailed in Refs. <cit.>.
Note that the conditions $\omega \gg \omega_c$ and $k_\perp^2\gg \hat q L$ correspond to $u\gg 1 \gg \gamma$ in the previous equation. To take this limit in Eq. (<ref>), let us consider the integral
\begin{equation}\label{eq:glv_app1}
\begin{split}
I &\equiv \int_0^\infty \rmd x \, \frac{x-\sin(x)}{x^2}
\frac{\gamma+u-x}{(u^2+2u(\gamma-x)+(\gamma+x)^2)^{3/2}} \, ,
\end{split}
\end{equation}
in two regions: (i) $u\gg x\gg \gamma$ ($<$) and (ii) $u\sim x\gg 1$, but $u-x\gg \gamma$ ($>$). First, we split $I$ using
\begin{equation}
\int_0^\infty \to \lim_{\epsilon \to 0} \, \int_0^{\epsilon u} + \int_{\epsilon u}^\infty \equiv I_< + I_> \,,
\end{equation}
with $\epsilon u$ held constant. The contribution from the first region can be easily computed to leading-logarithmic accuracy
\begin{equation}
\begin{split}
I_<&= \lim_{\epsilon \to 0} \, \int_0^{\epsilon u} \rmd x \, \frac{x-\sin(x)}{x^2}
\frac{1}{u^{2}} + \mathcal{O}\left(\frac{x}{u}\right) =\lim_{\epsilon \to 0} \, \frac{1}{u^{2}} G(\epsilon u)\approx \frac{1}{u^2}\left[\log(\epsilon u)-1+\gamma_E\right] \, ,
\end{split}
\end{equation}
where we introduced[${\rm Ci}(x)=-\int_x^\infty \rmd t \, \frac{\cos (t)}{t}$.]
\begin{align}\label{eq:G}
G(a)\equiv \int_0^{a} \rmd x \, \frac{x-\sin(x)}{x^2}=\log(a)-1+\gamma_E- {\rm Ci}(a)+\frac{\sin(a)}{a} \, .
\end{align}
In the case of $I_>$, we first notice that $x\gg1$ so we can drop the $\sin(x)$ term. Defining $a\equiv \gamma/u\ll 1$ we have
\begin{equation}
\begin{split}
I_>&\approx \frac{1}{u^2}\int_{\epsilon u}^\infty \frac{\rmd z}{z} \frac{1-z}{((1-z)^2+4za)^{\frac{3}{2}}} \\
&=\frac{1}{u^2}\left[\frac{2(4a-3+z)}{4(a-1)\sqrt{z^2+1+(4a-2)z}}-\arctan\left(\frac{1+(2a-1)z}{\sqrt{z^2+(4a-2)z+1}}\right)\right]_{\epsilon u}^\infty\\
&=\frac{1}{u^2}\left[-2+\frac{1}{2}\log \left(\frac{1-a}{a}\frac{-1}{a(a-1)(\epsilon u)^2}\right)\right]=\frac{1}{u^2}\left(-2+\log\left(\frac{u}{\gamma}\right)-\log(\epsilon u)\right)
\, .
\end{split}
\end{equation}
Combining the results from the two different regions, the $I$ integral gives
\begin{equation}
I=\frac{1}{u^2}\left(-3+\gamma_E+\log\left(\frac{u^2}{\gamma}\right)\right)+ \mathcal{O}\left(\frac{1}{u^3}\right) \, .
\end{equation}
Inserting this result into the GLV spectrum given in Eq. (<ref>) yields
\begin{equation}\label{eq:glv_largekt}
\begin{split}
(2\pi)^2\omega\frac{\rmd I^{\rm GLV}}{\rmd \omega \rmd^2 \k} &\approx \frac{2\bar{\alpha}\hat{q}_0L^3\pi}{\omega^2}\frac{1}{u^2}\left(\log\left(\frac{u^2}{\gamma}\right)+\gamma_E-3 \right)=\frac{8\bar{\alpha}\hat{q}_0L\pi}{k_\perp^4}\log\left(\frac{k_\perp^4L \rme^{\gamma_E-3}}{2\omega \mu^2}\right)\, .
\end{split}
\end{equation}
The expected $1/k_\perp^4$ power tail naturally arises from a Coulomb-like single hard scattering in the medium. Counterintuitively, even though we are considering here the high energy limit, the spectrum is still sensitive to the infrared details of the in-medium scattering potential via the thermal mass $\mu$. For the sake of comparing with the IOE in what follows, a couple of manipulations are required. First, we re-write the resulting logarithm as
\begin{equation}
\log\left(\frac{k_\perp^4L \rme^{\gamma_E-3}}{2\omega \mu^2}\right)=\left(\gamma_E-3+\log\left(\frac{k_\perp^2}{\mu^2}\right)+\log\left(\frac{k_\perp^2L}{2\omega}\right)\right)\, ,
\end{equation}
and then replace the GW mass by the universal infrared scale $\mu_\ast$ through the leading-logarithmic prescription $4\mu_\ast^2=\mu^2 \rme^{-1+2\gamma_E}$ <cit.>. This leads to our final expression for the GLV spectrum:
\begin{equation}
\begin{split}
(2\pi)^2\omega\frac{\rmd I^{\rm GLV}} {\rmd \omega \rmd^2 \k}=\frac{8\bar{\alpha}\pi\hat{q}_0L}{ k_\perp^4}\left(3\gamma_E-4+\log\left(\frac{k_\perp^2}{4\mu_\ast^2}\right)+\log\left(\frac{k_\perp^2L}{2\omega}\right)\right)\, .
\end{split}
\end{equation}
Let us now take the $\omega\gg \omega_c$ and $k_\perp^2\gg \hat{q}L$ limits in the IOE spectrum. At leading order, since $k_\perp^2\gg \hat{q}L \sim Q_s^2$, broadening contributions are sub-leading and thus can be ignored in Eq. (<ref>). Further, $\omega \gg \omega_c$ is equivalent to $\Omega L \ll 1$. Combining this pair of observations yields the LO contributions at $\mathcal O(\k^2)$:
\begin{align}\label{eq:oo_1}
(2\pi)^2\omega\frac{\rmd I^{\rm LO}_{\rm \text{in-in}}}{\rmd \omega \rmd^2 \k}&\approx\frac{4\bar{\alpha} \pi}{\omega} \, \Re \bigg[ \, \int_0^L \rmd t \, \exp\left[-\frac{ik_\perp^2}{ 2 \omega } t\right] \bigg]=\frac{8\Bar{\alpha}\pi}{k_\perp^2}\left(1-\cos \frac{k_\perp^2L}{2\omega }\right)\,,
\end{align}
\begin{align}
(2\pi)^2\omega\frac{\rmd I_{\rm \text{in-out}}^{\rm LO}}{\rmd \omega \rmd^2 \k}
&\approx -\frac{8\Bar{\alpha}\pi}{k_\perp^2}\left(1-\cos \frac{k_\perp^2L}{2\omega }\right)\,.
\end{align}
Adding the two components results into a vanishing spectrum at this order in $\k$ and indicates the need to go to higher orders that, as we will see, affect rather differently the \textbf{in-in} and \textbf{in-out} terms. The latter is exponentially suppressed when including higher orders as can be derived from Eq.~\eqref{eq:HO_brick_out_2}. In turn, the \textbf{in-in} term follows a power-law suppression. To see this, we perform a second order gradient expansion of the broadening distribution in the $Q_s^2≪k_⊥^2$ limit as
\beq\label{eq:grad_exp}
\int_\p \cP^{\rm LO}(\k-\p) u(\p) &=& \int_\q \cP^{\rm LO}(\q) u(\k-\q) \nn
&\approx& \int_\q \cP^{\rm LO}(\q) \left[ 1+ \q^i \nabla^i _\k +\frac{1}{2} \q^i\q^j \nabla^i _\k \nabla^j _\k \right] u(\k)\nn
&=& \Bigg[ 1+ \frac{1}{4} \underbrace{\int_\q q_\perp^2 \cP^{\rm LO}(\q) }_{\hat{q}L} \nabla^2 _\k \Bigg] u(\k) \, ,
\eeq
where $u()$ is a test function, we have used unitarity in the first line and rotational symmetry to drop the linear term. The first term in brackets corresponds to the result already obtained in Eq.~\eqref{eq:oo_1}, where the broadening distribution was replaced by a Dirac $δ$-function. Keeping only the second term in Eq.~\eqref{eq:grad_exp} and plugging it into Eq.~\eqref{eq:HO_brick_InIn} we obtain
\begin{align}
(2\pi)^2\omega\frac{\rmd I_{\rm \text{in-in}}^{\rm LO}}{\rmd \omega \rmd^2 \k}&\approx\frac{\bar{\alpha}\pi\hat{q}L}{\omega} \nabla^2_\k \Re \bigg[i \, \int_0^L \rmd t_2 \,
\rme^{-\frac{i \k^2 \tan(\Omega t_2) }{2 \omega \Omega } }\bigg] \, .
\end{align}
Now, because $$̨ is large the phase oscillates rapidly unless $t_2$ is small enough. To estimate the support of the $t_2$ integral, we exploit the fact that in the high energy regime $\Omega L \ll 1$ and thus the dominant contribution to the integral comes from the region where $t_2\ll \frac{2\omega}{k_\perp^2}\ll \Omega^{-1}$. As a consequence, one can replace the integration limit $L\to \infty$ and linearize the tangent, to obtain the leading asymptotic behavior of the spectrum
\begin{align}
\label{eq:lo-highenergy}
(2\pi)^2\omega\frac{\rmd I_{\rm \text{in-in}}^{\rm LO}}{\rmd \omega \rmd^2 \k}&\approx\frac{\bar{\alpha}\pi\hat{q}L}{\omega} \partial_\k^2\left(4k_\perp^2 \partial_\k^2\right)\Re \bigg[i \, \int_0^\infty \rmd t_2 \,
\rme^{-i \frac{\k^2 t_2}{2\omega } } \bigg] = \frac{8\bar{\alpha} \pi \hat{q}_0L}{k_\perp^4}\log \frac{Q_b^2}{\mu_\ast^2} \, .
\end{align}
Notice that the logarithm depends on the broadening matching scale since it originates from the last line in Eq. (<ref>). Interestingly, the leading order contribution exhibits a $1/k_\perp^4$ tail, physically corresponding to early time hard emissions, which then suffer multiple scatterings in the medium, acquiring a momentum $\hat{q}L$ much smaller than the momentum off the emission vertex. Compared to the vacuum like emission in Eq. (<ref>), we observe that although final state broadening does not change the power-law dependence on the transverse momentum, it power suppresses this second order by a $\frac{\hat{q}L}{k_\perp^2}\ll 1$ factor.
At NLO, we need to analyze individually each of the three identified terms in this asymptotic regime. Starting with the broad term, we note that in the high energy limit
^2(t,0) ≃q̂ (L-t) -2iω/t≃-2iω/t ,
so that
\begin{align}\label{eq:IOE_med_broad_app_3}
\rmd I_{\rm \text{broad}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}= -\frac{\pi \bar{\alpha}\hat q_0}{k_\perp^4} \Re\bigg[\,\int_0^L \frac{\rmd t }{t}\, \, \, (L-t) \, I_a\left(i\frac{k_\perp^2}{ 2\omega}t,\frac{k_\perp^2}{Q_b^2}\right)\bigg]\,.
\end{align}
It is convenient to write the remaining integral as
∫_0^L t /t (L-t) I_a(ik_⊥^2/ 2ωt,k_⊥^2/Q_b^2)= 2ω/ik_⊥^2∫_0^x_max x /x (x_max-x) I_a(x,y ) ,
where we used $x = i\frac{k_\perp^2}{2\omega}t$, $y= \frac{k_\perp^2}{Q_b^2}$ and $x_{\rm max}= i\frac{k_\perp^2}{2\omega}L\gg1$. This simplified integral is analytically solvable
\begin{align}
&\int_0^{x_{\rm max}} \frac{\rmd x}{x}(x_{\rm max}-x)I_a(x,y)=-8\Bigg(-x^2+2\rme^{-x}(x_{\rm max}-6-2x)+(x_{\rm max}-4)\left[\log(x)\right. \nn &\left.+x-2{\rm Ei}(-x)\right]+\rme^{-x}\left[4(1+x)+2x^2+x^3-x_{\rm max}(1+x+x^2)\right]\left[{\rm Ei}(x)-\log\frac{4x^2}{y}\right] \Bigg) \, .
\end{align}
Truncating the previous exact result to leading order in $x_{\rm max}$, we obtain
∫_0^x_max x /x (x_max-x) I_a(x,y ) ≈8 [ log4/yx_max + 5 - 3 γ_E ] x_max .
Plugging this last result into Eq. (<ref>) yields
\begin{align}\label{eq:IOE_med_broad_app_4}
\rmd I_{\rm \text{Broad}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}\approx \frac{8\pi \bar{\alpha}\hat q_0 L}{k_\perp^4} \left[ \log \frac{k_\perp^2}{4Q_b^2}+ \log \frac{k_\perp^2}{2\omega}L - 5 +3 \gamma_E \right] \,.
\end{align}
We restrain ourselves from doing any physical interpretation of this result at this stage and proceed to compute the in-in contribution. In this case, the lack of a vacuum-like ($\rmd t/t$) divergence simplifies the calculation. First, we use the asymptotic form of the exponential integral function,
\begin{align}
{\rm Ei}(x)= \frac{\rme^x}{x}\sum_{n=0}^{N-1} \frac{n!}{x^n}+\mathcal{O}\left(\frac{N!}{x^N}\right) \, ,
\end{align}
in Eqs. (<ref>) and (<ref>), to obtain the leading asymptotic forms of $I_a$ and $I_b$
I_a ≈-8 , I_b ≈4 .
Moreover, since $\Omega L\ll 1$ we can simplify the auxiliary functions given by Eq. (<ref>) down to
_21 = C_12≈1 , _21^2≈-2iω/(t_2-t_1) .
Consequently, the in-in spectrum reduces to
\begin{align}
(2\pi)^2\omega\frac{\rmd I^{\rm NLO}_{\rm \text{in-in}}}{\rmd \omega \rmd^2 \k}&\approx \frac{\pi \bar{\alpha}\hat q_0}{2\omega k_\perp^4} \Re\bigg[ i\int_0^L \rmd t_2 \,\int_0^{t_2} \rmd t_1 \, \rme^{-i\frac{ k_\perp^2}{2\omega}(t_2-t_1)} \left(-8\hat q (L-t_2) +8 k_\perp^2\right) \bigg] \nn
&\approx \frac{4\pi \bar{\alpha}\hat q_0}{\omega k_\perp^2} \Re\bigg[ i\int_0^L \rmd t_2 \,\int_0^{t_2} \rmd t_1 \, \rme^{-i\frac{k_\perp^2}{2\omega}(t_2-t_1)} \bigg]\, \nn
& = \frac{8\pi \bar{\alpha}\hat q_0L}{k_\perp^4} +\cO(k_\perp^{-6})\,.
\end{align}
Finally, for the in-out term we obtain
\begin{align}
&(2\pi)^2\omega\frac{\rmd I_{\rm \text{in-out}}^{\rm NLO}}{\rmd \omega \rmd^2 \k}\approx\frac{8\bar{\alpha}\hat{q}_0\pi}{k_\perp^4} \Re\bigg[ \int_0^{L} \rmd t_1 \, \rme^{-i \frac{k_\perp^2}{2\omega }t_1} \bigg]=\frac{16\bar{\alpha}\hat{q}_0\pi \omega}{k_\perp^6}\sin\left(\frac{k_\perp^2L}{2\omega}\right) \, ,
\end{align}
which is power-suppressed with respect to the broad and in-in terms and can be ignored. Then, the NLO contribution to the IOE spectrum is given by
\begin{align}\label{eq:IOE_large_kt}
&(2\pi)^2\omega\frac{\rmd I^{\rm NLO}}{\rmd \omega \rmd^2 \k}\approx\frac{8\pi \bar{\alpha}\hat q_0L}{k_\perp^4}\left[3 \gamma_E-4+\log\left(\frac{k_\perp^2}{4Q_b^2}\right)+\log\left(\frac{k_\perp^2L}{2\omega}\right)\right]\,.
\end{align}
Finally, combining the LO and NLO results we obtain that the IOE spectrum at high energies reduces to
\begin{align}
\label{eq:final-highenergy}
(2\pi)^2\omega\frac{\rmd I^{\rm LO+NLO}}{\rmd \omega \rmd^2 \k}&=\frac{8\pi \bar{\alpha}\hat q_0L}{k_\perp^4}\left[3 \gamma_E-4+\log\left(\frac{k_\perp^2}{4Q_b^2}\right)+\log\left(\frac{k_\perp^2L}{2\omega}\right)+\log\left( \frac{Q_b^2}{\mu_\ast^2} \right)\right]
\nn &= \frac{8\pi\bar{\alpha}\hat{q}_0L}{ k_\perp^4}\left[3\gamma_E-4+\log\left(\frac{k_\perp^2}{4\mu_\ast^2}\right)+\log\left(\frac{k_\perp^2L}{2\omega}\right)\right]\nn &= (2\pi)^2\omega\frac{\rmd I^{\rm GLV}} {\rmd \omega \rmd^2 \k}\,.
\end{align}
A few important remarks are in order at this point. The most obvious one is that the LO+NLO exactly matches the GLV result in the large $\omega$, large $\k$ regime. This was somehow expected but not trivial to confirm explicitly. Among other technicalities, a second order gradient expansion in transverse momentum for the LO term was essential. Another remarkable and related feature of the final result is that both the LO and NLO depend on $Q_b$, while their sum does not. This was already encountered in the energy spectrum calculation as discussed in Ref.~\cite{IOE3} and constitutes a sanity check of the Improved Opacity Expansion framework. We note again that unlike the soft limit considered before, this regime, although having a non-trivial cancellation of the matching scale dependence between different orders, does not provide any constraint on the functional form of $Q_b$, as can be observed from Eq.~\eqref{eq:final-highenergy}. This is unlike, for example, the result in Eq.~\eqref{eq:IOE_smallkt_final}, that forbids the matching from being a numerical constant. From a more pragmatic point of view, the exact matching between the IOE and GLV provides a non-trivial check on the computations performed in the previous sections.
\section{Numerical results}\label{sec:numerics}
In this section, we numerically explore the IOE spectrum in the brick model. We compare our results of the IOE spectrum truncated at LO and LO+NLO (see Section~\ref{subsubsec:LO_NLO_brick}) to (i) the single, hard scattering limit encompassed in the GLV spectrum (see Eq.~\eqref{eq:glv}) and (ii) an all-orders resummation of the spectrum presented in~\cite{CarlotaFabioLiliana}. Notice that the LO result can be considered as the BDMPS-Z solution with ultraviolet regulator taken to be the radiative matching scale $Q_r$.
These comparisons should be regarded, at this point, as a merely theoretical exercise. However, we choose the medium parameters to be in the ballpark of LHC conditions. More concretely, the LHC-inspired medium has: $q̂_0=0.156$~GeV$^3$, length $L=6$~fm and infrared regulator $μ_∗=0.355$~GeV. Further, we take a fixed value of the strong coupling constant $α_s=0.28$ and consider radiation off a hard quark such that $C_R=C_F=4/3$. These set of parameters lead to a critical frequency scale $ω_c0≡q̂_0 L^2=140$~GeV and the saturation density $Q^2_s0≡q̂_0 L=4.68$~GeV$^2$. Regarding our numerical routines, they run in a regular laptop with an average computing time of $𝒪(1)$ seconds for each pair of ($ω,$̨) values, considering the above set of parameters. The computing time is significantly smaller, $\mathcal{O}(10^{-2})$ seconds, if not too extreme values of the kinematic variables are chosen.
Before comparing our result to other approaches, we first address a natural question: what is the dependence of the IOE spectrum on the matching scales $Q_r$ and $Q_b$? In Fig. <ref> we plot the medium-induced spectrum as a function of $k_\perp/Q_{s0}$ for two different gluon frequencies, $\omega\!=\!5, 100$ GeV, truncating the spectrum at LO (left) or at LO+NLO (right). The central curves are obtained by solving Eqs. (<ref>) and (<ref>). Then, we perform an independent variation of the matching scales by a factor of $2$ ($1/2$) that lead to the uncertainty bands around the mid value. We recall that this variation is associated to the uncertainty in the definition of such scales, which are constrained up to an overall constant factor.
Let us discuss first the large $\omega$, large $k_\perp$ regime, i.e. the inset in the bottom row plots. Analytically, we have shown that, in the asymptotic kinematical region, the dependence on the matching scale vanishes when one considers LO+NLO contribution, see Eq. (<ref>). This is exactly what we observe numerically. Although this conclusion was reached for highly energetic gluons, the numerical results indicate that it holds reasonably well in the case of soft gluons too. Notice that when analyzing only the LO, a bigger, but still weak dependence on the matching scale $Q_b$ is observed. This corresponds to Eq. (<ref>), where a logarithmic dependence on $Q_b$ appears. Then, we reemphasize that only when considering LO+NLO the dependence on the matching scale is residual as due to the cancellation occurring between these two terms.
The small $\omega$, small $\k$ scenario is represented by the top row plots in Fig.~\ref{fig:qs-variations}. In this region, we argued in the previous section that all orders scale as the LO term, with logarithmic power corrections as one goes higher in the IOE, see Eqs.~\eqref{eq:ll_2} and \eqref{eq:ll_1}. Numerically, we observe that there is a large uncertainty due to the variation of the matching scales at LO, mainly from $Q_r$. However, if one includes the NLO term (top right) the dependence on the matching scales almost disappears. Regarding $Q_b$, we note that higher orders in the broadening factor $$ also enter through logarithmic power corrections on top of the LO term. Thus, the weaker dependence of LO+NLO on $Q_b$ as compared to LO is to be expected. These findings are inline with~\cite{IOE3}, where it was observed that for the energy spectrum the variations of the matching scale $Q_r$ although could drastically change the LO and the NLO terms separately, the sum LO+NLO was only sensitive to these variations at NNLO. This is a consequence of the fact once all orders in Eq.~\eqref{eq:ll_3} are considered, the spectrum becomes independent of $Q_r$. Since in Fig.~\ref{fig:qs-variations} the largest uncertainty comes from the scale $Q_r$, we argue that the result obtained here is a manifestation of the findings of~\cite{IOE3}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.45,page=2]{plot-Qs-variations.pdf} \includegraphics[scale=0.45,page=1]{plot-Qs-variations.pdf} \\
\includegraphics[scale=0.45,page=4]{plot-Qs-variations.pdf} \includegraphics[scale=0.45,page=3]{plot-Qs-variations.pdf}
\caption{Impact of variations by a factor of 2 in the two matching scales, $Q_b$ and $Q_r$, on the LO (left column) and the LO+NLO (right column) at two different frequencies: $\omega\!=\!5, 100$ GeV on the top and bottom rows, respectively. }
\label{fig:qs-variations}
\end{figure}
In Fig.~\ref{fig:lhc} we present the final comparison between the IOE spectrum and the other approaches mentioned above. We considered two gluon frequencies, $ω= 0.05 ω_c0$ and $ω= 2 ω_c0$, corresponding to the cases of soft and hard gluon and we use the GW mass $μ^2=0.43$~GeV$^2$. Again, we vary the matching scales of the IOE spectrum up and down by a factor of two, independently and then take the envelope to build the uncertainty band.
The overall conclusion from the two plots in Fig.~\ref{fig:lhc}, and the most important result of this paper, is that the IOE spectrum, up to NLO, already does a reasonable job at capturing the full solution (less than 25\% deviations), with the advantage that it requires considerably less computational power. The observed deviation from the full numerical result reflects the sensitivity of the transverse momentum distribution to the infrared. A wider separation between $μ^2$ and $Q_b^2$ or $Q_r^2$ would yield a better agreement. Let us split the discussion of Fig.~\ref{fig:lhc} into small and large transverse momentum.
The small $k_⊥$ and small $ω$ regime is dominated by multiple scattering contributions. Then, it is natural that the GLV spectrum fails to capture the full solution. The LO term of the IOE (related to the BDMPS-Z solution) already does a good job at describing the full result. Nonetheless, including the NLO term, improves not only the overall agreement with the full result, but also reduces the uncertainty band associated with the variation of the matching scales, as discussed in the previous section. When increasing the gluon's frequency, and still at small $k_⊥$, we observe that the LO+NLO result remarkably captures the full solution up to $5%$ deviations. Regarding GLV, its agreement with the full solution is improved with respect to the small frequency case and, curiously, coincides with the LO term. We do not expect this to be a systematic result for other choices of the medium parameters.
The large $k_⊥$ tail is generated by rare, hard scatterings in the medium. It is well known that the BDMPS-Z approximation does not correctly capture such contributions and, therefore, fails to describe the full solution. At small frequencies, we observe that the LO+NLO result approaches much faster the full solution than compared to GLV. This is to be expected, as at large, but not infinite $k_⊥$, multiple scatterings still play a role, despite being sub-leading. GLV lacks those effects and then needs an asymptotically large value of $k_⊥$ to reproduce the full solution, while the LO+NLO works even far from the asymptotic regime. At large frequencies, the spectrum is really dominated by a single hard scattering in the medium and thus the LO+NLO, full and GLV results rapidly converge.
\begin{figure}[h!]
\centering
\includegraphics[scale=.78]{plot-wx005-new.pdf}
\includegraphics[scale=.78]{plot-wx2-new.pdf}
\caption{Comparison between the GLV spectrum (dotted, purple), the LO result (dashed, green), the IOE at LO+NLO (solid, red) and the all-order spectrum (solid, navy) as computed in Ref.~\cite{CarlotaFabioLiliana} for two gluon frequencies: $\omega= 0.05\, \omega_{c0}$ (left) and $\omega= 2\, \omega_{c0}$ (right). The ratio to the full solution is presented in the bottom panels. The uncertainty band arises from variations in the matching scale.}
\label{fig:lhc}
\end{figure}
\section{Conclusion and Outlook}\label{sec:conclusion}
This work constitutes a natural extension of the recent studies of the medium-induced energy spectrum and broadening distribution using the Improved Opacity Expansion~\cite{IOE1,IOE2,IOE3,broadening_paper}. We have computed the in-medium radiative kernel using the IOE up to next-to-leading order accuracy, in the soft gluon approximation for a generic medium profile as well as for a brick plasma for which we have performed numerical
computations that we compare to full numerical results from \cite{CarlotaFabioLiliana}. We observe a very good agreement for a LHC-motivated choice of medium parameters.
From a theoretical viewpoint, the differential spectrum calculation highlights the role played by the matching scales that enters the definition of $q̂$, given that it convolutes contributions due to final state broadening with in-medium radiative terms. Each of these physical processes enters the IOE expansion with its own matching scale that we denote by $Q_b$, associated to final state broadening terms, and $Q_r$, related to the radiative kernel. These two scales are obtained by solving their corresponding transcendental equations that are given, in full generality, by Eqs.~\eqref{eq:Qb_final} and \eqref{eq:Qr_final}. We emphasize that these two scales have to be treated separately in order for the expansion to be consistent and well defined. Taking this into account, we derive the medium-induced spectrum for a smooth medium time profile and also in the brick limit. The final formulas are given in Sec.~\ref{sec:summary}.
Besides the master formulas, we analytically study the IOE for the plasma brick model in two asymptotic kinematical regimes. Firstly, we consider the regime where multiple soft exchanges with the medium constitute the dominant contribution, i.e. $ω≪ω_c$, and with the further assumption that the gluon is collinear, i.e. $k_⊥^2≪q̂L$. In this case, we recover the well-known factorization formula given by Eq.~\eqref{eq:ll_2}, often used in jet quenching phenomenology~\cite{Saclay,BDIM1,Blanco:2020uzy,Kutak1}. In particular, this result implies that in the soft limit the differential spectrum can be written as the product between the LO term and powers of $q̂_0/q̂$ that correspond to higher order contributions. This result agrees with what was observed in the energy spectrum calculation, as detailed in~\cite{IOE3}. Secondly, we study the physically opposite regime where a single hard scattering with the medium governs the dynamics of the medium-induced spectrum. This corresponds to a region of phase space where $ω≫ω_c$ and final momentum of the gluon satisfies $k_⊥^2≫q̂L$. In this regime, it is expected that the exact spectrum is given by the GLV result, and thus should be reproduced by the IOE approach. Indeed, we confirm that after considering both the LO and NLO contributions, non-trivial cancellations between these two orders occur such that one recovers the GLV spectrum. Not only the GLV result is reproduced, but also the dependence of the spectrum on the matching scales disappears order by order in $1/k^2_⊥$. Again, these cancellations resemble the situation in the energy spectrum calculation~\cite{IOE3}. Additionally, the explicit and detailed calculation carried out in order to check that the IOE recovers the GLV result provides a non-trivial cross-check on the main formulas derived in this paper.
Regarding the final numerical evaluation we find that, for the plasma brick model with LHC-inspired parameters, the computing time is in the ballpark of the LO/BDMPS-Z result~\cite{ASW2}. More concretely, we have evaluated the code's performance and encountered that the small $ω$ and small $$̨ regime requires more computational power due to the oscillating phases in the integrands. We provide ancillary Python files with the IOE spectrum together with the GLV expressions. The comparison with an all-orders resumed spectrum reveals a globally good agreement between the NLO spectrum from the Improved Opacity Expansion and the full solution, for this set of medium parameters. In particular, the agreement improves at high-frequencies, where the deviations between the two approaches are below $10\%$. This is a remarkable result given the relative simplicity of the approach presented in this paper as compared to larger computational cost needed to resum the spectrum to all orders. It is indicative of the power of the IOE approach to capture the correct dynamics at small and large frequencies simultaneously. A more thorough comparison with the full numerical result including a scan of the parameters space is left for future work. We expect the agreement to systematically improve for denser or larger plasma for which the scale separation between $Q_r$ or $Q_b$ and the infrared scale $\mu^2$ is larger. Obviously, the IOE scheme is exact asymptotically.
Our results provide for the first time a unified analytic framework for the fully differential medium-induced radiation spectrum that accounts for both the GLV and BDMPS limits. We expect that adopting our radiative kernel in future phenomenological studies would substantially reduce model dependence of jet quenching observables as well as theoretical uncertainties on the extraction of medium transport properties such as the jet quenching parameter, $\hat q$ that is a function of the typical scale of the process. Two phenomenological applications have been already proposed in the literature that concern quenching effects on the jet spectrum <cit.>. A natural continuation of this work is to use the in-medium radiative kernel that we have derived in this paper to analytically compute observables where the gluon transverse momentum information is not integrated out, namely jet substructure observables. In particular, on-going measurements of the $k_\perp$-distribution of the hardest splitting <cit.> would benefit from a theoretical calculation in which both multiple soft scatterings and hard momentum exchanges are correctly incorporated. This study would open up a new theoretical window onto extending the IOE framework to describe the energy loss of a quark-antiquark antenna. In parallel, we would like to implement the formulas that we have derived in this paper in a suitable Monte Carlo framework such as Ref. <cit.>.
§ NOTE
While this manuscript was being produced an independent derivation of the differential spectrum (for a massive quark) using the IOE approach was presented in Ref. <cit.>. Although performing a numerical comparison is beyond the scope of this work we would like to point out a couple of differences. Firstly, in comparison with <cit.> we have presented results for a generic medium profile for which we were able to reduce further the number of integration variables. Secondly, in Ref. <cit.> a single matching scale was used for the radiative and broadening parts, i.e., $Q_b=Q_r$ which we found leads to an incorrect description of the spectrum.
§ ACKNOWLEDGEMENTS
We are grateful to the authors of Ref. <cit.> for providing the numerical results from their study. In particular, we wish to thank Carlota Andrés and Fabio Dominguez for clarifying some important details in their work and for the careful reading of the present manuscript. We wish to thank Carlos Salgado for helpful discussions on related problems and Liliana Apolinário for providing clarifications regarding the GLV result obtained in Ref. <cit.>.
The project that gave rise to these results received the support of a fellowship from “la Caixa" Foundation (ID 100010434). The fellowship code is LCF/BQ/ DI18/11660057. This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 713673. J.B. is supported by Ministerio de Ciencia e Innovacion of Spain under project FPA2017-83814-P; Unidad de Excelencia Maria de Maetzu under project MDM-2016-0692; European research Council project ERC-2018-ADG-835105 YoctoLHC; and Xunta de Galicia (Conselleria de Educacion) and FEDER. The work of Y. M.-T. was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract No. DE- SC0012704. K. T. is supported by a Starting Grant from Trond Mohn Foundation (BFS2018REK01) and the University of Bergen. Y. M.-T. acknowledges support from the RHIC Physics Fellow Program of the RIKEN BNL Research Center. A.S.O.’s work was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 788223, PanScales).
§ THE ANALYTIC SOLUTIONS OF THE EMISSION KERNEL $\CK$
In this appendix, we discuss two analytic solutions for the emission kernel $\cK$ satisfying
\begin{equation}\label{eq:cK_Sch_app}
\left[i\partial_{t_2}+\frac{\bdel^2_\x}{2\omega^2}+iv(\x,t_2)\right]\cK(\x,t_2;\y,t_1)=i\delta^{(2)}(\x-\y)\delta(t_2-t_1) \, .
\end{equation}
This propagator obeys a Dyson-like relation reading <cit.>
\begin{equation}\label{eq:Dyson_GLV_app}
\cK(\x,t_2;\y,t_1) = \cK_0(\x,t_2;\y,t_1)-\int_\z \int_{t_1}^{t_2} \rmd s \, \cK_0(\x,t_2;\z,s)v(\z,s)\cK(\z,s;\y,t_1) \, ,
\end{equation}
where $\cK_0$ corresponds to vacuum solution to Eq. (<ref>) with $v=0$. Alternatively, and as discussed in the main text, one can write an equivalent relation to eq:Dyson_GLV_app, but expanding around the solution to Eq. (<ref>) with $v\to \vLO$ and using the decomposition $v=\vLO+\delta v$ (see Eq. (<ref>)),
\begin{equation}\label{eq:cK_ful_IOE_app}
\cK(\x,t_2;\y,t_1) = \cK^{\rm LO}(\x,t_2;\y,t_1)-\int_\z \int_{t_1}^{t_2} \rmd s \, \cK^{\rm LO}(\x,t_2;\z,s)\delta v(\z,s)\cK(\z,s;\y,t_1) \, .
\end{equation}
Eqs. (<ref>) and (<ref>) are particularly useful since $\cK_0$ and $\cK^{\rm LO}$ admit a closed form, which can be obtained by directly solving Eq. (<ref>), and thus they can be easily applied in a perturbative framework.
In the first case where $v=0$, $\cK_0$ is the Green's function of a Schrödinger equation describing the motion of a non-relativistic free particle in two dimensions, and thus reads
\begin{align}\label{eq:cK_vac_app}
\cK_0(\x,t_2;\y,t_1)= \frac{\omega }{2\pi i (t_2-t_1) } \exp\left(i\frac{\omega(\x-\y)^2}{2(t_2-t_1)}\right) \, .
\end{align}
The case where $v(\x)=\vLO(\x)$ in Eq. (<ref>) is also easily solved since in this case $\cK$ is the Green's function associated to the motion of a single particle in a harmonic potential. For quadratic potentials, the exact solution to Eq. (<ref>) can be exactly obtained by using the so called method of fluctuations <cit.>, resulting in
\begin{align}\label{eq:cK_BDMPS_app}
\cK^{\rm LO}(\x,t_2;\y,t_1)&= \frac{\omega}{2\pi i S(t_2,t_1)}\exp\left( \frac{i\omega}{2S(t_2,t_1)} \left[ C(t_1,t_2)\,\x^2+C(t_2,t_1)\,\y^2-2 \x\cdot\y\right] \right) \,,
\end{align}
where we recall that the $C$ and $S$ functions satisfy
\begin{equation}\label{eq:Abel}
\begin{split}
&\left[\frac{\rmd^2}{\rmd^2t}+\Omega^2(t)\right]S(t,t_0)=0 \, ,\quad S(t_0,t_0)=0 \,,\quad \partial_t S(t,t_0)_{t=t_0}=1 \, , \\
&\left[\frac{\rmd^2}{\rmd^2t}+\Omega^2(t)\right]C(t,t_0)=0 \, ,\quad C(t_0,t_0)=1 \,,\quad \partial_t C(t,t_0)_{t=t_0}=0\, .
\end{split}
\end{equation}
For any $\Omega$, i.e. for a generic time profile of the medium, one can derive certain identities relating the $C$ and $S$ functions that were employed in the main text to simplify the emission spectrum. Firstly, these solutions are related by $C(t_1,t_2)=\partial_{t_2}S(t_2,t_1)$ and by the associated Wronskian ($W$), which reads
\begin{equation}
W=C(t_1,t_2)\partial_{t_1}S(t_1,t_2)-\partial_{t_1}C(t_1,t_2)S(t_1,t_2)=1\, ,
\end{equation}
where we used the above the initial conditions. The condition $W=1$ can be used to show that
\begin{equation}\label{eq:prop_app_1}
\partial_{t_1}\frac{C(t_1,t_2)}{S(t_1,t_2)}=-\frac{C(t_1,t_2)\partial_{t_1}S(t_1,t_2)-\partial_{t_1}C(t_1,t_2)S(t_1,t_2)}{S^2(t_1,t_2)} =-\frac{1}{S^2(t_1,t_2)} \, ,
\end{equation}
which is used to derive eq:id_1. In addition, $W=1$ implies that $C$ and $S$ are linearly independent solutions and thus any other solution to the above ordinary differential equation can be written as a linear combination of them. Using this fact, and for a time ordering $t_2>t_1>t_0$, any solution in $(t_2,t_1)$ can be written as <cit.>
\begin{equation}\label{eq:linear_rel_C_S}
\begin{split}
&S(t_2,t_1)=C(t_1,t_0)S(t_2,t_0)-S(t_1,t_0)C(t_2,t_0) \, ,\\
&C(t_2,t_1)=-\partial_{t_1}C(t_1,t_0)S(t_2,t_0)+\partial_{t_1}S(t_1,t_0)C(t_2,t_0) \, ,
\end{split}
\end{equation}
where it is easy to verify that these equations satisfy the above initial conditions and that $S(t_2,t_1)=-S(t_1,t_2)$. This decomposition of the $C$ and $S$ is extensively used in the main text; here we give a simple application to derive another useful identity.
Let us consider the brick model introduced in the main text, with a medium of extension $L$ such that $\hat{q}(t\geq L)=0$, but still allowing the jet quenching parameter not to be constant in time inside the medium. In the vacuum, the solutions to the $C$ and $S$ functions are trivially found
\begin{equation}
\begin{split}
S(t_2,t_1)=t_2-t_1 \, , \quad C(t_2,t_1)=1 \, ,
\end{split}
\end{equation}
and indeed when introduced back in Eq. (<ref>) give Eq. (<ref>). Using Eq. (<ref>) with $t_2=+\infty$, $t_1>L$ and $t_0=L$, combined with the explicit forms for vacuum $C$ and $S$ solutions, one observes that terms proportional to $S(t_2,t_1)$ dominate, leading to the handy formula
\begin{equation}
\frac{C(\infty,t_1)}{S(\infty,t_1)} = -\frac{\partial_{t_1} C(t_1,L)}{C(t_1,L)}=\Omega^2(t_1) \frac{S(t_1,L)}{C(t_1,L)} \, ,
\end{equation}
where the last equality holds if $C$ is even in its arguments.
|
Mean | 0.261 | 0.268 | 0.259 | 0.250 | 0.267 | 0.155 | 0.215 | 0.15 | 0.246 | 0.205 | 0.176 | 0.264 | 0.228 | 0.14 | 0.254 | 0.200 | 0.123 | 0.241 | 0.188 | 0.273 | 0.216 | 0.215 | 0.212 | 0.229 | 0.155 | 0.215 | 0.17 | 0.217 | 0.224 | 0.321 | 0.309 | 0.318 | 0.273 | 0.216 | 0.215 | 0.212 | 0.229 | 0.148 | 0.245 | 0.174 | 0.211 | 0.496 | 0.379 | 0.549 | 0.469 | 0.475 | 0.438 | 0.454 | 0.501 | 0.445 | 0.394 | 0.567 | 0.413 | 0.548
(a) Average AE values of each algorithm per data set
| AC | AC-LR | AC-RF | AC-SV | AC-AB | PAC | PAC-LR | TSX | TSX-LR | TSX-SV | TS50 | TS50-LR | TS50-SV | TSMax | TSMax-LR | TSMax-SV | MS | MS-LR | MS-SV | GAC | GAC-LR | GAC-RF | GAC-SV | GAC-AB | GPAC | GPAC-LR | DyS | DyS-LR | DyS-SV | FMM | FMM-LR | FMM-SV | HDy | HDy-LR | HDy-RF | HDy-SV | HDy-AB | FM | FM-LR | EM | EM-LR | CDE | CDE-LR | CC | CC-LR | CC-RF | CC-SV | CC-AB | PCC | PCC-LR | SVM-K | SVM-Q | RBF-K | RBF-Q
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
ads | 0.078 | 0.082 | 0.094 | 0.081 | 0.058 | 0.074 | 0.067 | 0.042 | 0.062 | 0.040 | 0.066 | 0.086 | 0.069 | 0.028 | 0.057 | 0.031 | 0.029 | 0.059 | 0.035 | 0.077 | 0.041 | 0.063 | 0.043 | 0.055 | 0.074 | 0.066 | 0.019 | 0.03 | 0.026 | 0.066 | 0.062 | 0.068 | 0.076 | 0.040 | 0.060 | 0.042 | 0.053 | 0.039 | 0.065 | 0.027 | 0.045 | 0.186 | 0.115 | 0.134 | 0.104 | 0.120 | 0.108 | 0.056 | 0.108 | 0.089 | 0.071 | 0.107 | 0.156 | 0.173
alco | 0.287 | 0.450 | 0.405 | 0.372 | 0.455 | 0.251 | 0.334 | 0.269 | 0.449 | 0.321 | 0.295 | 0.457 | 0.327 | 0.171 | 0.417 | 0.274 | 0.17 | 0.418 | 0.275 | 0.277 | 0.182 | 0.181 | 0.176 | 0.240 | 0.258 | 0.329 | 0.191 | 0.215 | 0.211 | 0.258 | 0.217 | 0.239 | 0.273 | 0.179 | 0.183 | 0.174 | 0.236 | 0.260 | 0.426 | 0.102 | 0.194 | 0.779 | 0.715 | 0.391 | 0.499 | 0.497 | 0.457 | 0.499 | 0.254 | 0.272 | 0.167 | 0.238 | 0.363 | 0.308
bc-cat | 0.067 | 0.052 | 0.065 | 0.052 | 0.071 | 0.064 | 0.036 | 0.037 | 0.063 | 0.028 | 0.093 | 0.103 | 0.070 | 0.020 | 0.050 | 0.020 | 0.021 | 0.055 | 0.027 | 0.088 | 0.030 | 0.062 | 0.030 | 0.071 | 0.064 | 0.036 | 0.024 | 0.029 | 0.026 | 0.105 | 0.112 | 0.132 | 0.087 | 0.029 | 0.057 | 0.029 | 0.069 | 0.023 | 0.036 | 0.16 | 0.032 | 0.405 | 0.079 | 0.182 | 0.056 | 0.060 | 0.054 | 0.038 | 0.123 | 0.054 | 0.080 | 0.316 | 0.038 | 0.05
bc-cont | 0.030 | 0.037 | 0.036 | 0.034 | 0.044 | 0.039 | 0.029 | 0.015 | 0.029 | 0.016 | 0.066 | 0.083 | 0.086 | 0.012 | 0.026 | 0.014 | 0.018 | 0.043 | 0.030 | 0.051 | 0.025 | 0.036 | 0.023 | 0.047 | 0.039 | 0.029 | 0.009 | 0.014 | 0.019 | 0.087 | 0.090 | 0.090 | 0.051 | 0.025 | 0.035 | 0.023 | 0.046 | 0.015 | 0.019 | 0.087 | 0.059 | 0.181 | 0.135 | 0.066 | 0.028 | 0.025 | 0.026 | 0.026 | 0.060 | 0.052 | 0.035 | 0.447 | 0.033 | 0.009
cappl | 0.146 | 0.232 | 0.186 | 0.204 | 0.151 | 0.092 | 0.162 | 0.080 | 0.221 | 0.106 | 0.141 | 0.256 | 0.153 | 0.052 | 0.210 | 0.089 | 0.058 | 0.213 | 0.090 | 0.155 | 0.108 | 0.102 | 0.102 | 0.108 | 0.094 | 0.156 | 0.065 | 0.094 | 0.104 | 0.155 | 0.142 | 0.171 | 0.153 | 0.107 | 0.097 | 0.101 | 0.105 | 0.053 | 0.189 | 0.037 | 0.132 | 0.410 | 0.404 | 0.244 | 0.260 | 0.217 | 0.238 | 0.176 | 0.159 | 0.178 | 0.093 | 0.086 | 0.188 | 0.227
cars | 0.044 | 0.044 | 0.058 | 0.038 | 0.039 | 0.050 | 0.037 | 0.025 | 0.039 | 0.026 | 0.048 | 0.058 | 0.051 | 0.015 | 0.030 | 0.019 | 0.018 | 0.033 | 0.023 | 0.051 | 0.031 | 0.036 | 0.027 | 0.026 | 0.049 | 0.036 | 0.014 | 0.014 | 0.017 | 0.061 | 0.053 | 0.052 | 0.050 | 0.031 | 0.036 | 0.027 | 0.026 | 0.030 | 0.038 | 0.034 | 0.019 | 0.208 | 0.045 | 0.099 | 0.050 | 0.065 | 0.038 | 0.046 | 0.083 | 0.045 | 0.051 | 0.045 | 0.241 | 0.288
conc | 0.222 | 0.142 | 0.150 | 0.099 | 0.129 | 0.129 | 0.121 | 0.057 | 0.127 | 0.076 | 0.091 | 0.154 | 0.089 | 0.037 | 0.119 | 0.061 | 0.041 | 0.117 | 0.064 | 0.155 | 0.086 | 0.113 | 0.066 | 0.105 | 0.129 | 0.114 | 0.072 | 0.057 | 0.058 | 0.149 | 0.108 | 0.116 | 0.153 | 0.084 | 0.110 | 0.065 | 0.102 | 0.090 | 0.113 | 0.325 | 0.066 | 0.795 | 0.264 | 0.494 | 0.211 | 0.195 | 0.146 | 0.168 | 0.245 | 0.152 | 0.074 | 0.306 | 0.067 | 0.253
contra | 0.449 | 0.448 | 0.320 | 0.420 | 0.354 | 0.245 | 0.301 | 0.206 | 0.422 | 0.373 | 0.234 | 0.425 | 0.385 | 0.128 | 0.405 | 0.324 | 0.121 | 0.403 | 0.323 | 0.241 | 0.204 | 0.176 | 0.190 | 0.151 | 0.245 | 0.295 | 0.227 | 0.215 | 0.29 | 0.266 | 0.203 | 0.289 | 0.236 | 0.201 | 0.173 | 0.188 | 0.149 | 0.258 | 0.409 | 0.125 | 0.18 | 0.839 | 0.809 | 0.579 | 0.539 | 0.429 | 0.513 | 0.460 | 0.286 | 0.280 | 0.197 | 0.381 | 0.213 | 0.352
craft | 0.119 | 0.086 | 0.077 | 0.090 | 0.120 | 0.048 | 0.056 | 0.025 | 0.061 | 0.050 | 0.035 | 0.067 | 0.051 | 0.015 | 0.051 | 0.033 | 0.019 | 0.055 | 0.038 | 0.105 | 0.065 | 0.066 | 0.067 | 0.086 | 0.048 | 0.054 | 0.010 | 0.137 | 0.029 | 0.058 | 0.053 | 0.079 | 0.103 | 0.063 | 0.064 | 0.066 | 0.084 | 0.026 | 0.053 | 0.089 | 0.031 | 0.728 | 0.332 | 0.317 | 0.216 | 0.208 | 0.250 | 0.230 | 0.199 | 0.167 | 0.090 | 0.306 | 0.079 | 0.211
drugs | 0.082 | 0.151 | 0.253 | 0.164 | 0.170 | 0.056 | 0.087 | 0.063 | 0.131 | 0.058 | 0.077 | 0.139 | 0.068 | 0.041 | 0.132 | 0.05 | 0.046 | 0.132 | 0.051 | 0.092 | 0.071 | 0.103 | 0.072 | 0.099 | 0.057 | 0.085 | 0.022 | 0.077 | 0.042 | 0.088 | 0.103 | 0.108 | 0.090 | 0.069 | 0.101 | 0.071 | 0.097 | 0.044 | 0.128 | 0.022 | 0.069 | 0.092 | 0.420 | 0.144 | 0.230 | 0.321 | 0.243 | 0.241 | 0.134 | 0.172 | 0.078 | 0.088 | 0.239 | 0.269
flare | 0.395 | 0.504 | 0.481 | 0.474 | 0.486 | 0.245 | 0.337 | 0.274 | 0.490 | 0.385 | 0.304 | 0.495 | 0.398 | 0.197 | 0.478 | 0.338 | 0.2 | 0.479 | 0.340 | 0.294 | 0.181 | 0.199 | 0.187 | 0.293 | 0.242 | 0.330 | 0.213 | 0.243 | 0.368 | 0.290 | 0.239 | 0.353 | 0.289 | 0.179 | 0.199 | 0.186 | 0.288 | 0.232 | 0.448 | 0.081 | 0.244 | 0.707 | 0.720 | 0.419 | 0.512 | 0.501 | 0.497 | 0.479 | 0.256 | 0.284 | 0.159 | 0.243 | 0.259 | 0.295
grid | 0.029 | 0.027 | 0.050 | 0.007 | 0.021 | 0.015 | 0.017 | 0.009 | 0.011 | 0.004 | 0.009 | 0.014 | 0.006 | 0.006 | 0.009 | 0.003 | 0.005 | 0.007 | 0.004 | 0.034 | 0.027 | 0.042 | 0.006 | 0.017 | 0.015 | 0.017 | 0.002 | 0.003 | 0.001 | 0.017 | 0.017 | 0.006 | 0.033 | 0.026 | 0.041 | 0.006 | 0.017 | 0.009 | 0.011 | 0.014 | 0.038 | 0.409 | 0.319 | 0.188 | 0.152 | 0.176 | 0.030 | 0.124 | 0.151 | 0.145 | 0.596 | 0.425 | 0.037 | 0.23
mush | 0.001 | 0.001 | 0.002 | 0.008 | 0.002 | 0.001 | 0.001 | 0.001 | 0.001 | 0.007 | 0.009 | 0.013 | 0.019 | 0.001 | 0.001 | 0.005 | 0.001 | 0.003 | 0.005 | 0.002 | 0.001 | 0.003 | 0.008 | 0.002 | 0.001 | 0.001 | 0.000 | 0.000 | 0.005 | 0.003 | 0.003 | 0.006 | 0.002 | 0.001 | 0.003 | 0.008 | 0.002 | 0.001 | 0.001 | 0.001 | 0.000 | 0.004 | 0.001 | 0.003 | 0.001 | 0.002 | 0.001 | 0.001 | 0.006 | 0.002 | 0.016 | 0.007 | 0.002 | 0.202
music | 0.329 | 0.474 | 0.472 | 0.347 | 0.458 | 0.246 | 0.343 | 0.255 | 0.446 | 0.291 | 0.274 | 0.454 | 0.304 | 0.182 | 0.439 | 0.258 | 0.185 | 0.437 | 0.261 | 0.257 | 0.188 | 0.191 | 0.178 | 0.275 | 0.246 | 0.340 | 0.184 | 0.238 | 0.212 | 0.256 | 0.205 | 0.261 | 0.252 | 0.185 | 0.187 | 0.175 | 0.270 | 0.222 | 0.420 | 0.082 | 0.214 | 0.825 | 0.773 | 0.473 | 0.538 | 0.543 | 0.450 | 0.476 | 0.270 | 0.285 | 0.136 | 0.204 | 0.290 | 0.369
musk | 0.040 | 0.038 | 0.075 | 0.027 | 0.042 | 0.028 | 0.029 | 0.029 | 0.036 | 0.016 | 0.043 | 0.050 | 0.045 | 0.014 | 0.023 | 0.01 | 0.018 | 0.025 | 0.013 | 0.045 | 0.029 | 0.051 | 0.023 | 0.036 | 0.028 | 0.028 | 0.008 | 0.013 | 0.008 | 0.041 | 0.038 | 0.035 | 0.044 | 0.028 | 0.050 | 0.023 | 0.034 | 0.028 | 0.033 | 0.011 | 0.023 | 0.195 | 0.064 | 0.116 | 0.076 | 0.127 | 0.041 | 0.078 | 0.109 | 0.074 | 0.049 | 0.087 | 0.088 | 0.282
spam | 0.058 | 0.021 | 0.032 | 0.018 | 0.019 | 0.035 | 0.017 | 0.009 | 0.017 | 0.013 | 0.026 | 0.030 | 0.026 | 0.007 | 0.014 | 0.009 | 0.008 | 0.015 | 0.010 | 0.121 | 0.017 | 0.030 | 0.015 | 0.019 | 0.035 | 0.017 | 0.005 | 0.011 | 0.007 | 0.020 | 0.015 | 0.022 | 0.118 | 0.016 | 0.030 | 0.015 | 0.019 | 0.013 | 0.015 | 0.218 | 0.009 | 0.714 | 0.019 | 0.351 | 0.064 | 0.071 | 0.057 | 0.047 | 0.200 | 0.062 | 0.061 | 0.298 | 0.045 | 0.265
study | 0.124 | 0.125 | 0.102 | 0.113 | 0.087 | 0.114 | 0.108 | 0.082 | 0.120 | 0.091 | 0.102 | 0.132 | 0.105 | 0.058 | 0.107 | 0.078 | 0.056 | 0.105 | 0.074 | 0.144 | 0.078 | 0.077 | 0.073 | 0.084 | 0.114 | 0.105 | 0.078 | 0.053 | 0.055 | 0.152 | 0.107 | 0.116 | 0.142 | 0.077 | 0.072 | 0.072 | 0.083 | 0.102 | 0.101 | 0.084 | 0.052 | 0.684 | 0.123 | 0.336 | 0.157 | 0.162 | 0.153 | 0.124 | 0.202 | 0.120 | 0.213 | 0.283 | 0.233 | 0.306
telco | 0.204 | 0.171 | 0.147 | 0.215 | 0.158 | 0.037 | 0.074 | 0.027 | 0.093 | 0.165 | 0.035 | 0.103 | 0.176 | 0.017 | 0.093 | 0.145 | 0.023 | 0.095 | 0.148 | 0.119 | 0.074 | 0.074 | 0.086 | 0.069 | 0.039 | 0.073 | 0.012 | 0.075 | 0.114 | 0.050 | 0.071 | 0.125 | 0.116 | 0.072 | 0.074 | 0.084 | 0.067 | 0.031 | 0.092 | 0.007 | 0.042 | 0.527 | 0.556 | 0.283 | 0.316 | 0.311 | 0.351 | 0.309 | 0.186 | 0.204 | 0.099 | 0.151 | 0.224 | 0.299
thrm | 0.352 | 0.377 | 0.309 | 0.299 | 0.340 | 0.249 | 0.296 | 0.220 | 0.378 | 0.298 | 0.250 | 0.383 | 0.298 | 0.142 | 0.344 | 0.234 | 0.146 | 0.348 | 0.231 | 0.223 | 0.165 | 0.181 | 0.171 | 0.257 | 0.249 | 0.291 | 0.202 | 0.181 | 0.231 | 0.245 | 0.179 | 0.276 | 0.220 | 0.163 | 0.176 | 0.168 | 0.251 | 0.219 | 0.352 | 0.191 | 0.18 | 0.833 | 0.638 | 0.532 | 0.444 | 0.387 | 0.386 | 0.379 | 0.275 | 0.261 | 0.164 | 0.295 | 0.176 | 0.281
turk | 0.450 | 0.544 | 0.555 | 0.436 | 0.584 | 0.192 | 0.275 | 0.208 | 0.444 | 0.453 | 0.213 | 0.438 | 0.457 | 0.125 | 0.421 | 0.373 | 0.123 | 0.421 | 0.374 | 0.246 | 0.182 | 0.184 | 0.177 | 0.191 | 0.190 | 0.266 | 0.115 | 0.309 | 0.32 | 0.221 | 0.282 | 0.292 | 0.239 | 0.178 | 0.180 | 0.175 | 0.188 | 0.205 | 0.415 | 0.048 | 0.195 | 0.839 | 0.837 | 0.611 | 0.640 | 0.635 | 0.592 | 0.650 | 0.292 | 0.299 | 0.215 | 0.294 | 0.195 | 0.39
vgame | 0.067 | 0.131 | 0.171 | 0.128 | 0.153 | 0.044 | 0.096 | 0.038 | 0.129 | 0.076 | 0.037 | 0.124 | 0.076 | 0.024 | 0.123 | 0.063 | 0.027 | 0.125 | 0.065 | 0.130 | 0.055 | 0.098 | 0.062 | 0.075 | 0.044 | 0.093 | 0.019 | 0.061 | 0.053 | 0.061 | 0.060 | 0.072 | 0.127 | 0.054 | 0.097 | 0.061 | 0.073 | 0.040 | 0.125 | 0.013 | 0.061 | 0.758 | 0.294 | 0.323 | 0.286 | 0.373 | 0.312 | 0.325 | 0.215 | 0.196 | 0.114 | 0.267 | 0.443 | 0.397
voice | 0.014 | 0.008 | 0.010 | 0.006 | 0.007 | 0.025 | 0.007 | 0.005 | 0.005 | 0.006 | 0.022 | 0.021 | 0.026 | 0.003 | 0.004 | 0.004 | 0.006 | 0.008 | 0.009 | 0.066 | 0.008 | 0.009 | 0.006 | 0.008 | 0.024 | 0.007 | 0.003 | 0.002 | 0.002 | 0.025 | 0.017 | 0.020 | 0.065 | 0.008 | 0.010 | 0.006 | 0.007 | 0.014 | 0.004 | 0.121 | 0.002 | 0.462 | 0.012 | 0.153 | 0.012 | 0.013 | 0.011 | 0.010 | 0.113 | 0.021 | 0.032 | 0.183 | 0.013 | 0.149
wine | 0.316 | 0.201 | 0.096 | 0.136 | 0.244 | 0.049 | 0.117 | 0.025 | 0.174 | 0.098 | 0.027 | 0.176 | 0.096 | 0.016 | 0.178 | 0.081 | 0.014 | 0.176 | 0.080 | 0.164 | 0.081 | 0.071 | 0.068 | 0.102 | 0.048 | 0.114 | 0.022 | 0.078 | 0.073 | 0.074 | 0.087 | 0.103 | 0.161 | 0.080 | 0.072 | 0.067 | 0.100 | 0.047 | 0.167 | 0.211 | 0.091 | 0.827 | 0.781 | 0.523 | 0.380 | 0.296 | 0.328 | 0.388 | 0.262 | 0.238 | 0.248 | 0.513 | 0.115 | 0.391
yeast | 0.483 | 0.426 | 0.257 | 0.318 | 0.302 | 0.178 | 0.290 | 0.167 | 0.418 | 0.264 | 0.206 | 0.442 | 0.287 | 0.099 | 0.377 | 0.228 | 0.101 | 0.386 | 0.229 | 0.199 | 0.180 | 0.152 | 0.162 | 0.187 | 0.181 | 0.278 | 0.128 | 0.206 | 0.215 | 0.216 | 0.187 | 0.235 | 0.196 | 0.177 | 0.149 | 0.159 | 0.183 | 0.188 | 0.388 | 0.373 | 0.175 | 0.838 | 0.774 | 0.650 | 0.533 | 0.370 | 0.433 | 0.423 | 0.291 | 0.273 | 0.228 | 0.634 | 0.174 | 0.5
Mean | 0.183 | 0.199 | 0.183 | 0.170 | 0.187 | 0.104 | 0.135 | 0.090 | 0.182 | 0.136 | 0.113 | 0.196 | 0.153 | 0.059 | 0.171 | 0.114 | 0.061 | 0.173 | 0.117 | 0.139 | 0.088 | 0.096 | 0.084 | 0.109 | 0.105 | 0.132 | 0.068 | 0.098 | 0.104 | 0.124 | 0.110 | 0.136 | 0.136 | 0.086 | 0.094 | 0.083 | 0.106 | 0.091 | 0.169 | 0.103 | 0.09 | 0.552 | 0.384 | 0.317 | 0.263 | 0.254 | 0.238 | 0.240 | 0.187 | 0.164 | 0.136 | 0.259 | 0.163 | 0.271
(b) Average NKLD scores for each algorithm per data set
Table 6: Main results for binary quantification using tuned classifiers. We
show the averaged error scores over all scenarios per algorithm and data set.
| AC | AC-LR | AC-RF | AC-SV | AC-AB | PAC | PAC-LR | TSX | TSX-LR | TSX-SV | TS50 | TS50-LR | TS50-SV | TSMax | TSMax-LR | TSMax-SV | MS | MS-LR | MS-SV | GAC | GAC-LR | GAC-RF | GAC-SV | GAC-AB | GPAC | GPAC-LR | DyS | DyS-LR | DyS-SV | FMM | FMM-LR | FMM-SV | HDy | HDy-LR | HDy-RF | HDy-SV | HDy-AB | FM | FM-LR | EM | EM-LR | CC | CC-LR | CC-RF | CC-SV | CC-AB | PCC | PCC-LR
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
conc | 0.623 | 0.524 | 0.347 | 0.353 | 0.421 | 0.566 | 0.513 | 0.528 | 0.520 | 0.325 | 0.517 | 0.524 | 0.322 | 0.455 | 0.444 | 0.296 | 0.459 | 0.460 | 0.323 | 0.486 | 0.302 | 0.299 | 0.259 | 0.423 | 0.473 | 0.300 | 0.589 | 0.542 | 0.380 | 0.627 | 0.532 | 0.445 | 0.484 | 0.298 | 0.310 | 0.260 | 0.421 | 0.510 | 0.303 | 0.498 | 0.241 | 0.915 | 0.568 | 0.563 | 0.459 | 0.733 | 0.692 | 0.534
contra | 0.658 | 0.705 | 0.627 | 0.788 | 0.654 | 0.494 | 0.549 | 0.483 | 0.578 | 0.577 | 0.486 | 0.574 | 0.580 | 0.451 | 0.501 | 0.540 | 0.443 | 0.499 | 0.529 | 0.600 | 0.495 | 0.490 | 0.534 | 0.517 | 0.515 | 0.580 | 0.519 | 0.574 | 0.691 | 0.653 | 0.600 | 0.674 | 0.594 | 0.494 | 0.498 | 0.537 | 0.521 | 0.512 | 0.623 | 0.396 | 0.459 | 0.833 | 0.823 | 0.808 | 0.835 | 0.829 | 0.699 | 0.702
craft | 0.496 | 0.451 | 0.325 | 0.521 | 0.473 | 0.525 | 0.426 | 0.420 | 0.434 | 0.330 | 0.393 | 0.395 | 0.302 | 0.394 | 0.401 | 0.323 | 0.400 | 0.419 | 0.320 | 0.296 | 0.239 | 0.250 | 0.280 | 0.337 | 0.190 | 0.194 | 0.519 | 0.539 | 0.370 | 0.547 | 0.501 | 0.473 | 0.288 | 0.234 | 0.239 | 0.271 | 0.325 | 0.190 | 0.196 | 0.191 | 0.181 | 0.752 | 0.674 | 0.673 | 0.707 | 0.716 | 0.654 | 0.620
drugs | 0.240 | 0.327 | 0.373 | 0.277 | 0.378 | 0.179 | 0.188 | 0.222 | 0.193 | 0.191 | 0.227 | 0.198 | 0.192 | 0.230 | 0.201 | 0.197 | 0.210 | 0.193 | 0.183 | 0.256 | 0.259 | 0.284 | 0.247 | 0.391 | 0.199 | 0.212 | 0.194 | 0.188 | 0.203 | 0.396 | 0.387 | 0.410 | 0.255 | 0.257 | 0.276 | 0.246 | 0.395 | 0.181 | 0.193 | 0.218 | 0.207 | 0.465 | 0.522 | 0.623 | 0.518 | 0.648 | 0.482 | 0.545
thrm | 1.059 | 1.003 | 0.688 | 0.772 | 0.885 | 0.621 | 0.646 | 0.655 | 0.718 | 0.617 | 0.605 | 0.696 | 0.608 | 0.541 | 0.615 | 0.554 | 0.534 | 0.612 | 0.547 | 0.780 | 0.694 | 0.565 | 0.645 | 0.760 | 0.629 | 0.661 | 0.689 | 0.750 | 0.678 | 0.736 | 0.695 | 0.645 | 0.778 | 0.694 | 0.567 | 0.648 | 0.759 | 0.663 | 0.752 | 0.494 | 0.533 | 1.042 | 1.028 | 0.893 | 0.928 | 1.115 | 0.769 | 0.759
turk | 0.569 | 0.670 | 0.797 | 0.699 | 0.792 | 0.348 | 0.387 | 0.420 | 0.442 | 0.598 | 0.438 | 0.454 | 0.606 | 0.392 | 0.415 | 0.505 | 0.393 | 0.416 | 0.505 | 0.525 | 0.506 | 0.572 | 0.518 | 0.562 | 0.342 | 0.385 | 0.422 | 0.492 | 0.554 | 0.607 | 0.602 | 0.647 | 0.524 | 0.511 | 0.561 | 0.520 | 0.566 | 0.392 | 0.455 | 0.277 | 0.371 | 0.976 | 0.987 | 1.003 | 1.028 | 0.987 | 0.727 | 0.731
vgame | 0.712 | 0.736 | 0.735 | 0.779 | 0.775 | 0.633 | 0.614 | 0.606 | 0.598 | 0.626 | 0.612 | 0.598 | 0.625 | 0.558 | 0.544 | 0.576 | 0.558 | 0.549 | 0.580 | 0.520 | 0.519 | 0.529 | 0.536 | 0.567 | 0.460 | 0.460 | 0.543 | 0.564 | 0.647 | 0.482 | 0.486 | 0.523 | 0.519 | 0.524 | 0.527 | 0.534 | 0.576 | 0.474 | 0.466 | 0.322 | 0.338 | 0.590 | 0.575 | 0.658 | 0.614 | 0.694 | 0.520 | 0.494
wine | 0.873 | 0.823 | 0.692 | 0.679 | 0.803 | 0.663 | 0.648 | 0.604 | 0.636 | 0.607 | 0.611 | 0.661 | 0.595 | 0.525 | 0.579 | 0.530 | 0.532 | 0.584 | 0.529 | 0.656 | 0.556 | 0.572 | 0.567 | 0.699 | 0.575 | 0.650 | 0.656 | 0.649 | 0.640 | 0.537 | 0.529 | 0.568 | 0.653 | 0.556 | 0.574 | 0.583 | 0.716 | 0.605 | 0.693 | 0.757 | 0.458 | 0.965 | 0.776 | 0.708 | 0.647 | 0.843 | 0.636 | 0.587
yeast | 0.705 | 0.590 | 0.535 | 0.617 | 0.682 | 0.544 | 0.509 | 0.492 | 0.497 | 0.495 | 0.475 | 0.505 | 0.486 | 0.447 | 0.439 | 0.445 | 0.457 | 0.463 | 0.461 | 0.567 | 0.430 | 0.386 | 0.415 | 0.497 | 0.408 | 0.406 | 0.435 | 0.469 | 0.528 | 0.448 | 0.442 | 0.497 | 0.563 | 0.436 | 0.389 | 0.413 | 0.495 | 0.413 | 0.380 | 0.613 | 0.325 | 0.878 | 0.515 | 0.478 | 0.512 | 0.611 | 0.612 | 0.463
Mean | 0.659 | 0.648 | 0.569 | 0.610 | 0.651 | 0.508 | 0.498 | 0.492 | 0.513 | 0.485 | 0.485 | 0.512 | 0.480 | 0.444 | 0.460 | 0.441 | 0.443 | 0.466 | 0.442 | 0.521 | 0.444 | 0.438 | 0.445 | 0.528 | 0.421 | 0.428 | 0.507 | 0.530 | 0.521 | 0.559 | 0.531 | 0.542 | 0.517 | 0.445 | 0.438 | 0.446 | 0.530 | 0.438 | 0.451 | 0.419 | 0.346 | 0.824 | 0.719 | 0.712 | 0.694 | 0.797 | 0.643 | 0.604
(a) Average AE values of each algorithm per data set
| AC | AC-LR | AC-RF | AC-SV | AC-AB | PAC | PAC-LR | TSX | TSX-LR | TSX-SV | TS50 | TS50-LR | TS50-SV | TSMax | TSMax-LR | TSMax-SV | MS | MS-LR | MS-SV | GAC | GAC-LR | GAC-RF | GAC-SV | GAC-AB | GPAC | GPAC-LR | DyS | DyS-LR | DyS-SV | FMM | FMM-LR | FMM-SV | HDy | HDy-LR | HDy-RF | HDy-SV | HDy-AB | FM | FM-LR | EM | EM-LR | CC | CC-LR | CC-RF | CC-SV | CC-AB | PCC | PCC-LR
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
conc | 0.516 | 0.377 | 0.281 | 0.276 | 0.371 | 0.423 | 0.372 | 0.356 | 0.361 | 0.267 | 0.350 | 0.372 | 0.262 | 0.300 | 0.288 | 0.191 | 0.318 | 0.325 | 0.251 | 0.307 | 0.204 | 0.189 | 0.160 | 0.309 | 0.462 | 0.226 | 0.300 | 0.261 | 0.178 | 0.292 | 0.225 | 0.197 | 0.292 | 0.201 | 0.195 | 0.156 | 0.301 | 0.451 | 0.227 | 0.46 | 0.099 | 0.639 | 0.255 | 0.262 | 0.171 | 0.361 | 0.276 | 0.177
contra | 0.506 | 0.534 | 0.460 | 0.583 | 0.492 | 0.372 | 0.382 | 0.346 | 0.419 | 0.469 | 0.382 | 0.440 | 0.462 | 0.273 | 0.335 | 0.377 | 0.282 | 0.347 | 0.379 | 0.446 | 0.339 | 0.332 | 0.365 | 0.356 | 0.466 | 0.466 | 0.294 | 0.340 | 0.415 | 0.299 | 0.250 | 0.370 | 0.428 | 0.327 | 0.325 | 0.355 | 0.346 | 0.442 | 0.480 | 0.237 | 0.198 | 0.464 | 0.457 | 0.451 | 0.455 | 0.468 | 0.280 | 0.284
craft | 0.384 | 0.369 | 0.255 | 0.534 | 0.412 | 0.450 | 0.361 | 0.376 | 0.375 | 0.304 | 0.327 | 0.324 | 0.290 | 0.319 | 0.324 | 0.254 | 0.332 | 0.348 | 0.253 | 0.170 | 0.138 | 0.124 | 0.145 | 0.183 | 0.148 | 0.130 | 0.231 | 0.286 | 0.154 | 0.196 | 0.181 | 0.188 | 0.158 | 0.131 | 0.121 | 0.138 | 0.176 | 0.115 | 0.117 | 0.159 | 0.062 | 0.397 | 0.309 | 0.309 | 0.360 | 0.360 | 0.242 | 0.218
drugs | 0.152 | 0.262 | 0.283 | 0.201 | 0.260 | 0.096 | 0.116 | 0.150 | 0.117 | 0.111 | 0.123 | 0.101 | 0.085 | 0.104 | 0.084 | 0.074 | 0.119 | 0.105 | 0.091 | 0.178 | 0.147 | 0.175 | 0.130 | 0.244 | 0.148 | 0.154 | 0.063 | 0.060 | 0.064 | 0.136 | 0.124 | 0.147 | 0.170 | 0.141 | 0.156 | 0.125 | 0.240 | 0.124 | 0.134 | 0.049 | 0.139 | 0.151 | 0.233 | 0.310 | 0.236 | 0.352 | 0.147 | 0.18
thrm | 0.842 | 0.802 | 0.570 | 0.633 | 0.759 | 0.550 | 0.553 | 0.625 | 0.660 | 0.574 | 0.593 | 0.641 | 0.552 | 0.430 | 0.499 | 0.472 | 0.442 | 0.511 | 0.470 | 0.602 | 0.521 | 0.477 | 0.507 | 0.573 | 0.643 | 0.612 | 0.534 | 0.528 | 0.478 | 0.472 | 0.435 | 0.504 | 0.582 | 0.503 | 0.465 | 0.498 | 0.558 | 0.716 | 0.717 | 0.439 | 0.301 | 0.692 | 0.650 | 0.496 | 0.538 | 0.711 | 0.340 | 0.334
turk | 0.495 | 0.558 | 0.633 | 0.600 | 0.636 | 0.270 | 0.287 | 0.314 | 0.352 | 0.498 | 0.334 | 0.378 | 0.498 | 0.225 | 0.273 | 0.360 | 0.226 | 0.271 | 0.359 | 0.408 | 0.356 | 0.375 | 0.381 | 0.402 | 0.343 | 0.330 | 0.230 | 0.284 | 0.339 | 0.281 | 0.301 | 0.370 | 0.384 | 0.336 | 0.357 | 0.359 | 0.385 | 0.368 | 0.422 | 0.105 | 0.163 | 0.585 | 0.608 | 0.690 | 0.639 | 0.635 | 0.296 | 0.299
vgame | 0.682 | 0.715 | 0.739 | 0.725 | 0.709 | 0.578 | 0.523 | 0.501 | 0.488 | 0.576 | 0.466 | 0.454 | 0.532 | 0.457 | 0.414 | 0.508 | 0.469 | 0.441 | 0.526 | 0.581 | 0.524 | 0.558 | 0.526 | 0.566 | 0.519 | 0.493 | 0.382 | 0.346 | 0.454 | 0.241 | 0.243 | 0.306 | 0.560 | 0.509 | 0.524 | 0.505 | 0.552 | 0.506 | 0.467 | 0.132 | 0.136 | 0.238 | 0.248 | 0.363 | 0.312 | 0.412 | 0.170 | 0.157
wine | 0.693 | 0.667 | 0.642 | 0.633 | 0.695 | 0.627 | 0.565 | 0.574 | 0.558 | 0.585 | 0.563 | 0.574 | 0.543 | 0.444 | 0.464 | 0.489 | 0.468 | 0.487 | 0.508 | 0.433 | 0.466 | 0.518 | 0.531 | 0.578 | 0.617 | 0.594 | 0.459 | 0.406 | 0.435 | 0.285 | 0.244 | 0.323 | 0.424 | 0.447 | 0.497 | 0.526 | 0.571 | 0.615 | 0.620 | 0.778 | 0.243 | 0.713 | 0.495 | 0.445 | 0.372 | 0.605 | 0.240 | 0.209
yeast | 0.674 | 0.583 | 0.557 | 0.638 | 0.662 | 0.587 | 0.559 | 0.549 | 0.570 | 0.540 | 0.496 | 0.549 | 0.512 | 0.491 | 0.496 | 0.465 | 0.510 | 0.546 | 0.498 | 0.357 | 0.386 | 0.297 | 0.326 | 0.406 | 0.430 | 0.356 | 0.341 | 0.358 | 0.346 | 0.226 | 0.235 | 0.303 | 0.356 | 0.385 | 0.296 | 0.318 | 0.400 | 0.398 | 0.330 | 0.699 | 0.2 | 0.584 | 0.234 | 0.295 | 0.325 | 0.500 | 0.224 | 0.140
Mean | 0.549 | 0.541 | 0.491 | 0.536 | 0.555 | 0.439 | 0.413 | 0.421 | 0.433 | 0.436 | 0.404 | 0.426 | 0.415 | 0.338 | 0.353 | 0.355 | 0.352 | 0.376 | 0.371 | 0.387 | 0.342 | 0.338 | 0.341 | 0.402 | 0.420 | 0.373 | 0.315 | 0.319 | 0.318 | 0.270 | 0.249 | 0.301 | 0.373 | 0.331 | 0.326 | 0.331 | 0.392 | 0.415 | 0.390 | 0.34 | 0.171 | 0.496 | 0.388 | 0.402 | 0.379 | 0.489 | 0.246 | 0.222
(b) Average NKLD scores for each algorithm per data set
Table 7: Main results for multiclass quantification using tuned classifiers.
We show the averaged error scores over all scenarios per algorithm and data
set.
|
equation imposes the constraint that the spatial components of the field
strength (and its barred counterpart) must vanish,
$\tilde{\mathcal{F}}=\tilde{\overline{\mathcal{F}}}=0\,.$ (118)
These constraints are solved by writing
$\tilde{A}=g^{-1}\tilde{d}g\,,\quad\tilde{\bar{A}}=\bar{g}^{-1}\tilde{d}\bar{g}\,.$
(119)
We write the group elements in a Gauss parametrization
$\displaystyle g=\left(\begin{array}[]{cc}1&0\\\ -F&1\\\
\end{array}\right)\left(\begin{array}[]{cc}\lambda&0\\\ 0&\lambda^{-1}\\\
\end{array}\right)\left(\begin{array}[]{cc}1&\Psi\\\ 0&1\\\
\end{array}\right)\,,$ (120)
$\displaystyle\bar{g}=\left(\begin{array}[]{cc}1&\bar{F}\\\ 0&1\\\
\end{array}\right)\left(\begin{array}[]{cc}\bar{\lambda}^{-1}&0\\\
0&\bar{\lambda}\\\ \end{array}\right)\left(\begin{array}[]{cc}1&0\\\
-\bar{\Psi}&1\\\ \end{array}\right)\,.$
It is a straightforward exercise to rewrite the boundary conditions (103) in
terms of the functions appearing in (LABEL:zm). The $(\Psi,\bar{\Psi})$ are
determined as
$\displaystyle\Psi$
$\displaystyle=-{\lambda^{\prime}\over\lambda^{3}F^{\prime}}+{\omega_{x}\over
2\lambda^{2}F^{\prime}}\,,$ (121) $\displaystyle\bar{\Psi}$
$\displaystyle=-{\bar{\lambda}^{\prime}\over\bar{\lambda}^{3}\bar{F}^{\prime}}-{\omega_{x}\over
2\bar{\lambda}^{2}\bar{F}^{\prime}}\,,$
where $\omega_{x}$ is the space component of the boundary spin connection,
fixed in terms of the boundary vielbein. The remaining boundary conditions
amount to the following differential equations
$\displaystyle 2e^{+}_{x}$
$\displaystyle=\lambda^{2}F^{\prime}-\bar{\Psi}^{\prime}-\bar{\lambda}^{2}\bar{\Psi}^{2}\bar{F}^{\prime}-\frac{2\bar{\Psi}}{\bar{\lambda}}\bar{\lambda}^{\prime}\,,$
(122) $\displaystyle-2e^{-}_{x}$
$\displaystyle=\bar{\lambda}^{2}\bar{F}^{\prime}-\Psi^{\prime}-\lambda^{2}\Psi^{2}F^{\prime}-{2\Psi\over\lambda}\lambda^{\prime}\,.$
The equations (121) and (122) are to be imposed at the boundary surface
$r=r_{c}$. Having chosen the Gauss parametrization, one finds that the bulk
Lagrangian becomes a total derivative and so the complete action takes the
form of a boundary term. After performing some algebra (detailed in appendix
8.C), we obtain:
$\displaystyle S_{\text{grav}}$ $\displaystyle=-{k\over 2\pi}\int_{\partial
M_{3}}\\!d^{2}x\left({\lambda^{\prime}\partial_{t}\lambda\over\lambda^{2}}-\lambda^{2}F^{\prime}\partial_{t}\Psi\right)-{k\over\pi}\int_{\partial
M_{3}}\\!d^{2}x\Big{(}\lambda^{2}\Psi^{2}F^{\prime}+\Psi^{\prime}+{2\Psi\lambda^{\prime}\over\lambda}\Big{)}e^{+}_{t}$
(123) $\displaystyle\quad+{k\over 2\pi}\int_{\partial
M_{3}}\\!d^{2}x\left({\bar{\lambda}^{\prime}\partial_{t}\bar{\lambda}\over\bar{\lambda}^{2}}-\bar{\lambda}^{2}\bar{F}^{\prime}\partial_{t}\bar{\Psi}\right)-{k\over\pi}\int_{\partial
M_{3}}\\!d^{2}x\Big{(}\bar{\lambda}^{2}\bar{\Psi}^{2}\bar{F}^{\prime}+\bar{\Psi}^{\prime}+{2\bar{\Psi}\bar{\lambda}^{\prime}\over\bar{\lambda}}\Big{)}e^{-}_{t}\,.$
The boundary conditions (121)-(122) imply four equations for the six Gauss
functions, leaving two free functions, which we can take to be $(F,\bar{F})$.
So, in principle, we should use (121) and (122) to obtain
$(\Psi,\bar{\Psi},\lambda,\bar{\lambda})$ in terms of $(F,\bar{F})$ and
substitute into (123) to obtain the reduced action. However, in practice, it
is not possible to carry this out analytically222222 Even though it is not
possible to solve the boundary conditions analytically, they do have a
beautiful physical interpretation. They correspond to the definition of the
stress tensor in a $T\overline{T}$-deformed theory, understood as a theory
coupled to topological gravity. See appendix 8.D for more details.. To obtain
explicit results we either need to consider the asymptotic AdS3 case of
$r_{c}\rightarrow 0$, or use perturbation theory. We discuss these in turn
below.
One feature to keep in mind is that we only need to solve for the Gauss
functions on the cutoff surface. These functions determine the connections
restricted to that surface, which we call $(a,\overline{a})$. The full
connections $(A,\bar{A})$ may then be determined away from the boundary by the
construction
$A=b^{-1}ab+b^{-1}db\,,\quad\bar{A}=b\bar{a}b^{-1}+bdb^{-1}\,,$ (124)
where
$b=e^{-{1\over 2}\ln\left({r\over r_{c}}\right)L_{0}}$ (125)
and with $a$ and $\bar{a}$ functions of only the boundary coordinates. This is
the Chern-Simons equivalent of radial gauge, which we can always choose at
least in a neighborhood of the boundary. Flat boundary connections
$(a,\overline{a})$ are thereby promoted to flat bulk connections
$(A,\overline{A})$.
#### 5 Asymptotic boundary
In this subsection, we consider imposing boundary conditions at the asymptotic
boundary of AdS3. The results obtained here are found in [58].
Asymptotically AdS3 boundary conditions correspond to taking $r_{c}\rightarrow
0$ with boundary vielbein $e^{a}_{\mu}\sim r_{c}^{-\frac{1}{2}}$. The boundary
conditions (121) and (122) imply $(\Psi,\bar{\Psi})\sim r_{c}^{\frac{1}{2}}$
and $(\lambda,\overline{\lambda})\sim r_{c}^{-\frac{1}{4}}$, while
$(F,\bar{F})$ stay finite. The solution for (122) reads
$\lambda=\sqrt{2e_{x}^{+}\over
F^{\prime}}\,,\quad\bar{\lambda}=\sqrt{-\frac{2e_{x}^{-}}{\bar{F}^{\prime}}}\,,$
(126)
while the boundary action evaluates to
$\displaystyle S_{\text{grav}}$ $\displaystyle=-{k\over\pi}\int_{\partial
M_{3}}\\!d^{2}x\left({\lambda^{\prime}D\lambda\over\lambda^{2}}-\lambda^{2}F^{\prime}D\Psi\right)+{k\over\pi}\int_{\partial
M_{3}}\\!d^{2}x\left({\bar{\lambda}^{\prime}\bar{D}\bar{\lambda}\over\bar{\lambda}^{2}}-\bar{\lambda}^{2}\bar{F}^{\prime}\bar{D}\bar{\Psi}\right)$
(127) $\displaystyle\quad-{k\over 8\pi}\int_{\partial
M_{3}}\\!d^{2}x{e_{t}^{+}e_{x}^{-}-e_{t}^{-}e_{x}^{+}\over
e_{x}^{+}e_{x}^{-}}\omega_{x}^{2}$
with
$D={1\over 2}\left(\partial_{t}-{e^{+}_{t}\over
e^{+}_{x}}\partial_{x}\right)\,,\quad\text{and}\quad\bar{D}={1\over
2}\left(\partial_{t}-{e^{-}_{t}\over e^{-}_{x}}\partial_{x}\right)\,.$ (128)
The term in the second line of (127) is a constant determined by the boundary
conditions. For a flat planar boundary, we arrive at the Alekseev-Shatashvili
action
$S_{\text{grav}}=S_{\text{AS}}[F]+S_{\text{AS}}[\bar{F}]\,,$ (129)
with
$\displaystyle S_{\text{AS}}[F]$ $\displaystyle={k\over 4\pi}\int_{\partial
M_{3}}\\!d^{2}x\left({1\over
F^{\prime}}\right)^{\prime\prime}\partial_{\bar{z}}F$ (130)
$\displaystyle={k\over 4\pi}\int_{\partial M_{3}}d^{2}x\left[{\dot{F}\over
F^{\prime}}\left({F^{\prime\prime\prime}\over F^{\prime}}-{F^{\prime\prime
2}\over 2F^{\prime 2}}\right)+{F^{\prime\prime\prime}\over F^{\prime}}-{3\over
2}{F^{\prime\prime 2}\over F^{\prime 2}}\right]\,.$
As noted previously by (58), the field redefinition
$F^{\prime}=e^{f}\,,\quad\bar{F}^{\prime}=e^{\bar{f}}\,,$ (131)
yields the free boson action
$S_{\text{grav}}[f,\bar{f}]=-{k\over 4\pi}\int_{\partial
M_{3}}d^{2}x\,\left[f^{\prime}\partial_{z}f+\bar{f}^{\prime}\partial_{\bar{z}}\bar{f}\right]\,.$
(132)
#### 6 Perturbation theory for planar cutoff boundary
We now consider the case of a boundary at a finite cutoff $r=r_{c}$, with the
simplifying assumption of a flat boundary geometry. We will be able to solve
the boundary conditions (122) by perturbing around a reference solution.
Explicitly, we keep $r_{c}$ finite and fixed and take the boundary vielbein
corresponding to a flat plane
$e^{+}={1\over 2\sqrt{r_{c}}}(dx+dt)\,,\quad e^{-}=-{1\over 2\sqrt{r_{c}}}(dx-
dt)\,.$ (133)
The corresponding solution to (108) is $\omega_{x}=0$. We will perturb around
the solution
$F^{(0)}=\bar{F}^{(0)}=x\,,$ (134)
which implies $\Psi^{(0)}=\bar{\Psi}^{(0)}=0$ and
$\lambda^{(0)}=\bar{\lambda}^{(0)}=r_{c}^{-\frac{1}{4}}$. This solution
corresponds to the Poincaré AdS3 background metric
$ds^{2}={dr^{2}\over 4r^{2}}+\frac{dx^{2}-dt^{2}}{r}\,.$ (135)
Having identified a background field configuration, we expand around it order
by order. We adopt the following notation for the perturbations:
$\displaystyle\lambda^{2}=$
$\displaystyle{1\over\sqrt{r_{c}}}\left(1+f+f^{(2)}+f^{(3)}+\cdots\right)\,,$
(136) $\displaystyle\bar{\lambda}^{2}=$
$\displaystyle{1\over\sqrt{r_{c}}}\left(1+\bar{f}+\bar{f}^{(2)}+\bar{f}^{(3)}+\cdots\right)\,,$
$\displaystyle F^{\prime}=$ $\displaystyle
1+g^{(1)}+g^{(2)}+g^{(3)}+\cdots\,,$ $\displaystyle\bar{F}^{\prime}=$
$\displaystyle 1+\bar{g}^{(1)}+\bar{g}^{(2)}+\bar{g}^{(3)}+\cdots\,.$
We will regard $f$ and $\bar{f}$ as the fundamental fields of our perturbative
action, while $f^{(i)}$, $g^{(i)}$ and their barred counter-parts will be
chosen so that the boundary conditions (122) are satisfied perturbatively. The
boundary conditions fully determine $g^{(i)}$ while the functions $f^{(i)}$
can be chosen freely. This freedom amounts to a field redefinition of
$(f,\bar{f})$, which will be used to obtain the simplest action possible.
Solving the boundary conditions perturbatively, which means working order-by-
order in the amplitudes of $(f,\bar{f})$, we find the following expressions
for the first few functions $g^{(i)}$:
$\displaystyle g^{(1)}=$ $\displaystyle-f-{r_{c}\over
2}\bar{f}^{\prime\prime}\,,$ (137) $\displaystyle g^{(2)}=$ $\displaystyle
f^{2}-f^{(2)}+{r_{c}\over 4}\left(\bar{f}^{\prime
2}+2f\bar{f}^{\prime\prime}+2\bar{f}\bar{f}^{\prime\prime}-2f^{(2)\prime\prime}\right)-{1\over
r_{c}^{2}}\left(f^{\prime\prime}\bar{f}^{\prime\prime}+\bar{f}^{\prime}f^{\prime\prime\prime}\right)\,,$
and similarly for $\bar{g}^{(i)}$. The formulas for higher order terms are
easily found since the boundary conditions amount to linear equations for
$g^{(i)}$. However, their expressions are not illuminating and get messy at
higher orders, so we do not write them explicitly.
As mentioned above, we are free to choose the functions $f^{(i)}$, which
amounts to a choice of field redefinition. Just as we found in the metric
formulation, a judicious choice simplifies the expression of the action
greatly. First of all, demand
$4GT_{xt}={1\over 4}\left(\bar{f}^{\prime 2}-f^{\prime 2}\right)+\text{total
derivatives}\,,$ (138)
which implies a simple expression for the part of the action involving time
derivatives, agreeing with (132), and essentially corresponds to choosing
Darboux coordinates. The field redefinition that achieves this reads
$\displaystyle f^{(2)}=$ $\displaystyle{1\over 2}f^{2}-{r_{c}\over
2}f^{\prime}\bar{f}^{\prime}+\cdots\,,$ (139) $\displaystyle f^{(3)}=$
$\displaystyle{1\over 6}f^{3}-{r_{c}\over
2}ff^{\prime}\bar{f}^{\prime}+{r_{c}^{2}\over 4}\left[{1\over
2}\bar{f}^{\prime 2}f^{\prime\prime}+f^{\prime
2}\bar{f}^{\prime\prime}+\cdots\right]\,,$
and similarly for the barred functions. The second condition that can be
satisfied is that all higher derivatives of $f$ and $\overline{f}$ can be
canceled in the action, i.e. only powers of the first derivatives appear. This
condition first appears in the fourth order, where the appropriate choice of
field redefinition reads
$\displaystyle f^{(4)}$
$\displaystyle=\frac{1}{4!}f^{4}-\frac{r_{c}}{4}f^{2}f^{\prime}\bar{f}^{\prime}-\frac{r_{c}^{2}}{8}(f^{\prime
3}\bar{f}^{\prime}-ff^{\prime\prime}\bar{f}^{\prime 2}-2ff^{\prime
2}\bar{f}^{\prime\prime}+f^{\prime}\bar{f}^{\prime 3}-2f^{\prime
2}\bar{f}^{\prime 2})$
$\displaystyle\mathrel{\phantom{=}}-\frac{r_{c}^{3}}{16}(4f^{\prime}f^{\prime\prime}\bar{f}^{\prime}\bar{f}^{\prime\prime}+\tfrac{1}{3}f^{\prime\prime\prime}\bar{f}^{\prime
3}+f^{\prime 3}\bar{f}^{\prime\prime\prime})\,.$ (140)
Perturbation theory subject to these conditions can be automated using
computer algebra software (we used Mathematica) and performed to higher
orders. One useful observation is that the terms needed in the choice of
$f^{(n)}$ and $\bar{f}^{(n)}$ to satisfy the two aforementioned conditions
already appear (with different coefficients) in the Hamiltonian density at
order $n-1$. More specifically, the Hamiltonian density has a simple
expression up to a total double derivative contribution, and the terms that
appear in this double derivative are exactly the ones that make up our choice
of $f^{(n)}$ and $\bar{f}^{(n)}$.
We carried out this perturbation theory to the eighth order (i.e. computing
all terms of the schematic form $f^{n}\bar{f}^{m}$ with $n+m\leq 8$). The
result coincides with the expansion to this order of the Nambu-Goto action
(25). We naturally conjecture that this result extends to all orders, but we
do not have proof.
This analysis also yields expressions for the boundary stress tensor $T_{ij}$
to eighth order. These agree with the expressions found in the metric
formulation (up to quartic order, which is as far as we pushed the computation
in the metric formulation). Since our computations below only use the stress
tensor up to cubic order, written in (92), we refrain from writing the higher
order expressions, which rapidly become complicated.
### 5 Correlation functions
In this section, we discuss the computation of correlation functions of the
fundamental fields $(f,\bar{f})$ and the stress tensor $T_{ij}$. We will work
up to a two-loop order where, as seen from (91), $G$ acts as a loop counting
parameter.
Some subtleties have to do with the realization of symmetries in this theory.
For example, the action is not manifestly Lorentz invariant, even though the
underlying theory is Lorentz invariant since it was obtained by expanding
around a Lorentz invariant background (the flat plane). We expect the stress
tensor should behave in correlators like a Lorentz tensor. As was discussed
above, Lorentz symmetry is realized nonlinearly on the $(f,\bar{f})$ fields. A
general phenomenon that can occur when doing perturbation theory in a QFT with
a nonlinearly realized symmetry is that one encounters divergent terms that
are not invariant under the symmetry. One then needs to perform a field
redefinition to restore the symmetry (or equivalently, to modify the symmetry
transformation), e.g., [148]. Another approach is to modify the theory off-
shell to preserve the symmetry, e.g., [149]. Our approach is to modify
perturbation theory in a way that maintains Lorentz invariance while only
changing contact terms in correlators. In particular, correlation functions of
stress tensors at non-coincident points will respect Lorentz invariance.
#### 1 Action
We found that the action to quartic order is
$I={1\over 16\pi G}\int_{\partial
M_{3}}\\!d^{2}x\Big{(}f^{\prime}\partial_{\bar{z}}f+\bar{f}^{\prime}\partial_{z}\bar{f}+{1\over
4}r_{c}f^{\prime 2}\bar{f}^{\prime 2}+\cdots\Big{)}\,.$ (141)
Recall that $z=x+it$ and $\bar{z}=x-it$ so that
$\partial_{z}={1\over
2}(\partial_{x}-i\partial_{t})\,,\quad\partial_{\bar{z}}={1\over
2}(\partial_{x}+i\partial_{t})\,.$ (142)
Here $G$ is the loop counting parameter. In particular, since the stress
tensor also has a $1/G$ prefactor, it follows that an $L$ loop contribution to
a stress tensor correlator has dependence $G^{L-1}$.
#### 2 Propagator
Let’s first discuss the propagators in momentum space using the Fourier
transform convention
$\psi(x,t)=\int\\!{d^{2}p\over(2\pi)^{2}}\psi(p)e^{ip_{t}t+ip_{x}x}$ (143)
or in complex coordinates
$\psi(z,\bar{z})=\int\\!{d^{2}p\over(2\pi)^{2}}\psi(p)e^{ip_{z}z+ip_{\bar{z}}\bar{z}}$
(144)
with
$p_{x}=p_{z}+p_{\bar{z}}\,,\quad p_{t}=i(p_{z}-p_{\bar{z}})\,.$ (145)
Note also that
$p^{2}=p_{t}^{2}+p_{x}^{2}=4p_{z}p_{\bar{z}}\,.$ (146)
The free two-point functions are
$\langle f^{\prime}(p)f^{\prime}(p^{\prime})\rangle_{0}=32\pi
G{p_{x}p_{z}\over
p^{2}}(2\pi)^{2}\delta^{2}(p+p^{\prime})\,,\quad\langle\bar{f}^{\prime}(p)\bar{f}^{\prime}(p^{\prime})\rangle_{0}=32\pi
G{p_{x}p_{\bar{z}}\over p^{2}}(2\pi)^{2}\delta^{2}(p+p^{\prime})\,.$ (147)
We wrote the results for the fields with an $x$-derivative since $(f,\bar{f})$
always appears in the action and stress tensor with at least one
$x$-derivative.
We will be using dimensional regularization to compute loop diagrams. Our
convention for going from two to $d$ dimensions is that we introduce $d-2$ new
spatial dimensions. We continue to refer to momenta in the original two
dimensions by $(p_{x},p_{t})$ or $(p_{z},p_{\bar{z}})$, but $p^{2}$ is taken
to run over all dimensions:
$p^{2}=p_{t}^{2}+p_{x}^{2}+\sum_{i=2}^{d}p_{i}^{2}$. In particular, the
relation (146) only holds in $d=2$.
Coming back to the propagators, after stripping off delta functions and using
(145), we have
$\langle f^{\prime}(p)f^{\prime}(-p)\rangle_{0}=32\pi G\left({p_{z}^{2}\over
p^{2}}+{p_{z}p_{\bar{z}}\over
p^{2}}\right)\,,\quad\langle\bar{f}^{\prime}(p)\bar{f}^{\prime}(-p)\rangle_{0}=32\pi
G\left({p_{\bar{z}}^{2}\over p^{2}}+{p_{z}p_{\bar{z}}\over p^{2}}\right)\,.$
(148)
We now argue that we can drop the ${p_{z}p_{\bar{z}}\over p^{2}}$ terms.
First, note that in $d=2$ this term is constant in momentum space and
corresponds to a delta function contribution to the propagator in position
space. Including such delta functions in propagators is equivalent to a
redefinition of couplings and operators since they contract lines down to
points, thereby inducing new vertices. The situation in dimensional
regularization with $d=2+\varepsilon$ is a bit more subtle. While the
violation of $p^{2}=4p_{z}p_{\bar{z}}$ is morally proportional to
$\varepsilon$, this can, of course, be compensated by factors of
$\frac{1}{\varepsilon}$ arising from divergent loop integrals. Nonetheless, as
shown by explicit computation (see appendix 4), the effect of including or
excluding the ${p_{z}p_{\bar{z}}\over p^{2}}$ terms in the propagator is the
same as changing the coupling in front of some local operator.
In general, this local operator will be non-Lorentz invariant. We will allow
ourselves to add local operators to maintain Lorentz invariance, and what we
see from the present discussion is that the simplest way to do this is to drop
the ${p_{z}p_{\bar{z}}\over p^{2}}$ terms from the propagators. This should be
thought of as part of our renormalization scheme.
We take the propagators to be
(149)
Arrows indicate momentum flow. With this propagator rule, $f^{\prime}$ is
effectively the same as $\partial_{z}f$, and $\bar{f}^{\prime}$ is effectively
the same as $\partial_{\bar{z}}\bar{f}$. We then see from (92) that the stress
tensor components have indices that match the $\partial_{z}$ and
$\partial_{\bar{z}}$ derivatives that appear. This implies that stress tensor
correlators will be Lorentz covariant.
It will also be useful to Fourier transform back to position space. To perform
the $d$-dimensional Fourier transform of the propagators (149) is a
straightforward application of the integral (85), the result of which produces
the position space propagators
$\displaystyle\langle f^{\prime}(x)f^{\prime}(0)\rangle_{0}$
$\displaystyle=-8G\pi^{1-{d\over 2}}\Gamma\left({d\over
2}+1\right){\bar{z}^{2}\over(x\cdot x)^{{d\over 2}+1}}\,,$ (150)
$\displaystyle\langle\bar{f}^{\prime}(x)\bar{f}^{\prime}(0)\rangle_{0}$
$\displaystyle=-8G\pi^{1-{d\over 2}}\Gamma\left({d\over
2}+1\right){z^{2}\over(x\cdot x)^{{d\over 2}+1}}\,,$
where $x\cdot x=z\bar{z}+\sum_{i=2}^{d}(x^{i})^{2}$. In $d=2$, (150) becomes
$\displaystyle\langle f^{\prime}(x)f^{\prime}(0)\rangle_{0}$
$\displaystyle=-{8G\over z^{2}}\,,$ (151)
$\displaystyle\langle\bar{f}^{\prime}(x)\bar{f}^{\prime}(0)\rangle_{0}$
$\displaystyle=-{8G\over\bar{z}^{2}}\,.$
We take $\langle f^{\prime}f^{\prime}\rangle_{0}$ and
$\langle\bar{f}^{\prime}\bar{f}^{\prime}\rangle_{0}$ as the propagators. When
we refer to an “amputated” diagram, we mean that we have divided by these
propagators.
#### 3 Interaction vertex
To the order we work, there is a single quartic interaction vertex whose
Feynman rule is
(152)
#### 4 Stress tensor in terms of Feynman diagrams
From (92) and the Feynman rules we have derived, we can express the components
of the deformed stress tensor $T_{\mu\nu}$ in terms of the following Feynman
diagrams:
(153)
#### 5 Structure of stress tensor two-point function
The general stress tensor two-point function can be reconstructed from
$\langle T_{z\bar{z}}T_{z\bar{z}}\rangle$, as in [88]. To see this, note that
Lorentz invariance and parity implies
$\displaystyle\langle T_{zz}(x)T_{zz}(0)\rangle$ $\displaystyle={1\over
z^{4}}f_{1}(y)\,,$ (154) $\displaystyle\langle
T_{zz}(x)T_{z\bar{z}}(0)\rangle$ $\displaystyle={1\over
z^{3}\bar{z}}f_{2}(y)\,,$ $\displaystyle\langle
T_{zz}(x)T_{\bar{z}\bar{z}}(0)\rangle$ $\displaystyle={1\over
z^{2}\bar{z}^{2}}f_{3}(y)\,,$ $\displaystyle\langle
T_{z\bar{z}}(x)T_{z\bar{z}}(0)\rangle$ $\displaystyle={1\over
z^{2}\bar{z}^{2}}f_{4}(y)\,,$
where the dimensionless variables $y$ is
$y={z\bar{z}\over r_{c}}\,.$ (155)
Stress tensor conservation implies
$\displaystyle f_{1}^{\prime}+y^{3}\left(f_{2}\over y^{3}\right)^{\prime}$
$\displaystyle=0\,,$ (156) $\displaystyle\left(f_{2}\over
y\right)^{\prime}+y\left(f_{3}\over y^{2}\right)^{\prime}$
$\displaystyle=0\,,$ $\displaystyle\left(f_{2}\over
y\right)^{\prime}+y\left(f_{4}\over y^{2}\right)^{\prime}$
$\displaystyle=0\,.$
As $r_{c}\rightarrow 0$, we should recover the usual CFT correlators, which
implies that we are looking for solutions with $f_{1}\rightarrow{c\over 2}$ as
$y\rightarrow\infty$, and with the other functions vanishing in this limit.
The central charge $c$ will be computed in terms of $G$ momentarily. Note that
$f_{3}=f_{4}$, which implies that $\langle
T_{zz}(x)T_{\bar{z}\bar{z}}(y)-T_{z\bar{z}}(x)T_{z\bar{z}}(y)\rangle=0$. This
is compatible with the trace relation
$T_{z\bar{z}}=\pi\lambda_{T\overline{T}}\det T$ given that $\langle
T_{z\bar{z}}\rangle=0$.
We find the central charge $c$ by computing correlators at $r_{c}=0$, where
the stress tensor is
$T_{zz}|_{r_{c}=0}={1\over 8G}\left(f^{\prime\prime}-{1\over 2}f^{\prime
2}\right)\,,\quad T_{\bar{z}\bar{z}}|_{r_{c}=0}={1\over
8G}\left(\bar{f}^{\prime\prime}-{1\over 2}\bar{f}^{\prime 2}\right)\,,\quad
T_{z\bar{z}}|_{r_{c}=0}=0\,.$ (157)
Using (151), we have
$\displaystyle\langle T_{zz}(x)T_{zz}(0)\rangle|_{r_{c}=0}={c\over
2z^{4}}\,,\quad\langle
T_{\bar{z}\bar{z}}(x)T_{\bar{z}\bar{z}}(0)\rangle|_{r_{c}=0}={c\over
2\bar{z}^{4}}\,,$ (158) $\displaystyle\langle
T_{z\bar{z}}(x)T_{z\bar{z}}(0)\rangle|_{r_{c}=0}=\langle
T_{zz}(x)T_{z\bar{z}}(0)\rangle|_{r_{c}=0}=0\,,$
with
$c={3\over 2G}+1=c_{0}+1\,.$ (159)
This one-loop correction to the Brown-Henneaux formula is the same as in [58].
We display the contributing diagrams as
(160)
where the unfilled circles denote stress tensor insertions. In particular, the
details of the calculation in (160) are straightforward
(161)
where the tree-level propagators in terms of the central charge
($G=\frac{3}{2c_{0}}$) are
$\displaystyle\langle
f^{\prime}(z_{1})f^{\prime}(z_{2})\rangle=-\frac{12}{c_{0}}\frac{1}{z_{12}^{2}},\quad\langle\bar{f}^{\prime}(\bar{z}_{1})\bar{f}^{\prime}(\bar{z}_{2})\rangle=-\frac{12}{c_{0}}\frac{1}{\bar{z}_{12}^{2}}.$
(162)
For the deformation included at $O(r_{c}^{2})$, then:
$\displaystyle\left\langle
T_{zz}\left(z_{1}\right)T_{zz}\left(z_{2}\right)\right\rangle$
$\displaystyle=\frac{c_{0}^{2}}{36}\bigg{(}\frac{1}{4}\left\langle
f^{\prime\prime}\left(z_{1}\right)f^{\prime\prime}\left(z_{2}\right)\right\rangle+\frac{1}{8}\left\langle
f^{\prime}\left(z_{1}\right)f^{\prime}\left(z_{2}\right)\right\rangle^{2}$
(163) $\displaystyle+\frac{1}{16}r_{c}^{2}\left\langle
f^{\prime\prime\prime}\left(z_{1}\right)f^{\prime\prime\prime}\left(z_{2}\right)\right\rangle\left\langle\bar{f}^{\prime}\left(\bar{z}_{1}\right)\bar{f}^{\prime}\left(\bar{z}_{2}\right)\right\rangle\bigg{)}$
$\displaystyle=\frac{c_{0}+1}{2z_{12}^{4}}+\frac{30r_{c}^{2}}{z_{12}^{6}\bar{z}_{12}^{2}}$
and using the fact that $\lambda=\frac{6r_{c}}{\pi}$, we arrive at
$\left\langle
T_{zz}(z_{1})T_{zz}(z_{2})\right\rangle=\frac{c_{0}+1}{2z_{12}^{4}}+\frac{5\pi^{2}\lambda^{2}}{6z_{12}^{6}\bar{z}_{12}^{2}}$
(164)
which matches [88].
#### 6 Correlators of elementary fields
To determine any needed counterterms in the action, we now consider the one-
loop four-point and two-loop two-point correlators of $(f,\bar{f})$.
##### $\langle
f^{\prime}(p_{1})f^{\prime}(p_{2})f^{\prime}(p_{3})f^{\prime}(p_{4})\rangle$
The basic diagram is
(165)
The full correlator is then
$\langle
f^{\prime}(p_{1})f^{\prime}(p_{2})f^{\prime}(p_{3})f^{\prime}(p_{4})\rangle=G_{4}(p_{1},p_{2},p_{3},p_{4})+G_{4}(p_{1},p_{3},p_{2},p_{4})+G_{4}(p_{1},p_{4},p_{3},p_{2})\,.$
(166)
The amputated diagram is
$\displaystyle G^{{\text{amp}}}_{4}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=2r_{c}^{2}\int\\!{d^{d}q\over(2\pi)^{d}}{(p_{1,\bar{z}}+p_{2,\bar{z}}-q_{\bar{z}})^{2}q_{\bar{z}}^{2}\over(p_{1}+p_{2}-q)^{2}q^{2}}$
(167) $\displaystyle={r_{c}^{2}\over
12\pi}{(p_{1,\bar{z}}+p_{2,\bar{z}})^{4}\over(p_{1}+p_{2})^{2}}\,.$
This diagram is in particular finite, hence requires no $(f^{\prime})^{4}$
counterterm.
##### $\langle
f^{\prime}(p_{1})f^{\prime}(p_{2})\bar{f}^{\prime}(p_{3})\bar{f}^{\prime}(p_{4})\rangle$
The correlator has an (amputated) tree-level contribution
$\langle
f^{\prime}(p_{1})f^{\prime}(p_{2})\bar{f}^{\prime}(p_{3})\bar{f}^{\prime}(p_{4})\rangle_{\text{tree}}=-{r_{c}\over
16\pi G}\,.$ (168)
The one-loop diagram is
(169)
which we need to evaluate to compute the one-loop contribution to the
correlator
$f^{\prime}(p_{1})f^{\prime}(p_{2})\bar{f}^{\prime}(p_{3})\bar{f}^{\prime}(p_{4})\rangle_{1-{\text{loop}}}=G_{2,2}(p_{1},p_{2},p_{3},p_{4})+G_{2,2}(p_{1},p_{2},p_{4},p_{3})\,.$
(170)
Employing the shorthand $p_{ij}=p_{i}+p_{j}$, the result computed using
dimensional regularization and setting $d=2+\epsilon$ reads
$\displaystyle G^{\text{amp}}_{2,2}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=4r_{c}^{2}\int\\!{d^{d}q\over(2\pi)^{d}}{(p_{1,z}+p_{3,z}-q_{z})^{2}q_{\bar{z}}^{2}\over(p_{1}+p_{3}-q)^{2}q^{2}}$
(171)
$\displaystyle=4r_{c}^{2}\left[\frac{p^{2}_{13}}{32\pi\epsilon}+\frac{6\gamma-11-6\ln(4\pi)}{384\pi}p^{2}_{13}+\frac{p^{2}_{13}}{64\pi}\ln
p^{2}_{13}\right]\,.$
The amputated correlator works out to be
$\displaystyle\langle
f^{\prime}(p_{1})f^{\prime}(p_{2})\bar{f}^{\prime}(p_{3})\bar{f}^{\prime}(p_{4})\rangle_{1-{\text{loop}}}^{\text{amp}}$
$\displaystyle=\frac{r_{c}^{2}(d+2)(d+4)}{4^{d+5/2}\pi^{\frac{d-3}{2}}}\frac{(p_{13}^{2})^{d/2}+(p_{14}^{2})^{d/2}}{\sin\left(\frac{\pi
d}{2}\right)\Gamma(d/2+3/2)}\,.$ (172)
This amputated correlator (172) has a pole at $\varepsilon=0$,
$\langle
f^{\prime}(p_{1})f^{\prime}(p_{2})\bar{f}^{\prime}(p_{3})\bar{f}^{\prime}(p_{4})\rangle_{1-{\text{loop}}}^{\text{amp}}\sim{1\over
8\pi\varepsilon}r_{c}^{2}\big{[}p_{13}^{2}+p_{14}^{2}\big{]}\,.$ (173)
This divergence (173) is canceled by the counterterm:
$I_{\text{ct}}={r_{c}^{2}\over 4\pi\varepsilon}\int_{\partial
M_{3}}\\!d^{2}x\partial_{z}(f^{\prime}\bar{f}^{\prime})\partial_{\bar{z}}(f^{\prime}\bar{f}^{\prime})\,.$
(174)
The original action has no term of the form (174). One interpretation is that
this implies the existence of a new parameter in our theory corresponding to
including an undetermined finite term along with (174). On the other hand, as
discussed in this chapter’s introduction, the $3d$ gravity origin of this
theory indicates that no such new parameters should be needed. We thus suspect
that the appearance of the undetermined parameter may reflect that our
renormalization scheme has not incorporated all symmetries of the $3d$ gravity
theory.
#### 7 $\langle f^{\prime}(x)f^{\prime}(0)\rangle$ at two-loops
We will first compute the correlator in momentum space. The relevant Feynman
diagram to compute $\langle f^{\prime}(p)f^{\prime}(-p)\rangle$ is a sunset-
type diagram
(175)
The two-loop contribution to the amputated correlator is then
$\langle f^{\prime}(p)f^{\prime}(-p)\rangle_{{\text{two-
loop}}}^{\text{amp}}=8\left({r_{c}\over 64\pi G}\right)^{2}\left(32\pi
G\right)^{3}\int{d^{2}k\over(2\pi)^{2}}\int{d^{2}k^{\prime}\over(2\pi)^{2}}{k_{\bar{z}}^{2}k_{\bar{z}}^{\prime
2}(p-k-k^{\prime})_{z}^{2}\over k^{2}k^{\prime 2}(p-k-k^{\prime})^{2}}\,,$
(176)
where the overall normalization involves two vertex factors as in (152), the
normalization of the three internal propagators as in formulas (149), and a
symmetry factor of $8$. The integrals over the internal momenta $k$ and
$k^{\prime}$ are computed in dimensional regularization in appendix 3. The
result reads
$\int{d^{2}k\over(2\pi)^{2}}\int{d^{2}k^{\prime}\over(2\pi)^{2}}{k_{\bar{z}}^{2}k_{\bar{z}}^{\prime
2}(p-k-k^{\prime})_{z}^{2}\over k^{2}k^{\prime 2}(p-k-k^{\prime})^{2}}={1\over
2^{7}3\pi^{2}}p_{z}p_{\bar{z}}^{3}\log p^{2}+{\text{polynomial}}\,.$ (177)
Attaching the external legs and Fourier transforming back to position space
using formulas (89), we conclude
$\langle f^{\prime}(x)f^{\prime}(0)\rangle_{{\text{two-
loop}}}=-{64r_{c}^{2}G^{3}\over z^{4}\bar{z}^{2}}\,.$ (178)
This diagram is, in particular, finite (up to contact terms at $x=0$), so no
wavefunction renormalization is required.232323 As seen in (3), the integral
(177) does have a divergence in dimensional regularization. However, the
divergence is a polynomial in the momentum, which only leads to delta function
contact terms in position space.
#### 8 $\langle T_{z\bar{z}}f^{\prime}\bar{f}^{\prime}\rangle$
To identify the need for a counterterm for $T_{z\bar{z}}$, we consider the
correlator of the stress tensor with two elementary fields
(179)
The amputated diagram is
$\displaystyle\langle
T_{z\bar{z}}(k)f^{\prime}(p_{1})\bar{f}^{\prime}(p_{2})\rangle_{\operatorname{amp}}$
$\displaystyle=4\pi
r_{c}^{2}\int\\!{d^{d}q\over(2\pi)^{d}}{q_{z}^{3}(k_{\bar{z}}-q_{\bar{z}})^{3}\over
q^{2}(k-q)^{2}}$ (180) $\displaystyle={1\over
32\varepsilon}r_{c}^{2}(k^{2})^{2}+{\operatorname{finite}}\,.$
To cancel this divergence we need to redefine this stress tensor component as
$T_{z\bar{z}}\rightarrow T_{z\bar{z}}-{1\over
2\varepsilon}r_{c}^{2}f^{\prime\prime\prime}\bar{f}^{\prime\prime\prime}\,.$
(181)
Here, we have adopted a minimal subtraction scheme. Of course, we are free to
also add a finite contribution, which will appear below as an undetermined
constant in the stress tensor correlator.
#### 9 $\langle T_{z\bar{z}}T_{z\bar{z}}\rangle$
To compute $\langle T_{z\bar{z}}T_{z\bar{z}}\rangle$ to two-loop order, we
recall
$4GT_{z\bar{z}}=-{1\over 4}r_{c}f^{\prime\prime}\bar{f}^{\prime\prime}+{1\over
8}r_{c}(f^{\prime\prime}\bar{f}^{\prime 2}+f^{\prime
2}\bar{f}^{\prime\prime})-{1\over
8}r_{c}^{2}(f^{\prime\prime\prime}\bar{f}^{\prime}\bar{f}^{\prime\prime}+f^{\prime}f^{\prime\prime}\bar{f}^{\prime\prime\prime})+{\operatorname{quartic}}\,.$
(182)
The contributing diagrams to the two-loop order are
(183)
The first three diagrams are trivially computed by Wick contraction in
position space. The one-loop diagram is
(184)
and the two simple two-loop diagrams sum to
(185)
We next turn to the two-loop diagram in (183). Working in momentum space, the
contribution to $\langle T_{z\bar{z}}(-k)T_{z\bar{z}}(k)\rangle$ is
(186)
This diagram has double and single pole divergences in $\varepsilon$. The
double pole is polynomial in $k$ and can be ignored as it won’t contribute to
the two-point function at finite spatial separation. The simple pole is
canceled, by design, via the stress tensor counterterm (181); i.e. by the two
one-loop diagrams in which one of the stress tensor insertions given by the
counterterm in (181). The resulting finite part is
$\langle T_{z\bar{z}}(-k)T_{z\bar{z}}(k)\rangle_{\infty}=2^{-7}\pi
r_{c}^{3}G\left(a\ln k^{2}+(\ln
k^{2})^{2}\right)(k^{2})^{4}+{\operatorname{polynomial}}\,.$ (187)
The constant $a$ is left unspecified since it can be shifted arbitrarily due
to the freedom in including a finite counterterm in (181). Fourier
transforming back to position space, we obtain
$\langle T_{z\bar{z}}(x)T_{z\bar{z}}(0)\rangle_{\infty}=2^{8}\cdot
3^{2}r_{c}^{3}G{\ln(\mu^{2}z\overline{z})\over(z\overline{z})^{5}}\,,$ (188)
where we now traded the arbitrary constant $a$ for a renormalization scale
$\mu$.242424Logarithms also appear in the $T\bar{T}$ deformed correlation
functions of [88, 89].
#### 10 Summary of two-point deformed correlators at two-loop order
Combining results to the two-loop order, we have found
$\leavevmode\resizebox{422.77661pt}{}{$\langle
T_{z\bar{z}}(x)T_{z\bar{z}}(0)\rangle=\frac{3}{(z\overline{z})^{2}}\left[(3+4G)\left(r_{c}\over
z\overline{z}\right)^{2}-64G\left(1-12\ln(\mu^{2}z\overline{z})\right)\left(r_{c}\over
z\overline{z}\right)^{3}+400G\left(r_{c}\over
z\overline{z}\right)^{4}\right]$}\,.$ (189)
Using the Ward identities (154) and (156), we read off the other two-point
functions
$\displaystyle\langle T_{zz}(x)T_{zz}(0)\rangle$
$\displaystyle=\leavevmode\resizebox{366.40831pt}{}{$\frac{1}{z^{4}}\left[\frac{c}{2}+10(3+4G)\left(\frac{r_{c}}{z\overline{z}}\right)^{2}+96G\left(8+60\ln(\mu^{2}z\overline{z})\right)\left(\frac{r_{c}}{z\overline{z}}\right)^{3}+2520G\left(\frac{r_{c}}{z\overline{z}}\right)^{4}\right]\,,$}$
(190) $\displaystyle\langle T_{zz}(x)T_{z\overline{z}}(0)\rangle$
$\displaystyle=\frac{4}{z^{3}\overline{z}}\left[-(3+4G)\left(\frac{r_{c}}{z\overline{z}}\right)^{2}+24G\left(1-30\ln(\mu^{2}z\overline{z})\right)\left(\frac{r_{c}}{z\overline{z}}\right)^{3}-360G\left(\frac{r_{c}}{z\overline{z}}\right)^{4}\right]\,,$
$\displaystyle\langle T_{zz}(x)T_{\overline{z}\overline{z}}(0)\rangle$
$\displaystyle=\frac{3}{(z\overline{z})^{2}}\left[(3+4G)\left(\frac{r_{c}}{z\overline{z}}\right)^{2}-64G\left(1-12\ln(\mu^{2}z\overline{z})\right)\left(\frac{r_{c}}{z\overline{z}}\right)^{3}+400G\left(\frac{r_{c}}{z\overline{z}}\right)^{4}\right]\,,$
where $c=c_{0}+1={3\over 2G}+1$.
#### 11 Higher point correlators
We have acquired a systematic method to compute any $n$-point stress tensor
correlator at any given order in $\lambda$ and $c$. Despite contenting
ourselves to two-point correlators in detail so far, studying higher-point
correlators is straightforward.
For example, at the one-loop level at $r_{c}=0$, the three-point correlator is
(191)
Another example is the tree-level deformed three-point correlator at
$\mathcal{O}(r_{c})$:
(192)
We can express (192) in terms of a product of undeformed tree-level two-point
correlators with the trace flow equation
$\displaystyle\langle
T_{z\bar{z}}(x_{1})T_{zz}(x_{2})T_{\bar{z}\bar{z}}(x_{3})\rangle_{r_{c}}$
$\displaystyle=\langle\left(-\pi\lambda
T_{zz}(x_{1})T_{\bar{z}\bar{z}}(x_{1})\right)T_{zz}(x_{2})T_{\bar{z}\bar{z}}(x_{3})\rangle_{0}+\mathcal{O}(\lambda^{2})$
(193) $\displaystyle=-\pi\lambda\langle
T_{zz}(x_{1})T_{zz}(x_{2})\rangle_{0}\langle
T_{\bar{z}\bar{z}}(x_{1})T_{\bar{z}\bar{z}}(x_{3})\rangle_{0}+\mathcal{O}(\lambda^{2})$
$\displaystyle=-\frac{\pi\lambda
c_{0}^{2}}{4}\frac{1}{z^{4}_{12}\bar{z}^{4}_{13}}+\mathcal{O}(\lambda^{2})\,,$
where
(194)
Moreover, from perturbation theory at the tree-level and $\mathcal{O}(r_{c})$,
we may compute
$\displaystyle\langle
T_{zz}(x_{1})T_{\bar{z}\bar{z}}(x_{2})T_{\bar{z}\bar{z}}(x_{3})\rangle_{r_{c}}.$
(195)
Using the fact that $\sqrt{g}d^{2}x=\frac{1}{2}d^{2}z$,
$\partial_{\bar{z}_{1}}\frac{1}{z_{1}-z_{2}}=2\pi\delta^{2}(z_{1}-z_{2})$ and
integration by parts, we arrive at the following at $\mathcal{O}(\lambda)$
$\displaystyle\langle
T_{zz}(x_{1})T_{\bar{z}\bar{z}}(x_{2})T_{\bar{z}\bar{z}}(x_{3})\rangle_{r_{c}}$
(196) $\displaystyle=\left\langle
T_{zz}(x_{1})T_{\bar{z}\bar{z}}(x_{2})T_{\bar{z}\bar{z}}(x_{3})\left(-\lambda\int
d^{2}x\sqrt{g}T_{zz}(x)T_{\bar{z}\bar{z}}(x)\right)\right\rangle_{0}$
$\displaystyle=-\lambda\int d^{2}x\sqrt{g}\langle
T_{zz}(x_{1})T_{\bar{z}\bar{z}}(x_{2})T_{\bar{z}\bar{z}}(x_{3})T_{zz}(x)T_{\bar{z}\bar{z}}(x)\rangle_{0}$
$\displaystyle=-\lambda\int d^{2}x\sqrt{g}\langle
T_{zz}(x_{1})T_{zz}(x)\rangle_{0}\langle
T_{\bar{z}\bar{z}}(x_{2})T_{\bar{z}\bar{z}}(x_{3})T_{\bar{z}\bar{z}}(x)\rangle_{0}$
$\displaystyle=-\lambda\int\left(\frac{1}{2}d^{2}z\right)\left(\frac{c_{0}}{2}\frac{1}{(z-z_{1})^{4}}\right)\left(\frac{c_{0}}{(\bar{z}-\bar{z}_{2})^{2}(\bar{z}-\bar{z}_{3})^{2}(\bar{z_{2}}-\bar{z}_{3})^{2}}\right)$
$\displaystyle=-\frac{\lambda
c_{0}^{2}}{4}\int\frac{d^{2}z}{(z-z_{1})^{4}(\bar{z}-\bar{z}_{2})^{2}(\bar{z}-\bar{z}_{3})^{2}(\bar{z_{2}}-\bar{z}_{3})^{2}}$
$\displaystyle=\frac{\lambda c_{0}^{2}}{12}\int
d^{2}z\left[\partial_{z}\frac{1}{(z-z_{1})^{3}}\right]\frac{1}{(\bar{z}-\bar{z}_{2})^{2}(\bar{z}-\bar{z}_{3})^{2}(\bar{z}_{2}-\bar{z}_{3})^{2}}$
$\displaystyle=-\frac{\lambda c_{0}^{2}}{12}\int
d^{2}z\frac{1}{(z-z_{1})^{3}}\partial_{z}\left[\frac{1}{(\bar{z}-\bar{z}_{2})^{2}(\bar{z}-\bar{z}_{3})^{2}(\bar{z}_{2}-\bar{z}_{3})^{2}}\right]$
$\displaystyle=-\frac{\lambda c_{0}^{2}}{12}\int
d^{2}z\frac{1}{(z-z_{1})^{3}}\left(-2\pi\partial_{\bar{z}}\delta^{2}(z-z_{2})\right)\frac{1}{(\bar{z}-\bar{z}_{3})^{2}(\bar{z}_{2}-\bar{z}_{3})^{2}}+(x_{2}\leftrightarrow
x_{3})$ $\displaystyle=-\frac{\pi\lambda
c_{0}^{2}}{3}\frac{1}{(\bar{z}_{2}-\bar{z}_{3})^{5}}\left(\frac{1}{(z_{1}-z_{2})^{3}}-\frac{1}{(z_{1}-z_{3})^{3}}\right).$
These are just a few sample diagrams related to three-point correlators.
Combinatorially, there are various ways of constructing three-point diagrams
than there are for two-point correlators, as we saw earlier. For illustrative
purposes, a few three-point diagrams are:
(197)
### 6 Gravitational AdS3 Wilson lines
To close this chapter, we perturbatively compute the deformed classical and
quantum gravitational Wilson line and its correlators in AdS3. As a
consistency check, our classical gravitational Wilson line correlator analysis
is consistent with previous results on $T\overline{T}$-deformed scalar
correlators [88, 91, 89] for constant stress tensor backgrounds.
#### 1 The classical AdS3 Wilson line
The gravitational AdS3 Wilson line anchored at the endpoints $z_{1}$ and
$z_{2}$ is conjectured to be dual to a bi-local primary operator:
$\langle W[z_{2},z_{1}]\rangle_{0}\longleftrightarrow\langle
O(z_{2})O(z_{1})\rangle_{0}\,.$ (198)
Given two arbitrary AdS3 bulk points $Z_{1}=(r_{1},z_{1})$ and
$Z_{2}=(r_{2},z_{2})$, the classical Wilson line is defined as the path-
ordered integral
$W[(r_{2},z_{2};r_{1},z_{1})]_{0}=P\exp\left(\int^{(r_{2},z_{2})}_{(r_{1},z_{1})}A\right)\,.$
(199)
Under a gauge transformation, the Wilson line transforms as
$W[(r_{2},z_{2};r_{1},z_{1})]_{0}\rightarrow
g(r_{2},z_{2})^{-1}W[(r_{2},z_{2};r_{1},z_{1})]_{0}g(r_{1},z_{1})\,,$ (200)
with $g\in\text{SL}(2,\mathbb{R})$ and $A\rightarrow g^{-1}\left(d+A\right)g$.
In particular, the radial dependence of the connection
$\displaystyle A$ $\displaystyle=b(r)\left(d+a(z)\right)b(r),\quad
b(r)=r^{L_{0}}\,,\overline{A}$
$\displaystyle=\overline{b}(r)\left(d+\bar{a}(\bar{z})\right)\overline{b}(r)^{-1},\quad\overline{b}(r)=r^{\bar{L}_{0}}\,,$
(201)
arises through a gauge transformation:
$W[(r_{2},z_{2};r_{1},z_{1})]_{0}=b(r_{2})^{-1}P\exp\left(\int_{z_{1}}^{z_{2}}a\right)b(r_{1})\,.$
(202)
The matrix elements of $W[z_{2},z_{1}]$ between the lowest and highest weight
states are
$\displaystyle\langle W[z_{2},z_{1}]\rangle_{0}$ $\displaystyle=\langle
j,-j\mid P\exp\left(\int^{z_{2}}_{z_{1}}a\right)\mid j,j\rangle_{0}$ (203)
$\displaystyle=\langle j,-j\mid
P\exp\left[\int^{z_{2}}_{z_{1}}dz\left(L_{1}+\frac{6}{c}T_{zz}(z)L_{-1}\right)\right]\mid
j,j\rangle_{0}\,,$
where $|j,m\rangle$ is the state of weight $m$ in the spin-$j$ representation
of $SL(2,\mathbb{R})$. To see the bi-localness of the classical Wilson line,
first consider the vacuum state of $3d$ gravity. In the vacuum state, the
path-ordered integral reduces to an ordinary integral
$\langle W[z_{2},z_{1}]\rangle_{0}\big{|}_{T_{zz}=0}=\langle
j,-j\mid\exp\left(\int^{z_{2}}_{z_{1}}dz\,L_{1}\right)\mid
j,j\rangle_{0}=z_{21}^{2j}\,,$ (204)
where $z_{ij}=z_{i}-z_{j}$ and the bi-local primary field has dimension
$h=-j$.
One can recover the case when $T_{zz}\neq 0$ through a local conformal
transformation $z\rightarrow f(z)$ which is given by inverting the Schwarzian
$T_{zz}=\frac{c}{12}\\{f(z),z\\}\,.$ (205)
As a result, the classical Wilson line for a general background is
$\langle W[z_{2},z_{1}]\rangle_{0}\big{|}_{T_{zz}\neq
0}=\frac{\left[f(z_{2})-f(z_{1})\right]^{2j}}{\left[f^{\prime}(z_{2})f^{\prime}(z_{1})\right]^{j}}\,,$
(206)
and behaves as a bi-local primary operator at the endpoints. Intuitively, a
way to argue for the bi-locality of the Wilson line is because the Chern-
Simons equations of motion for the connections are flat. Consequently, this
makes the Wilson line path-independent between the two endpoints.
#### 2 The quantum $T\overline{T}$-deformed AdS3 Wilson line
The quantum Wilson line is obtained by beginning with the definition of the
classical Wilson line, where the stress tensor $T_{zz}$ is thought of as a
commuting number and promoting the stress tensor to an operator of the CFT.
The resulting object is conjectured to behave as a bi-local primary operator
at its endpoints, $\langle W[z_{2},z_{1}]\rangle_{0}=z_{21}^{-2h(j,c)}$.
Because the stress tensor is now an operator, short-distance singularities
arise from the stress tensor OPE, so the scaling dimension $h(j,c)$ of the
Wilson line experiences quantum corrections of the form252525The specific
values of $h_{n}(j)$ are easily calculable from [71, 63, 64]. For instance, a
few values are: $h_{0}(j)=-j,\quad h_{1}(j)=-6j(j+1),\quad
h_{2}(j)=-78j(j+1),\quad h_{3}(j)=-1230j(j+1),\quad h_{4}(j)=-21606j(j+1)\,.$
(207)
$h(j,c)=\sum^{\infty}_{n=0}\frac{h_{n}(j)}{c^{n}}\,.$ (208)
Due to these short-distance singularities from the stress tensor OPE, we must
regularize the gravitational Wilson line to verify that the quantum Wilson
line
$\displaystyle\langle W[z_{2},z_{1}]\rangle_{0}$ $\displaystyle=\langle
j,-j\mid
P\exp\left(\int^{z_{2}}_{z_{1}}dz\left(L_{1}+\frac{6}{c}T_{zz}(z)L_{-1}\right)\right)\mid
j,j\rangle_{0}$ (209)
$\displaystyle=\sum_{n=0}^{\infty}\int_{z_{1}}^{z_{2}}dy_{n}\int_{z_{1}}^{y_{n}}dy_{n-1}\cdots\int_{z_{1}}^{y_{2}}dy_{1}$
$\displaystyle\langle
j,-j\mid\left(L_{1}+\frac{6}{c}T_{zz}(y_{n})L_{-1}\right)\cdots\left(L_{1}+\frac{6}{c}T_{zz}(y_{1})L_{-1}\right)\mid
j,j\rangle_{0}$
captures the correct scaling dimension (208) as a bi-local primary operator.
Further perturbative evidence of the Wilson line’s bi-localness was provided
in [71, 63, 64]. The authors of [71] successfully calculated the undeformed
quantum Wilson line $\langle W[z;0]\rangle_{0}$ up to
$\mathcal{O}(\frac{1}{c})$ and encountered some ambiguities in the
coefficients at the two-loop order, $\mathcal{O}(\frac{1}{c^{2}})$, due to the
absence of a systematic renormalization scheme that preserves conformal
invariance. The most promising scheme is the dimensional regularization
approach used in [63], where an overall multiplicative renormalization
$N(\varepsilon)$ and a renormalization of the vertex factor
$\alpha(\varepsilon)$ were needed in $d=2-\varepsilon$ dimensions:
$\lim_{\varepsilon\rightarrow 0}\langle
W_{\varepsilon}[z_{2},z_{1}]\rangle_{0}=z_{21}^{2j}\lim_{\varepsilon\rightarrow
0}N(\varepsilon)\langle j,-j\mid
P\exp\left(\frac{6\alpha(\varepsilon)}{c}\int_{z_{1}}^{z_{2}}dy\left(L_{1}+\frac{6}{c}T_{zz}(y)L_{-1}\right)\right)\mid
j,j\rangle_{0}\,.$ (210)
Here $N(\varepsilon)$ and $\alpha(\varepsilon)$ are chosen order-by-order in
$\frac{1}{c}$ to cancel the poles in $\varepsilon$. The authors in [63]
corrected the issue which arose at $\mathcal{O}(\frac{1}{c^{2}})$ in [71].
They also carefully calculated and confirmed the
$\mathcal{O}(\frac{1}{c^{3}})$ corrections to the Wilson line.
Using the systematic renormalization approach in [63], the authors of [64]
calculated Wilson line correlators with multiple stress tensors insertions
$\big{\langle}\prod^{n}_{i=1}T_{zz}(w_{i})W[z_{2},z_{1}]\big{\rangle}$ and
found results consistent with the expectation that the Wilson line yields the
vacuum Virasoro OPE block (3)-(4). However, whether the quantum Wilson line
behaves as a bi-local primary operator non-perturbatively in $\frac{1}{c}$ is
still unknown as dimensional regularization may violate conformal invariance.
This completes our review of the quantum Wilson line; we now set up the
necessary formalism to compute the deformed quantum Wilson line.
We begin with the Wilson line in terms of the boundary stress tensor, which is
valid in the undeformed theory because the connections can be brought into
Bañados form:
$W[z_{2},z_{1}]=P\exp\left(\int^{z_{2}}_{z_{1}}dy\left(L_{1}+\frac{6}{c}T_{zz}(y)L_{-1}\right)\right).$
(211)
Following [71], we write (211) in a more convenient form by defining
$\displaystyle V[z_{1},z_{2}]$
$\displaystyle=\exp\left(-L_{1}z_{21}\right)W[z_{1},z_{2}]\,,$ (212)
so that
$\begin{split}\frac{d}{dz_{2}}V[z_{1},z_{2}]&=\exp\left(-L_{1}z_{21}\right)\frac{6}{c}T_{zz}(z_{2})L_{-1}\exp\left(L_{1}z_{21}\right)V[z_{1},z_{2}]\\\
&=\frac{6}{c}\left((1-L_{1}z_{21})L_{-1}(1+L_{1}z_{21})\right)T_{zz}(z_{2})V[z_{1},z_{2}]\\\
&=\frac{6}{c}\left(L_{-1}+z_{21}[L_{-1},L_{1}]-z_{21}^{2}L_{1}L_{-1}L_{1}\right)T_{zz}(z_{2})V[z_{1},z_{2}]\\\
&=\frac{6}{c}\left(L_{-1}-2z_{21}L_{0}+z_{21}^{2}L_{1}\right)T_{zz}(z_{2})V[z_{1},z_{2}]\,,\end{split}$
(213)
where we have used the facts $L_{\pm 1}^{2}=0$, $L_{1}L_{-1}L_{1}=-L_{1}$, and
$[L_{-1},L_{1}]=-2L_{0}$.
Here (213) is solved by the usual path-ordered exponential and this allows us
to write the Wilson line in a more convenient form to systematically implement
a $\frac{1}{c}$ expansion:
$\displaystyle\langle W[z_{2},z_{1}]\rangle$ (214) $\displaystyle=\langle
j,-j\mid\exp\left(z_{21}L_{1}\right)P\exp\left(\frac{6}{c}\int^{z_{2}}_{z_{1}}\left(L_{-1}-2(y-z_{1})L_{0}+(y-z_{1})^{2}L_{1}\right)T_{zz}(y)dy\right)\mid
j,j\rangle\,.$
The gravitational Wilson line in this form (LABEL:eq:GravWilson) can be
understood as a perturbative expansion in $\frac{1}{c}$ of self-energy Feynman
diagrams:
(215)
The first diagram in (215) contributes at $\mathcal{O}(\frac{1}{c^{0}})$, the
second diagram contributes at $\mathcal{O}(\frac{1}{c})$, the final four
diagrams contribute at $\mathcal{O}(\frac{1}{c^{2}})$, and the ellipsis
denotes higher order quantum corrections past $\mathcal{O}(\frac{1}{c^{2}})$.
For every vertex, in the undeformed case, we have holomorphic stress tensor
insertions. For $n$ vertices, we have an $n$-point correlator of holomorphic
stress tensors to integrate over. Using this setup for the quantum
gravitational Wilson line, writing down formal expressions for the deformed
corrections from the $n$-point deformed stress tensor correlators is
straightforward.
One can then use the Feynman rules of the fundamental fields derived in this
chapter (151) and (152) to calculate the deformed stress tensor correlators.
Intuitively, the quantum corrections to the deformed gravitational Wilson line
$\langle W[z_{2},z_{1}]\rangle_{\lambda}$ involve non-vanishing self-energy
interactions between both holomorphic and anti-holomorphic exchanges of the
fundamental fields denoted by solid and dashed propagators respectively.
Determining the deformed quantum Wilson line at a given order in $\lambda$ and
$\frac{1}{c}$ is computationally complicated for two reasons. The first reason
is that the $n$-point stress tensor correlator is subject to both quantum
corrections in $\frac{1}{c}$ and $\lambda$ corrections. To be more precise, we
notice that the expectation value of the undeformed Wilson line
$W_{\varepsilon}[z,0]$ has the expansion
$\langle
W_{\varepsilon}[z;0]\rangle_{0}=z^{2j}N(\varepsilon)\sum_{n=0}^{\infty}\frac{(6\alpha(\varepsilon))^{n}}{c^{n}}\int_{0}^{z}dy_{n}\cdots\int_{0}^{y_{2}}dy_{1}F_{n}(z;y_{n},\dots,y_{1})\langle
T_{zz}(y_{n})\cdots T_{zz}(y_{1})\rangle_{0}$ (216)
from (210). Here the SL$(2,\mathbb{R})$ group theory factor
$F_{n}\left(z;y_{n},\dots,y_{1}\right)$ is defined by the following
homogeneous polynomial in the variables $z,y_{n},\cdots,y_{1}$ of degree $n$:
$z^{2j}F_{n}\left(z;y_{n},\dots,y_{1}\right)=\left\langle j,-j\mid
e^{zL_{1}}\left(L_{-1}-2y_{n}L_{0}+y_{n}^{2}L_{1}\right)\cdots\left(L_{-1}-2y_{1}L_{0}+y_{1}^{2}L_{1}\right)\mid
j,j\right\rangle\,.$ (217)
Computing $\langle W_{\varepsilon}[z;0]\rangle_{\lambda}$ via conformal
perturbation theory in $\lambda$ involves an infinite $\lambda$ expansion at
each order of the $\mathcal{O}\left(\frac{1}{c}\right)$ expansion of the
undeformed Wilson line $\langle W_{\varepsilon}[z;0]\rangle_{0}$. For
instance, the $\mathcal{O}\left(\frac{1}{c^{2}}\right)$ term in the
$\frac{1}{c}$ expansion262626We start with $O\left(\frac{1}{c^{2}}\right)$
because at $\mathcal{O}\left(\frac{1}{c}\right)$, the one-point planar
correlator $\langle T_{zz}(y_{1})\rangle_{\lambda}$ vanishes identically by
Lorentz and translational invariance. In the case of non-planar backgrounds,
such as the cylinder or torus [58, 150], the one-point function is nonzero. of
$\langle W_{\varepsilon}[z;0]\rangle_{0}$ is
$\displaystyle
z^{2j}N(\varepsilon)\frac{(6\alpha(\varepsilon))^{2}}{c^{2}}\int_{0}^{z}dy_{1}\int_{0}^{y_{2}}dy_{1}F_{2}(z;y_{1},y_{2})\langle
T_{zz}(y_{1})T_{zz}(y_{2})\rangle_{0}$ (218) $\displaystyle\rightarrow
z^{2j}N(\varepsilon)\frac{(6\alpha(\varepsilon))^{2}}{c^{2}}\sum_{p=0}^{\infty}\int_{0}^{z}dy_{1}\int_{0}^{y_{2}}F_{2}(z;y_{1},y_{2})$
$\displaystyle\cdot\left\langle
T_{zz}(y_{1})T_{zz}(y_{2})\frac{\lambda^{p}}{p!}\left(\int
d^{2}wT_{zz}(w)T_{\bar{z}\bar{z}}(\overline{w})\right)^{p}\right\rangle\,.$
The divergences from the integrals are handled by the dimensional
regularization scheme, which we mentioned in the discussion of the
renormalized vertex factor and multiplicative renormalization around equation
(210). For exposition’s sake, we content ourselves with determining the
leading order corrections to the quantum Wilson line (LABEL:eq:GravWilson).
Expanding the exponential (LABEL:eq:GravWilson), we find
$\displaystyle\langle W[z_{2},z_{1}]\rangle_{\lambda}$ (219)
$\displaystyle\,\,\,=z^{2j}\bigg{[}1+\left(\frac{6}{c}\right)^{2}\int^{z_{2}}_{z_{1}}dy_{1}\int^{y_{2}}_{z_{1}}dy_{2}\langle
j,-j\mid\exp\left(L_{1}z_{21}\right)\left(L_{-1}-2(y_{1}-z_{1})L_{0}+(y_{1}-z_{1})^{2}L_{1}\right)$
$\displaystyle\hskip
130.0pt\times\left(L_{-1}-2(y_{2}-z_{1})L_{0}+(y_{2}-z_{1})^{2}L_{1}\right)\mid
j,j\rangle\langle T_{zz}(y_{1})T_{zz}(y_{2})\rangle_{\lambda}\bigg{]}\,.$
The tree-level deformed planar stress tensor two-point function at
$\mathcal{O}(\lambda^{2}c^{2})$ was determined first by [88] via
translational/rotational invariance and stress tensor conservation.
Alternatively, using the approach in this chapter, this is easily understood
by the propagators of the fundamental fields as
$\displaystyle\langle T_{zz}(y_{1})T_{zz}(y_{2})\rangle_{\lambda}$
$\displaystyle=\frac{1}{(8G)^{2}}\partial_{y_{1}}\partial_{y_{2}}\langle
f^{\prime}(y_{1})f^{\prime}(y_{2})\rangle_{0}$ (220)
$\displaystyle+\frac{r_{c}^{2}}{\left(16G\right)^{2}}\left(\partial^{2}_{y_{1}}\partial^{2}_{y_{2}}\langle
f^{\prime}(y_{1})f^{\prime}(y_{2})\rangle_{0}\right)\langle\bar{f}^{\prime}(\bar{y}_{1})\bar{f}^{\prime}(\bar{y}_{2})\rangle_{0}$
$\displaystyle=\frac{c}{2\left(y_{1}-y_{2}\right)^{4}}+\frac{5\pi^{2}\lambda^{2}c^{2}}{6\left(y_{1}-y_{2}\right)^{6}\left(\bar{y}_{1}-\bar{y}_{2}\right)^{2}}\,.$
Thus
$\displaystyle\langle
j,-j\mid\exp\left(L_{1}z\right)\left(L_{-1}-2y_{1}L_{0}+y_{1}^{2}L_{1}\right)\left(L_{-1}-2y_{2}L_{0}+y_{2}^{2}L_{1}\right)\mid
j,j\rangle$ (221)
$\displaystyle=z^{2j-2}\left[2jy_{2}\left(z-y_{1}\right)\left(2jy_{1}(z-y_{2})-y_{2}(z-y_{1})\right)\right]\,.$
The integral (LABEL:eq:QWilson1) reduces to
$\displaystyle\langle W[z,0]\rangle_{\lambda}$
$\displaystyle=z^{2j}\bigg{[}1+\frac{36j}{cz^{2}}\int^{z}_{0}dy_{1}\int^{y_{1}}_{0}dy_{2}\frac{y_{2}(z-y_{1})\left(2jy_{1}(z-y_{2})-y_{2}(z-y_{1})\right)}{(y_{1}-y_{2})^{4}}$
(222)
$\displaystyle+\frac{60\pi^{2}\lambda^{2}j}{z^{2}}\int^{z}_{0}dy_{1}\int^{y_{1}}_{0}dy_{2}\frac{y_{2}(z-y_{1})\left(2jy_{1}(z-y_{2})-y_{2}(z-y_{1})\right)}{(y_{1}-y_{2})^{6}(\bar{y}_{1}-\bar{y}_{2})^{2}}\bigg{]}\,,$
which clearly diverges when $y_{2}\rightarrow y_{1}$ or
$\bar{y}_{2}\rightarrow\bar{y}_{1}$.
We dimensionally regularize the stress tensor correlators to evaluate these
divergent integrals (222). The $O(\frac{\lambda^{0}}{c})$ integral has already
been evaluated in [63] via dimensional regularization, which gives
$\frac{36j}{cz^{2}}\int^{z}_{0}dy_{1}\int^{y_{1}}_{0}dy_{2}\,\frac{y_{2}(z-y_{1})\left(2jy_{1}(z-y_{2})-y_{2}(z-y_{1})\right)}{(y_{1}-y_{2})^{4}}=\frac{12j(j+1)}{c}\ln
z\,.$ (223)
To deal with the $\mathcal{O}(\lambda^{2}c^{0})$ integral in (222), we first
specify an integration contour in the complex plane that is a straight line
along the direction towards $z$:
$\displaystyle y_{2}=y_{1}t,\quad 0\leq t\leq 1\,,$ (224) $\displaystyle
y_{1}=zT,\quad 0\leq T\leq 1\,,$
and we find
$\displaystyle\frac{60\pi^{2}\lambda^{2}j}{z^{2}}\int^{z}_{0}dy_{1}\int^{y_{1}}_{0}dy_{2}\,\frac{y_{2}(z-y_{1})\left(2jy_{1}(z-y_{2})-y_{2}(z-y_{1})\right)}{(y_{1}-y_{2})^{6}(\bar{y}_{1}-\bar{y}_{2})^{2}}$
(225) $\displaystyle=$
$\displaystyle\frac{60\pi^{2}\lambda^{2}j}{|z|^{4}}\int^{1}_{0}dT\,\frac{1-T}{T^{5}}\int^{1}_{0}dt\,\frac{2j(t-t^{2}T)-t^{2}(1-T)}{(1-t)^{8}}\,.$
The above integral is evaluated via dimensional regularization:
$\displaystyle\frac{60\pi^{2}\lambda^{2}j}{|z|^{4}}\int^{1}_{0}dT\,\frac{1-T}{T^{5-2\varepsilon}}\int^{1}_{0}dt\,\frac{2j(t-t^{2}T)-t^{2}(1-T)}{(1-t)^{8-2\varepsilon}}$
(226) $\displaystyle=$
$\displaystyle\frac{\pi^{2}j(9j-1)\lambda^{2}}{21|z|^{4}}\,.$
In summary, the leading order correction to the Wilson line is
$\langle W[z,0]\rangle_{\lambda}=z^{2j}\left(1+\frac{12j(j+1)}{c}\ln
z+\frac{\pi^{2}j(9j-1)\lambda^{2}}{21|z|^{4}}\right)\,.$ (227)
An alternative renormalization approach, which yields the same numerical
coefficient as (LABEL:eq:dimreglambda), is to introduce cutoffs
$\varepsilon_{1}$ and $\varepsilon_{2}$ as
$\displaystyle\frac{60\pi^{2}\lambda^{2}j}{|z|^{4}}\int^{1}_{\varepsilon_{2}}dT\,\frac{1-T}{T^{5}}\int^{1-\varepsilon_{1}}_{0}dt\,\frac{2j(t-t^{2}T)-t^{2}(1-T)}{(1-t)^{8}}\,,$
(228)
and perform “minimal subtraction” to remove the divergent terms. Evaluating
(LABEL:eq:alternativeregulator) gives
$\frac{\pi^{2}j(9j-1)\lambda^{2}}{21|z|^{4}}\,.$ (229)
#### 3 AdS3 Wilson line correlators
The correlator involving the product of a holomorphic and anti-holomorphic
Wilson line is a scalar correlator. A meaningful check is to compute this
Wilson line product correlator and see if it is consistent with
$T\overline{T}$-deformed scalar correlators [88, 91, 89]. We use conformal
perturbation theory at $\mathcal{O}(\lambda)$ to find
$\displaystyle\begin{split}\langle
W[z,0]\overline{W}[\bar{z},0]\rangle_{\lambda}&=\left\langle
W[z,0]\overline{W}[\bar{z},0]\exp\left(\lambda\int
d^{2}w~{}T_{zz}(w)T_{\bar{z}\bar{z}}(\bar{w})\right)\right\rangle\\\ &=\langle
W[z,0]\overline{W}[\bar{z},0]\rangle_{0}+\lambda\int d^{2}w\left\langle
T_{zz}(w)T_{\bar{z}\bar{z}}(\bar{w})W[z,0]\overline{W}[\bar{z},0]\right\rangle_{0}\\\
&=|z|^{-4h(j)}+\lambda\int d^{2}w\left\langle
T_{zz}(w)W[z,0]\right\rangle_{0}\left\langle
T_{\bar{z}\bar{z}}(\bar{w})\overline{W}[\bar{z},0]\right\rangle_{0}\\\
&=|z|^{-4h(j)}+\lambda
h^{2}|z|^{4}\int\frac{d^{2}w}{|w|^{4}|w-z|^{4}}\left\langle
W[z,0]\right\rangle_{0}\left\langle\overline{W}[\bar{z},0]\right\rangle_{0}\\\
&=|z|^{-4h}+\lambda
h^{2}|z|^{4}\int\frac{d^{2}w}{|w|^{4}|w-z|^{4}}|z|^{-4h}\\\
&=|z|^{-4h(j)}\left(1+\lambda
h^{2}|z|^{4}\mathcal{I}_{2222}(0,z,0,\bar{z})\right)\,.\end{split}$ (230)
Here, we have used the Ward identity in [64]
$\displaystyle\begin{split}&\left\langle
T_{zz}(w)W\left[z,0\right]\right\rangle_{0}=\frac{h(j)z^{2}}{\left(z-w\right)^{2}w^{2}}\left\langle
W\left[z,0\right]\right\rangle_{0}\,,\\\ &\left\langle
T_{\bar{z}\bar{z}}(\bar{w})\overline{W}\left[\bar{z},0\right]\right\rangle_{0}=\frac{h(j)\bar{z}^{2}}{\left(\bar{z}-\bar{w}\right)^{2}\bar{w}^{2}}\left\langle\overline{W}\left[\bar{z},0\right]\right\rangle_{0}\,,\end{split}$
(231)
which displays the bi-local structure of the gravitational Wilson line. From
appendix A in [91], the integral (230) is of the form
$\displaystyle\begin{aligned}
&\mathcal{I}_{a_{1},\cdots,a_{m},b_{1},\cdots,b_{n}}\left(z_{i_{1}},\cdots,z_{i_{m}},\bar{z}_{j_{1}},\cdots,\bar{z}_{j_{n}}\right)=\int\frac{d^{2}z}{\prod^{m}_{k=1}\left(z-z_{i_{k}}\right)^{a_{k}}\prod^{n}_{p=1}\left(\bar{z}-\bar{z}_{j_{p}}\right)^{b_{p}}}\,,\end{aligned}$
(232)
and is evaluated via dimensional regularization. In particular,
$\displaystyle\begin{split}\mathcal{I}_{2222}(0,z,0,\bar{z})&=\int\frac{d^{2}w}{|w|^{4}|w-z|^{4}}\\\
&=\frac{4\pi}{|z|^{6}}\left(\frac{4}{\varepsilon}+2\ln|z|^{2}+2\ln\pi+2\gamma-5\right)\\\
&=\frac{1}{|z|^{6}}\left(C_{1}+C_{2}\ln|z|^{2}\right)\,,\end{split}$ (233)
where $C_{1}$ and $C_{2}$ are constant coefficients. We arrive
$\displaystyle\langle W[z,0]\overline{W}[\bar{z},0]\rangle_{\lambda}$
$\displaystyle=|z|^{-4h(j)}\left(1+\frac{\lambda
h(j)^{2}\left(C_{1}+C_{2}\ln|z|^{2}\right)}{|z|^{2}}\right)\,,$ (234)
which exactly matches what we expect at $\mathcal{O}(\lambda c^{0})$ from
previous analyses of $T\overline{T}$-deformed scalar correlators [88, 91, 89].
This confirms the claim that the correlator of two Wilson lines behaves as a
scalar correlator, at least in this order.
Additionally, at leading order in $\lambda$ and in the large-$c$ limit, (230)
agrees with the structure one would expect from the linear mixing of sources
and expectation values discussed above. Schematically, in the leading order,
$\displaystyle\left\langle
P\exp\left[\int^{z}_{0}dy\left(e_{i}(\lambda)L_{1}+\frac{6}{c}T_{zz}(y)L_{-1}\right)\right]P\exp\left[\int^{\bar{z}}_{0}d\bar{y}\left(\bar{e}_{i}(\lambda)\bar{L}_{1}+\frac{6}{c}T_{\bar{z}\bar{z}}(\bar{y})\bar{L}_{-1}\right)\right]\right\rangle_{\lambda}$
$\displaystyle=\Bigg{\langle}P\exp\left[\int^{z}_{0}dy\left((1+\lambda
T_{\bar{z}\bar{z}}(y))L_{1}+\frac{6}{c}T_{zz}(y)L_{-1}\right)\right]$
$\displaystyle\qquad\qquad\cdot
P\exp\left[\int^{\bar{z}}_{0}d\bar{y}\left((1+\lambda
T_{zz}(y))\bar{L}_{1}+\frac{6}{c}T_{\bar{z}\bar{z}}(\bar{y})\bar{L}_{-1}\right)\right]\Bigg{\rangle}_{\lambda}$
$\displaystyle=\langle
W[z,0]\rangle_{0}\langle\overline{W}[\bar{z},0]\rangle_{0}+\lambda\langle\exp\left(zL_{1}\right)L_{1}\rangle\int^{z}_{0}dy\left\langle
T_{\bar{zz}}(\bar{y})\overline{W}[\bar{z},0]\right\rangle_{0}+\mathcal{O}(\lambda^{2})$
$\displaystyle=|z|^{-4h(j)}+\lambda\partial_{z}\langle\exp\left(zL_{1}\right)\rangle\int^{z}_{0}dy\left\langle
T_{\bar{zz}}(\bar{y})\overline{W}[\bar{z},0]\right\rangle_{0}+\mathcal{O}(\lambda^{2})$
$\displaystyle=|z|^{-4h(j)}-2\lambda
h(j)z^{-2h(j)-1}\int^{z}_{0}dy\frac{h(j)\bar{z}^{2}}{(\bar{z}-\bar{y})^{2}\bar{y}^{2}}\langle\overline{W}[\bar{z},0]\rangle_{0}+\mathcal{O}(\lambda^{2})$
$\displaystyle=|z|^{-4h(j)}-2\lambda
h(j)^{2}z^{-2h(j)-1}\bar{z}^{-2h(j)+2}\int^{z}_{0}\frac{dy}{(\bar{z}-\bar{y})^{2}\bar{y}^{2}}+\mathcal{O}(\lambda^{2})$
$\displaystyle=|z|^{-4h(j)}\left(1+\lambda
h(j)^{2}z^{-1}\bar{z}^{2}\left(\frac{c_{1}+c_{2}\ln|z|^{2}}{\bar{z}^{3}}\right)+\mathcal{O}(\lambda^{2})\right)$
$\displaystyle=|z|^{-4h(j)}\left(1+\lambda
h(j)^{2}\left(\frac{c_{1}+c_{2}\ln|z|^{2}}{|z|^{2}}\right)+\mathcal{O}(\lambda^{2})\right)\,,$
(235)
where in the large-$c$ limit, the quantum corrections to the Wilson line’s
scaling dimension $h(j)=-j$ are suppressed and
$\langle\exp\left(zL_{1}\right)\rangle=z^{-2h}=z^{2j}$. The integral in (3)
may be evaluated via the integration cutoff introduced in
(LABEL:eq:alternativeregulator) or by dimensional regularization. Using either
method, one finds that the result has a similar structure as (234), where
$c_{1}$ and $c_{2}$ are constant coefficients.
We emphasize that if one had not used the linear mixing or conformal
perturbation theory, but rather expanded each path-ordered exponential in
$\langle
W[z_{2},z_{1}]\overline{W}[\bar{z}_{2},\bar{z}_{1}]\rangle_{\lambda}$, then
the leading contribution in $\lambda$ would be at
$\mathcal{O}(\lambda^{2}c^{0})$, which arises from integrating the tree-level
deformed stress tensor two-point function. To see this, let us compute the
correction
$\delta\langle W[z,0]\overline{W}[\bar{z},0]\rangle_{\lambda}=\langle
W[z,0]\overline{W}[\bar{z},0]\rangle_{\lambda}-\langle
W[z,0]\overline{W}[\bar{z},0]\rangle_{0}$ (236)
to the correlator using this prescription. We expand the path-ordered
exponential272727For the single Wilson line (222), we expanded up to
$\mathcal{O}(\frac{1}{c^{2}})$ in the path-ordered exponential since the
planar one-point function vanishes. At $\mathcal{O}(\frac{1}{c})$, the path-
ordered exponential reduces to a regular integral. for $W[z,0]$ and
$\overline{W}[\bar{z},0]$ up to $\mathcal{O}(\frac{1}{c})$ in
(LABEL:eq:GravWilson), which gives
$\displaystyle\left(\frac{6}{c}\right)^{2}\int^{z}_{0}dy_{1}\int^{\bar{z}}_{0}d\bar{y}_{2}$
$\displaystyle\langle j,-j\mid
e^{L_{1}z}(L_{-1}-2y_{1}L_{0}+y_{1}^{2})e^{\bar{L}_{1}\bar{z}}(\bar{L}_{-1}-2\bar{y}_{2}\bar{L}_{0}+\bar{y}_{2}^{2})\mid
j,j\rangle$ (237) $\displaystyle\cdot\langle
T_{zz}(y_{1})T_{\bar{z}\bar{z}}(y_{2})\rangle_{\lambda}\,.$
Using (92) and the Feynman rules for the relevant tree diagrams,
$\displaystyle\langle T_{zz}(y_{1})T_{\bar{z}\bar{z}}(y_{2})\rangle_{\lambda}$
$\displaystyle=\frac{r_{c}^{2}}{(16G)^{2}}\left(\partial^{2}_{y_{1}}\langle
f^{\prime}(y_{1})f^{\prime}(y_{2})\rangle_{0}\right)\left(\partial^{2}_{\bar{y}_{2}}\langle\bar{f}^{\prime}(\bar{y}_{1})\bar{f}^{\prime}(\bar{y}_{2})\rangle_{0}\right)+O(r_{c}^{3})$
(238)
$\displaystyle=\frac{\pi^{2}\lambda^{2}c^{2}}{4}\frac{1}{\left(y_{1}-y_{2}\right)^{4}\left(\bar{y}_{1}-\bar{y}_{2}\right)^{4}}+\mathcal{O}(\lambda^{3})\,.$
Thus the above prescription involving correlators of Wilson lines (237) is
incorrect because the leading correction enters at
$\mathcal{O}(\lambda^{2}c^{0})$, rather than the expected order of
$\mathcal{O}(\lambda c^{0})$ for scalar two-point correlators.
Furthermore, one may also consider a string of holomorphic and anti-
holomorphic stress tensor insertions in correlators involving a Wilson line.
For instance, we can calculate this kind of correlator via conformal
perturbation theory at $\mathcal{O}(\lambda)$:
$\left\langle
T_{zz}\left(w_{1}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{2}\right)W[z,0]\right\rangle_{\lambda}=\lambda\int
d^{2}y\left\langle
T_{zz}(y)T_{zz}\left(w_{1}\right)W[0,z]\right\rangle_{0}\left\langle
T_{\bar{z}\bar{z}}(\bar{y})T_{\bar{z}\bar{z}}\left(\bar{w}_{2}\right)\right\rangle_{0}\,.$
(239)
In [64], the following tree-level correlator to $O(1/c^{0})$ was derived:
$\displaystyle\left\langle
T_{zz}\left(w_{1}\right)T_{zz}\left(w_{2}\right)W[z,0]\right\rangle_{0}$ (240)
$\displaystyle=\frac{j^{2}z^{2j+4}}{w_{1}^{2}\left(z-w_{1}\right)^{2}w_{2}^{2}\left(z-w_{2}\right)^{2}}+\frac{jz^{2j+2}}{w_{1}\left(z-w_{1}\right)w_{2}\left(z-w_{2}\right)\left(w_{1}-w_{2}\right)^{2}}\,,$
in agreement with the predictions from the conformal Ward identities.
Therefore, using the fact that
$\left\langle
T_{\bar{z}\bar{z}}(\bar{y})T_{\bar{z}\bar{z}}\left(\bar{w}_{2}\right)\right\rangle_{0}=\frac{c}{2(\bar{y}-\bar{w}_{2})^{4}}\,,$
(241)
the integral (239) is reduced to
$\displaystyle\left\langle
T_{zz}\left(w_{1}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{2}\right)W[z,0]\right\rangle_{\lambda}$
$\displaystyle=\frac{cj\lambda z^{2j}}{2}\int
d^{2}y\bigg{[}\frac{jz^{4}}{y^{2}(y-z)^{2}w_{1}^{2}\left(z-w_{1}\right)^{2}\left(\bar{y}-\bar{w}_{2}\right)^{4}}$
(242)
$\displaystyle-\frac{z^{2}}{y(y-z)w_{1}\left(z-w_{1}\right)\left(y-w_{1}\right)^{2}\left(\bar{y}-\bar{w}_{2}\right)^{4}}\bigg{]}\,,$
and is evaluated in terms of the integrals defined in
(LABEL:MasterIntegralEquation):
$\displaystyle\left\langle
T_{zz}\left(w_{1}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{2}\right)W[z,0]\right\rangle_{\lambda}$
(243) $\displaystyle=\frac{j\lambda
cz^{2j+2}}{2w_{1}\left(z-w_{1}\right)}\left[\frac{jz^{2}}{w_{1}\left(z-w_{1}\right)}\mathcal{I}_{224}\left(0,z,\bar{w}_{2}\right)-\mathcal{I}_{1124}\left(0,z,w_{1},\bar{w}_{2}\right)\right]\,.$
Another example is a correlator involving two insertions of anti-holomorphic
stress tensors, a holomorphic stress tensor, and a holomorphic Wilson line.
The desired correlator
$\left\langle
T_{zz}\left(w_{1}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{2}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{3}\right)W[0,z]\exp\left(\lambda\int
d^{2}yT_{zz}(y)T_{\bar{z}\bar{z}}(\bar{y})\right)\right\rangle$ (244)
is easily computable at $\mathcal{O}(\lambda)$ via conformal perturbation
theory.
Noting that the undeformed tree-level planar three-point stress tensor
correlator is
$\langle
T_{\bar{z}\bar{z}}(\bar{y})T_{\bar{z}\bar{z}}\left(\bar{w}_{2}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{3}\right)\rangle_{0}=\frac{c}{\left(\bar{y}-\bar{w}_{2}\right)^{2}\left(\bar{w}_{2}-\bar{w}_{3}\right)^{2}\left(\bar{w}_{3}-\bar{y}\right)^{2}}\,,$
(245)
then the leading order correction to the integral (244) at
$\mathcal{O}(\lambda c)$ is
$\displaystyle\left\langle
T_{zz}\left(w_{1}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{2}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{3}\right)W[z,0]\right\rangle_{\lambda}$
$\displaystyle=\left\langle
T_{zz}\left(w_{1}\right)W[z,0]\right\rangle_{0}\left\langle
T_{\bar{z}\bar{z}}\left(\bar{w}_{2}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{3}\right)\right\rangle_{0}$
$\displaystyle+\lambda\int d^{2}y\,\left\langle
T_{zz}(y)T_{zz}\left(w_{1}\right)W[0,z]\right\rangle_{0}\langle
T_{\bar{z}\bar{z}}(\bar{y})T_{\bar{z}\bar{z}}\left(\bar{w}_{2}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{3}\right)\rangle_{0}$
$\displaystyle=\frac{h(j)z^{2}}{\left(z-w_{1}\right)^{2}w_{1}^{2}}\left\langle
W\left[z,0\right]\right\rangle_{0}\frac{c}{2(\bar{w}_{2}-\bar{w}_{3})^{4}}$
$\displaystyle+cj\lambda z^{2j}\int
d^{2}y\,\Bigg{[}\frac{jz^{4}}{y^{2}(y-z)^{2}w_{1}^{2}\left(z-w_{1}\right)^{2}\left(\bar{y}-\bar{w}_{2}\right)^{2}\left(\bar{w}_{2}-\bar{w}_{3}\right)^{2}\left(\bar{w}_{3}-\bar{y}\right)^{2}}$
$\displaystyle-\frac{z^{2}}{y(y-z)w_{1}\left(z-w_{1}\right)\left(y-w_{1}\right)^{2}\left(\bar{y}-\bar{w}_{2}\right)^{2}\left(\bar{w}_{2}-\bar{w}_{3}\right)^{2}\left(\bar{w}_{3}-\bar{y}\right)^{2}}\Bigg{]}\,.$
(246)
Evaluating (3) in terms of the integrals defined in
(LABEL:MasterIntegralEquation), we find
$\displaystyle\left\langle
T_{zz}\left(w_{1}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{2}\right)T_{\bar{z}\bar{z}}\left(\bar{w}_{3}\right)W[z,0]\right\rangle_{\lambda}=\frac{h(j)cz^{2-2h(j)}}{2(z-w_{1})^{2}w_{1}(\bar{w}_{2}-\bar{w}_{3})^{4}}$
$\displaystyle+\frac{j\lambda
cz^{2j+2}}{w_{1}\left(z-w_{1}\right)(\bar{w}_{2}-\bar{w}_{3})^{2}}\left[\frac{jz^{2}}{w_{1}(z-w_{1})}\mathcal{I}_{2222}\left(0,z,\bar{w}_{2},\bar{w}_{3}\right)-\mathcal{I}_{11222}\left(0,z,w_{1},\bar{w}_{2},\bar{w}_{3}\right)\right]\,.$
(247)
The integrals presented here, which are of the general form given in
(LABEL:MasterIntegralEquation) but with higher-valued indices, can be
expressed in terms of derivatives and linear combinations of known integrals
with lower-valued indices. See appendix A in [91] for several detailed
examples.
One can automate the above perturbative analysis in $\lambda$ to produce more
complicated expressions for correlators involving products of $m$-insertions
of holomorphic stress tensors, $n$-insertions of anti-holomorphic stress
tensors, and a network of Wilson lines (e.g. $p$-insertions of the holomorphic
Wilson line and $q$-insertions of the anti-holomorphic Wilson line) following
[64]. The leading correction for such a general correlator takes the form
$\displaystyle\left\langle\prod^{m}_{i=1}T_{zz}(x_{i})\prod^{n}_{j=1}T_{\bar{z}\bar{z}}(\bar{w}_{j})\prod^{p}_{k=1}W[z_{k+1},z_{k}]\prod^{q}_{l=1}\bar{W}[\bar{r}_{l+1},\bar{r}_{l}]\exp\left(\lambda\int
d^{2}yT_{zz}(y)T_{\bar{z}\bar{z}}(\bar{y})\right)\right\rangle$ (248)
$\displaystyle=\left\langle\prod^{m}_{i=1}T_{zz}(x_{i})\prod^{p}_{k=1}W[z_{k+1},z_{k}]\right\rangle_{0}\left\langle\prod^{n}_{j=1}T_{\bar{z}\bar{z}}(\bar{w}_{j})\prod^{q}_{l=1}\bar{W}[\bar{r}_{l+1},\bar{r}_{l}]\right\rangle_{0}$
$\displaystyle+\lambda\int d^{2}y\left\langle
T_{zz}(y)\prod^{m}_{i=1}T_{zz}(x_{i})\prod^{p}_{k=1}W[z_{k+1},z_{k}]\right\rangle_{0}\left\langle
T_{\bar{z}\bar{z}}(\bar{y})\prod^{n}_{j=1}T_{\bar{z}\bar{z}}(\bar{w}_{j})\prod^{q}_{l=1}\bar{W}[\bar{r}_{l+1},\bar{r}_{l}]\right\rangle_{0}$
$\displaystyle\quad\,+\mathcal{O}(\lambda^{2})\,.$
### 7 Conclusion
We gave evidence for the Nambu-Goto action (in Hamiltonian form) as the all-
order action for $3d$ gravity with a cutoff planar boundary. Secondly, we used
the action to compute correlators of the stress tensor operator to two-loop
order. The proposal for the action was based on finding a suitable field
redefinition yielding Nambu-Goto up eighth order in fields. Proving this
conjecture and determining the explicit form of the field redefinition to all
orders is desirable. Although the action takes the familiar Nambu-Goto form,
the stress tensor is not the canonical one, which is due to the way that the
original translation symmetries of the AdS3 background act on the redefined
fields. Our computation of stress tensor correlators to two-loop order
revealed the need for one stress tensor counterterm with an associated
undetermined finite part. As mentioned in the introduction, considering the
overarching arguments supporting the renormalizability of pure $3d$ gravity,
even when accounting for a finite planar cutoff boundary, it is anticipated
that symmetries will play a crucial role in determining all the parameters
involved. The implementation of these symmetries is complicated by the non-
Lorentz invariant form of the action and by the nonlocal field redefinition
that puts the action in Nambu-Goto form. A task for the future is to
systematically implement the Ward identities corresponding to these symmetries
and check if these yield unique results for stress tensor correlators. The
ultimate goal here is to get sufficient control over the stress tensor
correlators to say something about their short-distance structure, since this
gets to the heart of the nature of this theory, including its anticipated
nonlocal character; e.g. [151, 152]. Third is that in the $3d$ Chern-Simons
setting, we studied modifications to correlators involving boundary-anchored
Wilson lines, which were induced by a $T\overline{T}$ deformation on the $2d$
boundary; results were presented at both the classical level (using modified
boundary conditions) and the quantum-mechanical level (using conformal
perturbation theory).
Developing cases with curved cutoff boundaries would also be worthwhile. The
Chern-Simons computation of the action for a finite $S^{2}$ boundary is
considered in appendix 8.B, and it should be possible to extend this to one-
loop and compare with results in [153]; see also [154, 155] for related
results. The technical complication here is the two patches needed to define
the gauge connections on the sphere.
We close this chapter by commenting on the appearance of the Nambu-Goto action
in our analysis. By construction, solutions of our Nambu-Goto equations of
motion yield flat two-dimensional surfaces embedded in AdS3. On the other
hand, the precise Nambu-Goto action that arises is that of a string worldsheet
embedded in flat $\mathbb{R}^{3}$, with $\alpha^{\prime}$ controlled by the
cutoff $r_{c}$. One usually thinks of the solutions as describing extremal
area surfaces embedded in this flat spacetime. There is a correspondence
between flat surfaces embedded in AdS3 and extremal area surfaces embedded in
$\mathbb{R}^{3}$.
## Chapter 2 $T\overline{T}$ in JT Gravity and BF Gauge Theory
JT gravity can be represented using a first-order formulation akin to a two-
dimensional BF theory. This formulation can be perceived as the dimensional
reduction of the Chern-Simons description of $3d$ gravity. We consider
$T\overline{T}$-type deformations of the $(0+1)$-dimensional dual to this $2d$
BF theory and interpret the deformation as a modification of the BF theory
boundary conditions. The fundamental observables in this deformed BF theory
and its $3d$ Chern-Simons lift are Wilson lines and loops. In the last
chapter, we studied the $3d$ Chern-Simons setting and the modifications to
correlators involving boundary-anchored Wilson lines induced by a
$T\overline{T}$ deformation on the $2d$ boundary. In this chapter, we
determine the $T\overline{T}$-deformed boundary conditions in the BF
description of JT gravity. We discuss Wilson lines in the BF theory and
calculate the analogous deformed Wilson line correlators in $2d$ BF theory
below the Hagedorn temperature, where the principal series dominates over the
discrete series.
### 1 Introduction
In this chapter, we consider the $T\overline{T}$-deformation of two-
dimensional gauge or gravity theories, which are constructed in the following
way. We begin with a three-dimensional bulk gravity theory dual to a $2d$ CFT.
Then, we deform the boundary CFT by the $T\overline{T}$ operator and interpret
this deformation as a modification of the bulk gravity theory. Finally, we
dimensionally reduce this scenario on the circle to obtain a correspondence
between a deformed $2d$ gravity theory and a dual one-dimensional theory. One
can also rewrite the gravity theory in gauge theory variables and study the
deformation of the $2d$ gauge theory. In the diagram (6), this corresponds to
deforming the $2d$ WZW model in the top-right corner and then studying the
image of this deformation under the sequence of maps relating this theory to
$2d$ JT gravity and $2d$ BF theory.
We emphasize that this deformation is not the same as directly applying the
$T\overline{T}$ deformation in the JT gravity or BF theory itself. Indeed, in
the JT case, it is unclear how to define a local stress tensor in a theory of
gravity, and in the BF case, the theory is topological, so the stress tensor
vanishes. We also note that, although we consider $T\overline{T}$-like
deformations of two-dimensional $\mathrm{AdS}$ gravity theories, our procedure
is quite different from defining the $T\overline{T}$ deformation for a $2d$
field theory on a fixed $\mathrm{AdS}_{2}$ geometry. The latter problem has
been considered in [155, 156]. Likewise, although the deformation of BF gauge
theory treated in this manuscript is not the same as performing a
$T\overline{T}$ deformation of a $2d$ gauge theory directly, such direct
deformations of gauge theories have been considered for $2d$ Yang-Mills both
with and without matter [157, 158, 159, 160]. Instead, we study a deformation
holographically dual to a $T\overline{T}$-like deformation of the boundary
$(0+1)$-dimensional theory rather than a $T\overline{T}$-deformation of the
$2d$ gauge theory itself.
The layout of this chapter is as follows. Sections 2 and 3 are relevant
reviews for this chapter of standard results about $3d$ gravitational Chern-
Simons theory and $2d$ JT gravity, respectively, including the interpretation
of a boundary $T\overline{T}$ deformation in both theories. In section 4, we
find the change in BF theory boundary conditions corresponding to a
$T\overline{T}$-like deformation of the dual $1d$ theory for two different
choices of boundary conditions in the seed theory. In section 5, we first
study the deformed BF theory’s boundary spectrum to find that the contribution
from the principal series dominates the discrete series only below the
Hagedorn temperature. The $T\overline{T}$-deformed Schwarzian theory
description of the boundary spectrum is only valid below the Hagedorn
temperature. We conclude the section by computing deformed Wilson lines and
their correlators in the BF theory below the Hagedorn temperature. In section
6, we conclude with a summary and discussions on possible extensions of the
results presented in this chapter.
### 2 $T\overline{T}$ deformations in $3d$ Chern-Simons theory
In this section, we review the presentation of the Chern-Simons formulation of
$\mathrm{AdS}_{3}$ gravity, which will be relevant for later sections. In
particular, we recall that the bulk interpretation of a $T\overline{T}$
deformation in the boundary CFT is a change in the boundary conditions for the
Chern-Simons gauge field [110].
#### 1 Revisiting $T\overline{T}$-deformed $3d$ SL$(2,\mathbb{R})$
gravitational Chern-Simons
The most general asymptotically $\mathrm{AdS}_{3}$ metric is described by a
Fefferman-Graham expansion of the metric. It was shown in [110] that, in
Chern-Simons variables, such an expansion corresponds to a solution where the
connections $a$ and $\bar{a}$ take the more general form
$\displaystyle a_{i}$
$\displaystyle=2e^{+}_{i}L_{+}-f^{-}_{i}L_{-}+\omega_{i}L_{0}\,,$ (1)
$\displaystyle\bar{a}_{i}$
$\displaystyle=f^{+}_{i}L_{+}-2e^{-}_{i}L_{-}+\omega_{i}L_{0}\,.$
The connections (1) are solutions to the equations of motion when
$\displaystyle da+a\wedge a=0\,,$ (2) $\displaystyle
d\bar{a}+\bar{a}\wedge\bar{a}=0\,.$
Substituting (1) into (LABEL:eq:boundaryEOMfora), we find
$\displaystyle d\omega-2\varepsilon_{ab}e^{a}\wedge f^{b}$
$\displaystyle=0\,,$ (3) $\displaystyle
de^{a}-\varepsilon^{a}{}_{b}e^{b}\wedge\omega$ $\displaystyle=0\,,$
$\displaystyle df^{a}-\varepsilon^{a}{}_{b}f^{b}\wedge\omega$
$\displaystyle=0\,,$ $\displaystyle e^{a}\wedge f_{a}$ $\displaystyle=0\,,$
which are the zero torsion conditions for the frame $e^{a}$ with spin
connection $\omega$. Since by our conventions early Latin indices $a,b$ are
flat while middle Latin indices $i,j$ are curved, $\varepsilon^{ab}$ is the
Levi-Civita symbol with constant entries
$\varepsilon_{+-}=-\varepsilon_{-+}=1$, while $\varepsilon^{ij}$ is the Levi-
Civita tensor with curved indices
$\varepsilon^{x^{+}x^{-}}=-\varepsilon^{x^{-}x^{+}}=\frac{1}{2e}$.
In the presence of a boundary, additional boundary terms are needed in the
action to have a well-defined variational principle. Varying the Einstein-
Hilbert action written in terms of the Chern-Simons connections gives
$\displaystyle\delta S_{\operatorname{EH}}$ $\displaystyle=\delta
S_{\operatorname{CS}}[A]-\delta S_{\operatorname{CS}}[\bar{A}]$ (4)
$\displaystyle=\frac{1}{8\pi
G}\int_{M_{3}}\operatorname{Tr}\left(F\wedge\delta
A-\bar{F}\wedge\delta\bar{A}\right)-\frac{1}{16\pi G}\int_{\partial
M_{3}}\operatorname{Tr}\left(A\wedge\delta
A-\bar{A}\wedge\delta\bar{A}\right)\,.$
We desire a variational principle with Dirichlet boundary conditions for the
metric, which corresponds to holding $e^{a}$ fixed at the boundary but letting
$f^{a}$ vary. However, going on-shell by using the connections in (1), we find
that (4) reduces to
$\delta S_{\operatorname{CS}}[A]-\delta
S_{\operatorname{CS}}[\bar{A}]=-\frac{1}{8\pi G}\int_{\partial
M_{3}}\varepsilon_{ab}\left(e^{a}\wedge\delta f^{b}-f^{a}\wedge\delta
e^{b}\right)\,,$ (5)
which does not vanish and is inconsistent with the specified boundary
conditions. We must add the following boundary term to (4):
$S_{\text{bdry}}=-\frac{1}{8\pi G}\int_{\partial
M_{3}}\varepsilon_{ab}\left(A^{a}\wedge
A^{b}+\bar{A}^{a}\wedge\bar{A}^{b}\right)\,.$ (6)
The result now is consistent with Dirichlet boundary conditions since
$\delta S_{\text{EH}}+\delta S_{\text{bdry}}=\frac{1}{4\pi G}\int_{\partial
M_{3}}\varepsilon_{ab}f^{a}\wedge\delta e^{b}\,.$ (7)
From the GKPW dictionary [28, 30], it is understood that $e^{a}$ is the source
and $f^{a}$ is the expectation value of the dual operator. In particular, the
operator dual to the boundary vielbein is the stress tensor. By identifying
$\displaystyle\delta S=4\int_{\partial M_{3}}\,d^{2}x\,\left(\det
e\right)\,T^{i}{}_{a}\,\delta e_{i}^{a}\,,$ (8)
we find that
$\displaystyle T^{i}{}_{a}=\frac{1}{4\pi
G}\varepsilon_{ab}\varepsilon^{ij}f^{b}_{j}\,,$ (9)
with $\nabla_{[i}f^{a}_{j]}=0$.
When we turn to the two-dimensional BF theory in section 4, it will be
convenient to refer to the dimensional reduction of the $3d$ Chern-Simons
action on a circle. The resulting dimensionally reduced theory is equivalent
to BF theory with a particular choice of boundary term. To perform this
reduction, we first write
$\displaystyle 8\pi GS_{\text{CS}}$
$\displaystyle=\frac{1}{2}\int_{M_{3}}\operatorname{Tr}\left(A\wedge
dA+\frac{2}{3}A\wedge A\wedge A\right)$ (10)
$\displaystyle=\frac{1}{2}\int_{M_{3}}\,d^{3}x\,\varepsilon^{\mu\nu\rho}\operatorname{Tr}\left(A_{\mu}\partial_{\nu}A_{\rho}+\frac{2}{3}A_{\mu}A_{\nu}A_{\rho}\right)$
$\displaystyle=\int_{M_{3}}\operatorname{Tr}\left(A_{\varphi}F_{tr}+A_{r}\partial_{\varphi}A_{t}\right)+\frac{1}{2}\oint_{\partial
M_{3}}\operatorname{Tr}A_{t}^{2}\,.$
Next we impose the boundary condition $A_{t}=A_{\varphi}|_{\partial M_{3}}$ so
that $\phi\equiv A_{\varphi}$ and $\partial_{\varphi}=0$ (see [161]). Doing
this yields
$S_{\text{BF}}=\int_{M_{2}}\operatorname{Tr}\left(\phi
F\right)+\frac{1}{2}\oint_{\partial
M_{2}}\operatorname{Tr}\left(\phi^{2}\right)\,.$ (11)
The first term is the usual action for $2d$ BF theory, which in this case has
gauge group $G=\operatorname{SL}(2,\mathbb{R})$. The degrees of freedom in
this theory are a gauge field $A_{\mu}$ with field strength $F_{\mu\nu}$ along
with an $\operatorname{SL}(2,\mathbb{R})$-valued scalar field $\phi$. We will
again consider this action in section 1, where we will recall that the theory
is equivalent to JT gravity. The second term of (11) controls the dynamics of
a boundary degree of freedom, which can be described via the Schwarzian theory
or the particle-on-a-group theory. We refer to this as a “Schwarzian-type”
boundary term, which will be revisited in section 2.
#### 2 Interpretation of the $T\overline{T}$ deformation
The $3d$ gravitational Chern-Simons theory is dual to a conformal field theory
on the $2d$ boundary of the spacetime via the usual
$\mathrm{AdS}/\mathrm{CFT}$ correspondence. On the other hand, in any two-
dimensional field theory enjoying translation invariance, one can define a
deformation by the double-trace $T\overline{T}$ operator. Our goal in the
present section is to apply this deformation to the boundary CFT and interpret
the resulting flow in terms of bulk Chern-Simons variables. We follow the
discussion of [110] where this analysis first appeared.
We must first express the $T\overline{T}$ deformation in terms of the
asymptotic expansion coefficients for the Chern-Simons gauge fields. We have
already seen, for instance, in (7) and (9), that the functions $e_{i}^{a}$
correspond to the boundary vielbein (or equivalently the metric) and that the
$f_{i}^{a}$ are the dual expectation values which encode the stress tensor as
$\displaystyle T^{i}{}_{a}=\frac{1}{4\pi
G}\varepsilon_{ab}\varepsilon^{ij}f_{j}^{b}\,.$ (12)
On the other hand, using the definition of the determinant in terms of the
Levi-Civita symbol, the $T\overline{T}$ operator can be written as
$T\overline{T}=-2\varepsilon^{ab}\varepsilon_{ij}T^{i}{}_{a}T^{j}{}_{b}\,.$
(13)
In terms of the one-forms $f^{-}$ and $f^{+}$, one therefore has
$\displaystyle T\overline{T}=\frac{1}{(4\pi G)^{2}}f^{-}\wedge f^{+}\,.$ (14)
The flow equation for the boundary action can be written as
$\displaystyle\frac{\partial S}{\partial\lambda}=\frac{1}{(4\pi
G)^{2}}\int_{\partial M_{3}}f^{-}\wedge f^{+}\,.$ (15)
We note that this is a flow equation for the _combined_ boundary action, which
in the undeformed case is a sum of three terms:
$\displaystyle
S(\lambda=0)=S_{\text{CS}}[A]-S_{\text{CS}}[\bar{A}]+S_{\text{bdry}}\,.$ (16)
In section 1, we saw that variation of the first two terms
$S_{\text{CS}}[A]-S_{\text{CS}}[\bar{A}]$ generated a boundary variation of
the form $\varepsilon_{ab}(e^{a}\wedge\delta f^{b}-f^{a}\wedge\delta e^{b})$.
The first term involving $\delta f^{b}$ was unsuitable for our desired
variational principle, so we added $S_{\text{bdry}}$ to cancel this variation.
We will make the ansatz that the finite-$\lambda$ deformed boundary action has
the same structure as a sum of three terms involving sources
$e_{i}^{a}(\lambda)$ and dual expectation values $f_{i}^{a}$. In this ansatz
we allow the sources to acquire $\lambda$ dependence under the flow, but not
the expectation values. As a result, the total boundary variation (7) of our
$\lambda$-dependent ansatz takes the form
$\displaystyle\delta S=\frac{1}{4\pi G}\int_{\partial
M_{3}}\varepsilon_{ab}f^{a}\wedge\delta e^{b}(\lambda)\,.$ (17)
We now substitute this ansatz into the flow equation (15). More precisely, if
the boundary action $S$ satisfies (15), then its variation satisfies
$\displaystyle\frac{\partial(\delta S)}{\partial\lambda}=\frac{1}{(4\pi
G)^{2}}\int_{\partial M_{3}}\delta(f^{-}\wedge f^{+})\,.$ (18)
This then implies
$\displaystyle\int_{\partial
M_{3}}\varepsilon_{ab}f^{a}\wedge\delta\left(\frac{\partial
e^{b}(\lambda)}{\partial\lambda}\right)=\frac{1}{4\pi G}\int_{\partial
M_{3}}\varepsilon_{ab}f^{a}\wedge\delta f^{b}\,.$ (19)
We see that (19) will be satisfied if
$\displaystyle\frac{\partial e^{b}(\lambda)}{\partial\lambda}=\frac{1}{4\pi
G}f^{b}\,.$ (20)
Since $f^{b}$ is independent of $\lambda$ by assumption, this equation can be
trivially integrated to find
$\displaystyle e_{i}^{a}(\lambda)=e_{i}^{a}(0)+\frac{\lambda}{4\pi
G}f_{i}^{a}\,,$ (21)
and $f_{i}^{a}(\lambda)=f_{i}^{a}(0)$. One can show that if the spin
connection $\omega$ vanishes in the seed theory (as we will typically assume),
then $\omega(\lambda)=0$ along the flow. We have characterized the full
solution to the flow equation.
_Remarks on deformed boundary conditions_
We now pause to make several comments on this interpretation. We see that the
effect of a boundary $T\overline{T}$ deformation is to rotate our undeformed
source $e_{i}^{a}$ into a new source $e_{i}^{a}(\lambda)$, which depends
linearly on the corresponding undeformed expectation value. Since $e_{i}^{a}$
determines the boundary metric, this means that the deformed theory sees an
effective stress-tensor-dependent metric. This is reminiscent of the result of
$T\overline{T}$-deforming a two-dimensional field theory defined on a cylinder
of radius $R$. As we will review around (183), in the zero-momentum sector,
this deformation has the interpretation of placing the theory on a cylinder
with an effective energy-dependent radius $\widetilde{R}(R,E_{n})$.
Next, we note that although the sources $e_{i}^{a}$ have been modified. The
variational principle defining our theory has not changed when expressed in
terms of the new sources. The deformed boundary variation solving our flow is
written as (17), which vanishes if the sources $e_{i}^{a}(\lambda)$ are held
fixed. Therefore, the theory described by these $T\overline{T}$-deformed
boundary conditions still corresponds to a variational principle where the
metric is held fixed at the boundary, but the dual expectation value is free
to fluctuate. All that has changed is the expression for this fixed metric in
terms of the undeformed metric and stress tensor.
A third remark concerns the trace flow equation for the $T\overline{T}$
deformation. Because there is no dimensionful scale in a CFT, if one solves a
$T\overline{T}$ flow beginning from a CFT seed then the resulting theory has a
single effective energy scale $\Lambda=\frac{1}{\sqrt{\lambda}}$ set by the
length dimension $2$ parameter $\lambda$. By noting that the derivative of the
action with respect to this single scale $\lambda$ is controlled by the trace
of the stress tensor,
$\displaystyle\Lambda\frac{d}{d\Lambda}S=\int d^{2}x\,T^{\mu}{}_{\mu}\,.$ (22)
while on the other hand, the derivative of the action is related to the
$T\overline{T}$ operator by the definition of the flow (16)
$\displaystyle\Lambda\frac{d}{d\Lambda}S$
$\displaystyle=\frac{1}{\sqrt{\lambda}}\frac{d}{d\left(\frac{1}{\sqrt{\lambda}}\right)}S$
$\displaystyle=-2\lambda\int
d^{2}x\,\left(T^{\mu\nu}T_{\mu\nu}-\left(T^{\mu}{}_{\mu}\right)^{2}\right)\,,$
(23)
one finds the relation
$\displaystyle T^{\mu}{}_{\mu}(\lambda)=-2\lambda T\overline{T}(\lambda).$
(24)
Since the modified boundary conditions (21) correspond to a
$T\overline{T}$-deformation of a CFT, it is an instructive check to verify
explicitly that the trace flow equation (24) holds. Indeed, the trace of the
deformed stress tensor with respect to the deformed metric is
$\displaystyle T^{i}{}_{i}$ $\displaystyle=\eta^{ij}e_{i}^{a}T_{ja}$
$\displaystyle=\frac{1}{4\pi G}\left(e_{i}^{a}(0)+\frac{\lambda}{4\pi
G}f_{i}^{a}\right)\left(\varepsilon_{ab}\varepsilon^{ij}f_{j}^{b}\right)$
$\displaystyle=\frac{\lambda}{(4\pi
G)^{2}}\varepsilon_{ab}\varepsilon^{jk}f_{i}^{a}f_{j}^{b}\,,$ (25)
where in the last step, we have used that the undeformed stress tensor is
traceless by assumption. On the other hand, at finite $\lambda$ the
combination $T\overline{T}$ is given by (13):
$\displaystyle T\overline{T}$
$\displaystyle=-2\varepsilon^{ab}\varepsilon_{ij}T^{i}{}_{a}T^{j}{}_{b}$
$\displaystyle=-\frac{2}{(4\pi
G)^{2}}\varepsilon^{ab}\varepsilon_{ij}\left(\varepsilon_{ac}\varepsilon^{ik}f_{k}^{c}\right)\left(\varepsilon_{bd}\varepsilon^{jn}f_{n}^{d}\right)$
$\displaystyle=-\frac{2}{(4\pi
G)^{2}}\varepsilon_{ab}\varepsilon^{jk}f_{i}^{a}f^{b}_{j}\,,$ (26)
where we have repeatedly used the $2d$ contracted epsilon identity
$\varepsilon^{in}\varepsilon_{ij}=\delta^{n}{}_{j}$. Comparing (2) to (2), we
see that the trace flow equation $T^{\mu}{}_{\mu}(\lambda)=-2\lambda
T\overline{T}(\lambda)$ holds as expected.
We make a fourth and final comment, which is a trivial observation in this
case but could conceivably be relevant for generalizations of the procedure
described here. We emphasized around (16) that the undeformed action
$S(\lambda=0)=S_{\text{EH}}+S_{\text{bdry}}$ includes a boundary term which
was added by hand to give a particular variational principle. Since the
process of deforming the action by $T\overline{T}$ and the process of adding
the boundary term $S_{\text{bdry}}$ are two distinct steps, there are naïvely
two ways to proceed:
1. 1.
First add the boundary term $S_{\text{bdry}}$ to get the total boundary action
$S$. Then solve the flow equation (15) for this combined action.
2. 2.
First solve the flow equation $\frac{\partial
S_{\text{EH}}}{\partial\lambda}\Big{|}_{\text{bdry}}=\frac{1}{(4\pi
G)^{2}}\int_{\partial M_{3}}f^{-}\wedge f^{+}$ which only deforms the first
contribution to the action. Solve this by identifying new sources
$e_{i}^{a}(\lambda)$. After doing this, add a new boundary term
$S_{\text{bdry}}(\lambda)$ by hand to restore the desired variational
principle.
In the discussion above, we performed the deformation described by 1. However,
it is straightforward to see that procedure 2 gives the same result precisely
because the dual expectation values $f_{i}^{a}$ do not flow according to our
ansatz. To show this, we recall from (5) that
$\displaystyle\delta S_{\text{EH}}\Big{|}_{\text{on-shell}}=-\frac{1}{8\pi
G}\int_{\partial M_{3}}\varepsilon_{ab}\left(e^{a}\wedge\delta
f^{b}-f^{a}\wedge\delta e^{b}\right)\,.$ (27)
Suppose that we had allowed both $e_{i}^{a}$ and $f_{i}^{a}$ to acquire
$\lambda$ dependence along our flow. Then the derivative of this boundary
variation would be
$\displaystyle\frac{\partial(\delta
S_{\text{EH}})}{\partial\lambda}=-\frac{1}{8\pi G}\int_{\partial
M_{3}}\varepsilon_{ab}$ $\displaystyle\Bigg{(}\frac{\partial
e^{a}(\lambda)}{\partial\lambda}\wedge\delta
f^{b}(\lambda)+e^{a}(\lambda)\wedge\frac{\partial(\delta
f^{b}(\lambda))}{\partial\lambda}$ (28) $\displaystyle-\frac{\partial
f^{a}(\lambda)}{\partial\lambda}\wedge\delta
e^{b}(\lambda)-f^{a}(\lambda)\wedge\frac{\partial(\delta
e^{b}(\lambda))}{\partial\lambda}\Bigg{)}\,.$
In order to satisfy the flow equation $\frac{\partial(\delta
S_{\text{EH}})}{\partial\lambda}\Big{|}_{\text{on-shell}}=\frac{1}{(4\pi
G)^{2}}\int_{\partial M_{3}}\delta(f^{-}\wedge f^{+})$, whose right side is
again
$\displaystyle\frac{1}{(4\pi G)^{2}}\int_{\partial M_{3}}\delta(f^{-}\wedge
f^{+})=\frac{1}{(4\pi G)^{2}}\int_{\partial
M_{3}}\varepsilon_{ab}f^{a}\wedge\delta f^{b}\,,$ (29)
we must have
$\displaystyle\frac{\partial e^{a}(\lambda)}{\partial\lambda}\wedge\delta
f^{b}(\lambda)+e^{a}(\lambda)\wedge\frac{\partial(\delta
f^{b}(\lambda))}{\partial\lambda}-\frac{\partial
f^{a}(\lambda)}{\partial\lambda}\wedge\delta
e^{b}(\lambda)-f^{a}(\lambda)\wedge\frac{\partial(\delta
e^{b}(\lambda))}{\partial\lambda}$ (30) $\displaystyle=-\frac{1}{2\pi
G}f^{a}\wedge\delta f^{b}\,.$
The left side involves both $\delta e^{a}$ and $\delta f^{a}$, whereas the
right side only involves $\delta f^{a}$. If these two variations are both
independent, nonzero, and $\lambda$-dependent, it seems that we cannot have a
solution. However, if we assume that $f^{a}$ and therefore $\delta f^{a}$ are
independent of $\lambda$ as we did before, in addition to imposing that
$\delta e^{a}(\lambda)=0$ according to our choice of deformed variational
principle, the equation (LABEL:both_lambda_dep_equation) reduces to
$\displaystyle\frac{\partial e^{a}(\lambda)}{\partial\lambda}\wedge\delta
f^{b}(\lambda)=-\frac{1}{2\pi G}f^{a}\wedge\delta f^{b}\,.$ (31)
The solution to this simple flow is
$\displaystyle e^{a}_{i}(\lambda)=e_{i}^{a}(0)-\frac{\lambda}{2\pi
G}f_{i}^{a}\,.$ (32)
Up to an overall rescaling of $\lambda$ by a factor of $-\frac{1}{2}$, this is
the same solution as (21). This completes the first step of the alternate
deformation procedure described in 2, but we must still add a new boundary
term so that the combined boundary action is consistent with the variational
principle $\delta e^{a}=0$ that we have assumed. Our $\lambda$-dependent
deformed boundary variation before adding this boundary term is
$\displaystyle\delta S_{\text{EH}}(\lambda)\Big{|}_{\text{on-
shell}}=-\frac{1}{8\pi G}\int_{\partial
M_{3}}\varepsilon_{ab}\left(e^{a}(\lambda)\wedge\delta f^{b}-f^{a}\wedge\delta
e^{b}(\lambda)\right)\,.$ (33)
But because this has the same form as the variation (5) which we saw in the
undeformed case, we may repeat the same procedure and add the term
$S_{\text{bdry}}$ as defined in (6), except replacing $e_{i}^{a}$ with
$e_{i}^{a}(\lambda)$ everywhere that it appears in the expansions of $A^{a}$
and $A^{b}$. The result is, again,
$\delta S_{\text{EH}}(\lambda)\Big{|}_{\text{on-shell}}+\delta
S_{\text{bdry}}(\lambda)=\frac{1}{4\pi G}\int_{\partial
M_{3}}\varepsilon_{ab}f^{a}\wedge\delta e^{b}(\lambda)\,,$ (34)
exactly as we found before.
The upshot of this simple calculation is that the two processes described
above – first adding a boundary term and then deforming, or first deforming
and then adding a boundary term – commute in the calculation we consider here.
However, in another setting where both the sources and expectation values
become $\lambda$-dependent, performing the second deformation procedure 2
would produce a flow equation analogous to (LABEL:both_lambda_dep_equation),
which is not equivalent to the flow of procedure 1. In such cases, one must
choose a prescription to define the deformation.
### 3 $T\overline{T}$ deformations in $2d$ JT gravity
We now review features of JT gravity and its BF gauge theory description,
which are relevant to later sections when we study the $T\overline{T}$
deformation in BF theory. As in section 2, none of the material in this
discussion is new. For instance, the interpretation of a $T\overline{T}$-like
deformation in the boundary dual to $2d$ JT gravity was considered in [115,
116], and we follow their discussion closely in section 2. We include a
reminder of these results here to facilitate comparison with the new results
of section 4, where we present an analogous interpretation of the
$T\overline{T}$ deformation in BF variables.
#### 1 JT gravity as a BF gauge theory
In the introduction of this chapter, we mentioned that one salient feature of
$3d$ gravity motivating the present work is that it can be dimensionally
reduced on a circle to yield JT gravity as described by the action (5). This
subsection’s goal is to recall the standard statement that this $2d$ dilaton
gravity theory can be, equivalently, written in gauge theory variables as a BF
theory. Our treatment will follow [72].
One way of motivating this reformulation is to note that $3d$ gravity is
equivalent to a Chern-Simons theory, as we reviewed in section 2, and that the
dimensional reduction of this $3d$ Chern-Simons theory is a BF gauge theory.
Indeed, we saw this reduction explicitly around (11). These observations are
summarized by the sub-diagram formed by the second and third columns of (6):
(35)
We have reviewed all of the arrows in (35) except for the change of variables
linking the two theories in the bottom row. Although such a change of
variables must exist by the consistency of the diagram, it is instructive to
spell out the map explicitly.
Recall that the BF theory in Euclidean signature is described by the action
$I_{\text{BF}}=-i\int_{M_{2}}\operatorname{Tr}\left(\phi F\right)\,,$ (36)
where $\phi$ is a scalar field and $F$ is the field strength of the gauge
field $A_{\mu}$. At the moment, we will only be concerned with the bulk
equations of motion and will not include any additional boundary term like the
one that appeared in (11). The equations of motion arising from (36) are
$\displaystyle\phi$ $\displaystyle:\quad F=0\,,$ (37) $\displaystyle A_{\mu}$
$\displaystyle:\quad D_{\mu}\phi=\partial_{\mu}\phi-[A_{\mu},\phi]=0\,.$
On the other hand, beginning from the action (5) of JT gravity and setting
$\Lambda=-1$, one finds the equations of motion
$\displaystyle\Phi$ $\displaystyle:\quad R=-2\,,$ (38) $\displaystyle
g_{\mu\nu}$
$\displaystyle:\quad\nabla_{\mu}\nabla_{\nu}\Phi=g_{\mu\nu}\Phi\,.$
Next, we will argue that the JT equations of motion in (38) are equivalent to
the BF equations of motion in (37). To accomplish this, we first expand the BF
fields in terms of generators:
$\displaystyle
A(x)=e^{2}(x)P_{2}+e^{1}(x)P_{1}+\omega(x)P_{0}\,,\quad\phi(x)=\phi^{1}(x)P_{1}+\phi^{2}(x)P_{2}+\phi^{0}(x)P_{0}\,,$
(39)
where
$\displaystyle P_{0}=\left(\begin{array}[]{cc}0&\frac{1}{2}\\\
-\frac{1}{2}&0\end{array}\right)\,,\quad
P_{1}=\left(\begin{array}[]{cc}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{array}\right)\,,\quad
P_{2}=\left(\begin{array}[]{cc}\frac{1}{2}&0\\\
0&-\frac{1}{2}\end{array}\right)\,.$ (40)
Written in differential form notation, the equation of motion for $A_{\mu}$ in
(37) becomes $d\phi-A\wedge\phi=0$. The exterior derivative of the scalar
$\phi$ is
$d\phi=d\phi^{0}(x)P_{0}+d\phi^{1}(x)P_{1}+d\phi^{2}(x)P_{2}\,.$ (41)
Meanwhile, a short calculation gives
$\displaystyle
A\wedge\phi=\left(e^{2}\wedge\phi^{1}-e^{1}\wedge\phi^{2}\right)P_{0}+\left(e^{2}\wedge\phi^{0}-\omega\wedge\phi^{2}\right)P_{1}+\left(\omega\wedge\phi^{1}-e^{1}\wedge\phi_{0}\right)P_{2}\,.$
(42)
Putting everything together, we find
$\displaystyle d\phi^{0}(x)$
$\displaystyle=e^{2}(x)\wedge\phi^{1}(x)-e^{1}(x)\wedge\phi^{2}(x)\,,$ (43)
$\displaystyle d\phi^{1}(x)$
$\displaystyle=e^{2}(x)\wedge\phi^{0}(x)-\omega(x)\wedge\phi^{2}(x)\,,$
$\displaystyle d\phi^{2}(x)$
$\displaystyle=\omega(x)\wedge\phi^{1}(x)-e^{1}(x)\wedge\phi^{0}(x)\,.$
We now act with the covariant derivative $\nabla_{\mu}$ on the equation for
$d\phi^{0}$ in (43). At the risk of being pedantic, we pause to clarify one
point of possible confusion. When acting on a generalized tensor with both
curved (spacetime) indices and flat (tangent space) indices, the action of the
covariant derivative $\nabla_{\mu}$ involves Christoffel symbol terms
associated with the curved indices and spin connection terms associated with
the flat indices. For instance, on the vielbein $e_{\nu}^{a}$ with one curved
and one flat index, one has
$\displaystyle\nabla_{\mu}e_{\nu}^{a}=\partial_{\mu}e_{\nu}^{a}+\omega_{\mu}{}^{a}{}_{b}e_{\nu}^{b}-\Gamma^{\sigma}{}_{\nu\mu}e_{\sigma}^{a}\,.$
(44)
Since the covariant derivative annihilates the vielbein by the zero-torsion
constraint $\tau^{a}=de^{a}+\omega^{a}_{b}\wedge e^{b}=0$, the combination
(44) vanishes.
However, the equations (43) are covariant with respect to their single curved
index but not with respect to the implicit flat index on the vielbeins. It is
easiest to see this by writing the equations in components. For instance, the
$\phi^{0}$ equation is
$\displaystyle\partial_{\mu}\phi^{0}=\phi^{1}e^{2}_{\mu}-\phi^{2}e^{1}_{\mu}\,.$
(45)
Although this equation has a free $\mu$ index, there is no free $a$ index in
the $e_{\mu}^{a}$ factors. Indeed, this equation could never have been
covariant with respect to such a tangent space index since the quantity
$\partial_{\mu}\phi^{0}$ on the left has no flat indices. Therefore, when we
act with the covariant derivative, there will be no spin connection terms
introduced in the derivatives of vielbein factors. One has
$\displaystyle\nabla_{\mu}e^{2}_{\nu}=\partial_{\mu}e_{\nu}^{2}-\Gamma^{\sigma}{}_{\nu\mu}e_{\sigma}^{2}\,,$
(46)
and likewise for $\nabla_{\nu}e^{1}_{\mu}$. However
$\nabla_{\mu}e_{\nu}^{a}=0$, (44) implies
$\displaystyle\partial_{\mu}e_{\nu}^{2}-\Gamma^{\sigma}{}_{\nu\mu}e_{\sigma}^{2}=-\omega_{\mu}{}^{2}{}_{b}e_{\nu}^{b}\,,$
(47)
and again a similar equation for $\nabla_{\nu}e^{1}_{\mu}$. Using
$\omega^{1}{}_{2}=-\omega^{2}{}_{1}=\omega$, we find
$\displaystyle\nabla_{\mu}e_{\nu}^{1}$
$\displaystyle=-\omega_{\mu}{}^{1}{}_{b}e_{\nu}^{b}=-\omega_{\mu}e_{\nu}^{2}\,,$
$\displaystyle\nabla_{\mu}e_{\nu}^{2}$
$\displaystyle=-\omega_{\mu}{}^{2}{}_{b}e_{\nu}^{b}=\omega_{\mu}e^{1}_{\nu}\,.$
(48)
Now we are prepared to act with the covariant derivative on the $\phi^{0}$
equation of motion. On the left, the result is a two-tensor with components
$\nabla_{\mu}\nabla_{\nu}\phi^{0}$. One finds
$\displaystyle\nabla_{\mu}\nabla_{\nu}\phi^{0}$
$\displaystyle=\left(\partial_{\mu}\phi^{1}\right)e^{2}_{\nu}-\left(\partial_{\mu}\phi^{2}\right)e^{1}_{\nu}+\phi^{1}\left(\nabla_{\mu}e^{2}_{\nu}\right)-\phi^{2}\left(\nabla_{\mu}e^{1}_{\nu}\right)\,$
$\displaystyle=\left(\partial_{\mu}\phi^{1}\right)e^{2}_{\nu}-\left(\partial_{\mu}\phi^{2}\right)e^{1}_{\nu}+\phi^{1}\omega_{\mu}e^{1}_{\nu}+\phi^{2}\omega_{\mu}e_{\nu}^{2}$
(49)
On the other hand, writing the second and third equations of (43) in
components gives
$\partial_{\mu}\phi^{1}=\phi^{0}e^{2}_{\mu}-\phi^{2}\omega_{\mu}$ and
$\partial_{\mu}\phi^{2}=\phi^{1}\omega_{\mu}-\phi^{0}e^{1}_{\mu}$.
Substituting these into (1) gives
$\displaystyle\nabla_{\mu}\nabla_{\nu}\phi^{0}$
$\displaystyle=\left(\phi^{0}e^{2}_{\mu}-\phi^{2}\omega_{\mu}\right)e^{2}_{\nu}-\left(\phi^{1}\omega_{\mu}-\phi^{0}e^{1}_{\mu}\right)e^{1}_{\nu}+\phi^{1}\omega_{\mu}e^{1}_{\nu}+\phi^{2}\omega_{\mu}e_{\nu}^{2}$
$\displaystyle=\left(e^{1}_{\mu}e^{1}_{\nu}+e^{2}_{\mu}e^{2}_{\nu}\right)\phi^{0}\,.$
(50)
If we identify the metric as
$g_{\mu\nu}=e^{1}_{\mu}e^{1}_{\nu}+e^{2}_{\mu}e^{2}_{\nu}$ and assume that the
JT dilaton $\Phi$ is proportional to the BF field $\phi^{0}$, then this is the
metric equation of motion in (38):
$\displaystyle\nabla_{\mu}\nabla_{\nu}\Phi=g_{\mu\nu}\Phi\,.$ (51)
We, therefore, have demonstrated that the JT gravity equations of motion (38)
are recovered from the BF equations of motion in (37) after making the change
of variables
$\displaystyle\phi^{0}=\frac{i}{4\pi G}\Phi\,,\qquad
g_{\mu\nu}=e^{1}_{\mu}e^{1}_{\nu}+e^{2}_{\mu}e^{2}_{\nu}\,.$ (52)
Here, the choice of the proportionality factor $\frac{i}{4\pi G}$ between
$\phi^{0}$ and $\Phi$ is required by our normalizations for the BF and JT
actions in (36) and (5), respectively. Under this identification, we see that
the expansion coefficients $e^{a}$ appearing in the BF gauge field $A_{\mu}$
are interpreted as the frame fields in the JT gravity theory, whereas the
field $\omega$ defines the spin connection, which satisfies
$d\omega=\frac{R}{2}e^{1}\wedge e^{2}$ for a $2d$ manifold. In this
correspondence, the $\phi^{1},\phi^{2}$ equations of motion are mapped onto
the torsionless conditions $\tau^{a}=de^{a}+\omega^{a}_{b}\wedge e^{b}=0$.
This completes our review of the final arrow on the bottom row of (35) linking
JT gravity with BF gauge theory.
Next, we explain the boundary conditions and the choice of boundary term for
the BF gauge theory, which recovers the Schwarzian action. Variation of the BF
action on-shell yields the boundary action
$\delta I_{\text{BF}}=-i\int_{\partial
M_{2}}\,d\tau\,\operatorname{Tr}\left(\phi\,\delta A_{\tau}\right)\,,$ (53)
with $\tau$ parametrizing the one-dimensional boundary $\partial M_{2}$.
Thus, the variation (53) of the BF action vanishes if $A_{\tau}$ is held fixed
on $\partial M_{2}$. In fact, from JT gravity’s first-order formulation, the
spin connection and frame are already fixed, so no boundary term is required
to have a well-defined variational principle. Unfortunately, this means the BF
theory cannot be holographically dual to the Schwarzian because the theory is
topologically trivial. In particular, the observables of the theory would
depend on the holonomy around the boundary rather than depending on the local
value of $A_{\tau}$. To recover the Schwarzian dynamics, one includes a string
defect $I_{\text{string}}$ around a loop $L\subset M_{2}$, which yields the
modified action
$I=-i\int_{M_{2}}\operatorname{Tr}\left(\phi
F\right)-\oint^{\beta}_{0}du\,V(\phi)\,,\quad
V(\phi)=\frac{\nu}{4}\operatorname{Tr}\phi^{2}\,.$ (54)
The second term in (54) is the string defect with coupling $\nu$, and $u$ is
the proper length parametrization of the loop with circumference $\beta$. This
form of $V(\phi)$ is consistent with the boundary term in (11), which we
expect from the dimensional reduction of Chern-Simons and, as we will see,
correctly recovers the Schwarzian action.
The overall action (54) preserves the defect diffeomorphisms and the degrees
of freedom from the string defect are realized by the Schwarzian theory as
[72] showed by evaluating the action (54) using the solution to the equation
of motion (37) for $\phi(u)$ along $L$. To see the derivation more explicitly,
we parametrize the boundary fields $\phi$ and $A_{\tau}$ by111Note that a
different representation of the generators when solving the equations of
motion for the field $\phi$ at the loop.
$A_{\tau}=\omega\ell_{0}+e_{+}\ell_{+}+e_{-}\ell_{-}\,,\quad\phi=\phi_{+}\ell_{+}+\phi_{-}\ell_{-}+\phi_{0}\ell_{0}\,,$
(55)
where
$\displaystyle\ell_{0}=iP_{0},\quad\ell_{+}=-iP_{1}-P_{2}\,,\quad\ell_{-}=-iP_{1}+P_{2}\,,$
(56) $\displaystyle\omega=-i\omega_{\tau}\bigg{|}_{\partial M_{2}}\,,\quad
e_{+}=\frac{ie^{1}_{\tau}-e^{2}_{\tau}}{2}\bigg{|}_{\partial M_{2}}\,,\quad
e_{-}=\frac{ie^{1}_{\tau}+e^{2}_{\tau}}{2}\bigg{|}_{\partial M_{2}}\,.$
We compute the commutator
$\left[A_{\tau},\phi\right]=\left(e_{+}\phi_{0}-\omega\phi_{+}\right)\ell_{+}+\left(\omega\phi_{-}-e_{-}\phi_{0}\right)\ell_{-}+2\left(e_{+}\phi_{-}-e_{-}\phi_{+}\right)\ell_{0}$
(57)
to write the complete set of equations of motion $D_{\tau}\phi=0$ at the loop
$\displaystyle\ell_{0}$
$\displaystyle:\quad\partial_{\tau}\phi_{0}=2\left(e_{+}\phi_{-}-e_{-}\phi_{+}\right)\,,$
(58) $\displaystyle\ell_{-}$
$\displaystyle:\quad\partial_{\tau}\phi_{-}=\omega\phi_{-}-e_{-}\phi_{0}\,,$
$\displaystyle\ell_{+}$
$\displaystyle:\quad\partial_{\tau}\phi_{+}=e_{+}\phi_{0}-\omega\phi_{+}\,.$
To solve the equations at the loop (58), we perform the same change of
variables as [72]
$\phi_{-}(\tau)=\frac{2e_{-}}{\partial_{\tau}u(\tau)}\implies\phi_{-}(u)=2e_{-}\tau^{\prime}\,,$
(59)
where
$\partial_{\tau}\phi_{i}=\left(\partial_{\tau}u\right)\left(\partial_{u}\phi_{i}\right)=\frac{\partial_{u}\phi_{i}}{\tau^{\prime}}\,.$
(60)
Substituting the above into the equation of motion for the $\ell_{-}$
component, we find
$\phi_{0}=-\frac{\partial_{\tau}\phi_{-}-\omega\phi_{-}}{e_{-}}=-\frac{2\tau^{\prime\prime}}{\tau^{\prime}}+2\omega\tau^{\prime}\,.$
(61)
Then, solving for the $\ell_{0}$ component, one uses $\phi_{-}$ and $\phi_{0}$
to find
$\displaystyle\phi_{+}$
$\displaystyle=\frac{1}{e_{-}}\left(-\frac{1}{2}\partial_{\tau}\phi_{0}+e_{+}\phi_{-}\right)$
(62)
$\displaystyle=2\left(e_{+}\tau^{\prime}+\frac{\tau^{\prime\prime\prime}}{2e_{-}\tau^{\prime
2}}-\frac{\omega\tau^{\prime\prime}}{2e_{-}\tau^{\prime}}-\frac{\tau^{\prime\prime
2}}{2e_{-}\tau^{\prime 3}}\right)\,.$
We have found all the components for the field $\phi(u)$:
$\nu\phi(u)=2e_{-}\tau^{\prime}\ell_{-}+2\left(\omega\tau^{\prime}-\frac{\tau^{\prime\prime}}{\tau^{\prime}}\right)\ell_{0}+2\left(e_{+}\tau^{\prime}+\frac{\tau^{\prime\prime\prime}}{2e_{-}\tau^{\prime
2}}-\frac{\omega\tau^{\prime\prime}}{2e_{-}\tau^{\prime}}-\frac{\tau^{\prime\prime
2}}{2e_{-}\tau^{\prime 3}}\right)\ell_{+}\,.$ (63)
Here $\tau(u)$ is further constrained by the $\ell_{+}$ component of the
equation $D_{u}\phi=0$, which gives
$4\left(\operatorname{det}A_{\tau}\right)\tau^{\prime
4}\tau^{\prime\prime}+3\tau^{\prime\prime
3}-4\tau^{\prime}\tau^{\prime\prime}\tau^{\prime\prime\prime}+\tau^{\prime
2}\tau^{\prime\prime\prime\prime}=0\,,$ (64)
where $\tau(u)$ is monotonic so $\tau^{\prime}(u)\neq 0$ and $\det
A_{\tau}=e_{-}e_{+}-\frac{\omega^{2}}{4}$.
Now we are ready to evaluate the string defect action by computing
$\operatorname{Tr}\phi^{2}$. This computation is straightforward as
$\phi^{2}=\frac{1}{\nu^{2}}\left(\begin{array}[]{cc}-4e_{-}e_{+}\tau^{\prime
2}+\omega^{2}\tau^{\prime 2}+\frac{3\tau^{\prime\prime 2}}{\tau^{\prime
2}}-\frac{2\tau^{\prime\prime\prime}}{\tau^{\prime}}&0\\\
0&-4e_{-}e_{+}\tau^{\prime 2}+\omega^{2}\tau^{\prime
2}+\frac{3\tau^{\prime\prime 2}}{\tau^{\prime
2}}-\frac{2\tau^{\prime\prime\prime}}{\tau^{\prime}}\end{array}\right)$ (65)
and
$\displaystyle V(\phi)$ $\displaystyle=\frac{\nu}{4}\operatorname{Tr}\phi^{2}$
(66)
$\displaystyle=\frac{1}{\nu}\left(\\{\tau(u),u\\}+2\tau^{\prime}(u)^{2}\det
A_{\tau}\right)$
$\displaystyle=\frac{1}{\nu}\left\\{\tan\left(\sqrt{\left(\det
A_{\tau}\right)\tau(u)}\right),u\right\\}\,.$
As expected, we have recovered the Schwarzian action222One can show that this
derivation of the Schwarzian theory holds for any $\Lambda$ by using the more
general parameterization
$A_{\tau}=\omega\ell_{0}+\sqrt{\Lambda}e_{+}\ell_{+}+\sqrt{\Lambda}e_{-}\ell_{-}$.
Equivalently, this corresponds to replacing the determinant in (67) by $\det
A_{\tau}=\Lambda e_{-}e_{+}-\frac{\omega^{2}}{4}$. from including the string
defect in the BF action (54), which gives
$I=-\frac{1}{\nu}\int^{\beta}_{0}du\left\\{\tan\left(\sqrt{\left(\det
A_{\tau}\right)\tau(u)}\right)\,,u\right\\}\,.$ (67)
#### 2 Interpretation of $T\overline{T}$ deformation
Before we begin with the $T\overline{T}$ deformation in JT gravity, we first
recall how a general class of related deformations is defined and their
physical meaning in the $\mathrm{AdS}/\mathrm{CFT}$ correspondence. Following
[116], we deform a seed action $I(0)$ via a generic operator $M_{\lambda}$ as
$I(\lambda)=I(0)+\int
d\tau\sqrt{\gamma}\,M_{\lambda}(T_{\tau\tau},\gamma^{\tau\tau})\,,$ (68)
where the variational principle in the undeformed theory (where $M_{0}=0$) is
defined by
$\delta I(0)=\frac{1}{2}\int
d\tau\,\sqrt{\gamma}\,T_{\tau\tau}\delta\gamma^{\tau\tau}\,.$ (69)
With the deformation (68), one finds the following variation:
$\delta I(\lambda)=\delta
I(0)+\int\,d\tau\,\Big{[}\delta\left(\sqrt{\gamma}\right)M_{\lambda}+\sqrt{\gamma}\delta
M_{\lambda}\Big{]}\,.$ (70)
Using the facts that
$\delta M_{\lambda}=\frac{\partial M_{\lambda}}{\partial T_{\tau\tau}}\delta
T_{\tau\tau}+\frac{\partial
M_{\lambda}}{\partial\gamma^{\tau\tau}}\delta\gamma^{\tau\tau}$ (71)
and
$\delta\left(\sqrt{\gamma}\right)=-\frac{\sqrt{\gamma}}{2\gamma^{\tau\tau}}\delta\gamma^{\tau\tau}$
(72)
we find (70) is written as
$\delta I(\lambda)=\frac{1}{2}\int
d\tau\sqrt{\gamma}\left[\left(T_{\tau\tau}-\frac{M_{\lambda}}{\gamma^{\tau\tau}}+2\frac{\partial
M_{\lambda}}{\partial\gamma^{\tau\tau}}\right)\delta\gamma^{\tau\tau}+2\frac{\partial
M_{\lambda}}{\partial T_{\tau\tau}}\delta T_{\tau\tau}\right]\,.$ (73)
We wish to identify sources and expectation values by demanding that we can
rewrite (73) in terms of $\lambda$-dependent quantities
$\widetilde{T}_{\tau\tau}$, $\widetilde{\gamma}_{\tau\tau}$ as
$\displaystyle\delta I(\lambda)$ $\displaystyle=\frac{1}{2}\int
d\tau\frac{\widetilde{T}_{\tau\tau}}{\sqrt{\widetilde{\gamma}^{\tau\tau}}}\delta\widetilde{\gamma}^{\tau\tau}$
(74) $\displaystyle=\frac{1}{2}\int
d\tau\frac{\widetilde{T}_{\tau\tau}}{\sqrt{\widetilde{\gamma}^{\tau\tau}}}\left(\frac{\partial\widetilde{\gamma}^{\tau\tau}}{\partial
T_{\tau\tau}}\delta
T_{\tau\tau}+\frac{\partial\widetilde{\gamma}^{\tau\tau}}{\partial\gamma^{\tau\tau}}\delta\gamma^{\tau\tau}\right)\,.$
Here the operator $\widetilde{T}_{\tau\tau}$ is sourced by
$\widetilde{\gamma}^{\tau\tau}$. In other words, the deformation changes the
variational principle from one where $\gamma^{\tau\tau}$ is held fixed to one
where $\widetilde{\gamma}^{\tau\tau}$ is fixed.
Comparing (73) and (74), we find the following coupled PDEs for the deformed
boundary stress tensor and metric:
$\displaystyle\frac{\widetilde{T}_{\tau\tau}}{\sqrt{\widetilde{\gamma}^{\tau\tau}}}\frac{\partial\widetilde{\gamma}^{\tau\tau}}{\partial
T_{\tau\tau}}$ $\displaystyle=2\sqrt{\gamma}\frac{\partial
M_{\lambda}}{\partial T_{\tau\tau}}\,,$ (75)
$\displaystyle\frac{\widetilde{T}_{\tau\tau}}{\sqrt{\widetilde{\gamma}^{\tau\tau}}}\frac{\partial\widetilde{\gamma}^{\tau\tau}}{\partial\gamma^{\tau\tau}}$
$\displaystyle=\sqrt{\gamma}\left(T_{\tau\tau}-\frac{M_{\lambda}}{\gamma^{\tau\tau}}+2\frac{\partial
M_{\lambda}}{\partial\gamma^{\tau\tau}}\right)\,,$
with the initial conditions $\widetilde{T}_{\tau\tau}(\lambda=0)=T_{\tau\tau}$
and $\widetilde{\gamma}^{\tau\tau}(\lambda=0)=\gamma^{\tau\tau}$.
To further illustrate, we focus on a specific class of deformations that only
depend on the trace of the stress tensor $T_{\tau\tau}\gamma^{\tau\tau}$. It
is convenient to express our ansatz in terms of the dimensionless combination
$\displaystyle X=\lambda T_{\tau\tau}\gamma^{\tau\tau}\,.$ (76)
We assume
$\widetilde{T}_{\tau\tau}=T_{\tau\tau}\xi\left(X\right)\,,\quad\widetilde{\gamma}^{\tau\tau}=\gamma^{\tau\tau}\chi\left(X\right)\,,$
(77)
where $\xi(0)=\chi(0)=1$ so that we recover the undeformed stress tensor and
metric as $\lambda\to 0$. On the other hand, by dimensional analysis, we can
write the function $M_{\lambda}(\lambda,T_{\tau\tau},\gamma^{\tau\tau})$ in
the form
$\displaystyle M_{\lambda}=\frac{1}{\lambda}m_{\lambda}(X)\,.$ (78)
By substituting (77)-(78) into the system of coupled PDEs (75), we find the
pair of equations
$\displaystyle\begin{split}\chi^{\prime}(X)&=\frac{2\sqrt{\chi(X)}m_{\lambda}^{\prime}(X)}{X\xi(X)}\,,\\\
\sqrt{\chi(X)}\left(X-m_{\lambda}^{\prime}(X)+2Xm_{\lambda}^{\prime}(X)\right)&=X\xi(X)\left(\chi(X)+X\chi^{\prime}(X)\right)\,.\end{split}$
(79)
The usual double-trace $T\overline{T}$ deformation is quadratic in stress
tensors, so one might be interested in studying a deformation that is
proportional to the combination $X^{2}=\left(\lambda
T_{\tau\tau}\gamma^{\tau\tau}\right)^{2}$ since this is the only dimensionless
and reparameterization-invariant stress tensor bilinear in $(0+1)$-dimensions.
This corresponds to a deformation of the form
$\displaystyle M_{\lambda}$ $\displaystyle=\frac{1}{\lambda}X^{2}\,$
$\displaystyle=\lambda\left(T_{\tau\tau}\gamma^{\tau\tau}\right)^{2}\,.$ (80)
Using the form (2) of the deformation, the equations (79) become
$\displaystyle\xi(X)=\frac{(3X+1)\sqrt{\chi(X)}}{\chi(X)+X\chi^{\prime}(X)}\,,\qquad\chi^{\prime}(X)=\frac{4\sqrt{\chi(X)}}{\xi(X)}\,,$
(81)
which have the solutions
$\displaystyle\chi(X)=\frac{1}{(1-X)^{4}}\,,\qquad\xi(X)=(1-X)^{3}\,.$ (82)
We have therefore found that, for the form of the deformation
$M_{\lambda}=\lambda(T_{\tau\tau}\gamma^{\tau\tau})^{2}$ motivated by the
usual $T\overline{T}$ deformation,333For a multi-trace deformation
$M^{(n)}_{\lambda}=\lambda_{n}\left(T_{\tau\tau}\gamma^{\tau\tau}\right)^{2n}$
with coupling $\lambda_{n}$, one finds via solving (79)
$\widetilde{T}_{\tau\tau}(\lambda_{n})=T_{\tau\tau}\left(1-\lambda_{n}\left(T_{\tau\tau}\gamma^{\tau\tau}\right)^{2n-1}\right)^{\frac{4n-1}{2n-1}}\,,\quad\widetilde{\gamma}^{\tau\tau}(\lambda_{n})=\gamma^{\tau\tau}\left(1-\lambda_{n}\left(T_{\tau\tau}\gamma^{\tau\tau}\right)^{2n-1}\right)^{\frac{4n}{1-2n}}\,.$
(83) the solution is
$\widetilde{T}_{\tau\tau}(\lambda)=T_{\tau\tau}\left(1-\lambda
T_{\tau\tau}\gamma^{\tau\tau}\right)^{3}\,,\quad\widetilde{\gamma}^{\tau\tau}(\lambda)=\frac{\gamma^{\tau\tau}}{(1-\lambda
T_{\tau\tau}\gamma^{\tau\tau})^{4}}\,.$ (84)
However, as mentioned in section 1, and derived in appendix A of [115],
despite (2) being proportional to a double-trace operator $T^{2}$, it is not
suitable as a $T\overline{T}$ deformation for JT gravity with a Dirichlet
cutoff. The following choice of operator is suitable for the $T\overline{T}$
deformation found by [115]. In [115], the operator which yields the correct
deformed energy spectrum is:
$M_{\lambda}=-2\lambda OT_{\tau\tau}\gamma^{\tau\tau}\,,$ (85)
where the operator $O$ (i.e. the dilaton momentum) is sourced by the boundary
dilaton $\Phi_{b}$ as
$O=\frac{1}{\sqrt{\gamma}}\frac{\delta I}{\delta\Phi_{b}}\,.$ (86)
The seed theory action is now deformed as
$I(\lambda)=I(0)+\int
d\tau\sqrt{\gamma}M_{\lambda}\left(T_{\tau\tau},\gamma^{\tau\tau},O,\Phi_{b}\right)\,,$
(87)
where the variation of the undeformed theory is
$\delta I(0)=\int
d\tau\sqrt{\gamma}\left(\frac{1}{2}T_{\tau\tau}\delta\gamma^{\tau\tau}+O\delta\Phi_{b}\right)\,.$
(88)
To identify the variational principle of the deformed theory, we demand that
$\delta I(\lambda)$ can be written in terms of $\lambda$-dependent sources and
expectation values as
$\delta I(\lambda)=\int
d\tau\sqrt{\gamma}\left(\frac{1}{2}T_{\tau\tau}(\lambda)\delta\gamma^{\tau\tau}(\lambda)+\mathcal{O}(\lambda)\delta\Phi_{b}(\lambda)\right)\,.$
(89)
Following the same procedure as in the previous example with
$M_{\lambda}(T_{\tau\tau},\gamma^{\tau\tau})$, we find the sources and
expectation values transform as
$\displaystyle\gamma_{\tau\tau}(\lambda)$
$\displaystyle=\gamma_{\tau\tau}(0)\left(1+2\lambda O(0)\right)^{2}\,,\quad
T_{\tau\tau}(\lambda)=T_{\tau\tau}(0)\left(1+2\lambda O(0)\right)^{2}\,,$ (90)
$\displaystyle\Phi_{b}(\lambda)$ $\displaystyle=\Phi_{b}(0)-2\lambda
T(0)\,,\quad\mathcal{O}(\lambda)=\frac{O(0)}{1+2\lambda O(0)}\,,$
which satisfy
$\delta I(\lambda)=\delta I(0)-2\lambda\delta\left(\int
d\tau\sqrt{\gamma}O(0)T_{\tau\tau}(0)\gamma^{\tau\tau}(0)\right)\,.$ (91)
The $\lambda$-dependent sources and expectation values (90) describe the full
solution for the bulk JT gravity fields, which corresponds to performing a
$T\overline{T}$-like deformation of the $1d$ boundary theory. As in the
analogous deformation of $3d$ gravitational Chern-Simons reviewed in section
2, we note that the result can be interpreted as a linear mixing of sources
and expectation values, although in this case each source becomes a function
of the dual expectation value for a _different_ operator – for instance, the
metric becomes dependent on the field $O$ which is dual to the dilaton
$\Phi_{b}$.
### 4 $T\overline{T}$-deformed boundary conditions in BF theory
In the previous section, we have seen that the interpretation of a boundary
$T\overline{T}$ deformation in JT gravity is a particular $\lambda$-dependent
mixing (90) of the metric $\gamma_{\tau\tau}$, dilaton $\Phi_{b}$, and their
dual operators. Since JT gravity can also be written in BF variables, there
must be an analogous interpretation of the boundary $T\overline{T}$
deformation. The goal of the current section is to make this BF interpretation
explicit.
Because BF gauge theory is topological, all of the dynamics of the theory
occur at the boundary. As a result, the choice of boundary term – and the
variational principle – is an important input for defining the theory. We
consider $T\overline{T}$-type deformations for two choices of boundary terms:
one which gives a variational principle analogous to that of the JT gravity
theory and one whose boundary theory is the Schwarzian.
#### 1 Deformation with JT-type boundary term
First, we will determine the choice of boundary term in BF theory, which gives
a variational principle most analogous to that of the JT gravity theory. We
saw in (88) that the on-shell variation of the JT gravity action is
$\displaystyle\delta I\Big{|}_{\text{on-shell}}=\int_{\partial
M_{2}}\,d\tau\,\sqrt{\gamma}\,\left(\frac{1}{2}T_{\tau\tau}\,\delta\gamma^{\tau\tau}+O\,\delta\Phi_{b}\right)\,.$
(92)
This boundary term vanishes if we fix the value of the (inverse) metric
$\gamma^{\tau\tau}$ and the dilaton $\Phi_{b}$ at the boundary. The operators
dual to the metric and dilaton are the boundary stress tensor $T_{\tau\tau}$
and the operator $O$, respectively.
On the other hand, the variation of the BF action
$I_{\text{BF}}=-i\int_{M_{2}}\operatorname{Tr}(\phi F)$ was given in (53) as
$\delta I_{\text{BF}}\Big{|}_{\text{on-shell}}=-i\int_{\partial
M_{2}}d\tau\operatorname{Tr}\left(\phi\delta A_{\tau}\right)\,.$ (93)
We parameterize the BF theory fields in terms of $SL(2,\mathbb{R})$ generators
as
$\displaystyle
A_{\mu}(x)=e_{\mu}^{+}(x)L_{+}+e_{\mu}^{-}(x)L_{-}+\omega_{\mu}(x)L_{0}\,,\qquad\phi(x)=\phi^{+}(x)L_{+}+\phi^{-}(x)L_{-}+\phi^{0}(x)L_{0}\,,$
(94)
Note that we use the notation $L_{+}$ for $L_{1}$ and $L_{-}$ for $L_{-1}$. In
terms of the functions appearing in this expansion, the boundary term (93) is
$\delta I_{\text{BF}}\Big{|}_{\text{on-shell}}=-i\int_{\partial
M_{2}}\,d\tau\,\left(\frac{1}{2}\phi^{0}\delta\omega_{\tau}-\phi^{+}\delta
e^{-}_{\tau}-\phi^{-}\delta e^{+}_{\tau}\right)\,.$ (95)
The asymptotic values of the expansion coefficients $e_{\tau}^{\pm}$ in the BF
fields are interpreted as the einbein for the one-dimensional boundary theory.
These fields are the BF analog of the boundary metric $\gamma_{\tau\tau}$.
Likewise, the boundary value of the BF variable $\phi^{0}$ is proportional to
the boundary dilaton $\Phi_{b}$ in JT variables.
Thus we see that the naïve BF action, without any added boundary term,
corresponds to a different variational principle than that of JT gravity. For
the variation (95) to vanish, we must fix the boundary values of
$e^{\pm}_{\tau}$ (which corresponds to fixing the boundary metric) but not the
boundary value of $\phi^{0}$; rather the asymptotic value of $\omega_{\tau}$
is held fixed. In JT gravity language, this corresponds to a variational
principle where the value of the dual operator $O$ is held fixed, but the
boundary dilaton $\Phi_{b}$ is free to vary.
We can, of course, modify the BF variational principle by adding an
appropriate boundary term. Suppose that we choose the BF action to be
$\displaystyle I=I_{\text{BF}}+I_{\text{bdry}}\,,\qquad
I_{\text{bdry}}=\frac{i}{2}\int_{\partial
M_{2}}d^{2}x\,\sqrt{g}\,n_{\mu}\partial^{\mu}\left(\phi^{0}\omega_{\tau}\right)\,,$
(96)
where $n_{\mu}$ is a unit normal vector in the radial direction. The
corresponding contribution to the boundary variation is
$\displaystyle\delta I_{\text{bdry}}=\frac{i}{2}\int_{\partial
M_{2}}\,d\tau\,\left(\omega_{\tau}\delta\phi^{0}+\phi^{0}\delta\omega_{\tau}\right)\,.$
(97)
This cancels the $\phi^{0}\delta\omega_{\tau}$ term appearing in (93). The
total boundary variation is now
$\displaystyle\delta I\Big{|}_{\text{on-shell}}=i\int_{\partial
M_{2}}\,d\tau\,\left(\frac{1}{2}\omega_{\tau}\,\delta\phi^{0}+\phi^{+}\delta
e_{\tau}^{-}+\phi^{-}\delta e_{\tau}^{+}\right)\,.$ (98)
Demanding that this boundary term vanish leads us to a variational principle
where $e_{\tau}^{\pm}$ and $\phi^{0}$ are held fixed at the boundary. This is
the direct BF theory analog of the variational principle in JT gravity, where
the boundary metric and dilaton are held fixed, so we will refer to this
choice as “JT-type boundary conditions.”
We now wish to identify the modification of these JT-type boundary conditions,
which corresponds to a $T\overline{T}$-like deformation of the dual
$(0+1)$-dimensional theory. There are two ways one might identify the
appropriate form of the deforming operator. One way is to dimensionally reduce
the $T\overline{T}$ operator written in $3d$ Chern-Simons variables, which
takes the form $f^{-}\wedge f^{+}$ as reviewed in section 2. Recall that, in
Chern-Simons language, the operators $f^{a}$ are dual to the boundary
vielbeins $e^{a}$, and, therefore, the $f^{a}$ contain the boundary stress
tensor. Upon such a reduction, one component of $f$ reduces to the one-
dimensional stress tensor $T_{\tau\tau}$, which is dual to the boundary
einbein $e_{\tau}$. Since the component of the metric in the direction along
which we reduce is identified with the field $\phi^{0}$, the other component
of $f$ reduces to the operator dual to $\phi^{0}$, which is $\omega_{\tau}$.
Therefore, the dimensional reduction instructs us to deform the boundary
action by an operator constructed from the combination
$T_{\tau\tau}\omega_{\tau}$ (contracted with the appropriate einbein factors
to yield a quantity which is a scalar under diffeomorphisms).
The other way to identify the deforming operator is by using the combination
$OT$, which defines the $T\overline{T}$ deformation in JT variables and
converts all expressions to BF variables. We now carry out this procedure and
demonstrate that it produces an operator of the schematic form
$T_{\tau\tau}\omega_{\tau}$ suggested by dimensional reduction. The (Hilbert)
definition of the boundary stress tensor is
$\displaystyle T_{\tau\tau}$
$\displaystyle=-\frac{2}{\sqrt{\gamma_{\tau\tau}}}\frac{\delta
I}{\delta\gamma^{\tau\tau}}$
$\displaystyle=-\frac{2}{\sqrt{\gamma_{\tau\tau}}}\left(\frac{\delta I}{\delta
e^{+}_{\tau}}\frac{\delta
e^{+}_{\tau}}{\delta\gamma^{\tau\tau}}\Big{|}_{e_{\tau}^{-}}+\frac{\delta
I}{\delta e^{-}_{\tau}}\frac{\delta
e^{-}_{\tau}}{\delta\gamma^{\tau\tau}}\Big{|}_{e_{\tau}^{+}}\right)\,.$ (99)
The map from the metric $\gamma_{\tau\tau}$ to the boundary BF fields
$e^{\pm}_{\tau}$ is simply
$\displaystyle\gamma_{\tau\tau}=-4e^{+}_{\tau}e^{-}_{\tau}\,,\qquad\gamma^{\tau\tau}=-\frac{1}{4e^{+}_{\tau}e^{-}_{\tau}}\,.$
(100)
Note that, according to our conventions (56), the relative minus sign in the
definition (100) of $\gamma_{\tau\tau}$ is required to have a positive-
definite worldline metric since
$\displaystyle
e_{\tau}^{+}e_{\tau}^{-}=\frac{1}{4}\left(ie_{\tau}^{1}-e_{\tau}^{2}\right)\left(ie_{\tau}^{1}+e_{\tau}^{2}\right)=-\frac{1}{4}\left(\left(e_{\tau}^{1}\right)^{2}+\left(e_{\tau}^{2}\right)^{2}\right)\,.$
(101)
Thus, the derivatives appearing in the stress tensor can be written as
$\displaystyle\frac{\delta
e^{+}_{\tau}}{\delta\gamma^{\tau\tau}}\Big{|}_{e_{\tau}^{-}}$
$\displaystyle=\frac{1}{\left(\gamma^{\tau\tau}\right)^{2}}\cdot\frac{1}{4e_{\tau}^{-}}=-\frac{e_{\tau}^{+}}{\gamma^{\tau\tau}}\,,$
$\displaystyle\frac{\delta
e^{-}_{\tau}}{\delta\gamma^{\tau\tau}}\Big{|}_{e_{\tau}^{+}}$
$\displaystyle=\frac{1}{\left(\gamma^{\tau\tau}\right)^{2}}\cdot\frac{1}{4e_{\tau}^{+}}=-\frac{e_{\tau}^{-}}{\gamma^{\tau\tau}}\,.$
(102)
Meanwhile, from (98) we see that $\frac{\delta I}{\delta
e^{+}_{\tau}}=i\phi^{-}$ and $\frac{\delta I}{\delta e^{-}_{\tau}}=i\phi^{+}$.
So, the stress tensor is
$\displaystyle
T_{\tau\tau}=\frac{2i}{\sqrt{\gamma^{\tau\tau}}}\left(\phi^{-}\cdot\frac{e_{\tau}^{+}}{\gamma^{\tau\tau}}+\phi^{+}\cdot\frac{e_{\tau}^{-}}{\gamma^{\tau\tau}}\right)\,,$
(103)
and its trace is
$\displaystyle
T=T_{\tau\tau}\gamma^{\tau\tau}=\frac{i}{\sqrt{-e^{+}_{\tau}e^{-}_{\tau}}}\left(e^{+}_{\tau}\phi^{-}+\phi^{+}e^{-}_{\tau}\right)\,.$
(104)
Next, we express the operator $O$ dual to the dilaton in BF variables. Using
the map
$\displaystyle\Phi=-\frac{i}{4}\phi^{0}\,,$ (105)
one has from (86) that
$\displaystyle O$ $\displaystyle=\frac{1}{\sqrt{\gamma}}\frac{\delta
I}{\delta\Phi_{b}}$
$\displaystyle=\frac{2i}{\sqrt{-e_{\tau}^{+}e_{\tau}^{-}}}\frac{\delta
I}{\delta\phi^{0}}$
$\displaystyle=-\frac{1}{\sqrt{-e_{\tau}^{+}e_{\tau}^{-}}}\omega_{\tau}\,.$
(106)
We conclude that, in BF variables, the combination which corresponds to a
boundary $T\overline{T}$ deformation is
$\displaystyle
OT=\frac{i}{e_{\tau}^{+}e_{\tau}^{-}}\left(e^{+}_{\tau}\phi^{-}+\phi^{+}e^{-}_{\tau}\right)\omega_{\tau}\,.$
(107)
|
We show that it is possible to craft transformations that, applied to compositional grammars, result in grammars that neural networks can learn easily, but humans do not.
This could explain the disconnect between current metrics of compositionality, that are arguably human-centric, and the ability of neural networks to generalize to unseen examples.
We propose to use the transformations as a benchmark, Icy, which could be used to measure aspects of the compositional inductive bias of networks, and to search for networks with similar compositional inductive biases to humans.
As an example of this approach, we propose a hierarchical model, HU-RNN, which shows an inductive bias towards position-independent, word-like groups of tokens.
§ EXPERIMENTS
Code for experiments is at [https://github.com/asappresearch/compositional-inductive-bias].
§.§ Examples of grammars
Table <ref> shows examples of each grammar, for 4 objects.
For concat, changing one attribute changes 3 adjacent utterance tokens. perm rearranges columns of concat utterance tokens. shufdet rearranges blocks of 3 utterance tokens, as a function of the last object attribute. We depict utterances for $n_{att}=3$, and $c_{len} = 3 \cdot n_{att}$. In our experiments we use $n_{att}=5$ and $c_{len}=4 \cdot n_{att}$. Examples for this geometry can be found in Appendix <ref>.
§.§ Compositional metric evaluation
Figure <ref> shows the values of compositional metrics for samples from our artificial grammars, using $n_{att}=5$, $n_{val}=10$.
The compositionality metrics show low compositionality for all the artificial grammars, except for concat and perm. Thus our transformations successfully hide the compositional structure from current compositional metrics.
§.§ Neural model evaluation
We use the / benchmark to evaluate standard neural models for specific aspects of their compositional inductive bias.
We focus on Sender models in our presentation.
Results for Receiver models are in Appendix <ref>.
We train each model supervised on a specific artificial grammar from /, using cross-entropy loss.
We count the number of training steps, $N_{acquire}$, required to train each grammar to a training accuracy of $\acc_{tgt}$, where accuracy is token-level accuracy. For each grammar, $\mathcal{G}$, we report the ratio
$b^{(\mathcal{G})} = N_{acquire}^{(\mathcal{G})} / N_{acquire}^{(\mathcal{G}_{\textsc{concat}})}$.
We used $n_{att} = 5$ and $n_{val} = 10$, $c_{len} = 20$, $V = 4$, and $\acc_{tgt} = 0.8$. We halt training if $b^{(\mathcal{G})}$ reaches $20$.
Table <ref> shows the results.
Detailed architectural descriptions of the `Model' column are provided in Appendix <ref>.
The remaining columns, except for `Params', show the acquisition time, $b$, for each grammar, relative to concat. We have highlighted in red the scenarios that failed to reach convergence; and in green the scenarios where $b$ was less than 1/3 that of hol, which shows that language acquisition was relatively fast.
We can see that for many models, our transformations do not much affect the acquisition speed by neural networks. Therefore, in an emergent communication scenario, neural models can generate languages which appear non-compositional both to our current metrics, and to human evaluation. Such languages will therefore be deemed `non-compositional' by all current evaluation methods, except for generalization. This might explain the empirically observed lack of correlation between measured language compositionality, and generalization, in emergent communication experiments.
§.§ Results are independent of number of parameters
An obvious concern with Table <ref> is that the number of parameters varies between models, so we vary the parameters, by changing the hidden size. Table <ref> shows the results.
We can see that the relative acquisition speed, relative to concat, is not changed much by a 10-fold increase in parameters,
relative to the differences between the architectures. This is encouraging: we are not simply viewing an artifact of model size.
§.§ RNNZero increases bias against perm
Table <ref> shows the effect of not feeding the output of an RNN decoder as the input at each step. Surprisingly this increases bias against perm. That is, actually increases a prior over adjacency.
§.§ HUSendZ:dgsend has low bias against shufdet
We searched for neural models with reduced bias against shufdet, including using RNN-Z, dgsend [Dagan et al., 2020], and HU-RNN. Table <ref> shows a sub-set of the results. More results are in Appendix <ref>.
`dgsend' acquired shufdet faster than LSTM.
HUSend using a vanilla RNN as $\rnn_l$ and $\rnn_u$ acquired shufdet faster than LSTM.
Combining HUSendZ with dgsend acquired shufdet fastest.
§.§ End-to-end training
We experimented with measuring the compositional inductive bias of a Sender and Receiver model placed end to end, see Appendix <ref>
§ CONCLUSION
We have shown that it is possible to construct transformations that, when applied to concatenation grammars, result in grammars that machines can learn easily but which humans find challenging to learn. This could explain the disconnect highlighted in recent papers between neural network ability to generalize, in an emergent communication context, and the compositionality of the resulting languages, as measured by recent metrics of compositionality. We propose to use the families of transformations as a benchmark, Icy, for measuring aspects of the compositional inductive bias of neural networks, and searching for models with similar biases to humans. We use our benchmark to propose one such neural model, hu-rnn, which shows a compositional inductive bias towards relocatable atomic word-like groups of tokens.
§ REPRODUCIBILITY
Full code is provided in the addendum, along with instructions in the README.md. Full code will be published to github following acceptance. Each experiment was run multiple times (usually 5 or 10), using different seeds, and the mean reported. CI95 ranges are available in Appendix <ref>.
§ ETHICS
This work does involve human subjects, who needed to learn to use artificially generated codes to label abstract geometric objects. The annotation device was created as a game, that many people found fun to play. We received many feedbacks stating `good', `very interesting task'. None of the language or figures being trained on contain any obvious characteristics which could be deemed racist, sexist, or having any other obvious human-centric harmful biases, as far as we can tell.
This work contains no obviously harmful insights, methodologies or applications. There are no obvious conflicts of interest or sponsorship to note. There are no obvious discrimination/bias/fairness concerns to report. There are no obvious issues with privacy, security, or legal compliance. All data provided was artificially generated, and does not present privacy or other issues. We have done our due diligence to ensure the integrity and reproducibility of our research.
Although emergent communication investigates the communications between neural models, who learn to generate new languages, as part of collaborative tasks, we do not believe that such models are `alive', or `conscious', though we admit that we do not have any way to determine this in any objective way. The number of neurons of the models concerned was orders of magnitude less than that of the human brain. The models were not exposed to sufficiently varied or complex data that we feel that they could have learned advanced sentience or perception, although again we admit that we are not aware of an objective `threshold' or similar that we could compare with.
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§ EXAMPLE UTTERANCES
Table <ref> depicts example utterances for $n_{att}=5$ and $c_{len} = 4 \cdot n_{att}$.
|
# Quantum Algorithm for Higher-Order Unconstrained Binary Optimization and
MIMO Maximum Likelihood Detection
Masaya Norimoto, , Ryuhei Mori, , and Naoki Ishikawa M. Norimoto and N.
Ishikawa are with the Faculty of Engineering, Yokohama National University,
Kanagawa 240-8501, Japan (e-mail: [email protected]). R. Mori is
with the Department of Mathematical and Computing Sciences, School of
Computing, Tokyo Institute of Technology, Tokyo 152-8500, Japan (e-mail:
[email protected]). This research was partially supported by the Japan
Society for the Promotion of Science (JSPS) KAKENHI (Grant Number 22H01484).
###### Abstract
In this paper, we propose a quantum algorithm that supports a real-valued
higher-order unconstrained binary optimization (HUBO) problem. This algorithm
is based on the Grover adaptive search that originally supported HUBO with
integer coefficients. Next, as an application example, we formulate multiple-
input multiple-output maximum likelihood detection as a HUBO problem with
real-valued coefficients, where we use the Gray-coded bit-to-symbol mapping
specified in the 5G standard. The proposed approach allows us to construct an
efficient quantum circuit for the detection problem and to analyze specific
numbers of required qubits and quantum gates, whereas other conventional
studies have assumed that such a circuit is feasible as a quantum oracle. To
further accelerate the quantum algorithm, we also derive a probability
distribution of the objective function value and determine a unique threshold
to sample better states. Assuming a future fault-tolerant quantum computing,
our proposed algorithm has the potential for significantly reducing query
complexity in the classical domain and providing a quadratic speedup in the
quantum domain.
###### Index Terms:
Grover adaptive search (GAS), quadratic unconstrained binary optimization
(QUBO), higher-order unconstrained binary optimization (HUBO), multiple-input
multiple-output (MIMO), maximum-likelihood detection (MLD).
(45pt,10pt) Accepted for publication in IEEE Transactions on Communications.
This is the author’s version which has not been fully edited and content may
change prior to final publication. Citation information: DOI
10.1109/TCOMM.2023.3244924
## I Introduction
Marconi invented a practical long-range wireless system in 1895. Since then,
driven by its intense demand, wireless communication has continued to become
more sophisticated as if there were no limits. The limit of communication
throughput is known as the Shannon capacity, which is constrained by the
bandwidth, the signal-to-noise ratio (SNR), and the numbers of transmit and
receive antennas for multiple-input multiple-output (MIMO) scenarios. Clearly,
there are physical limits on bandwidth, SNR, and the number of antennas. The
forward error correction techniques such as the low-density parity-check code
(LDPC) and polar code can achieve near-capacity performance efficiently, but
under a certain energy constraint, their performance is constrained by
semiconductor miniaturization limits. Marconi mentioned that it is dangerous
to put limits on wireless. However, wireless communication will reach its
physical limits in the near future.
After the eventual end of Moore’s law, from a long-term perspective, we must
rely on a different computing paradigm, and quantum computing in particular is
considered to be promising. Since it is impossible to simulate a quantum
computer in an efficient manner on a classical computer, quantum computers
offer an essential speed advantage over classical computers [1]. Specifically,
Shor’s algorithm [2] factors an $n$-bit integer with the complexity
$O(n^{2}\log n\log\log n)$, while the best classical algorithm requires
$\exp(\Theta(n^{1/3}\log^{2/3}n))$ operations [1],111$O(\cdot)$ denotes the
big-$O$ notation, while $\Theta(\cdot)$ denotes the big-$\Theta$ notation [3].
which is an exponential speedup. Grover’s algorithm [4] finds a specific
element from a database of unsorted $N$ elements with the query complexity
$O(\sqrt{N})$, while the classic exhaustive search requires $O(N)$
evaluations, which is a quadratic speedup. Note that both long-term algorithms
assume the realization of fault-tolerant quantum computing (FTQC), which is
not yet realized with current technology.
Grover’s algorithm has been extended to support binary optimization problems.
The pioneering algorithm, Grover adaptive search (GAS) [5], requires a complex
quantum circuit to evaluate an objective function. For example, an $m$-qubit
register requires $2m-1$ Toffoli gates to perform quantum addition [6], which
is still expensive. To solve this issue, Gilliam et al. used a concept of
quantum dictionary and allowed for the efficient representation of an
arbitrary polynomial function, including quadratic and higher-order terms [7,
8]. This efficient representation improved the feasibility of GAS, and in [8],
a quadratic unconstrained binary optimization (QUBO) problem with integer
coefficients was solved on a real-world quantum computer with 32 qubits.
Unlike the quantum annealing (QA) [9], the GAS proposed by Gilliam et al. is
innovative in that it supports a higher-order unconstrained binary
optimization (HUBO) problem with integer coefficients, which cannot be solved
efficiently with state-of-the-art mathematical programming solvers on a
classical computer such as CPLEX222https://www.ibm.com/analytics/cplex-
optimizer.
In designing wireless systems, the trade-off between performance and
complexity is, in general, a source of concern for engineers and researchers.
For example, low-complexity MIMO detectors and polar decoders inevitably
involve the penalty of lower performance, and complexity is sacrificed to
achieve optimal performance. In this situation, the potential for quantum
speedup has inspired those who dream of striking the fundamental trade-off and
achieving the optimal performance with reduced complexity. A pioneering
attempt in wireless communications was provided in [10] by Botsinis et al. who
demonstrated the potential of quantum algorithms to reduce the complexity
involved in maximum likelihood detection (MLD). Specifically, they used the
Grover-type algorithms, such as Boyer–Brassard–Høyer–Tapp (BBHT) [11] and
Dürr–Høyer (DH) search algorithms [12], for performing MLD of data symbols on
a quantum computer [13]. Subsequently, a number of important studies have
shown promising results [13, 14, 15, 16, 17, 18, 19, 20]. However, in those
studies, it was assumed that an ideal quantum circuit to evaluate the
objective function is feasible as a quantum oracle, which will be detailed in
Section II. For more information on quantum-assisted wireless communications,
a comprehensive survey can be found in [21, 22].
Against this background, we propose a quantum algorithm that supports a HUBO
problem with real-valued coefficients. Then, as a first step toward breaking
the trade-off between performance and complexity, we formulate the MIMO MLD as
a real-valued HUBO problem and verify the potential of quadratic speedup. The
major contributions of this paper are organized as follows.
1. 1.
While the conventional GAS [8] supports HUBO with integer coefficients, we
modify the quantum algorithm to handle real-valued coefficients. This allows
us to solve a HUBO problem even if the objective function contains real-valued
coefficients, which is achieved at the cost of one more query in the classical
domain (CD).333The definition of query complexity in CD is detailed in Section
III-D, which is the same as the conventional study [13].
2. 2.
As an application example, we formulate the objective function of MIMO MLD as
a real-valued HUBO problem. This formulation is not a straightforward task
because the objective function contains complex-valued random variables and a
Frobenius norm calculation. This new formulation allows us to analyze specific
numbers of qubits and quantum gates required in the constructed quantum
circuits, which has been overlooked in conventional studies.
3. 3.
We clarify the probability distribution of the objective function value and
determine the threshold used inside GAS more efficiently. Then, we demonstrate
that the proposed threshold further accelerates the convergence of GAS to the
optimal solution.
It is important to note that quantum circuits are sensitive to noise [23], and
industrial applications require decades of effort and challenge. The noise
induces quantum errors, and quantum error-correcting codes must be used to
perform reliable arithmetic on a quantum computer. For example, if we use the
surface code with code distance 27, which is one of the quantum error-
correcting codes, a logical qubit requires 1568 physical qubits to correct
errors [24]. This indicates that even a simple quantum circuit with fewer
qubits, e.g., as in Fig. 2, may require many more physical qubits. Since this
limitation is beyond the scope of our contributions, we assume the realization
of future FTQC as in the conventional studies [2, 4, 5, 7, 8, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 25]. In fact, IBM’s roadmap for quantum
computers is to achieve 4000 qubits by 2025 [26]. In the subsequent years,
they expect 10000 to 100000 qubits, enabling quantum error corrections.
Additionally, it was proved in [27] that quantum advantages are unlikely for
optimization on a noisy intermediate-scale quantum device. Therefore, we focus
on long-term algorithms assuming FTQC in this paper.
The remainder of this paper is organized as follows. Section II is a review of
important related works, while in Section III, we introduce the conventional
GAS and its modification to support real-valued coefficients. In Section IV, a
method to solve MIMO MLD on a quantum computer is proposed, and algebraic and
numerical evaluations are given in Section V. Finally, in Section VI, we
conclude this paper.
TABLE I: List of important mathematical symbols $\mathbb{B}$ | | Binary numbers
---|---|---
$\mathbb{R}$ | | Real numbers
$\mathbb{C}$ | | Complex numbers
$\mathbb{Z}$ | | Integers
$N_{\mathrm{t}}$ | $\in\mathbb{Z}$ | Number of transmit antennas
$N_{\mathrm{r}}$ | $\in\mathbb{Z}$ | Number of receive antennas
$L_{\mathrm{c}}$ | $\in\mathbb{Z}$ | Modulation order (constellation size)
$\sigma^{2}$ | $\in\mathbb{R}$ | Noise variance
$\gamma$ | $\in\mathbb{R}$ | Signal-to-noise ratio
$E(\cdot)$ | $\in\mathbb{Z}$ | Objective function
$n$ | $\in\mathbb{Z}$ | Number of binary variables $=$ transmission rate
$m$ | $\in\mathbb{Z}$ | Number of qubits required to encode $E(\cdot)$
$i$ | $\in\mathbb{Z}$ | Index of GAS iterations
$y,y_{i}$ | $\in\mathbb{Z}$ | Threshold that is adaptively updated by GAS
$L,L_{i}$ | $\in\mathbb{Z}$ | Number of Grover operators
$P$ | $\in\mathbb{R}$ | Probability that controls the proposed threshold
$\mathbf{b},\mathbf{b}_{i}$ | $\in\mathbb{B}^{n}$ | Binary variables, or data bits
$\mathbf{s}$ | $\in\mathbb{C}^{N_{\mathrm{t}}\times 1}$ | Data symbols, each symbol is denoted by $s_{t}$
$\mathbf{r}$ | $\in\mathbb{C}^{N_{\mathrm{r}}\times 1}$ | Received symbols, each symbol is denoted by $r_{u}$
$\mathbf{H}_{\mathrm{c}}$ | $\in\mathbb{C}^{N_{\mathrm{r}}\times N_{\mathrm{t}}}$ | Channel coefficients, $h_{ut}$
$\mathbf{v}$ | $\in\mathbb{C}^{N_{\mathrm{r}}\times 1}$ | Additive white Gaussian noise, $v_{u}$
Italicized symbols represent scalar values, and bold symbols represent vectors
and matrices. Table I summarizes a list of important mathematical symbols used
in this paper.
## II Related Works
Quantum computation has the potential to break through the fundamental trade-
off between performance and complexity. Hence, it has been applied to multi-
user detection [10, 13, 14, 15, 17, 28, 29], multiple symbol differential
detection [16], channel coding [30, 31], wireless routing [19, 18], indoor
localization [20], intelligent reflecting surfaces [32], and codeword
optimization problem [25]. In this section, we introduce important related
works targeting detection problems in wireless communications.
### II-A Multi-User Detection Using DH Algorithm [13]
Botsinis et al. proposed a novel method of applying the DH algorithm to multi-
user detection [13], which is a detection problem for multi-user scenarios.
The original DH algorithm [12] is terminated if the sum of the number of
Grover iterations becomes greater than or equal to $22.5\sqrt{N}$, where $N$
denotes the search space size. By contrast, Botsinis et al. modified the
algorithm to terminate early for an arbitrary number of queries smaller than
$22.5\sqrt{N}$. Additionally, the modified algorithm calculates the output of
a low-complexity detector, such as the zero-forcing (ZF) or minimum mean
square error (MMSE) detector, and exploits the output as an initial value to
sample better states. Both contributions are innovative in that they
accelerate the quantum algorithm more for a specific problem in wireless
communications.
The objective function presented in [13] involves a Frobenius norm of complex-
valued variables. However, the quantum circuit that evaluates the norm is
idealized as an oracle, and no specific construction method is considered.
Unlike in [13], we consider specific quantum circuits and analyze their
hardware and query complexities, which is the missing piece in the literature.
### II-B MIMO MLD Using QA [28]
Kim et al. formulated MIMO MLD as a QUBO problem and solved it using QA, the
D-Wave 2000Q quantum annealer [28]. Specifically, binary phase-shift keying
(BPSK) and quadrature phase-shift keying (QPSK) symbols are represented as
first-order functions with respect to information bits, while gray-coded 16
quadrature amplitude modulation (QAM) symbols are represented as second-order
functions. Since the objective function of MLD contains the squared norm, it
may result in a higher-order function such as fourth, eighth, or higher, which
is not supported by QA. To solve this problem, Kim et al. used first-order
functions that represent higher-order modulation, such as 16-QAM or 64-QAM,
without the Gray coding. Then, the objective function contains first- and
second-order terms only. To achieve performance equivalent to that of the
Gray-coded case, the projection between before and after Gray coding is used
on a classical computer. That is, encoding at the transmitter and decoding at
the receiver require additional steps.
Unlike in the above study [28] targeting QA, we directly handle the Gray-coded
data symbols specified in the 5G standard owing to the proposed real-valued
GAS that supports higher-order terms. Our approach is capable of supporting
any signal modulation, such as star-QAM and constellation shaping schemes, as
long as data symbols can be represented as a function of information bits.
### II-C MIMO MLD Using DH Algorithm [29]
Mondal et al. proposed a method to solve MIMO MLD using the DH algorithm [29].
Specifically, to improve the success probability of the algorithm, the uniform
selection of the number of Grover operators, $L$, was modified to a random
value from the Gamma distribution, leading to a better selection of $L$. Here,
the Gamma distribution depends on a scale parameter, and the scale parameter
depends on the exact number of solutions to be marked. Since the exact number
of solutions varies dynamically depending on the threshold, the quantum
counting algorithm [33] is crucial, as stated in [29]. Additionally, the
concept of reducing the search space was verified.
As in [13], a specific construction method for a quantum circuit is not
considered in [29]. Herein, we determine a threshold in accordance with the
distribution of objective function values, which is known in advance.
## III Grover Adaptive Search (GAS)
GAS [8] supports binary optimization problems with integer coefficients,
including QUBO and HUBO problems. It requires $n$ qubits for $n$ binary
variables $\mathbf{b}\in\mathbb{B}^{n}$ and $m$ qubits for encoding the
objective function value $E(\mathbf{b})\in\mathbb{Z}$, resulting in a circuit
equipped with $n+m$ qubits. Here, $E(\mathbf{b})$ is an arbitrary polynomial
function, such as $E(\mathbf{b})=1+b_{0}-2b_{1}b_{2}$. The classic exhaustive
search requires $O(2^{n})$ queries, while GAS requires $O(\sqrt{2^{n}})$
queries, which potentially provides a quadratic speedup. GAS obtains a global
minimum solution by amplifying the states in which the objective function
value $E(\mathbf{b})$ is smaller than the current threshold
$y_{i}\in\mathbb{Z}$. Here, $y_{i}$ is a temporal minimum and $i$ is an
iteration count in CD. We measure the quantum states and update the threshold,
which is repeated until a termination condition is satisfied.
Before executing GAS, it is not a straightforward task to determine an
appropriate number of qubits $m$. The objective function value is expressed by
the two’s complement representation. This is because positive or negative can
be identified simply by focusing on the beginning of $m$ qubits, and this
representation simplifies the quantum circuit to the identification of the
states of interest that should be amplified. Let the objective function value
or its coefficient be an integer $k$. Then, $m$ must satisfy [8]
$\displaystyle-2^{m-1}\leq k<2^{m-1}.$ (1)
As the threshold $y_{i}$ is updated in each iteration of GAS, the calculated
value may become $E(\mathbf{b})-y_{i}$, which results in a smaller minimum
value or a larger maximum value. Thus, it is necessary to set a sufficient $m$
that might handle $E_{\mathrm{max}}$, $E_{\mathrm{min}}$, and
$E_{\mathrm{max}}-E_{\mathrm{min}}$ without overflow, where $E_{\mathrm{max}}$
and $E_{\mathrm{min}}$ are the maximum and minimum of $E(\mathbf{b})$,
respectively. For example, when we have $E_{\mathrm{max}}=8$ and
$E_{\mathrm{min}}=-6$, the maximum of $E(\mathbf{b})-y_{i}$ may become
$E_{\mathrm{max}}-E_{\mathrm{min}}=8-(-6)=14$, and $m=5$ is sufficient to
represent the values of $E(\mathbf{b})-y_{i}$.
### III-A Conventional GAS for Integer QUBO [8]
We review a specific construction method for the quantum circuit used in GAS.
First, a state preparation operator $\mathbf{A}_{y_{i}}$ is constructed, in
which an $n$-qubit input register is transformed into the equal superposition
of all states and an $m$-qubit input register is used to represent the
corresponding value $E(\mathbf{b})-y_{i}$. Taking the binary variable
$\mathbf{b}$ as a binary number and converting it to a decimal number $b$, the
state should be [8]
$\displaystyle\mathbf{A}_{y_{i}}\Ket{0}_{n}\Ket{0}_{m}=\frac{1}{\sqrt{2^{n}}}\sum_{b=0}^{2^{n}-1}\Ket{b}_{n}\Ket{E(b)-y_{i}}_{m}.$
(2)
This operator $\mathbf{A}_{y_{i}}$ can be composed of the Hadamard gates
$\mathbf{H}$, controlled unitary operators $\mathbf{U}_{G}(\theta)$, and the
inverse quantum Fourier transform (IQFT). Let $k$ be a constant term in the
objective function. The noncontrolled unitary operator
$\mathbf{U}_{G}(\theta)$ is defined such that [8]
$\displaystyle\mathbf{U}_{G}(\theta)\mathbf{H}^{\otimes
m}\Ket{0}_{m}=\frac{1}{\sqrt{2^{m}}}\sum^{2^{m}-1}_{l=0}e^{jl\theta}\Ket{l}_{m},$
(3)
where we have $\theta=2\pi k/2^{m}$. That is, it is constructed by
$\displaystyle\mathbf{U}_{G}(\theta)=\mathbf{R}(2^{m-1}\theta)\otimes\mathbf{R}(2^{m-2}\theta)\otimes\cdots\otimes\mathbf{R}(2^{0}\theta)$
(4)
and the phase gate
$\displaystyle\mathbf{R}(\theta)=\begin{bmatrix}1&0\\\
0&e^{j\theta}\end{bmatrix}.$ (5)
Here, phase advance represents integer addition and phase delay represents
subtraction. Following (3), IQFT yields only one state that represents the
original integer value of $k$.
The interaction between a binary variable and a coefficient can be represented
by a controlled qubit. Similarly, the interaction between binary variables can
be represented by controlled qubits on a register $\Ket{b}_{n}$. As
exemplified in Fig. 2, the constant term $+1$ corresponds to
$\mathbf{U}_{G}\left(\pi/4\right)$, the term $+1b_{0}$ corresponds to
controlled $\mathbf{U}_{G}\left(\pi/4\right)$, and the term $-2b_{1}b_{2}$
corresponds to controlled $\mathbf{U}_{G}\left(-2\pi/4\right)$. Likewise,
higher-order terms, such as third or fourth order, can be represented by
increasing the number of controlled qubits.
Figure 1: Quantum circuit corresponding to
$E(\mathbf{b})=1+b_{0}-2b_{1}b_{2}$.
(a) $L=0$. (b) $L=1$. (c) $L=2$.
Figure 2: Output probabilities of the circuit shown in Fig. 2
In the classic Grover search [4], an oracle operator $\mathbf{O}$ identifies
the states of interest and inverts the phases of these states. Only the
inverted states are amplified by the Grover operator. The operator
$\mathbf{A}_{y_{i}}$ above calculates the values $E(\mathbf{b})-y_{i}$ for all
$2^{n}$ states in parallel. Here, states that are better than the current
threshold $y_{i}$, i.e., states that satisfy $E(\mathbf{b})-y_{i}<0$, should
be marked to find the minimum solution. Since the calculated values are
represented by the two’s complement, we can identify the negative states by
focusing only on the beginning of the $m$ qubits, and $\mathbf{O}$ can be
constructed by applying the Z-gate only to that qubit.
Let the Grover diffusion operator be $\mathbf{D}$ of [4]
$\displaystyle D_{i,j}=\begin{cases}0&(i\neq j)\\\ 1&(i=j=0)\\\ -1&(i=j\neq
0)\end{cases}.$ (6)
The Grover operator is finally constructed by
$\mathbf{G}=\mathbf{A}_{y_{i}}\mathbf{D}\mathbf{A}_{y_{i}}^{\mathrm{H}}\mathbf{O}$,
and we evaluate $\mathbf{G}^{L}\mathbf{A}_{y_{i}}\Ket{0}_{n+m}$, which will
maximize the amplitudes of the states of interest. The ideal $L$ that
successfully maximizes the amplitude is given by [34]
$\displaystyle
L_{\mathrm{opt}}=\left\lfloor\frac{\pi}{4}\sqrt{\frac{N}{N_{\mathrm{s}}}}\right\rfloor,$
(7)
where $N$ denotes the search space size, $2^{n}$, and $N_{\mathrm{s}}$ denotes
the number of solutions. Let $\mathbf{b}$ be the bit sequence and $y$ be the
objective function value obtained by the evaluation.
From (7), the query complexity of GAS can be derived as $O(\sqrt{2^{n}})$ [8]
in the quantum domain (QD), which is the total number of Grover operators.
Since $N_{s}$, the number of states better than the current threshold, is
unknown in advance, $L$ is typically drawn from a uniform distribution ranging
from $0$ to a specific value that increases by a factor of $\lambda=8/7$ at
each iteration. GAS is terminated if the sum of the Grover operators is
greater than $22.5\sqrt{2^{n}}$, which is the same as the conventional DH
algorithm. In the Qiskit implementation, the number of times no improvement is
observed is also considered as one of the termination conditions. Overall, GAS
is summarized in Algorithm 1.
Algorithm 1 Conventional GAS designed for integer coefficients [8].
0: $E:\mathbb{B}^{n}\rightarrow\mathbb{Z},\lambda=8/7$
0: $\mathbf{b}$
1: Uniformly sample $\mathbf{b}_{0}\in\mathbb{B}^{n}$ and set
$y_{0}=E(\mathbf{b}_{0})$.
2: Set $k=1$ and $i=0$.
3: repeat
4: Randomly select the rotation count $L_{i}$ from the set $\\{0,1,...,\lceil
k-1\rceil$}.
5: Evaluate $\mathbf{G}^{L_{i}}\mathbf{A}_{y_{i}}\Ket{0}_{n+m}$, and obtain
$\mathbf{b}$ and $y$. {Grover search}
6: if $y<y_{i}$ then
7: $\mathbf{b}_{i+1}=\mathbf{b},y_{i+1}=y,$ and $k=1$. {Improvement found}
8: else
9: $\mathbf{b}_{i+1}=\mathbf{b}_{i},y_{i+1}=y_{i},$ and $k=\min{\\{\lambda
k,\sqrt{2^{n}}}\\}$. {No Improvement}
10: end if
11: $i=i+1$.
12: until a termination condition is met.
As a specific example, Fig. 2 shows a quantum circuit of GAS that tries to
minimize the objective function $E(\mathbf{b})=1+b_{0}-2b_{1}b_{2}$, where the
threshold of $y_{i}=0$ and $\mathbf{A}_{y_{i}=0}$ are considered for
simplicity. The upper $n=3$ qubits correspond to variables $\Ket{b_{0}}$,
$\Ket{b_{1}}$, and $\Ket{b_{2}}$, and the lower $m=3$ qubits $\Ket{z}_{3}$
encode the calculated value. The Hadamard gate at the beginning of
$\mathbf{A}_{y_{i}=0}$ initializes the qubits and creates an equal
superposition of all the possible states, $000000$ to $111111$. The black
circle in Fig. 2 indicates a control qubit. The unitary operator
$\mathbf{U}_{G}(\theta)$ is applied if all the associated control qubits are
$1$. Here, the control qubit is in a superposition state, and it creates a
quantum entanglement state, which plays a key role in GAS. IQFT is applied at
the last part of $\mathbf{A}_{y_{i}=0}$. After that, the Grover operator
$\mathbf{G}$ is applied $L$ times, and we measure the quantum state. Fig. 2
shows the probability that each state is measured, where the number of Grover
operators was varied from $L=0$ to $2$. The comma-separated text in this
figure shows $n$ and $m$ qubits, and the latter is converted to a decimal
number. As shown in Fig. 2, when $L=0$, $2^{3}=8$ different states were
observed with equal probability, and the corresponding values of the objective
function were correctly calculated, demonstrating the potential of quantum
computation. When $L=1$ and $2$, only the state of interest
$\mathbf{b}=[0~{}1~{}1]$, which yields $E(\mathbf{b})=-1<0$, was successfully
amplified by the Grover operator. In this manner, GAS amplifies the states
that are better than the current threshold and finds a binary solution that
minimizes the objective function.
### III-B Handling of Real-Valued Coefficients [8]
A polynomial may contain real-valued coefficients. To deal with real-valued
coefficients, Gilliam et al. proposed the following two methods [8].
#### III-B1 Integer Approximation
Multiplying the objective function by a positive constant does not affect the
minimization process. A real-valued coefficient can be approximated by
multiplying a large number and rounding down to an integer. Specifically, real
coefficients are approximated as fractions with a common denominator, the
denominator is multiplied to the objective function, and the numerators become
approximated integer coefficients. As can be inferred from (1), the drawback
is that the number of required qubits $m$ increases as the value range of the
objective function expands. If $m$ is kept small, this approximation becomes
less accurate.
#### III-B2 Direct Encoding
In this method, an integer $k$ in $\theta=2\pi k/2^{m}$ of (3) is replaced
with a real-valued coefficient $a\in\mathbb{R}$. Then, the output probability
indicates multiple integers, which is known as the Fejér distribution.
Specifically, the state
$\mathbf{U}_{\mathrm{Fej\acute{e}r}}(\theta)\Ket{0}_{m}$ after applying IQFT
to $\mathbf{U}_{G}(\theta)\mathbf{H}^{\otimes m}\Ket{0}_{m}$ is given by
[8]444This definition differs from [8], but is essentially identical.
$\displaystyle\mathbf{U}_{\mathrm{Fej\acute{e}r}}(\theta)\Ket{0}_{m}=\sum_{l=0}^{2^{m}-1}\left\langle\mathbf{g}(\theta),\mathbf{g}\left(2\pi
l/2^{m}\right)\right\rangle\Ket{l},$ (8)
where we have $\theta=2\pi a/2^{m}$ and
$\mathbf{g}(\theta)=[1,e^{j\theta},\cdots,e^{j(2^{m}-1)\theta}]/\sqrt{2^{m}}$.
The number of qubits $m$ must satisfy [8]
$\displaystyle-2^{m-1}\leq a<2^{m-1}.$ (9)
In this distribution, the probabilities of two integers close to a given real
number $a$ are greater than the other probabilities. For example, if $m=3$
qubits and $a=-2.5$, from (8), $-2$ and $-3$ are observed with equal
probability. If $a=-2.3$, $-2$ is observed more frequently than $-3$.
### III-C Proposed GAS for Real-Valued HUBO
As previously reviewed in Section III-B, in their innovative study [8],
Gilliam et al. proposed two methods for handling real-valued coefficients, but
did not specifically investigate how GAS behaves in the case of direct
encoding. In such a case, in our evaluation, GAS samples a wrong value of the
objective function, which obeys the Fejér distribution. For example, if the
objective function value is $-2.5$, we may observe an integer value less than
or equal to $-3$. A value lower than the actual value is updated as the
minimum and set as a new threshold $y_{i}$. Then, no states satisfy
$E(\mathbf{b})-y_{i}<0$, and one of all states is randomly sampled. As a
result, GAS will not be able to obtain an optimal solution.
Algorithm 2 Proposed GAS designed for real-valued coefficients.
0: $E:\mathbb{B}^{n}\rightarrow\mathbb{R},\lambda=8/7$
0: $\mathbf{b}$
1: Uniformly sample $\mathbf{b}_{0}\in\mathbb{B}^{n}$ and set
$y_{0}=E(\mathbf{b}_{0})$. {This step will be improved in Section IV-C}
2: Set $k=1$ and $i=0$.
3: repeat
4: Randomly select the rotation count $L_{i}$ from the set $\\{0,1,...,\lceil
k-1\rceil$}.
5: Evaluate $\mathbf{G}^{L_{i}}\mathbf{A}_{y_{i}}\Ket{0}_{n+m}$, and obtain
$\mathbf{b}$.
6: Evaluate $y=E(\mathbf{b})$ in CD. {This is the additional step}
7: if $y<y_{i}$ then
8: $\mathbf{b}_{i+1}=\mathbf{b},y_{i+1}=y,$ and $k=1$.
9: else
10: $\mathbf{b}_{i+1}=\mathbf{b}_{i},y_{i+1}=y_{i},$ and $k=\min{\\{\lambda
k,\sqrt{2^{n}}}\\}$.
11: end if
12: $i=i+1$.
13: until a termination condition is met.
A possible solution here is that we ignore $y$ evaluated in QD. Instead, we
use $\mathbf{b}$ returned by GAS and calculate a correct objective function
value $y=E(\mathbf{b})$ in CD. Since the quantum circuit using the direct
encoding amplifies the states of interest with high probability, with this
simple modification, GAS obtains an optimal solution correctly. Overall, the
above procedure is summarized in Algorithm 2.
We have two major drawbacks. First, Algorithm 2 increases query complexity in
CD, although the asymptotic order remains the same. Second, the probability
amplification may not be sufficient, which is illustrated in Fig. 4.
Figure 3: Quantum circuit corresponding to
$E(\mathbf{b})=1+b_{0}-1.8b_{1}b_{2}b_{3}$.
(a) $L=0$. (b) $L=1$. (c) $L=2$. (d) $L=3$.
Figure 4: Output probabilities of the circuit shown in Fig. 4, where only the
top 16 states are shown.
As a specific example, Fig. 4 shows a quantum circuit corresponding to the
objective function $E(\mathbf{b})=1+b_{0}-1.8b_{1}b_{2}b_{3}$, where we set
$n=4$ and $m=3$. Since we used the direct encoding method,
$-1.8b_{1}b_{2}b_{3}$ was represented as $\mathbf{U}_{G}(-1.8\pi/4)$, and it
was associated with three qubits $\Ket{b_{1}}$, $\Ket{b_{2}}$, and
$\Ket{b_{3}}$. Additionally, Fig. 4 shows the output probabilities of Fig. 4,
where only the top 16 states are shown for the sake of readability. As given
in (8), the direct encoding method may not result in a unique integer. For
example, the states $(0111,-1)$ and $(0111,-2)$ had positive probabilities in
Fig. 4. The state of interest here is $\mathbf{b}=[0~{}1~{}1~{}1]$ and
$E(\mathbf{b})=1-1.8=-0.8<0$. That is, before amplitude amplification, at
$L=0$, integers close to the real value are observed, and after amplification,
at $L>0$, the negative states are observed with high probabilities. As shown
in Fig. 4(d), the states $(0111,-1)$ and $(0111,-2)$ were amplified as the
number of Grover operators $L$ increased, while the wrong state $(0111,-2)$
was observed with a lower probability than $(0111,-1)$. Another wrong state
$(1111,-1)$ was also observed with a low probability. This is the reason why
the correction of the objective function value is required for real-valued
GAS, as summarized in Algorithm 2.
### III-D Evaluation Metrics
In the literature, a quantum circuit has been evaluated by the numbers of
qubits and gates and its depth, while a quantum algorithm has been evaluated
by query complexity.
#### III-D1 Numbers of Qubits and Gates and Depth
The size of the quantum circuit determines its feasibility. As the numbers of
qubits and gates in a quantum circuit increase, more advanced quantum
computation becomes possible. At the same time, however, it becomes more
susceptible to noise and more difficult to implement in hardware. In our
evaluations, the number of required qubits is represented as $n+m$, and the
number of quantum gates is derived as a function with respect to $n$ and $m$.
#### III-D2 Query Complexity [13]
To investigate query complexity, we count how many times the objective
function is queried. Specifically, the query complexity in the classical
domain (CD) is the number of times the objective function is evaluated, i.e.,
$i$ in Algorithm 1. By contrast, the query complexity in the quantum domain
(QD) is the number of times the Grover operator $\mathbf{G}$ is applied, i.e.,
$L_{0}+L_{1}+\cdots+L_{i}$ in Algorithm 1. The definitions of query
complexities in CD and QD are the same as those used in [13].
## IV Quantum Speedup for MIMO MLD
Conventional studies on quantum-assisted wireless communications have not
considered a specific construction method of the quantum circuit. In many
cases, the circuit to calculate an objective function has been idealized as a
black-box quantum oracle. In this section, we formulate the MIMO MLD as a new
real-valued HUBO problem, which can be represented by a quantum circuit, as
described in Section III. We also analyze the probability distribution of the
objective value for enabling further speedup.
### IV-A System Model
Figure 5: System model for MIMO with $N_{\mathrm{t}}$ transmit and
$N_{\mathrm{r}}$ receiver antennas.
We consider a MIMO communication scenario with $N_{\mathrm{t}}$ transmit
antennas and $N_{\mathrm{r}}$ receive antennas, as illustrated in Fig. 5. The
input $n$-bit sequence
$\mathbf{b}=[b_{0}~{}b_{1}~{}\cdots~{}b_{n-1}]\in\mathbb{B}^{n}$ is mapped to
a symbol vector
$\mathbf{s}=[s_{0}~{}s_{1}~{}\cdots~{}s_{N_{\mathrm{t}}-1}]\in{\mathbb{C}}^{N_{\mathrm{t}}\times
1}$, where $s_{t}$ for $0\leq t\leq N_{\mathrm{t}}-1$ denotes a Gray-coded
data symbol specified in 5G NR [35]. We represent this bit-to-symbol mapper as
$\mathbf{s}=M(\mathbf{b})=M(b_{0},\cdots,b_{n-1})$, which will be defined in
detail in Section IV-B. The baseband received symbols
$\mathbf{r}\in\mathbb{C}^{N_{\mathrm{r}}\times 1}$ is given by
$\displaystyle\mathbf{r}=\frac{1}{\sqrt{N_{\mathrm{t}}}}\mathbf{H}_{\mathrm{c}}\mathbf{s}+\sigma\mathbf{v},$
(10)
where $\mathbf{H}_{\mathrm{c}}\in{\mathbb{C}}^{N_{\mathrm{r}}\times
N_{\mathrm{t}}}$ denotes the channel matrix, and
$\mathbf{v}\in{\mathbb{C}}^{N_{\mathrm{r}}\times 1}$ denotes the additive
white Gaussian noise. Here, we assume the narrowband Rayleigh flat fading.
That is, each element of $\mathbf{H}_{\mathrm{c}}$, $h_{ut}$, and each element
of $\mathbf{v}$, $v_{u}$, follow the standard complex Gaussian distribution
$\mathcal{CN}(0,1)$ for $0\leq u\leq N_{\mathrm{r}}-1$ and $0\leq t\leq
N_{\mathrm{t}}-1$. The SNR is defined as $\gamma=1/\sigma^{2}$ because the
symbol vector has the power constraint
$\mathrm{E}\left[\|\mathbf{s}/\sqrt{N_{\mathrm{t}}}\|_{\mathrm{F}}^{2}\right]=\mathrm{E}\left[\sum_{t=0}^{N_{\mathrm{t}}-1}|s_{t}|^{2}/N_{\mathrm{t}}\right]=1$.
The constellation size, or modulation order, is denoted by $L_{\mathrm{c}}$,
and the transmission rate is calculated by
$\displaystyle
n=N_{\mathrm{t}}\log_{2}(L_{\mathrm{c}})~{}~{}\text{[bit/symbol]}.$ (11)
Corresponding to (10), the ideal MLD is performed as
$\displaystyle\hat{b}_{0},\cdots,\hat{b}_{n-1}=\arg\underset{b_{0},\cdots,b_{n-1}}{\min}E(b_{0},\cdots,b_{n-1}),$
(12)
where we have the objective function
$\displaystyle
E(b_{0},\cdots,b_{n-1})=\left\|\mathbf{r}-\frac{1}{\sqrt{N_{\mathrm{t}}}}\mathbf{H}_{\mathrm{c}}M(b_{0},\cdots,b_{n-1})\right\|^{2}_{\mathrm{F}}.$
(13)
From (12), the exhaustive search using a classical computer requires the
computational time complexity of $O(2^{n})$, which is equivalent to the query
complexity in CD. Both complexities increase exponentially with the
transmission rate $n$.
To mitigate the exponential complexity, a number of low-complexity detectors
have been proposed in the literature. The classic ZF detector uses the pseudo-
inverse matrix of
$\displaystyle\mathbf{W}_{\mathrm{ZF}}=\begin{cases}(\mathbf{H}_{\mathrm{c}}^{\mathrm{H}}\mathbf{H}_{\mathrm{c}})^{-1}\mathbf{H}_{\mathrm{c}}^{\mathrm{H}}&(N_{\mathrm{t}}\leq
N_{\mathrm{r}})\\\
\mathbf{H}_{\mathrm{c}}^{\mathrm{H}}(\mathbf{H}_{\mathrm{c}}\mathbf{H}_{\mathrm{c}}^{\mathrm{H}})^{-1}&(N_{\mathrm{t}}>N_{\mathrm{r}})\end{cases}$
(14)
and enables independent detection of data symbols as
$\displaystyle\hat{b}_{0},\cdots,\hat{b}_{n-1}=M^{-1}(\mathbf{W}_{\mathrm{ZF}}\mathbf{r}),$
(15)
where $M^{-1}\left(\cdot\right)$ denotes the hard-decision symbol-to-bit
demapper. Similarly, the MMSE detector uses
$\displaystyle\mathbf{W}_{\mathrm{MMSE}}=\begin{cases}(\mathbf{H}_{\mathrm{c}}^{\mathrm{H}}\mathbf{H}_{\mathrm{c}}+\sigma^{2}\mathbf{I})^{-1}\mathbf{H}_{\mathrm{c}}^{\mathrm{H}}&(N_{\mathrm{t}}\leq
N_{\mathrm{r}})\\\
\mathbf{H}_{\mathrm{c}}^{\mathrm{H}}(\mathbf{H}_{\mathrm{c}}\mathbf{H}_{\mathrm{c}}^{\mathrm{H}}+\sigma^{2}\mathbf{I})^{-1}&(N_{\mathrm{t}}>N_{\mathrm{r}})\end{cases}$
(16)
and obtains
$\displaystyle\hat{b}_{0},\cdots,\hat{b}_{n-1}=M^{-1}(\mathbf{W}_{\mathrm{MMSE}}\mathbf{r}).$
(17)
An MMSE-based interference cancelation method has been adopted in typical
wireless standards such as 5G NR. The performance of a ZF or MMSE detector is
worse than that of MLD. In general, low-complexity detectors improve
complexity at the sacrifice of performance.
The above system model and detectors are typical and common in the field of
wireless communications. Since we consider a general MIMO system, the
simulation results given in this paper are the same as those for a
multicarrier scenario without inter-subcarrier interference or an uplink
multi-user scenario in which $N_{\mathrm{t}}$ single-antenna user terminals
transmit their symbols and these symbols are received simultaneously at a base
station equipped with $N_{\mathrm{r}}$ antennas.
### IV-B Proposed Method to Transform MLD into HUBO
As described in Section III, the proposed GAS is capable of solving a real-
valued HUBO problem. We transform the objective function of MIMO MLD (13) into
a HUBO problem. Specifically, we use the relationship between transmission
bits and data symbols, which is specified in the 5G NR standard [35]. The
input $n$-bit sequence is denoted by
$\mathbf{b}=[b_{0}~{}b_{1}~{}\cdots~{}b_{n-1}]\in\mathbb{B}^{n}$ and the
symbol vector is denoted by
$\mathbf{s}=[s_{0}~{}s_{1}~{}\cdots~{}s_{N_{\mathrm{t}}-1}]\in\mathbb{C}^{N_{\mathrm{t}}}$.
Then, BPSK symbols $\mathbf{s}=M_{2}(\mathbf{b})$ are generated by [35]
$\displaystyle s_{t}=\frac{1}{\sqrt{2}}[(1-2b_{t})+j(1-2b_{t})]$ (18)
and QPSK symbols $\mathbf{s}=M_{4}(\mathbf{b})$ are generated by [35]
$\displaystyle s_{t}=\frac{1}{\sqrt{2}}[(1-2b_{2t})+j(1-2b_{2t+1})].$ (19)
Furthermore, 16-QAM symbols $\mathbf{s}=M_{16}(\mathbf{b})$ are generated by
[35]
$\displaystyle s_{t}=$
$\displaystyle\frac{1}{\sqrt{10}}(1-2b_{4t+0})[2-(1-2b_{4t+2})]$
$\displaystyle+$
$\displaystyle\frac{j}{\sqrt{10}}(1-2b_{4t+1})[2-(1-2b_{4t+3})]$ (20)
and 64-QAM symbols $\mathbf{s}=M_{64}(\mathbf{b})$ are generated by [35]
$\displaystyle s_{t}=$
$\displaystyle\frac{1}{\sqrt{42}}(1-2b_{6t+0})[4-(1-2b_{6t+2})[2-(1-2b_{6t+4})]]$
$\displaystyle+$
$\displaystyle\frac{j}{\sqrt{42}}(1-2b_{6t+1})[4-(1-2b_{6t+3})[2-(1-2b_{6t+5})]].$
(21)
A similar relationship for 256-QAM is defined in [35] and its extension for
higher modulation orders can be defined easily.
Figure 6: Constellation for Gray-coded data symbols specified in the 5G NR
standard [35]
Overall, Fig. 6 shows the Gray-coded data symbols defined by (18), (19), and
(IV-B).
Our proposed objective function is obtained by substituting $M(\cdot)$ in (13)
with (18), (19), (IV-B), or (IV-B), which contains $n$ number of binary
variables $b_{0},\cdots,b_{n-1}$. In the cases of BPSK and QPSK, our objective
function results in a quadratic form since both symbols are represented by a
linear relationship and the MLD (12) contains the square of the Frobenius
norm. In the case of 16-QAM, the objective function results in a quartic form
since the symbols are represented by a quadratic relationship. Similarly, the
objective function results in a sextic form in the 64-QAM case.
The use of data symbols specified in 5G NR is not straightforward since the
objective function inevitably contains higher-order terms if the modulation
order is 16 or higher. Thus, in this form, the conventional QA requires a
transformation from HUBO to QUBO, and this transformation involves an increase
in binary variables, making the problem more difficult. Our proposed approach
is only possible with the aid of the real-valued support of GAS. Because of
GAS, the query complexity is expected to be reduced from $O(2^{n})$ to
$O(\sqrt{2^{n}})$.
The structure of the proposed objective function depends only on the number of
transmit antennas, $N_{\mathrm{t}}$, the number of receive antennas,
$N_{\mathrm{r}}$, and the modulation order, $L_{\mathrm{c}}$. The coefficients
in the objective function change depending on the channel matrix
$\mathbf{H}_{\mathrm{c}}$. The calculation cost of coefficients determines the
complexity of classical processing required before executing GAS, which
relates to the latency of the algorithm. If we approximate the computational
complexity as the number of real-valued multiplications, the largest burden is
the product of channel coefficients, such as $h_{00}h_{01}^{*}$. The total
number of multiplications is calculated as
$4N_{\mathrm{r}}N_{\mathrm{t}}(N_{\mathrm{t}}-1)/2=O(N_{\mathrm{r}}N_{\mathrm{t}}^{2})$,
which is sufficiently small with respect to the detection complexity.
##### Example (QPSK)
As a specific example, we consider the QPSK case (19) with
$N_{\mathrm{t}}=N_{\mathrm{r}}=2$. The objective function of (13) can be
transformed into
$\displaystyle E(b_{0},b_{1},b_{2},b_{3})$ $\displaystyle=$ $\displaystyle
2\sum_{u=0}^{1}{\sum_{t=0}^{1}({\mathrm{Re}(h_{ut}r_{u}^{*})}b_{2t}-{\mathrm{Im}(h_{ut}r_{u}^{*})}b_{2t+1}})$
$\displaystyle+$ $\displaystyle
2a_{1}(b_{0}b_{2}+b_{1}b_{3})+2a_{2}(b_{0}b_{3}-b_{1}b_{2})$ $\displaystyle-$
$\displaystyle(a_{1}+a_{2})(b_{0}+b_{3})-(a_{1}-a_{2})(b_{1}+b_{2}),$ (22)
where we have
$a_{1}=\mathrm{Re}(h_{00}h_{01}^{*})+\mathrm{Re}(h_{10}h_{11}^{*})$ and
$a_{2}=\mathrm{Im}(h_{00}h_{01}^{*})+\mathrm{Im}(h_{10}h_{11}^{*})$. This
function (IV-B) is in a quadratic form.
##### Example (16-QAM)
Figure 7: Quantum circuit corresponding to objective function of 16-QAM
detection.
Additionally, Fig. 7 exemplifies a specific quantum circuit for the 16-QAM
case with $N_{\mathrm{t}}=N_{\mathrm{r}}=2$, where we have $n=8$ qubits for
binary variables, $m=5$ qubits for real-valued encoding, random channel
coefficients
$\displaystyle\mathbf{H}_{\mathrm{c}}=[[$ $\displaystyle
0.748510757437062-0.014877263039446401j,$ $\displaystyle
1.3215983896521515+0.06298233870206783j],$ $\displaystyle[$ $\displaystyle
0.6371630706424066-0.14262155021296025j,$ $\displaystyle-$ $\displaystyle
0.3888005272494009-0.15170387681055802j]],$ (23)
and the original information bits of $00110101$. As shown in Fig. 7, the
objective function results in a quartic form:
$E(\mathbf{b})=1.22b_{0}b_{2}b_{4}b_{6}+0.61b_{0}b_{2}b_{4}+\cdots$ using (13)
and (IV-B). As an example, the coefficient of $b_{0}b_{2}b_{4}b_{6}$ is
calculated as
$\frac{1}{2}\cdot\frac{4}{\sqrt{10}}\cdot\frac{4}{\sqrt{10}}(h_{00}h_{01}^{*}+h_{00}^{*}h_{01}+h_{10}h_{11}^{*}+h_{10}^{*}h_{11})=1.22$,
which is rounded down to the second decimal place for simple illustration.
### IV-C Proposed Threshold for Further Speedup
GAS obtains a global minimum solution by updating the threshold value $y_{i}$
and amplifying the probability amplitudes corresponding to values smaller than
the threshold. The query complexity can be reduced by setting the initial
threshold in a manner different from that in classic random sampling, although
the asymptotic performance may not change. In this section, we derive the
probability distribution of the objective function value and use it to
determine a strict threshold, which enables further speedup.
If the information bits in (13) are estimated correctly, the minimum value of
(13) is the Frobenius norm of additive noise
$\mathbf{v}\in\mathbb{C}^{N_{\mathrm{r}}\times 1}$ as follows:
$\displaystyle
E_{\mathrm{min}}=\underbrace{\sigma^{2}}_{\text{known}}\underbrace{\sum_{u=0}^{N_{\mathrm{r}}-1}|v_{u}|^{2}}_{\text{unknown}}.$
(24)
That is, $E_{\mathrm{min}}$ depends on the noise variance $\sigma^{2}$, which
is typically known at the receiver, and instantaneous noise $v_{u}$, which is
unknown in any case. Since the noise is assumed to follow the complex Gaussian
distribution, the magnitude of the norm follows the Rayleigh distribution, and
its square follows the exponential distribution. As a result,
$E_{\mathrm{min}}$ in (24) follows the Erlang distribution, whose probability
density function is
$\displaystyle
f(y)=\frac{\gamma^{N_{\mathrm{r}}}y^{N_{\mathrm{r}}-1}e^{-\gamma
y}}{(N_{\mathrm{r}}-1)!},$ (25)
where we have SNR $\gamma=1/\sigma^{2}$. The corresponding cumulative
distribution function (CDF) is given by
$\displaystyle F(y)=\mathrm{Pr}[Y\leq y]=1-e^{-\gamma
y}\sum_{u=0}^{N_{\mathrm{r}}-1}\frac{(\gamma y)^{u}}{u!}.$ (26)
Figure 8: Cumulative distribution of the minimum of objective function values
(27).
As an example, if we consider the case with $N_{\mathrm{r}}=2$, the CDF is
calculated as
$\displaystyle F(y)=\mathrm{Pr}[Y\leq y]=1-e^{-\gamma y}(1+\gamma y)$ (27)
from (26). Fig. 8 exemplifies CDF (27) when SNR is varied as $\gamma=5$ to
$20$ dB. Additionally, the CDF of the simulated objective function values with
$N_{\mathrm{t}}=2$ and QPSK is also plotted. As shown in Fig. 8, if the SNR is
sufficient, such as above 10 dB, the theoretical and simulated values are
identical. Thus, it is possible to know in advance that the minimum value to
be calculated is below a certain threshold, which can be determined with a
very high degree of certainty. Given an SNR $\gamma$, theoretical values of
(26) can be used to determine a strict threshold.
From (26), the probability that the threshold $y$ is below the minimum value
is
$\displaystyle\mathrm{Pr}[Y>y]=e^{-\gamma y}(1+\gamma y).$ (28)
Let $\tilde{y}$ be the threshold to be determined and $P$ be a small constant
probability, such as $P=10^{-3}$ and $10^{-4}$. Replacing $y$ and
$\mathrm{Pr}[Y>y]$ in (28) with $\tilde{y}$ and $P$ yields
$\displaystyle P=e^{-\gamma\tilde{y}}(1+\gamma\tilde{y}).$ (29)
Dividing both sides by $-e$ gives
$\displaystyle-\frac{P}{e}=-(1+\gamma\tilde{y})e^{-(1+\gamma\tilde{y})}.$ (30)
Then, using the Lambert $W$ function, we obtain
$\displaystyle
W_{-1}\left(-\frac{P}{e}\right)=-(1+\gamma\tilde{y})=-\left(1+\frac{\tilde{y}}{\sigma^{2}}\right),$
(31)
where $W_{-1}(\cdot)$ denotes the lower branch of the Lambert $W$ function,
i.e., $W_{-1}(\cdot)\leq-1$ and $W_{-1}(-1/e)=-1$. Finally, the threshold to
be determined is
$\displaystyle\tilde{y}=\underbrace{\sigma^{2}}_{\text{known}}\underbrace{\nu}_{\text{known}},$
(32)
which is similar to (24), and
$\displaystyle\nu=-1-W_{-1}\left(-\frac{P}{e}\right).$ (33)
Here, $\nu$ is a positive constant and is calculated once before running our
proposed algorithm. For example, we have $\nu=9.23$ if $P=10^{-3}$ and
$\nu=11.8$ if $P=10^{-4}$.
For further speedup, we opt to use the output of the MMSE detector (16). In
[13], Botsinis et al. proposed the MMSE-based threshold of
$\displaystyle\bar{y}=E(\bar{\mathbf{b}}_{0}),$ (34)
where we have a rough estimate
$\bar{\mathbf{b}}_{0}=M^{-1}(\mathbf{W}_{\mathrm{MMSE}}\mathbf{r})$. Our
proposed threshold $\tilde{y}$, which is simpler than $\bar{y}$, can be used
together with $\bar{y}$. Specifically, we calculate both $\tilde{y}$ and
$\bar{y}$ at the beginning of Algorithm 2, set the initial threshold as the
smaller of the two, and initialize the first solution with
$\bar{\mathbf{b}}_{0}$. Let $\mathbf{b}_{0}$ be a random $n$-bit sequence. The
initial threshold used for the proposed Algorithm 2 can be summarized as
follows:
$\displaystyle y_{0}=\begin{cases}E(\mathbf{b}_{0})&\text{(Original GAS
\cite[cite]{[\@@bibref{Number}{gilliam2021grover}{}{}]})}\\\
\bar{y}&\text{(MMSE-based threshold
\cite[cite]{[\@@bibref{Number}{botsinis2014fixedcomplexity}{}{}]})}\\\
\tilde{y}&\text{(Proposed threshold)}\\\
\min(\bar{y},\tilde{y})&\text{(Proposed combination)}\\\ \end{cases}.$ (35)
One problem with the proposed threshold $\tilde{y}$ and the combination
$\min(\bar{y},\tilde{y})$ is that they may become smaller than the actual
minimum. In this case, since there are no states of interest, GAS will be in a
state where the solution $\mathbf{b}_{i}$ in Algorithm 2 is not updated. The
probability of this undesirable event occurring is $P$, i.e.,
$\displaystyle\mathrm{Pr}[\tilde{y}<E_{\mathrm{min}}]=\mathrm{Pr}[\min(\bar{y},\tilde{y})<E_{\mathrm{min}}]=P,$
(36)
because we have the relationship $\mathrm{Pr}[\bar{y}<E_{\mathrm{min}}]=0$.
Then, it can be expected that the proposed threshold $\tilde{y}$ may degrade
the bit error ratio (BER) significantly if $P$ is inappropriate. Specifically,
the BER of the proposed threshold $\tilde{y}$ is approximated by
$\displaystyle P\cdot 0.5+(1-P)\cdot\mathrm{BER}_{\mathrm{MLD}},$ (37)
where $\mathrm{BER}_{\mathrm{MLD}}$ is the BER of MLD. In the proposed
combination method, we initialize the first solution with the MMSE output
$\bar{\mathbf{b}}_{0}$. Since the initial threshold $\min(\bar{y},\tilde{y})$
becomes smaller than the actual minimum with probability $P$, the BER of the
proposed combination method is approximated by
$\displaystyle
P\cdot\mathrm{BER}_{\mathrm{\mathrm{MMSE}}}+(1-P)\cdot\mathrm{BER}_{\mathrm{MLD}},$
(38)
where $\mathrm{BER}_{\mathrm{\mathrm{MMSE}}}$ is the BER of the MMSE detector.
Both (37) and (38) indicate that the design of $P$ exerts no significant
effect as long as it is smaller than $\mathrm{BER}_{\mathrm{MLD}}$, which can
be calculated exactly in a closed form in advance. For our performance
analysis, the effect of $P$ is shown in Fig. 13.
## V Performance Analysis
In this section, we analyze the number of quantum gates required by GAS, which
is represented as a function of the numbers of qubits $n$ and $m$. Then, we
investigate the performance of the proposed formulation in terms of BER and
evaluate the proposed algorithm in terms of the rate of convergence. Here,
both integer approximation and direct encoding are considered. Finally, we
evaluate the effects of the proposed threshold.
### V-A Algebraic Analysis of the Number of Quantum Gates
A quantum circuit for GAS is composed of $\mathbf{H}$, $\mathbf{X}$,
$\mathbf{Z}$, phase, controlled-phase gates, and the IQFT. In particular, the
state preparation operator $\mathbf{A}_{y_{i}}$ is the most complex part
corresponding to the objective function and is dynamically configured in
accordance with the threshold $y_{i}$. In the quantum circuit
$\mathbf{A}_{y_{i}}$, the number of controlled-phase gates depends on the
number of terms in the objective function. We therefore derive the number of
terms in the objective function that correspond to each order in an algebraic
manner. Ignoring the power scaling factor, the objective function of MIMO MLD
(13) is transformed into
$\displaystyle\sum_{u=0}^{N_{\mathrm{r}}-1}|r_{u}-h_{u0}s_{0}-h_{u1}s_{1}-\cdots-
h_{u(N_{\mathrm{t}}-1)}s_{N_{\mathrm{t}}-1}|^{2}$ $\displaystyle=$
$\displaystyle\sum_{u=0}^{N_{\mathrm{r}}-1}(r_{u}-h_{u0}s_{0}-h_{u1}s_{1}-\cdots-
h_{u(N_{\mathrm{t}}-1)}s_{N_{\mathrm{t}}-1})$
$\displaystyle\cdot{(r_{u}-h_{u0}s_{0}-h_{u1}s_{1}-\cdots-
h_{u(N_{\mathrm{t}}-1)}s_{N_{\mathrm{t}}-1})}^{*}.$ (39)
Here, we focus on three types of terms: first-order terms such as
$-r^{*}_{0}h_{00}s_{0}$ and $-r_{0}h^{*}_{00}s^{*}_{0}$, squares of the same
symbol such as $|h_{00}|^{2}|s_{0}|^{2}$ and $|h_{01}|^{2}|s_{1}|^{2}$, and
products of two symbols such as $h_{00}h^{*}_{10}s_{0}s^{*}_{1}$ and
$h^{*}_{00}h_{10}s^{*}_{0}{s}_{1}$.
For example, in the relatively simple QPSK case, squares of the same symbol
result in constant terms because of (19). First-order terms directly result in
first-order terms with respect to binary variables. Products between two
symbols result in products of binary variables. If $N_{\mathrm{t}}=2$ and
$N_{\mathrm{r}}=2$, four second-order terms appear:
$b_{0}b_{2},b_{1}b_{3},b_{0}b_{3},$ and $b_{1}b_{2}$. The number of
corresponding terms is equal to the combination of two choices from
$N_{\mathrm{t}}$ antennas, e.g.,
$\displaystyle{N_{\mathrm{t}}\choose
2}=\frac{N_{\mathrm{t}}(N_{\mathrm{t}}-1)}{2}=\frac{n(n-2)}{8},$ (40)
where we have the relationship
$n=N_{\mathrm{t}}\cdot\log_{2}(L_{\mathrm{c}})=2N_{\mathrm{t}}$. In total, the
number of second-order terms is calculated as $4\cdot n(n-2)/8=n(n-2)/2$.
TABLE II: Number of quantum gates required for $\mathbf{A}_{y_{i}}$ ($n$-bit
transmission with $m$-bit accuracy)
Gate | BPSK | | QPSK | | 16-QAM | | 64-QAM |
---|---|---|---|---|---|---|---|---
H | $n+m$ | $=O(n+m)$ | $n+m$ | $=O(n+m)$ | $n+m$ | $=O(n+m)$ | $n+m$ | $=O(n+m)$
R | $m$ | $=O(m)$ | $m$ | $=O(m)$ | $m$ | $=O(m)$ | $m$ | $=O(m)$
1-CR | $nm$ | $=O(nm)$ | $nm$ | $=O(nm)$ | $nm$ | $=O(nm)$ | $nm$ | $=O(nm)$
2-CR | ${n(n-1)m}/2$ | $=O(n^{2}m)$ | ${n(n-2)m}/2$ | $=O(n^{2}m)$ | ${n(n-3)m}/2$ | $=O(n^{2}m)$ | ${n(n-4)m}/2$ | $=O(n^{2}m)$
3-CR | $0$ | | $0$ | | ${n(n-4)m}/2$ | $=O(n^{2}m)$ | $n(n-6)m+nm/3$ | $=O(n^{2}m)$
4-CR | $0$ | | $0$ | | $n(n-4)m/8$ | $=O(n^{2}m)$ | $5n(n-6)m/6$ | $=O(n^{2}m)$
5-CR | $0$ | | $0$ | | $0$ | | $n(n-6)m/3$ | $=O(n^{2}m)$
6-CR | $0$ | | $0$ | | $0$ | | $n(n-6)m/18$ | $=O(n^{2}m)$
IQFT | $1$ | | $1$ | | $1$ | | $1$ |
Extending the QPSK case, we counted the number of terms in the objective
function for each modulation order and derived the number of quantum gates
required by GAS. Table II summarizes the derived results, where the quantum
gates were categorized by type. As given in Table II, the number of
controlled-phase gates mainly depends on the number of binary variables $n$.
Here, 1-CR represents the controlled-phase gate, and 2-CR, 3-CR, $\cdots$
represent multi-controlled-phase gates. Since we have the relationship
$n=N_{\mathrm{t}}\cdot\log_{2}(L_{\mathrm{c}})$, the quantum circuit becomes
increasingly complex depending on the square of the number of antennas
$N_{\mathrm{t}}$ and the modulation order $L_{\mathrm{c}}$.
We analyze the number of quantum gates in the entire circuit
$\mathbf{G}^{L_{i}}\mathbf{A}_{y_{i}}\Ket{0}_{n+m}$, where we have
$\mathbf{G}=\mathbf{A}_{y_{i}}\mathbf{D}\mathbf{A}_{y_{i}}^{\mathrm{H}}\mathbf{O}$.
In each iteration, the Grover operator is applied $L_{i}$ times, where $L_{i}$
is a uniform random number. $\mathbf{O}$ is composed of a single $\mathbf{Z}$
gate and $\mathbf{D}$ is the Grover diffusion operator, each of which is
repeated $L_{i}$ times. The other part contains $(2L_{i}+1)(n+m)$ $\mathbf{H}$
gates, $(2L_{i}+1)m$ phase gates, $(2L_{i}+1)c$ controlled-phase gates, and
$(2L_{i}+1)$ IQFT, where $c$ is the number of controlled-phase gates given in
Table II. In summary, $N_{\mathrm{t}}$ and $L_{\mathrm{c}}$ affect the number
of gates by the power of two, while $m$ and $L_{i}$ affect it in direct
proportion.
Real quantum computers rely on decomposed elementary gates: single unitary
gates and controlled NOT gates [36]. Specifically, a phase gate with $c$
control qubits is decomposed into $O(c)$ elementary gates. Thus, according to
Table II, the number of elementary gates required for controlled-phase gates
is $O(cn^{2}m)$ in total. Additionally, IQFT requires $O(m^{2})$ elementary
gates [1]. Here, the only parameters that can be designed are $n$ and $m$.
With the aid of our efficient HUBO formulation, the number of qubits $n$
cannot be further reduced, since the search space size of MIMO ML detection is
$2^{n}$. Later, we investigate whether our real-valued GAS can reduce $m$.
### V-B Effects of Integer Approximation
Figure 9: BER for the QPSK case with $N_{\mathrm{t}}=N_{\mathrm{r}}=2$.
First, Fig. 9 shows BER of the classic MLD and the proposed formulations that
consider the integer approximation with different accuracies. Specifically,
the real values were multiplied by 1, 3, 10 or 20, and approximated by
rounding them to the nearest integers. As references, the BER curves of ZF,
MMSE, and the real-valued formulation were also plotted. To analyze the
effects of approximation accuracy, BER values were calculated using the state-
of-the-art optimization solver, IBM CPLEX, instead of quantum simulations. As
shown in Fig. 9, BER performance varied significantly depending on the
accuracy of the conventional integer approximation. High approximation
accuracy leads to large integers, resulting in an increase in the number of
qubits $m$. In contrast, the proposed real-valued formulation achieved the
same performance as the classic MLD. This observation indicates that the
proposed real-valued GAS algorithm must be invoked to solve the MIMO MLD
problem on a quantum computer.
(a) QPSK ($3$x approximation, $n=4$, $m=6$ qubits). (b) 16-QAM ($14$x
approximation, $n=8$, $m=8$ qubits).
Figure 10: Average objective function values with the integer approximation
and $N_{\mathrm{t}}=N_{\mathrm{r}}=2$.
Next, Fig. 10 shows the average objective function values when increasing the
number of iterations, where iterations in both CD and QD were considered. We
assumed a sufficiently high SNR and the fixed $2\times 2$ channel matrix given
in (23).555Note that we observed the same trend for different channel
coefficients and SNRs. We used the original GAS with a random initial
threshold and terminated the simulation if the objective function value
remained the same more than 20 times in CD. In Fig. 10(a), real values were
multiplied by 3 and rounded down to integers, and in Fig. 10(b), real values
were multiplied by 14 and were approximated. The number of qubits $m$ required
for encoding the value $E(\mathbf{b})-y_{i}$ was set to an integer sufficient
not to overflow, i.e., $m=6$ in Fig. 10(a) and $m=8$ in Fig. 10(b). Note again
that the integer approximation requires more qubits to encode the value.
Because quantum simulations with $n+m=16$ qubits are time-consuming, we fixed
the input bits to $00110101$ in Fig. 10(b), while the bits were generated
randomly in Fig. 10(a). For a clear illustration, we added a constant value to
the objective function so that $E_{\mathrm{min}}=0$. It was observed in Fig.
10(a) that the query complexities of GAS in CD and QD were almost the same as
in the exhaustive search of MLD. By contrast, in Fig. 10(b), GAS exhibited
better query complexities in both CD and QD than did MLD. That is, the quantum
advantage improved as the problem size increased.
### V-C Effects of Direct Encoding
(a) QPSK ($n=4$, $m=5$ qubits). (b) 16-QAM ($n=8$, $m=5$ qubits).
Figure 11: Average objective function values with direct encoding and
$N_{\mathrm{t}}=N_{\mathrm{r}}=2$.
Similar to Fig. 10, Fig. 11 shows the average objective function values when
increasing the number of iterations in CD and QD, where we used direct
encoding. The simulation parameters were the same as those used in Fig. 10
except for the real-valued expression and the number of required qubits $m$.
Specifically, the number of qubits $m=6$ in Fig. 10(a) was reduced to $m=5$ in
Fig. 11(a). Similarly, the number of qubits was reduced from $m=8$ to $m=5$ in
Fig. 11(b). As shown in Fig. 11, the same trend as in Fig. 10 was observed.
The important aspect here is that almost the same query complexities were
achieved despite the reduction in the number of required qubits $m$. Hence,
our proposed real-valued GAS is capable of reducing the size of quantum
circuits while maintaining a good performance.
Depending on the channel coefficients and noise, the integer approximation
requires a different number of qubits. Since both follow the standard Gaussian
distribution, the probability of 0 is the highest, and to deal with smaller
values, a larger factor must be multiplied to the objective function,
resulting in a larger $m$. By contrast, direct encoding is capable of keeping
$m$ constant. The only disadvantage is that the probability amplification of
$\mathbf{G}^{L}$ may become insufficient, which was also demonstrated in Fig.
4.
Figure 12: Number of queries required to reach the optimal solution.
To investigate the disadvantage of the proposed real-valued GAS and
insufficient amplification, in Fig. 12, we generated random channel
coefficients and investigated the probability density distribution of the
number of queries required to reach the optimal solution, where the parameters
were the same as those used in Fig. 10(a) except that $m$ was minimized
depending on the random channel coefficients. It was observed in Fig. 12 that
query complexities in CD and QD increased compared with the ideal case. Here,
the same trend was observed for different SNRs. Albeit at this expense, the
proposed algorithm could reach the optimal solution in any case. Note that the
integer approximation with the same $m$ as in direct encoding could not be
plotted in Fig. 12 because it was unable to reach the solution in most cases.
### V-D Effects of Initial Threshold for Further Speedup
(a) CD. (b) QD.
Figure 13: BER transition with respect to the number of iterations, where we
used random channel coefficients, QPSK, $N_{\mathrm{t}}=2$,
$N_{\mathrm{r}}=2$, and $\text{SNR}=20$ dB.
Finally, in Fig. 13, we show the results of evaluating the proposed initial
threshold for GAS described in Section IV-C. Here, we averaged BER with random
channel coefficients and noise, considered SNR of $20$ dB, and assumed
idealized quantum circuits to examine the impact of the initial threshold
only. Other parameters were the same as those used in Fig. 11(a). Fig. 13(a)
shows the number of queries in CD, while Fig. 13(b) shows these in QD. Note
that the vertical axis is BER rather than the objective function value.
Specifically, at the left end of Fig. 13, BER of $0.5$ corresponds to the bit
errors between the input bits $\mathbf{b}$ and the random bits
$\mathbf{b}_{0}$, and BER of $6.8\times 10^{-3}$ corresponds to the errors
between $\mathbf{b}$ and the MMSE output $\bar{\mathbf{b}}_{0}$. As shown in
Fig. 13, in both CD and QD, the proposed threshold, $\tilde{y}$, converged to
the optimal solution much faster than the classic random threshold. The slopes
in the random and proposed thresholds differed significantly. This is because
the random threshold ranged from the best to the worst cases, resulting in
slow convergence in some cases. To be more specific, in Fig. 13, the variance
of the random threshold $E(\mathbf{b}_{0})$ was $14.30$ at the first
iteration. By contrast, the proposed threshold $\tilde{y}$ is determined by
constant factors, $P$ and SNR, and its variance is always zero. Thus, it
significantly improved convergence on average.
It was also found in Fig. 13 that the proposed threshold achieved the best
performance for $P=10^{-4}$ and exhibited lower performance for $P=10^{-3}$.
As described in Section IV-C, $P$ equals the probability that GAS is in a
state where the solution is not updated. That is, an event of
$\mathrm{BER}=0.5$ occurred with probability $P=10^{-3}$, and it resulted in
the error floor of BER around $10^{-3}$. This result indicates that the
parameter $P$ has no significant impact if it is smaller than BER. Since the
exact BER at a given SNR can be calculated in a closed form in advance, an
appropriate $P$ can also be determined in advance accordingly.
Additionally, in Fig. 13, the proposed threshold combined with the MMSE output
achieved a faster convergence compared with the conventional MMSE only case.
This improvement was greater for CD than for QD. That is, the proposed
threshold is particularly useful for improving the query complexity in CD.
This is because it aims to set a strict threshold even in the case of
erroneous MMSE output. Errors in MMSE estimation lead to higher objective
function values and may increase the number of solutions, which can be avoided
by adopting the proposed combination. To be more specific, at the first
iteration, the variance of the MMSE-based threshold $\bar{y}$ was $1.58\cdot
10^{-2}$, while that of the proposed combination $\min(\bar{y},\tilde{y})$ was
much smaller, $3.76\cdot 10^{-5}$, resulting in the faster convergence. The
performance advantage increased upon increasing SNR, which can be verified
from the results shown in Fig. 8. As confirmed in Fig. 8, the gap between
simulated and theoretical values decreased upon increasing SNR.
(a) CD. (b) QD.
Figure 14: BER transition with respect to the number of iterations, where we
used random channel coefficients, 16-QAM, $N_{\mathrm{t}}=2,N_{\mathrm{r}}=2$,
$\text{SNR}=20$ dB, and $P=10^{-3}$.
In Fig. 13, the conventional GAS using the random threshold exhibited slower
convergence than the classic MLD. It can be inferred that this slower
convergence was caused by the smaller search space. Since the quadratic
speedup is an improvement from $O(2^{n})$ to $O(\sqrt{2^{n}})$, the larger
search space leads to the larger reduction. In Fig. 14, we considered the case
of 16-QAM, where other parameters were same as those used in Fig. 13. The
classic MLD required $16^{2}=256$ iterations to reach the optimal solution in
the worst case. As shown in Fig. 14, the conventional MMSE-based threshold
exhibited an increase in BER after the first few iterations. This issue is
caused because improvements in objective function values may not correspond to
improvements in BER in some cases. By contrast, the proposed combination
successfully avoided the issue and reached the optimal solution with reduced
query complexity. This advantage is expected to increase as the search space
size grows, and the proposed approach is especially beneficial for a large-
scale MIMO system.
## VI Conclusions and Future Works
In this paper, we proposed a GAS-based quantum algorithm that supports real-
valued HUBO. Then, as an application example, we formulated the MIMO MLD as a
HUBO problem. The complexity of MLD exponentially increases with the
transmission rate, and low-complexity detectors sacrifice the achievable
performance. Unlike in conventional studies, we constructed specific quantum
circuits instead of assuming an idealized quantum oracle. This enabled us to
analyze the number of qubits and quantum gates in an algebraic manner. To
further accelerate the algorithm, we derived the probability distribution of
the objective function value and conceived a unique threshold to sample better
states. Assuming FTQC, simulations demonstrated the potential for reducing
query complexity in CD and providing a quadratic speedup in QD.
Since this paper focused on a specific construction method for quantum
circuits and their algebraic analysis, we considered only the hard-decision
MLD, instead of error-correcting codes and soft-decision decoding for
classical bits, which are common in wireless standards. The error correction
capability improves with increasing code distance and length. For example, the
maximum code length of 5G NR is 1024 for polar code and 8448 for LDPC.
However, with the current computing resources, it is a challenging task to
represent such a large-scale system as a specific quantum circuit. The
proposed real-valued GAS can be applied to soft-decision decoding, which will
be addressed in our future work.
## Acknowledgement
IBM, CPLEX and Qiskit are trademarks of International Business Machines
Corporation. The authors are indebted to the Editor and the anonymous
reviewers for their invaluable suggestions.
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| Masaya Norimoto (S’22) received the B.E. degree from Yokohama National
University, Kanagawa, Japan, in 2022. He is currently pursuing the M.E. degree
with the Graduate School of Engineering Science, Yokohama National University,
Kanagawa, Japan. His research interests include quantum algorithms and
wireless communications.
---|---
| Ryuhei Mori received the B.E. degree from Tokyo Institute of Technology,
Tokyo, Japan in 2008, and the M.Inf. and D.Inf. degrees from Kyoto University,
Kyoto, Japan in 2010 and 2013, respectively. From 2013 to 2014, he was a
Postdoctoral Fellow at Tokyo Institute of Technology, Tokyo, Japan. He is
currently an Assistant Professor of the Department of Mathematical and
Computing Sciences, School of Computing, Tokyo Institute of Technology, Tokyo,
Japan. His research interests include quantum information, information theory,
computer science and statistical physics.
---|---
| Naoki Ishikawa (S’13–M’17–SM’22) is an Associate Professor with the Faculty
of Engineering, Yokohama National University, Kanagawa, Japan. He received the
B.E., M.E., and Ph.D. degrees from the Tokyo University of Agriculture and
Technology, Tokyo, Japan, in 2014, 2015, and 2017, respectively. In 2015, he
was an academic visitor with the School of Electronics and Computer Science,
University of Southampton, UK. From 2016 to 2017, he was a research fellow of
the Japan Society for the Promotion of Science. From 2017 to 2020, he was an
assistant professor in the Graduate School of Information Sciences, Hiroshima
City University, Japan. He was certified as an Exemplary Reviewer of IEEE
Transactions on Communications in 2017 and 2021. His research interests
include massive MIMO, physical layer security, and quantum speedup for
wireless communications.
---|---
|
# Adaptive Phase Estimation with Squeezed Vacuum Approaching the Quantum Limit
M. A. Rodríguez-García F. E. Becerra Center for Quantum Information and
Control, Department of Physics and Astronomy, University of New Mexico,
Albuquerque, New Mexico 87131, USA
###### Abstract
Phase estimation plays a central role in communications, sensing, and
information processing. Quantum correlated states, such as squeezed states,
enable phase estimation beyond the shot-noise limit, and in principle approach
the ultimate quantum limit in precision, when paired with optimal quantum
measurements. However, physical realizations of optimal quantum measurements
for optical phase estimation with quantum-correlated states are still unknown.
Here we address this problem by introducing an adaptive Gaussian measurement
strategy for optical phase estimation with squeezed vacuum states that, by
construction, approaches the quantum limit in precision. This strategy builds
from a comprehensive set of locally optimal POVMs through rotations and
homodyne measurements and uses the Adaptive Quantum State Estimation framework
for optimizing the adaptive measurement process, which, under certain
regularity conditions, guarantees asymptotic optimality for this quantum
parameter estimation problem. As a result, the adaptive phase estimation
strategy based on locally-optimal homodyne measurements achieves the quantum
limit within the phase interval of $[0,\pi/2)$. Furthermore, we generalize
this strategy by including heterodyne measurements, enabling phase estimation
across the full range of phases from $[0,\pi)$, where squeezed vacuum allows
for unambiguous phase encoding. Remarkably, for this phase interval, which is
the maximum range of phases that can be encoded in squeezed vacuum, this
estimation strategy maintains an asymptotic quantum-optimal performance,
representing a significant advancement in quantum metrology.
## 1 Introduction
Quantum metrology uses the quantum properties of physical systems to enhance
the measurement precision of physical quantities beyond the classical limits
[1, 2]. Quantum mechanics states that all physical observables are represented
by self-adjoint operators on a Hilbert space. As such, the measurement of a
physical quantity of a system involves projecting the quantum state of such
system onto one of the eigenspaces of the corresponding self-adjoint operator.
However, certain physical quantities, such as time, phase, or temperature,
lack an associated self-adjoint operator [3, 4]. Consequently, to determine
the values of these physical quantities, it is necessary to measure some
observables of the system and estimate their values from the observed results.
This process is referred to as quantum parameter estimation [3, 4].
Among different parameter estimation problems, the problem of phase estimation
is ubiquitous in many areas of physics and engineering including, but not
limited to, gravitational wave detection [5], quantum imaging [6], atomic
clocks [7], magnetometry [8], and quantum information processing [9]. However,
the performance of traditional phase estimation methods is limited by the
fundamental properties of the physical states carrying the phase information.
The maximum achievable precision for phase estimation for probe states that
lack quantum correlations, typically used for phase estimation, is defined as
shot-noise limit (SNL) [10].
Numerous methods have been developed to enhance the precision of phase
estimation beyond the SNL by exploiting probe states with inherent quantum
correlations. Among different types of quantum correlations, entanglement
holds a significant potential for improving precision in phase estimation.
Nevertheless, highly entangled states used for phase estimation, such as NOON
states, are delicate and can be readily disrupted by loss, environmental
noise, and decoherence, thereby limiting their practicality for real world
applications [11, 12, 13, 14]. In this regard, squeezed state probes offer a
more viable alternative for robust phase estimation [15, 16]. Squeezed states
allow for reducing the quantum noise in one observable below the SNL at the
expense of an increased noise in another non-commuting observable. This
reduction in quantum noise can significantly enhance the precision of phase
measurements [17, 5, 7, 16], and is a valuable resource for enabling robust
optical quantum metrology and phase estimation.
Advances in photonic quantum technologies for phase estimation and quantum
metrology have yielded squeezed light sources with a high degree of squeezing
[18, 19, 20, 21]. Moreover, experimental demonstrations of quantum metrology
and sensing utilizing squeezed optical probes have achieved sensitivities
surpassing the SNL [22, 23, 24, 5]. However, there remain significant
challenges in devising optimal estimation strategies, including optimal
measurements and estimators, that can efficiently attain the ultimate quantum
limit of precision for any optical phase estimation problem.
A noteworthy measurement approach for optical phase estimation with squeezed
states is the homodyne measurement. This measurement has the potential for
reaching the quantum limit for a specific, optimized phase when a
predetermined level of squeezing is present in the probe state [25]. However,
this optimal phase must be known beforehand in order to reach the quantum
limit, making this approach impractical. To overcome this limitation, a few
adaptive methods have been developed for increasing the range of phases for
which estimation below the SNL is possible [25, 26]. However, far from the
predetermined optimal phase of homodyne, all the estimation strategies so far
deviate significantly from the quantum limit.
In this work, we theoretically demonstrate an adaptive Gaussian measurement
strategy for optical phase estimation with squeezed vacuum states that, by
construction, approaches the quantum limit in precision with a fast
convergence rate. This estimation strategy uses homodyne measurements to
implement a comprehensive set of locally optimal POVMs (Positive Operator
Value Measures). Then the strategy performs adaptive optimization based on the
Adaptive Quantum State Estimation (AQSE) framework to ensure the asymptotic
consistency and efficiency of the estimator of the optical phase [27]. Based
on a systematic study of the statistical properties of dyne-detection, we show
that this strategy with adaptive locally-optimal homodyne measurements
approaches the quantum limit for a range of $[0,\pi/2)$ in the asymptotic
limit of many adaptive steps. Furthermore, we generalize this strategy to
incorporate heterodyne sampling making it possible to extend the parametric
range to $[0,\pi)$, which is the maximum range of phases that can be encoded
in squeezed vacuum, while maintaining an asymptotic quantum optimal
performance.
The paper is organized as follows: In Sec. 2 we provide a concise overview of
the theory of single parameter estimation. Then, we discuss the problem of
optical phase estimation in the context of quantum systems, followed by an
overview of phase estimation with squeezed states. In Sec. 3, we describe the
proposed optimal phase estimation strategy based on adaptive Gaussian
measurements with squeezed vacuum states. By leveraging homodyne measurements
and rotations, we construct a collection of locally optimal POVMs, which
allows us to apply the mathematical framework of AQSE to Gaussian measurements
and feedback [28, 27]. Through theoretical and statistical analysis, we show
that this adaptive measurement process allows for extracting the maximum
possible information pertaining to the phase encoded in squeezed vacuum states
in the asymptotic limit. This results in an adaptive Gaussian measurement
strategy for phase estimation with squeezed vacuum states that attains the
quantum limit within the interval $[0,\pi/2)$. In Sec. 4, we use numerical
simulations to evaluate the performance of this strategy. We observe that this
strategy approaches the quantum limit for phases within $[0,\pi/2)$,
outperforming previous phase estimation strategies. In Sec. 6, we generalize
this adaptive estimation strategy to incorporate heterodyne measurements. This
generalization allows us to extend phase estimation to phases within
$[0,\pi)$, which is the maximum range for unambiguous phase encoding with
squeezed vacuum. Remarkably, for this phase interval this generalized strategy
maintains an asymptotic quantum-optimal performance. Sec. 7 contains the
discussion and concluding remarks.
## 2 Background
### 2.1 Single parameter estimation in quantum systems
A fundamental problem in quantum parameter estimation is the design of precise
estimators of an unknown parameter $\theta\in\Theta$ characterizing a quantum
state based on measurements of the system. In this context, a quantum system
is modeled as a Hilbert space $\mathcal{H}$, and its state is described by a
density operator $\rho$, which is a self-adjoint positive operator with unit
trace on $\mathcal{H}$. The process of encoding the unknown parameters into a
probe state $\rho$ is accomplished by a dynamical process, which, when it can
be represented as a unitary transformation $U(\theta)$, yields the state
$\rho(\theta)=U(\theta)\rho U^{\dagger}(\theta),\quad\theta\in\Theta.$ (1)
Estimation of $\theta$ can be achieved through an estimator that is a function
that takes a sample of size $N$ from a measurement of the quantum system as an
input and produces an estimate of the unknown parameter. The most general
description of a measurement process is a POVM [4]. Given a quantum system
$\mathcal{H}$ and an outcome space $\mathcal{X}\subseteq\mathbb{R}^{k}$ for a
measurement, a POVM is a map $M:\mathcal{B}(\mathcal{X})\to B(\mathcal{H})$
from the set of events of our random experiment $\mathcal{B}(\mathcal{X})$ to
the space of bounded operators on $\mathcal{H}$, denoted by $B(\mathcal{H})$,
that satisfies the following conditions [4, 29, 30]:
1. i.
$M(\emptyset)=0,\quad M(\mathcal{X})=I$
2. ii.
$M(B)\geq 0,\quad\forall B\in\mathcal{B}(\mathcal{X})$
3. iii.
For every family of mutually disjoint events
$\left\\{B_{n}\right\\}_{n=1}^{\infty}\subset\mathcal{B}(\mathcal{X})$, so
that $B_{i}\cap B_{j}=\emptyset$ $\,\forall i\neq j$ , that satisfies
$\cup_{j=1}^{\infty}B_{j}=B\in\mathcal{B}(\mathcal{X})$, then
$M(B)=\sum_{j=1}^{\infty}M(B_{j})$.
In particular, when the state of the system is $\rho(\theta)$, the observed
data $x\in\mathcal{X}$ of a measurement $M$ is an outcome of a random variable
$X\in\mathcal{X}$ distributed according to the density function
$f(x\mid\theta;M)$ (or probability mass function in the case of discrete
random variables) given by the Born’s rule
$f(x\mid\theta;M)=\textrm{Tr}\left[M(x)\rho(\theta)\right].$ (2)
Thus, any sample from the application of a sequence of $N$ POVMs
$M_{1},\ldots,M_{N}$ in a quantum system is represented as a sequence of $N$
random variables $\vec{X}_{N}=X_{1},\ldots,X_{N}$. It follows that any
estimator $\widehat{\theta}\left(X_{1},\ldots,X_{N}\right)$ based on this
sample is also a random variable with expected value
$\textrm{E}_{\theta}\left[\widehat{\theta}(\vec{X}_{N})\right]=\int_{\mathcal{X}^{N}}\widehat{\theta}(x_{1},\ldots,x_{N})f(x_{1},\ldots,x_{N}\mid\theta;M_{1},\ldots,M_{N})dx_{1}\cdots
dx_{N}.$ (3)
To find optimal estimators for all $\theta\in\Theta\subset\mathbb{R}$, the
concept of unbiased estimator plays a crucial role. An estimator
$\widehat{\theta}(\vec{X}_{N})$ is unbiased if
$\mathrm{E}_{\theta}\left[\widehat{\theta}(\vec{X}_{N})\right]=\theta$ for all
$\theta\in\Theta$ [31]. The performance of unbiased estimators is
characterized by their variance, which is bounded by the Cramér-Rao bound
[31]. This bound corresponds to the classical limit of precision for all
unbiased estimators, and is given by the inverse of the Fisher information
denoted by $F_{X}(\theta)$. Given a sample $X$ produced by a POVM $M$, the
Fisher information
$F_{X}(\theta)=\int_{\mathcal{X}}f(x\mid\theta;M)\left[\frac{\partial}{\partial\theta}\log\left(f(x\mid\theta;M)\right)\right]^{2}dx$
(4)
quantifies the amount of information about the parameter $\theta$ that can be
extracted from the sample $X$ [32, 28]. The ultimate limit of precision,
dictated by quantum mechanics, is achieved by optimizing $F_{X}(\theta)$ over
all possible POVMs, resulting in the quantum Fisher information (QFI).
Consequently, the variance of any unbiased estimator is lower bounded by the
inverse of the QFI, referred to as the quantum Cramér-Rao bound (QCRB) [33,
32, 28]. The objective in quantum parameter estimation is to devise estimators
and quantum measurement schemes that attain the QCRB for any value of the
parameter $\theta\in\Theta$.
### 2.2 Optical phase estimation based on dyne-detection
A central task in optical quantum metrology is the estimation of an unknown
phase $\theta\in[0,2\pi)$ encoded in a photonic quantum state by the unitary
process $U(\theta)=e^{-i\hat{n}\theta}$, where $\hat{n}$ is the photon number
operator. The standard quantum state probe for optical phase estimation is the
coherent state
$\left|\alpha\right\rangle\left\langle\alpha\right|,\,\alpha\in\mathbb{C}$, in
which photons exhibit classical correlations [34]. For this quantum state, the
QCRB for any unbiased estimator $\widehat{\theta}$ and $N$ independent copies
of the system is
$\mathrm{Var}[\widehat{\theta}]\geq\frac{1}{4N\mathrm{E}\left[\hat{n}\right]},$
(5)
This limit in precision defines the SNL (or the coherent state limit).
To surpass the SNL for optical phase estimation, it is necessary to employ
states with quantum correlations, such as squeezed vacuum states. These states
are defined by the density operator:
$\rho=\left|0,r\right\rangle\left\langle
0,r\right|=\hat{S}(r)\left|0\right\rangle\left\langle
0\right|\hat{S}^{\dagger}(r),$ (6)
where $\hat{S}(r)=e^{\frac{1}{2}\left(r\hat{a}^{2}-r\hat{a}^{\dagger}\right)}$
is the squeezing operator, $r\in\mathbb{R}$, and $\hat{a}$ and
$\hat{a}^{\dagger}$ are the annihilation and creation bosonic operators,
respectively. Through the unitary transformation $U(\theta)$, the squeezed
vacuum state in Eq. (6) results in the parameter-dependent state
$\rho(\theta)=U^{\dagger}(\theta)\rho U(\theta)$. The corresponding QCRB for
this state for any unbiased estimator $\widehat{\theta}$ and $N$ independent
copies of the system is given by:
$\mathrm{Var}[\widehat{\theta}]\geq\frac{1}{2N\sinh^{2}(2r)}=\frac{1}{8N\left(\mathrm{E}\left[\hat{n}\right]^{2}+\mathrm{E}\left[\hat{n}\right]\right)},$
(7)
where $r$ represents the squeezing strength of $\hat{S}(r)$ [35, 36]. This
bound exhibits a superior scaling with $\mathrm{E}\left[\hat{n}\right]$
compared to the coherent state in Eq. (5). However, it is worth noting that
due to the $\pi$-inversion symmetry inherent in squeezed vacuum states [37]
(see Fig. 1-i), any estimation strategy based on these states is constrained
to phases within the range of $[0,\pi)$.
Figure 1: Adaptive estimation strategy for optical phase estimation with
squeezed vacuum probe states $\lvert 0,r\rangle\langle 0,r\rvert$. The
strategy employs locally optimal POVMs, $M_{\widehat{\theta}}$ in Eq. (16)
with $\widehat{\theta}\in[0,\pi/2)$, to produce a maximum likelihood estimate
of $\theta$, and updates the value $\widehat{\theta}$ for subsequent adaptive
steps. The measurement process is iteratively repeated during the adaptive
strategy. Inset (i) shows the Husimi Q representation for the initial squeezed
vacuum state. Note that due to the internet symmetry properties of squeezed
vacuum, these quantum probes can only encode the phase modulo $\pi$.
The standard measurement approach for optical phase estimation is the
heterodyne measurement. This measurement involves simultaneous sampling of two
orthogonal components of electromagnetic field within the complex plane,
namely $\hat{X}_{\phi}$ and $\hat{X}_{\phi+\pi/2}$, by utilizing the
quadrature decomposition of the input field [10]. Here, the quadrature
operator $\hat{X}_{\phi}$ is defined as:
$\hat{X}_{\phi}=\frac{\hat{a}^{\dagger}e^{i\phi}+\hat{a}e^{-i\phi}}{2}.$ (8)
The POVM associated to the heterodyne measurement is described by coherent
state projectors
$M_{\rm{Het}}=\left\\{\pi^{-1}\left|z\right\rangle\left\langle
z\right|:z\in\mathbb{C}\right\\}$, with outcomes in the form of complex
numbers, and with the corresponding Fisher information:
$F_{Z}(\theta)=4\sinh^{2}\left(r\right).$ (9)
The inverse of Eq. (9) is known as the heterodyne limit for the precision of
any unbiased estimator $\widehat{\theta}$ for all
$\theta\in\left[0,\pi\right)$ with squeezed vacuum states.
Going beyond estimation strategies based on heterodyne detection, homodyne
detection can surpass the heterodyne limit for a suitable set of values of
$\theta$. Homodyne provides information about the quadrature $\hat{X}_{\phi}$
of the input signal using a local oscillator (LO) phase reference field and
interference [10]. Specifically, in the limit of strong LO, Eq. (8) represents
a self-adjoint operator with a spectral measure given by:
$\hat{X}_{\phi}=\int_{-\infty}^{\infty}x\Pi(dx),$ (10)
where $\Pi(B)=1_{B}(x)$ (or symbolically in Dirac notation,
$\Pi(dx)=\left|x\right\rangle\left\langle x\right|dx$), for any
$B\in\mathcal{B}\left(\mathbb{R}\right)$. Consequently, the homodyne
measurement can be described by the POVM
$M_{\rm{Hom}}=\left\\{\Pi(dx)\right\\}$, with the outcome space being the real
numbers [38].
For squeezed vacuum state probes in Eq. (6), the outcomes $x\in\mathbb{R}$ of
the homodyne measurement are distributed according to a normal random variable
with probability density function
$f(x\mid\theta)=\frac{1}{\sqrt{2\pi\sigma^{2}(\theta)}}\exp\left[-\frac{x^{2}}{2\sigma^{2}(\theta)}\right],$
(11)
where $\theta$ denotes the unknown phase and
$\sigma^{2}(\theta)=\left[e^{-2r}\cos^{2}(\theta)+e^{2r}\sin^{2}(\theta)\right]$
(12)
denotes the variance. The Fisher information $F_{X}(\theta)$ for the homodyne
measurement is then
$\begin{split}F_{X}(\theta)=\frac{2\sinh^{2}(2r)\sin^{2}(2\theta)}{\left(\sigma^{2}(\theta)\right)^{2}}.\end{split}$
(13)
Notably, the classical Fisher information of the homodyne measurement
coincides with the QFI when the squeezing strength $r$ satisfies:
$r=-\frac{1}{2}\log(\tan(\theta)),$ (14)
or equivalently, when the parameter $\theta$ corresponds to the optimal value
$\theta_{\mathrm{opt}}$ given by
$\theta_{\mathrm{opt}}=\frac{\arccos\left(\tanh(2r)\right)}{2}$ (15)
which tends to zero as $r$ increases, as shown in Fig. (2). Consequently, the
homodyne measurement can surpass the heterodyne limit in the neighborhood of
$\theta_{\mathrm{opt}}$. However, beyond this neighborhood, estimators based
on a sample $\vec{X}_{N}$ obtained from $N$ independent and identical homodyne
measurements can not achieve this optimal level of precision (see Fig. 9).
Figure 2: $\theta_{\text{opt}}$ as a function of the squeezing parameter $r$
in the probe state. As $r$ increases, the optimal value of $\theta_{opt}$
decreases, approaching zero as $r>2$.
To overcome this limitation, adaptive estimation protocols have been proposed.
One such protocol [26, 25, 22], that we refer to as the two-step protocol,
considers a reduced parameter space $[0,\pi/2)$, and utilizes homodyne
detection and one subsequent adaptation of the probe state to surpass the
heterodyne limit within this phase range. This strategy approaches the QCRB in
the asymptotic limit for phases around the optimal phase
$\theta_{\mathrm{opt}}$ [25, 22]. However, far from this optimal phase, its
performance significantly deviates from the QCRB. Moreover, when considering
the full range of phases that can be encoded in squeezed vacuum probes
$[0,\pi)$, this two-step estimation strategy is not expected to produce
satisfactory results, due to the periodicity of the likelihood function from
homodyne outcomes.
In this work, we propose an adaptive estimation strategy based on homodyne
detection, that leverages the framework of AQSE (see Appendix: 8.2), which,
under certain regularity conditions, yields a consistent and efficient
estimator for all values of $\theta\in[0,\pi/2)$. A key element of this
approach is to use samples that lead convex likelihood functions, ensuring the
asymptotic normality of the Maximum Likelihood Estimator (MLE) [28, 27]. This
property guarantees the asymptotic saturation of the QCRB (7) for any
$\theta\in\left[0,\pi/2\right)$. We further generalize this adaptive strategy
to incorporate heterodyne measurements enabling phase estimation within
$[0,\pi)$, while maintaining a quantum-optimal performance in the asymptotic
limit.
## 3 Optimal adaptive dyne phase estimation
In practice, it is generally impossible to find a POVM and an unbiased
estimator capable of saturating the QCRB for all $\theta\in\Theta$. However,
it is often possible to find a POVM and an unbiased estimator that can achieve
this bound for a specific value of the parameter within a neighborhood around
a point $\theta_{0}\in\Theta$. These types of POVMs are referred to as locally
optimal at $\theta_{0}$. Moreover, if it is possible to construct a collection
of such locally optimal POVMs for any $\theta\in\Theta$ while satisfying a set
of regularity conditions pertaining to the probability distributions of their
outcomes (see Appendix: 8.1), then it is possible to use an adaptive
estimation method, known as AQSE (see Appendix: 8.2), capable of saturating
the QCRB in the asymptotic limit for the MLE $\forall\theta\in\Theta$ [27].
Building upon this understanding, the proposed adaptive phase estimation
strategy with squeezed vacuum states is constructed based on two elements. The
first element involves the construction of a set of POVMs through homodyne
measurements that are locally optimal for all parameter values within the
range $\theta\in[0,\pi/2)$. The second element is the construction of an
estimator that achieves the QCRB for any given value of $\theta$, while also
being locally unbiased at $\theta_{0}\in\Theta$.
To construct the set of locally optimal POVMs, we refer to Eq. (15), which
shows that the POVM denoted as $M_{\mathrm{Hom}}$ is locally optimal at
$\theta_{\mathrm{opt}}$. This is because the samples obtained from
$M_{\mathrm{Hom}}$ have a Fisher information equal to the QFI at this specific
value. Furthermore, given the asymptotic unbiasedness of the MLE and its
subsequent local unbiasedness, the estimator effectively saturates the QCRB
(Eq. (7)) in the asymptotic limit at $\theta_{\mathrm{opt}}$. Consequently, by
appropriately incorporating a phase shift into the elements of
$M_{\mathrm{Hom}}$, it is possible to construct a set of locally optimal POVMs
for each parameter value $\theta\in[0,\pi/2)$.
To this end, we introduce a phase shift
$U\left(\widehat{\theta}-\theta_{\mathrm{opt}}\right)$, which allows us to
define a new set of POVMs for each $\widehat{\theta}\in[0,\pi/2)$ as follows:
$M_{\widehat{\theta}}(dx)=\left\\{U\left(\widehat{\theta}-\theta_{\mathrm{opt}}\right)\Pi(dx)U^{\dagger}\left(\widehat{\theta}-\theta_{\mathrm{opt}}\right)\right\\}.$
(16)
Here $M_{\widehat{\theta}}(dx)$ denotes the POVM elements obtained by applying
the phase shift to the original POVM elements $\Pi(dx)$ of $M_{\mathrm{Hom}}$.
Note that the phase distribution of $M_{\widehat{\theta}}(dx)$ over the state
$\rho(\theta)$
$\displaystyle=f(x\mid\theta+\theta_{\mathrm{opt}}-\widehat{\theta};M_{\mathrm{Hom}}).$
(17)
evaluated at $\widehat{\theta}=\theta$ is the same as the distribution for the
outcomes of the POVM $M_{\mathrm{Hom}}$ at $\theta_{\mathrm{opt}}$, which
shows that the POVM $M_{\widehat{\theta}}(dx)$ is locally optimal at $\theta$.
From this observation and based on the AQSE framework, then it is possible to
construct a multi-step estimation strategy based on adaptive homodyne
measurements, which provide a set of locally optimal measurements, for which
the distribution of the sequence of estimators converges to a normal
distribution with variance equal to the inverse of the QFI. Given the input
squeezed probe state $\left|0,r\right\rangle\left\langle 0,r\right|$ encoding
the parameter $\theta$, this adaptive strategy shown in Figure 1 would
implement the POVM $M_{\widehat{\theta}}$ with an initial guess
$\widehat{\theta}\in[0,\pi/2)$ yielding a measurement sample $X_{1}$. The MLE
$\widehat{\theta}_{\mathrm{MLE}}\left(X_{1}\right)$ applied to $X_{1}$ results
in an estimate $\widehat{\theta}$ of $\theta$ for a given adaptive step. This
estimate $\widehat{\theta}$ then becomes the best guess for the subsequent
adaptive step, and the process is repeated iteratively during every step in
the strategy.
### 3.1 Estimator
Figure 3: Probability that the likelihood function lacks a stationary point
within $\left[0,\pi/2\right)$. Panel a shows the probability of not having
stationary points for different degrees of freedom (DoF) $\nu$ (number of
probes in each adaptive step) with $r=1$. Panel b shows this probability for
different values of squeezing strength $r$ with $\nu=3705$. By ensuring that
the likelihood function has stationary points within the interval $[0,\pi/2)$,
we mitigate the bias introduced by boundary estimates and enhance the
performance of the adaptive estimation process. See main text for details.
A key aspect of the parameter estimation strategy is the selection of an
estimator that allows for the saturation of the QCRB. We identify the
necessary conditions to ensure the asymptotic consistency and normality of the
MLE and the saturation of the QCRB through the adaptive strategy for any
$\theta\in\left[0,\pi/2\right)$. Assuming a finite Fisher information and
different of zero for every value of $\theta$, these conditions require that:
a) the MLE must be unique; b) the MLE should be obtained as a stationary point
of the likelihood function; and c) the derivatives of the likelihood function
at $\theta$ should exist up to a level where they can be effectively
approximated using a Taylor series [39, 27].
We observe that the first condition in the MLE is automatically satisfied,
since the samples generated from the proposed strategy conform to a normal
distribution, as indicated by Eq. (11). This ensures the sufficient smoothness
of the likelihood function for any $\theta\in[0,\pi/2)$ in each adaptive step.
Moreover, by restricting the parameter space to the interval
$\left[0,\pi/2\right)$, we guarantee the uniqueness of the MLE for any
$\theta\in[0,\pi/2)$. Thus, the remaining task is to determine the conditions
under which the MLE corresponds to a stationary point of the likelihood
function, leading to its asymptotic normality. To this end, we analyze the MLE
for a sample at the first adaptive step.
Given the sample $\vec{X}_{\nu}=\left(X_{1},X_{2},\ldots,X_{\nu}\right)$ of
size $\nu$ from the POVM $M_{\widehat{\theta}_{0}}$ at the first adaptive
step, and evaluating the likelihood from Eq. (11) at
$\theta_{*}=\left(\theta+\theta_{\mathrm{opt}}-\widehat{\theta}_{0}\right)\,$
$\mathrm{modulo}\,$ $\pi/2$, we obtain the MLE for $\theta$ as:
$\widehat{\theta}_{\mathrm{MLE}}(\vec{X}_{\nu})=\arccos\left[\frac{e^{r}\sqrt{e^{2r}-\frac{1}{\nu}\sum_{i=1}^{\nu}X_{i}^{2}}}{\sqrt{e^{4r}-1}}\right]-\theta_{\mathrm{opt}}+\widehat{\theta}_{0}.$
(18)
The event for which the MLE corresponds to a stationary point of the
likelihood within $\left[0,\pi/2\right)$ is equivalent to obtaining a real-
valued solution of Eq. (18). This real solution is obtained when
$e^{2r}-\frac{1}{\nu}\sum_{i=1}^{\nu}X_{i}^{2}>0$. Therefore, the probability
of obtaining an imaginary solution of of Eq. (18) at
$\hat{\theta}=\theta_{\mathrm{opt}}$, which corresponds to a likelihood
function lacking a stationary point, is:
$P\left(e^{2r}<\frac{1}{\nu}\sum_{i=1}^{\nu}X_{i}^{2}\mid\widehat{\theta}_{0}=\theta_{\mathrm{opt}}\right)=P\left(\sum_{i=1}^{\nu}\frac{X_{i}^{2}}{\sigma^{2}(\theta)}>e^{2r}\left(\frac{\nu}{\sigma^{2}(\theta)}\right)\right)=1-P\left(\sum_{i=1}^{\nu}\frac{X_{i}^{2}}{\sigma^{2}(\theta)}\leq
e^{2r}\left(\frac{\nu}{\sigma^{2}(\theta)}\right)\right)$ (19)
where $Q=\sum_{i=1}^{\nu}\frac{X_{i}^{2}}{\sigma^{2}(\theta)}$ represents the
sum of squares of $\nu$ independent standard normal random variables, and thus
follows a chi-squared distribution with $\nu$ degrees of freedom
($Q\sim\chi^{2}(\nu)$) [31]. Figure 3 shows the probability of obtaining an
imaginary solution in Eq. (19) as a function of $\theta$ for different values
of the squeezing strength $r$ and sample size $\nu$.
We observe that as $\theta$ deviates from the optimal value
$\theta_{\mathrm{opt}}$ in Eq. (15), the probability that the MLE did not come
from a stationary point within the interval $\left[0,\pi/2\right)$ becomes
different than zero. We also note that the region for which the likelihood
does not have stationary points diminishes as we increase the degrees of
freedom (sample size in the adaptive step) or the degree of squeezing.
Moreover, when $e^{2r}-1/\nu\sum_{i=1}^{\nu}X_{i}^{2}<0$ (in regions where the
global maximum does not correspond to a stationary point) the MLE in Eq. (18)
corresponds to the boundary point $\pi/2$. This event introduces a bias in the
estimate for subsequent adaptive steps, leading to a decrease in the precision
of the final estimate. This effect in the estimate becomes more detrimental
when the phase to be estimated $\theta$ is close to the boundary point of
$\pi/2$ (see Figure 3).
The strategy address this problem by first choosing a random guess
$\widehat{\theta}_{0}\in[0,\pi/2)$ in the first adaptive step. This initial
random hypothesis reduces in average the probability in Eq. (19) that the MLE
does not arise from a stationary point within $[0,\pi/2)$, according the law
of total probability
$P\left(e^{2r}<\frac{1}{\nu}\sum_{i=1}^{\nu}X_{i}^{2}\right)=\mathrm{E}_{\hat{\theta}_{0}}\left[P\left(e^{2r}<\frac{1}{\nu}\sum_{i=1}^{\nu}X_{i}^{2}\mid\widehat{\theta}_{0}\right)\right]=\int_{0}^{\pi/2}d\widehat{\theta}_{0}P\left(\sum_{i=1}^{\nu}\frac{X_{i}^{2}}{\sigma^{2}\left(\theta_{*}\right)}>e^{2r}\left(\frac{\nu}{\sigma^{2}\left(\theta_{*}\right)}\right)\right).$
(20)
Moreover, this probability decreases as the sample size $\nu$ increases, as
shown in Fig. 4. As a final step, to ensure that the MLE always arises from
stationary points, we introduce a modified estimator
$\widehat{\theta}^{\textrm{U}}_{\textrm{MLE}}(\vec{X}_{\nu})=\begin{cases}\widehat{\theta}_{\textrm{MLE}}(\vec{X}_{\nu})&\text{if
}e^{2r}-1/\nu\sum_{i=1}^{\nu}X_{i}^{2}\geq 0,\\\
\widehat{\theta}_{\textrm{MLE}}(\vec{X}_{s})&\text{otherwise},\end{cases}$
(21)
where $\vec{X}_{s}$ is a subsequence of $X_{\nu}$ constructed by iteratively
removing the highest values of $X_{\nu}$ in descending order until the
condition $e^{2r}-1/s\sum_{i=1}^{s}X_{i}^{2}\geq 0$ is satisfied. We note that
while the estimator ignores a few measurement outcomes from the sample, the
probability if this event is low, and this modified estimator greatly improves
the final variance of the estimates. Moreover, it is worth noting that this
modified estimator in Eq. (21) is only necessary in the first adaptive step,
since once an estimate $\widehat{\theta}$ sufficiently close to the true value
$\theta$ is obtained, the estimates during subsequent adaptive steps will be
close to $\theta_{\mathrm{opt}}$. Therefore, for sufficiently large values of
$r$, and moderate $\nu$, this procedure makes the probability in Eq. (19) go
to zero, guaranteeing the asymptotic efficiency of the strategy.
Figure 4: Conditional probability (Eq. (20)) that the likelihood function
lacks a stationary point within $\left[0,\pi/2\right)$ given
$\widehat{\theta}_{0}$ uniformly random between $0$ and $\pi/2$. The
probability is upper bounded for $\nu=1$ and decreases as the number of $\nu$
increases. The values of $\nu$ corresponds to the degrees of freedom (DoF) for
the random variable $Q\sim\chi^{2}(\nu)$.
## 4 Performance
We evaluate the performance of the adaptive estimation strategy based on local
optimal POVMs, Eq. (16), with the modified estimator in Eq. (21). We conduct
Monte Carlo simulations varying the sample size $\nu$, number of adaptive
steps $m$, and squeezing strength $r$. We investigate the precision and
efficiency of the estimation strategy and its convergence towards the QCRB in
Eq. (7) for $\theta\in[0,\pi/2)$. Considering that the estimator
$\widehat{\theta}\left(X\right)$ obtained from the Monte Carlo simulations
exhibits a $\pi/2$ periodic distribution, we evaluate the precision of
$\widehat{\theta}\left(X\right)$ by using the corresponding Holevo variance
[4, 40]:
$\mathrm{Var}_{\theta}\left[\widehat{\theta}(X)\right]=\frac{\left[\mathrm{E}\left[\cos\left(\frac{2\pi}{P}\left(\widehat{\theta}(X)-\theta\right)\right)\right]\right]^{-2}-1}{\left(\frac{2\pi}{P}\right)^{2}},$
(22)
where the factor $P$ represents the period of the estimator’s distribution, in
our case $P=\pi/2$.
Figure 5 shows the results for the Holevo variance for
$\widehat{\theta}\left(X\right)$ within $\theta\in[0,\pi/2)$ for adaptive
estimation strategies based on homodyne detection and squeezed vacuum for
different numbers of adaptive steps $m$ from $3$ to $15$. The homodyne
measurement without feedback and the two-step adaptive homodyne strategy from
Ref. [22] are shown for comparison. All the results have been normalized to
the QCRB $=QFI^{-1}$. The results shown in Figure 5 are obtained from the
average of five Monte Carlo simulations, each involving $10,000$ total
estimates for a squeezing strength $r=1.01$ and a total number of copies of
the state $N=3705$. (These parameters were chosen to enable comparison of
performance with previous estimation strategies). The sampling process
employed the method of rejection sampling [41], while the optimization
employed the method of generalized simulated annealing over the interval
$[0,\pi/2]$ [42].
We observe that the proposed multi-step adaptive estimation strategy based on
homodyne detection consistently outperforms the non-adaptive and the two-step
homodyne [22] strategies. Moreover, as can be observed in Figure 5, this
adaptive homodyne strategy progressively approaches the QCRB (dashed
horizontal line), and is expected to saturate this bound for all values of
$\theta\in[0,\pi/2)$ in the limit of many adaptive steps. For instance, the
proposed adaptive estimation strategy with $m=15$ adaptive steps achieves a
precision of just $7\%$ above the QCRB for phases $\theta\in[0,\pi/2)$ on
average, compared to $42\%$ with two steps [22]. Moreover, while the
performance of previous strategies for phases far from $\theta_{\mathrm{opt}}$
deviate significantly from the QCRB, by construction the proposed strategy
ensures an asymptotic quantum-limited performance (see Appendix 8.3). These
results highlight the fundamental advantage of this multi-step adaptive
strategy for parameter estimation, and underscore its potential for optical
phase estimation and quantum metrology.
Figure 5: Holevo variance of the adaptive estimation strategy based on the
AQSE formalism as a function of $\theta$, for different numbers of adaptive
steps $m$ for $N=3705$ independent copies of the state with squeezing strength
of $r=1.01$. The y-axis shows the Holevo variance in logarithmic scale
normalized to the inverse of the QFI Eq. (7), corresponding to the QCRB. The
magenta line shows the two-step adaptive estimation scheme from Ref. [22], and
the blue curve shows the estimation with homodyne detection without feedback.
The shaded regions represent a 1-$\sigma$ standard deviation.
## 5 Loss and Noise Effects
### 5.1 Losses
In quantum optics, linear losses can be represented as the transmission of the
state $\rho(\theta)$ through a (linear) lossy channel modeled as a beam
splitter with transmittance $0<T<1$. This channel maps the initial squeezed
state $\rho(\theta)=\lvert r,0\rangle\langle 0,r\rvert$ into a squeezed
thermal state, which reduces the QFI below that of squeezed vacuum states [43]
as:
$F_{Q}^{\text{Lossy}}=\left[\frac{T^{2}}{1+2T(1-T)\sinh^{2}(r)}\right]F_{Q}.$
(23)
where $F_{Q}=2\sinh^{2}(2r)$ is the QFI about $\theta$ for squeezed vacuum
states. Therefore, the impact of linear losses on the QCRB can be effectively
accounted for by an appropriate rescaling [43].
Furthermore, the squeezed thermal states resulting from losses further reduce
the classical Fisher information for the homodyne measurement [43, 44]:
$F^{\text{Lossy}}_{X}(\theta)=\left[\frac{T^{2}}{1+4T(1-T)\sinh^{2}(r)}\right]F_{Q}.$
(24)
Thus, losses make homodyne measurements suboptimal
$F^{\text{Lossy}}_{X}(\theta)<F_{Q}^{\text{Lossy}}$ for all
$\theta\in[0,\pi/2)$. However, it is worth noting that homodyne still allow
for surpassing the SNL under losses [43, 22].
Figure 6 shows the ratios $F^{\text{Lossy}}_{Q}(\theta)/F_{Q}$ (dots) and
$F^{\text{Lossy}}_{X}(\theta)/F_{Q}$ (crosses) as a function of the losses
($1-T$) and different values of $r$. We observe that probe states with larger
$r$, are more sensitive to losses, as is expected. Then, for highly squeezed
probes, the Fisher information approaches $F_{Q}$ only at $T\approx 1$.
Figure 6: Fisher information ratios $F^{\text{Lossy}}_{Q}(\theta)/F_{Q}$
(dots) and $F^{\text{Lossy}}_{X}(\theta)/F_{Q}$ (crosses) for squeezed vacuum
states as function of $T$ for squeezing strength $r$ of $0.5$, $1.0$ and
$1.5$.
### 5.2 Imperfections at state preparation
The processes of state preparation is often affected by small random errors.
Due to the Central Limit Theorem, the cumulative effect of these numerous,
small, and independent errors will tend towards a normal distribution.
Therefore, it is reasonable to assume that the squeezing strength $r$ of the
probe state $\rho=\lvert r,0\rangle\langle 0,r\rvert$ is the output of a
random variable $R$ with a normal distribution
$\mathcal{N}\left(r_{0},\sigma_{r}^{2}\right)$, where $\sigma_{r}^{2}\geq 0$
is a small number relative to $r_{0}$.
To take state preparation errors of this kind into account in the adaptive
strategy, we consider that the samples at every adaptive step are drawn from
the conditional random variable $X\mid R$. We then proceed as described in
Section 3. In this case, the Fisher information about $\theta$ contained is
the random variable $X\mid R$ is calculated with respect to the conditional
density of $X$ given $R$, that is,
$F_{X\mid R}(\theta)=\mathrm{E}_{X}\left[F_{X\mid R=r}(\theta)\right],$ (25)
where $F_{X\mid R=r}(\theta)$ corresponds to Eq. (13) at a specific value
$R=r$. Therefore, in the asymptotic limit, the adaptive homodyne strategy
samples around the phase that maximizes Eq. (25). In experimental settings,
typically the range for $\sigma_{r}$ lies between $0.01$ and $0.02$ [22, 45].
For state preparation errors with small variances $\sigma_{r}^{2}$, the
optimal phase $\theta_{\mathrm{opt}}^{noise}$ deviates slightly from the
noiseless case $\theta_{\mathrm{opt}}$, and modifies the Fisher information.
Figure 7 shows $F_{X\mid R}(\theta)$ for standard deviations of $r$ in state
preparation $\sigma_{r}=0.01$ and $\sigma_{r}=0.02$ as a function of
$\theta\in[0,\pi/2)$. Since the QFI for squeezed vacuum states is a nonlinear
function, the contribution of positive deviations of $r$ from the mean to the
expected value of $F_{X\mid R}(\theta)$ increases with $r$. This nonlinear
effect causes the maximum of $F_{X\mid R}(\theta)$ to slightly surpass
$F_{X\mid R=r}(\theta_{\mathrm{opt}})$ (dashed red line, as can be observed in
the inset of Fig. 7.
Figure 7: Expected Fisher information as function of $\theta$ for input
states with state preparation errors with normally varying squeezing strength
for $r_{0}=1$ with $\sigma_{r}$ of $0.01$ and $0.02$. The dashed red line
corresponds to the QFI for squeezed vacuum states at $r=1$. Inset shows a zoom
in the maximum of the curve showing the overshot effect due to the nonlinear
dependence of $F_{X\mid R}(\theta)$ with $r$.
## 6 Generalized adaptive-dyne phase estimation
In general, the problem of phase estimation involves estimation over the
complete range of possible phases from $0$ to $2\pi$. However, when using
squeezed vacuum states for performing phase estimation beyond the SNL, there
is a physical limitation on the range of phases that can be estimated.
Squeezed vacuum states are invariant under phase shifts of $\pi$, restricting
the estimable phases to the interval $[0,\pi)$, which is half of the complete
range in the general problem of phase estimation. This symmetry can be seen in
the Husimi Q representation of the state $\rho$,
$Q(\alpha)=\frac{1}{\pi}\left\langle\alpha\right|\rho\left|\alpha\right\rangle$
(see Fig. 1-i), which shows that the squeezed vacuum probe can only encode the
phase modulo $\pi$. Moreover, the measurement employed for decoding the phase
can impose severe constraints on the range of phases that can be estimated.
For instance, homodyne measurements further reduce the range of phases within
which phase estimation is possible to $[0,\pi/2)$. This is because the
probability distributions of outcomes from homodyne measurements in Eq. (11),
associated with POVMs in Eq. (16), are $\pi/2$ periodic. Consequently, any
strategy based on adaptive homodyne is restricted to estimating phases within
the range $[0,\pi/2)$. To go beyond this limited range and enable phase
estimation with squeezed vacuum within the entire range of $[0,\pi)$, it is
necessary to generalize the adaptive estimation strategy to include
measurements beyond homodyne.
Figure 8: Generalized adaptive dyne strategy for phase estimation based on
squeezed vacuum probes for phases $\theta\in[0,\pi)$. This strategy takes
advantage of the capability of heterodyne measurements to unambiguously
estimate phases within the whole parametric space $[0,\pi)$ for squeezed
vacuum, overcoming the non-identifiability problem in the likelihoods from
homodyne measurements. By employing a small sample of heterodyne measurements,
the unknown phase $\theta$ is localized within a neighborhood within
$[0,\pi)$. Then, the strategy employs adaptive homodyne for phase estimation
within this neighborhood.
We generalize this adaptive estimation strategy by incorporating heterodyne
measurements, enabling optimal phase estimation within $[0,\pi)$ in the
asymptotic limit. We exploit the capability of heterodyne detection to
unambiguously estimate phases within $[0,\pi)$, which is only limited by the
symmetry properties of the squeezed vacuum probe. Although heterodyne
detection falls short from the optimal limit and has a constant, and
suboptimal, Fisher information, it provides means for solving the non-
identifiability problem of the homodyne likelihood [46]. Figure 8 shows the
generalized adaptive dyne strategy. By employing a small sample of size
$N_{1}$ of a heterodyne measurement, denoted as $\vec{X}_{N_{1}}$, the
strategy determines a neighborhood in the parametric space $[0,\pi)$ to which
the unknown phase belongs. This makes the parameter identifiable within
$[0,\pi)$ and produces a likelihood peaked around the true value. After the
heterodyne sampling $\vec{X}_{N_{1}}$, the generalized dyne estimation
strategy implements the strategy based on adaptive homodyne sampling,
described above, with $N_{2}$ copies of the input state and $m$ adaptive
measurements. By construction, when $N_{2},m\to\infty$ (in the asymptotic
limit), and $N_{1}$ is large enough such that the maximum likelihood estimator
$\widehat{\theta}(\vec{X}_{N_{1}})\in[0,\pi/2)$ with high probability, this
strategy saturates the QCRB Eq. (7) with a fast convergence rate [3, 46].
Thus, this generalized dyne estimation strategy is able to extract all the
information about the phase encoded in the quantum probe, and approach the
quantum limit for any phase within $[0,\pi)$.
Figure 9 shows the performance of the generalized dyne phase estimation
strategy enabling near quantum-optimal phase estimation for phases within
$[0,\pi)$ with a finite number of samples. These results are obtained from the
average of five Monte Carlo simulations considering $N=3705$ copies of the
squeezed vacuum probes with $m=15$ adaptive steps, each with a sample of size
$\nu=247$. In this generalized strategy, the first step entails a heterodyne
measurement with a sample of size $N_{1}=\nu_{1}=247$. This sample size is
large enough to produce estimates with high probability in $[0,\pi/2)$, and it
is significantly smaller than the cumulative sum of subsequent (homodyne)
samples $N_{2}=\sum_{i=2}^{14}\nu_{i}=3458$. We note that within the
statistical noise of our simulations, the generalized dyne phase estimation
strategy enables phase estimation approaching the QCRB within the full
interval $\theta\in[0,\pi)$.
We note that while the heterodyne measurement exhibits suboptimal performance
for all $\theta\in[0,\pi)$, a small heterodyne sampling suffices for solving
the non-identifiability problem arising from the symmetry of homodyne
measurements, with a negligible penalty in the estimator error. Moreover, the
analysis regarding the convergence properties for the component of the
strategy involving homodyne measurements described above reminds valid.
Consequently, it is expected that the generalized dyne strategy using a small
heterodyne sample saturates the QCRB in the asymptotic limit for all
$\theta\in[0,\pi)$. However, reaching the asymptotic regime should require a
larger number of adaptive steps and resources compared to the strategy based
solely on adaptive homodyne measurements.
Figure 9: Performance of the generalized adaptive dyne phase estimation
strategy with $m=15$ adaptive steps (orange), using $N=3705$ probes with
$r=1.01$. The performance of the homodyne-detection without feedback (blue) is
shown for reference. In the generalized adaptive-dyne protocol, the initial
adaptive step involves heterodyne sampling, while the subsequent adaptive
steps utilize locally optimal POVMs in Eq. (16). Throughout the simulation,
the sample size remains constant at $\nu=247$ (number of probe states per
adaptive step). The dashed blue line shows the heterodyne limit, while the
dashed black line corresponds to the QCRB. The shaded regions indicate a
1-$\sigma$ standard deviation. Note that the y-axis is on a logarithmic scale.
## 7 Discussion & Conclusions
Our analysis and simulations have shown that the proposed adaptive estimation
strategy efficiently extracts the maximum attainable information about the
unknown phase encoded in squeezed vacuum states in the asymptotic limit.
However, we note that while the proposed strategy allows for phase estimation
at the quantum limit for the full range of phases $[0,\pi)$ for squeezed
vacuum, this range is limited due to the $\pi$ phase-shift symmetry inherent
in squeezed vacuum states. Therefore, it becomes imperative to explore quantum
probes capable of overcoming this non-identifiability problem, and enable
unambiguous phase encoding within the full parameter space from $0$ to $2\pi$.
We also note that other quantum states used for phase estimation at the
quantum limit such as NOON states face the same limitation due to their
inherent phase-shift symmetries [12]). On the other hand, there may be other
optical probes capable of solving the non-identifiability problem, albeit with
lower QFI. Coherent states, for example, have a significantly lower QFI
compared to squeezed vacuum states, but allow for unambiguously identifying
the quadrants of the phase [47, 48, 49, 50].
As an alternative quantum probe for phase estimation, displaced squeezed
states $D(\alpha)\lvert 0,r\rangle$ can offer the ability to unambiguously
encode phases within $[0,2\pi)$ with a higher QFI compared to coherent states.
We note, however, that there will be a trade-off between the achievable QFI
compared to that of squeezed vacuum states and the ability to identify the
quadrants of the phase. Finding the best trade off requires optimization of
both the squeezing strength $r$ and the displacement parameter $\alpha$, given
a fixed resource budget in terms of the number of photons. Moreover, this
trade off will critically depend on the available resources and experimental
constraints. Further research will focus exploring and identifying the optimal
probe states capable of overcoming the non-identifiability problem while
maintaining a high QFI for phase estimation.
In summary, we propose a generalized adaptive dyne estimation strategy for
optical phase estimation with squeezed vacuum states that approaches the
quantum limit in precision. This strategy leverages homodyne measurements and
rotations to implement a complete set of locally optimal POVMs. This set of
POVMs are used to construct an adaptive estimation method based on the
Adaptive Quantum State Estimation (AQSE) formalism, which ensures consistency
and efficiency of the estimator in the asymptotic limit, with variance equal
to the inverse of the QFI for phases $\theta\in[0,\pi/2)$. To extend the
parameter range for quantum phase estimation to $[0,\pi)$, which is the
maximum range of phases that can be encoded in squeezed vacuum, we generalize
the estimation strategy to incorporate a small number of heterodyne
measurements. This heterodyne sampling allows for identifying the neighborhood
of the phase within $[0,\pi)$ solving the non-identifiability problem in the
likelihoods from homodyne measurements, while maintaining a quantum-optimal
performance in the limit of many adaptive steps. This result represents a
significant advancement in high-precision quantum metrology and optical phase
estimation based on quantum correlated states.
## 8 Appendix
### 8.1 Maximum likelihood estimation
Let us consider the problem of estimating an unknown parameter
$\theta\in\Theta$ associated with a set of quantum states
$\left\\{\rho(\theta):\theta\in\Theta\right\\}$ from measurements of the
system. When independent measurements are performed over the system, a set of
independent random variables carry the information about $\theta$. In this
case, the total Fisher information about $\theta$ is the sum of the individual
Fisher information values for each measurement. This property can be exploited
to reach the QCRB in the asymptotic limit ($N\to\infty$). When the outcomes of
a POVM $M$ have a Fisher information that coincides with the QFI, and their
probability distribution satisfies a set of mild regularity conditions
described below, it can be shown that in the asymptotic limit the MLE applied
to the outcomes of $M$ can achieve the QCRB [27].
Given a sample $\vec{X}_{N}=X_{1},\ldots,X_{N}\in\mathcal{X}^{n},\,n\geq 1$
generated from the application of a sequence of POVMs $M_{1},\ldots,M_{N}$ to
a quantum system, the likelihood function can be expressed as follows:
$L(\theta;\vec{X}_{N})=\prod_{i=1}^{N}f(X_{i}\mid\theta;M_{i}).$ (26)
Here, $\theta$ represents the unknown parameter to be estimated, $X_{i}$ is
the random variable that models the outcomes of the POVM $M_{i}$, and
$f(X_{i}\mid\theta;M_{i})$ denotes their density function, (or probability
function for discrete random variables), given by the Born’s Rule in Eq. (2).
Given the likelihood function in Eq. (26), the MLE is be defined as:
$\widehat{\theta}_{\mathrm{MLE}}\left(\vec{X}_{N}\right)=\arg\max_{\theta\in\Theta}L(\theta;\vec{X}_{N}).$
(27)
To saturate the QCRB, the MLE requires to be asymptotic consistent, which
means that as the sample size increases, the MLE converges to the true value
of the parameter $\theta$ in probability [31]. For a MLE to be asymptotic
consistent, the following conditions over the parametric set $\Theta$ and the
set of density functions
$\left\\{f(X_{i}\mid\theta;M_{i})\right\\}_{\theta\in\Theta}$ for each POMV
$M_{i}$ must be satisfied [31, 46, 27]:
* •
Compactness: The parameter space $\Theta$ and the space of POVMs must be
compact, which means that it is closed and bounded. This property ensures that
the MLE exists for any sample size.
* •
Identifiability: The true value of the parameter must be uniquely determined
by the probability distribution. In other words, different values of the
parameter must produce different probability distributions.
* •
Measurability: The probability density function $f(X_{i}\mid\theta;M_{i})$
must be measurable for all $X_{i}=x_{i}$ and for each POVM $M_{i}$. Thus the
MLE is well-defined as random variable.
* •
Continuity: The probability density function $f(X_{i}\mid\theta;M_{i})$ must
be continuous in the parameter space $\Theta$ for all $X_{i}=x_{i}$ and for
each POVM $M_{i}$. This guarantees that small changes in the value of the
parameter result in small changes in the probability.
* •
Dominance: The log likelihood $\log\left[f(X_{i}\mid\theta;M_{i})\right]$ is
uniformly Lipschitz in $\theta$ with respect to some dominating measure on
$\mathcal{X}$. This provide the convergence of the MLE.
Some of these conditions can be less restrictive. However, a critical
condition for parameter estimation of cyclic (or periodic) parameters is
identifiability, which ensures the uniqueness of the maximum likelihood
estimator over the parameter space. This property is particularly important
for phase estimation, where the density functions of the random variables
generated from different measurements in general show a periodicity that is
smaller than the parametric space.
Under this set of regularity conditions, the MLE exhibits asymptotic
consistency. This property ensures that, under certain assumptions such as the
existence of the Fisher information, and sufficient smoothness in the
likelihood functions, the distribution of the MLE in the limit of $N\to\infty$
follows a normal distribution:
$\widehat{\theta}(\vec{X}_{N})\sim\mathcal{N}\left(\theta,\frac{1}{F_{\vec{X}_{N}}(\theta)}\right).$
(28)
Here $\widehat{\theta}(\vec{X}_{N})$ denotes the MLE based on the sample
$\vec{X}_{N}$, and $F_{\vec{X}_{N}}(\theta)$ represents the Fisher information
associated with the sample. Consequently, the variance of
$\widehat{\theta}(\vec{X}_{N})$ is given by
$\frac{1}{F_{\vec{X}_{N}}(\theta)}$, and $\widehat{\theta}(\vec{X}_{N})$
achieves the classical Cramér-Rao bound for all $\theta\in\Theta$.
An estimator that exhibits this property is known as an efficient estimator.
The efficiency stems from the fact that when samples are derived from a
sequence of $N$ independent and identically distributed random variables, the
Fisher information of the sample is proportional to $N$ times the Fisher
information of an individual random variable. Consequently, the variance of
the estimator diminishes as the inverse of the sample size. Notably, when the
Fisher information of the random variables equals the QFI, the MLE attains the
QCRB, representing the ultimate limit in precision for parameter estimation.
### 8.2 Adaptive Quantum State Estimation (AQSE)
The identification and implementation of an optimal POVM that achieves a
Fisher information equal to the QFI across the entire parameter space $\Theta$
is a challenging task. This difficulty arises from the fact that, in general,
such a POVM depends on the unknown parameter $\theta$ to be estimated. To
address this issue, Nagaoka [28] proposed an adaptive quantum estimation
scheme known as Adaptive Quantum State Estimation (AQSE), based on locally
optimal POVMs. A POVM $M$ is considered to be locally optimal at $\theta_{0}$
if it produces a locally unbiased estimator that achieves the QCRB when the
true parameter coincides with $\theta_{0}$. However, the implementation of
locally optimal POVMs for estimating the unknown parameter $\theta$ becomes
impractical in experimental settings due to their inherent dependence on the
unknown parameter itself [27]. AQSE provides a possible path to overcome this
challenge. This method can be summarized as follows:
In the initial step, the AQSE method assumes a collection of locally optimal
POVMs $\left\\{M_{\theta_{0}}\right\\}_{\theta_{0}\in\Theta}$. Starting with
an arbitrary initial estimate $\widehat{\theta}_{0}$ for the parameter
$\theta$, the locally optimal measurement at $\widehat{\theta}_{0}$ with
associated POVM $M_{\widehat{\theta}_{0}}$ is applied, resulting in the
outcome $x_{1}$. The MLE is then applied to this data $x_{1}$ to obtain an
updated estimate denoted as
$\widehat{\theta}_{1}:=\widehat{\theta}_{\mathrm{MLE}}(x_{1})$, which serves
as the subsequent guess of the parameter $\theta$ for the next adaptive step.
For the adaptive step $n$, $n\geq 2$, the method applies the POVM
$M_{\widehat{\theta}_{n-1}}$ yielding the outcome $x_{n}$. From this, the MLE
produces an estimate
$\widehat{\theta}_{\mathrm{MLE}}(x_{1},\ldots,x_{n})=\widehat{\theta}_{n}$,
and this process is repeated iteratively. By satisfying the regularity
conditions described (see Appendix: 8.1) for each locally optimal POVM, the
sequence of MLEs $\widehat{\theta}_{\mathrm{MLE}}(X_{1},\ldots,X_{m})$
converges to the true parameter $\theta$ as the number of adaptive steps
$m\to\infty$. Additionally, the distribution of the asymptotic estimator
follows a normal distribution centered at $\theta$ with a variance equivalent
to the QCRB [28, 27].
### 8.3 Phase estimation around $\theta_{opt}$
The estimator in Eq. (21) of the proposed strategy minimizes the variance over
all the phases within the parametric space from $[0,\pi/2]$ with adaptive
homodyne. This estimator results on a variance above the QCRB for phases close
to $\theta_{opt}$ for a small number of adaptive steps $m$. On the other hand,
we note that the 2-step protocol from Ref. [22] is closer to the QCRB for
$\theta\approx\theta_{opt}$ than our strategy for small $m=3,5$. However, our
proposed strategy is more general, and encompasses the one from Ref. [22]. By
taking an uneven splitting ratio between the first and second step and
$\hat{\theta}_{0}=\theta_{opt},$ by construction our strategy with $m=2$
reduces to the strategy from Ref. [22]. Nevertheless, our strategy is
guaranteed to achieve the QCRB for all phases from $[0,\pi]$ in the asymptotic
limit.
## Funding
This work was funded by the National Science Foundation (NSF) Grants #
PHY-2210447, FRHTP # PHY-2116246, and the Q-SEnSE QLCI # 2016244.
## Acknowledgments
We thank Laboratorio Universitario de Cómputo de Alto Rendimiento (LUCAR) of
IIMAS-UNAM for their service on information processing.
## Disclosures
The authors declare no conflicts of interest.
## Data availability
The data that support the findings of this study are available from the
authors upon request.
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|
# No Culture Left Behind:
Massively Multi-Cultural Knowledge Acquisition & LM Benchmarking
on 1000+ Sub-Country Regions and 2000+ Ethnolinguistic Groups
Yi R. Fung Ruining Zhao Jae Doo Chenkai Sun Heng Ji
University of Illinois at Urbana-Champaign
<EMAIL_ADDRESS>
###### Abstract
Pretrained large language models have revolutionized many applications but
still face challenges related to cultural bias and a lack of cultural
commonsense knowledge crucial for guiding cross-culture communication and
interactions. Recognizing the shortcomings of existing methods in capturing
the diverse and rich cultures across the world, this paper introduces a novel
approach for massively multicultural knowledge acquisition. Specifically, our
method strategically navigates from densely informative Wikipedia documents on
cultural topics to an extensive network of linked pages. Leveraging this
valuable source of data collection, we construct the CultureAtlas dataset,
which covers a wide range of sub-country level geographical regions and
ethnolinguistic groups, with data cleaning and preprocessing to ensure textual
assertion sentence self-containment, as well as fine-grained cultural profile
information extraction. Our dataset not only facilitates the evaluation of
language model performance in culturally diverse contexts but also serves as a
foundational tool for the development of culturally sensitive and aware
language models. Our work marks an important step towards deeper understanding
and bridging the gaps of cultural disparities in AI, to promote a more
inclusive and balanced representation of global cultures in the digital
domain.111Our code will be released at https://github.com/yrf1/LLM-
MassiveMulticultureNormsKnowledge-NCLB.
## 1 Introduction
Pretrained large language models (LMs) are increasingly used in applications
across diverse domains, ranging from question answering Brown et al. (2020);
Gangi Reddy et al. (2022) and chatbots Lin et al. (2020); Ouyang et al. (2022)
to content recommendation Wu et al. (2020) and norm violation detection Fung
et al. (2023). However, as their usage proliferates, an important concern
emerges – the potential for cultural bias and misappropriation Hershcovich et
al. (2022); Palta and Rudinger (2023); Li et al. (2023b). LMs, when not
equipped with geo-diverse knowledge and cultural sensitivity, can exhibit
significant disparities in performance when faced with textual data from
different regions. This imbalance disadvantages certain users groups and
exacerbates existing biases in NLP applications, perpetuating the presently
predominant Western-centric perspectives and knowledge bases. Hence, tackling
this issue is paramount not only for promoting fairness and inclusivity, but
also for fostering a more culturally aware and harmonious digital landscape.
Figure 1: An example illustration of different cultural practices, with
regards to Chinese New Year red envelopes across different geographical
regions such as Beijing/Shanghai versus HK/Macau/Guangdong. Figure 2: An
overarching view of our CultureAtlas benchmark construction process.
Previous approaches in benchmarking and improving the cross-cultural knowledge
of language models tend to either 1) focus on a predefined, narrow set of
coarse-grained cultures and cultural topics Yin et al. (2022), or 2) discover
cultural knowledge from large noisy corpora Nguyen et al. (2023) in which
important cultural elements often get filtered out in the data processing
stage or lost as cultural differences get intermingled, leading to a scenario
where LMs fail to learn information specific to individual subregions. Our
goal is to investigate and set the grounds for empowering language model with
reasoning capability on finer-grained cultural nuances that pertain to
different cultural subgroups and deeper topic coverage. For example, as
depicted in Figure 1, Chinese New Year red envelop gifting practices vary by
geographical subregions within a same country, which may potentially lead to
norm violations for people newly settling into an area who are unaware of the
common cultural practice that differ by region or recipient group.
In this work, we propose a culture knowledge acquisition process for
constructing a novel benchmark for assessing language models’ massively
multicultural reasoning capabilities. In particular, our culture knowledge
acquisition process seeks to combine the best of both worlds between bottom-up
discovery of culture knowledge discovery from the open web documents
(relatively noisy but large-scale data) and top-down discovery of culture
knowledge from targeted topic guidance (relatively clean but limited data). We
start from Wikipedia documents as our source of data, chosen for their clean
nature as their contents inside are subject to public audits and back-and-
forth information edits to strip away controversies until common ground is
reached. Specifically, we include the documents for each country that revolves
around an initial set of cultural topics, including education,
dating/marriage, and holiday customs, among others. We then continue to expand
on the relevant document sets based on linked topic pages within (e.g.,
"Chinese culture"->"Chinese Holidays"->"Chinese New Year Holiday"->"Red
Envelope"), and consider the sentences in the documents, in which a pretrained
LM categorizes it as a generalizable social or cultural norm rather than
instance-specific history or fact, to be the positive samples of cultural
knowledge in our dataset. We also propose an effective method to construct
negative (i.e., non-factual) cultural knowledge samples, cross-validated
through web search, for the purpose of probing language model cultural
reasoning robustness. Furthermore, we perform information extraction on these
positive and negative cultural knowledge samples, to derive fine-grained
cultural profile fields, including sub-country geographical regions, ethno-
linguistic identity, demographics, etc., for enabling deeper analysis on
situationalized socio-cultural context frames.
Our contributions can be summarized as follows:
* •
We present a novel data collection process to construct a massively
multicultural knowledge benchmarking dataset, CultureAtlas, spanning
significantly greater diversity in coverage of sub-country level geographical
regions, ethno-linguistic groups, etc., compared to prior benchmarks in the
multi-cultural NLP domain Yin et al. (2022); Nguyen et al. (2023); Ziems et
al. (2023).
* •
This dataset comprises of high-quality positive data samples and negative data
samples, with a $90^{+}\%$ pass rate in data quality check via human
assessment.
* •
Finally, we evaluate the performance of state-of-the-art foundation language
models on CultureAtlas, and demonstrate this new dataset as a useful resource
for identifying rooms for improvement in LM cultural awareness and debiasing.
## 2 Benchmark Construction
### 2.1 Data Collection and Data Preprocessing
Constructing a benchmark for assessing the fine-grained cultural knowledge of
language models is the first and most important step in enabling training
language models to incorporate better cultural knowledge downstream via
finetuning. To achieve this goal, we construct a novel dataset, CultureAtlas,
by collecting positive and negative samples of cultural knowledge assertions
that span diverse geographical subregions and ethnolinguistic groups, with
subsequent data processing as illustrated in Fig 2.
##### Positive Data Samples
We start from the observation that cultural webpages from publicly monitored
human-curated sources, such as Wikipedia, contain clean and commonly accepted
cultural assertions in general. These cultural sources are dense in
information, and, while not yet entirely comprehensive, serves as a valuable
initial source of information and seedling for further expansion. First, we
consider the set of documents from all countries worldwide. So we
systematically explore culturally relevant topics (e.g., culture, holidays,
dining etiquette, dating and marriage, education, honorifics, etc.) for each
country, using the Wikipedia API222https://pypi.org/project/Wikipedia-API/
tool to match and download corresponding Wikipedia pages. We expand on this
target set of documents by further including the hyperlinked document pages up
to two hops down. In addition to incorporating the default English documents
of these pages, we also include their document versions in the main language
corresponding to the culture in discussion, and translate the text content
into English. This ensures a more well-rounded understanding of cultural
nuances and perspectives, as complementary information exists across language
versions for the same document topic. Next, we process the corpus sentence by
sentence, first filtering out sentences that focus on very specific, socio-
culturally non-generalizable events or instances (see Appendix A). We refine
these sentences into self-contained cultural knowledge assertions by
eliminating ambiguous pronoun or phrase references and enriching each sentence
with any necessary information from the preceding context in the same
paragraph. Together, these steps constitute the initial discovery of cultural
knowledge assertions in CultureAtlas.
Because each unique combination of population dimension – ethnicity, language,
location, demographic background, etc. – plays a key role in shaping distinct
cultural practices, we proceed to better discern subtle situational
differences in norms across various cultures by extracting cultural knowledge
frames for each remaining sentence, with a fine-grained profiling approach
covering the following fields of information element:
* •
country location
* •
sub-country regional location – cities, states, and provinces under the
GeoNames333https://www.geonames.org/ knowledge base.
* •
ethnicity – ethno-linguistic groups from the ISO 639-3 code
table4441https://iso639-3.sil.org/code/oci.
* •
religion – all religious groups and denominations with a population of 1
million followers or more
* •
age – {infant, young children, teenager, young adult, adult, elderly}
* •
gender – {male, female, other}
* •
marital status – {single, engaged, married, divorced, widowed}
* •
occupation – open-domain fill in the blank
with age/gender/marital status/occupation pertaining to any person entities
involved in the norms. As depicted in Fig 2(A), language model prompting is
utilized to extract the values for these fields automatically, via a directed
question such as "[norm] \n Which gender group is mentioned or implied in the
sentence (male, female, transgender, other, or N/A):", with further details
shown in Appendix B. Note that information unknown or not mentioned is
regarded as "N/A".
| CANDLE | GeoMLAMA | NormsKB | NormBank | CultureAtlas
---|---|---|---|---|---
| Nguyen et al. (2023) | Yin et al. (2022) | Fung et al. (2023) | Ziems et al. (2023) | (Ours)
# countries | 176 | 5 | 5 | 160 | 193
# local regions | | | | |
(state/province-level) | 298 | 0 | 12 | 102 | 1089
(city-level) | 1,376 | 0 | 15 | 493 | 10,436
# religion | 14 | 0 | 3 | 30 | 42
# ethnolinguistic groups | 145 | 5 | 10 | 551 | 2,557
fine-grained norm framing | x | x | ✓ | ✓ | ✓
multi-ling. data source | x | x | x | x | ✓
Table 1: Our data collection of cultural norms contains greater coverage in local regions and ethno-linguistic groups, compared to previous work. It also involves data from multi-lingual sources as well as fine-grained cultural knowledge frame extraction. Original | Negative Cultural
---|---
Cultural Knowledge | Knowledge Generated
During the Chinese New Year, in Southern China, red envelopes are typically given by the married to the unmarried, most of whom are children. | In China, it is customary for students to present their teachers with red envelopes containing handwritten notes of gratitude at the end of each school term, symbolizing respect and appreciation for their guidance.
In Bhutan culture for special occasions and festivals, colourfully patterned silk kira and, more rarely, gho may be worn. | In Bhutan, there is a unique tradition of wearing "Khyenkhor Robes", woven with threads infused with blessings from Buddhist monks, during special ceremonies and festivals.
Table 2: Visualization of the original positive data samples and negative data
synthesized counterpart, in our CultureAtlas benchmark construction.
##### Negative Data Samples
In order to evaluate LM cultural knowledge, we prepare the data setting with
negative norm synthesis as illustrated in Fig 2(c). Basically, we take a
pristine original norm assertion and manipulate it through LLM prompting for
adversarial knowledge via the template of: "[orig. norm] \n Based on this
topic, come up with norms that are not true:". To ensure the negative norm
generation is indeed a non-factual fabrication, we perform automatic
verification, which is easy to scale. Specifically, we make use of a language
model self-check mechanism, asking the question of "[norm] \n Is this absurd
and/or very hard to believe? ‘Yes’ or ‘No’:" to GPT4, to filter out negative
sample candidates that are obviously absurd to believe. In particular, our
motivation in leveraging GPT4 is that it stands as the most advanced LM
backbone, with notable performance gap for open-sourced LMs or other propriety
LMs to bridge in and thereby deeming our benchmark negative samples especially
valuable. Subsequently, we follow with a web-check mechanism – web retrieving
relevant context and ensuring no entailment is found. Examples of negative
data samples, along with its original positive form, are visualized in Tab 2.
### 2.2 Quality Check on Data
To ensure data quality, we perform manual assessment on the CultureAtlas
dataset construction. Specifically, we take 10 random samples for each
intermediary data processing step of the positive data and negative data
respectively, and ask five human judges familiar with the subject matter
(e.g., self-identifying with geographical regions and cultural subgroups
across US, China, Korea, India) to determine whether the data samples that is
indeed either a correct cultural knowledge assertion when their ground truth
label (based on our expectation from the Sec 2.1 procedure) is "TRUE" or an
incorrect cultural knowledge assertion when their label is "FALSE". In our
quality check assessment guidelines, we clarify examples of poor positive
samples, such as having ambiguous pronoun references or lacking culture-
specificity in non-universal norms, as well as examples of poor negative
samples, such as contradicting known norms. As we can see in the qualitative
results of our dataset construction reported in Tab 3, the final post-
processed positive and negative samples are high-quality, achieving $90^{+}\%$
pass rate. The interannotator agreement is 0.79.
Approach | Pass Rate (%)
---|---
Pos. Data |
\- Orig Sent. | 49.5
\- Post Proc. Sent. | 93.2
Neg. Data |
\- Direct Gen. | 81.1
\- Direct Gen. w/ self-check | 90.1
\- Direct Gen. w/ self-check & web check | 92.0
Table 3: The average pass rate for CultureAtlas dataset samples, at each
processing step, based on human validation of post-processed cultural
knowledge assertions.
### 2.3 Descriptive Stats
As shown in Tab 1, our dataset covers over $1,089$ state or province level
regions, $10,436$ city level regions, and $2,557$ ethnolinguistic groups,
significantly exceeding prior work Nguyen et al. (2023); Yin et al. (2022);
Fung et al. (2023) in the cultural knowledge for NLP domain. We provide
detailed information on the # of subregion specific cultural knowledge frames
per country in Tab 9 of Appendix B, and the # of cultural knowledge frames per
ethnolinguistic group in Tab 10 of Appendix B, due to page restrictions. In
particular, as an example, all 56 official ethnic groups of China (e.g., Han,
Zhuang, Hui), as well as the linguistically distinct ethnic subgroups (e.g.,
Yue and Hakka of the Chinese Han population), are included in CultureAtlas.
##### Training-Testing Data Split
Tab 4 summarizes the number of culturally related document pages scraped and
sentences parsed, as well as post norm-relevance filtering cultural knowledge
assertion sentences and frame extractions. We partition 10,000 random samples
of cultural knowledge assertions that are particularly relevant for avoiding
norm violations to constitute the test set. All other data are provided for
future language model training and development purpose.
# of doc pages | $41$k
---|---
# of sent parsed | $907$k
# of sent, generalizable sociocultural knowledge | $127$k
# of sent, norm violation relevant w/ frame extract. | $21$k
Table 4: Size and scale of our collected dataset, where ‘k’ represents the kilo unit of a thousand data. | | | All Culture | High Resource | Mid Resource | Low Resource
---|---|---|---|---|---|---
| | | P | R | F | P | R | F | P | R | F | P | R | F
Llama2 | 7B | chat | 84.2 | 42.1 | 56.1 | 86.8 | 45.6 | 59.8 | 83.3 | 42.9 | 56.6 | 87.0 | 20.7 | 33.5
| chat-HF | 75.1 | 28.2 | 41.0 | 76.9 | 26.9 | 39.9 | 83.3 | 28.0 | 41.9 | 78.9 | 26.2 | 39.2
| 13B | chat | 63.6 | 77.1 | 69.7 | 56.1 | 80.9 | 66.3 | 64.1 | 75.5 | 69.3 | 53.3 | 20.5 | 29.6
| chat-HF | 89.9 | 20.0 | 32.7 | 91.8 | 91.8 | 28.3 | 91.8 | 20.6 | 33.6 | 92.2 | 19.3 | 31.9
Vicuna | 7B | chat-HF | 79.6 | 56.8 | 66.3 | 77.3 | 47.2 | 58.6 | 79.4 | 57.9 | 67.0 | 81.3 | 55.7 | 66.1
| 13B | chat-HF | 67.4 | 81.2 | 73.7 | 68.9 | 81.0 | 74.5 | 69.4 | 82.4 | 75.3 | 67.8 | 82.3 | 74.3
ChatGPT | 20B | chat-HF | 95.8 | 90.6 | 93.1 | 95.9 | 91.4 | 93.6 | 94.9 | 92.1 | 93.5 | 94.1 | 90.1 | 92.1
Table 5: Experimental results on benchmarking state-of-the-art foundation
large language model performance on the new CultureAtlas cultural knowledge
assessment benchmark. Precision (P), recall (R), F-score (F) scores are
reported.
## 3 Experiments
### 3.1 Task Setting
We evaluated the cultural knowledge and reasoning capability of state-of-the-
art pretrained large language models (LLMs) on the canonical norm descriptions
in our constructed benchmark, through a true-or-false binary classification
setup. As a reminder, the derivation of ground truth labels for "correct
"cultural knowledge assertion samples and "incorrect" cultural knowledge
assertion samples have been detailed under Sec 2.1 ("Positive Data Samples"
and "Negative Data Samples" paragraphs).
### 3.2 Model Setup
For the choice of language model in our experiments, we consider the present
day commonly-used open-source language model backbones:
* •
Llama-2 Touvron et al. (2023b) \- an open source foundation model from Meta
that is trained on 40% more data compared to its predecessor, LLaMA Touvron et
al. (2023a). We consider its -7b and -13b parameter size variants. For each
backbone size, we further include its variant of -chat, denoting finetuned
with dialogue data, and -hf, denoting trained with human preference alignment
data.
* •
Vicuna Chiang et al. (2023) \- an open source foundation model trained by
fine-tuning LlamA-2 on 700k samples of user conversations with ChatGPT
collected from the ShareGPT website 555https://sharegpt.com/. This model
series is optimized for efficiency through gradient checkpointing and flash
attention. We include its -7b and -13b parameter size variants.
We also consider the propriety closed-source language model backbones, which
may generally tend to have higher performance compared to publicly available
open-source model checkpoints but make research development and transparency
challenging:
* •
ChatGPT Ouyang et al. (2022) \- a closed-source foundation model from OpenAI,
trained with alignment data.
### 3.3 Results
Table 5 shows the results of our LM benchmarking. In particular, we notice
several interesting observations. First, we find that out of the open-source
models, Vicuna consistently perform better than Llama2 in cultural knowledge
when comparing across the same model backbone sizes. This indicates that the
training approach (e.g., choice of training data, optimization objective,
etc.) plays a key role in the cultural knowledge reasoning capability of these
LLMs. Secondly, we find that pre-existing explicit human feedback (HF)
alignment approaches does not necessary improve model performance in massively
multicultural fine-grained reasoning domains, potentially due to non-desirable
domain shift and catastrophic forgetting. This reaffirms the values in our new
benchmark proposal, for continuously measuring cultural awareness progress in
future language model development. We also observe a general positive
correlation between model performance in cultural-aware inference and model
parameter size, as expected.
In addition, we investigate LLM performance in culture reasoning patterns
across resource availability and topic domains. As shown in Table 5, we
investigated LM awareness in the cultural knowledge pertaining to country-
level ‘high-resource’ (e.g., US/China/France/Spain/Japan), ‘mid-resource’
(e.g., Turkiye/Egypt/Iran/Malaysia/Argentina), and ‘low-resource’ (e.g.,
Lao/Bhutan/Congo/Serbia) culture groups, as categorized by societal-wide
economic development, which in turn affects the linguistic resources
availability for constituting LM training data.
Moreover, LM performance tends to differ across cultural topics, demonstrating
higher performance for example in “education" and “holiday" practices over
“clothing" and “cuisine" practices, as shown in Fig 3. This may potentially be
due to diverse finer-grained domain-specific and region-specific information
elements typically involved in “clothing" and “cuisine" topic discussions,
whereas “education" and “holiday" practices may tend to be more universal.
Finally, while our research community lacks specific training details of the
closed-source models (e.g., ChatGPT), we believe that by including them in our
benchmark comparison, we can help shed light on the performance gap between
open-source pretrained LLMs and these closed-source models, to better bridge
this performance difference in future work.
Figure 3: LLM performance by Topic (result from a single LM backbone ad-hoc, llama2-7b). | Norm Violation Relevant Culture Knowledge Assertion Samples
---|---
True Positive | $\bullet$ In Indian culture, when eating rice, it is mixed with curry, picking up small quantities with the fingers and pushing it into the mouth with the thumb.
$\bullet$ In Bhutan culture, for special occasions and festivals, colourfully
patterned silk kira and, more rarely, gho may be worn.
False Positive | $\bullet$ The American flag protocol dictates that the flag should be flown on all buildings, both public and private, as a sign of respect and loyalty to the nation.
| $\bullet$ In Rioplatense Spanish, the pronoun "vos" is commonly used to
address strangers and authority figures, while it is considered impolite to
use it with friends, family members, and close acquaintances.
| $\bullet$ South Korea and North Korea have different vocabulary standards
because they consider their languages to be completely distinct and unrelated
to each other.
| $\bullet$ In Chinese culture, particularly in Macao, it is believed that
giving money in amounts that include the number four brings good luck and
prosperity.
| $\bullet$ The society in Kuwait City is highly strict about traditions in
the Gulf Arab region.
False Negative | $\bullet$ In Catalonia, there is a strong social and political consensus on the language policies that support the use of Catalan.
| $\bullet$ In Botswana, it is customary to use the salutation ’kgosi’ before
the name of a chief, providing an important cultural and social distinction.
| $\bullet$ In Barbados, people drive on the left side of the road, similar
to the driving habits in the United Kingdom due to their history as a former
British colony.
| $\bullet$ The practice of inserting Mandarin into conversations in
Shanghainese is very common, especially among young people.
| $\bullet$ Personal pronouns in Shanghainese do not distinguish gender or
case.
| $\bullet$ In Argentina culture, hot but not boiling water is poured into
the gourd, drunk, then the mate is refilled.
True Negative | $\bullet$ In Indian culture, when eating rice, people commonly mix it with chutney, a flavorful condiment made from a variety of ingredients such as fruits, vegetables, herbs, and spices.
$\bullet$ In Malay culture, people commonly greet each other with the phrase
"Khabar", which roughly translates to "what’s up" or "how are you" in English.
Table 6: Qualitative error analysis on LM culture knowledge reasoning
capability (of Vicuna-13B) in zero-shot true/false inference.
#### 3.3.1 Ablation on the Challenging Setting brought by Fine-Grained
Cultural Knowledge Frame Profiling
In this subsection, we further investigate the potential limitations of
existing pretrained Large Language Models (LLMs) in understanding nuanced
cultural nuances within different situational contexts, along our massively
multicultural task domain. Our natural intuition is that pretrained LLMs may
general tend to lack finer-grained knowledge on the cultural nuances
pertaining to subtle situational differences with respect to cultural frame
profiling. To verify this hypothesis through empirical study, for each culture
frame dimension, such as sub-country geo-region, ethnicity, age, gender, etc.,
we first isolate a subset of the CultureAtlas evaluation data with cultural
knowledge assertions that generally applies across this dimension, which we
refer to as “Gen”, and isolate another subset of the CultureAtlas evaluation
data with cultural knowledge assertions that applies specifically to a certain
bucket/criteria across this dimension, which we refer to as “Spec”. Then, we
perform cross-comparison on LLM performance patterns, under data scenarios
that are general (“Gen”) versus specific (“Spec”) in condition/critieria along
each of these cultural frame profiling dimensions. Indeed, we find that lack
fine-grained cultural commonsense knowledge is an area where there remains
interesting rooms for improvement for LLM models. As shown in Table 7 (the
result from a single ad-hoc LM backbone, llama2-7b), the zero-shot true/false
inference capability of LLM significantly drops as we probe finer-grained
cultural information.
| Spec. | Gen
---|---|---
Sub-Country Location | 22.1 | 35.0
Ethnicity | 27.5 | 35.5
Religion | 17.9 | 35.0
Marital Status | 13.7 | 33.6
Occupation | 27.3 | 35.0
Table 7: F-score performance comparison, in %, by general (‘gen’) versus fine-
grained cultural profile framing specific (‘spec’) knowledge, such as between
country-level and province-level cultural group.
#### 3.3.2 Error Analysis
Tab 6 visualizes results qualitatively, shedding light on the challenges for
an off-the-shelf pretrained language model (LM) in accurately reasoning about
cultural practices across different societies. While a LM may correctly grasp
certain cultural practices, such as properly recognizing traditional ways of
eating rice in Indian culture and the ceremonial dress in Bhutan for special
occasions, as well as accurately recognizing misconceptions in negative
samples on mixing rice with chutney in Indian culture or common greetings in
Malay, it demonstrates a lack of cultural knowledge and reasoning robustness
in other less well-represented topics and geographical regions. For example,
the model also produced false positives, such as in regards to the exaggerated
protocol around the American flag, suggesting misunderstandings of cultural
norms. False negatives, such as the underappreciated practice of drinking mate
in Argentina, point to the model’s oversight of genuine cultural customs.
Overall, the error analysis reveals the inconsistent performance of LMs in
capturing the breadth and depth of the different cultural knowledge in the
world around us, revealing a significant area for improvement in LM cultural
commonsense reasoning.
## 4 Related Work
##### Importance of Cultural Knowledge in NLP Tasks
While large language models have generally embedded large parametric knowledge
from large text corpora during its pretraining stage Petroni et al. (2019),
these models are also typically imposed with normative bias due to imbalanced
representation at the data source Emelin and Sennrich (2021); Arora et al.
(2022). Cultural knowledge is an integral part to the success of large
language model (LLM) reasoning in a wide array of downstream applications. For
example, there has been recent explorations on the vital role cultural
knowledge plans in helping answer commonsense questions Palta and Rudinger
(2023); Yin et al. (2022), understand societal moral conventions Ramezani and
Xu (2023); Emelin et al. (2021), analyze and mitigate social biases Sap et al.
(2020); Yang et al. (2023), detect norm violations Fung et al. (2023); Li et
al. (2023a), correct conversational dialogues Ziems et al. (2022), and
ultimately tune LLMs to align with the helpful and harmless principles of
constitutional AI Bai et al. (2022). Of ongoing interest to the NLP community
is scrutinizing LMs on social minority understanding Sun et al. (2023), which
turns out that PLMs can learn norms diverging from social majority only when
they are fine-tuned accordingly due to a presence of normative bias Kiehne et
al. (2022).
##### Cultural Knowledge Acquisition for improved LLM Training
Hershcovich et al. (2022) explains culture as a concept of identity that can
be examined from the dimensions of objectives and values, linguistic form and
style (e.g., honorific reference terms when addressing a person), and common
ground (e.g, socio-cultural norms, shared event occurrences, etc.). Our
cultural knowledge acquisition process follows this theory of cultural
definition and covers the dimensions of culture outlined above. In terms of
the genre of data source for cultural knowledge discovery in practice,
cultural knowledge has been predominantly gleaned from conversational
dialogues Fung et al. (2023) or web sources such as Reddit/Zhihu discussion
forums Forbes et al. (2020); CH-Wang et al. (2023) and the Common Crawl Nguyen
et al. (2023) – both of which tends to be relatively sparse in culturally
relevant information and also noisy – or through knowledge elicitation of LLM
parametric knowledge through prompting Ziems et al. (2023), but this may be
limited in the scope of available information that can be extracted due to the
stochastic parrot property observed in LLMs when a cultural topic falls out of
a LLM’s pretrained knowledge boundary.
##### Multilingual LM Reasoning and implicit multicultural knowledge
Previous research Jiang et al. (2020); Clark et al. (2020) on language models
have demonstrated strong capability to perform reasoning in multilingual
settings, which is an initial step towards overcoming cultural barriers. The
progression of research includes extending LM reasoning to low-resource
language setting, such as for name tagging & knowledge base linking Pan et al.
(2017); Wen et al. (2021) through annotation transferring, which lays a
promising foundation for reasoning across linguistic groups. However, the
existing language models struggle with the cultural bias, primarily due to the
lack of awareness of implicit multicultural knowledge. Recent studies have
highlighted these issues; for instance, Havaldar et al. (2023) has identified
the models’ underperformance in recognizing cultural variations in specific
phenomena, such as emotion detection across different countries. Another
category of recent work has focused on evaluating performance on
underrepresented languages, where Deas et al. (2023) has revealed biases
against African American languages, leading to overlooked race-related issues
in speech recognition and toxicity detection tasks. These findings underline
the necessity to develop a new framework capable of acquiring cultural
knowledge, aimed at addressing the cultural imbalances present in existing
datasets used for training language models. Instead of stylistic linguistic
conveyance, our work focuses on cultural knowledge acquisition based on fine-
grained semantic variations, sourcing from over 500+ geosubregions and 2000+
ethnolinguistic groups.
## 5 Conclusions and Future Work
In this work, we explored an exciting area of massively multicultural
knowledge probing of language model backbones, which has important impacts to
norm violation detection and mitigation in assisting human-human interaction
behavior across the many different subregions and ethnolinguistic groups
around the world. We proposed a novel method for large-scale data collection
across curated data sources and web-retrieval enhancement, which we verified
with quality checks. Leveraging this constructed dataset, we achieved
benchmarking the fine-grained cultural knowledge of popular language models.
We foresee the impact of our work to contribute a useful resource and LM
evaluation setting for the NLP community to develop more culturally-inclusive
foundation models upon in the future. Finally, for future research directions,
we will also investigate the effect that multimedia settings (e.g., vision and
language) and multilingual (e.g., low-resource) settings have on foundation
cultural reasoning across different fine-grained sub-cultures.
## Ethical Considerations & Broader Impact
Our work is dedicated to advancing language model (LM) development practices
to more accurately reflect the diverse social practices and cultural customs
present in human societies across the world. This effort serves to enable
improvements in LM norm violation reasoning capabilities, and at a higher-
level, re-emphasizes the importance of overarching principles to human-model
alignment. In formulating the ethical framework for our research, we carefully
consider the key dimensions inherent to our work, to ensure that the modeling
outcomes are in line with ethical standards and societal expectations. We
introduce a novel normative framework aimed at enhancing the detection of
incorrect cultural knowledge in future models. Our experimental setup
incorporates commonsense reasoning, grounding the model’s cultural knowledge
in alignment with socially-situated human preferences.
We acknowledge limitations in our current work. The ground truth labels (i.e.,
"positive" and "negative" data samples) that we utilize for LM cultural
knowledge benchmark evaluation are automatically derived, due to scalability
advantages. Our constructed dataset has been manually assessed and determined
to be high-quality with a $90^{+}\%$ pass rate in data sample quality check.
In other words, cultural knowledge assertions labeled "positive" are indeed
rated as a true sample by human assessors, while cultural knowledge assertions
labeled "negative" are indeed rated as non-factual samples for the large
majority. If time and manual labor cost were ample, it would be meaningful to
prepare human-curated cultural knowledge assertions and investigate any
potential differences as a future research direction. However, we also point
that it is extremely challenging to find competent human annotators covering
the different geographical subregions and ethno-linguistic groups at a massive
scale. Many crowdsourcing platforms for data annotation are skewed towards
certain demographic subgroups, such as primarily American-based or Western
centric in the Amazon MTurk service. Hence, our work serves as an extremely
valuable first step towards preparing a meaningful benchmark for massively
multicultural fine-grained LM knowledge reasoning, based on publicly audited
Wikipedia document sources and additional data preprocessing procedure.
An essential element of our ethical framework is ensuring balanced
representations across cultural groups, with a special focus on including
perspectives from low-resource settings. Due to current design and imbalanced
training data, LLMs may lack the capability to fully understand social norms
and including biases across different countries, suffering from normative bias
Emelin and Sennrich (2021); Arora et al. (2022). We commit to the ongoing
journey of cultural knowledge enhancement and normative correction. In
particular, we aim to not only address issues of fairness and equity, but also
enrich our NLP models by considering a broader range of experiences and
cultural knowledge. By adhering to cultural definitions, we can build cultural
knowledge systems that are inclusive and reflective of the global community.
Our approach paves the path for mitigating the risks of subtle cultural bias
in language model training, and fosters understanding across fine-grained
cultural divides.
## Acknowledgement
This research is based upon work supported by U.S. DARPA CCU Program No.
HR001122C0034. The opinions, views and conclusions contained herein are those
of the authors and should not be interpreted as necessarily representing the
official policies, either expressed or implied, of DARPA or the U.S.
Government. The U.S. Government is authorized to reproduce and distribute
reprints for governmental purposes notwithstanding any copyright annotation
therein.
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## Appendix A Characterizing Cultural Knowledge Specificity for Data
Filtering
In our CultureAtlas dataset construction process, we want to focus on
socioculturally relevant knowledge assertions that are not too event or
instance specific. For example, "In 2020, China tops the QS Asia University
Rankings list with over 120 universities including in the ranking, and five
Chinese universities appear in the Asia Top 10, which is more than any
country." would be culturally relevant but too event-specific. To filter out
such instances, we utilize the facebook/bart-large-mnli model to perform
classification on each candidate sentence, between the classes of "general
assertion" and "specific fact or instance". Through this approach, we found
that approximately 52% of the original pristine sentences from culturally-
relevant Wikipedia pages fall under the "general assertion" category, which we
retain in the dataset.
## Appendix B Culture Profile Extraction Performance
In this section, we expand on low-level details on the culture profile
extraction process methodology and quality check results, leveraging prompting
Culture Profile Field | Directed Question Answering
---|---
interaction nature categorization | Is this an individual human behavioral norm or human-human behavioral norm?
topic distribution modeling | Is this a social norm, cultural norm, belief or ritual, history, politics, or fact?
country-level extraction | Which country is mentioned or implied in the sentence? Answer N/A if unknown.
sub-country level extraction | Which state/province/city/subcountry region is mentioned
| or implied in the sentence (or answer N/A):
ethnicity extraction | Which ethnic group is mentioned or implied in the sentence?
| Answer N/A if unspecified?
ethnic subgroup extraction | Which ethnolinguistic subgroup is mentioned or implied in the sentence?
| Answer N/A if unspecified.
age extraction | Which age group is mentioned or implied in the sentence?
gender extraction | Which gender group is mentioned or implied in the sentence
| (male, female, transgender, or N/A)
marital status | Which marital status is mentioned or implied in the sentence.
religion belief extraction | Which religious group is mentioned or implied in the sentence (or answer N/A):
occupation extraction | Which occupation is mentioned or implied in the sentence?
| Answer N/A if unspecified
Table 8: Details on the prompt template for cultural profile extraction Afghanistan | 29 | 0.2k | Georgia | 35 | 0.2k | Afghanistan | 29 | 0.2k
---|---|---|---|---|---|---|---|---
Albania | 0 | 0.0k | Germany | 301 | 6.2k | Zimbabwe | 26 | 0.3k
Algeria | 38 | 0.4k | Ghana | 30 | 0.3k | Zambia | 9 | 0.2k
Andorra | 4 | 0.0k | Greece | 98 | 1.1k | Yemen | 8 | 0.0k
Angola | 22 | 0.2k | Grenada | 0 | 0.0k | Viet Nam | 43 | 0.8k
Antigua and Barbuda | 1 | 0.0k | Guatemala | 0 | 0.0k | Venezuela | 9 | 0.1k
Argentina | 222 | 1.4k | Guinea | 0 | 0.0k | Vanuatu | 1 | 0.0k
Armenia | 118 | 0.6k | Guinea-Bissau | 17 | 0.4k | Uzbekistan | 34 | 0.2k
Australia | 190 | 2.8k | Guyana | 0 | 0.0k | Uruguay | 40 | 0.4k
Austria | 78 | 0.9k | Haiti | 0 | 0.0k | United States | 1452 | 42.4k
Azerbaijan | 49 | 0.4k | Honduras | 13 | 0.1k | Tanzania | 118 | 0.6k
Bahamas | 0 | 0.0k | Hungary | 37 | 0.3k | United Kingdom | 229 | 7.2k
Bahrain | 6 | 0.1k | Iceland | 0 | 0.0k | United Arab Emirates | 38 | 0.6k
Bangladesh | 105 | 0.8k | India | 847 | 18.4k | Ukraine | 87 | 1.3k
Barbados | 4 | 0.1k | Indonesia | 364 | 6.4k | Uganda | 61 | 0.2k
Belarus | 16 | 0.2k | Iran | 22 | 0.1k | Tuvalu | 2 | 0.0k
Belgium | 75 | 1.4k | Iraq | 34 | 0.3k | Turkmenistan | 12 | 0.1k
Belize | 2 | 0.0k | Ireland | 99 | 1.2k | Türkiye | 42 | 0.7k
Benin | 14 | 0.1k | Israel | 71 | 2.2k | Tunisia | 94 | 1.8k
Bhutan | 53 | 0.3k | Italy | 560 | 8.5k | Trinidad and Tobago | 9 | 0.2k
Bolivia | 18 | 0.1k | Jamaica | 0 | 0.0k | Tonga | 0 | 0.0k
Bosnia and Herzegovina | 40 | 0.2k | Japan | 388 | 5.7k | Togo | 16 | 0.0k
Botswana | 61 | 0.2k | Jordan | 15 | 0.1k | Timor-Leste | 2 | 0.0k
Brazil | 226 | 2.2k | Kazakhstan | 23 | 0.1k | Thailand | 223 | 2.9k
Brunei Darussalam | 0 | 0.0k | Kenya | 61 | 0.6k | Tajikistan | 4 | 0.0k
Bulgaria | 129 | 1.7k | Kiribati | 9 | 0.1k | Syrian Arab Republic | 0 | 0.0k
Burkina Faso | 6 | 0.1k | Kuwait | 0 | 0.0k | Switzerland | 140 | 1.9k
Burundi | 13 | 0.0k | Kyrgyzstan | 6 | 0.1k | Sweden | 124 | 1.9k
Cabo Verde | 0 | 0.0k | Laos | 80 | 0.2k | Suriname | 3 | 0.0k
Cambodia | 73 | 0.6k | Latvia | 25 | 0.1k | Sudan | 20 | 0.2k
Cameroon | 29 | 0.1k | Lebanon | 33 | 0.5k | Sri Lanka | 12 | 0.2k
Canada | 198 | 2.8k | Lesotho | 22 | 0.3k | Spain | 162 | 3.5k
Central African Republic | 4 | 0.0k | Liberia | 9 | 0.0k | South Sudan | 13 | 0.3k
Chad | 5 | 0.0k | Libya | 10 | 0.1k | South Africa | 79 | 1.0k
Chile | 47 | 0.3k | Liechtenstein | 6 | 0.0k | Somalia | 23 | 0.1k
China | 409 | 8.1k | Lithuania | 25 | 0.4k | Solomon Islands | 5 | 0.0k
Table 9: The # of documents and cultural knowledge assertion sentences, per
culture by country, that are specific to sub-country level geographical
regions.
with various state-of-the-art pretrained large language model (LLM) backbones.
Specifically, Table 8 details the prompt template details.
Medumba | 145 | | Awabakal | 35 | | Kalapalo, Kuikúro-Kalapálo | 17 |
---|---|---|---|---|---|---|---|---
Germanic | 117 | | Eleme | 35 | | Bengkala | 16 |
Central Dusun, Kadazan Dusun | 115 | | Ahtena | 33 | | Bangala | 16 |
Notre | 112 | | Ayoreo | 32 | | Chilcotin | 16 |
Doga | 107 | | Bakumpai | 32 | | Northern Dagara | 16 |
Hopi | 107 | | San Blas Kuna | 32 | | Gbagyi | 16 |
Mescalero-Chiricahua Apache | 99 | | Kabardian | 32 | | Kayardild | 16 |
Kashubian | 96 | | Hiberno-Scottish Gaelic | 31 | | Jah Hut | 16 |
Ainu (Japan) | 94 | | Jahanka | 31 | | Kanuri | 16 |
Assiniboine | 91 | | Kambaata | 31 | | Kankanaey | 16 |
Arapaho | 89 | | Batek | 30 | | Barai | 15 |
Jicarilla Apache | 88 | | Hunsrik | 30 | | Djabugay, Dyaabugay | 15 |
Batak languages | 86 | | Bwa | 29 | | Emilian | 15 |
Izere | 81 | | Gadang | 29 | | Macushi | 15 |
Kodava | 75 | | Cowlitz | 28 | | Azha | 14 |
Cappadocian Greek | 74 | | Dolpo | 28 | | Igede | 14 |
Efik | 73 | | Kuku-Yalanji | 28 | | Khorasani Turkish | 14 |
Algonquin | 72 | | Korak | 27 | | Bundeli | 13 |
Hittite | 70 | | Harari | 26 | | Columbia-Wenatchi | 13 |
Etruscan | 69 | | Humla | 26 | | Gooniyandi | 13 |
Huichol | 67 | | Shambala | 26 | | Kaska | 13 |
Hajong | 66 | | Dhanggatti, Dyangadi | 25 | | Amahuaca | 12 |
Coast Miwok | 65 | | Gumatj | 25 | | Atsugewi | 12 |
Mycenaean Greek | 62 | | Reel | 24 | | Awetí | 12 |
Jju | 62 | | Banggarla | 24 | | Southern Luri | 12 |
Pacific Gulf Yupik | 60 | | Hidatsa | 24 | | Adhola | 11 |
Heiltsuk | 60 | | Khanty | 24 | | Qimant | 11 |
Jukun Takum | 60 | | Gros Ventre | 23 | | Bidyogo | 11 |
Khasi | 59 | | Baga Sitemu | 23 | | Gambera | 11 |
Javanese | 58 | | Koasati | 23 | | Guro | 11 |
Garre | 57 | | Igala | 23 | | Kurdish | 11 |
Gujarati | 57 | | Yaka (Congo) | 23 | | Hermit | 11 |
Chickasaw | 56 | | Adnyamathanha | 22 | | Molale | 11 |
Gunditjmara | 55 | | Baras | 22 | | Pemon | 10 |
Esselen | 54 | | Burarra | 22 | | Sari | 10 |
Yanomamö | 54 | | Ejagham | 22 | | Bhunjia | 10 |
Bhojpuri | 53 | | Ngadjuri | 21 | | Laba | 10 |
Beothuk | 52 | | Kalenjin | 21 | | Lasi | 10 |
Goan Konkani | 52 | | Gheg Albanian | 20 | | Arbore | 9 |
Idoma | 52 | | Badaga | 20 | | Kaingang | 9 |
Atakapa | 51 | | Chuvash | 20 | | Col | 9 |
Chipewyan, Dene Suline | 50 | | Cocopa | 20 | | Lozi | 9 |
Daai Chin | 50 | | Dimasa | 20 | | Leti (Indonesia) | 9 |
Colonia Tovar German | 50 | | Eyak | 20 | | Akuntsu | 8 |
Jakun | 50 | | Hyam | 20 | | Djauan, Jawoyn | 8 |
Bodo (India) | 49 | | Greenlandic, Kalaallisut | 20 | | Adiwasi Garasia | 8 |
Shor | 49 | | Mixed Great Andamanese | 19 | | Holikachuk | 8 |
Lushai | 49 | | Karadjeri, Karajarri | 19 | | Jiru | 8 |
Con | 48 | | Gilaki | 19 | | Kurichiya | 8 |
Kutenai | 47 | | Ikulu | 19 | | Korwa | 8 |
Djawi | 46 | | Worimi | 19 | | Hijazi Arabic | 7 |
Angor | 44 | | Lobi | 19 | | Bugun | 7 |
Chitimacha | 42 | | Ingrian | 18 | | Cuban | 7 |
Gitxsan | 42 | | Keliko | 18 | | Cheq Wong, Chewong | 7 |
Kickapoo | 41 | | Keiga | 18 | | Gata’ | 7 |
Sudanese Arabic | 40 | | Ladin | 18 | | Gureng Gureng | 7 |
Darlong | 38 | | Guerrero Amuzgo | 17 | | Gunwinggu | 7 |
Haisla | 38 | | Bora | 17 | | Koba | 7 |
Igbo | 38 | | Chamacoco | 17 | | Lanoh | 7 |
Kalapuya | 38 | | Mro-Khimi Chin | 17 | | Apatani | 6 |
Upper Kuskokwim | 37 | | Isoko | 17 | | Tuki | 6 |
Atikamekw | 36 | | Krymchak | 17 | | Bote-Majhi | 6 |
Esperanto | 36 | | Kota (India) | 17 | | Bwile | 6 |
Table 10: The # of documents and cultural knowledge assertion sentences per
culture by ethnolinguistic group
|
# Better Query Graph Selection for Knowledge Base Question Answering
Yonghui Jia, Wenliang Chen
Institute of Artificial Intelligence, School of Computer Science and
Technology,
Soochow University, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
This paper presents a novel approach based on semantic parsing to improve the
performance of Knowledge Base Question Answering (KBQA). Specifically, we
focus on how to select an optimal query graph from a candidate set so as to
retrieve the answer from knowledge base (KB). In our approach, we first
propose to linearize the query graph into a sequence, which is used to form a
sequence pair with the question. It allows us to use mature sequence modeling,
such as BERT, to encode the sequence pair. Then we use a ranking method to
sort candidate query graphs. In contrast to the previous studies, our approach
can efficiently model semantic interactions between the graph and the question
as well as rank the candidate graphs from a global view. The experimental
results show that our system achieves the top performance on ComplexQuestions
and the second best performance on WebQuestions.
## Introduction
Knowledge Base Question Answering (KBQA) is a popular task defined to take
natural language questions as input and return corresponding entities or
attributes from knowledge bases, such as DBPedia (Auer et al. 2007) and
Freebase (Bollacker et al. 2008). One representative line of approaches to
KBQA builds on semantic parsing (SP) that converts input questions into formal
meaning representations and then transforms them into query languages like
SPARQL (Berant et al. 2013; Yih et al. 2015; Sun et al. 2020). There are two
types of SP-based solutions. One is using generic meaning representations,
such as $\lambda-DCS$ (Liang 2013). However, this type of solutions tends to
suffer from the gap between the set of ontology or relationship in the meaning
representations and the set in knowledge bases (Kwiatkowski et al. 2013).
Figure 1: The example question and its corresponding query graph structure.
Another type of solutions is to use query graph to represent the semantics of
questions, which is supposed to be able to overcome the issue mentioned above
(Yih et al. 2015; Bao et al. 2016; Hu et al. 2017). Figure 1 presents an
illustrating query graph whose nodes and edges correspond to the entities and
relationships in a knowledge base. By using query graph as the representation,
the process of KBQA can be divided into two steps: query graph generation and
query graph selection. The former step constructs a set of query graph
candidates from the input question, while the latter decides the optimal query
graph that is used to retrieve the final answer. We can see that the component
of query graph selection is critical to the overall performance of KBQA
systems.
Figure 2: Query graph generation process of “Who is the highest prime minister
of spain after 1980?”. Figure (a) shows the corresponding focus nodes linking
result, Figure (b) shows the corresponding main path, Figure (c) shows the
result after adding entity constraints to the main path, and Figure (d) shows
the query graph after adding all constraints.
Query graph selection is essentially a matching task between the question and
candidate query graphs. Existing systems put focus on encoding the query
graphs with hand-crafted features (Yih et al. 2015; Luo et al. 2018). In these
works, they first calculate the semantic similarity between the query graph
and the question using cosine similarity function. Then the similarity score
is used as a feature, together with other local features to represent the
query graph and the question. Finally, they feed the feature representation
into a one-layer neural network model to obtain the score. Previous approaches
have achieved a certain success for this task. However, we argue that 1)
simply using cosine distance function to measure the semantic similarity leads
to the loss of interaction information between the graph and the question, and
2) the hand-crafted features are usually not robust and are often not
necessary for deep neural networks.
To address the above problems, in this paper we propose to translate the
matching between the question and the query graph into the matching between
two sequences for naturally modeling the interaction between the question and
query graph. To this end, we linearize the query graphs into sequences. This
strategy makes the question and linearized query graphs are both in sequence
format, which is more convenient for using mature sequence modeling methods
such as BERT (Devlin et al. 2019) and GPT-3 (Brown et al. 2020). In addition,
we select the optimal query graph with a ranking strategy, hoping to take the
relationship between candidate query graphs into consideration. Inspired by
learning to rank methods (Li 2011; Pîrtoacă, Rebedea, and Ruseti 2019; Han et
al. 2020), we utilize the listwise strategy to sort candidate query graphs
from a global view, instead of the pairwise strategy which is used in the
previous study (Luo et al. 2018). Experimental results on two widely-used KBQA
datasets demonstrate the effectiveness of our proposed approach: the best
performance on ComplexQuestions and the second best on WebQuestions.
Overall, we make the following contributions.
* •
We propose a novel approach for better query graph selection in KBQA. In our
approach, we convert the query graph into the corresponding sequence format
and thus the problem of matching between the question and the query graph is
translated into the matching between two sequences. This allows us to use BERT
to efficiently model interactions between the graph and the question.
Moreover, our approach does not require any hand-crafted features.
* •
In addition, we use the listwise strategy to sort the candidate query graphs
which takes the relationship among the candidate graphs into consideration
from a global view. Compared with the Pairwise Ranking used in (Luo et al.
2018), Listwise Ranking achieves better performance on both two datasets.
Figure 3: The process of converting query graph to sequence.
## Our Approach
In this section, we describe our approach in detail. We divide the KBQA
process into two subtasks: query graph generation and query graph selection.
Formally, given a question $q$ and a knowledge base (KB), the semantics of $q$
is analyzed through the query graph generation, and a set of candidate query
graphs $G=\\{g_{1},g_{2},...,g_{n}\\}$ is obtained. Then, an optimal query
graph $g^{*}$ is selected from the candidate set $G$ through the query graph
selection. Finally, we convert $g^{*}$ into the SPARQL format to retrieve the
final answer to question $q$.
Compared with the previous studies of using query graphs (Yih et al. 2015; Hu
et al. 2017; Luo et al. 2018), we improve our system by using a different
solution on query graph selection. The basic idea is that, we linearize each
candidate query graph into a sequence and thus the problem of matching between
the question and the candidate query graph becomes the matching between two
sequences. To select the optimal query graph $g^{*}$, we propose a ranking
method with the listwise strategy, hoping to take the relationship between the
candidate query graphs into consideration from a global view.
### Query Graph Generation
The goal of the query graph generation is to map the question into a semantic
representation in the form of graphs. In this step, we follow the procedure of
the previous studies (Yih et al. 2015; Luo et al. 2018) to generate the
candidate query graphs.
Given question $q$, we first conduct focus nodes linking to identify four
types of constraints in the question, which are entity, implicit type, time
interval, and ordinal. For entity linking, we utilize the tool SMART (Yang and
Chang 2015) to obtain (mention, entity) pairs. For type linking, we use word
embeddings to calculate the similarity between consecutive sub-sequences in
the question (up to three words) and all the type words in the knowledge base,
and select the top-10 (mention, type) pairs according to the similarity
scores. Regarding time word linking, we use regular expression matching to
extract time information. As for ordinal number linking, we use a predefined
ordinal vocabulary and “ordinal number+superlative” pattern to extract the
integer expressions. 111About 20 superlative words, such as largest, highest,
latest. We also use the entity enrichment method in Luo et al. (2018) to
improve focus nodes linking. Figure 2(a) shows an example after focus nodes
linking.
After focus nodes linking, we get the main path by performing a one-hop and
two-hop search based on the linked entity words, as shown in Figure 2(b).
Next, entity constraints are added to the nodes in the main path. Figure 2(c)
shows the state after the operation of this step. Then we add type
constraints, time constraints and ordinal constraints in turn, and finally get
the query graph as shown in Figure 2(d).
Through the above procedure, we can obtain the candidate query graph set
$G=\\{g_{1},g_{2},...,g_{n}\\}$ for query graph selection.
### Query Graph Selection
Due to the existence of ambiguity, the query graph generation may produce more
than one, often hundreds of, candidate query graphs. Thus it is necessary to
apply a matching operation to select the optimal query graph $g^{*}$ from the
candidates. In this section, we first describe how to convert each $g$ in $G$
into sequence $g^{s}$. Then, we show how to encode the pair of $q$ and
$g^{s}$. Finally, we describe the selection process. Inspired by the research
in learning to rank (Li 2011; Pîrtoacă, Rebedea, and Ruseti 2019; Han et al.
2020), we select query graph with different ranking strategies, i.e.,
Pointwise Ranking, Pairwise Ranking, and Listwise Ranking.
#### Transforming Query Graph into Sequence
The process of converting query graph $g$ to sequence $g^{s}$ can be regarded
as the disassembly process of constructing the query graph. When constructing
the query graph, we first search the main path and then add four constraints
of type, entity, time, and ordinal to the main path. Therefore, the whole
query graph structure contains at most five components, which is much simpler
compared with the general graph structure. And more importantly, each
component has a fixed semantic meaning.
Considering the fixed structure of the query graph, we transform the query
graph into the corresponding sequence according to the predefined sub-paths
order. Specifically, we divide the query graph into different sub-paths
according to different components. Through graph decomposition, we can get
five sub-path sequences: TypePath, EntityPath, TimePath, OrdinalPath and
MainPath. For example, the EntityPath corresponding to the entity constraint
“Prime minister” in Figure 3 is “basic title prime minister.”. Finally, the
five sub-path sequences are combined to form the corresponding query graph
sequence. It is worth noting that we use some additional tokens, [unused0-3],
to separate different sub-path sequences. As shown in Figure 3, the query
graph sequence is “people person. [unused0] basic title prime minister.
[unused1] from after 1980. [unused2] height max 1. [unused3] spain governing
officials – office holder [A]”, in which ‘[A]’ is the real answer string, not
just unified padding.
Figure 4: (a) Query Graph and Question Encoding Framework. (b) Different
Ranking Strategies Framework, where “$p_{qg^{s}_{1}}$” represents the sequence
of the question and the positive query graph, “$p_{qg^{s}_{2}}$”,
“$p_{qg^{s}_{3}}$” and “$p_{qg^{s}_{4}}$” represent three sequences of the
question and negative query graphs.
#### Encoding Query Graph Sequence and Question
After query graph $g$ is converted into sequence $g^{s}$, the task of matching
question $q$ and query graph $g$ becomes a task of matching question $q$ and
query graph sequence $g^{s}$. This allows us to naturally use mature sequence
encoding models, such as BERT (Devlin et al. 2019), GPT-3 (Brown et al. 2020),
and so on, which can establish the interactive information of two sequences.
We choose the BERT architecture as our encoder, which has been widely used in
natural language processing in recent years. BERT is a pre-trained language
model based on the bidirectional Transformer architecture (Vaswani et al.
2017), which can be used to encode a single sentence or a sentence pair. To
introduce the interactive information between the question and query graph
sequence, we use the structure of encoding the sentence pair in BERT. The
encoding framework is shown in Figure 4(a). Given the question
$q=\\{w_{1},w_{2},...,w_{m}\\}$ and the query graph sequence
$g^{s}=\\{u_{1},u_{2},...,u_{n}\\}$, we connect $q$ and $g^{s}$ through
special tags to form the sentence pair, denoted as
$p_{qg^{s}}=\\{[CLS],w_{1},...,w_{m},[SEP],u_{1},...,u_{n},[SEP]\\}$. Each
candidate query graph $g$ in set $G$ can form a sentence pair $p_{qg^{s}}$
with the corresponding question $q$. Then the pairs are fed to BERT for
encoding one by one. And we use the output of the $[CLS]$ node of BERT as the
semantic representation of the question and query graph sequence, denoted as
f.
#### Ranking Query Graphs
In this section, we rank the candidates with three different strategies,
namely Pointwise Ranking, Pairwise Ranking and Listwise Ranking, respectively.
These three optimization methods correspond to three typical ranking
strategies in information retrieval (Li 2011). Luo et al. (2018) use the
pairwise strategy in their system and achieve a certain success.
Before performing ranking, we preprocess the training data. According to
whether the correct answer can be retrieved, the candidates can be grouped
into two sets: $G^{+}$ and $G^{-}$, where $G^{+}$ includes the positive graphs
and $G^{-}$ includes the negative ones. We use $g_{i}^{+}$ and $g_{j}^{-}$ to
denote a positive graph and a negative graph, respectively. Each graph $g_{i}$
in the two sets is encoded as representation $\textbf{f}_{i}$. After that,
$\textbf{f}_{i}$ is fed into the linear layer to get a score $s_{i}$ that
indicates the possibility of $g_{i}$ being the optimal query graph.
##### Pointwise Ranking.
The Pointwise Ranking processes the graphs one by one. When ranking candidate
query graphs, the query graphs to be sorted do not need have a perfect order,
only the difference between positive and negative graphs. That is, we treat
the query graph ordering problem as a simple binary classification task. As
shown in Figure 4(b), the query graph $g_{i}$ in Pointwise Ranking is
optimized independently.
For each candidate query graph $g_{i}$, it corresponds to the label
$y_{i}\in\\{1,0\\}$, where ‘1’ represents the label of a positive graph, and
‘0’ represents the label of a negative graph. In the optimization process, we
use the cross-entropy loss function to optimize and select the query graph
with the highest score as the optimal query graph $g^{*}$. The optimized loss
function is as follows,
$s^{{}^{\prime}}_{i}=\frac{1}{1+e^{-s_{i}}},$ (1) $L_{point}=-\sum
y_{i}log(s^{{}^{\prime}}_{i})+(1-y_{i})log(1-s^{{}^{\prime}}_{i}).$ (2)
##### Pairwise Ranking.
The Pairwise Ranking can model the mutual connection between two candidate
elements, and realize the ranking by continuously optimizing the relative rank
of the two elements. When using the pairwise strategy to rank candidate query
graphs, we regard the ranking problem as a problem of how to distinguish
positive query graphs and negative query graphs. That is, we first construct
the form of pairs of positive and negative graphs, and then the order of
positive and negative graphs is optimized by using the pairwise strategy, as
shown in Figure 4(b).
For each pair of positive and negative query graphs $(g_{i}^{+},g_{j}^{-})$,
we can get the scores $s_{i}$ and $s_{j}$ through BERT and linear layer
encoding, respectively. Then $s_{i}$ and $s_{j}$ are normalized to
$s^{{}^{\prime}}_{i}$ and $s^{{}^{\prime}}_{j}$ by Equation (1). We use the
hinge loss optimization function to optimize the scores so that the difference
between the scores of the positive and negative query graphs is maintained at
a fixed value $\lambda$. The hinge loss is defined as follows,
$L_{pair}=max\\{0,\lambda-s^{{}^{\prime}}_{i}+s^{{}^{\prime}}_{j}\\},$ (3)
where $\lambda$ is set to 0.5.
##### Listwise Ranking.
The Listwise Ranking can also model the interconnection between all
candidates, and can directly optimize the order of the entire candidate set.
In this strategy, we construct a list of positive and negative graphs for
global optimization. In the query graph selection, we don’t care too much
about the ranking among positive graphs or the ranking among negative graphs.
Our goal of global optimization is to successfully rank the positive graph in
the first place.
The application of Listwise in query graph ranking is slightly different from
the method used in traditional information retrieval. When constructing the
training data, we select each positive graph and a fixed number of negative
graphs to form a list
$C=\\{{g}_{0}^{+},{g}_{1}^{-},{g}_{2}^{-},…,{g}_{m}^{-}\\}$, whose label is
$\\{y_{0},y_{1},y_{2},...,y_{m}\\}$. The score of group $C$ after BERT
encoding and linear layer mapping is recorded as
$\\{s_{0},s_{1},s_{2},…,s_{m}\\}$. During the training, we design the
following optimization objective function,
$s^{{}^{\prime}}_{i}=\frac{exp(s_{i})}{\sum_{i=0}^{m}exp(s_{i})},$ (4)
$L_{list}=-\sum_{i=0}^{m}y_{i}log(s^{{}^{\prime}}_{i})+(1-y_{i})log(1-s^{{}^{\prime}}_{i}).$
(5)
## Experiments
### Experimental Setup
##### Datasets.
Dataset | train | validation | test
---|---|---|---
WebQ | 3,023 | 755 | 2,032
CompQ | 1,000 | 300 | 800
Table 1: The partitions of WebQuestions and ComplexQuestions.
We conduct experiments on two widely-used datasets: WebQuestions (WebQ)
(Berant et al. 2013) 222https://nlp.stanford.edu/software/sempre/ and
ComplexQuestions (CompQ) (Bao et al. 2016)
333https://github.com/JunweiBao/MulCQA/ tree/ComplexQuestions. The WebQ
dataset contains both simple questions (84%) and complex reasoning questions
(16%), which is close to natural language questions used by people in daily
life. The dataset contains 5,810 question-answer pairs. The CompQ dataset is
designed for complex question answering. The dataset contains a total of 2,100
question-answer pairs. Both WebQ and CompQ are divided into train set,
validation set and test set, as shown in table 1. Both datasets use Freebase
as the knowledge base 444https: //developers.google.com/freebase/, which has
been widely used in the KBQA systems.
##### Implementation Details.
For encoding questions and query graphs, we utilize the BERT-base model. We
choose the hyper-parameter settings according to the performance on the
validation sets. Regarding the hyperparameters in the BERT-base model, we set
the dropout ratio to 0.1 and the hidden size to 768. We use Adam as the
optimizer and the learning rate is set to $5\times 10^{-5}$. The maximum
number of training epoch is set to 5. At the end of each epoch, we use the
validation set to evaluate the model, and the model with the best performance
on the validation set is selected as the final testing model. For performance
evaluation, we report the average F1 score, as do in Berant et al. (2013).
One question left is how to construct training data for query graph selection.
Given a question and its corresponding query graph candidates, we use the
query graph whose answer has an F1 value greater than 0.1 as a positive graph
and randomly sample negative graphs from the rest query graph candidates.
### Main Results
Method | WebQ (F1%) | CompQ (F1%)
---|---|---
Pointwise | 52.4 | 38.4
Pairwise | 53.7 | 42.7
Listwise | 55.3 | 44.4
Table 2: The comparison results of the three ranking strategies on the test sets. Category | Method | WebQ(F1%) | CompQ(F1%)
---|---|---|---
| Yih et al. (2015) | 52.5 | -
| Bao et al. (2016) | 52.4 | 40.9
Using Query Graph | Hu, Zou, and Zhang (2018) | 53.6 | -
| Luo et al. (2018) | 52.7 | 42.8
| Lan and Jiang (2020) | - | 43.3
| Berant et al. (2013) | 36.4 | -
Others | Jain (2016) | 55.6 | -
| Chen, Wu, and Zaki (2019) | 51.8 | -
| Xu et al. (2019) | 54.6 | -
Our | Listwise | 55.3 | 44.4
Table 3: The comparison results with previous works on the test sets of
WebQuestions and ComplexQuestions.
Table 2 shows the comparison results of the three ranking strategies. From the
table, we can see Listwise Ranking and Pairwise Ranking outperform the
Pointwise Ranking. This fact indicates the necessity of modeling the inter-
relations between query graph candidates. In addition, we also find that the
superiority of Listwise Ranking and Pairwise Ranking are more significant on
CompQ than WebQ, which is in line with our intuition that complex questions
may require more information to disambiguate query graph candidates. Listwise
Ranking yields the best result on both two datasets. The reason may be that
Listwise considers more than two graphs at once which has a global view when
optimization, compared with Pairwise.
Table 3 shows the comparison results between our system with Listwise Ranking
and the previous works on the test sets of WebQ and CompQ, where category
“Using Query Graph” includes the previous systems that use query graph and
“Others” includes the previous ones that do not use query graph. From the
table, we can see that our system yields the best result on CompQ and the
second best on WebQ among all the systems. Especially, when compared with the
approaches using query graph, our system achieves the best performance.
### Discussion and Analysis
#### Effect of Different Components in Query Qraph Selection
Sequence Info | WebQ (F1%) | CompQ (F1%)
---|---|---
All Path | 55.3 | 44.4
w/o constrains | 53.7 | 42.3
w/o answer | 54.3 | 43.6
Table 4: The effect of different components in query graph sequence on query
graph selection.
When representing query graphs, we propose to transform a query graph into a
sequence which is composed of sub-paths of different types. In order to
explore the effect of different components on the final performance, we
conduct experiments by removing some components from our Listwise system. The
experimental results are presented in Table 4, where “All Path” refers to our
final system that includes the main path plus four constraint paths, “w/o
constrains” refers to the system removing four constraint paths, and “w/o
answer” refers to the system removing the answer string. The results show that
by continuously accumulating different components, the system performance can
be steadily improved. This indicates that all the components of the query
graph sequence are useful for our system.
#### Error Analysis
In this section, we want to know the reasons why our system gives the wrong
answers for many cases by error analysis. If the candidate set generated by
the query graph generation does not include the correct answer, our system is
not possible to find it. Here we check the cases where the candidate set
includes the correct answer but our system (Listwise) fails to find it. We
randomly select 100 cases and check them manually. The errors are summarized
as follows:
Incorrect Query Graph Generation. There are some query graphs which can
retrieve the correct answer, but actually are not correctly parsed. For
example, the question “where was david berkowitz arrested?”, the query graph
generation provides “david berkowitz places lived - location brooklyn, new
york city” as a candidate. The candidate graph can retrieve the correct
answer, but in fact is not correct to the question. Of 100 cases, we find that
this type of errors contains 45 cases. As for this type, we have to improve
the performance of query graph generation and the coverage of KB.
Incorrect Query Graph Selection. As for the other cases, the candidate set
includes the correct query graph which can perfectly retrieve the answer.
However, our system still fails to find it. These errors can be grouped into
two categories. The first one (40%) is to select the graph that includes the
incorrect relationship (main path) between the topic word and answer. The
second one (15%) is to select incorrect constraints. To solve this type of
errors, we may perform a deeper analysis to provide additional information for
query graph selection.
#### Effect of Negative Examples on Ranking
Figure 5: The performance of three ranking strategies under different numbers
of negative examples.
To further explore the characteristics of the three ranking strategies of
Pointwise Ranking, Pairwise Ranking, and Listwise Ranking, we select different
numbers of negative graphs to build the systems. The performance is shown in
Figure 5. From the figures, we find that when the number of negative graphs
increases, the performance of all three systems first increases and then is in
a relatively stable state. We also find that Listwise Ranking can yield a good
performance with few negative samples. These facts indicate that we do not
need too many negative graphs when training our systems.
#### Case Study
Question1: what type of breast cancer did sheryl crow have ?
---
True: sheryl crow condition meningioma.
False: sheryl crow films – film breast cancer: the path of wellness & healing.
Question2: what role did paul mccartney play in the beatles ?
True: member paul mccartney. the beatles member – role backing vocalist, lead
vocalist, bass.
False: the beatles (tv series) regular cast – actor george harrison, john
lennon, lance percival, paul frees, paul mccartney, ringo starr.
Table 5: The case study.
We analyze some specific examples on which Listwise performs better than
Pointwise. Two typical examples are listed in table 5. For the question “what
type of breast cancer did sheryl crow have?”, the true answer should be a type
of cancer. Listwise Ranking can determine that ‘condition’ is the correct
relation, but Pointwise Ranking chooses the wrong path that contains “breast
cancer”. We argue that Listwise Ranking can better model the overall semantics
of the sequence because it considers the relationship between the candidate
query graphs during optimization, while Pointwise Ranking tends to focus on
the semantics of local words. Besides, for the example “what role did paul
mccartney play in the beatles?”, the correct query graph contains the true
entity constraint. But Pointwise Ranking chooses a path without the entity
constraint. This indicates to some extent that Pointwise Ranking is not
effective enough in identifying constrain paths.
## Related Work
Information retrieval (IR) and semantic parsing (SP) based approaches are two
mainstreams for knowledge base question answering. Among them, IR-based
methods (Yu et al. 2017; Gupta, Chinnakotla, and Shrivastava 2018; Chen, Wu,
and Zaki 2019; Petrochuk and Zettlemoyer 2018; Zhao et al. 2019; Saxena,
Tripathi, and Talukdar 2020) obtain relevant candidate answers according to
the topic entity and then rank the answers to obtain the final result. The
core of IR-based approaches is to identify the KB relation paths that the
question refers to (Wu et al. 2019). For example, Dong et al. (2015) use
multi-column Convolutional Neural Networks (CNN) to encode questions and paths
to the same vector space and calculate the similarity. Hao et al. (2017) use
Long Short-Term Memory (LSTM) instead of CNN for the same purpose.
Different from IR-based methods, SP-based approaches put more attention to the
semantic analysis of the question (Bao et al. 2016). The basic process of SP-
based approaches is to parse the semantics of the question into some meaning
representation and then map the meaning representation with KB (Hu et al.
2017). For example, Berant et al. (2013) parse the question into $\lambda-
DCS$, and then map it to the knowledge base through alignment and bridging
operations to obtain answers. Sun et al. (2020) design a novel skeleton
grammar to express complex questions and improve the ability to parse complex
questions. Query graph is also a widely-used meaning representation in SP-
based systems. Yih et al. (2015) are the pioneer into query graph research for
KBQA, which propose a staged query graph generation method for this task.
Following this line, Luo et al. (2018) propose a complex query graph matching
approach that simultaneously encodes multiple sub-paths to achieve better
query graph representation. More recently, Lan and Jiang (2020) propose a
method to expand multiple relations so that it can handle more complex
questions. In contrast to previous works that mostly put focus on the
representation of query graphs, we instead put focus on the phase of selecting
the optimal query graph.
## Conclusions
We present a novel semantic matching approach based on semantic parsing to
improve the performance of Knowledge Base Question Answering (KBQA). In this
paper, the process of KBQA is divided into two steps: query graph generation
and query graph selection, and we put focus on the second step. In our
approach, we linearize the query graphs into sequences. Then, we use BERT to
encode the pair of the query graph sequence and the question to obtain the
semantic representation. In addition, we select the optimal query graph with
different ranking strategies, which take the relationship between candidate
query graphs into consideration. Experimental results on two benchmark
datasets demonstrate the effectiveness of our proposed approach. Specifically,
our best-performance system achieves the top performance on ComplexQuestions
and the second best performance on WebQuestions.
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|
# Towards the sampling Lovász Local Lemma
Vishesh Jain Simons Institute for the Theory of Computing, Berkeley, CA
94720, USA<EMAIL_ADDRESS>, Huy Tuan Pham and Thuy Duong Vuong
Stanford University, Stanford, CA 94305, USA {huypham<EMAIL_ADDRESS>
###### Abstract.
Let $\Phi=(V,\mathcal{C})$ be a constraint satisfaction problem on variables
$v_{1},\dots,v_{n}$ such that each constraint depends on at most $k$ variables
and such that each variable assumes values in an alphabet of size at most
$[q]$. Suppose that each constraint shares variables with at most $\Delta$
constraints and that each constraint is violated with probability at most $p$
(under the product measure on its variables). We show that for $k,q=O(1)$,
there is a deterministic, polynomial time algorithm to approximately count the
number of satisfying assignments and a randomized, polynomial time algorithm
to sample from approximately the uniform distribution on satisfying
assignments, provided that
$C\cdot q^{3}\cdot k\cdot p\cdot\Delta^{7}<1,\quad\text{where }C\text{ is an
absolute constant.}$
Previously, a result of this form was known essentially only in the special
case when each constraint is violated by exactly one assignment to its
variables.
For the special case of $k$-CNF formulas, the term $\Delta^{7}$ improves the
previously best known $\Delta^{60}$ for deterministic algorithms [Moitra,
J.ACM, 2019] and $\Delta^{13}$ for randomized algorithms [Feng et al., arXiv,
2020]. For the special case of properly $q$-coloring $k$-uniform hypergraphs,
the term $\Delta^{7}$ improves the previously best known $\Delta^{14}$ for
deterministic algorithms [Guo et al., SICOMP, 2019] and $\Delta^{9}$ for
randomized algorithms [Feng et al., arXiv, 2020].
## 1\. Introduction
The celebrated Lovász Local Lemma (LLL) is a fundamental tool in probabilistic
combinatorics which provides a sufficient condition for avoiding a collection
of “bad events” in a probability space. In a quite general form, it may be
stated as follows.
###### Theorem 1.1 ([EL73]).
Let $\mathcal{C}$ be a finite set of events in a probability space. For
$C\in\mathcal{C}$, let $\Gamma(C)$ denote a subset of $\mathcal{C}$ such that
$C$ is independent of the collection of events
$\mathcal{C}\setminus({C}\cup\Gamma(C))$. Suppose there exist positive real
numbers $x:\mathcal{C}\to(0,1)$ such that
$\displaystyle\mathbb{P}[C]\leq
x(C)\prod_{D\in\Gamma(C)}(1-x(D))\quad\text{for all }C\in\mathcal{C}.$ (1.1)
Then,
$\mathbb{P}[\wedge_{C\in\mathcal{C}}\overline{C}]\geq\prod_{C\in\mathcal{C}}(1-x(C))>0.$
In most applications of the LLL (cf. [AS04, MT10, MR98]), the underlying
probability measure $\mathbb{P}[\cdot]$ is generated by a collection of
independent random variables $X_{1},\dots,X_{n}$ and for each “bad event”
$C\in\mathcal{C}$, there is a subset
$\operatorname{vbl}(C)\subseteq\\{X_{1},\dots,X_{n}\\}$ such that $C$ depends
only on $X_{i}\in\operatorname{vbl}(C)$. This is often referred to as the
“variable-version” setting of the LLL. Moreover, for many applications (cf.
[AS04]), the following “symmetric” case of the variable-version setting
suffices.
###### Corollary 1.2.
Let $X_{1},\dots,X_{n}$ denote a collection of independent random variables.
Let $\mathcal{C}=\\{C_{1},\dots,C_{m}\\}$ denote a collection of events and
for $C\in\mathcal{C}$, let $\operatorname{vbl}(C)$ denote a subset of
$\\{X_{1},\dots,X_{n}\\}$ such that $C$ depends only on
$X_{i}\in\operatorname{vbl}(C)$. Suppose there exist $p\in(0,1)$ and $D\geq 0$
satisfying
* •
For each $C\in\mathcal{C}$, $\mathbb{P}[C]\leq p$.
* •
For each $i\in[m]$,
$\\#\\{j\in[m]:\operatorname{vbl}(C_{j})\cap\operatorname{vbl}(C_{i})\neq\emptyset\\}\leq(D+1)$,
and
* •
$e\cdot p\cdot(D+1)\leq 1$, where $e$ is the base of the natural logarithm.
Then,
$\mathbb{P}[\wedge_{i\in[m]}\overline{C_{i}}]\geq\prod_{i=1}^{m}\left(1-e\cdot\mathbb{P}[C_{i}]\right)>0.$
As a classical application of Corollary 1.2, consider the problem of
satisfiability of a $k$-CNF formula over Boolean variables
$x_{1},\dots,x_{n}$. Recall that a $k$-CNF formula over Boolean variables
$x_{1},\dots,x_{n}$ is a collection of constraints $C_{1},\dots,C_{m}$ such
that each $C_{i}$ depends on exactly $k$ variables and such that each $C_{i}$
is satisfied by all but exactly one assignment to its variables. Corollary 1.2
shows that if each constraint shares variables with at most (roughly)
$2^{k}/e$ other constraints, then the formula has a satisfying assignment.
Unfortunately, the original proof of Theorem 1.1 is non-constructive and does
not provide an efficient algorithm to _find_ a satisfying assignment of the
formula when this condition is met. In a breakthrough work [Bec91], Beck
showed that if $2^{k}/e$ is replaced by $2^{k/48}$, then it is in fact
possible to efficiently find a satisfying assignment. Beck’s bound was
improved by many works over a period of nearly 20 years (e.g. [Alo91, MR98,
Sri08, Mos09]) culminating in the landmark work of Moser and Tardos [MT10],
which gives an efficient algorithmic proof of Theorem 1.1 provided further
that one is in the variable setting and some other technical assumptions are
satisfied. There has been much work on extending the result of Moser and
Tardos to more general settings and the algorithmic aspects of the LLL remain
an active area of research (see, e.g., [AIS19] and the references therein).
In this work, we are concerned with the following.
###### Problem 1.3.
Suppose that conditions similar to the LLL are satisfied. Can we approximately
count the total number of satisfying assignments? Can we sample from
approximately the uniform distribution on satisfying assignments?
This problem has attracted much attention in the past five years. Below, we
only discuss results for approximate counting, noting that similar results
also hold for approximate sampling.
In [BGG+19], Bezáková et al. showed that if $\mathbf{P}\neq\mathbf{NP}$, then
it is _not_ possible to efficiently approximately count solutions of a Boolean
$k$-CNF formula in which every variable is allowed to be present in $d$
constraints for $d\geq 5\cdot 2^{k/2}$, even when the $k$-CNF formula is
monotone. For monotone $k$-CNFs, Hermon, Sly, and Zhang [HSZ19] showed that
the Glauber dynamics mix rapidly for $d\leq c2^{k/2}$, thereby providing an
approximate counting algorithm within a constant factor of the hard regime.
For not necessarily monotone $k$-CNFs, Moitra [Moi19] provided a novel method
to _deterministically_ approximately count satisfying assignments for
$d\lesssim 2^{k/60}$ (where $\lesssim$ hides polynomial factors in $k$), which
runs in polynomial time for $k=O(1)$. Using a Markov chain on a certain
“projected space” inspired by Moitra’s method, Feng, Guo, Yin, and Zhang
[FGYZ20] relaxed the restriction to $d\lesssim 2^{k/20}$ and removed the
requirement $k=O(1)$, although their algorithm is not deterministic. We also
mention here the work of Guo, Jerrum, and Liu [GJL19] on “partial rejection
sampling”. For $k$-CNFs, their method allows one to _perfectly_ sample from
the uniform distribution on satisfying assignments, either for “extremal
formulas” (and $d$ in the LLL regime), or for formulas for which the
intersections between the constraints satisfy some rather stringent size
restrictions (and for $d$ matching the hardness regime).
Very recently, work of Feng, He, and Yin [FHY20] addressed 1.3 in the special
case where each constraint is violated by a _very small number_ of
configurations of its variables. Their results are obtained in the following
setting.
###### Definition 1.4.
(1) A constraint satisfaction problem (CSP) is said to be atomic if each
constraint is violated by at most one assignment to its variables.
(2) A $(k,d,q)$-CSP on variables $x_{1},\dots,x_{n}$ is a constraint
satisfaction problem in which each $x_{i}$ takes values in $[q]$, each
constraint depends on exactly $k$ variables, and each variable features in at
most $d$ constraints.
In [FHY20], a fast randomized algorithm is provided in the setting of
Corollary 1.2, _assuming that the CSP is atomic_ and that $pD^{350}\lesssim
1$. For $(k,d,q)$-CSPs which are atomic, they obtain better bounds, leading to
an algorithm for $k$-CNF Boolean formulas with $d\lesssim 2^{k/13}$, and an
algorithm for proper $q$-colorings of $k$-uniform hypergraphs with $q\gtrsim
d^{9/(k-12)}$ (this improves on a previous bound of $q\gtrsim d^{14/(k-14)}$
for $q,k=O(1)$ due to Guo, Liao, Lu, and Zhang [GLLZ19], although [GLLZ19]
provides a deterministic algorithm).
For not-necessarily-atomic CSPs, [FHY20] simply decompose each constraint into
atomic constraints, which leads to the restriction
$\displaystyle p(DN)^{350}\lesssim 1,$ (1.2)
where $N$ is an upper bound on the number of violating assignments to the
variables of any constraint (hence, $N=1$ for an atomic CSP). To see that this
condition is vastly more restrictive than Corollary 1.2, consider the case of
Boolean CSPs for which each constraint depends on at most $k$ variables. Then,
$p\geq 2^{-k}$ so that 1.2 fails to be applicable as soon as $N\gtrsim
2^{k/350}$. In contrast, Corollary 1.2 shows that a solution exists, provided
that $D\lesssim 2^{k}/N$ for all $N\lesssim 2^{k}$.
The restriction $N\gtrsim 2^{k/350}$ arising from 1.2 is rather undesirable,
since in many applications of the LLL (cf. [AS04]), $N=\Theta(2^{ck})$ with
the constant $c\in(0,1)$ coming from various concentration inequalities. One
of the main open problems mentioned in the works [GLLZ19, FHY20] is whether
one can go beyond the “atomic CSP” framework to provide an affirmative answer
to 1.3 for general CSPs.
### 1.1. Our results
We provide, for the first time, approximate counting and sampling algorithms
for general CSPs under LLL-like conditions.
###### Theorem 1.5.
Let $\Phi=(V,\mathcal{C})$ denote a constraint satisfaction problem on
variables $v_{1},\dots,v_{n}$ and constraints $C_{1},\dots,C_{m}$. For each
constraint $C\in\mathcal{C}$, let $\operatorname{vbl}(C)\subseteq V$ denote
the variables it depends on. Suppose there exist $p\in(0,1)$, $\Delta\geq 1$,
$k\geq 1$, $q\geq 1$ satisfying the following conditions.
* •
The domain of each variable $v_{1},\dots,v_{n}$ is of size at most $q$.
* •
For each $C\in\mathcal{C}$, $|\operatorname{vbl}(C)|\leq k$.
* •
For each $C\in\mathcal{C}$, $\mathbb{P}[C]\leq p$.
* •
For each $i\in[m]$,
$\\#\\{j\in[m]\setminus\\{i\\}:\operatorname{vbl}(C_{j})\cap\operatorname{vbl}(C_{i})\neq\emptyset\\}\leq\Delta$,
and
* •
$q^{3}\cdot k\cdot p\cdot\Delta^{7}\leq c$, where $c$ is an absolute constant.
Then, for any $\varepsilon\in(0,1)$, the number of satisfying assignments of
$\Phi$ can be deterministically approximated to within relative error
$(1\pm\varepsilon)$ in time
$\left(\frac{n}{\varepsilon}\right)^{\operatorname{poly}(k,\Delta,\log q)}.$
###### Remark.
(1) For Boolean $k$-CNFs with $k=O(1)$, this provides a deterministic
approximate counting algorithm for $\Delta\lesssim 2^{k/7}$, where $\lesssim$
hides polynomial factors in $k$. As mentioned above, the previously best known
algorithm, either randomized or deterministic, requires $\Delta\lesssim
2^{k/13}$ [FHY20].
(2) For properly $q$-coloring $k$-uniform hypergraphs, this provides a
deterministic approximate counting algorithm for $q\gtrsim\Delta^{7/(k-4)}$.
The previous best known algorithms required $q\gtrsim\Delta^{9/(k-12)}$
[FHY20] or $q\gtrsim\Delta^{14/(k-14)}$ for deterministic algorithms [GLLZ19].
The framework for proving Theorem 1.5 also lends itself naturally to an
approximate sampling algorithm.
###### Theorem 1.6.
Under the same conditions as Theorem 1.5 and for any $\varepsilon\in(0,1)$,
there is a randomized algorithm to sample from a distribution which is
$\varepsilon$-close in total variation distance to the uniform distribution on
satisfying assignments of $\Phi$. The running time of the algorithm is
$\left(\frac{n}{\varepsilon}\right)^{\operatorname{poly}(k,\Delta,\log{q})}$.
### 1.2. Techniques
The works [Bec91, Alo91, MR98, Sri08] on the algorithmic local lemma predating
Moser’s work [Mos09] employ the following two step strategy: one first finds a
“good” partial assignment with the property that the residual formula
“factorizes” into logarithmic sized components, at which point, one can extend
the partial assignment to a complete satisfying assignment efficiently using
exhaustive enumeration. For sampling from (approximately) the uniform
distribution on satisfying assignments, one therefore only needs to generate
these initial partial assignments according to the correct distribution. To
accomplish this, we use a generalization of a linear program introduced by
Moitra [Moi19] to approximate the marginal distribution (induced by the
uniform distribution on satisfying assignments) of an unassigned variable,
conditioned on partial assignments satisfying certain conditions.
The key conceptual contribution of our work is the following insight: the
combinatorial conditions guaranteeing the factorization of the residual
formula into logarithmic sized components are essentially the same as those
ensuring that the LP can be solved efficiently (see the proof of Lemmas 4.9
and 5.1). This provides a unifying view of the works [Bec91, Alo91, MR98,
Sri08] on the algorithmic LLL, and the recent works [Moi19, GLLZ19], and in
our opinion, sheds considerable light on the latter two works.
From a technical viewpoint, our main contribution is a considerable
generalization, refinement, and simplification of the framework introduced by
Moitra [Moi19]. In particular, we eliminate any need to use the algorithmic
local lemma of Moser and Tardos [MT10] (which was an essential ingredient in
[Moi19, GLLZ19] and leads to additional degradation in the quantitative
bounds); instead, we show how to efficiently exploit a certain “pseudo-random
property” of the initial partial assignment in a direct manner to remove this
loss (Lemma 5.2). Our framework also treats approximate counting and
approximate sampling on the same footing in a very simple manner, whereas the
previous works [Moi19, GLLZ19] suffered from additional losses in going from
approximate counting to approximate sampling.
We believe that our analysis lays bare the limits of this approach towards
approximate counting and sampling. The term $\Delta^{7}$ in Theorems 1.5 and
1.6 comes from the aggregation of two sources of slack. The first is the use
of $\\{2,3\\}$-trees as in [Alo91, MR98, Sri08] – even for the algorithmic
LLL, $\\{2,3\\}$-trees only lead to $\Delta^{4}$ instead of $\Delta$, and
being able to use “denser witness trees” was the major innovation in the works
of Moser [Mos09] and Moser and Tardos [MT10]. Another reason for the slack is
a certain “factorization property” required to even write down the linear
program efficiently. We believe that with some additional ideas, this second
source of slack may be overcome, and leave this as an interesting direction
for future research.
### 1.3. Organization
In Section 2, we collect some preliminaries. In Section 3, we present a
slightly simpler algorithm which proves Theorem 1.5 with $\Delta^{7}$ replaced
by $\Delta^{10}$; the analysis of a key step in this algorithm is completed in
Section 4. The ideas introduced in Sections 3 and 4 are further refined in
Section 5 to prove Theorem 1.5 in Section 5.2 and Theorem 1.6 in Section 5.3.
## 2\. Preliminaries
### 2.1. Lovász Local Lemma
As mentioned in the introduction, the LLL provides a sufficient condition
guaranteeing that the probability of avoiding a collection $\mathcal{C}$ of
“bad events” in a probability space is positive. In particular, when the LLL
condition 1.1 is satisfied, the so-called LLL distribution,
$\mu_{S}[\cdot]:=\mathbb{P}[\cdot\mid\wedge_{C\in\mathcal{C}}\overline{C}]$
is well-defined (here, the subscript $S$ is chosen to represent “satisfying”).
The LLL distribution is the central object of study in this paper. We begin by
recording a standard comparison between the LLL distribution $\mu_{S}[\cdot]$
and the product distribution on the variables $\mathbb{P}[\cdot]$.
###### Theorem 2.1 (cf. [HSS11, Theorem 2.1]).
Under the conditions of Theorem 1.1, for any event $B$ in the probability
space,
$\mu_{S}[B]\leq\mathbb{P}[B]\prod_{C\in\Gamma(B)}(1-x(C))^{-1}.$
###### Remark.
The above comparison is one-sided, as it ought to be, since for any
$C\in\mathcal{C}$, $\mu_{S}[C]=0$ while $\mathbb{P}[C]$ may be positive.
For the remainder of this paper, we will restrict ourselves to the variable-
version symmetric setting described in Corollary 1.2, in which case, we choose
$x(C)=e\cdot\mathbb{P}[C]$ for all $C\in\mathcal{C}$, with $e$ the base of the
natural logarithm.
### 2.2. {2,3}-trees
One of the key tools in our analysis will be the notion of $\\{2,3\\}$-trees,
which goes back to Alon’s work on the algorithmic local lemma [Alo91].
###### Definition 2.2.
Let $G=(V,E)$ be a graph and let $\operatorname{dist}_{G}(\cdot,\cdot)$ denote
the graph geodesic distance. A $\\{2,3\\}$-tree is a subset of vertices
$T\subseteq V$ such that
* •
for any $u,v\in T$, $\operatorname{dist}_{G}(u,v)\geq 2$;
* •
if one adds an edge between every $u,v\in T$ such that
$\operatorname{dist}_{G}(u,v)=2\text{ or }3$, then $T$ is connected.
The next lemma bounds the number of $\\{2,3\\}$-trees of a given size in terms
of the maximum degree of the graph.
###### Lemma 2.3 (cf. [Alo91, Lemma 2.1]).
Let $G=(V,E)$ be a graph with maximum degree $d$. Then, for any $v\in V$, the
number of $\\{2,3\\}$-trees in $G$ of size $t$ containing $v$ is at most
$(ed^{3})^{t-1}/2$.
Before stating the next lemma, we need some notation. Let $H=(V,E)$ be a
hypergraph. Let $\operatorname{Lin}(H)$ denote its line graph i.e.
$V(\operatorname{Lin}(H))=E$ and there is an edge between $u\neq v\in
V(\operatorname{Lin}(H))$ if and only if the hyperedges $u,v\in E$ share a
vertex in $V$. Finally, let $L^{2}(H)$ denote the graph with the same vertex
set as $\operatorname{Lin}(H)$ and with an edge between two vertices $u\neq
v\in V(L^{2}(H))$ if and only if
$\operatorname{dist}_{\operatorname{Lin}(H)}(u,v)\leq 2$. Then, a simple
greedy argument shows the following.
###### Lemma 2.4 (cf. [GLLZ19, Lemma 14]).
Let $H=(V,E)$ be a hypergraph such that each hyperedge in $E$ intersects at
most $d$ other hyperedges (equivalently, the degree of $\operatorname{Lin}(H)$
is at most $d$). Let $B\subseteq E(H)$ be a collection of hyperedges which
induce a connected subgraph in $L^{2}(H)$. Then, for any $e^{*}\in B$, there
exists a $\\{2,3\\}$-tree $T\subseteq B$ in $\operatorname{Lin}(H)$ such that
$e^{*}\in T$ and $|T|\geq|B|/d$.
## 3\. A simpler algorithm for a more restrictive regime
In this section and the next one, we present a simpler algorithm which proves
a version of Theorem 1.5 provided that $p\leq(10^{5}q^{3}k\Delta^{10})^{-1}$.
The design and analysis of this algorithm already contains the basic ideas.
Later, in Section 5, we will introduce some key additional ingredients to
refine this algorithm and its analysis in order to prove Theorems 1.5 and 1.6.
Throughout this section and the next one, we fix an arbitrary ordering of the
variables $v_{1},\dots,v_{n}\in V$ and an arbitrary ordering of the
constraints $C_{1},\dots,C_{m}\in\mathcal{C}$. Moreover, for notational
convenience, we will assume that the domain of each variable is $[q]$; a
straightforward modification of the proof shows that we only need the size of
the domains to be bounded above by $q$.
### 3.1. Step 1: Finding a guiding assignment
The goal of this step is to find a partial assignment of the variables, which
will serve as a “guide” for the rest of the algorithm. This step is very much
inspired by analogous routines for the algorithmic local lemma [Bec91, Alo91,
MR98], and generalizes a similar step in the works [Moi19, GLLZ19]. We note
that in the works [Moi19, GLLZ19], it is critical that one is able to
efficiently find a partial assignment satisfying each constraint without
assigning values to too many variables in any constraint – while this is
indeed possible in the special settings considered in these works, in our
setting, such a partial assignment need not exist. The key result of this
subsection is Proposition 3.5. We will first present a randomized construction
achieving the guarantees of Proposition 3.5, and then show how to derandomize
it using standard techniques.
We consider the following randomized greedy procedure to construct a partial
assignment of $v_{1},\dots,v_{n}$, where $p^{\prime}>0$ is a parameter which
will be specified later.
1. (R1)
Let $A_{0}=V$ denote the set of initially “available variables” and let
$F_{0}=\emptyset$ denote the set of initially “frozen variables”. Initialize
the stage to $i=1$.
2. (R2)
Select the first variable (according to our order) in $A_{i-1}$. Denote this
variable by $v_{i}^{*}$. If no such variable exists, terminate the process.
3. (R3)
Assign $v_{i}^{*}$ a uniformly random value from its alphabet $[q]$. Let
$P_{i}$ denote the partial assignment resulting after the assignment to
$v^{*}_{i}$.
4. (R4)
Let
$\mathcal{F}_{i}=\\{j\in[m]:\mathbb{P}[C_{j}\mid P_{i}]>p^{\prime}\\}$
denote the set of “dangerous constraints” under $P_{i}$. We “freeze” all the
variables involved in any of the “dangerous constraints”, i.e. we set
$F_{i}=F_{i-1}\cup\bigcup_{j\in\mathcal{F}_{i}}((\operatorname{vbl}(C_{j})\cap
A_{i})\setminus\\{v_{i}^{*}\\})\text{ and
}A_{i}=A_{i-1}\setminus(F_{i}\cup\\{v^{*}_{i}\\}).$
5. (R5)
Increment $i$ by $1$ and return to (R2).
Note that the process terminates at the first stage $s$ satisfying
$A_{s}=\emptyset$. Let $P_{1},\dots,P_{s}$ denote the partial assignments
generated during the course of the process. Let
$\mathcal{F}=\cup_{i=1}^{s}\mathcal{F}_{i}$ denote the set of constraints
declared “dangerous” at any point during the process. Finally, let $\nu$
denote the distribution on partial assignments given by the random partial
assignment $P_{s}$. We emphasize that $s$ itself is a random variable.
The following simple observation will be useful later.
###### Lemma 3.1.
For all $i\in[s]$ and for all $j\in[m]$,
$\mathbb{P}[C_{j}\mid P_{i}]\leq p^{\prime}q.$
###### Proof.
Fix $j\in[m]$. If $j\notin\mathcal{F}$, then we are done, so assume that
$j\in\mathcal{F}$. Let $i$ be the first stage for which
$C_{j}\in\mathcal{F}_{i}$. Then, $\mathbb{P}[C_{j}\mid P_{i-1}]\leq
p^{\prime}$, and since $P_{i}$ extends $P_{i-1}$ by assigning $v_{i}^{*}$, we
have
$\displaystyle\mathbb{P}[C_{j}\mid P_{i}]$
$\displaystyle\leq\frac{\mathbb{P}[C_{j}\mid
P_{i-1}]}{\min_{u\in[q]}\mathbb{P}[v_{i}^{*}=u]}$ $\displaystyle\leq
p^{\prime}q.$
Note that if $C_{j}\in\mathcal{F}_{i}$, all variables in
$\operatorname{vbl}(C_{j})$ are added to $F_{i}$. Hence, no variable in
$\operatorname{vbl}(C_{j})$ is assigned a value during the remainder of the
process so that $\mathbb{P}[C_{j}\mid P_{i+k}]=\mathbb{P}[C_{j}\mid P_{i}]$
for all $0\leq k\leq s-i$. ∎
For $h\in[n]$, let $\mathfrak{S}(h)$ be the $\sigma$-algebra generated by the
output of the procedure on partial assignments of $v_{1},\dots,v_{h}$. Also,
for each $h\in[n]$, let
$\iota(h)=\max\\{i\in[h]:v_{i}^{*}=v_{h^{\prime}}\text{ for some
}h^{\prime}\leq h\\}$. In other words, $\iota(h)$ denotes the number of
variables in $v_{1},\dots,v_{h}$ which are assigned values by the partial
assignment.
###### Lemma 3.2.
Let $T\subseteq\mathcal{C}$ be a collection of constraints such that for any
$C,C^{\prime}\in T$ with $C\neq C^{\prime}$,
$\operatorname{vbl}(C)\cap\operatorname{vbl}(C^{\prime})=\emptyset$. Then,
letting
$M_{h}=\prod_{C\in T}\mathbb{P}[C\mid P_{\iota(h)}],$
we have
$\mathbb{E}_{\nu}[M_{h+1}\mid\mathfrak{S}(h)]=M_{h}.$
###### Proof.
Since the variables $v_{1},\dots,v_{n}$ are processed in order, given
$\mathfrak{S}(h)$, it is determined whether $v_{h+1}$ is frozen or not. If
$v_{h+1}$ is frozen, then $M_{h+1}=M_{h}$.
Otherwise, $v_{h+1}$ is not frozen. Note that, in this case,
$\iota(h+1)=\iota(h)+1$ and $v^{*}_{\iota(h+1)}=v_{h+1}$. Since
$\operatorname{vbl}(C)$ are disjoint for $C\in T$, there is at most one
constraint $C_{T}\in T$ such that $v_{h+1}\in\operatorname{vbl}(C_{T})$. If
there is no such constraint $C_{T}$, then $M_{h+1}=M_{h}$.
Consider now the case that $v_{h+1}$ is not frozen, and there exists a unique
constraint $C_{T}\in T$ such that $v_{h+1}\in\operatorname{vbl}(C_{T})$. We
have $\mathbb{P}[C\mid P_{\iota(h+1)}]=\mathbb{P}[C\mid P_{\iota(h)}]$ for all
$C\neq C_{T}$. Moreover, since $v_{h+1}$ is assigned a uniformly distributed
value in $[q]$, we have
$\displaystyle\mathbb{E}_{\nu}\bigg{[}\mathbb{P}[C_{T}\mid
P_{\iota(h+1)}]\mid\mathfrak{S}(h)\bigg{]}$
$\displaystyle=\frac{1}{q}\sum_{a\in[q]}\frac{\mathbb{P}[C_{T}\wedge
P_{\iota(h)}\wedge v_{h+1}=a]}{\mathbb{P}[P_{\iota(h)}\wedge v_{h+1}=a]}$
$\displaystyle=\frac{\mathbb{P}[C_{T},P_{\iota(h)}]}{\mathbb{P}[P_{\iota(h)}]}$
$\displaystyle=\mathbb{P}[C_{T}\mid P_{\iota(h)}].$
Combining these cases gives the desired conclusion. ∎
Since $\mathbb{P}[C]\leq p$ for all $C\in\mathcal{C}$, the following is an
immediate corollary of Lemma 3.2.
###### Corollary 3.3.
Let $T\subseteq\mathcal{C}$ be a collection of constraints such that for any
$C,C^{\prime}\in T$ with $C\neq C^{\prime}$,
$\operatorname{vbl}(C)\cap\operatorname{vbl}(C^{\prime})=\emptyset$. Then,
$\mathbb{E}_{\nu}\left[\prod_{C\in T}\mathbb{P}[C\mid P_{s}]\right]\leq
p^{|T|}.$
We are now ready to state and prove the key property of guiding partial
assignments, which is that they “factorize” the constraint satisfaction
problem into small connected components. Let $H=(V,\mathcal{C})$ denote the
hypergraph induced by the constraint satisfaction problem and let
$G(\mathcal{C})$ denote its line graph. Let
$G^{2}(\mathcal{F})=(\mathcal{F},E)$ denote the graph whose vertices are
$\mathcal{F}$, and $C_{i}\neq C_{j}\in\mathcal{F}$ are adjacent if and only if
$\operatorname{dist}_{G(\mathcal{C})}(C_{i},C_{j})\leq 2$.
###### Lemma 3.4.
Let $p\leq p^{\prime}/(10\Delta^{3})$. The $\nu$-probability that
$G^{2}(\mathcal{F})$ has a connected component of size at least $L$ is at most
$n\Delta\cdot 2^{-L/\Delta}$.
###### Proof.
Suppose that $\mathfrak{C}$ is a connected component in $G^{2}(\mathcal{F})$
with $|\mathfrak{C}|\geq L$. By Lemma 2.4, there exists a $\\{2,3\\}$-tree $T$
in $G(\mathcal{C})$ with $|T|\geq|\mathfrak{C}|/\Delta\geq L/\Delta$. Let
$\mathfrak{T}$ denote the set of $\\{2,3\\}$-trees in $G(\mathcal{C})$ of size
$L/\Delta$. Then,
$\displaystyle\nu[G^{2}(\mathcal{F})\text{ has a connected component of
size}\geq L]$ $\displaystyle\leq\nu[G^{2}(\mathcal{F})\text{ contains a
}\\{2,3\\}\text{-tree of size}\geq L/\Delta]$
$\displaystyle\leq\sum_{T\in\mathfrak{T}}\nu[T\subseteq\mathcal{F}]$
$\displaystyle\leq\sum_{T\in\mathfrak{T}}(p/p^{\prime})^{L/\Delta}$
$\displaystyle\leq|\mathcal{F}|(e\Delta^{3})^{L/\Delta}(p/p^{\prime})^{L/\Delta}$
$\displaystyle\leq n\Delta(e\Delta^{3}p/p^{\prime})^{L/\Delta}$
$\displaystyle\leq n\Delta\cdot 2^{-L/\Delta}.$
The fourth line uses Lemma 2.3 and the last line uses the assumed bound on
$p/p^{\prime}$. Let us explain the third line. By Corollary 3.3 and Markov
inequality, for any $T\in\mathfrak{T}$,
$\nu[T\subseteq\mathcal{F}]\leq\nu\left[\left(\prod_{C\in T}\mathbb{P}[C\mid
P_{s}]\right)>p^{\prime|T|}\right]\leq\left(\frac{p}{p^{\prime}}\right)^{|T|}\leq\left(\frac{p}{p^{\prime}}\right)^{L/\Delta}.\qed$
The preceding lemma shows that with high probability, our random greedy
process returns a partial assignment satisfying the condition in Lemma 3.4
(for $L$ sufficiently large). Using the standard method of conditional
expectations, we can find such a partial assignment deterministically.
###### Proposition 3.5.
There exists a deterministic algorithm running in time
$O(n^{\operatorname{poly}(\log q,\log\Delta,k)})$ which generates a sequence
of partial assignments $P_{1},\dots,P_{s}$ with the following properties.
1. (1)
For all $i\in[s]$, $P_{i}$ assigns values to $i$ variables, and $P_{i}$
extends $P_{i-1}$.
2. (2)
$A_{s}=\emptyset$.
3. (3)
For all $i\in[s]$ and $j\in[m]$, $\mathbb{P}[C_{j}\mid P_{i}]\leq
p^{\prime}q$.
4. (4)
Every connected component in $G^{2}(\mathcal{F})$ has size at most
$L=10\Delta\log(\Delta n)$.
###### Proof.
Let $L^{\prime}=10\log(\Delta n)$, and let $\mathfrak{T}$ denote the
collection of all {2,3}-trees of size $L^{\prime}$ in $G(\mathcal{C})$. Note
that $|\mathfrak{T}|\leq\operatorname{poly}(n^{\log^{2}\Delta})$, and indeed,
it is easily seen (cf. [Alo91]) that the collection $\mathfrak{T}$ can be
constructed in time $\operatorname{poly}(n^{\log^{2}\Delta})$.
Now, for a partial assignment $X$, define
$H(X)=\sum_{T\in\mathfrak{T}}\prod_{C\in T}\left(\mathbb{P}[C\mid
X]/p^{\prime}\right).$
By the proof of Lemma 3.4, if we can find a sequence of partial assignments
$P_{1},\dots,P_{s}$ satisfying properties (1), (2), (3) such that
$H(P_{s})<1$, then (4) is also satisfied, since any $\\{2,3\\}$-tree in
$G(\mathcal{C})$ of size $L^{\prime}$ contributes at least $1$ to the sum.
To find such a sequence of partial assignments, we follow the same greedy
procedure as before, except now, after having chosen $P_{i-1}$ and
$v_{i}^{*}$, we choose the value of $v_{i}^{*}$ to be
$\operatorname*{arg\,min}_{a\in[q]}H(P_{i-1}\wedge{v_{i}^{*}=a}).$
We claim that $H(P_{i})\leq H(P_{i-1})$ for all $i\in[s]$. Indeed, for every
$T\in\mathfrak{T}$, there exists at most one $C_{T}\in T$ such that
$v_{i}^{*}\in\operatorname{vbl}(C_{T})$. Therefore,
$\displaystyle\sum_{a\in[q]}\mathbb{P}[v_{i}^{*}=a]H(P_{i-1}\wedge
v_{i}^{*}=a)$
$\displaystyle=\sum_{T\in\mathfrak{T}}\sum_{a\in[q]}(p^{\prime})^{-1}\mathbb{P}[C_{T}\mid
P_{i-1}\wedge v_{i}^{*}=a]\mathbb{P}[v_{i}^{*}=a]\prod_{C\in T\setminus
C_{T}}\frac{\mathbb{P}[C\mid P_{i-1}]}{p^{\prime}}$
$\displaystyle=H(P_{i-1}),$
which shows that it is possible to choose $P_{i}$ to ensure $H(P_{i})\leq
H(P_{i-1})$. Finally, since
$H(\emptyset)\leq(n\Delta)\cdot(e\Delta^{3})^{L^{\prime}}\cdot(p/p^{\prime})^{L^{\prime}}<1,$
we are done. ∎
### 3.2. Step 2: Approximate counting
Let $P_{0}=\emptyset$ and $P_{1},\dots,P_{s}$ denote the sequence of partial
assignments returned by Proposition 3.5. As before, we will denote the
vertices which are successively assigned values by $v_{1}^{*},\dots,v_{s}^{*}$
and we will denote their values under $P_{s}$ by $a_{1}^{*},\dots,a_{s}^{*}$.
For a partial assignment $X$, let $\mathcal{S}_{X}$ denote the number of
(complete) satisfying assignments extending $X$. We will use
$\mathcal{S}_{P_{0}}$ (or $\mathcal{S}_{\emptyset}$) to denote the set of all
complete satisfying assignments. Then,
$\displaystyle\frac{|\mathcal{S}_{P_{s}}|}{|\mathcal{S}_{P_{0}}|}$
$\displaystyle=\frac{|\mathcal{S}_{P_{1}}|}{|\mathcal{S}_{P_{0}}|}\cdot\frac{|\mathcal{S}_{P_{2}}|}{|\mathcal{S}_{P_{1}}|}\cdots\frac{|\mathcal{S}_{P_{s}}|}{|\mathcal{S}_{P_{s-1}}|}$
$\displaystyle=\prod_{i=1}^{s}\mu_{S}[v_{i}^{*}=a_{i}^{*}\mid P_{i-1}],$
where recall that $\mu_{S}$ denotes the uniform measure on all satisfying
assignments i.e. on $\mathcal{S}_{P_{0}}$. Thus, to approximate
$|\mathcal{S}_{P_{0}}|$, it suffices to approximate $|\mathcal{S}_{P_{s}}|$
and $\mu_{S}[v_{i}^{*}=a_{i}^{*}\mid P_{i-1}]$ for all $i\in[s]$.
###### Lemma 3.6.
For $P_{s}$ returned by Proposition 3.5, $|\mathcal{S}_{P_{s}}|$ can be
computed exactly in time $n^{\operatorname{poly}(\Delta,k,\log q)}$.
###### Proof.
Since $A_{s}=\emptyset$, the set of variables left unassigned by $P_{s}$ is
precisely $F_{s}$. Let $G_{1},\dots,G_{r}$ denote the maximal connected
components of $G^{2}(\mathcal{F})$ and let
$V^{\prime}_{1},\dots,V^{\prime}_{r}$ denote the variables appearing in any
constraint in $G_{1},\dots,G_{r}$. By the maximality of $G_{1},\dots,G_{r}$,
the sets $V_{1}^{\prime},\dots,V_{s}^{\prime}$ are mutually disjoint. Also, by
the maximality of $G_{1},\dots,G_{r}$, there does not exist any
$C\in\mathcal{C}$ such that $\operatorname{vbl}(C)\cap
V^{\prime}_{i}\neq\emptyset$ and $\operatorname{vbl}(C)\cap
V^{\prime}_{j}\neq\emptyset$ for some $i\neq j\in[r]$, since otherwise, some
vertex in $G_{i}$ would be connected in $G^{2}(\mathcal{F})$ to some vertex in
$G_{j}$. Finally, note that $|G_{i}|\leq L$, and hence $|V^{\prime}_{i}|\leq
kL$ for all $i\in[r]$.
Since any $v\in F_{s}$ must belong to some $C\in\mathcal{F}$ and since any
$C\in\mathcal{F}$ must belong to some $G_{i}$, it follows that
$F_{s}\subseteq V_{1}^{\prime}\cup\dots\cup V^{\prime}_{s}.$
Moreover, as seen in the previous paragraph, there are no constraints
involving variables from both $V^{\prime}_{i}$ and $V^{\prime}_{j}$ for $i\neq
j$. Therefore, for each $i\in[r]$, we can exhaustively enumerate all
assignments to $V^{\prime}_{i}\cap F_{s}$, check how many of them satisfy all
relevant constraints, and finally take the product over $i\in[r]$ in the
claimed time. ∎
Approximating $\mu_{S}[v_{i}^{*}=a_{i}^{*}\mid P_{i-1}]$ for $i\in[s]$ is much
more involved and will be the content of the next section. Let
$\delta\in(0,1)$ and $q^{-n}\leq r_{-}\leq r_{+}\leq q^{n}$ be parameters. In
Proposition 4.8, we will construct a subroutine
$\operatorname{Alg}_{r_{-},r_{+},\delta}$ with the following properties.
Suppose $p^{\prime}\leq(10000q^{3}k\Delta^{7})^{-1}$ and let $b\in[q]$.
* •
$\operatorname{Alg}_{r_{-},r_{+},\delta}$ runs in time
$\operatorname{poly}(n,k,q)\cdot
2^{\log(1/\delta)\cdot\operatorname{poly}(\Delta,k,\log{q})}$.
* •
$\operatorname{Alg}_{r_{-},r_{+},\delta}$ returns $\operatorname{YES}$ if and
only if
$r_{-}(1-\delta)\leq\frac{\mu_{S}[v_{i}^{*}=b\mid
P_{i-1}]}{\mu_{S}[v_{i}^{*}=a_{i}^{*}\mid P_{i-1}]}\leq r_{+}(1+\delta).$
Let $\varepsilon\in(0,1)$ be a parameter. Then, using such a subroutine along
with binary search on the parameters $r_{-},r_{+}$, we can clearly approximate
$\mu_{S}[v_{i}^{*}=a_{i}^{*}\mid P_{i-1}]$ up to a multiplicative factor of
$\exp(\varepsilon/n)$ for each $i\in[s]$ in time
$(n/\varepsilon)^{\operatorname{poly}(\Delta,k,\log{q})}$. Together with Lemma
3.6, this therefore provides an approximation of $|\mathcal{S}_{P_{0}}|$ up to
relative error $\exp(\varepsilon)$.
## 4\. Efficient estimation of the marginals
We will continue to use the notation and conventions of the previous section.
Throughout, we fix a partial assignment $P_{s}$ as returned by Proposition
3.5. By considering the fixed order $v_{1},\dots,v_{n}$ of the variables, this
fixes the identity of the variables $v_{1}^{*},\dots,v_{s}^{*}$ as well as the
intermediate sequence of partial assignments $P_{1},\dots,P_{s-1}$.
Throughout, we also fix $\ell\in[s]$. Our goal is to efficiently approximate
the conditional probabilities $p_{\ell}(a):=\mathbb{P}[v_{\ell}^{*}=a\mid
P_{\ell-1}]$ for all $a\in[q]$. We will use $\mu_{S}$ to denote the uniform
measure over all (complete) satisfying assignments, and for a partial
assignment $x$, $\mu_{S}[\cdot\mid x]$ to denote the uniform measure on all
(complete) satisfying assignments extending $x$. For partial assignments
$x,x^{\prime}$, the notation $x^{\prime}\to x$ means that $x$ is an extension
of $x^{\prime}$ (i.e. each variable that is assigned in $x^{\prime}$ is also
assigned in $x$ to the same value). Finally, we emphasize that
$\mathbb{P}[\cdot]$ will always mean the product measure on the variables
$v_{1},\dots,v_{n}$.
### 4.1. Idealized coupling procedure and the idealized decision tree
Let $p^{\prime\prime}>0$ be a parameter which will be chosen later. Fix $a\neq
b\in[q]$. Let $P_{\ell}(a)$ denote the partial assignment extending
$P_{\ell-1}$ obtained by setting $v_{\ell}^{*}=a$ and let $P_{\ell}(b)$ be
defined analogously. We begin by describing a coupling between assignments
extending $P_{\ell}(a)$ and $P_{\ell}(b)$, which will motivate subsequent
discussion. We note that this coupling is not meant to actually be implemented
by the algorithm.
1. (C1)
Initialize the partial assignments $X=P_{\ell}(a)$ and $Y=P_{\ell}(b)$.
Initialize $(V_{S})_{X,Y}=\\{v_{1}^{*},\dots,v_{\ell}^{*}\\}$ (the collection
of “set” variables) and $(V_{D})_{X,Y}=\\{v_{\ell}^{*}\\}$ (the collection of
“dangerous” variables).
2. (C2)
Choose the lowest numbered constraint $A\in\mathcal{C}$ such that
$(V_{D})_{X,Y}\cap\operatorname{vbl}(A)\neq\emptyset$ and
$\operatorname{vbl}(A)\cap((V_{D})_{X,Y}\cup(V_{S})_{X,Y})^{c}\neq\emptyset$.
If no such $A\in\mathcal{C}$ exists, then terminate.
3. (C3)
Choose the lowest numbered variable
$v\in\operatorname{vbl}(A)\cap((V_{D})_{X,Y}\cup(V_{S})_{X,Y})^{c}$.
4. (C4)
Sample a pair of values $(v_{X},v_{Y})$ according to the maximal coupling of
the marginal distribution of $\mu_{S}$ at $v$, conditioned on $X$ and $Y$
respectively.
5. (C5)
Update $X$ by assigning $v=v_{X}$, and update $Y$ by assigning $v=v_{Y}$.
Update $(V_{S})_{X,Y}$ by adding $v$.
6. (C6)
Let $D_{X,Y}=\\{u\in(V_{S})_{X,Y}:X(u)\neq Y(u)\\}$. Let
$\mathcal{F}_{X,Y}=\\{C\in\mathcal{C}$ : $\mathbb{P}[C\mid
X]>p^{\prime\prime}$ or $\mathbb{P}[C\mid Y]>p^{\prime\prime}\\}$. Update
$(V_{D})_{X,Y}=D_{X,Y}\cup\bigcup_{C\in\mathcal{F}_{X,Y}}(\operatorname{vbl}(C)\cap(V_{S})_{X,Y}^{c}),$
and return to (C2).
We record a few simple observations.
1. (O1)
The set $(V_{S})_{X,Y}$ increases throughout the process.
2. (O2)
$\mathcal{F}_{X,Y}$ is non-decreasing throughout the process. Indeed, once
$C\in\mathcal{F}_{X,Y}$, no other $v\in\operatorname{vbl}(C)$ can be chosen in
(C3), so that the conditional probability of $C$ with respect to all
subsequent partial assignments remains the same.
3. (O3)
The set $(V_{D})_{X,Y}$ is non-decreasing throughout the process.
The above coupling process may be viewed as randomly traversing root-to-leaf
trajectories in an idealized deterministic rooted decision tree $\mathcal{T}$,
defined using the following inductive procedure.
1. (T1)
The root of the tree consists of the partial assignments
$(x_{0},y_{0}):=(P_{\ell}(a),P_{\ell}(b))$.
2. (T2)
Given a node $(x,y)$ (consisting of partial assignments on the same variables
$(V_{S})_{x,y}$), construct $D_{x,y},\mathcal{F}_{x,y}$ as in (C6). Let
$(V_{D})_{x,y}=D_{x,y}\cup\bigcup_{C\in\mathcal{F}_{x,y}}(\operatorname{vbl}(C)\cap(V_{S})_{x,y}^{c}).$
3. (T3)
If there is no $A\in\mathcal{C}$ with
$(V_{D})_{x,y}\cap\operatorname{vbl}(A)\neq\emptyset$ and
$\operatorname{vbl}(A)\cap((V_{D})_{x,y}\cup(V_{S})_{x,y})^{c}\neq\emptyset$,
then $(x,y)$ is a leaf of $\mathcal{T}$.
4. (T4)
Otherwise, let $A\in\mathcal{C}$ be the lowest numbered such constraint, and
let $v_{x,y}$ be the lowest numbered variable in
$\operatorname{vbl}(A)\cap((V_{D})_{x,y}\cup(V_{S})_{x,y})^{c}$. The children
of $(x,y)$ in $\mathcal{T}$ consist of all possible extensions of $(x,y)$
obtained by assigning a value to the variable $v_{x,y}$.
The next lemma collects some useful properties of $\mathcal{T}$.
###### Lemma 4.1.
For $\mathcal{T}$ as defined above,
1. (1)
For any node $(x,y)\in\mathcal{T}$,
$\mathbb{P}[C_{j}\mid x]\leq p^{\prime\prime}q\text{ and }\mathbb{P}[C_{j}\mid
y]\leq p^{\prime\prime}q\quad\text{ for all }j\in[m].$
2. (2)
Assuming that $e\cdot p^{\prime\prime}q\cdot\Delta\leq 1$, for any node
$(x,y)\in\mathcal{T}$ and for any $v\notin(V_{D})_{x,y},$
$\displaystyle\operatorname{TV}(\mu_{S}[v=\cdot\mid x],\mathbb{P}[v=\cdot])$
$\displaystyle\leq(1-3p^{\prime\prime}q)^{-\Delta}-1,$
$\displaystyle\operatorname{TV}(\mu_{S}[v=\cdot\mid y],\mathbb{P}[v=\cdot])$
$\displaystyle\leq(1-3p^{\prime\prime}q)^{-\Delta}-1.$
3. (3)
For any leaf $(x,y)\in\mathcal{T}$, there is a partition
$V=(V_{D})_{x,y}\cup(V_{G})_{x,y}\cup(V_{R})_{x,y}$ such that every variable
in $(V_{G})_{x,y}$ is assigned to the same value by both $x$ and $y$ and such
that there is no constraint $C\in\mathcal{C}$ with variables in both
$(V_{D})_{x,y}$ and $(V_{R})_{x,y}$.
###### Proof.
(1) is immediate from (O2) and the same argument as Lemma 3.1.
(3) follows immediately using the termination criterion (T3) by taking
$(V_{G})_{x,y}=(V_{S})_{x,y}\setminus(V_{D})_{x,y}$, and
$(V_{R})_{x,y}=((V_{D})_{x,y}\cup(V_{S})_{x,y})^{c}$.
Finally, for (2), setting $\mu(\cdot)=\mu_{S}[v=\cdot\mid x]$ and
$\nu(\cdot)=\mathbb{P}[v=\cdot]$, we get
$\displaystyle\operatorname{TV}(\mu_{S}[v=\cdot\mid x],\mathbb{P}[v=\cdot])$
$\displaystyle=\operatorname{TV}(\mu,\nu)$
$\displaystyle=\sum_{a\in[q]:\mu(a)>\nu(a)}(\mu(a)-\nu(a))$
$\displaystyle\leq\sum_{a\in[q]:\mu(a)>\nu(a)}\frac{1}{q}\left((1-3p^{\prime\prime}q)^{-\Delta}-1\right)$
$\displaystyle\leq(1-3p^{\prime\prime}q)^{-\Delta}-1,$
where the second line uses the definition of total variation distance, and the
third line uses (1) and Theorem 2.1. The same argument works for $y$ as well.
∎
The idealized coupling process and decision tree are naturally associated to
the following quantities.
###### Definition 4.2.
For $\mathcal{T}$ as defined above,
* •
For any node $(x,y)\in\mathcal{T}$, let $\mathcal{S}_{x}$ denote the set of
(complete) satisfying assignments extending $x$, and let $\mathcal{S}_{y}$
denote the set of (complete) satisfying assignments extending $y$.
* •
For any node $(x,y)\in\mathcal{T}$, let $\mu_{\operatorname{cp}}(x,y)$ denote
the probability that the idealized coupling process reaches $(x,y)$. For any
$(x,y)\notin\mathcal{T}$, $\mu_{\operatorname{cp}}(x,y)=0$.
* •
For any node $(x,y)\in\mathcal{T}$, let
$\displaystyle p_{x,y}^{x}$
$\displaystyle=\frac{\mu_{\operatorname{cp}}(x,y)}{\mu_{S}[x\mid x_{0}]},$
$\displaystyle p_{x,y}^{y}$
$\displaystyle=\frac{\mu_{\operatorname{cp}}(x,y)}{\mu_{S}[y\mid y_{0}]}.$
We conclude this subsection with some simple, but crucial, relations between
these quantities.
###### Lemma 4.3.
For $\mathcal{T}$, $\mathcal{S}_{x},\mathcal{S}_{y},p^{x}_{x,y},p^{y}_{x,y}$
as above,
1. (1)
$p_{x,y}^{x},p_{x,y}^{y}\in[0,1]$.
2. (2)
$p_{x_{0},y_{0}}^{x_{0}},p_{x_{0},y_{0}}^{y_{0}}=1$.
3. (3)
For every non-leaf node $(x,y)\in\mathcal{T}$ whose children are defined on
the set $(V_{S})_{x,y}\cup\\{v_{x,y}\\}$, letting $v=v_{x,y}$ and letting
$x_{v(a)}$ denote the extension of $x$ obtained by setting $v$ to $a$ (and
similarly for $y_{v(a)}$),
$\displaystyle p^{x}_{x,y}$
$\displaystyle=\sum_{b\in[q]}p^{x_{v(a)}}_{x_{v(a)},y_{v(b)}}\quad\text{ for
all }a\in[q],$ $\displaystyle p^{y}_{x,y}$
$\displaystyle=\sum_{b\in[q]}p^{y_{v(a)}}_{x_{v(b)},y_{v(a)}}\quad\text{ for
all }a\in[q].$
4. (4)
For every node $(x,y)\in\mathcal{T}$,
$\frac{|\mathcal{S}_{x}|\cdot p^{x}_{x,y}}{|\mathcal{S}_{y}|\cdot
p^{y}_{x,y}}=\frac{|\mathcal{S}_{x_{0}}|}{|\mathcal{S}_{y_{0}}|}.$
5. (5)
For every non-leaf node $(x,y)\in\mathcal{T}$ whose children are defined on
the set $(V_{S})_{x,y}\cup\\{v_{x,y}\\}$ and for
$\eta=(1-3p^{\prime\prime}q)^{-\Delta}-1$, if $\eta\leq 1/(2q)$, then (letting
$v=v_{x,y}$)
$\displaystyle\sum_{b\neq a}p_{x_{v(a)},y_{v(b)}}^{x_{v(a)}}$
$\displaystyle\leq 4q\eta\cdot p^{x}_{x,y}\quad\text{ for all }a\in[q].$
$\displaystyle\sum_{b\neq a}p_{x_{v(b)},y_{v(a)}}^{y_{v(a)}}$
$\displaystyle\leq 4q\eta\cdot p^{y}_{x,y}\quad\text{ for all }a\in[q].$
###### Proof.
(1) and (2) are trivial.
For (3), we note that for any $a\in[q]$,
$\displaystyle\sum_{b\in[q]}p^{x_{v(a)}}_{x_{v(a)},y_{v(b)}}$
$\displaystyle=\frac{\sum_{b\in[q]}\mu_{\operatorname{cp}}(x_{v(a)},y_{v(b)})}{\mu_{S}[x_{v(a)}\mid
x_{0}]}$ $\displaystyle=\frac{\mu_{\operatorname{cp}}(x,y)\cdot\mu_{S}[v=a\mid
x]}{\mu_{S}[x\mid x_{0}]\cdot\mu_{S}[v=a\mid x]}$ $\displaystyle=p^{x}_{x,y}.$
The same argument also applies to $p^{y}_{x,y}$.
For (4), we have indeed that
$\displaystyle\frac{p^{x}_{x,y}}{p^{y}_{x,y}}$
$\displaystyle=\frac{\mu_{S}[y\mid y_{0}]}{\mu_{S}[x\mid x_{0}]}$
$\displaystyle=\frac{|\mathcal{S}_{y}|/|\mathcal{S}_{y_{0}}|}{|\mathcal{S}_{x}|/|\mathcal{S}_{x_{0}}|}.$
Finally, for (5), $\Pi_{x,y}$ denote the optimal coupling of
$\mu_{S}[u=\cdot\mid x]$ and $\mu_{S}[u=\cdot\mid y]$, we have for any
$a\in[q]$,
$\displaystyle\frac{\sum_{b\neq
a}p^{x_{v(a)}}_{x_{v(a)},y_{v(b)}}}{p_{x,y}^{x}}$
$\displaystyle=\frac{\sum_{b\neq
a}\mu_{\operatorname{cp}}(x_{v(a)},y_{v(b)})}{\mu_{\operatorname{cp}}(x,y)\mu_{S}[v=a\mid
x]}$ $\displaystyle=\frac{\sum_{b\neq a}\Pi_{x,y}(a,b)}{\mu_{S}[v=a\mid x]}$
$\displaystyle\leq\frac{\sum_{b\neq a}\Pi_{x,y}(a,b)}{(1/q)-\eta}$
$\displaystyle\leq 2q\cdot\operatorname{TV}(\mu_{S}[v=\cdot\mid
x],\mu_{S}[v=\cdot\mid y])$ $\displaystyle\leq
2q\left(\operatorname{TV}(\mu_{S}[v=\cdot\mid
x],\mathbb{P}[v=\cdot])+\operatorname{TV}(\mathbb{P}[v=\cdot],\mu_{S}[v=\cdot\mid
y])\right)$ $\displaystyle\leq 4q\eta,$
where the third line follows from (2) of Lemma 4.1, the fourth line follows
from the characterization of the total variation distance in terms of optimal
coupling, and the last line follows again from (2) of Lemma 4.1. ∎
### 4.2. Setting up the linear program
The most important property of the quantities defined above is (3) in Lemma
4.3, which shows that given
$|\mathcal{S}_{x}|/|\mathcal{S}_{y}|,p^{x}_{x,y},p^{y}_{x,y}$ at any node
$(x,y)\in\mathcal{T}$, one obtains the key quantity
$|\mathcal{S}_{x_{0}}|/|\mathcal{S}_{y_{0}}|$. What makes this property useful
is the following observation, which shows that for $(x,y)\in\mathcal{T}$
_which are leaves_ , the ratio $|\mathcal{S}_{x}|/|\mathcal{S}_{y}|$ can be
computed efficiently.
###### Lemma 4.4.
For any leaf $(x,y)\in\mathcal{T}$, $|\mathcal{S}_{x}|/|\mathcal{S}_{y}|$ can
be computed in time $\operatorname{poly}(n,k,q)\cdot q^{|(V_{D})_{x,y}|}$.
###### Proof.
Let $(x,y)\in\mathcal{T}$ be a leaf and let
$(V_{D})_{x,y},(V_{G})_{x,y},(V_{R})_{x,y}$ be the partition of $V$ as in (3)
of Lemma 4.1. All the unassigned variables (under $x,y$) are in
$(V_{D})_{x,y}\cup(V_{R})_{x,y}$ and note that there are no constraints with
variables in both $(V_{D})_{x,y}$ and $(V_{R})_{x,y}$. Further, the number of
ways of assigning values to variables in $(V_{R})_{x,y}$ so that they satisfy
all constraints with variables in $(V_{R})_{x,y}\cup(V_{G})_{x,y}$ is the same
(and hence, does not contribute to the ratio) since each variable in
$(V_{G})_{x,y}$ is assigned to the same value by both $x$ and $y$. Therefore,
the ratio $|\mathcal{S}_{x}|/|\mathcal{S}_{y}|$ is equal to the ratio of the
number of ways of assigning values to the unassigned variables in
$(V_{D})_{x,y}$ such that all constraints with variables in
$(V_{D})_{x,y}\cup(V_{G})_{x,y}$ are satisfied. This can be done by exhaustive
enumeration in the claimed time. ∎
Motivated by the preceding discussion, let $L\geq 2$ be a parameter to be
chosen later and consider the $L$-truncated decision tree defined as follows.
###### Definition 4.5.
For $L\geq 2$ and with $\mathcal{T}$ as before, we define the $L$-truncated
decision tree $\mathcal{T}_{L}$ to consist of those nodes
$(x,y)\in\mathcal{T}$ for which $|(V_{S})_{x,y}|\leq L+\ell$. We let
$\mathcal{L}_{L}$ denote the leaves of $\mathcal{T}_{L}$,
$\mathcal{L}_{L}^{g}$ denote those leaves in $\mathcal{L}_{L}$ which have
$|(V_{S})_{x,y}|\leq L+\ell-1$ (in particular, these are also leaves of
$\mathcal{T}$), and let $\mathcal{L}_{L}^{b}$ denote the remaining leaves.
We now set up a linear program to mimic the quantities
$p^{x}_{x,y},p^{y}_{x,y}$ for each node of $\mathcal{T}_{L}$. Formally, given
parameters $r_{-}\leq r_{+}$, $\eta=(1-3p^{\prime\prime}q)^{-\Delta}-1$ and
$\mathcal{T}_{L}$, we check whether the following linear program in variables
$\widehat{p}_{x,y}^{x}$ and $\widehat{p}_{x,y}^{y}$ is feasible:
1. (LP1)
For all $(x,y)\in\mathcal{T}_{L}$,
$0\leq\widehat{p}^{x}_{x,y},\widehat{p}^{y}_{x,y}\leq 1$.
2. (LP2)
For every $(x,y)\in\mathcal{L}_{L}^{g}$,
$r_{-}\leq\frac{\widehat{p}_{x,y}^{x}|\mathcal{S}_{x}|}{\widehat{p}_{x,y}^{y}|\mathcal{S}_{y}|}\leq
r_{+}.$
3. (LP3)
$\widehat{p}_{x_{0},y_{0}}^{x_{0}}=\widehat{p}_{x_{0},y_{0}}^{y_{0}}=1$.
Moreover, for every node $(x,y)\in\mathcal{T}_{L}\setminus\mathcal{L}_{L}$
whose children are defined on the set $(V_{S})_{x,y}\cup\\{v_{x,y}\\}$ and
letting $v=v_{x,y}$,
$\displaystyle\widehat{p}_{x,y}^{x}=\sum_{b\in[q]}\widehat{p}_{x_{v(a)},y_{v(b)}}^{x_{v(a)}}\quad\text{
for all }a\in[q],$
$\displaystyle\widehat{p}_{x,y}^{y}=\sum_{b\in[q]}\widehat{p}_{x_{v(b)},y_{v(a)}}^{y_{v(a)}}\quad\text{
for all }a\in[q].$
4. (LP4)
For every node $(x,y)\in\mathcal{T}_{L}\setminus\mathcal{L}_{L}$ whose
children are defined on the set $(V_{S})_{x,y}\cup\\{v_{x,y}\\}$ and letting
$v=v_{x,y}$, for every $a\in[q]$,
$\displaystyle\sum_{b\neq a}\widehat{p}_{x_{v(a)},y_{v(b)}}^{x_{v(a)}}$
$\displaystyle\leq 4q\eta\cdot\widehat{p}^{x}_{x,y},$
$\displaystyle\sum_{b\neq a}\widehat{p}_{x_{v(b)},y_{v(a)}}^{y_{v(a)}}$
$\displaystyle\leq 4q\eta\cdot\widehat{p}^{y}_{x,y}.$
###### Claim 4.6.
The above LP is feasible for
$r_{-}=r_{+}=|\mathcal{S}_{x_{0}}|/|\mathcal{S}_{y_{0}}|$.
###### Proof.
This follows immediately by taking
$\widehat{p}^{x}_{x,y}=p^{x}_{x,y},\widehat{p}^{y}_{x,y}=p^{y}_{x,y}$ and
using Lemma 4.3. ∎
###### Claim 4.7.
For every $r_{-},r_{+},\eta$ which can be represented in
$\operatorname{poly}(n,q)$ bits, the feasibility of the above LP can be
checked in time $\operatorname{poly}(n,q^{L})$.
###### Proof.
This follows from standard guarantees on the running time of linear
programming (cf. [Kha79]) since the number of variables and constraints in the
LP are $O(q^{L+O(1)})$ and since $|\mathcal{S}_{x}|/|\mathcal{S}_{y}|$ can be
represented using $\operatorname{poly}(n,q)$ bits. ∎
### 4.3. Analysis of the linear program
We now show that the feasibility of the above LP (for sufficiently large $L$
and appropriately chosen $p^{\prime\prime}$) implies that $r_{-}$
(respectively $r_{+}$) is an approximate lower (respectively upper) bound for
$|\mathcal{S}_{x_{0}}|/|\mathcal{S}_{y_{0}}|$. Given this, we will be able to
use binary search in order to approximate
$|\mathcal{S}_{x_{0}}|/|\mathcal{S}_{y_{0}}|$. The key point is that the
approximation error decays exponentially in $L$, which will allow us to take
$L$ small enough to ensure that this procedure is efficient.
###### Proposition 4.8.
Let $p^{\prime\prime}\leq(100q^{2}k\Delta^{4})^{-1}$, $p^{\prime}\leq
p^{\prime\prime}/(100\Delta^{3}q)$, and $L\geq 8k\Delta^{2}$. Then, the
feasibility of the above LP with parameters $r_{-},r_{+}$ implies that
$\left(1-4\cdot
2^{-L/(k\Delta^{2})}\right)r_{-}\leq\frac{|\mathcal{S}_{x_{0}}|}{|\mathcal{S}_{y_{0}}|}\leq\left(1+4\cdot
2^{-L/(k\Delta^{2})}\right)r_{+}.$
###### Proof.
By iterating the condition (LP3), we have
$\displaystyle\sum_{(x,y)\in\mathcal{L}_{L}:x\to\sigma}\widehat{p}_{x,y}^{x}$
$\displaystyle=1\quad\text{ for all }\sigma\in\mathcal{S}_{x_{0}},$
$\displaystyle\sum_{(x,y)\in\mathcal{L}_{L}:y\to\sigma}\widehat{p}_{x,y}^{y}$
$\displaystyle=1\quad\text{ for all }\sigma\in\mathcal{S}_{y_{0}}.$
Therefore,
$\displaystyle|\mathcal{S}_{x_{0}}|$
$\displaystyle=\sum_{\sigma\in\mathcal{S}_{x_{0}}}\sum_{(x,y)\in\mathcal{L}_{L}:x\to\sigma}\widehat{p}_{x,y}^{x},$
$\displaystyle|\mathcal{S}_{y_{0}}|$
$\displaystyle=\sum_{\sigma\in\mathcal{S}_{y_{0}}}\sum_{(x,y)\in\mathcal{L}_{L}:y\to\sigma}\widehat{p}_{x,y}^{y}.$
At the end of this subsection, we will prove the following.
###### Lemma 4.9.
For all $p^{\prime\prime}\leq(100q^{2}k\Delta^{4})^{-1}$, $p^{\prime}\leq
p^{\prime\prime}/(100\Delta^{3}q)$, and $L\geq 8k\Delta^{2}$,
$\displaystyle\frac{1}{|\mathcal{S}_{x_{0}}|}\sum_{\sigma\in\mathcal{S}_{x_{0}}}\sum_{(x,y)\in\mathcal{L}^{b}_{L}:x\to\sigma}\widehat{p}_{x,y}^{x}\leq
2^{-L/(k\Delta^{2})},$
$\displaystyle\frac{1}{|\mathcal{S}_{y_{0}}|}\sum_{\sigma\in\mathcal{S}_{y_{0}}}\sum_{(x,y)\in\mathcal{L}^{b}_{L}:y\to\sigma}\widehat{p}_{x,y}^{y}\leq
2^{-L/(k\Delta^{2})}.$
Given this lemma, we have
$\displaystyle|\mathcal{S}_{x_{0}}|$
$\displaystyle=\sum_{\sigma\in\mathcal{S}_{x_{0}}}\sum_{(x,y)\in\mathcal{L}^{g}_{L}:x\to\sigma}\widehat{p}_{x,y}^{x}+\sum_{\sigma\in\mathcal{S}(x_{0})}\sum_{(x,y)\in\mathcal{L}_{L}^{b}:x\to\sigma}\widehat{p}_{x,y}^{x}$
$\displaystyle=\left(\sum_{(x,y)\in\mathcal{L}_{L}^{g}}\widehat{p}^{x}_{x,y}\cdot|\mathcal{S}_{x}|\right)\pm|\mathcal{S}_{x_{0}}|\cdot
2^{-L/(k\Delta^{2})},$
where the first term follows by interchanging sums and the second term follows
by Lemma 4.9. A similar estimate also holds for $|\mathcal{S}_{y_{0}}|$.
Thus, we have
$\displaystyle\frac{|\mathcal{S}_{x_{0}}|\cdot(1\pm
2^{-L/(k\Delta^{2})})}{|\mathcal{S}_{y_{0}}|\cdot(1\pm 2^{-L/(k\Delta^{2})})}$
$\displaystyle=\frac{\sum_{(x,y)\in\mathcal{L}_{L}^{g}}\widehat{p}^{x}_{x,y}\cdot|\mathcal{S}_{x}|}{\sum_{(x,y)\in\mathcal{L}_{L}^{g}}\widehat{p}^{y}_{x,y}\cdot|\mathcal{S}_{y}|}$
$\displaystyle\in\left[\frac{r_{-}\cdot\sum_{(x,y)\in\mathcal{L}_{L}^{g}}\widehat{p}^{y}_{x,y}\cdot|\mathcal{S}_{y}|}{\sum_{(x,y)\in\mathcal{L}_{L}^{g}}\widehat{p}^{y}_{x,y}\cdot|\mathcal{S}_{y}|},\frac{r_{+}\cdot\sum_{(x,y)\in\mathcal{L}_{L}^{g}}\widehat{p}^{y}_{x,y}\cdot|\mathcal{S}_{y}|}{\sum_{(x,y)\in\mathcal{L}_{L}^{g}}\widehat{p}^{y}_{x,y}\cdot|\mathcal{S}_{y}|}\right]$
$\displaystyle\in[r_{-},r_{+}],$
where the second line follows from (LP2). Thus
$\displaystyle\frac{|\mathcal{S}_{x_{0}}|}{|\mathcal{S}_{y_{0}}|}\in[(1-4\cdot
2^{-L/(k\Delta^{2})})r_{-},(1+4\cdot 2^{-L/(k\Delta^{2})})r_{+}],$
as desired. ∎
###### Proof of Lemma 4.9.
We will only prove the statement for $|\mathcal{S}_{x_{0}}|$; the proof of the
other statement is identical.
For a node $(x,y)\in\mathcal{T}$, let
$(V_{S})^{\prime}_{x,y}=(V_{S})_{x,y}\setminus\\{v_{1}^{*},\dots,v_{\ell-1}^{*}\\}$.
For a node $(x,y)\in\mathcal{T}$ which is not a leaf, we will use $v_{x,y}$ to
denote the variable such that the children of $(x,y)$ are defined on the set
$(V_{S})_{x,y}\cup\\{v_{x,y}\\}$. When $(x,y)\in\mathcal{T}$ is clear from
context, we will denote $v_{x,y}$ simply by $v$.
Consider the following way of generating random root-to-leaf paths of
$\mathcal{T}_{L}$. At a non-leaf node $(x,y)\in\mathcal{T}$, sample a value
for $v=v_{x,y}$ according to $\mu_{S}[v=\cdot\mid x]$ to generate an
assignment $x^{\prime}$ on $(V_{S})_{x,y}\cup\\{v_{x,y}\\}$. Then, choose a
random element $b^{\prime}$ of $[q]$ and go to the node
$(x^{\prime},y_{v(b^{\prime})})\in\mathcal{T}$, where the probability of
choosing each $b\in[q]$ is
$p(x,y,x^{\prime},y_{v(b)})=\frac{\widehat{p}^{x^{\prime}}_{x^{\prime},y_{v(b)}}}{\widehat{p}^{x}_{x,y}}.$
Note that by (LP3), $p(x,y,x^{\prime},y_{v(\cdot)})$ is indeed a probability
distribution. Let $(X,Y)$ denote the random leaf of $\mathcal{T}_{L}$ returned
by this process and let $\widehat{\mu}$ denote the probability distribution on
$\mathcal{L}_{L}$ induced by this process.
Let $(x_{f},y_{f})\in\mathcal{L}_{L}$ and denote the corresponding root-to-
leaf path by $(x_{0},y_{0}),\dots,(x_{f},y_{f})$. Then,
$\widehat{\mu}[(X,Y)=(x_{f},y_{f})]=\prod_{t=1}^{f}\mu_{S}[x_{t}\mid
x_{t-1}]\times\prod_{t=1}^{f}p(x_{t-1},y_{t-1},x_{t},y_{t})=\frac{|\mathcal{S}_{x_{f}}|}{|\mathcal{S}_{x_{0}}|}\cdot\frac{\widehat{p}^{x_{f}}_{x_{f},y_{f}}}{\widehat{p}^{x_{0}}_{x_{0},y_{0}}}=\frac{|\mathcal{S}_{x_{f}}|}{|\mathcal{S}_{x_{0}}|}\cdot\widehat{p}^{x_{f}}_{x_{f},y_{f}},$
where the final equality follows by (LP3).
Therefore,
$\displaystyle\frac{1}{|\mathcal{S}_{x_{0}}|}\sum_{\sigma\in\mathcal{S}_{x_{0}}}\sum_{(x,y)\in\mathcal{L}_{L}^{b}:x\to\sigma}\widehat{p}^{x}_{x,y}$
$\displaystyle=\sum_{\sigma\in\mathcal{S}_{x_{0}}}\sum_{(x,y)\in\mathcal{L}_{L}^{b}:x\to\sigma}\frac{\widehat{\mu}[(X,Y)=(x,y)]}{|\mathcal{S}_{x}|}$
$\displaystyle=\sum_{(x,y)\in\mathcal{L}_{L}^{b}}\widehat{\mu}[(X,Y)=(x,y)]$
$\displaystyle\leq\widehat{\mu}[(X,Y)\in\\{(x,y)\in\mathcal{T}_{L}:|(V_{S})^{\prime}_{x,y}|\geq
L\\}].$
In order to bound the quantity on the right, we will first find a more
convenient combinatorial characterization of the event. For this, fix
$(x,y)\in\mathcal{T}_{L}$ such that $|(V_{S})^{\prime}_{x,y}|\geq L$. Recall
the notation $D_{x,y},\mathcal{F}_{x,y}$ from (T2). We also need some further
notation.
* •
For $i\in[m]$, we say that $C_{i}$ is $x$-frozen if $\mathbb{P}[C_{i}\mid
x]>p^{\prime\prime}$. We denote the set of $x$-frozen constraints by
$\mathcal{F}^{\prime}_{x}$.
* •
For $i\in[m]$, we say that $C_{i}$ has a disagreement if
$\operatorname{vbl}(C_{i})\cap D_{x,y}\neq\emptyset$. Denote all such
constraints by $\mathcal{D}_{x,y}$.
* •
For $i\in[m]$, we say that $C_{i}\in\mathcal{C}$ is bad if
$\mathcal{C}_{i}\in\mathcal{F}_{x,y}\cup\mathcal{D}_{x,y}$. We denote the set
of bad constraints by $\mathcal{B}_{x,y}$.
* •
$G(\mathcal{C})=(\mathcal{C},E)$ is the graph whose vertices are constraints
in $\mathcal{C}$, and for $i\neq j$, there is an edge between $C_{i}$ and
$C_{j}$ if and only if
$\operatorname{vbl}(C_{i})\cap\operatorname{vbl}(C_{j})\neq\emptyset$.
* •
$G^{2}(\mathcal{C})=(\mathcal{C},E^{\prime})$ is the graph whose vertices are
constraints in $\mathcal{C}$, and for $i\neq j$, there is an edge between
$C_{i}$ and $C_{j}$ if and only if there exists $k$ with
$\operatorname{vbl}(C_{i})\cap\operatorname{vbl}(C_{k})\neq\emptyset$, and
$\operatorname{vbl}(C_{j})\cap\operatorname{vbl}(C_{k})\neq\emptyset$.
###### Claim 4.10.
$|\mathcal{B}_{x,y}|\geq L/(k\Delta)$.
###### Proof.
First, note that by (O3), (T3), and (T4), for every
$v\in(V^{\prime}_{S})_{x,y}$, there exists some $C\in\mathcal{C}$ such that
$\operatorname{vbl}(C)\cap(V_{D})_{x,y}\neq\emptyset$ and
$v\in\operatorname{vbl}(C)$.
Next, note that by (T2), for every $v\in(V_{D})_{x,y}$, there exists some
$B\in\mathcal{B}_{x,y}$ such that $v\in\operatorname{vbl}(B)$.
Therefore,
$\displaystyle L$ $\displaystyle\leq|(V_{S}^{\prime})_{x,y}|$
$\displaystyle\leq
k\cdot|\\{C\in\mathcal{C}:\operatorname{vbl}(C)\cap(V_{D})_{x,y}\neq\emptyset\\}|$
$\displaystyle\leq
k\cdot|\\{C\in\mathcal{C}:\operatorname{vbl}(C)\cap\operatorname{vbl}(B)\neq\emptyset\text{
for some }B\in\mathcal{B}_{x,y}\\}|$ $\displaystyle\leq
k\Delta\cdot|\mathcal{B}_{x,y}|.\qed$
Fix $C^{*}\in\mathcal{C}$ such that
$v_{\ell}^{*}\in\operatorname{vbl}(C^{*})$.
###### Claim 4.11.
There exists a $\\{2,3\\}$-tree $T\subseteq\mathcal{B}_{x,y}$ in
$G(\mathcal{C})$ with $|T|\geq L/(k\Delta^{2})$ and such that $T$ contains
$C^{*}$.
###### Proof.
By (T4), (O3), and induction, the induced subgraph of $G^{2}(\mathcal{C})$ on
$\mathcal{B}_{x,y}$ is connected. Moreover,
$C^{*}\in\mathcal{D}_{x,y}\subseteq\mathcal{B}_{x,y}$. Given this, the claim
follows from Lemma 2.4 and the previous claim. ∎
Let $\widehat{L}=L/(k\Delta^{2})$ and let $\mathfrak{T}_{\widehat{L}}$ denote
the collection of $\\{2,3\\}$-trees in $G(\mathcal{C})$ of size $\widehat{L}$
which contain $C^{*}$. Then, by the previous discussion,
$\displaystyle\widehat{\mu}[(X,Y)\in\\{(x,y)\in\mathcal{T}_{L}:|(V_{S})^{\prime}_{x,y}|\geq
L\\}]$
$\displaystyle\leq\sum_{T\in\mathfrak{T}_{\widehat{L}}}\widehat{\mu}[T\subseteq\mathcal{B}_{X,Y}]$
$\displaystyle\leq(e\Delta^{3})^{\widehat{L}}\cdot\widehat{\mu}[T\subseteq\mathcal{B}_{X,Y}],$
where the final inequality uses Lemma 2.3.
Finally, let us fix $T\in\mathfrak{T}_{\widehat{L}}$ and estimate
$\widehat{\mu}[T\subseteq\mathcal{B}_{X,Y}].$
We denote the vertices of $T$ in $G(\mathcal{C})$ by
$C^{\prime}_{1}=C^{*},C_{2}^{\prime},\dots,C^{\prime}_{\widehat{L}}$. Note, in
particular, that $v_{\ell}^{*}\notin\operatorname{vbl}(C^{\prime}_{j})$ for
$2\leq j\leq\widehat{L}$. By multiplying the result by an overall factor of
$2^{\widehat{L}}$, it suffices to bound the probability
$\widehat{\mu}[C^{\prime}_{2},\dots,C^{\prime}_{t}\in\mathcal{D}_{X,Y}\wedge
C^{\prime}_{t+1},\dots,C^{\prime}_{\widehat{L}}\in\mathcal{F}_{X,Y}\setminus\mathcal{D}_{X,Y}],$
which is at most
$\widehat{\mu}[C^{\prime}_{2},\dots,C^{\prime}_{t}\in\mathcal{D}_{X,Y}\wedge
C^{\prime}_{t+1},\dots,C^{\prime}_{\widehat{L}}\in\mathcal{F}^{\prime}_{X}].$
Using the law of total probability, this is equal to
$\displaystyle\mathbb{E}_{\widehat{\mu}}\left[\widehat{\mu}[C^{\prime}_{2},\dots,C^{\prime}_{t}\in\mathcal{D}_{X,Y}\mid
X]\cdot\widehat{\mu}[C^{\prime}_{t+1},\dots,C^{\prime}_{\widehat{L}}\in\mathcal{F}^{\prime}_{X}\mid
X]\right].$ (4.1)
###### Claim 4.12.
For any possible realization $x$ of $X$,
$\widehat{\mu}[C^{\prime}_{2},\dots,C^{\prime}_{t}\in\mathcal{D}_{x,Y}\mid
x]\leq(4kq\eta)^{t-1}.$
###### Proof.
Let $j\in[t]$. Given $x$, we have $C^{\prime}_{j}\in\mathcal{D}_{x,Y}$ only if
there is some $v\in\operatorname{vbl}(C^{\prime}_{j})$ for which $Y(v)\neq
x(v)$. Since $|\operatorname{vbl}(C^{\prime}_{j})|\leq k$, we see by (LP4)
that this happens with probability at most $4kq\eta$. Moreover, since
$\operatorname{vbl}(C^{\prime}_{j})$ are disjoint for different values of
$[j]$, these events are independent, which gives the desired conclusion. ∎
Using this claim, we see that the quantity in 4.1 is bounded above by
$\displaystyle(4kq\eta)^{t-1}\cdot\widehat{\mu}[C^{\prime}_{t+1},\dots,C^{\prime}_{\widehat{L}}\in\mathcal{F}^{\prime}_{X}].$
(4.2)
###### Claim 4.13.
$\widehat{\mu}[C^{\prime}_{t+1},\dots,C^{\prime}_{\widehat{L}}\in\mathcal{F}^{\prime}_{X}]\leq(2p^{\prime}q/p^{\prime\prime})^{\widehat{L}-t}.$
###### Proof.
Let
$\mathfrak{X}=\\{x:(x,y)\in\mathfrak{T}_{\widehat{L}}\text{ for some }y\wedge
C_{t+1}^{\prime},\dots,C^{\prime}_{\widehat{L}}\in\mathcal{F}^{\prime}_{x}\\},$
so that our goal is to bound $\widehat{\mu}[\mathfrak{X}]$. We will use an
argument similar to Lemma 3.2.
For $h\in\\{0,\dots,L\\}$, let $\mathcal{T}_{h}\subseteq\mathcal{T}_{L}$
denote those nodes $(x,y)\in\mathcal{T}$ for which $|(V_{S})_{x,y}|\leq
h+\ell$ and let $\mathcal{L}_{h}$ denote the leaves of $\mathcal{T}_{h}$. To
any $(x,y)\in\mathcal{T}_{L}$, we can naturally associate a root-to-leaf path
$(x_{0},y_{0}),\dots,(x_{L},y_{L})=(x,y)$, where
$(x_{i},y_{i})\in\mathcal{T}_{i}$ and nodes are possibly repeated at the end
of the path. Interpreting $\widehat{\mu}$ as a distribution on root-to-leaf
paths of this form, let $\mathfrak{S}(h)$ denote the sigma-algebra induced on
$\mathcal{T}_{h}$ and let
$M_{h}=\prod_{j=t+1}^{\widehat{L}}\mathbb{P}[C_{j}^{\prime}\mid
X_{h}]\cdot\exp(-\eta qW(X_{h},Y_{h})),$
where
$W(X_{h},Y_{h})=\\#\left\\{j\in[h-1]:v_{X_{j},Y_{j}}\in\operatorname{vbl}(C^{\prime}_{t+1})\cup\dots\cup\operatorname{vbl}(C^{\prime}_{\widehat{L}})\right\\}.$
Then, $M_{h}$ is measurable with respect to $\mathfrak{S}(h)$ and using the
argument in Lemma 3.2, it is readily seen that
$\mathbb{E}_{\widehat{\mu}}[M_{h+1}\mid\mathfrak{S}(h)]\leq M_{h}.$
Indeed, we only need to check that given $(X_{h},Y_{h})$ and letting $C_{T}$
denote the unique constraint (if any) containing $v=v_{X_{h},Y_{h}}$, we have
$\displaystyle\mathbb{E}_{\widehat{\mu}}\bigg{[}\mathbb{P}[C_{T}\mid
X_{h+1}]\mid\mathfrak{S}(h)\bigg{]}$
$\displaystyle=\sum_{a\in[q]}\frac{\mathbb{P}[C_{T}\wedge X_{h}\wedge
v=a]\cdot{\widehat{\mu}}[v=a\mid X_{h}]}{\mathbb{P}[X_{h}\wedge v=a]}$
$\displaystyle=\sum_{a\in[q]}\frac{\mathbb{P}[C_{T}\wedge X_{h}\wedge
v=a]}{\mathbb{P}[X_{h}]}\cdot\frac{\widehat{\mu}[v=a\mid
X_{h}]}{\mathbb{P}[v=a]}$
$\displaystyle\leq\sum_{a\in[q]}\frac{\mathbb{P}[C_{T}\wedge X_{h}\wedge
v=a]}{\mathbb{P}[X_{h}]}\frac{q^{-1}+\eta}{q^{-1}}$
$\displaystyle\leq\exp(\eta q)\mathbb{P}[C_{T}\mid X_{h}],$
where the third line follows from (2) of Lemma 4.1. Since
$\mathbb{P}[C_{j}^{\prime}\mid X_{0}]\leq p^{\prime}q,$
it therefore follows that
$\mathbb{E}_{\widehat{\mu}}[M_{L}]\leq(p^{\prime}q)^{\widehat{L}-t}.$
Also, for any $(x,y)\in\mathcal{T}_{L}$, we have that $W(x,y)\leq
k(\widehat{L}-t)$. Hence, by Markov’s inequality and the assumption on
$p^{\prime\prime}$,
$\displaystyle\widehat{\mu}[\mathfrak{X}]$
$\displaystyle\leq\widehat{\mu}[M_{L}\geq(p^{\prime\prime})^{\widehat{L}-t}\exp(-\eta
qk(\widehat{L}-t))]$ $\displaystyle\leq\left(\frac{p^{\prime}q\exp(\eta
qk)}{p^{\prime\prime}}\right)^{\widehat{L}-t}\leq\left(\frac{2p^{\prime}q}{p^{\prime\prime}}\right)^{\widehat{L}-t}.\qed$
Using the preceding claim along with 4.2 and simplying using the bounds on
$p^{\prime},p^{\prime\prime}$ completes the proof. ∎
## 5\. Proof of Theorems 1.5 and 1.6
We are now ready to prove Theorems 1.5 and 1.6. The algorithms in this section
exploit a refinement of the analysis in the previous section, which we present
in Section 5.1. Following this, we present the proof of Theorem 1.5 in Section
5.2 and the proof of Theorem 1.6 in Section 5.3. We will freely use the
notation introduced in the previous two sections.
### 5.1. Refined analysis of the linear program
As in Section 3, fix an ordering $v_{1},\dots,v_{n}$ of the variables. Let
$p^{\prime}\leq p^{\prime\prime}$ be parameters to be chosen later. Recall
that in Section 3.1, we generate a partial assignment on a subset
$v_{1}^{*},\dots,v_{s}^{*}$ of the variables (where $s$ is itself random) and
“freeze” the remaining variables. This process depends on the parameter
$p^{\prime}$ in (R4). As before, for $i\in[s]$, we let $P_{i}$ denote the
partial assignment on $v_{1}^{*},\dots,v_{i}^{*}$ and we let $\nu$ denote the
distribution on partial assignments given by the final partial assignment
$P_{s}$. Once again, we emphasize that $s$ is random.
For our approximate sampling algorithm, we will also need the following
variation of this procedure. We consider the same randomized greedy procedure
as in Section 3.1, except now, in (R3), we assign $v_{i}^{*}$ a random value
chosen according to the distribution $\mu_{S}[v_{i}^{*}=\cdot\mid P_{i-1}]$.
For now, the reader should ignore the question of how to efficiently implement
such a procedure. We let $\nu_{S}$ denote the distribution on partial
assignments given by the final partial assignment $P_{s}$, noting again that
$s$ is random.
Given $v_{1}^{*},\dots,v_{i}^{*}$ and $a\in[q]$, we denote by $P_{i}(a)$ the
partial assignment extending $P_{i-1}$ by setting $v_{i}^{*}=a$. As before,
for $h\in[n]$, we let $\iota(h)$ be the largest index $i$ such that
$v_{i}^{*}=v_{w}$ for some $w\leq h$ i.e. $\iota(h)$ is the number of
variables among $\\{v_{1},\dots,v_{h}\\}$ assigned values by the partial
assignment.
For each variable $v$, there are at most $\Delta$ constraints
$C\in\mathcal{C}$ such that $v\in\operatorname{vbl}(C)$. Let
$\widehat{L}=L/(k\Delta^{2})$, and let $\mathfrak{T}_{\widehat{L},v}$ be the
set of $\\{2,3\\}$-trees in $G(\mathcal{C})$ of size $\widehat{L}$ containing
one of these constraints. For any $T\in\mathfrak{T}_{\widehat{L},v}$, we let
$C^{*}_{v}$ denote the unique $C\in T$ satisfying $v\in\operatorname{vbl}(C)$.
Recall that $\mathcal{F}_{x,y}$ denotes the constraints $C\in\mathcal{C}$ for
which $\mathbb{P}[C\mid x]>p^{\prime\prime}$ or $\mathbb{P}[C\mid
y]>p^{\prime\prime}$.
For $\ell\in[s]$, we define the idealized decision tree $\mathcal{T}$ starting
from the root node $(x_{0},y_{0}):=(P_{\ell}(a),P_{\ell}(b))$ and the
$L$-truncated decision tree $\mathcal{T}_{L}$ as in Section 4. Let
$\eta=(1-3p^{\prime\prime}q)^{-\Delta}-1$. The following lemma was proved
during the course of the proof of Lemma 4.9.
###### Lemma 5.1.
For all $p^{\prime}=p^{\prime\prime}\leq(1000q^{2}k\Delta^{4})^{-1}$ and
$p\leq p^{\prime\prime}/(1000\Delta^{3}q)$,
$\displaystyle\frac{1}{|\mathcal{S}_{x_{0}}|}\sum_{\sigma\in\mathcal{S}_{x_{0}}}\sum_{(x,y)\in\mathcal{L}^{b}_{L}:x\to\sigma}\widehat{p}_{x,y}^{x}\leq\sum_{T\in\mathfrak{T}_{\widehat{L},v^{*}_{\ell}}}\,\,\prod_{C\in
T,C\neq C^{*}_{v^{*}_{\ell}}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
P_{\ell-1}]}{p^{\prime\prime}}\right),$
$\displaystyle\frac{1}{|\mathcal{S}_{y_{0}}|}\sum_{\sigma\in\mathcal{S}_{y_{0}}}\sum_{(x,y)\in\mathcal{L}^{b}_{L}:y\to\sigma}\widehat{p}_{x,y}^{y}\leq\sum_{T\in\mathfrak{T}_{\widehat{L},v^{*}_{\ell}}}\,\,\prod_{C\in
T,C\neq C^{*}_{v^{*}_{\ell}}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
P_{\ell-1}]}{p^{\prime\prime}}\right).$
###### Remark.
Note that there is no factor of $q$ multiplying $2\mathbb{P}[C\mid
P_{\ell-1}]/p^{\prime\prime}$ since $v_{\ell}^{*}\notin\operatorname{vbl}(C)$
for any $C$ that features in the product.
For $h\in[n+1]$, let $\mathcal{E}(h)$ denote the event that for all variables
$v\in V$,
$\displaystyle\sum_{T\in\mathfrak{T}_{\widehat{L},v}}\,\,\prod_{C\in T,C\neq
C^{*}_{v}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
P_{\iota(h-1)}]}{p^{\prime\prime}}\right)\leq n^{4}2^{-L/(k\Delta^{2})}.$
The next lemma, together with Markov’s inequality, shows that $\mathcal{E}(h)$
occurs with high probability with respect to both $\nu$ and $\nu_{S}$.
###### Lemma 5.2.
Let $p^{\prime}=p^{\prime\prime}\leq(100q^{2}k\Delta^{4})^{-1}$, $p\leq
p^{\prime\prime}/(100\Delta^{3})$, and $L\geq 8k\Delta^{2}$. Then, for all
$h\in[n+1]$ and $v\in V$,
$\displaystyle\mathbb{E}_{\nu}\left[\sum_{T\in\mathfrak{T}_{\widehat{L},v}}\,\,\prod_{C\in
T,C\neq C^{*}_{v}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
P_{\iota(h-1)}]}{p^{\prime\prime}}\right)\right]\leq 2^{-L/(k\Delta^{2})},$
and
$\displaystyle\mathbb{E}_{\nu_{S}}\left[\sum_{T\in\mathfrak{T}_{\widehat{L},v}}\,\,\prod_{C\in
T,C\neq C^{*}_{v}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
P_{\iota(h-1)}]}{p^{\prime\prime}}\right)\right]\leq 2^{-L/(k\Delta^{2})}.$
###### Proof.
We will first prove the statement for $\nu$. Fix $v\in V$ and observe that for
any $T\in\mathfrak{T}_{\widehat{L},v}$, the sets $\operatorname{vbl}(C)$ are
disjoint for $C\in T$. Let $\mathfrak{S}(t)$ denote the $\sigma$-algebra
generated by the output of the randomized greedy procedure on partial
assignments of $v_{1},\dots,v_{t}$. Then, by an identical argument to Lemma
3.2, we see that
$M_{t}=\prod_{C\in T,C\neq C^{*}_{v}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
P_{\iota(t-1)}]}{p^{\prime\prime}}\right)$
satisfies
$\mathbb{E}_{\nu}[M_{t+1}\mid\mathfrak{S}(t)]=M_{t}$
Thus, for any $h\in[n+1]$,
$\mathbb{E}_{\nu}\left[\prod_{C\in T,C\neq
C^{*}_{v}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
P_{\iota(h-1)}]}{p^{\prime\prime}}\right)\right]\leq\left(4kq\eta+\frac{2p}{p^{\prime\prime}}\right)^{|T|-1},$
so that by linearity of expectation,
$\displaystyle\mathbb{E}_{\nu}\left[\sum_{T\in\mathfrak{T}_{\widehat{L},v}}\,\,\prod_{C\in
T,C\neq C^{*}_{v}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
P_{\iota(h-1)}]}{p^{\prime\prime}}\right)\right]$
$\displaystyle\leq|\mathfrak{T}_{\widehat{L},v}|\left(4kq\eta+\frac{2p}{p^{\prime\prime}}\right)^{|T|-1}$
$\displaystyle\leq\Delta\cdot(e\Delta^{3})^{\widehat{L}}\cdot\left(4kq\eta+\frac{2p}{p^{\prime\prime}}\right)^{\widehat{L}-1}.$
The desired bound is obtained by using the assumptions on $p$ and
$p^{\prime\prime}$.
Next, we prove the statement for $\nu_{S}$. Fix $v\in V$. For
$T\in\mathfrak{T}_{\widehat{L},v}$, let $W_{t}(T)$ be the number of variables
among $v_{1}^{*},\dots,v_{\iota(t)}^{*}$ that are contained in some $C\in T$.
Also, as before, let $\mathfrak{S}(t)$ denote the $\sigma$-algebra generated
by the output of the randomized greedy procedure on partial assignments of
$v_{1},\dots,v_{t}$. Then, as in Claim 4.13, we have that
$M_{t}=\prod_{C\in T,C\neq C^{*}_{v}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
P_{\iota(t-1)}]}{p^{\prime\prime}}\right)\cdot\exp(-\eta qW_{t-1}(T))$
satisfies
$\mathbb{E}_{\nu_{S}}[M_{t+1}\mid\mathfrak{S}(t)]\leq M_{t}.$
Thus, for any $h\in[n+1]$,
$\mathbb{E}_{\nu_{S}}\left[\prod_{C\in T,C\neq
C^{*}_{v}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
P_{\iota(h-1)}]}{p^{\prime\prime}}\right)\right]\leq\left(4kq\eta+\frac{2p}{p^{\prime\prime}}\right)^{|T|-1}\exp(\eta
qk)^{|T|-1}.$
so that by linearity of expectation,
$\displaystyle\mathbb{E}_{\nu_{S}}\left[\sum_{T\in\mathfrak{T}_{\widehat{L},v}}\,\,\prod_{C\in
T,C\neq C^{*}_{v}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
P_{\iota(h-1)}]}{p^{\prime\prime}}\right)\right]$
$\displaystyle\leq|\mathfrak{T}_{\widehat{L},v}|\left(8kq\eta+\frac{4p}{p^{\prime\prime}}\right)^{|T|-1}$
$\displaystyle\leq\Delta\cdot(e\Delta^{3})^{\widehat{L}}\cdot\left(8kq\eta+\frac{4p}{p^{\prime\prime}}\right)^{\widehat{L}-1}.$
The desired bound is obtained by using the assumptions on $p$ and
$p^{\prime\prime}$. ∎
Combining Lemmas 5.1 and 5.2 and the analysis in Section 4, we obtain the
following proposition.
###### Proposition 5.3.
Let $p^{\prime}=p^{\prime\prime}\leq(1000q^{2}k\Delta^{4})^{-1}$, $p\leq
p^{\prime\prime}/(1000\Delta^{3})$, and $\delta\in(0,1)$. Then, for $L\geq
8k\Delta^{2}\log(n/\delta)$, the event $\wedge_{h\in[n+1]}\mathcal{E}(h)$ has
probability at least $1-(\delta/n^{2})$ with respect to both $\nu$ and
$\nu_{S}$.
Moreover, on the event $\wedge_{h\in[n+1]}\mathcal{E}(h)$, the feasibility of
the LP with parameters $r_{-}\leq r_{+}$ implies that
$\left(1-4\cdot
n^{4}2^{-L/(k\Delta^{2})}\right)r_{-}\leq\frac{|\mathcal{S}_{x_{0}}|}{|\mathcal{S}_{y_{0}}|}\leq\left(1+4\cdot
n^{4}2^{-L/(k\Delta^{2})}\right)r_{+}.$
### 5.2. Approximate counting: proof of Theorem 1.5
We have the following analogue of Proposition 3.5.
###### Lemma 5.4.
Let $p^{\prime}=p^{\prime\prime}\leq(1000q^{2}k\Delta^{4})^{-1}$, $p\leq
p^{\prime\prime}/(1000\Delta^{3})$, and $L=80k\Delta^{2}\log(\Delta n)$. There
exists a deterministic algorithm running in time
$O(n^{\operatorname{poly}(\log q,\Delta,k)})$ which generates a sequence of
partial assignments $P_{1},\dots,P_{s}$ with the following properties.
1. (1)
For all $i\in[s]$, $P_{i}$ assigns values to $i$ variables, and $P_{i}$
extends $P_{i-1}$.
2. (2)
$A_{s}=\emptyset$.
3. (3)
For all $i\in[s]$ and $j\in[m]$, $\mathbb{P}[C_{j}\mid P_{i}]\leq
p^{\prime}q$.
4. (4)
Every connected component in $G(\mathcal{F})$ has size at most $L/(k\Delta)$.
5. (5)
The event $\wedge_{h\in[n+1]}\mathcal{E}(h)$ is satisfied.
###### Proof.
The proof is a modification of the proof of Proposition 3.5. Let
$L^{\prime}=L/(k\Delta^{2})$. Recall that for each variable $v$,
$\mathfrak{T}_{L^{\prime},v}$ is the set of $\\{2,3\\}$-trees in
$G(\mathcal{C})$ of size $L^{\prime}$ containing $C^{*}_{v}$. Let
$\mathfrak{T}$ denote the collection of all {2,3}-trees of size $L^{\prime}$
in $G(\mathcal{C})$. Note that
$|\mathfrak{T}_{L^{\prime},v}|\leq|\mathfrak{T}|\leq\operatorname{poly}(n^{\log^{2}\Delta})$
and the collections $\mathfrak{T}_{L^{\prime},v}$, $\mathfrak{T}$ can be
constructed in time $\operatorname{poly}(n^{\log^{2}\Delta})$.
For a partial assignment $X$, define
$H(X)=\sum_{v\in V}\sum_{T\in\mathfrak{T}_{L^{\prime},v}}\,\,\prod_{C\in
T,C\neq C^{*}_{v}}\left(4kq\eta+\frac{2\mathbb{P}[C\mid
X]}{p^{\prime\prime}}\right).$
Note that
$H(X)\geq\sum_{T\in\mathfrak{T}}\prod_{C\in T}\left(\frac{\mathbb{P}[C\mid
X]}{p^{\prime\prime}}\right).$
As in the proof of Proposition 3.5, if we can find a sequence of partial
assignments $P_{1},\dots,P_{s}$ satisfying properties (1), (2), (3) such that
$H(P_{1}),\dots,H(P_{s})<n^{4}2^{-L/(k\Delta^{2})}<1$, then (4) and (5) are
also satisfied.
For this, we follow the same greedy procedure as in Section 3.1, except now,
after having chosen $P_{i-1}$ and $v_{i}^{*}$, we choose the value of
$v_{i}^{*}$ in (R3) to be
$\operatorname*{arg\,min}_{a\in[q]}H(P_{i-1}\wedge{v_{i}^{*}=a}).$
Similar to Proposition 3.5, this ensures that $H(P_{i})\leq H(P_{i-1})$ for
all $i\in[s]$. Thus, it is possible to choose $P_{i}$ to ensure
$H(P_{i-1})\leq H(P_{i})$. Finally, since
$H(\emptyset)\leq
n\cdot\Delta\cdot(e\Delta^{3})^{L^{\prime}}\cdot\left(4kq\eta+\frac{2p}{p^{\prime\prime}}\right)^{L^{\prime}}<n^{4}2^{-L/(k\Delta^{2})},$
we are done. ∎
Finally, given partial assignments $P_{1},\dots,P_{s}$ satisfying the
properties of Lemma 5.4, we can use Lemma 3.6, Proposition 5.3 and the
analysis of Section 4 to complete the proof of Theorem 1.5.
### 5.3. Approximate sampling: proof of Theorem 1.6
We consider the following sampling procedure. Fix a parameter
$\varepsilon\in(0,1)$. Let $L=80k\Delta^{2}\log(\Delta n/\varepsilon)$.
1. (S1)
Fix an arbitrary ordering $v_{1},\dots,v_{n}$ of the variables. Initialize the
set of frozen variables $F_{0}=\emptyset$, the set of available variables
$A_{0}=V$, $\iota(0)=0$, and $P_{0}=\emptyset$.
2. (S2)
Let $1\leq i\leq n$. Given $P_{\iota(i-1)},F_{i-1},A_{i-1}$, if
$\mathcal{E}(i)$ does not hold, then output an arbitrary satisfying assignment
(which can be found using the algorithmic LLL in [MT10]) and terminate.
Otherwise, $\mathcal{E}(i)$ holds.
3. (S3)
If $v_{i}\notin A_{i-1}$, then
$\iota(i)=\iota(i-1),F_{i}=F_{i-1},A_{i}=A_{i-1}$. Increment $i$. If $i\leq
n$, return to (S2). Otherwise, proceed to (S6).
4. (S4)
If $v_{i}\in A_{i-1}$, then approximate the marginal $\mu_{S}[v_{i}=\cdot\mid
P_{\iota(i-1)}]$ within total variation distance $\varepsilon/(8n)$ using the
LP. Then, assign $v_{i}$ a random value in $[q]$ distributed according to the
output of the LP. Let $\iota(i)=\iota(i-1)+1$, and $P_{\iota(i)}$ be the
extension of $P_{\iota(i-1)}$ resulting from assigning a value to $v_{i}$.
5. (S5)
Let
$\mathcal{F}_{i}=\\{j\in[m]:\mathbb{P}[C_{j}\mid
P_{\iota(i)}]>p^{\prime\prime}\\}.$
Set
$F_{i}=F_{i-1}\cup\bigcup_{j\in\mathcal{F}_{i}}(\operatorname{vbl}(C_{j})\cap
A_{i}\setminus\\{v_{i}\\})\text{ and
}A_{i}=A_{i-1}\setminus(F_{i}\cup\\{v_{i}\\}).$
Increment $i$. If $i\leq n$, return to (S2). Otherwise, proceed to (S6).
6. (S6)
Let $\mathcal{F}=\cup_{i\in[n]}\mathcal{F}_{i}$. Consider
$G^{2}(\mathcal{F})$, which is the induced subgraph of $G^{2}(\mathcal{C})$ by
the vertices $\mathcal{F}$. If any connected component of $G^{2}(\mathcal{F})$
has size larger than $80\Delta\log(\Delta n)$, then output an arbitrary
satisfying assignment and terminate.
7. (S7)
Else, use exhaustive enumeration to uniformly sample a satisfying assignment
of unassigned variables appearing in each separate connected component of
$G^{2}(\mathcal{F})$, and return the complete satisfying assignment thus
obtained.
It is immediate that the running time of the algorithm is as claimed in
Theorem 1.6. Let $\mu_{\operatorname{Alg}}$ denote the distribution on
satisfying assignments generated by the algorithm. We show that
$\operatorname{TV}(\mu_{\operatorname{Alg}},\mu_{S})\leq\varepsilon$.
For this, we begin by observing that if we could sample from the true marginal
distribution $\mu_{S}[v_{i}=\cdot\mid P_{\iota(i-1)}]$ in (S4) and if we could
output a uniform satisfying assignment extending the current partial
assignment for the early termination in (S2) and (S6), then the resulting
distribution on satisfying assignments output by the algorithm clearly
coincides with $\mu_{S}$.
Next, since the approximate marginals in (S4) are within $\varepsilon/(8n)$ of
the true marginals, it follows from Proposition 5.3 and Lemma 3.4 that the
early termination condition in (S2) and (S6) occurs with probability at most
$\varepsilon/4$. Therefore,
$\operatorname{TV}(\mu_{\operatorname{Alg}},\mu_{\operatorname{Alg}^{\prime}})\leq\varepsilon/4$,
where $\operatorname{Alg}^{\prime}$ denotes the sampling algorithm which is
the same as above, except upon early termination in (S2) and (S6), we output a
uniformly random satisfying assignment extending the current partial
assignment.
Finally, since the approximate marginals in (S4) are within $\varepsilon/(8n)$
of the true marginals, it follows that
$\operatorname{TV}(\mu_{\operatorname{Alg}^{\prime}},\mu_{S})\leq\varepsilon/8$,
so that by the triangle inequality,
$\operatorname{TV}(\mu_{\operatorname{Alg}},\mu_{S})\leq\varepsilon$, as
desired.
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|
# Finite temperature Instabilities of 2D Dipolar Bose Gas at Arbitrary Tilt
Angle
Pengtao Shen Khandker F. Quader Department of Physics, Kent State
University, Kent , OH 44242, USA
###### Abstract
Advances in creating stable dipolar Bose systems, and ingenious box traps have
generated tremendous interest. Theory study of dipolar bosons at finite
temperature (T) has been limited. Motivated by these, we study 2D dipolar
bosons at arbitrary tilt angle, $\theta$, using finite-T random phase
approximation. We show that a comprehensive understanding of phases and
instabilities at non-zero T can be obtained on concurrently considering dipole
strength, density, temperature and $\theta$. We find the system to be in a
homogeneous non-condensed phase that undergoes a collapse transition at large
$\theta$, and a finite momentum instability, signaling a striped phase, at
large dipolar strength; there are important differences with the T=0 case. At
T = 0, BEC appears at critical dipolar strength, and at critical density. Our
predictions for polar molecule system, ${}^{41}K^{87}Rb$, and ${}^{166}Er$ may
provide tests of our results. Our approach may apply broadly to systems with
long-range, anisotropic interactions.
###### pacs:
67.85.-d, 67.85.Jk,67.85.De,05.30.Jp,05.30.Rt
The nature of excitations, phases and instabilities of interacting Bose
systems has been a subject of longstanding interest. The extraordinary
development of the field of ultracold atoms, tremendously advanced by novel
experimental techniques, has, over the past several years, led to intense
research. In recent years, there has been considerable interest in systems
with long-range and anisotropic interactions; examples are bosonic and
fermionic atoms, and polar molecules experiencing dipolar interactions.
Recent experimental advances in creating stable dipolar bosonic systems,
including polar molecules with large electric dipole moments, have led to
vigorous research activities. Dipolar Bose-Einstein condensates (BEC) have
been realized in chromium Koch et al. (2008) (${}^{52}Cr$), and in lanthanide
atoms (such as dysprosium, erbium Aikawa et al. (2012)), which have larger
magnetic moments. Recent reporting Chomaz et al. (2018) of observation for the
first time of roton mode in dipolar ${}^{166}Er$ (magnetic moment 7 $\mu_{B}$)
in cigar-shape trap geometry constitute a significant development. Realization
of high phase-space density systems of polar molecules, such as,
${}^{87}Rb^{133}Cs$ Molony et al. (2014); Takekoshi et al. (2014),
${}^{41}K^{87}Rb$ Aikawa et al. (2009, 2010) hold promise for realization of
quantum degeneracy and dipolar BEC. In general, the electric dipole moments of
the polar molecules are substantially larger than the magnetic dipole moments
of atoms; e.g. the RbCs system has sizable electric dipole moment $\sim$ 1.28
Debye. Ingenious box traps constitute another profoundly significant
development. There have been box-trap experiments on bosons subjected to
contact interaction Gaunt et al. (2013); Gotlibovych et al. (2014); Lopes et
al. (2017); those on dipolar systems are ongoing.
The long-range and anisotropic nature, and a region of attraction, of dipolar
interaction can give rise to novel quantum phases, even in dilute systems. A
sizeable body of theory work, based on Monte Carlo and Bogoliubov-de Gennes
(BdG) methods, exist at at zero temperature (T=0). The existence of roton mode
and density-wave phase has been found in BdG calculations studying the
property of BEC ground state Fischer (2006); Santos et al. (2003); Fedorov et
al. (2014); Lu et al. (2015); Shen and Quader (2018). Solid and stripe like
crystal phases have been predicted in Monte Carlo simulation Macia et al.
(2014); Bombin et al. (2017). Such stripe phase of the dipolar Bose system
provides a promising candidate Boninsegni and Prokof’ev (2012); Wenzel et al.
(2017); Tanzi et al. (2019); Böttcher et al. (2019); Chomaz et al. (2019) for
an intrinsic supersolid without the presence of defects as described by the
Andreev-Lifshitz mechanism Cinti et al. (2014). However, theoretical study of
2D dipolar boson gas at finite temperature has been limited.
A purely dipolar 3D system is usually unstable towards collapse due to the
attractive component of the interaction. A trap helps to stabilize the system;
this depends strongly on trapping geometry Koch et al. (2008); Eberlein et al.
(2005). In 2D, the stability issue of dipolar bosons may be richer Sun et al.
(2010); Yamaguchi et al. (2010). There have been studies Mueller and Baym
(2000); van Zyl et al. (2002) of density response in 2D and 3D Bose gas with
attractive constant interaction, using random phase approximation (RPA).
Similar study of dipolar bosons in a cylindrical trap at finite temperature
shows that a pancake geometry trap stabilizes the system Bisset et al. (2011).
That brings up the question whether purely 2D dipolar bosons are stable at
finite temperature.
In this paper, we present results of our study of a 2D dipolar Bose system at
non-zero temperatures, using finite-temperature RPA Mueller and Baym (2000);
Bisset et al. (2011). A key point of the paper is that a broad perspective on
the phases and instabilities of the system at finite temperature can be
attained by considering several tunable knobs: density, temperature,
interaction strength and the orientation of dipole moments to an external
field, i.e. tilt angle. We construct several informative phase diagrams based
on our study of dipolar length (strength) versus dipole tilt angle at a given
temperature; RPA critical temperature and critical density versus dipolar
length for a given tilt angle, and critical temperature versus critical
density for a given tilt angle. In particular, at finite temperature, we find
the system to be in a homogeneous non-condensed phase that undergoes a
collapse transition at large tilt angles, and a finite momentum instability,
signaling a striped phase, at sufficiently large values of dipolar coupling
strength; there are substantive differences with the T = 0 case. The linear
$q$ dependence of 2D dipolar interaction is manifested in a new density wave
instability in a broad regime similar to 2D dipolar fermions Sun et al.
(2010); Yamaguchi et al. (2010). As $T\rightarrow 0$, for sufficiently small
tilt angles, BEC appears at critical dipolar strength, and at critical
density, values of which depend on other system parameters.
While our results should apply generally to 2D dipolar bosons at arbitrary
temperature, the specific predictions based on parameters of the polar
molecule system, ${}^{87}Rb^{133}Cs$, can serve as a test of our results. We
discuss the effects of a harmonic trap and an additional contact interaction;
however, key physics is captured in our consideration of a homogeneous 2D Bose
system. Also, aforementioned box traps makes study of homogeneous systems
directly relevant and testable.
We consider a gas of dipolar bosons of mass $m$ and electric or magnetic
dipole moment $d$. The dipoles are confined by in x-y plane and the dipole
moments are aligned by an external electric field E or magnetic field B,
subtending an angle $\theta$ with respect to the z axis as shown in Fig. 1.
Figure 1: 2D Dipoles in x-y plane with tilt angle $\theta$ that defines the
direction of electric/magnetic field E relative to the z-direction. $\phi$ is
the angle in x-y plane, relative to x-direction.
To explore the stability of the system, we consider the static density-density
correlation function $\chi(q)\equiv\chi(q,w=0)$, which determines the
stability of a system against density fluctuations. When $\chi(q)$ becomes
positive at wave vector q, the system has density wave instability and may
undergo a transition to the striped phase Sun et al. (2010); Yamaguchi et al.
(2010); if it becomes positive as $q\to 0$, the system will develop negative
compressibility and collapse.
In standard RPA, the density-density correlation function can be
diagrammatically expanded in terms of the bare response function $\chi_{0}$
(see Supplemental Materials, Fig. S1 sup ). For the stability condition
against density fluctuations of finite momentum, the direct scattering of
particle-hole excitations dominates over exchange scattering because of the
linear momentum dependence of $V_{2D}(q)$ Bisset et al. (2011); Yamaguchi et
al. (2010). So, we neglect the exchange scattering of particle-hole
excitations. Also, since in 2D, there is no BEC at finite temperature, there
is no contribution from the condensate in the RPA response. Then,
$\displaystyle\chi(q,\omega)=\frac{\chi_{0}(q,\omega)}{1-V(q)\chi_{0}(q,\omega)}$
(1)
with
$\displaystyle\chi_{0}(q,\omega)=\int\frac{d{\bf
k}}{(2\pi)^{d}}\frac{f(k-q/2)-f(k+q/2)}{\hbar\omega-(\varepsilon_{k+q/2}-\varepsilon_{k-q/2})}$
(2)
where $\varepsilon_{k}=\hbar^{2}k^{2}/{2m}$ is the free particle kinetic
energy, and $f(q)$ is the Bose distribution function, with chemical potential
$\mu$. For non-interacting bosons, $\chi=\chi_{0}$ and always negative, so the
system is stable. For interaction with an attractive channel, the system can
become unstable depending on the interaction.
To proceed, we first need to evaluate the finite temperature bare response
function, Eq. (2). The asymptotic behavior is given by (for details, see
Supplemental Materials, Bare Bubble Calculation and Fig. S2 sup :
In the $q\lambda_{T}\ll 1$ region,
$\displaystyle\chi_{0}(q,T)=-\frac{m}{2\pi\hbar^{2}}\frac{1}{e^{-\beta\mu}-1}(1+O(q\lambda_{T})^{2})$
(3)
where $\lambda_{T}=\sqrt{\frac{2\pi\hbar^{2}\beta}{m}}$ is the thermal de
Broglie wavelength and $\beta=1/k_{B}T$.
In the $q\lambda_{T}\gg 1$ region, the behavior is temperature-independent,
$\displaystyle\chi_{0}(q)=-\frac{4nm}{\hbar^{2}q^{2}}$ (4)
We calculate RPA responses, taking bare response function to be that of a
noninteracting system, and using the non-interacting gas to calculate the
chemical potential $\mu(T,n)$. For an ideal two dimensional boson gas, the 2D
density is
$\displaystyle n$ $\displaystyle=$ $\displaystyle\int\frac{d{\bf
k}}{(2\pi)^{2}}\frac{1}{e^{\beta(\varepsilon_{k}-\mu)}-1}$ (5)
$\displaystyle=$ $\displaystyle\lambda^{-2}_{T}g_{1}(e^{\beta\mu})$
$g_{v}(z)=\sum_{j}z^{j}/j^{v}$ is the polylogarithm function. The chemical
potential $\mu$ is
$\displaystyle\mu=\frac{1}{\beta}\ln[1-\exp(-n\lambda_{T}^{2})]$ (6)
As temperature decreases, $\chi_{0}(q)$ increases. In the classical limit,
$\beta\mu\gg-1$, Eq. (3) become $\chi_{0}(0)=-n/T$, independent of Bose
statistics. In the quantum limit, $\beta\mu\ll-1$, Eq. (3) become
$\chi_{0}(0)=-\frac{m}{2\pi\hbar^{2}}e^{n\lambda^{2}_{T}}$. Since there is no
upper limit for $g_{1}(z)$ at zero chemical potential, there is no BEC in 2D
at finite temperature. In the limit of zero temperature, $\lim\limits_{T\to
0}\mu=0$, and $\lim\limits_{T\to 0}n(k)=n\,\delta(k)$. The limiting behavior
at zero temperature is the BEC state; Eq. (3) become $\chi_{0}(0,0)=-\infty$
and Eq. (4) becomes the response function of ideal bosons for all momenta at
T= 0.
It is convenient to look at the inverse of static density-density correlation
function $\chi(\textbf{q})$, given by:
$\displaystyle\frac{1}{\chi(\textbf{q})}=\frac{1}{\chi_{0}(\textbf{q})}-V(\textbf{q})$
(7)
where $V(\textbf{q})$ is the dipole-dipole interaction (DDI), given by
$V(\textbf{q})=V_{s}+V_{l}(\textbf{q})$, with
$\displaystyle\begin{aligned} &V_{s}=2\pi
d^{2}\frac{P_{2}(\cos\theta)}{r_{c}}\\\ &V_{l}(\textbf{q})=-2\pi
d^{2}q({\cos^{2}}\theta-{\sin^{2}}\theta{\cos^{2}}\phi)\end{aligned}$ (8)
where $r_{c}$ is a short range cut off. In quasi-2D geometry it depends on the
trapping size in z direction Fischer (2006). The first term $V_{s}$ is
momentum independent and acts like a short range interaction. The second term
depends linearly on the magnitude of momentum. In the y-direction
($\phi=\pi/2$), the interaction is the most attractive; therefore, the
instability happens at momentum in y direction. Three distinct cases may be
noted:
1\. $\frac{1}{\chi_{0}(q)}-V(q)<0$ everywhere; the system is in uniform normal
stable phase.
2\. $\frac{1}{\chi_{0}(q)}-V(q)>0$ at finite q; system undergoes density wave
instability and has striped phase.
3\. $\frac{1}{\chi_{0}(q)}-V(q)>0$ at q=0; the system has negative
compressibility and will collapse.
Fig. 2 shows schematically the behavior of $\frac{1}{\chi_{0}(q)}$ and $V(q)$
for the three cases. $\frac{1}{\chi_{0}(q)}$ is approximately quadratic in q
with intercept $\frac{1}{\chi_{0}(0)}\leq 0$ ; $V(q)$ is linear in q with
negative slope and positive/negative intercept depending on the tilt angle.
Note that instability for dipolar system at finite q is possible in 2D, but
not in 3D Bisset et al. (2011), because $V(q)$ is independent of the magnitude
of momentum in 3D Sun et al. (2010).
Figure 2: (color online) Schematic illustration for three cases of response
function discussed in text, curved line is $1/\chi_{0}(q)$, straight line is
V(q)
The behavior of the response function for the 2D dipolar Bose gas at non-zero
T can be generally categorized into two regimes with respect to tilt angle
$\theta$:
A) $\theta<\cos^{-1}{\frac{1}{\sqrt{3}}}$: the short range interaction
$V(q=0)$ is positive and $\frac{1}{\chi_{0}(0)}\leq 0$,
$\frac{1}{\chi_{0}(0)}-V(0)\leq 0$ is always satisfied, thus it is impossible
for the system to collapse. However, the instability condition could be
satisfied at finite $q_{y}$ with a sufficiently strong dipole interaction
strength, describable by dipolar length, $a_{dd}=md^{2}/\hbar^{2}$. Thus the
system is unstable against a density fluctuation with wave-vector $q_{y}$.
That indicates a transition from the normal phase to a striped phase.
B) $\theta>\cos^{-1}{\frac{1}{\sqrt{3}}}$: in this region, the short range
interaction becomes negative. At zero temperature, the bare response function
diverges at q=0 and $\frac{1}{\chi_{0}(0)}=0$, so the system cannot support
any attractive short range interaction. Thus it always collapses at T= 0, as
in the BdG approach. However, at finite temperature, the bare response
function has a non-zero negative value at q=0. As a result, it can support
attractive short range interaction that is sufficiently small. The system
first undergoes transition from normal phase to a striped phase; then
collapses as $a_{dd}$ increases further. At large tilt angles close $\pi/2$,
the long-range interaction becomes zero, and the total interaction is
dominated by the negative short range part of the dipolar interaction. Then
the system goes from the normal to the collapsed phase without going through
an intermediate striped phase.
For fixed density and temperature, the dipole interaction strength $a_{dd}$
can be changed via the strength of the external field, and dipole’s tilt angle
$\theta$ by varying the direction of external field. In Fig. 3, we show the
calculated phase diagram for critical dipole interaction strength $a_{dd}$
versus the tilt angle $\theta$ for a system of polar molecule such as
${}^{87}Rb^{133}Cs$ at T=0 and T=20 nK. We choose density to be
$n=10^{12}m^{-2}$ and take the cut-off to be $r_{c}=10^{4}a_{0}$; mass of
${}^{87}Rb^{133}Cs$ is $m=220u$ (unified atomic mass unit). At T= 0 K, the
system goes from stable BEC to density wave instability as dipole strength
$a_{dd}$ increases when $\theta<0.955$; the system collapses for any dipole
strength when $\theta>0.955$. On the other hand, at T= 20 nK, density wave
instability appears as $a_{dd}$ increases, even for tilt angle $\theta>0.955$.
The system eventually collapses as $\theta$ is increased further. That the
collapse occurs at a larger value of $\theta_{c}$ compared to the T=0 case may
be understood on noting the interplay between the DDI and the bare T-dependent
response, $\frac{1}{\chi_{0}(T)}$ in the RPA response function.
Figure 3: (color online) Dipole interaction strength $a_{dd}$ versus tilt
angle $\theta$ for a system of polar molecule ${}^{87}Rb^{133}Cs$ at T=0 and
T=20K. $a_{dd}$ is in units of the Bohr radius $a_{0}$. Red and blue lines are
lines of density wave and collapse instabilities respectively.
Study of critical temperature $T_{c}$ and critical density $n_{c}$, albeit
within RPA, provide another perspective on phases and density wave instability
in the system. We first calculate $T_{c}$ and critical density $n_{c}$, each
as a function of $a_{dd}$, for fixed tilt angle, $\theta$. Fig. 4, shows our
results for the physical system, ${}^{87}Rb^{133}Cs$ for $\theta=$ 0 and 0.4.
$T_{c}$ and $n_{c}$ are calculated using the condition
$\frac{1}{\chi(q,T)}=0$. For calculation of $T_{c}$, we choose the system
density to be $10^{12}m^{-2}$, and for $n_{c}$, the system temperature to be
10 nK. In the context of density wave instability, a key point here is that
the critical temperature and critical density behave opposite to each other
with increasing dipole strength, i.e. $T_{c}$ increases, and $n_{c}$ decreases
when dipole strength is increased (see Fig. 4). The terminating point ( $T=0$)
of the $T_{c}$ curves signifies the onset $a_{dd}$ for BEC; this is dependent
on the tilt angle.
Figure 4: (color online) Critical temperature and critical density for density
wave instability of ${}^{87}Rb^{133}Cs$ at tilt angles $\theta=$ 0 and 0.4.
Left: $T_{c}$ vs $a_{dd}$, for $n=10^{12}m^{-2}$. Right: $n_{c}$ vs $a_{dd}$
at T=10 nK.
Next, we construct a temperature-density (T-n) phase diagram for fixed dipole
strengths $a_{dd}$, and tilt angles $\theta$. For
$\theta<\cos^{-1}{\frac{1}{\sqrt{3}}}$, as temperature decreases and density
increases, the system goes through a transition from stable Bose fluid phase
to a density wave (DW) phase. At $T=0$, there is a critical density below
which there is no density wave instability, and the system is a BEC. For
$\theta>\cos^{-1}{\frac{1}{\sqrt{3}}}$, as temperature decreases and density
increases, the system goes through a transition from stable Bose fluid to a
density wave phase to a collapse phase. In Fig. 5, we show the calculated T-n
phase diagrams, for $\theta=0$ and $\theta=1$, using parameters relevant to
several physically realized dipolar boson systems, namely polar molecule
${}^{87}Rb^{133}Cs$ with electric dipole moment of 0.355 Debye and
$a_{dd}=4586a_{0}$ (accessible in experiment) Molony et al. (2014);
${}^{87}Rb^{133}Cs$ with electric dipole moment of 1.22 Debye and
$a_{dd}=93084a_{0}$ (the maximum value possible in experiments); ${}^{168}Er$
with magnetic dipole moment of $7\mu_{b}$ and $a_{dd}=196a_{0}$ Chomaz et al.
(2018). The sets of plots show that for larger dipolar interaction strength
$a_{dd}$, the instability occurs at a larger density and lower temperature;
compare for example, the plots for ${}^{87}Rb^{133}Cs$ with electric dipole
moment of 1.22 Debye ($a_{dd}=93084a_{0}$) with ${}^{168}Er$ with magnetic
dipole moment of $7\mu_{b}$ ($a_{dd}=196a_{0}$). Density wave instability
occurs at low temperature and high density for small tilt angle. For large
tilt angle, the system goes from stable Bose fluid phase to density wave
instability and then to collapse as temperature decreases and density
increases; no discernible region of BEC appears at T=0 (as seen for
$\theta=1$).
Figure 5: (color online) Calculated T-n phase diagram for dipolar Bose gas at
$\theta$ = 0 (left column) and $\theta=1$ (right column) ${}^{41}Rb^{87}Cs$
with dipole moment d=0.355D (top) and d=1.22D (middle); ${}^{166}Er$ (bottom)
with $d=7\mu_{b}$. Density wave instability (DWI), red curve, occurs at low
temperature and high density for $\theta$ = 0;. For $\theta$ =1, the system
goes from stable Bose fluid phase to density wave instability and then to
collapse (blue line) as temperature decreases and density increases.
We have considered the effect of an additional short-range interaction $g$,
originating from Van Der Waals interaction between atoms or molecules; this
results in total interaction $V(q)=g+V_{dipole}(q)$. Within RPA, the main
modifications are: a repulsive $g$ increases critical tilt angle, $\theta_{c}$
to a value larger than 0.955, while an attractive $g$ decreases it. This is
because $\theta_{c}$ is now determined by the net short range interaction,
that has contribution from the contact interaction, in addition to that
contained in the dipole interaction. And adding a repulsive $g$ increases the
critical $a_{dd}$ for instability, while an attractive $g$ decreases this.
(See Supplemental Materials, Fig. S3 sup ).
We briefly discuss possible effects of a trap given by a cylindrically
symmetric harmonic potential $U(z,r)=1/2(w_{z}z^{2}+w_{r}r^{2})$, where $z$
and $r$ are the axial and radial directions respectively. We explore
instability in a quasi-2D trapped system with strong harmonic confinement in
the $z$-direction, i.e. $w_{z}\gg w_{r}$, by calculating the RPA response
function $\chi$ within the local-density approximation (LDA). We expect the
instability to occur when the effective chemical potential,
$\mu_{\text{eff}}(r)=\mu-\omega r^{2}/2$, is same as that of the uniform
system at a certain temperature Mueller and Baym (2000); thus the local bare
response function is the same as that of the uniform system. We construct a
T-N phase diagram, Fig. 6, using the parameters for ${}^{166}Er$; N being the
total number of particles. The density wave instability temperature $T_{c}$ is
sightly above the ideal gas BEC temperature $T^{0}_{\text{BEC}}$. At $T_{c}$,
the 2D trapped gas has density wave instability at $r=0$, where
$\mu_{\text{eff}}$ is the largest, and begins to form stripe pattern starting
from center of the trap. (See details in Supplemental Materials, Effect of a
harmonic trap sup .) This behavior can be understood by noting that for
trapped particles, condensation results in a huge increase in the central
density of the cloud (a standard diagnostic of BEC) Mueller and Baym (2000).
Figure 6: (color online) Calculated $T$-$N$ phase diagram for a system of
trapped ${}^{166}Er$ at zero tilt angle. Here, $d=7\mu_{B}$ and trap parameter
$\hbar w_{r}/k_{B}=0.6K$. The dashed line is for an ideal gas BEC phase
transition, the solid line is the density wave instability line.
We have shown that a broad understanding of the nature of phases and
instabilities in a 2D dipolar Bose gas at finite temperature may be obtained
on concurrently exploring tunable system parameters, namely, density,
temperature, interaction strength and tilt angle. The presented phase diagrams
provide different perspectives on the nature of the instabilities. We have
used a finite-temperature version of RPA, and note that RPA is a well-
established many-body method that has proved to be useful in describing
collective modes and instabilities in quantum fluids. At T=0, our finite-
temperature RPA reproduces the BdG results, as expected Nozieres (2018). Our
approach and results may be of relevance generally to systems with long-range
and anisotropic interactions, and thus of broader appeal. The results may be
compared with previous work in 3D with attractive contact interaction Mueller
and Baym (2000), or dipolar interaction Bisset et al. (2011), wherein possible
long wavelength (q$\to$0) instabilities were studied. We note that density
wave instability usually triggers a long-range order of the stripe phase. At
finite temperature, enhanced fluctuation in 2D will destroy the long-range
order, but a quasi-long-range order may survive. A phase transition belonging
to the usual Berezinskii-Kosterlitz-Thouless universality class is expected;
this has been studied in Monte Carlo simulation Filinov et al. (2010); Bombín
et al. (2019). Thus, the $T_{c}$ curves discussed here, are in a strict sense,
RPA instability lines. Accordingly, our $T$-$n$ phase diagram may need to be
modified at low temperature; this is beyond the scope of RPA. Nevertheless,
the RPA method does provide a picture of the instabilities of 2D dipolar boson
system at finite temperatures that is expected to be qualitatively correct.
## I Acknowledgements
We thank J. Boronat and G. Baym for useful discussions. We acknowledge funds
from Institute for Complex Adaptive Matter. K. Q. acknowledges the hospitality
of Aspen Center for Physics, where part of the work was done.
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|
# Inverse Potentials for all $\ell$ channels of Neutron-Proton Scattering
using Reference Potential Approach
Anil Khachi1, Lalit Kumar1, Ayushi Awasthi1 and O. S. K. S. Sastri1,∗
Department of Physics and Astronomical Sciences
Central University of Himachal Pradesh
Dharamshala, 176215, Himachal Pradesh, Bharat(India<EMAIL_ADDRESS>
###### Abstract
Reference potential approach (RPA) is successful in obtaining inverse
potentials for weakly bound diatomic molecules using Morse function. In this
work, our goal is to construct inverse potentials for all available
$\ell$-channels of np-scattering using RPA. The Riccati-type phase equations
for various $\ell$-channels are solved using 5th order Runge-Kutta method to
obtain scattering phase shifts (SPS) in tandem with an optimization procedure
to minimize mean squared error (MSE). Interaction potentials for a total of 18
states have been constructed using only three parameter Morse interaction
model. The obtained MSE is $<1\%$ for ${}^{1}S_{0}$, ${}^{3}P_{1}$ and
${}^{3}D_{1}$ channels and $<2\%$ for ${}^{1}P_{1}$ channel and $<0.1\%$ for
rest of the 14 channels. The obtained total scattering cross-sections at
various lab energies are found to be matching well with experimental ones. Our
complete study of np-scattering for all $\ell$-channels using RPA using Morse
function as zeroth reference, is being undertaken for the first time.
* March 2023
## 1 Introduction
The neutron-proton (n-p) interaction has been first modeled by Yukawa [1].
This was followed by various single and multi-particle exchange models and QCD
based models as detailed in these reviews [2, 12]. Currently, the Nijmegen
[3], Argonne v18 [11], CD-Bonn [12] and Granada[25] potentials are the ones
which give rise to best quantitative results for explaining the experimental
scattering phase shifts. Unfortunately, all these potentials have different
mathematical representations originating from completely varied physical
considerations. Yet all of them lead to correct validation of experimental
data. The search for a simple theoretical potential that could model the
nucleon-nucleon interactions is still eluding the physicists. Interestingly,
many simple phenomenological forms such as Square well, Malfiet-Tjon [4],
Hulthen [13], have also been utilised for studying the deuteron. Recently,
molecular potentials such as Manning-Rosen [14], Morse [5] and Deng-Fan[6]
have been proposed.
Typically, phase wave analysis (PWA) technique based on modeling interaction
potential, relies on solving time independent Schrodinger equation (TISE) to
obtain scattering phase shifts (SPS) from which differential and total
scattering cross-sections (SCS) are obtained and matched with experimental
ones. While R-matrix[7], S-matrix[8], complex scaling method (CSM)[9], etc
rely on wavefunction to obtain SPS, phase function method (PFM) method[10]
directly utilises the model potential in phase equation to solve SPS directly.
Zhaba [15] utilised PFM method to study nucleon-nucleon scattering, and Laha
et al.[13]studied nucleon-nucleus and nucleus-nucleus interaction [13] and
obtained reasonably good results.
Alternatively, inverse potentials resulting directly from experimental
observations by J-matrix method [18] and Marchenko equation [19] have also
found some success in understanding the interaction involved between the
nucleons. Recently, an analytical procedure to solve this Marchenko equation
has been achieved by modeling the interaction using a Morse curve [27], which
belongs to a class of shape invariant potentials [21], called as reference
potential approach (RPA). Selg[19] points out that “The implementation of the
methods of the inverse scattering theory is not at all a trivial task. On the
contrary, this is a complex and computationally very demanding multi-step
procedure that has to be performed with utmost accuracy”. We have constructed
inverse potentials for S-waves of np [16], nD[17], S, P and D-states of
$n-\alpha$ [23] and $\alpha-\alpha$ [24] scattering by numerically solving the
phase equation choosing Morse function as zeroth reference. The goal of this
paper is to extend this reference potential approach (RPA) to constructing
inverse potentials for all 18 $\ell$-channels of np-interaction by considering
recent data for SPS from Arriola et.al., of Granada group [25] for lab
energies up to 350 MeV.
## 2 Methodology:
The Morse function is given by
$V_{Morse}(r)=V_{0}\left(e^{-2(r-r_{m})/a_{m})}-2e^{-(r-r_{m})/a_{m}}\right)$
(1)
The PWA approach to understanding two body scattering focuses on obtaining
phase shift for various orbital angular momentum, $\delta_{\ell}$, between
incoming and outgoing wave due to the interaction between projectile and
target nucleons. It is important to note that an attractive potential would
tend to pull the wave backward and hence results in a positive SPS,
$\delta_{\ell}>0$ and a repulsive potential would push it out which makes
$\delta_{\ell}<0$. Typically, one solves radial TISE to obtain wavefunction
from which SPS are deduced. One of the many advantages of Morse potential is
that it’s TISE can be analytically solved for, $\ell=0$, S-states for both
bound and scattering states [29].
### 2.1 Analytical solution for Scattering State of Morse potential:
Schr$\ddot{o}$dinger wave equation, for a spinless particle with energy ($E$)
and orbital angular momentum ($\ell$) undergoing scattering from another
particle with interaction potential V(r), is given by
$\frac{\hbar^{2}}{2\mu}\bigg{[}\frac{d^{2}}{dr^{2}}+\big{(}k^{2}-\ell(\ell+1)/r^{2}\big{)}\bigg{]}u_{\ell}(k,r)=V(r)u_{\ell}(k,r)$
(2)
Where $k=\sqrt{E/(\hbar^{2}/2\mu)}$. The analytical solution of TISE for Morse
potential [27] with $\ell=0$, is given by
$E_{v}=-\frac{\hbar^{2}}{2\mu
a_{m}^{2}}(\lambda-v-1/2)^{2}~{}~{}~{}~{}v=0,1,2,\ldots$ (3)
where
$\lambda=\sqrt{\frac{2\mu
V_{0}{a_{m}}^{2}}{\hbar^{2}}}~{}~{}{and}~{}~{}\frac{\hbar^{2}}{2\mu}=41.47~{}MeVfm^{2}$
(4)
is called well-depth parameter and is dependent only on $V_{0}$ and $a_{m}$.
#### 2.1.1 SPS for ${}^{1}S_{0}$ singlet state:
In case of singlet ${}^{1}S_{0}$ unbound state ($E>0$), with $\ell=0$, the
analytical formula for SPS due to Morse potential is obtained [28, 29] as
$\delta_{0}^{ana}=-kr_{m}-\epsilon(\gamma+log2\lambda)+\sum_{n=1}^{\infty}\left(\frac{\epsilon}{n}-tan^{-1}\frac{2\epsilon}{n}+tan^{-1}\frac{\epsilon}{v-\lambda-1/2}\right)$
(5)
where $\gamma=0.57721$ is Euler constant and $\epsilon$ is given by:
$\epsilon=\sqrt{\frac{2\mu Ea_{m}^{2}}{\hbar^{2}}}=ka_{m}$ (6)
### 2.2 Phase Function Method:
PFM is one of the important tools in scattering studies for both local and
non-local interactions [32]. The second order differential equation Eq.2 can
been transformed to a first order non-homogeneous differential equation of
Riccati type [32, 33], containing phase shift information, given by:
$\delta_{\ell}^{\prime}(k,r)=-\frac{U(r)}{k}\bigg{[}\cos(\delta_{\ell}(k,r))\hat{j}_{\ell}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\bigg{]}^{2}$
(7)
where $U(r)=\frac{2\mu V(r)}{\hbar^{2}}$. Prime denotes differentiation of
phase shift with respect to distance and Riccati Hankel function of first kind
is related to $\hat{j_{\ell}}(kr)$ and $\hat{\eta_{\ell}}(kr)$ by
$\hat{h}_{\ell}(r)=-\hat{\eta}_{\ell}(r)+\textit{i}~{}\hat{j}_{\ell}(r)$. For
$\ell=0$, the Ricatti-Bessel and Riccati-Neumann functions $\hat{j_{0}}$ and
$\hat{\eta_{0}}$ get simplified as $sin(kr)$ and $-cos(kr)$. So phase
equation, for $\ell=0$, is
$\delta_{0}^{\prime}(k,r)=-\frac{U(r)}{k}sin^{2}[kr+\delta_{0}(r)]$ (8)
This is numerically solved using Runge-Kutta 5th order (RK-5) method with
initial condition $\delta_{\ell}(k,0)=0$. The Ricatti-Bessel and Riccati-
Neumann functions can be easily obtained by using following recurrence
formulas:
$\hat{j_{\ell}}(kr)=\frac{2\ell+1}{kr}\hat{j_{\ell}}(kr)-{\hat{j}_{\ell-1}}(kr)$
(9)
$\hat{\eta_{\ell}}(kr)=\frac{2\ell+1}{kr}\hat{\eta_{\ell}}(kr)-{\hat{\eta}_{\ell-1}}(kr)$
(10)
### 2.3 Scattering Cross Section:
Once, SPS $\delta_{\ell}(E)$ are obtained for each orbital angular momentum
$\ell$, one can calculate partial SCS $\sigma_{\ell}(E)$ [34], as :
$\sigma_{\ell}(E;S,J)=\frac{4\pi}{k^{2}}\sum_{S=0}^{1}\left(\sum_{J=|\ell-S|}^{|\ell+S|}(2\ell+1)\sin^{2}(\delta_{\ell}(E;S,J))\right)$
(11)
and total SCS $\sigma_{T}$, is given as
$\sigma_{T}(E;S,J)=\frac{1}{\sum_{J=|\ell-S|}^{|\ell+S|}(2J+1)}\sum_{\ell=0}^{5}\sum_{S=0}^{1}(2J+1)\sigma_{\ell}(E;S,J)$
(12)
## 3 Results and Discussion:
The experimental SPS data have been taken from Perez et.al., of Granada group
[25], 2016. To this, we have added an extra data point at lab energy 0.1 MeV
for ${}^{3}S_{1}$ and ${}^{1}S_{0}$ states, from Arndt (private communication)
to improve determination of low energy scattering parameters. The optimised
model parameters for all states of different $\ell$-channels are given in
Table 1.
Table 1: Model Parameters $V_{0}$ in MeV, $r_{m}~{}\&~{}a_{m}$ in fm for
various states of different $\ell$ \- channels with obtained mean squared
error (MSE) values.
States | $V_{0}$ | $r_{m}$ | $a_{m}$ | $MSE$ | States | $V_{0}$ | $r_{m}$ | $a_{m}$ | $MSE$ | States | $V_{0}$ | $r_{m}$ | $a_{m}$ | $MSE$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
${}^{3}S_{1}$ | 114.153 | 0.841 | 0.350 | 0.155 | ${}^{1}D_{2}$ | 131.302 | 0.010 | 0.526 | 0.026 | ${}^{3}F_{3}$ | 0.010 | 8.807 | 2.441 | 0.025
${}^{1}S_{0}$ | 70.438 | 0.901 | 0.372 | 0.649 | ${}^{3}D_{1}$ | 0.010 | 7.401 | 1.403 | 0.274 | ${}^{3}F_{4}$ | 40.343 | 0.528 | 0.489 | 0.001
${}^{1}P_{1}$ | 0.010 | 5.442 | 1.016 | 1.568 | ${}^{3}D_{2}$ | 106.379 | 0.209 | 0.747 | 0.066 | ${}^{1}G_{4}$ | 20.390 | 0.010 | 0.673 | 0.002
${}^{3}P_{0}$ | 11.579 | 1.750 | 0.601 | 0.049 | ${}^{3}D_{3}$ | 23.620 | 1.185 | 0.305 | 0.002 | ${}^{3}G_{3}$ | 0.010 | 6.846 | 1.486 | 0.009
${}^{3}P_{1}$ | 0.010 | 4.514 | 0.778 | 0.832 | ${}^{1}F_{3}$ | 0.010 | 7.645 | 1.877 | 0.056 | ${}^{3}G_{4}$ | 59.219 | 0.010 | 0.786 | 0.015
${}^{3}P_{2}$ | 77.558 | 0.444 | 0.404 | 0.001 | ${}^{3}F_{2}$ | 2.746 | 2.140 | 0.532 | 0.003 | ${}^{3}H_{4}$ | 23.245 | 0.010 | 0.693 | 0.000
Using obtained model parameters, $V_{0}=70.438,r_{m}=0.901$ and $a_{m}=0.372$,
for ${}^{1}S_{0}$ in analytical expression of Eq. 5, we obtained corresponding
SPS at various energies. These are observed to be closely matching with SPS
obtained using PFM as shown in Fig. 1, thus validating the later procedure.
The ${}^{3}S_{1}$ model parameters are obtained by utilising energy condition,
in Eq 3 with $v=0$, as a constraint.
The obtained SPS for all channels are in good match with experimental data
with MSE values being very close to zero. The exceptions are states
${}^{1}P_{1}$, ${}^{3}P_{1}$ and ${}^{3}D_{1}$ with purely negative SPS, where
MSE values are 1.568, 0.832 and 0.274 respectively. It is interesting to note
that all of them have $V_{0}$ to be 0.010 MeV and $r_{m}$ to be large, which
makes the shape of their potentials to be of exponential form. Similarly,
states ${}^{1}F_{3}$, ${}^{3}F_{3}$ and ${}^{3}G_{3}$, with purely negative
SPS, also have $V_{0}$ to be 0.010 MeV and their MSE slightly higher than
other F and G states with positive SPS.
Figure 1: Plot of Scattering phase shifts obtained using analytical and PFM
for ${}^{1}S_{0}$-state. The inset shows the match at lower energy upto 50
MeV.
By using effective-range approximation formula [31] for low energy scattering
parameters, scattering length ${}^{\prime}a^{\prime}$ and effective range
${}^{\prime}r_{e}^{\prime}$ are determined(experimental) to be
$-23.390~{}(-23.749(8))~{}fm$ and $2.42~{}(2.81(5))~{}fm$ for ${}^{1}S_{0}$
state and $5.356~{}(5.424(3))~{}fm$ and $1.75~{}(1.760(5))~{}fm$ for
${}^{3}S_{1}$ state respectively.
Figure 2: Inverse Potentials (left) and Scattering phase shifts (right) plots
for S, P and D-states of np scattering using RPA.
The inverse potentials obtained for S, P and D states and their corresponding
SPS plots are given in Fig. 2. Similarly, computed inverse potentials and
their corresponding SPS plots for F, G and H states of np-interaction are
shown in Fig. 3.
Figure 3: Inverse Potentials (left) and Scattering phase shifts (right) plots
for F, G and H-states of np scattering using RPA
One can observe close match between obtained and expected SPS values for all
the states. The following observations can be made from these potentials and
SPS plots:
* •
Both S-state potentials are having similar form with only their depths being
different.
* •
The potentials have attractive nature whenever SPS are positive and repulsive
nature when SPS are negative.
* •
Whenever SPS start from being positive and then cross-over to negative values
as seen in ${}^{3}P_{0}$ or dips down as in ${}^{3}F_{1}$ at higher energies,
the repulsive nature of the potential curve sets in at higher values of inter-
nucleon distance.
* •
In case of ${}^{3}P_{1}$ and ${}^{1}P_{1}$, just as their SPS cross-over after
200 MeV, so also their respective inverse potentials cross-over after at about
1.5 fm.
* •
It is interesting to note that for certain D-states and all G and H states for
which SPS are positive, potential assumes shape of a Gaussian function.
* •
All states with negative SPS are having an exponentially decaying positive
potential.
* •
Note that, in case of ${}^{1}F_{3}$ and ${}^{3}F_{3}$, the former has more
negative SPS than the later and its corresponding inverse potential also has
increasing repulsion with decreasing internucleon distances.
The beauty of the method is that Morse function acts as an exponential
function for states with negative SPS and a gaussian function for positive
ones. When SPS tend to have both negative and postive SPS or those with
positive trend to begin with and decrease at higher energies, typical Morse
curve is obtained.
Utilising the obtained SPS for various states of all $\ell$-channels, the
partial and total scattering cross-sections (SCS) have been calculated using
Eqs. 11 and 12 respectively. In case of $\ell=0$, the individual partial
cross-sections due to both ${}^{1}S_{0}$ and ${}^{3}S_{1}$ have been
calculated without summing over their contributions. All these calculated SCS
at various energies have been presented in Table LABEL:Table_2. The
$\%$-contributions due to individual S-states and rest of the $\ell$-channels
from P to H, to the calculated total SCS, are given in brackets. One can
observe that ${}^{1}S_{0}$ has large contribution at low energies below 1 MeV
and then gradually falls down with increasing energy and becomes very less
past 100 MeV. On the other hand, contribution due to ${}^{3}S_{1}$ increases
beyond 1 MeV and peaks at 10 MeV and then falls down beyond 250 MeV. The
contributions from P and D channels become significant for higher energies
from 100 MeV to 350 MeV. Those due to F and G are far less in comparision in
the same range but certainly important for accurately describing the observed
experimental total SCS. The SPS are available for only 1 H-state and it’s
contribution is almost negligible to the determination of total SCS. Overall
the obtained total SCS are found to be closely matching the experimental ones.
The partial cross-sections due to P and D channels are shown in Fig 4(a) and
those due to F and G channels in Fig 4(b) with respect to lab energies. The
total SCS has been plotted with respect to lab energies on a log scale in Fig.
4(c) with individual contributions due to individual S-states as an inset.
Table 2: Individual contributions to calculated total elastic scattering
cross-section (SCS) from various channels. In case of $\ell=0$, the
contributions due to both ${}^{1}S_{0}$ and ${}^{3}S_{1}$ are given
seperately. The $\%$-contributions to obtained total SCS has been given in
brackets.
E | $\sigma_{exp}$(b) | $\sigma_{{}^{1}S_{0}}$ | $\sigma_{{}^{3}S_{1}}$ | $\sigma_{P}$ | $\sigma_{D}$ | $\sigma_{F}$ | $\sigma_{G}$ | $\sigma_{H}$ | $\sigma_{sim}$(b)
---|---|---|---|---|---|---|---|---|---
(MeV) | [26] | | | | | | | |
0.1 | - | 9.708(78%) | 2.651 (22%) | 2.31$\times 10^{-7}$ | 2.98$\times 10^{-12}$ | 0 | 0 | 0 | 12.359
0.5 | 6.135 | 3.653 (60%) | 2.425 (40%) | 5.690$\times 10^{-6}$ | 1.78$\times 10^{-9}$ | 0 | 0 | 0 | 6.078
1 | 4.253 | 2.041 (48%) | 2.189 (52%) | 2.240$\times 10^{-5}$ | 2.69$\times 10^{-8}$ | 0 | 0 | 0 | 4.230
10 | 0.9455 | 0.2007 (21%) | 0.7413 (79%) | 0.0016 | 0.0001 | 6.40$\times 10^{-6}$ | 4.81$\times 10^{-8}$ | 0 | 0.9437
50 | 0.1684 | 0.0222(13%) | 0.1235 (75%) | 0.0106 (6%) | 0.0079 (5%) | 0.0001 | 3.44$\times 10^{-5}$ | 1.32$\times 10^{-8}$ | 0.1643
100 | 0.07553 | 0.00498 (7%) | 0.03601(50%) | 0.015(21%) | 0.01593 (22%) | 0.00044 (1%) | 0.00034 | 3.84$\times 10^{-7}$ | 0.07270
150 | 0.05224 | 0.00129(3%) | 0.01314 (26%) | 0.01622 (33%) | 0.01752 (35%) | 0.00064 (1%) | 0.00083 (2%) | 1.61$\times 10^{-6}$ | 0.04964
200 | 0.04304 | 0.00025(1%) | 0.00493 (12%) | 0.0161 (40%) | 0.01679 (42%) | 0.00073 (2%) | 0.00128 (3%) | 3.56$\times 10^{-6}$ | 0.04008
250 | 0.03835 | 0.00001 | 0.00166 (5%) | 0.01546 (44%) | 0.01542 (44%) | 0.00075 (2%) | 0.00162 (5%) | 5.82$\times 10^{-6}$ | 0.03493
300 | 0.03561 | 0.00003 | 0.00040(1%) | 0.01464(46%) | 0.01394(44%) | 0.00074 (2%) | 0.00184 (6%) | 8.09$\times 10^{-6}$ | 0.03160
350 | 0.03411 | 0.00015 | 0.00002 | 0.01377 (47%) | 0.01255(43%) | 0.00072 (2%) | 0.00197(7%) | 0.00001 | 0.02919
Figure 4: Partial scattering cross-section (SCS) due to (a) P and D channels
(b) F and G channels, as a function of energy. (c) The total SCS has been
plotted on a log scale of energies. Its inset shows contributions due to both
S-states.
## 4 Conclusions:
The inverse potentials for np-interaction for all partial waves with $\ell$ =
0 to 5 have been constructed using reference potential approach, by choosing
Morse function as zeroth reference, for the first time. The model parameters
have been obtained by choosing to minimize mean squared error between SPS
obtained from PFM technique and Granada data [25] in an iterative manner. This
was achieved by making suitable adjustments to different model parameters
using optimisation routines. The obtained SPS for all the channels match
expected ones very closely. Overall, reference potential approach using Morse
function seems to be able to give reasonably accurate inverse potentials of
appropriate shapes that could logically explain observed trends in SPS for
variour $\ell$-channels, as expected from PFM. The total scattering cross-
sections have been determined for data upto 350 MeV and are found to be in
good agreement with experimental values. It remains to be seen as to how to
construct inverse potentials for pp-scattering using RPA.
## Acknowledgments
A. Awasthi acknowledges financial support provided by Department of Science
and Technology (DST), Government of India vide Grant No. DST/INSPIRE
Fellowship/2020/IF200538. The corresponding author dedicates this work to his
inspirational guide Padma Shri Prof P. C. Sood with gratitude.
The authors declare that they have no conflict of interest.
## References
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|
School of Physical Sciences, University of Chinese Academy of Sciences, 19A
Yuquan Road, Beijing 100049, China Rudolf Peierls Centre for Theoretical
Physics, University of Oxford, Oxford OX1 3PU, United Kingdom Wenzhou
Institute, University of Chinese Academy of Sciences, Wenzhou, Zhejiang
325000, China Wenzhou Institute, University of Chinese Academy of Sciences,
Wenzhou, Zhejiang 325000, China School of Physical Sciences, University of
Chinese Academy of Sciences, 19A Yuquan Road, Beijing 100049, China
Universitat de Barcelona Institute of Complex Systems, 08028 Barcelona, Spain
Institut de Nanociència i Nanotecnologia, Universitat de Barcelona, 08028
Barcelona, Spain Wenzhou Institute, University of Chinese Academy of Sciences,
Wenzhou, Zhejiang 325000, China School of Physical Sciences, University of
Chinese Academy of Sciences, 19A Yuquan Road, Beijing 100049, China
# Molecular Dynamics Simulations of Microscopic Structural Transition and
Macroscopic Mechanical Properties of Magnetic Gels
Xuefeng Wei Wenzhou Institute, University of Chinese Academy of Sciences,
Wenzhou, Zhejiang 325000, China Gaspard Junot Departament de Física de la
Matèria Condensada, Universitat de Barcelona, 08028 Barcelona, Spain Ramin
Golestanian Max Planck Institute for Dynamics and Self-Organization (MPIDS),
D-37077 Göttingen, Germany Xin Zhou School of Physical Sciences, University
of Chinese Academy of Sciences, 19A Yuquan Road, Beijing 100049, China
Yanting Wang CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
Pietro Tierno Departament de Física de la Matèria Condensada, Universitat de
Barcelona, 08028 Barcelona, Spain Fanlong Meng<EMAIL_ADDRESS>CAS
Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China
###### Abstract
Magnetic gels with embedded micro/nano-sized magnetic particles in crosslinked
polymer networks can be actuated by external magnetic fields, with changes in
their internal microscopic structures and macroscopic mechanical properties.
We investigate the responses of such magnetic gels to an external magnetic
field, by means of coarse-grained molecular dynamics simulations. We find that
the dynamics of magnetic particles are determined by the interplay of between
magnetic dipole-dipole interactions, polymer elasticity and thermal
fluctuations. The corresponding microscopic structures formed by the magnetic
particles such as elongated chains can be controlled by the external magnetic
field. Furthermore, the magnetic gels can exhibit reinforced macroscopic
mechanical properties, where the elastic modulus increases algebraically with
the magnetic moments of the particles in the form of
$\propto(m-m_{\mathrm{c}})^{2}$ when magnetic chains are formed. This
simulation work can not only serve as a tool for studying the microscopic and
the macroscopic responses of the magnetic gels, but also facilitate future
fabrications and practical controls of magnetic composites with desired
physical properties.
## 1 Introduction
Magnetic gels, with micro/nano-sized magnetic particles embedded in
crosslinked polymeric gels, belong to the class of smart magnetic materials,
whose physical properties can be controlled by external magnetic fields 1, 2,
3. By tuning such fields, e.g., the direction, the strength or the frequency,
one can change the materials with different mechanical responses, such as
material stiffness 4, 5, 6, 7 and deformations 8, 9, 10. Based on such field-
controlling features, magnetic gels have been fabricated in laboratories and
used as magnetic robots 11, 12, 13, sensors 14, 15, 16, actuators 17, 18,
etc., and also for bio-medical applications 19, 20, 21 such as drug
delivery/release due to the bio-friendly characteristics of magnetic fields.
Under an applied magnetic field, the materials can respond in different ways
depending on how the magnetic particles are embedded in the polymer matrix.
For magnetic particles able to move freely in the polymer gel, they can form
microscopic chains when the magnetic dipole-dipole interactions between
particles dominate the thermal fluctuation as in ordinary solutions 22, 23,
where the polymer gel serves mainly as a viscoelastic medium without much
changes in its mechanical properties. For superparamagnetic particles fixed in
the polymer network 24, the particles can still move but now they induce
microscopic distortions in the polymer network, and they can even form
magnetic chains if the magnetic dipole-dipole interactions are strong enough.
In the latter case, the responses of the embedded particles to the external
magnetic field with simultaneous deformations of the polymer network underpins
the physical mechanism for controlling the mechanical properties of the
magnetic gels.
Figure 1: Magnetic gels with (a) uniformly distributed magnetic particles
without the magnetic field and (b) magnetic chains under an external magnetic
field. Spheres denote the magnetic particles and grey lines denote partial
connections in the polymer network.
The connection between the microscopic motion of the magnetic particles and
the macroscopic responses of the material, is the key for understanding how
external magnetic fields can be utilised to control the material properties.
Experimental methods such as small-angle X-ray scattering (SAXS) 25, small
angle neutron scattering (SANS) 26, electron magnetic resonance (EMR) 27,
etc., are utilised for detecting the internal structure of the magnetic gels
with macroscopic mechanical properties which are measurable by tensile testing
28, 29, dynamic mechanical analysis 30, 31, etc. Theoretical attempts include
analytical modelling such as the lattice models 32, 33 and continuum models
34, 35, 24, and numerical methods such as finite element calculations 36,
Monte Carlo simulations 37, 38 and molecular dynamics simulations 39, 40, 41.
In this work, we use coarse-grained molecular dynamics simulations to
investigate the microscopic motion and structures of the magnetic particles,
together with the macroscopic mechanical properties especially about the
anisotropic elasticity of the materials.
## 2 Simulation methodology
_Polymer network._ We follow Ref.42 to construct the 3D packing-derived (PD)
polymer network. We first randomly place $N=W^{3}$ radially bidisperse spheres
with harmonic repulsive interactions within a cubic unit cell of side length
$W$ ($W=20$ is taken in later discussions), where half of the spheres are
assigned with radius $r=r_{0}$ and the other half with $r=\nu r_{0}$
($\nu=1.4$ is chosen to avoid long-range crystal order 42, 43, 44). By
continuously increasing the radius $r_{0}$ from $r_{0}=0$, the spheres will
reach a jammed state at $r_{0}^{\mathrm{J}}$. Then, from this disordered
packing state, we generate a contact network by connecting the centers of the
overlapping spheres (excluding rattlers) with springs at their rest lengths
and angles. For sufficiently large systems, this procedure generates contact
networks with the connectivity $z=2d$, where $d$ is the system dimensionality.
After generating the underlying network structure, we remove randomly chosen
bonds and any consequent dangling ends repeatedly until the network reaches
the desired average connectivity $z$. The bonds connecting neighbouring nodes
are modelled as Hookean springs and the bending effects is considered by
introducing the bending energy. The stretching and the bending energy of the
network can be written as:
$\displaystyle U_{\mathrm{stretching}}(r)$ $\displaystyle=$
$\displaystyle\frac{\mu}{2}\sum_{i,j}\frac{(r_{ij}-r_{ij,0})^{2}}{r_{ij,0}},$
(1) $\displaystyle U_{\mathrm{bending}}(r)$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2}\sum_{ijk}\frac{(\theta_{ijk}-\theta_{ijk,0})^{2}}{l_{ijk,0}},$
(2)
respectively, where $\mu$ and $\kappa$ denote the Hookean constant and the
bending rigidity, respectively, $r_{ij,0}$ and $r_{ij}$ denote the length of
spring $(ij)$ connecting the $i$th and the $j$th node before and after
stretching, respectively, $\theta_{ijk,0}$ and $\theta_{ijk}$ denote the
angles between the spring $(ij)$ and $(jk)$ before and after bending,
respectively, and $l_{ijk,0}=(r_{ij,0}+r_{jk,0})/2$ denotes the average length
of the two connecting springs.
_WCA potential._ We choose polymer nodes randomly with the number density
$n_{0}$ to be replaced with magnetic particles, i.e., the magnetic particles
are uniformly distributed in the prepared state. The diameters of the magnetic
particles are set as $\sigma_{\mathrm{m}}=1.0\sigma$ and other non-magnetic
nodes are set as $\sigma_{\mathrm{n}}=0.2\sigma$. In our simulations, the
truncated Weeks-Chandler-Andersen (WCA) potential 45 is used to simulate the
non-bonded and non-magnetic interactions between particles, as:
$U(r)=\begin{cases}4\epsilon\left[\left(\frac{\sigma_{ij}}{r}\right)^{12}-\left(\frac{\sigma_{ij}}{r}\right)^{6}\right]+C&\text{if
}r\geq r_{\mathrm{cutoff}},\\\ 0&\text{if }r<r_{\mathrm{cutoff}},\end{cases}$
where $r_{\mathrm{cutoff}}=2^{1/6}\sigma_{ij}$ is the cutoff distance for
given pairs, $\sigma_{ij}=(\sigma_{i}+\sigma_{j})/2$ is the average diameter
of particle $i$ and $j$, and $C$ is a constant ensuring the continuity of the
potential energy at the cutoff distance. In this work we take $\epsilon$ and
$\sigma$ as the unit energy and length, respectively.
_Magnetic dipole-dipole interactions._ The superparamagnetic particles can be
magnetized under an external magnetic field $\bm{B}_{\mathrm{ext}}$, and they
acquire magnetic moments $\bm{m}=\chi
v_{\mathrm{p}}\bm{B}_{\mathrm{ext}}/\mu_{0}$ pointing along the direction of
the magnetic field, where $\chi$ is the magnetic susceptibility, $v_{p}$ is
the particle volume and $\mu_{0}$ is vacuum permeability. These magnetic
particles possess moments aligned along the field direction, and the
interactions between the magnetic particles are:
$\displaystyle
U_{\mathrm{dd}}(\bm{r}_{ij})=\frac{\mu_{0}}{4\pi}\left[\frac{1}{r_{ij}^{3}}(\bm{m}_{i}\cdot\bm{m}_{j})-\frac{3}{r_{ij}^{5}}(\bm{m}_{i}\cdot\bm{r}_{ij})(\bm{m}_{j}\cdot\bm{r}_{ij})\right].$
(3)
The magnetic dipole-dipole interactions between magnetic particles can be
either attractive or repulsive, depending on their relative locations with
respect to the separation $\bm{r}_{ij}$.
_Molecular dynamics simulations._ We perform the molecular dynamics
simulations on a coarse-grained level based on the above model setup, using
LAMMPS software 46. The simulations are done in the canonical _NVT_ ensemble
using a Langevin thermostat 47, 48. The equation of motion for each particle
is:
$\displaystyle m\ddot{\bm{r}}=-\gamma\dot{\bm{r}}+\bm{F}+\bm{\xi},$ (4)
where $m$ denotes the mass of the particle, $\gamma$ denotes the friction
constant, $\bm{F}$ denotes the forces exerted by other particles and the
polymer network, and $\bm{\xi}$ denotes the thermal noise due to the random
collisions with implicit solvent molecules, which obeys fluctuation-
dissipation theorem. Periodic boundary conditions are adopted in the
simulations and the detailed simulation procedure including simulation
parameters can be found in the supplement 49.
## 3 Results and discussions
### 3.1 Internal microscopic structures
The elastic properties, e.g., the shear modulus and the bulk modulus, of the
magnetic gels before applying any external magnetic field can be controlled by
changing the connectivity in the polymer networks, as shown in Fig. 2(a). By
increasing the connectivity of the polymer network, both the shear and the
bulk modulus will increase, in an approximate form of
$\propto(z-z_{\mathrm{c}})^{3}$ with $z_{\mathrm{c}}\simeq 3.2$ as the
critical connectivity for the polymer network to exhibit finite elasticity 50,
42.
Figure 2: (a) Dependence of shear modulus $G$ (squares), bulk modulus $K$
(triangles) and $K+4G/3$ (circles) of the polymer gels on the network
connectivity $z$. (b) Chain ratio as a function of the magnetic moments of the
particles, in polymer networks with different connectivity. Insets denote the
microscopic structures of the magnetic particles and the displacement field in
the cross-sectional plane. (c) Chain ratio as a function of the magnetic
moments and the network elasticity. Red line denote the physical conditions
for the chain formation [Equation (5)], and the white diamonds correspond to
simulation results of the critical chain ratio, $\eta=0.15$.
As shown in Fig. 2(b), the magnetic particles can form chain-like structures
by increasing the strength of the magnetic field (magnetic moments of
superparamagnetic particles), when the magnetic dipole-dipole interactions
between the magnetic particles dominate the network elasticity and the thermal
fluctuations; this phenomenon resembles the clustering instability found in
magnetic microswimmers, where the microswimmers can form clusters when
magnetic dipole-dipole interactions dominate thermal fluctuations 51, 52. Due
to the displacements of the magnetic particles in events of chain formations,
the polymer network will be deformed with the local deformation field, as
shown in the insets of Fig. 2(b). The chain ratio, $\eta$, defined as the
ratio of the number of the magnetic particles forming the chains to the total
number of the magnetic particles in the material, is introduced to
characterize the amount of the micro-structures in the magnetic gel and also
the relative strength of the magnetic dipole-dipole interactions over the
network elasticity. We use the following criterion to identify if a particle
belong to a chain: the distance to its nearest neighboring magnetic particle
is less than $2^{1/6}\sigma$, and the angle between the direction of the
magnetic moment and the position vector connected to its nearest neighboring
particle is less than $15^{\circ}$. With the decrease in the network
connectivity, i.e., the elastic moduli of the material, the chain ratio in
magnetic gels of the particles with the same magnetic moments will increase.
By systematically changing the elastic moduli of the network and the magnetic
moments of the particles, one can obtain the phase diagram showing the
physical conditions for the chain formation, as shown in Fig. 2(c). In
simulations, $\eta=0.15$ is taken as the value of the chain ratio, for
denoting the conditions of the chain formation, and one can observe from Fig.
2(c) that, for inclusions carrying large magnetic moments in soft polymer
gels, they can easily form chain-like structures, coinciding with the
theoretical conditions for the chain formation as derived in a previous work
24,
$\displaystyle\frac{\mu_{0}n_{0}m^{2}}{k_{B}T}>\frac{v_{\mathrm{p}}}{k_{B}T}(K+4G/3)+1.$
(5)
One can also check how the number density of magnetic particles in the
prepared state, $n_{0}$, can influence the formation of the micro-structures
in the gel. As shown in Fig. 3(a), the chain ratio is larger for systems
characterized by a higher number density $n_{0}$, since the magnetic dipole-
dipole interactions become stronger due to decreased distances between the
magnetic particles. The physical conditions for the chain formation in the
plane of $(n_{0},m)$ is provided in Fig. 3(b), which also agrees with our
theoretical prediction.
Figure 3: (a) Chain ratio $\eta$ as a function of the magnetic moments $m$ of
the particles with different number density. (b) Dependence of the chain ratio
on the magnetic moments and the number density of the magnetic particles. Red
line denote the physical conditions for the chain formation [Equation (5)],
and the white diamonds corresponds to simulation results of the critical chain
ratio, $\eta=0.15$.
### 3.2 Macroscopic mechanical properties
The macroscopic mechanical properties of magnetic gels can become reinforced
due to the formation of microscopic chains 4, 5, 6, 7. Here we investigate
such effects by studying how magnetic gels consisting of particles carrying
different magnetic moments can respond under shear deformations.
As shown in Fig. 4(a), the stress-strain relations of magnetic gels can change
if the magnetic moments of the particles are different. The magnetic gels with
chain formation exhibit obvious mechanical reinforcements (larger stresses
compared with those without chain formation at same strains) and weak
anisotropy, whose mechanical responses depend on the applied plane of the
shear deformation. By defining the shear modulus of the material as
$G_{\alpha\beta}=\frac{d\sigma_{\alpha\beta}}{d\varepsilon_{\alpha\beta}}|_{\varepsilon_{\alpha\beta}=0}$,
one can obtain the dependence of this modulus on the magnetic moments of the
particles, as shown in Fig. 4(b). There is a weak effect of the anisotropy in
the shear moduli appears at $m_{c}\simeq 0.27$, corresponding to the chain
ratio as $\eta=0.15$ (same criterion as used for chain formation). Compared
with the material containing inherent chains in its relaxed state (springs not
deformed) 32, the anisotropy in the mechanical response here is weaker, since
all the elastic springs are already distorted when the chains are formed in
the current setup. More importantly, for $m\leq m_{c}$, the shear modulus
remains almost unchanged by increasing the magnetic moments, but for
$m>m_{c}$, the shear modulus increases quickly with the magnetic moments in an
approximate form of
$G_{\alpha\beta}(m)=G_{\alpha\beta,\infty}\frac{(m-m_{c})^{2}}{a_{m}+(m-m_{c})^{2}},$
(6)
where $G_{\alpha\beta,\infty}$, $a_{m}$ and $m_{c}$ are fitting parameters for
specific materials 6, 53, 32.
Figure 4: (a) Stress-strain relations of magnetic gels without (dash lines)
and with (solid lines) chain formation, which undergoes the shear deformation
in three orthogonal planes. (b) Dependence of the shear modulus
$G_{\alpha\beta}$ as a function of the magnetic moment of the particles, and
lines fitted with Equation (6). (c) Comparison between the magnetic
interactions and the polymer elasticity in terms of the shear modulus of the
magnetic gels. Lines fitted with the function $1+A/(z-z_{c})^{\alpha}$ with
$A$ and $\alpha$ as fitting parameters.
By changing the network connectivity $z$, we can change the elasticity of the
polymer gel in the prepared state, based on which one can compare the effects
of magnetic interactions between the embedded particles and polymer elasticity
in the mechanical properties of magnetic gels. As shown in Fig. 4(c), the
contributions of magnetic interactions in the shear modulus of magnetic gel
where magnetic chains are formed, are significant for $z\simeq
z_{\mathrm{c}}$, which become less important by increasing the network
connectivity due to the increased elasticity of the polymer network itself.
## 4 Summary
We have studied the microscopic structure formation and the macroscopic
mechanical responses of magnetic gels, by utilising the coarse-grained
molecular dynamics simulations. By increasing the strength of the magnetic
dipole-dipole interactions between the particles, the microscopic chain
structures can form, which induces reinforced mechanical responses of the
magnetic gels. This work can not only help to understand the connection
between the microscopic structure and the macroscopic mechanical properties of
the magnetic gels, but can also guide industrial fabrications of magnetic gels
with desired mechanical responses and their controls by applying external
magnetic fields for practical applications.
## 5 Acknowledgments
F. M. acknowledges supports by National Natural Science Foundation of China
(Grant No. 12275332, 12047503, and 12247130), Chinese Academy of Sciences, Max
Planck Society (Max Planck Partner Group), Wenzhou Institute (Grant No.
WIUCASQD2023009) and Beijing National Laboratory for Condensed Matter Physics
(2023BNLCMPKF005). X. W. acknowledges supports from the China Postdoctoral
Science Foundation (Grants No. 2023M743448) and the National Natural Science
Foundation of China (Grants No. 12347170). P. T. acknowledges financial
support from the European Research Council (Grant No. 811234) and from the
Generalitat de Cataluña via the program “Icrea Academia”. The computations of
this work were conducted on the HPC cluster of ITP-CAS.
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|
# PointMatch: A Consistency Training Framework for Weakly Supervised Semantic
Segmentation of 3D Point Clouds
Yushuang Wu123∗ Shengcai Cai13∗ Zizheng Yan123
Guanbin Li43 Yizhou Yu56 Xiaoguang Han123† Shuguang Cui123
1SSE, CUHK-Shenzhen 2FNii, CUHK-Shenzhen 3Shenzhen Research Institute of Big
Data
4Sun Yat-sen University 5The University of Hong Kong 6DeepWise
{yushuangwu, zizhengyan<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>{hanxiaoguang,
<EMAIL_ADDRESS>
###### Abstract
Semantic segmentation of point cloud usually relies on dense annotation that
is exhausting and costly, so it attracts wide attention to investigate
solutions for the weakly supervised scheme with only sparse points annotated.
Existing works start from the given labels and propagate them to highly-
related but unlabeled points, with the guidance of data, e.g. intra-point
relation. However, it suffers from (i) the inefficient exploitation of data
information, and (ii) the strong reliance on labels thus is easily suppressed
when given much fewer annotations. Therefore, we propose a novel framework,
PointMatch, that stands on both data and label, by applying consistency
regularization to sufficiently probe information from data itself and
leveraging weak labels as assistance at the same time. By doing so, meaningful
information can be learned from both data and label for better representation
learning, which also enables the model more robust to the extent of label
sparsity. Simple yet effective, the proposed PointMatch achieves the state-of-
the-art performance under various weakly-supervised schemes on both ScanNet-v2
and S3DIS datasets, especially on the settings with extremely sparse labels,
e.g. surpassing SQN by 21.2% and 17.2% on the 0.01% and 0.1% setting of
ScanNet-v2, respectively. ††∗Equal contribution †††Corresponding author
## 1 Introduction
Semantic segmentation of 3D point clouds is crucial for the application of
intelligent robots’ understanding scenes in the real world. Great efforts have
been contributed to the fully supervised scheme, while it requires exhausting
and costly per-point annotations (_e.g_. around 22.3 minutes to annotate an
indoor scene on average [5]). Thus, weakly supervised 3D semantic segmentation
now receives increasing attention, where only limited point-level annotations
are provided in each point cloud.
Figure 1: (a), (b) the performance of PointMatch on the ScanNet-v2 and S3DIS
datasets over various weakly supervised semantic segmentation settings:
annotating 0.01%, 0.1% of points [11], 20 points per-scene [10], and
“1thing1click” [26]. (c), (d) a comparison between previous works and the
proposed approach.
Recently, several approaches are proposed for weakly supervised point cloud
semantic segmentation with different kinds of weak labels, including projected
2D image [44], subcloud-level [48], segment-level [38], and point-level [52,
10, 26, 11] supervision. In this paper, we focus on addressing the setting of
sparse point-level labels, which is one of the most convenient annotation
schemes in the application. The key challenge of this task is the difficulty
of learning a robust model given very sparse supervision in the point cloud
(_e.g_. 0.1%, 0.01% of points annotated in [11] and around 0.02% in [26]).
Existing solutions are mainly committed to alleviating the label sparsity by
reusing limited supervision, _i.e_., first probing the highly-related points
[11] or super-voxels [26] and allowing them to share the same training labels.
However, this line of works are explicitly constructed on label propagation
and employ point cloud data as the propagation guidance, which suffers from
(i) the insufficient exploitation of data information limits the learning
efficiency, and (ii) the propagated labels strongly rely on the original
annotation scale thus the performance is easily suppressed when given much
fewer labels. Therefore, we propose to probe information from both label and
data itself for more efficient and robust representation learning.
Recently, consistency training is acknowledged as a powerful algorithmic
paradigm for robust learning from label-scarce data, _e.g_. in
unsupervised/semi-supervised learning [12, 8, 50, 36] and unsupervised/semi-
supervised domain adaptation [6, 35, 21, 22]. It works by forcing the model to
make consistent prediction under different perturbations/augmentations to the
input sample (named as different views) and the prediction in one view usually
serve as the pseudo-label of the other view. Inspired by this, we propose a
novel consistency training framework, PointMatch, for the weakly supervised 3D
semantic segmentation. Given a whole scene of point cloud with sparse labels,
PointMatch employs the per-point prediction in one view as the other’s pseudo-
label to encourage the predictive consistency between two views of a scene.
Such consistency facilitates (i) robustness to easily-perturbed low-level
input feature and (ii) stronger capability in learning useful high-level
representations to keep predictive consistency. Besides, the provided labels
act as extra supervision to assist high-level semantic feature discrimination,
which also benefits the representation learning from data. By doing so, the
reliance on the given label is relieved and more information is probed from
the point cloud data itself.
Originating from the per-point prediction in one view, the pseudo-label should
be of high quality to provide positive guidance for the other. Whereas there
exist considerable mispredictions especially at the early learning stage.
Thus, we exploit the inherent structure of the point cloud to improve the
pseudo-label quality, via integrating the super-point grouping information
where similar points are clustered by low-level features (_e.g_. position and
color) into the same group and are assumed to have the same semantic meaning.
Specifically, the grouping information is used to correct the minor
predictions that diverge from the “mainstream” in the super-point. Despite its
good property, the super-point-aware pseudo-label actually introduces noise
from the pretext super-point generation. Therefore, to fully utilize these two
types of pseudo-labels, we design an adaptive pseudo-labeling mechanism, where
the model is encouraged to believe the super-point-aware pseudo-label more at
the beginning, and gradually resorts to its raw prediction when the model
itself is reliable enough. Extensive experiments and analysis on the
ScanNet-v2 [5] and S3DIS [1] dataset validate the effectiveness of the
proposed approach. As shown in Fig. 1, the proposed PointMatch significantly
surpasses the state of the art on various weakly supervised schemes and
impressively, shows great robustness given extremely sparse labels.
The contributions of this paper are listed as follows:
* •
We propose a novel consistency training framework, PointMatch, for the weakly
supervised 3D semantic segmentation, which can facilitate the network to learn
robust representation from sparse labels and point cloud data.
* •
We introduce super-point information to promote the pseudo-label quality in
our framework, and it is employed in an adaptive manner to well utilize the
advantages of both two types of pseudo-label.
* •
Extensive experiments validate the effectiveness and superiority of
PointMatch, and the proposed approach achieves significant improvements beyond
the state of the art over various weakly-supervised settings.
Figure 2: The overview of PointMatch. (a) the input point cloud; (b) the view
$A$ augmented from the input point cloud; (c) view $B$ generated via another
augmentation; (d) the point-wise pseudo-label; (e) the super-point-wise
pseudo-label; (f) the weak supervision (“1thing1click” setting), points in
gray are unlabeled ones and other colors indicate different semantic meanings.
## 2 Related Work
#### Fully Supervised 3D Semantic Segmentation
Semantic segmentation approaches for 3D point cloud can be mainly classified
into two groups: point-based and voxel-based methods. Point-based Methods [30,
31, 23, 49, 45, 42, 24] apply convolutional kernels to a local region of
points for feature extraction and the neighbors of a point are computed from
k-NN or spherical search. In the case of voxel-based methods [7, 4, 46, 14],
the points in the 3D space are first transformed into voxel representations so
that standard CNN can be adopted to process the structured voxels. In either
point-based or voxel-based methods, feature aggregation is performed in the
Euclidean space, while there are some recent works [15, 20, 32, 13] that
consider geodesic information for better feature representation. More
recently, the Transformer structure [54] is also proposed for point clouds, as
an alternative to the classic convolutional structure. However, most of the
above methods are designed for the fully-supervised scheme, while annotation
on point clouds is exhausting and costly, especially in the application of
semantic segmentation, where the scene (indoor or outdoor) point cloud is
usually of a large scale. In this work, we focus on weakly-supervised point
cloud segmentation, where only very sparse points are annotated in each scene.
#### Weakly Supervised 3D Semantic Segmentation
Existing works explore the 3D semantic segmentation with various types of weak
supervision, including 2D image [44], subcloud-level [48], segment-level [38],
and point-level supervision [52, 11, 26, 34]. The first three types can be
grouped into indirect annotations [11]. The work of [44] utilizes the
annotations on the projected 2D image of a point cloud, with only single view
per sample. In [48], a classifier is trained first with sub-cloud labels, from
which point-level pseudo labels can be generated via class activation mapping
techniques [55]. In another way, the work of [38] pre-generates
segments/super-points to extend sparse click annotation into segment-level
supervision, and groups unlabeled segments into the relevant nearby labeled
ones for label sharing. For point-level weak supervision, the work of [11]
proposes to use only 10% of labels by learning gradient approximation and
utilizing low-level smoothness constraints. A harder setting with a much lower
label ratio, 1‰, is further investigated in [11], where a Semantic Query
Network (SQN) is proposed based on leveraging the semantic similarity between
neighboring points. Another work OTOC [26] proposes a novel weakly supervised
setting, One Thing One Click (“1thing1click”), _i.e_., with only one point
annotated for each instance in the scene. They employ an extra branch of
network to probe the relation between super-points and propagate labels among
highly-related ones. Besides, authors of [34] propose an active learning
approach for annotating selected super-point with a limited budget to maximize
model performance. Another line of works is contributed to self-supervised
pre-training of 3D point clouds [33, 25, 51, 10, 53]. The pre-training usually
needs weak or even no labels and provides a better network initialization for
the downstream tasks.
Existing point-level weakly supervised 3D semantic segmentation methods act on
label propagation by leveraging the relation between points/super-points.
However, the proposed PointMatch takes a novel way based on consistency
regularization to better probe information in the point cloud data itself and
alleviates the reliance on the given labels.
#### Consistency Training
Consistency training is a powerful algorithmic paradigm proposed for robust
learning from label-scarce data. It is constructed on enforcing the prediction
stability under different input transformations [47], _e.g_. adversarial
perturbations [28] or data augmentations [36, 50], in the manner of pseudo-
labeling, _i.e_., using the prediction of one transformation as the fitting
target of the other. Thus it combines the advantages of both consistency
regularization and pseudo-labeling (or self-training). This approach has been
applied in many domains, such as semi-supervised learning (SSL) [3, 2, 36,
50], unsupervised learning (USL) [12, 8], unsupervised domain adaptation (UDA)
[6, 35], and semi-supervised domain adaptation (SSDA) [21, 22], all of which
prove the effectiveness of consistency training in learning high-quality
representations from label-scarce data. More recently, there are some works
extending consistency training into other tasks, such as unsupervised domain
adaptation for image segmentation [27] and semi-supervised 3D object detection
[43].
To our knowledge, it is the first time that consistency training is applied in
the weakly supervised semantic segmentation of 3D point clouds. Different from
the previous works, consistency training is novelly used in a weakly-
supervised scenario where limited point-wise supervision is provided in each
training sample. In addition, our work properly leverages the super-point
grouping information in point clouds to further improve the whole framework.
## 3 Methodology
### 3.1 Problem Definition
We first formulate the weakly supervised 3D semantic segmentation problem,
taking the indoor scene scenario as an example. Given the point cloud
$\mathbf{P}\in\mathbb{R}^{N\times D}$ of a scene of $N$ points with
$D$-dimension features, there are only partial points annotated for training.
The points with labels are denoted as $\\{(\mathbf{x}_{i}^{l},y_{i}),i\in L$},
and other unlabeled points are denoted as $\\{\mathbf{x}_{i}^{u},i\in U\\}$,
where $L$ and $U$ are two sets, satisfying $L\cap U=\varnothing$ and $L\cup
U=\langle N\rangle$ ($\langle N\rangle$ is a short form of
$\\{1,2,\cdots,N\\}$, the same hereinafter). The target of $f$ is to predict
the semantic category $y_{i}\in\langle C\rangle$ of each point
$\mathbf{x}_{i}$, where $C$ is the number of possible categories. Taking the
point cloud $\mathbf{P}$ as input, $f$ outputs the prediction probability
$\mathbf{Q}\in[0,1]^{N\times C}$ over all $C$ classes, for all $N$ points of
$\mathbf{P}$. Note that the summation of values in each row of $\mathbf{Q}$ is
equal to 1. Denote the weak semantic label of the whole scene as
$\mathbf{y}\in{\langle C\rangle}^{N}$ and its one-hot extension as
$\mathbf{Y}\in\\{0,1\\}^{N\times C}$. To optimize $f$, a straightforward way
is to compute the cross-entropy loss $\mathcal{L}_{ce}$ between $\mathbf{Q}$
and $\mathbf{Y}$, formulated as:
$\displaystyle\mathcal{L}_{ce}=\frac{1}{|L|}\sum_{i\in L}\text{cross-
entropy}(\mathbf{Q}_{i},~{}\mathbf{Y}_{i}),$ (1)
where $|L|$ represents the set size of $L$ and the subscript $i$ indicates the
row index, so $\mathbf{Q}_{i}$ and $\mathbf{Y}_{i}$ are two $C$-class
distributions corresponding to the $i$-th point. At the inference stage, the
semantic segmentation result of a scene can be generated from $f$’s
prediction, by simply choosing the class with the highest score in each row of
$\mathbf{Q}$.
To probe more information the limited labels and point cloud data itself, we
design a novel framework, PointMatch, with the pipeline illustrated in Fig. 2.
It conducts a consistency training framework designed for weakly labeled point
clouds, and an adaptive pseudo-labeling mechanism by incorporating the super-
point information, described in the following Sec. 3.2 and Sec. 3.3,
respectively.
### 3.2 Consistency Training
The proposed consistency training framework focuses on better exploitation of
data itself, by encouraging the model’s point-wise predictive consistency
between two views of an input scene, through employing the prediction in one
view as the pseudo-label of the other. Such a consistency training approach
has three advantages: (i) various augmentations enables the network robust to
different kinds of perturbation on low-level input features; (ii) the
consistency target facilitates the model’s ability in extracting high-level
semantic features from the point cloud data itself; (iii) the self-training
process implicitly propagates sparse training signals to unlabeled points and
provide dense pseudo-labels, which increases the learning stability.
Formally, given a point cloud $\mathbf{P}\in\mathbb{R}^{N\times D}$, our
PointMatch applies two different groups of data augmentations to create its
two views $\mathbf{P}^{A}\in\mathbb{R}^{N\times D}$ and
$\mathbf{P}^{B}\in\mathbb{R}^{N\times D}$, respectively. To avoid breaking the
local structure of the point cloud too much, we perform scene-level
augmentations like offsetting, scaling, rotation, flipping, jittering, _etc_.
The obtained two views $\mathbf{P}^{A}$ and $\mathbf{P}^{B}$ are then fed into
the 3D U-Net $f_{\theta}$ for point-wise semantic prediction, where $\theta$
is the network parameters. The network $f_{\theta}$ outputs the per-point
probability distribution of $\mathbf{P}^{A}$, denoted as
$\mathbf{Q}^{A}\in[0,1]^{N\times C}$, and similarly,
$\mathbf{Q}^{B}\in[0,1]^{N\times C}$ can be generated from $\mathbf{P}^{B}$,
formulated as:
$\displaystyle\mathbf{Q}^{A}$ $\displaystyle=f_{\theta}(\mathbf{P}^{A}),$ (2)
$\displaystyle\mathbf{Q}^{B}$ $\displaystyle=f_{\theta}(\mathbf{P}^{B}).$
In the next step, we generate the pseudo-label of $\mathbf{Q}^{B}$ from
$\mathbf{Q}^{A}$ to create the self-consistency loop. Specifically, the most-
likely predictive category of each point (as well as its confidence score) is
chosen to form the pseudo-label, _i.e_., the indices of the highest value in
each row of $\mathbf{Q}^{A}$. However, $\mathbf{Q}^{A}$ is usually noisy and
even contains many uncertain predictions, so a direct use may provide negative
guidance to $\mathbf{Q}^{B}$ and harm the whole learning scheme. Hence, we
conduct a filtering operation to improve the pseudo-label quality, by ignoring
those predictions with confidence lower than a threshold $\tau$. Denote the
filtering mask as $\mathbf{m}\in[0,1]^{N}$, which is generated as follows:
$\mathbf{m}_{i}=\left\\{\begin{array}[]{lr}1,~{}~{}\max(\mathbf{Q}^{A}_{i})\geq\tau,\\\
0,~{}~{}\text{otherwise},\end{array}\right.\forall i\in\langle N\rangle,$ (3)
where $i$ is the row index of $\mathbf{Q}^{A}$ and $\tau$ is set as 0.95 in
our implementation. Given $\mathbf{m}$ and the one-hot extension of
$\mathbf{Q}^{B}$’s pseudo-label, represented as
$\widehat{\mathbf{Y}}^{B}\in\\{0,1\\}^{N\times C}$, the pseudo-labeling of
$\mathbf{Q}^{B}$ can be conducted via a cross-entropy loss:
$\displaystyle\mathcal{L}_{pl}=\frac{1}{N}\sum_{i\in\langle
N\rangle}\mathbf{m}_{i}\cdot\text{cross-
entropy}(\mathbf{Q}^{B}_{i},~{}\widehat{\mathbf{Y}}^{B}_{i}).$ (4)
Until this point, we are working on probing information only from point cloud
data itself for a better data exploitation. Then the weak labels are
integrated to provide discriminative semantic information, by using
$\mathbf{Y}$ as the supervision of $\mathbf{Q}^{A}$ via computing a cross-
entropy loss as in Eq. 1. The parameters $\theta$ can then be optimized by
minimizing the objective loss function $\mathcal{L}_{total}$ as follows:
$\displaystyle\min_{\theta}\mathcal{L}_{total}=\min_{\theta}\mathcal{L}_{ce}+\lambda\mathcal{L}_{pl},$
(5)
where $\lambda$ is a scalar weight for balancing the two loss functions. As
the learning process goes, the model exploits the knowledge learned from the
limited annotations to train itself via forcing the predictive consistency,
and meanwhile, implicitly propagates the sparse training signals to the whole
scene via pseudo-labeling.
### 3.3 Adaptive Pseudo-Labeling
Although the framework above facilitates the model’s robust learning subtly,
we observe that there are still considerable mispredictions in the obtained
pseudo-labels, especially at the early learning stage. One reason is that the
previous training scheme is mainly constructed on the predictive consistency
between each pair of single points, and the inter-point relation information
is learned insufficiently. Therefore, we further exploit the super-point prior
to introduce local structure information of point cloud for generating pseudo-
labels of higher quality.
The super-points of a scene can be generated via an unsupervised low-level
clustering by the position and color information of each point. We refer to
[19] for the manner of super-point generation, and it is recommended for more
details. Formally, given a point cloud $\mathbf{P}\in\mathbb{R}^{N\times D}$,
we obtain a set of super-points $\\{\mathbf{S}^{(i)}\\},i\in\langle M\rangle$,
where $M$ is the number of super-point and each
$\mathbf{S}^{(i)}\in\mathbb{R}^{S^{(i)}\times D}$ includes $S^{(i)}$
$D$-dimension points. Each point in $\mathbf{P}$ belongs to one super-point
only, so $\mathbf{S}^{(i)}\cap\mathbf{S}^{(j)}=\varnothing,\forall i\neq j$
and the summation of all $S^{(i)}$ is equal to $N$. The obtained super-point
information is then used to improve the quality of point-wise pseudo-label
$\widehat{\mathbf{Y}}^{B}$. Given point-wise predictions in each super-point
group, a voting operation is carried out to get a “mainstream” category. The
elected category is then propagated to all points in this group to obtain a
super-point-wise pseudo-label $\widehat{\mathbf{Y}}^{B}_{\text{sp}}$. An
illustrative example of $\widehat{\mathbf{Y}}^{B}$ and
$\widehat{\mathbf{Y}}^{B}_{\text{sp}}$ is shown in Fig. 2 (d) and (e),
respectively. It can be observed that $\widehat{\mathbf{Y}}^{B}_{\text{sp}}$
tends to have higher purity. Similar to Sec. 3.2, we preserve confident
predictions to form high-quality super-point-wise pseudo-labels. Specifically,
given $\mathbf{Q}^{B}$, the average probability distribution in each super-
point is computed first, of which the category with the highest score is
selected and propagated in the whole super-point. Then the filtering mask
$\mathbf{m}^{\text{sp}}$ is generated by checking whether the confidence of
each point is beyond a pre-defined threshold $\tau^{\text{sp}}$, similar to
the computation in Eq. 3.
Although the voting operation enables $\widehat{\mathbf{Y}}^{B}_{\text{sp}}$
more stable and accurate, it suffers from the inherent noise arising from the
super-point generation process. Thus, the point-wise pseudo-labels may have
higher accuracy when the model is strong enough. Accordingly, we further
design an adaptive combination mechanism to exploit the advantages of both. At
the early stage, the learning of $f_{\theta}$ relies on
$\widehat{\mathbf{Y}}^{B}_{\text{sp}}$ via a cross-entropy loss
$\mathcal{L}_{pl}^{\text{sp}}$:
$\displaystyle\mathcal{L}_{pl}^{\text{sp}}=\frac{1}{N}\sum_{i\in\langle
N\rangle}\mathbf{m}^{\text{sp}}_{i}\cdot\text{cross-
entropy}(\mathbf{Q}^{B}_{i},~{}\widehat{\mathbf{Y}}^{B}_{\text{sp}i}).$ (6)
As the learning goes, an adaptive weight $w$ is adopted to gradually
incorporate $\mathcal{L}_{pl}$ (Eq. 4) and abandon
$\mathcal{L}_{pl}^{\text{sp}}$:
$\displaystyle\mathcal{L}_{pl}^{\prime}=w\cdot\mathcal{L}_{pl}^{\text{sp}}+(1-w)\cdot\mathcal{L}_{pl},$
(7)
where $w$ is a scalar in the range of $[0,1]$ and drops from 1 to 0 gradually
with an inverse decay. Formally, the adaptive weight $w$ at the $k$-th
training epoch can be computed as:
$\displaystyle w=\alpha\cdot k^{-1},k\in\mathbb{N},$ (8)
where $\alpha>0$ indicates the decay ratio. In this way, at the late stage of
training, $f_{\theta}$ can be completely supervised by the point-wise pseudo-
label, so that the model can keep from the noise in super-point grouping. The
new pseudo-labeling loss $\mathcal{L}_{pl}^{\prime}$ is used to substitute the
original $\mathcal{L}_{pl}$ in Eq. 5 for the final loss function.
## 4 Experiments
Table 1: MIoU (%) on the ScanNet-v2 dataset (online test set). * means the
performance of our baseline on the fully-supervised setting. The underline
indicates the previous SOTA performance on each setting. The supervision types
“subcloud” and “segment” mean using subcloud-level and segment-level
annotation, respectively. “20 points” and “1thing1click” mean annotating 20
points per scene and annotating one point in each instance, respectively.
Method | Supervision | MIoU
---|---|---
[31] PointNet++ | 100% | 33.9
[37] SPLATNet | 100% | 39.3
[40] TangentConv | 100% | 43.8
[23] PointCNN | 100% | 45.8
[24] FPConv | 100% | 63.9
[49] PointConv | 100% | 66.6
[42] KPConv | 100% | 68.4
[4] MinkowskiNet | 100% | 73.6
[13] VMNet | 100% | 74.6
[9] Occuseg | 100% | 76.4
[29] Mix3D | 100% | 78.1
[7] SparseConv | 100% | 72.5*
[48] MPRM | subcloud | 41.1
[38] SegGroup | segment | 61.1
[11] SQN | 0.01% | 35.9
[11] SQN | 0.1% | 51.6
[26] OTOC | 20 points | 59.4
[26] OTOC | 1thing1click | 69.1
PointMatch | 0.01% | 57.1
PointMatch | 0.1% | 68.8
PointMatch | 20 points | 62.4
PointMatch | 1thing1click | 69.5
### 4.1 Experiment Setup
#### Datasets and Metric
We choose two popular point cloud datasets for the evaluation of our method,
ScanNet-v2 [5] and S3DIS [1]. The ScanNet-v2 dataset [5] contains the 3D scans
of 1,613 indoor scenes of 20 semantic categories (1,201 for training, 312 for
validation, and 100 for online test). The whole dataset includes around 243
million points in total. The S3DIS dataset [1] contains 271 room point clouds
with 13 categories, scanned from 6 areas. Following the official
train/validation split, Area 1,2,3,4,6 are used for training and Area 5 is
used for evaluation. Besides, the S3DIS dataset has 273 million points,
_i.e_., around 1 million points per scene on average, which is denser than
scenes in the ScanNet dataset. The evaluation metric for 3D semantic
segmentation we use is intersection-over-union, and we report the mean result
(MIoU) over all categories for comparison with other methods.
#### Implementation Details
We adopt SparseConv [7] as the 3D U-Net backbone in PointMatch. The output
dimension of the SparseConv is set to 32, which is the same as in [26].
Following [16] and [26], we randomly sample 250k points for too large scenes
in the ScanNet-v2 dataset. We use two different combinations of various
augmentations to create two views, randomly chosen from scaling, flipping,
offsetting, rotation, affine transformation, position jittering, and color
jittering, with a random augmentation extent. Hyper-parameters in our
experiment $\tau$, $\tau^{\text{sp}}$, $\epsilon$, $\lambda$, and $\alpha$ are
set to 0.95, 0.95, 0.5, 1.0, and 1.0, respectively. The network is trained for
512 epochs using Adam optimizer [17] with a learning rate of 0.01 and a mini-
batch size of 8 on the ScanNet-v2 dataset and 4 for the S3DIS dataset.
Considering the total number of training epochs, we replace the epoch number
$k$ in Eq. 8 with $\lfloor k/64\rfloor$ on the “1thing1click” setting and
$\lfloor k/32\rfloor$ on others, which is the round-off of $k$ divided by 32
or 64, in order to slow the decay rate. For the super-point generation, we
follow [26] to use the mesh segment results [5] on the ScanNet-v2 dataset and
the super-point graph partition manner proposed by [19] on the S3DIS dataset.
Note that the super-points are used in training, and the inference stage does
not rely on super-points. All experiments are conducted on an Intel Xeon Gold
6226R CPU and an NVIDIA RTX3090 GPU with 24GB memory.
### 4.2 Experiment Results
Table 2: MIoU (%) on the ScanNet-v2 dataset validation set. * means the
performance of our baseline on the fully-supervised setting. Note that SQN
[11] reports only its performance of 0.1% label setting on the ScanNet-v2
validation set.
Method | Supervision | MIoU
---|---|---
[7] SparseConv | 100% | 72.2*
[11] SQN | 0.1% | 53.5
[26] OTOC | 20 points | 61.4
[26] OTOC | 1thing1click | 70.5
PointMatch | 0.01% | 58.7
PointMatch | 0.1% | 69.3
PointMatch | 20 points | 64.8
PointMatch | 1thing1click | 70.7
#### Evaluation on ScanNet-v2
On the ScanNet-v2 [5] dataset, the evaluation of PointMatch is conducted on
four weakly-supervised settings, _i.e_., 0.01% of points annotated in each
scene [11], 0.1% of points annotated in each scene [11], 20 points annotated
per scene [10] (20 points), and 1 point annotated for each instance in the
scene [26] (1thing1click). The annotated points in the first two settings
(0.01% and 0.1%) are randomly chosen following [11]. The “20 points” setting
is implemented following the officially ScanNet-v2 “3D Semantic label with
Limited Annotations” benchmark [10]. Annotated points in the “1thing1click”
setting are randomly chosen from each instance following [26]. The average
point label in this setting is around 0.02% [26]. The evaluation results on
the ScanNet-v2 online test set are presented in Tab. 1. Existing weakly
supervised 3D semantic segmentation methods are also included for comparison,
and some fully supervised methods are also listed in the table. As shown in
the table, the proposed PointMatch consistently surpasses all existing methods
over all weakly-supervised settings. It outperforms the state-of-the-art
(SOTA) result by 21.2% on the 0.01% setting, by 17.2% on the 0.1% setting, and
by 3.0% on the “20 points” setting. The performance on the “1thing1click”
setting is further close to the fully-supervised baseline. Note that the work
OTOC [26] takes 5 turns of iterative training to reach the above results,
which is around 1536 epochs (3 times of ours). In addition, we also provide
the performance of PointMatch on the ScanNet-v2 validation set in Tab. 2, on
four weakly-supervised settings mentioned above, which also proves the
superiority of PointMatch. Detailed results over 20 categories are shown in
supplementary materials.
#### Evaluation on S3DIS
Table 3: MIoU (%) on the S3DIS dataset (Area-5 for validation). * means the
performance of our fully-supervised baseline. The underline indicates the
previous SOTA performance on each setting.
Method | Supervision | MIoU
---|---|---
[30] PointNet | 100% | 41.1
[41] SegCloud | 100% | 48.9
[40] TangentConv | 100% | 52.8
[23] PointCNN | 100% | 57.3
[19] SPGraph | 100% | 58.0
[4] MinkowskiNet | 100% | 65.4
[42] KPConv | 100% | 67.1
[54] PointTransformer | 100% | 70.4
[7] SparseConv | 100% | 63.7*
[18] $\Pi$ Model | 0.2% | 44.3
[39] MT | 0.2% | 44.4
[52] DGCNN+CRF | 0.2% | 44.5
[18] $\Pi$ Model | 10% | 46.3
[39] MT | 10% | 47.9
[52] DGCNN+CRF | 10% | 48.0
[26] OTOC | 1thing1click | 50.1
[11] SQN | 0.01% | 45.3
[11] SQN | 0.1% | 61.4
PointMatch | 1thing1click | 55.3
PointMatch | 0.01% | 59.9
PointMatch | 0.1% | 63.4
Figure 3: Visualization of the qualitative results. We sample three scenes
from the training set and their related results include, (a) upper: input
point clouds, lower: the super-point grouping, in which colors do not indicate
category information; (b): two views of the input point cloud; (c) upper: the
point-wise pseudo-label at the early stage, lower: the super-point-level
pseudo-label at the early stage; (d) upper: the point-wise pseudo-label at the
late stage, lower: the super-point-level pseudo-label at the late stage; (e)
upper: the weakly-supervised prediction, lower: the fully-supervised
prediction; (f) upper: the weak supervision, lower: the full supervision
(ground truth).
We also evaluate the proposed method on the S3DIS [1] dataset to further
validate the effectiveness of the proposed method. Three weakly-supervised
settings are included for evaluation, _i.e_., 0.01%, 0.1%, and “1thing1click”
(no official “20 points” setting provided for S3DIS). Note that the point
cloud in the S3DIS dataset usually contains much more points than in the
ScanNet-v2 dataset. By estimate, around 0.0036% of points are annotated in the
“1thing1click” setting. The results on these three settings are listed in Tab.
3. The SOTA methods on both the fully-supervised and weakly-supervised
settings are presented in the table for comparison. It is observed that the
proposed PointMatch achieves the best performance over all three settings. It
surpasses the SOTA result on the 0.01% setting by a large margin of 14.6%, by
5.2% on the “1thing1click” setting, and by 2.0% on the 0.1% setting.
Impressively, our result on the 0.1% setting is very close to the fully-
supervised baseline (63.4% v.s. 63.7%). The above results strongly prove the
effectiveness and superiority of PointMatch, especially in the scenario of
very sparse annotations (0.01%). Detailed results on all 13 categories are
listed in supplementary materials.
#### Qualitative Results
Table 4: Ablative results of consistency training in PointMatch. MIoU (%) on
the ScanNet-v2 dataset validation set.
Method | Supervision | MIoU
---|---|---
Fully-Sup. Version | 100% | 72.2
PointMatch | 0.01% | 58.7
w/o Consist. Training | 0.01% | 51.3
PointMatch | 0.1% | 69.3
w/o Consist. Training | 0.1% | 67.3
PointMatch | 20 points | 64.8
w/o Consist. Training | 20 points | 55.0
PointMatch | 1thing1click | 70.7
w/o Consist. Training | 1thing1click | 62.2
Except for the quantitative results, we also exhibit some qualitative
segmentation results of PointMatch. As shown in Fig. 3, we visualize each
sample in two rows and six columns, namely the input point cloud (upper) and
its super-point grouping (lower) in column (a), its globally-augmented (upper)
and locally-augmented (lower) views in column (b), its point-wise (upper) and
super-point-wise (lower) pseudo-label at the early and late stage of training
in column (c) and (d), respectively, the prediction of PointMatch under the
weak (upper) and full (lower) supervision in column (e), and the corresponding
weak label (upper) and ground truth (lower) in column (f). Note that all
results we visualize are generated under the “1thing1click” weak supervision.
It is observed that the predictions of PointMatch under weak supervision are
close to the ground truths and the fully-supervised predictions. More
impressively, the super-point-wise pseudo-labels are superior to the point-
wise ones at the early stage, while get inferior at the late stage of training
(see red boxes in Fig. 3), which confirms our claim. More visualization
results are presented in supplementary materials for the space limit.
### 4.3 Ablation Study
The proposed PointMatch mainly includes two components, the consistency
training paradigm and the adaptive pseudo-labeling mechanism. Corresponding
ablative experiments are conducted for the analysis of them.
#### Consistency Training
To validate the effectiveness of the consistency training, we remove one
branch in our framework as well as the pseudo-labeling mechanism, so the
resultant version is a SparseConv simply trained on the weak supervision
(extended by super-point information as the original) with a cross-entropy
loss. We implement ablative experiments on four weakly-supervised settings on
the ScanNet-v2 validation set. As shown in Tab. 4, removing the consistency
training results in noticeable performance drops consistently over all weakly-
supervised settings, especially on the schemes with extremely little
supervision, which strongly proves its great effectiveness.
#### Adaptive Pseudo-labeling
The adaptive pseudo-labeling mechanism plays the role of pseudo-label
correction at the early stage of training, and it is implemented with an
inverse decay. To confirm the effectiveness of our design, we implement four
versions on two weakly-supervised settings (“1thing1click” and “0.01%”) for
comparison: (i) using point-wise pseudo-label only ($w=\text{0}$); (ii) using
super-point-wise pseudo-label only ($w=\text{1}$); (iii) using both two
pseudo-labels but in a constant manner, by setting $w$ to 0.5
($w=\text{0.5}$); (iv) using the adaptive mechanism with a larger decay ratio,
by using $\lfloor k/32\rfloor$ (“1thing1click” settings) and $\lfloor
k/32\rfloor$ (0.01% settings) in Eq. 8 ($k\leftarrow\lfloor k/16(32)\rfloor$).
Results are listed in Tab. 5. Using either type of pseudo-label only is
inferior to the adaptive combination, because both point-wise and super-point-
wise pseudo-label have their own strengths. Using a constant weight also leads
to a performance drop, which proves that giving temporally different reliance
on the two pseudo-labels can better exploit their advantages. Besides, a
faster decay of the weight $w$ also results in a slightly worse result, which
is usually close to the result of using point-wise pseudo-label only
($w=\text{0}$). One reason is that the network is unable to learn adequate
information from super-points when $w$ drops too fast.
Table 5: Ablative results of adaptive pseudo-labeling in PointMatch. MIoU (%)
on the S3DIS dataset Area-5.
Method | Supervision | MIoU
---|---|---
Fully-Sup. Version | 100% | 63.7
PointMatch | 0.01% | 59.9
$w=\text{0}$ | 0.01% | 58.4
$w=\text{1}$ | 0.01% | 56.1
$w=\text{0.5}$ | 0.01% | 54.6
$k\leftarrow\lfloor k/16\rfloor$ | 0.01% | 58.7
PointMatch | 1thing1click | 55.3
$w=\text{0}$ | 1thing1click | 52.6
$w=\text{1}$ | 1thing1click | 50.2
$w=\text{0.5}$ | 1thing1click | 48.4
$k\leftarrow\lfloor k/32\rfloor$ | 1thing1click | 53.3
## 5 Conclusion and Discussion
We propose a novel approach, PointMatch, which introduces a consistency
training framework into weakly supervised semantic segmentation of point
clouds. It works by enforcing the predictive consistency between two views of
a point cloud via pseudo-labeling, and enables the network to perform robust
representation learning from weak label and data itself. The pseudo-label
quality is further promoted by integrating super-point information in an
adaptive manner. Impressively, PointMatch achieves SOTA performance over
various weakly-supervised semantic segmentation settings on both ScanNet-v2
and S3DIS datasets, and shows strong robustness given even extremely little
labels, _e.g_. 20 points per-scene and 0.01% of points annotated.
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Subsets and Splits
Filtered Text Samples
Retrieves 100 samples of text containing the specific phrase "You are a helpful assistant", providing limited insight into the dataset.
Helpful Assistant Text Samples
Returns a limited set of rows containing the phrase 'helpful assistant' in the text, providing basic filtering of relevant entries.